6 l't ¡r r*

"RESIDUT ARITHMETIC IN

DIGITAL COIIPUTERS"

RAMESH CHANDRA DEBNATH/ M.SC.

BEING A THESIS SUBMITTED

FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

IN THE

DEPARTMENT OF ELECTRICAL ENGINEERING

THE UNIVERSITY OF ADELAIDE

IVIARCl-i I979

SUIq¡4ARY

The number systems used have great impact on the

speed of a computer, At present at least 4 number systems

(binary, negabinary, ternary and residue) are in use eitherfor general purposes or for special purposes. The calîry

plîopagation delay which j-s inherent to all weighted systems

(binary, negabi-nary and ternary) is a limiting factor on the

speed of computations. tlnlike the weighted system, the

residue system is carry free which is the m¿rln attractj-vefeature of this system and therefore its potentiality in a

general purpose computer, in the light of present day tech-

nology, has been investigated.

As the background of the investigation the binary and

negabinary arithmetic have been thoroughly studied. A ne',r

method for fractional negabinary division, and a correction

on negabinary multiplì-cation have been proposed. It has

been observed that the binary arithmetic operations are fas-

ter thair thcse ir. negabinary syslem.

The residue system has got. problems too. The sign

detectj-on, operations j-nvolving sign detection (comparison,

overflow detection, division) and the resj-due-to--decimal- con-

version are difficult. Moreover, the residue systemT being

an i.nteger: systetn, makes floati-ng poinc arithmetic operatlcns

very difficult. The def:ails of the existing algorithms,/

methods to solve those problems

tigated and drawbacks of those

di scussed.

have been thoroughly inves-

algoritlrms/methods have been

. This thesis proposes a new method on multiplj-cative

overflow detect-ion in the residue system, a technique for

the determination of the first approximation for residue

integer division, algorithms for floating point arithmetic

operations in the residue system and a method for the res-

idue--to-decimal conv.rsion.

In order to verify those proposed methods and algor-

ithms a 16-15 moduli- residue arithmetic unit has been

designed using LS1 components. The unit together with (in

this case) an SDK-80 microcomputer can perform all the

integer and floating poi-nt arithmetic operations.

It has been obse::ved that t'he binary system still- main-

tains its predominance over the residue system in general

purpose digital computers. However, the residue system is

faster where there is no operation involvi-ng sign detection,

and also the carry free property, in very large systems, is

of great i-mportance j-n minimj-sing the chip connectíons and

inter chip delays. In future, investigation should be carried

out on the possibility of using VLS1 components which may up-

set the present conclusions which are generally in favour of

the binary system.

ACKNOWI,EDGEMENT

The autholî likes t'o acknowledge the contributions ofthose who have, directly or indlrectly, assisted in thisresearch and preparat-i-on of this t.hesis. The author ispleased to thank Dr D.A. Pucknell, the project supervisor,

for his invaluabÌe grlid"rr." and encouragement.

The author likes to thank the Radio Research Board of

Australia and the Col-ombo Plan Authority for their financialassistance for this project.

The author would like to thank t.he technical staff of

the Department of Electrical Engineering, University of

Adelaide and specially Mr R.C. Nash for his helping hand inthe computer laboratory.

The author is pleased to thank Mr D.C. Pawsey, senior

lecturer and Mr C.K. Linke, laboratory manager, of the

Department of Electrj-cal Engineerirg, universj-ty of Adelaide,

for their positive help in making a pleasant stay inAusf:ra1ia.

The author likes to thank Mrs Mercaa Fuss for her

excellent typing.

Fina1ly, the

Rani Debnath, for

author wislies to thank his wife, Mrs Asha

'Ehroughout the research.her encouragement

PAPER WRITTEN BY THE AUTHOR

with D.A. Pucknell, "Orr multiplicatlve overflow detection

in residue number system". Electronics Letters, Vol.L4,

No.5, March 1.978, Þp. L29-130.

ERRATA: "On multiplicativenumber system", Electronj-cs

L978, p. 385.

overflor^¡ detection i-n residue

Letters, Vol.14, No.l-2, June

[x]

Fr (ArB)

rr (a. e)

Fr (AlB)

r (n)

Pol (r (n) )

X+V

rn( x, v,....)SX or S (X)

x-

V

NOTATIONS

The highest integer value of X,but less than X. (X may be anarithmetic expressi-on also. )

Floating point au].dition/subtraction with operands Aand B.

Floating point multiplicat1onwith operands A and B.

Floating point division withoperands À and B.

Shift T to n positions left.(_

Pol-arÍse r (n) .

Replace the value of X with V.(/ may be an arithmetic or alogical expression. )

fnputs X, Y ,

Sign of X.

Swap the values of X and Y.

Residue of X with respect tomodule m.

Absotute value of X.

Gate with inputs b, c, . . .lz anooutput ,L at the location 4. on theresidue arithmetic unit (net-t)board.

and G.7 are the co-efficientsA.

Exponent of X.

MantÍssa of X.

E (X)F"n, ,t,]

G (X) 2E(X )

lx l* or [x ]*

lxlGate (qb,c,...Þ,L)

Coef f (a) :> d2 t a-7

Exp (X)

Mnr (x)

:l

0,2

of

Contains.

L/

3/

CONTBNTS

INTRODUCTION

1.1 Introductory Note

1.2 Number Systems an overview

ÏNTRODUCTÏON TO BINARY AND NEGABINARYNUMBER SYSTEMS,

2.L Binary Arithmetic

2.I.1 Binary Addition and Subtraction2.I.2 Binary Multiplication2.t.3 Binary Division

2.2 The Negabinary Number System

2.2.1 Negabinary Number Representation( lnreger)

2.2.2 Negabinary Number Representation(fraction)

2.2.3 Polarisation of Negabinary Number2.2.4 Negabinary Addition/Subrraction2.2.5 Negabinary Multiplication2.2.6 Negabinary Divislon2.2.7 A Correction Proposed on Varrable

Shift Ncgabinary Multiplication2.2.8 Method Proposed on Negabinary

Fracti-onal Divisicn

2.3 Concl-usion

RESIDUE ARITHMETTC LITERATURE REVTEW

3.1 Introduction

3.2 Integer Residue ArithmetÍc an outl j-ne

3.2.1 Addition, Subl-raction andMultipl j-cation

3.2.2 Division

Page

B

B

9

l_1

t4

15

15

15

16

l_6

20

20

22

27

2B

2B

29

29

2c)

1

1

2

2/

23

3.3 Overflow

3.3.1 Overflow due to Addition/Sul:traction

3.3.2 Mul-tiplicative Overf low

3.4 Floating Poj.nt Arithmetic in aResidue Number System

3.4

33

34

34

35

3

1 The Representation of FloatingPoint Numbers

3.4.2 Floating Point Arithmetic

Numbers to

36

36

40

40

43

43

43

44

50

52

59

59

61

65

65

66

66

71,

73

4/

5 Transl-ation of ResidueDecimal Numbers

3.6 Conclusion

RESIDUE ARTTHMETTCTECHNIQUE PROPOSED

ALGORITHMS AND

4.L Tntroducti-on

4.2 Multiplicative Overf l-owAspects

Theoretical-

4 .2.L Logarithmj-c Distribution ofResidue Numbers

4.2.2 Design Ideas for OverflowDetection

4.2.3 Desi-gn Example on OverflowDetection

4 3 General-

4.3.14.3.2

4.4.L4.4.2

4.4.3

¿.Á.a

4.4.5

Division

A Division Processthe first approx-successive approxJ-m-

Theoretical ConceptFloating Point Representationof a Number

Floatiug Point Arithmetic withZero ExponentFloat-ing Point Ar:j thmetic withNon-zero ExponentHow to find the co-eff-tcientof a mantissa

How to findimation andations

4.4 Floating Point Ari--hmetic

4.4.6

4.5 Output

4.5. 1

4.5.2

5.6

5.7

9

5. 10

5.11

Translation

Theoretical- AspectsConversion Prrccedure

ErrorPoint

in the Proposed FloatingArithmetic Operations

.TotB

73

74

74

75

77

79

79

90

9B

108

118

r26

t44

L47

L49

151

153

155

L70

A1

A7

A9

5/

4.6 Conclusion

DESIGN OF A REÀL TIME RESTDUEARITHMETIC COIVIPUTER

1 Introduction

2 Addition, Subtraction and Multiplicat.ion( rnteger )

3 Comparison

4 Overflow Detection

5.5

5

5

5

5

I5

Integer Division

Floating Point Arithmetic

Output Circuit of the RAU

Conversion of Residue Numbers toDeci-mal Numbers

How the Residue ArithmeticComputer Works

FIow to Operate the RAC

Conclusion

6/ CONCLUSTON

BIBLIOGRÀPHY

Appendix- 1

Appendix-2

Appendl x-3

Some fmportant Proofs

Floating Point Val-ue

Table of mantlssa and1exponent "f B

Appen<lix-4

Appendix-5

Appendix-6

Error AnalysisPoint Àddi-tion

of Floatingand Multiptication

Mixed Radix Conversion

Program Listings

A16

A24

A26

À

1 INTRODUCTION

1.1 Introductory Note

The number system has a great impact on the speed of

computer arithmetic which, together wÍth other factors

access ti-¡ne of memories, memory hierarchy and computer

architecture, inflüence the speed of a computer as a whole.

scientific probrems such as the solution of rarge sets of

linear equations, problems embracing the field of structural

design, traffic f-icw, signal processing etc. are associated-

with a great deal of computation. A number system with its

associated arithmetic algori-thms can greatly effect the

speed of computat j-on.

The choice of a number system depends on the availab-ility of efficient arithmetic algorithms. Technology also

influences the choice a great deal. For example, a decimar

system would be preferable to a binary system if technology

could provide low cost reliable ten valued logic circuits.one nuirrber system can offer advantages on a particular cper-

ation, but may be unsuitable from the consideration of over-

all s;peed of computatl-on. For example, in a resi_due system,

mu]-tiplication is faster, however, other operations such as

si-gn detection, comparison and overflow detection are slow

processes" rt is, therefore, worthwhile to investigate which

of the existing number systems can offer the greatest overall

1

2

speed advantage in computer arithmetic for general purpose

computers. The rest of this chapter is an overvj-ew of the

existing number systems and their advantages and disad.van-tages.

Chapter 2 ð,iscusses the arithmetic operations in binary

and negabinary systems. The existing arithmetic algorithms

in the resldue system have been discussed in Chapter 3. A

few improved techr:iques and algorithms for residue arithmetic

have been proposed in Chapter 4 and the design of a residue

arithmetic computer, on the basis of the proposed techniques

and algorithms, has been shown in Chapter 5. Chapter 6 glves

conclusions on the comparative studj.es of binary and residue

systems.

1-.2 Number Systems - an overview

To represent all numbers, a number system must have

unique sets of representation. A number system may be a

weighted system or a nonweighted system. Some examples of

the weighted systems are the binary system, negabinary

system, and ternary system,' whereas the residue system is a

nonweighted svstem.

A nurnber in a weighted system can be expressed as

x

where a.í

of weights. Tf

are the

N

= t aiwii=o

number system is

is a mixed radix

T,IJí ATC

ca1led

sets of integers and øi are the sets

the successive power of radix r,rJ, the

a fixed radix system; otherwise itThe binary sys t.em (w= 2) , negablnarysystem.

system (w= - 2 ) and ternary system (r,0= 3 ) are

fixed radix systems. Examples of a mixed

been given in Appendix 5.

3

in the family of

radix system have

Number systems which have found use in digital com-

puters are the binary system, negabinary system I f], lZl,ternary system [ 3] and the residue system. Tn subsequent

sections brief discussions now follow on these number systems.

L .2 .1 e*ilery. NumÞ_er S stem

The binary number system is a tr¡/o digit (0,1) number

system having a radix *2. The decimal value of a binarynumber depends on the posj-tion of 1. Thus for example

(11011) 2 = L -24+I -23+0.22+!.2r+L.20

= (27) to

Due to easy converslon from binary number to octalnumber (ø=8) and vice versa, some digital computers such as

the TBM-7099, Data General Nova-l, Nova-2 and many others,use the octal system for easy input-output operation. The

conversion of binary number to octal number is done by for-mlng groups of three consecutive bits and expressing each

group as an octal digit (O thrcugh 7). Thus above binarynumber can be expressed in octal form as

(01-1, 011)2 = (::¡r = 3.BI+3.80 = (27]lto

The octal number is converted to binary number by

simply expressing the octal digits in binary forln. Thus

(75)a = (111,101)2.

4

Presently some computers fBM-360, IBM-370 and many

mi-croprocessors make use of hexadecimal numbers (w=t6). The

sixteen digits of hexadecimal numbers are 0,L,2,3,4,5,6,7,8,9,A,B,C,D,E,F. A binary number is converted to hexadecimal

form by taking groups of four consecutive bits and represen-

ting each group by a hexadecimal digit. Thus

(1101, 1111)z = (DF)ro = 13.16t+15.160 = (ZZZ¡ro

l{hen hexadecimar digits are represented in binary form, itgives equivalent binary number. For example

(94)rs = (1001, 101-0)2.

Another code which is often used is the blnary coded

decimal number (eCO). This number is really a decimal number,

the digits of which are expressed in binary form. Thus

(789) r o = (0111, 1000, 1001) uro

B.inary states (0,1) are easily realisaÌ:le in a physical

system and is one of the main advantages of binary systems

and this is the reason why almost all present day computers

use binary number systems. The obvious disadvantage which isinherent in a1-1 weighted systems is the carry propagation

delay which limits the speed of computation for examÞle, inaddition and especially in multiplication which is a process

of additj-ons a¡rd shifts. Another disadvantage is the sign

correction required in complement (1's or 2's) multipli-cation.These are the reasons why the potentiality of other number

systems should be investigated. The speed of arithmetic i,s

particrrlarly important in the growing field of real tj.me

processing in which a v¡jde range of digital computers/micro-

processors are being applied.

5

I .2. Z _NeggÞl-ns.ry Numþe! Svst_em

A negabinary

digit system. But

For example

(a) (1101I) -z

(b) (0100r) _z

like a binary system, is a twosystem,

unlike the binary system the radix j_s -2

1. (-2)a+I. (-2) t+0. (-Z)2+L. (-Z) t+1. (-2) 0

16-B+0-2+1 = (7)ro

0. (-z) u+1 . (-2) 3+O . (-z) 2+t. (-2) 0

o-8+o+o+1 = (-7)ro

The advantage of this system is that no sign correctiorr

is needed for multiplicatj-on. The disadvantages are: (i)

duri-ng add.ition dcuble carries are generated (to be dj-scussed

in Chapter 2) and therefore it is difficult to mainr.ain speedy

propagation of carries, (ii) from examples (a) and (b), it is

seen that when the most signifi-cant l is in an odd posi.tion,

the number is negative and if it is in an even position, the

number is positj-ve. Thj-s means that the sign is implicit.The detection of implJ-cit sign is not so easy as that of

explicit sign as in the case of the binary system, and (ij-i)

from the example (a) it is clear that to repr-esent (7) ro ,

this system needs 5 bits, whereas the binary system need.s only

4 bits (¡ bits for magnitude + 1 bit for sign). Furthermore,

some of the arithmetic operations are not easy. Chapter 2

will dj-scuss its arithmetic operations.

L.2.3 Ternary Number Systems

A ternary lnteger number can be expressecl es

vN

T

r=oai3t

6

where cLí (i=o, 1,, .....,N) are the coefficients and can

have any value of the set (0,\,2) in an ordinary ternarysystem or any value of the set (-1,0,1) in a symmetric system.

This number system, specially the symmetric one, offers poten-

tial advantages fc¡r the following reasons (i) sign conversion

is easy (simple inversion) (ii) roundíng is done by simpry

t::uncating the number to the required numbe.r of <ligits (i_ii)

since the ternary system needs 58% less c1-igits than equivalent

binary number, addition and multiplicatj.on will be faster l+),[s] and there w11l- be a potential reduc'r-ion Ín wi.ring cost Io].

The Russian

ary computer. Itstate logic.

'SETUN' is an

used magnetic

example of

core s lzzl

a symmetri-c tern-

to realize three

A handful of papers j_s avaitabl_e on

ithms lq), [s], [z] and rernary swirching

arithmetic algor-

functions IB ]- [12].

The ternary system sounds promising for the future and

interest is rapidly growlng in the hardv¡are realization of

three vatued logic, [o], Ir:] lz+1.

I.2.4 Res j-due Numbel Sy.stem

The residue number system is not a weighted system.

Therefore its arithmetj-c operations are carry free which isthe main attractive feature of the system. since carry does

not propagate from one resio.ue to the next one, the speed ofaddition, subtraction and multiplicatj.on is faster than the

eqnivalent binary system. Ì.{oreoveri in the case of multip-l-ication, the storing of the partial prorluct is not required.

7

However, sÍgn detection and operations involving sign detec-

tion (comparison, overflow detection and division) are not

so easy as.in the binary system. Furthermore, sj-nce residue

arithmetic is an integer arithmetic, floating point arith-

metic is very dÍfficult. So far as j-t is known there is

only one residue computer developed by RCA [Zg ]. Limited

uses of residue numbers are found in error detection and

correction algorithms lzs] lzll. The details of residue

arithmetic operatj-pns will be discussed in Chapter 3.

INTRODUCTION TO BINARY ANDNEGABINARY NUMBER SYSTEMS

As has been discussed earlier, the binary number system

is a two val-ued (ù,1) number system with its radix (base) 2;

whilst the negablnary system j-s a two valued system with

radix -2. Since two state physical entities are readily

avai-lable the binary system has an advantage over all other

-systenìs. Holl'ever, carry propagation delay is the main dis-

advant-age of the binary system," specially in multiplication,

which involves a series of additions and speed is limited by

the carry propagation <lelay in the additions. For complement

numbers, multiplication needs sign correction for negative

operand/operancls .

fn the negabinary system the si-gn is implicit therefore

multiplication does not need correction for negative operands.

However, this system has got a twin carry problem which iseven worse than that of the binary system, and also due to

the implicit nature of the sign, the sign detection is not so

easy a.s .in the binary system.

Later sections give a review of arithmetic in binary

and negabinary systems.

2.L Binary Arithmetic

2

Any of the

ground to b-inary

standarct books l:0,

arithmetic" Binary

^-'131 ) can give the back-

numbers are presented

B

9

either in signed 1's complement form, or signed magnitude

form or 2's complement form, However, whatever be the form

of representatiorr, the sign is always represented by the most

significant bit (tuse¡ .

2.1,.1 Bina-sy Àddi-Eicn and Su.btr.asti-on

Tables 2.I, 2-2 and 2.3 show the

addition/subtraction of numbers in 1's

magnitude form and 2's complement form

algorithms of binary

complement form, signed

respectively.

Table 2.I

A1gor:ithms for binary addition/subtractionin l's complement form

Method Operation Remark

(i)

( ii)

(ii i)

(iv)

Direct Addition

Direct Subtraction

Addition by Subtraction

Subtraction by Addition

Treat sign bit as a numberas long as the registercapacity for the nunber isnot exceeded.

Add (or subtract) end roundcarry (or borrow), wheneverit occurs, to/fron leastsignificant bit of surn (ordifference) .

7=X+V7=X-VZ = X -V7=X+V

7 is l's complement of Y

Example: è^ddition of (+9) ro ano - (6) ,o in 1's complement.

0,10011, 1001

End Round Carry -->1 0 0010I

0 0011 (sum) "

10

Table 2.2

AlEorithms for binary additionr/subtractionin signed magnÍtude form

Addition

Subtraction

Same

DifferentZ*=X*+V*z*+x*-V*

Signs of X and }/ Operation Sign of Sum

sign (x)

sign (x), if itdoes not overflow;tgn (Ð , if itoverflolvssj gn (x), if itdoes not overflow

sign (x), if itoverflows

sign (x)

*

s]-gn

sign

(z) =

(z) =

(z) =

(z) =

(z) =

(z) =

s]-gn

Same z X* Y* sign

sign

Different 7* - Y* ¡ Yx sign

In Table 2.2 negative number 7* (after operation) is in2's complement form. To change negative 7* to the signed

magnitude form the magnitude of Z*is 2's complemented.

Example: Addition of X +0101 (+5 ) and V -1001 (-e) .

0101100 1

I 1100overflow

Therefore, the sign of 7x is 1 [ =

and the sum is 2's complement form of -4

converted back to signeo magnitude form,

s j-gn ( x)

. This must now be

l

i.e. to -0100.

1l_

Table 2.3

Algorithms for binary addition/subtractionin 2's complement form

Itfethod 0peration Sign bit

(i)(ii)

(ii i)

Di.rect AdditionDirect SubtractionAddition by subtractionof 2t s complement

Subtraction by Adclition

Treat sign bit as a number bitas long as register capacityfor nunber does not exceed.

7=X+7=X-Z=X-

v

vv

(iv) 7=X+V

V is the 2's complement of V.

Exarnple: Additi-on : *(fO¡rand -(7)ro

(0,1010) z(t, t_oo1) z(0,00Lr) z 3

From the study of algorithms in Tabl-es 2.7, 2.2 and 2.3

it is obvj-ous that 2's complement addition/subtraction iseasier than in any other form of representation. However, itis more difficult to form than 1's complement representation.

2.L.2 Binary Multipli cation

Bi-nary multiplication is generally a repeated addition

and shifting process. Multiplication can be performed hy using

one of the following wide-.Iy used nethocls [¡O].(i) Direct Multiplication Method

(ii) Burks Goldstine Von Neumann Method

(if f ) Robertson's !-irst Ivlethod

(iv) Robertson's Second Method

(v) Booth's Method

(vi) A Short Cut Multi-plication Method.

L2

Binary multiplicatj-on with signed magnitude form ofnumber is much easier since it does not need sign correctj_on,.

the sign of the resul't is the Exclusj-ve oR function of the

signs of the operands. The magnitude of the result is simply

the product of two magnitudes (operands). rn the case ofsigned complement numbers, the system needs correction if any

onerlboth of the operands are negative. The Direct Multiplic-ation Method and short cut Multiplication luethod are most

suitable for signeÈ magnitude multiplication. A1l the other

methods are suitable for signed complement mul-tiplication and

these algorithms have the property of needing correction forsi-gned complement multiplj-cation.

A varj-ation of the above techniques is a multipledigits multiplication method [90]. In this merhod the two

LSB of the multiplier are examined at a time and a decision

of operation/no operation is taken accordÍng to the Table

2.4. This technique is sultable for signed magnitude

multiplication.

Table 2.4

Table for decision modes in multipledigits multiplication

MultipJ.ier digits Operation Shift Riqht

00

01

10

11

No operation

Add multiple

Add 2 ' (multiplicand)

Add 3 (multiplicand)

2

2

2

2

13

In quatenary techniques [¡Z] three consecutive bits are

examined and decision of operation/no operation j-s taken

according to the Table 2.5. An extra one bit of storage is

needed at the LSD position of multiplier which is shifted

two bits rlght af'cer each examination; while the partial

product j-s not shifted.

Table 2.5

Tabl-e for decision nLodes inquatenary multiplication

Multipl-ier bits Operation

000

001

0 l_0

100

101

110

111

No operationAdd 1 (muì.tipticand)Add 2 ' (multiplicand)Subtract 2 . (multiplicand)Subtract 1 (multiplicand)Subtract 1 ' (multiplicand)No operation

Exanrple: Quatenary Mu1tiplj-cation with

Multiplicand X = 1100 : Multiplier Y

Initial Partial Product: 00000000Extra storage

Multjplier Register: 1111 i 0011,11 1

I

00001 1

1111

11110100 Subtract X

11110100 No operation

10110100 Addition of16x

Simultaneous multiplication [¡0 ] is a method of hardware

multiplication in which the shifted multiplicand is simul-tan-

eously added to the previous partial product. This technique

is f ast but expensive. Plenty of f -iterature ref erences [:A ]

[52] are available on this technrque.

RoM mulriplicarion IS¡ ] , lS+] i=multiplication.

L4

anothe:: approach to

Integrated circuits for multiplicarion [SS I [SA ] are

al-so avai-lable. These I. Cs can be cascaded to accommodate

nrrmbers of large word length.

2.L.¡ sj¿ar.y Ð:feion

Binary division is commonly performed using one ofthree methods [¡O ]

Method (iii) Non-restoring Method. The ¡nost comrnonJ.y usecl

method is the non-restoring method. The algorithm for the

non-restoring method has been given in Table 2.6.

Tab1e 2-6

Algorithm for binary division by thenon-restoring method

Remainder

Partial- remainder

Quotient

Correction

Rn = 2-rL rn

rn= 2rn-r+ Í-2q")Vgn = 1 if rn-r and Y

Ç[n = o if rn-t and Y

O-(-1+2-n) + î qiI=I

Add (-1+z-o¡

areare

ofof

same signdifferent

sr-gn2-( i-r )

Array

method have

circuit for

divisions irnplementing non-restoring divisionbeen proposed in [Sg] [O¿ ]. An irrregrared

binary division [ZZ ] is also available.

2.2

2.2.1 N .ine¡¡¿ Number

General expression for a

number i-s

resentation (fnteqers

negative radix integer

a(-ß) 6"i

where ai. ß-1

For negabinary the expression 2.I becomes

ø (-2) a"í (-2)'

No lj-teratu::e on the representati-on

abinary number is avai.lable, however, the

of fractj-onal number j-n neqabina-rir system

0

1ß

o

ns|IJ

=

1

1

nTJ

i=o

15

(2. r)

of fractional neg-

general expression

is given by

Q -2\

where0<a-i1

Thus (11011") _z = tx(-2) a+1x (-2)t+Ox( -2)'+lx( -2)r+1x ( -2) o

= 16_8+0_2+1 = (7) r o

and (01111,) _z = 0x (-2) {+lx (-2) 3+l-x ( -2)2+lx( -2) t

+lx ( -2) o

= _B+4 _2+L = _ (5) ¡ o

From above two examples, it is seen that, if the non-

zero MSB is in an even position, then the number is posicive,'

and if the non-zero MSB is in an odd posltion, then the

number is negative.

The conve:-'slon of integer number of positive base to

negative base and vice versa has been shov¡n in [00] - [0e].

2.2.2 lIçqeþf¡_rff lrlumJcer Repres_entation (ffactÍoq)

n.L1=O

a(-2) CL

- i.{- r(-2) -i+l

16

where a"í f (0,1)

The representation of fractional negabinary number

needs two extra bits before negabinary point.thus

Thus

(11.0111)-z

2.2.3 Polarisation gE Ne$þin_AÐa Numbe.r

Similar to, the 'complement'of Binary Numbers there

exists a 'Polarisation' IZO] of Negabinary Numbcrs. The

procedure to polarise a negabinary number can be stated as

follows: Starting from the LSB complement immediate

next bit following a '1' .

Example: (a) polarisation of (5) ro = (0101) -z is

(11!1) - z - (5) ro

(b) Polarisation of (-3) ro = (11-01) - 2 l_s

(0111) -z (3) ro

2.2.4 Neqabinary traction

A truth table for negabinary additicn, ês presented in

[OS], has been shown in Table 2.7.

Lx(-2) r+1-x (-2) o+Ox ( -2) - t+tx (-Z¡ - z

+l-x ( -2) 3+1x (-Z¡ - u

f r aìrgl- lroJ ,o .

L7

Table 2.7

Logic table for negabinary addition

Number

xy Sum, S Carry Cr Co

0

0

0

1

0

0

0

I

0

1

I0

1

0

i

0

ii

00

Twin Carry

The twin carry gro\^¡s to unmanageable numbers as the addition

moves towards the MSB position. However, carry-borrow tech-

niques [Og ] ana llZl can minimize the twin carry accümulatj-on.

The technique can be summarised as: A carry generated in any

bit position is consldered to be a borrow to the next bitposition¡ a borrow generated in any bit position is consid-

ered to carry to the next bit position. Example:

+

(5) ro

(3) ro

(8) ro 1 0 0 0

Truth tables of addition and subtraction, according to

carry-borrow technique, have been shown in Tables 2.8 and

2.9 respectively.

1

I0

I0

0

0

0

it

L

L8

Table 2.8

Addition Truth - Table

Carry

Ci

GeneratedCatry

ci* r

GeneratedBorrow

Bi*tXYBorrow Sun

Bi S

0

0

0

0

0

I

0

0

1

I

0

I

0

1

0

0

0

0

0

0

0

0

0

0

0

I

I1

0

0

1

0

0

0

1

7

0

a

0

0

1

0

0

1

0

0

L

0

0

0

x

0

0

I0

0

L

0

0

I

0

0

0

1

1

I

0

0

0

I

1

L

0

0

0

0

0

0

I

I

L

I1

I

19

Tab1e 2.9

Subtraction Truth Table

Numbers Carry

Ci

GeneratedCarry

Ci+ r

GenerateclBorror'¡

Bi+lxyBorrow Sun

Bi S

0

0

0

0

0

0

0

0

1

0

0

0

0

1

0

1

1

0

0

0

0

0

l-

0

0

1

l-

1

0

0

1

0

0

0

1

1

0

I

0

0

1

0

0

L

0

0

1

0

0

0

1

0

0

1

0

0

1

0

0

1

0

0

0

1

L

1

0

0

0

1

1

1

0

0

0

0

0

0

1

1

I

1

1

1

Subtraction can also be performed by the method of

addition of the polarised subtrahend l'lZ). This technique

is like subtraction by addition of a cornplemented subtrahend

in binary subti'action.

The addition and

have been of a 'ripple

carry look-ahead adder

subtraction methods discussed so far

carry' type. For fast addition, a

has been proposed by Agrawal Izo].

20

2.2"5 Neqabinary Multiplica-uion

The methods of multiplication in negative base have

been discussed in [70] and 114). Hoh¡ever:, a variable-bit-

shift negabinary multiplication process IZg] is an interes-

ting method. In this method trn¡o consecutive l-east signif-

icant bits of the multiplier are examined at a tj-me and the

decisi-on of arithmetic operation,/no operation is t.aken

according to the Table 2.LO-

Table 2.IO

Table for decision modes innegabinary multiplication

T\so bits ofmultiplier

/i+ r V i

Arithmetic operati-onrequired

Number of rightshift

No operation

Add Multiplicancl

No operation

Subtract Multiplicand

However, this method does not work when two bj.ts of the mul-

tiplier are 10. Multiplication of a non-zero multiplicand

with the multiplier 10, will produce zero result according

to table 2.IOi whereas the actual result should be (-Z x

multiplicand). Therefore, this method needs correction. A

correction has J¡een proposed in section 2..2.7.

2.2.6 Negabinary Divi sion

2

2

1

2

0

1

0

1

0

0

1

1

Elaborate discussions on J-nteger di.vision in the neg-

qiven in lzo ] ano [zr ].ative h¡ase systern Ìtave been

Agrawal has

division in

2T

presented the method of negabinary j-nteger

IZS¡, and this has been described below.

(1, b, and q be the non-zero dividend, non-zero

divisor and quotient respectively. The at b and

exp::essed as

Let

poJ.arised

q can be

(L=

Li=o

m-n+ 1Lo

6-r,

4í*n- I6rr- I

4i+n 4i+n-1

6n 6rr-r

&í*r*l eí+n

6"

ai( -2)í

6i ( -z ¡i

c¡ i( -z)í

mç

1=om

b

q

The division is started with a leading zero added to a" and

the most signifj-cant bits (t"ISss) of b alì-gned to a.. The

quotient bit is evaluated after shifting the polarised

divisor one bit to the right. The quotient is taken to be I

if the addition of the polarised divisor is possible. cener-

ally any one of the following conditions exists after shif-

ting the polarised-divisor one bit right when the l-th, quotie:rt

is being evaluated-

(i) o a-i

6'o

4s

(ir)

(iii)

eí+r'-2

Tn- z

0,í d-s

6o

4i+n- 1

6o- r

i+n- 2

n-2

a

6

&g

6o

If condition (fii) exists, the polarised additj-on is always

possible. If condition (ii ) exists, the polarJ-sed addj-tion

i s never pos,sible . If condition ( j" ) exi sts , polarl. sed addit-

ion Ís possible when the rnagnitude of Pcri is equal to or

smaller than the dividend (or thej-s given by the multiplication ofgiven as:

22

partial remainder) . pcri

6 ny e.rí, where ctcrí 1s

Bit No

Value of {cri

i

o

i-l í-2 i-3

I 1

2r0111I

the rightmost bit is 1 when i is even; otherwise it is zero.

Þ'cra 6'Q.rl

However, the method described above is not applicable

to negabinary fractional divisj-on. Therefore, one method for

negabinary fractional division has been proposed in sectj_on

2.2.8.

2.2.7 å correction propgsed on variaþle shiftneqabinary multiplication

Except for one correction, the method j_s the same as

that di-scussed in section 2.2.5.

Table 2.It

Algorithm for negabinary multiplication

Two bits ofmultiplier

ví*L viOperation Right Shi,'t

No operation

Add Multiplicand

Add (-2. rrlultiplicand)

Subtract MultipJ-icand

[-Z'(muttiplicand) is obtained by shifting multiplicand one

position left. ]

2

2

2

2

0

1

0

1

0

0

1

1-

23

The correction is only for V i+ t V i = 10.

EXa¡plç.: MultÍplication of (3) ro by - (10) ro

(3) ro

(-10) ro

(0111) -2101001110

1110(100110)-2 = (-30)ro

According to the proposed correction, the multiplier

is a 2-shift multiplier and algorithm is same as bhat of

binary multiple digits multiplicatj-on (table 2.4) .

2.2.8 Method Prgpo.Ped on_ Nsggþj-nall¿ Fr.a.cjio! Divi sion

The proposeC method assumes that both divisor and

divj-dend are fractional and divisor is greater than dividend

(i.e. X<y). The quotient Q can be written as (from negabin-

ary representation of fractional number section 2-2.2) "

e = atG2)'+ao (-2)'+a-r, (-2)-'*o- r(-2)-'+ t&-n(-2)-n

= -or(2)t+a.o (2) o*r!, a-í(-Ð-í

Let -a (Z)r+æo (2) 0 be represented by two bits QtQo

Therefore

(Negabinary Addition)

ne - QtQs+.'i a-í(-2)-íI=I

If CL_i = 1

(i)

(ii)

If tl-i = ot

(iii)

then

(-r)-í o-í = -1 if:(-f)-''a-í = lif

'then

i is an odd numk¡er.

í is an even number.

(-1)-'a-í = o

24

In the negabinary system -l is repl:esented by two bits,1 1. Therefore, at feast two bits are needed to represent

the set (0,1,-1) in negabinary system.

In other words 0 can be written as

Q = QtQo Iot ( -2) r-Fao ( -Ð ol.z-í

-i+1 -l= QtQu + (-a72 *0"s2

-at2-1+r+ao2-t can be denoted by two bits

Q-i+t Q-í

ThenO-eteo+ Q-í+tQ-í

(fne negabinary point lies just after Qo bit)

n+I

i= InL=

Irs][øaJ ro

X=

l-

nt=]-

Fig 2.I shows the flow chart of negabinary fractionaldivision process, and it is similar to the non-restoring

fractional division j-n the binary system.

Divide 15IO

364

byB l0

v- 3

(00.110007) - z

B( 01. l-0I) - z

1,,

(

00. 1100101 . 101

00.011001

00. L1001

00. 0101 101_.101

00. 0c1 1 1

00.0111

00.1L0101 . 101

00.0111

00 . 111

01 . 10101. r_01"

00.000

25

Addition, since sign does not agreesign agrees with the sign of V. Therefore

QtQo = 11

l-eft shift

polarisation

subtractionsign disagrees with that of Y.

Q oQ't = 00

left shift

polarisat j-on

additionsign agrees with that of Y.

Q-tQ-z = 01

left shift

Therefore

Therefore

polarlsation

subtraction(for zero rentainder Q-zQ-z = L t,since i=3 is an odd humber)

Therefore Q = etQo+8oe-t+Q-tQ-z*Q-ze-z

5- B'

26

tN ( xrY)

i.-0

?5¡: SY

T <_ X_YT + X+ Y

7ST = SY

<_.t Iq<- 00q r9o

Yes ls

?

n

T <-'ir-

X +Y+l

x <- porlt'(il ]

Q+- k:0q

rlq-k

PotI l)X+T <--i+

X+Yi+ t

END

5

T:07

Jlsod1SYST

ls

2

e- ll<- 0l1

Yes

Yes

¡¡. q. e 00-l.l -l

dd

lso,,

<- ltqo-i

- ot

Ye 5

Yes

N

No

No

Yc s

Ycs

Fig.2.1 Fiow chort of negobinory f roctionol division proccess'

27

2.3 Conclusion

The details of binary and negablnary arithmeticmethods have been discussed in the preceding sections. Itis clear from this study that al-l the methods of binary

arithmetic can, with slight modification, be used in neg-

abinary arithmetic. However, the twj-n carry propagation isthe main disaclvantage of the negabinary system. The carry-

borrow technique of additionr/subtraction is serial in nature;

therefore, it is inherently s1ow. The negabinary carry look

ahead adder is complex and has no speed advantage over abinary adder. Polarisatíon, similar to complement in the

binary system, is a serial type of operation,' therefore, it

is a slow process. Negabinary multiplication does not need

correction, but additions of partlal products (pp) are

slower, since negabinary addition is slower than binary

addition. The proposed negabinary fractj-onal divisionmethod is similar to the non-restoring binary fractionaldivision, however, since polarisation and negabinary adoition

are sl-ow in comparison with the complement and binary add-

ition, the process of negabinarlr division is slower than thatin the binary system. Above all it can be concluded that the

negabinary system has no speed advantage over a binary system.

Therefore, other irumber systems which are carry free are well

worth investigating,

3 RESIDUE ARTTHMETIC

I.,ITERATURE REVIEW

3. 1 fnt oducti-on

Due to the carry free property of residue arithmetic,addition, subtraction and multiplication are all faster inthe residue system than in the binary system. However, sign

detection, operations involving sign detection (comparison,

overflow detection) and dlvision are slow processes. Also

since residue ariLhmetic is an integer arithmetic, residue

floating point arithmetic is very difficult. Similar1y out-put translation (resj-due to decimal) which, aÌthough less

frequently usecl, is still a problem. However, due to the

carry free properties, residue arithmetic systems are, well

worth investigating as an alternative to the conventional

(binary) system. A comprehensiwe knowledge of residue arith-metic can be gained from [ze ] , lZgf.

To be of use in any general purpose computer any number

system needs efficient aJgori-thms on sign detecti-on, overflcw

detection, division, iloating point arithmetic and output

translation. Later work in this thesis investigates suitable

i-mprovement in some of these areas for the residue system.

The remaining sections of this chapter discuss resj-due

addit-ion, sr.rbtracti.on. multiplication, existÍng algor-ithms

on division, overflcl detection, floating point arithmetic

and output transl-ation.

2B

29

3.2 Tnteûer Residue Arithmetic an outlÍne

3.2"L Acldition Subtraction and Multipl ication

rf (Prr, þzx¡ þsx,

are the residues cf the

the residue system of

addi ti on/ subt::act ion

There are three

must be considered in

, Þay) and

(Vrqo Pzq, PsA,

operands X and V respectì-vely in

, ñ), then

mn

pn,j)

moduli *i (í=7,2,3,

and multiplication can be expressed as

(f) Ai{dition/subtraction:

lXtVl| !M

where m1

lX'/lr,r

= lnr*.n rql,-r', lpr*.P rql^r,

lnr*tnrql^r., ,lnn* tpnql

M-trl1= I

(ii) Multiplication:

lPrx'Pyll^r, lPr*'P4jl^r,

lPrx'Pr,¡ln r', ,lPn*'PnU,l ^n

Combinational logic or look up tables in suitabl-e mem-

ories can be used to impl-ement addition-subtraction and mul-

tiplication processes. The modulus whj.ch needs the most time

in operation determines the speed of each overall operation.

?t) Di.vi sion

3.2.2a Jhç tvpes of division

types of division [ze ] each of which

contemplati-ng poss.ible implementations.

Category 2z Scaling : Division by factor,/fact.ors of M.

Category 3: General Division : Division by any arbitrary

numbers.

Category 1: Division Remainder Zero :

of division, the dividend

divisor.

General Division is of particular

a general purpose computer both divisor

arbitrary numbers.

3.2.2b Methods of General Divlsion

There are at least two methods for

in a residue system - General- Division I

General Division II I Ze] .

Division I and is restricted by

General Divi-sion I needs a look

numbers 2' (í=0,L,2, ...., N) in

highest. poi^rer to 2 contained in

is formed in memories, where the

30

Tn this category

is a multiple of

interest, since in

and dividend are

accomplishing thís

I ze] , þol and

a special requirement.

up table which contains the

residue form. (u is the

f$ -rl). rhe l.ook up table

address represents the

General Division II process is slower than General

exponent (power) to 2 and the

in residue form. This method

and. the divisor Y

contents gj-ve the value oÍ

assumes that the dividend

and nonzero. The detail.s

2í

x

are posi-Live

gi-ven below.

of

the procedures are

(i) Find c(y) =2k. (Starting with the highest arldress

number ( N) , the contenLs of each location of the table is

compared with V, until 2l' < y .2k*') .

2N:

-c(v) '

and c(y)

(fi) Find the first approximation, F

and G(/) are the numbers of powers of 2

can be found by division of Category 1. )

31

(since 2N

t 2N, r

(i_ii )

chart

Follow the

of Fig. 3. 1-

j-terative process shown with the flow

To determine

[¡n ], it requires on

by fpsqfithmic search technique

average

(3.1)

F it requires

(3 .2)

c(v)

t-he

\ tog, " modular operations [¡o]

After c(y) has been obtained, to find(on the average) additional

(n+2) modular operations

Iterative process needs on the average

(Where n is the number of moduli; c = logz W:/Z-L))

c2

(n+1) modular operatiorrs (3.3)

Therefore +-otal division process requires, on the

averagen; -Log "

c+n+ , * i (n+l-)

operationsmodular (3.4)

32

STA RT

L +-0j <--oQr-êo

i e- i+t

R.<- X- Y.F)

ls

¡itiveRj

L

qL+<_

L+ I

F

YesIsR o?

lsF=0?

F <_ F/2

X<- Ra *ãou

ENO

Fig. 3.1 Ftow Chort 0f Iterot,ive Process inresidue irrteger clivisioll .

33

The first approximation F can be det.ermined by fgg_ç_

{fv¿.çfpn technique [¡O] from the value of E(X) and E(y) "

The steps that are followed are:

(i) Convert X to its mixed radix form (Appendix-s)

and pick up the most significant nonzero diqit

(ii)

auv"

Find

table which

2i' ( í=L ,2 ,

2N < (n-r) <

n(lLux+r )np)

table.

(say).

ç(.(aux+l )np) and c (øu*. nt )

the value

where N is

R

from a look up

of

such that

). Also find

another Ìook up

contains

N),

2N*1. (np mi= I l-

and E (au*np) from

(iii) t¡ind x' x- c(lctu*+1 )np) and determine

radix conversion.

the sign

If sj-gnof

is

X' by another mj-xed

negative c(X) = G(a"np)

and E(X) = n(arr*nr).

+1)np) and E(X) =

i where i = I,2,3, ....N).

otherwise c(X) = c((a

E(lau*+r )np) (e (x)

Similarly find E ( Y) .

u

UX

The F j-s given by

2E(X)-E(Y) (in residue form)

If two mixed rad.'i-x converters and assocj-a.ted look up

F

tahles are

ati-ons.

3.3 Overflory

If m, (i=1,213, n)t_

residue s¡,:stem and the modul-i

then t-he range of the numbers

available, F can be obtained in 2n rr,odular oper-

are the moclul-j. of a par.-ticul-ar

are pairvrise ::elatively p::ime,

that cen be rrniquely represented

l-s

the

of

gr_ven

range

negative numbers

-1] and

34

by M = *.t_, *r. If one of the moduli is even, then- l=r L

of the positive numbers (posit_ive half) and that

{o to tt/z

An overflow is said to have

an arithmetic operation exceeds M

half or negatlve half.

(negative half) are defined by

{¡rt/z to M-1} respecrively.

occurred if the result of

of positiveor the range

3. 3. 1 "a:¿eff].ç¡g due tq ditio subi:raction

An additive overflow can occur if two operands X and

/ are of same signs. rf oper:ands are positive and overflow

occurs, the result falls j-n the negative half. For negative

operands, the result falls in the positive harf. Therefore

additive overflow can be easj-ly detected by detecting the

incorrect sj-gn of the result. Subtractive overflow can occur

if the signs of the operands differ. since subtraction can

be reduced to addition (X + (-y) ), subtractive overflow

detection is similar to the additive overflow detection.

3. 3. 2 ¡4qlÇjplrcel;LvC overf low

When multiplicative overf-'l-ow occurs, one of the two

fol-l-owing phenomena occurs (-r.) the resul-t j-s of incorrectsign (exceeds the range of posit:i-ve half or negative half ) or(i-t¡ the result is of correct sign, but incorreet magnitude

(exceeds M) .

The overflow of 1=irst, category can be detected by

checi<ing the sign of the product- (wheL-her the slgn is incor-

rect)," When overflow is of secon,d catecrory, the following

35

relation I za] is va1id.

G

x

Then G

lxY lr'rx

* c(v) (3.5a)

Fi::st -lJj]-* t* calculated by techniques of General

Division I.x

arithmic search technique.x

time will be needed to find it. However, íf the expresslon

(3.5a) j-s w::j-tten as (3.5b), the division -ltn-l-O can bex

avoided.

c( (xy) r"l)G (X) I c(v) (i.5b)

c((Xy)r,r), c(X) and c(f) can be obtained by fast technique

proposed in [30]. Tf two mixed radix converters are available,

the overflow of second category can be detection with

4n modular operations (3.5c)

Multi.plicative overflow can also be detected by using

a redundant modulus [Za]. The time required. j-s at the most

3 MRC (t',lixcd Radix Conversioir,' Appendix-S) with n+1 moduli

plus 1 modular cperation. This is equivalent to .3n+1

modular operations. (1MRc is equi-valent to (n-Ð modular

operations fAppendi*-s] with n moduli).

3.4 1 tin Point Ari

At present it j.s diffjcu.Lt to perfo.rrn floatirrg point

arit.hmetic in a residue number systenì. There is a lj-mit-ed

amount. of published work on this problem, for example, [¡¡ "]

I xY I r,r

36

has proposed 2 as t.he base of the exponent, while [:Z] has

proposed a base cf 10. However, ho details of floating

point arithmetic operations have been given. The algor-

ithms p::oposed in [¡t] are of particular interest, since

details of arithmetic operations have been presented with

j-lIustration. Furthermore, the format used is different

from the conventional format. The details of the algor-

ithms ha,ve been given in the f ollowing sections "

3.4.1 The representati-on o f ! : o_a_!i¡9. p-e.i¡tç. number:s

A floating number can be represented as

k Me

(u-f ) ì e- is an integer

(,s I o)

(,5 o)

M

^where kl . *t-¿

^

8._X -k ML

L

b

M

M

M

fimii= t

1

and In1 4 Ir2 ( In3 < <mn

3.4.2 Floatinq Point Aqithmetic

LetX

*o* and VU

nIImii= I

and 7 - Fr(x o v) =krL¿z Mo,

arithmetrc operations (addition,

multiplication or division) .

kØ,t

M7M

^awhere O denotes

subtraction,

3.4.2a AlcloJij-thms_ f o¿ floatinq point addj.tionsubtr et- ion

ØX U

(ft is assumed that and X, / are positive)

( 1) F.ind A¿

FI (X!Y )

(2) rf Lø =

If it is

k brv m-a

Let the

(3) rf Az =

k

(4)

ø rf Lø > 2,

I, test whether Á

true Fl(XtY) - X

3l

>Aa

otherwise scale [Zg ]

Uø..

,l

x

m2

result be

mm^x ^ u*'

*o ,* "

mn.

^Tf it is truetest whethe:: ¿.. =

lt

Otherwise, scale k

KR

0,

SO

' ^r-r* t

a

KR

I'ind k= = k* r K*

mm,5+1

^a

b)' +zll a a*O* and the result be KR.

and set Lz = øX ,ì Áz^x.

þuut.to'mn+ I l-S

and m' l

zeros,

is done using redundant mod.ulus rn¡+1.

chosen that (M-1) ¡, + (NI . m¡+t - 1)-¿

(5) Find the

n+1) of k

If o"í are

(6)

set ¿ = 0z

,L=0z

mj-xed radix digits (1"í (i = 1,213,

_ which is associated with moduli mi.z

Otherwl se ,

zero diqi t "

increase ^ z

increase Q-z

denote by

IfU_

& the mostu

n*l, scale k,

signifj-cant non-

bv m and^

+1

0 and.by

by

1

1

rf ^ -zn, set á

z

(7) If U < n,

which can

U ancl á. ,

Tf^<n,qotoz-1 and decrease

âz+n

T_= A ,+ LL+ t mi

permanent table by using

1,, and increase .52 by U.

If ^

> n, increase ¿z- z

rt, go to step 5 "

multiolv kt4

be read fron-r

decrease ¿

by

by

a

step 5.

by /Jz

by

3B

(B) If U

k is normal-ized.7

+1) from look up

+1) and determine

multiply k- by m¿z

n, test whether á. 0. If it is true.

rf^ I o,z

table. Find

( (u lmo. )

( (rulmo.)

I21

2

fetch

lk.l -the sign. Tf

and decrease

sign is

¿ bv1z-

negative

7

3.4.2b Alqorithms for floatins poin! multipl j- cat j-on

(1) Find ln"lr. and lkrln,.{i = n*1, ni2t n+p)

by method of base extension IZe].rl¡,tuttiptication is done using p numbers of redun-Ldant moduli to prevent loss of information due to

multiplicative overflow. The redundant moduli

are so chosen that p < n and

(M-1) 2 m m m m -1).n+pn+I n+2'|(¡r-z14

and 'iln+r ( Irln+! < .... <

(2) Set e.

(3) rf ^ xk

za

k

z

a

Øz

=Ø+e ^.x^x

a0, find k

fíndk -kzxentat-ion of mr

wise, increase

result bv m6,,1!a

l_s

. kA, and

where the

from look

find

residue repres-

up tabl-e. Other-

k and scale thea

if 1_La

. ITI¡ r

read

bv 1,

md q*'

k*

(4) Find the mixed radix

(i = 1, 2, ...., n+p)

arezeros, setu^z= 0

the most significant

(5) If U > n*2, replace k

mn to get k..

digits

ofkz

O=, -z

non-zero digit by au.

(Appendlx-5 ) o"i

If all the digits

0. Otherwise denote

t: +u-n. )

!Ía rn, *t jz

kz

by

and increase ^

by LL-n. If ,5, = n, Set á

by 1. Go to step 4.

39

) and increase

a

0z

and increase L 7

(6) rf u n*1, replace k

7

by (kz I

^z+r

n,

mz

¿ bv1z'bv 1.

table,

If sign

. rf ^

retcrr Icompute

= n,

(M-1)

lk. l

zsetá =0andincrease7

(in resiclue form) fronr

Øz

+ (M-1) and find the sisn.

is posit j-ve, replace k, by (n. l^0 r., , )

and j-ncrease o, by 1

and increase u, by 1

rf [ -7set ó to

zzero

3.4.2c Al-qorj-thrns for f1oat-14q p_g¿n! clivi sion

Let X and / be the dividend and divisor respectively.

(1)

(2)

(3)

rind lk* l*1by method of

Multiulv kx

up table.rI The value ofL

a) rfL >' x-b) lfA

where B

c) rf.

rfd)

and ifrl*1 (i = n*1, n+2,

base-extension.

n+p)

by A which can be read from a look

A depends on and óU

and ¿

and Á

and C ^x

^ x

U

a

q

a

0

0Jr^

^

oq I O,

A=M

[=B.CX

nIT mií= 13

a

and

:îTm1

+t i-= ^

A M

B . D where

l

x-Áa*tA

^

If^<^,x u.

L x

^ X

a

qMA

^ x

nf = Tmii=n+ó y-,5 ,+ t

decrease a. by 1 and increase ^-

byz'n

(4 ) l-ind k (kx. A I kt) by scal-ing.z

k = 0,(rrz

division i,s undefined)

ltO

3. 5 Translation of Residue Numbers to Decimal Numbers

A number j-n residue form j-s not readily understand-

able. Therefore the resiclue systern needs a provision for

translating residue numbers to decimal numbers. Since

output translation is l-ess frequently used, more time can

be al-lotted for translation than for arj-thmetic operations.

The Chinese Remainder Theorem [ZS] can be used for

output translation provided that the system has the fac-

iJ-ity of mod-M operation. The A(X) method [34] does nor

need mod-M operation, however, it needs super modulus to

determine a(X). and the method is applicable only if the

base tÁ) (in this case 10) is pairwise relatively prir".e with

mi. Benerjee and Brzozowski [35] pointed out that the

method of base extension [24] can Ue used provided (i) gcd

(ø,mi) =1 or (ii) gcd (w,mi) I 1 for only one modulus and

(¿l ) ffi; for that particular modulus.I

lgca stands for greatest common djvisorl

3.6 Conclusion

It is obvious from above discussions that addition/srii¡traction and multiplication are straight forward operat-

ions; and delay is equal to the delay of one modular cper-

atlon. The c)peration-s can easily be j-mplemented with memor-

i-es (took up tables). Therefore there is no real problem inpractical implementation of these operations.

Division , hov/ever,

tion and multipJ-ication.

is not so easy as

Determinatlon of

addition/subtrac-

the first approx-

imation F which needs f-n12

logz c + n+

4t

moCular operations,)

using logarithmic search technique is obviously a t.ime-

consuming operation. However, fast division technique can

red.uce that time (for determination of F) to 2n modular

operations. This

faster. Still an

technique can be

improvernent, is

irìêêrrs that the dlvision process will be

investigation, whether a fast division

:Lmplemented in a different way with speed

wc::thwhi1e.

Mu-t-tipl-1 cative overf low detectlon in redundant res-

i.clue system is faster (3n + 1 modular ope::ations) than in

non-redundant system (4n moclular operatjons). However, a

redundant system needs an extra modular operation which may

not be desl::able. The choice of redundant modulus may be

critical. This means that if the redundant modulus gets

large, implementation of operation with that modul-us may be

difficult and operation may get slow. Again, the introduc-

tion of a redundant residue decreases the available range

of tÌre system which may not be appreciated. Therefore an

alternate approach Ís to choose the non-redundant system

and carry out an investigation for faster overflow detection

technique.

As discussed earlier floating point multiplication

and divj.sion need p numbers cf redundant moduli to prevent

loss of information due to rnultiplicative overflow. This

compels the system to have the capacity of (n+p) modular

operation units anC the capacity of l.iRC hrith (n+p) modul-i.

If p is large e"g" (p=n) , then MRC with (n+p=2¡) may be

difficu-lt. If p is chosen to be smaLl-er, each redundant

42

modulus u¡ill get larger (to meet the requiremenL stated

before) and mechanisation will be dj.fficult. Therefore it

is worthwhile to investigate alternative floating point

aì-gorithms and particularly those which can be implemented

wlthÍn the intege.r- residue ar'-ithmetic env-ironment.

In output translation, the Chinese Remainder Theorem

needs arithmetic r..'-ì-th mod-M which is a costly affair. The

A(X) method is restr'icted by the requirement tha'f: (ø,m. ¡

must be pain^lise relatively prime. In conversion of resiclue

numbers to decimal numbers, the base is l-0 which is a com-

posite of 2 and 5; and a residue system generally has one

even moduJ-us to have well-defined positive half and negative

harf; therefore the above requirements cannot be fulfilled.

The methocl of using base extension covers a wide range of

situations. However, in a system where IJ) . *i and gcd

(w,m=) I I, the base extension method Ís not applicable.a

fn that situation an alternatj-ve method must be investigated.

The next chapter proposes a

to the above problems.

few algorithms relating

APPEND TO CI{APTER 3 OF THTSIS

"Residue Arithmetic in D'igital Computers"

R.C. Debnath

FURTHER RELEVANT PAPERS

Jul I ien [82 ] establ i shes that the residue operations I i ke

addition, subtraction, mult'iplication and scaling can be easì1y'impl emented wi th I ook-up tab'les j n currentìy avai I abl e ROMs . He

also contends that scaling by the metric vector estìmate process ìsei ther equal ìy or more h'igh'ly ef f i ci ent than the exact divi si on

techni que. Res i due arì thmet'i c operati ons ( addi tì on , subtracti on ,

mul ti pì i cat'ion and scal i ng ) ímp'lemented wi th I ook-up tab'les ì n h'igh

densjty ROM can allow for a high speed digital processor.

The conversion of a residue number to a b'inary number is not an

easy process. The method based on the Chinese Rema'inder Theorem requ'iresresidue arithrnetic operation with Modulo-M, and ìs therefore a costlymethod. A nrore efficjent method is to use the Mixed Radix Decodìng

technique [83] " This technìque needs less memory ìocatìons and can

also be ìmplemented wjth current'ly ava'ilable ROMs or PROMs.

Add'i ti onal Ref erences : -

G.A. Julìien, "Residue number scal'ing and other operat'ions usìng

ROM arrays", IEEE Trans. on Computers, Voì .C-27, N0.4, AprììI978, pp.325-336.

t82l

t83l . A. Baraniecka and G.A. Jul1ien, "0n decoding techn'iques forresi due number system real i zatì ons of di g'ita'l si gnaì process i ng

hardware", IEEE Trans. on Circqits and Systems, Vol.-CAS 25,

N0.11, No. 1978, pp.935-936.

44

from the following groupings of the numbers from 0 to M/2.

Category 1-

Category 2

Group-0 =Group-1 =Group-2 =Group-0 =Group-1 =)Group-2 =)

(0,1);

(2,3,

((uf ,

(0,1) ;

(2,3,

((+þ ,

(M þ -rt(M

1,f +r, vr/ 2)

tr\f -tM/2)

From the groupings of the numbers, it is clear that-

no overflow occurs for (Group-0) . {(Group-1) or (croup-2)};

but overffow will occur for (Group-2) " (Group-2). Howe\/er,

there is no certainty whether overflow will occur for(croup-l-) . { (croup-1) or (croup-2)}. Even if overf low

occurs, it is not clear whether the product exceeds P(>z\ or

N(%) or (M-1). This is due to poor resolutÍon of nurnbers ingroups 1 and 2. Therefore a 'Logarithmj-c Distribution' which

gives hiqh resolution of numbers in groups is proposed.

4 .2 . i- Loqarithmi c Di stribution of resldue numbers

The numbers 0 to M/2 (positive half ancl the highest

negatj-ve number) are distributed among several groups accor-

ding to the tabl-e 4 . L.

r\f *t,

If g(x) and g (V) are

pectively, then it can be

(apr (a)) I rhat

Case (i) : if g (X) +-

case (ii) ; if g (X) +

Case (ii.i) : if tj(X) +

the group numbers of X anC l/ res-

shown from Table 4. i I Appendix-l;

U , no over,'ll-ow cccurs;

U, overflow must occur.

LJ, or¡erf J-ow may or may not-

45

occuri hohrever, if it occurs, the product will be of

incorrect sign.

Tabl-e 4 . L

Distribution of Numbers

0

1

2

3III

^ _1"

..-6+ l,s- tlw.¿

rñ-2Ár2

^_^+ 3vI .¿

Numbers

0

1

-3W -2-^*t

-> w.2-L*u

--> w .2

1

1

-^+^-1

-^+(Á+r)

-^+ l^+2)

qLa-up.

Á+1

u-L

-^+^vr.2

w.2

w.2

-,5+(Á+t)

-¿+(A-t)

-+. VrI.2

¿ w-2I,I

-+

W 1

-1

M/2

^

fwn"r" a n*1, such that 2n S lr{ < 2n+l; in

the highest power ro 2

wL

Q4í' and ¿ q/2

contained in 2M.

+ 1. When A is odd

other words U is

When A is even,

, vr= (yþ, and

^

nurnber

rounded

number

tq*t) /zf

or

However, in most1,

ï'rí'CASCS

the

or

value of W.

1)L a Lt

whe.re L is the

is not an integer

integer value of

The lowest

¿-1) is ¡tre

lowest integer

in the (a +1. ¡t h

ëþnumber. fn those cases closest higher

wþ to be the,M-'l1)':-s chosen

jn the (¿-i)th

integer value

group (i

of L/2,

in (a-ir-l-)th group. The lor,vest' rrumber

46

group (k = I ,2 , , ó -3 ) is given by 2k .w. However, Íncase (iii-) in some situations, mult1plication of two high-

est numbers in two separate groups may be equal to orgreater than M (due to rounding error and choice of the

closest- higher integer value of (\iþ or ë-þ for w) . under

these circumstances a minor adjustment of the highest int-eger is needed. The higl-rest number of the higher group ofthe two is clecreased by a minimum nurnber¡ say o, such that-

the product of the decrease¿ num¡er and the ot.her highest

number becomes l-ess than M. The lowest number of next

higher group (w-ith respect to the higher group of the tivo)

is d.ecreased by the same number cx,.

is a rational

or ëþnumber)

Tf7,

olf is an integer

and even, then

number (i.e. M or M/2

(Á-l)th group is to be increased by 1

ber of the (,s-Z¡th group is increased

Example 1:

16x15x13x11 = 34320

M < 216 ; therefo::e rj =

= 185'256¡ therefore W =

76/2+1-9

17160

lowest number of

the highest nuin-

1.

and m = 11,

r6.

l-86.

m 15, m =

t.he

arrd

by

13If m = 16,4

construct a

3

'group2

table' .

I

Solution:

(i)

(ii)

(iii )

( rv)

(v)

(vi )

Jvl =

2t5 <

þ(l{

^M/2 :

Table 4.2

47

Table 4.2

Group table for Example 1.

0

l_

2

3

6

L2

24

47

93

186

372

744

1488

297 6

5952

1 1904

->

+

---+

+

-à---+---i

+

5

l-1

23

46

92

185

?77

743L487

2975

5 951

11903

77L60

Group

0

1

2

3

4

5

6

7

B

9

l_0

11

T2

13

L4

15

Numbers

Example 2z Form=2,3

construct a

m=3,andm=13,2\

'grou.p table' .

Solut.ion:

(i)

(ii )

(iri)

(iv)

(v)

(v-t )

Pt=2x3xl-3=78

2e < M < 2t,' therefore U = 7

ëf = 6'24;thereforeW=7

^ = (q+L)/2 = 4

M/2 = 55

Table 4.2(a)

4B

Table 4.2(a)

Group table for Example 2

Nr¿ntloers. qroup

0

1

2

3

4

5

6

0

1

2

4

7

L4

2B

3

6

13

27

55

-+

--'__>

---+

It is seen from the table 4.2(a) that the product of

6 (ttre highest number in 3td group) and 13 (the highest,thnumber in 4"" group) is equal to 78 (=M) . Therefore the

hi.ghest number of 4th group is replaced by 1,2 and the low-

est number of 5th group is replaced by 13. The correct

'group table' is shown in table 4.2(b).

Table 4 .2 (b)

Correct Çrol1Ð table for Example 2

Ntimbers GEalB

0

1

2

3

4

5

6

0

1

2

4

7

13

2B

-+_.>

-.->--b

3

6

I227

55

49

Example 3: Ifm=2,m=LL,andm=32t

consl-ruct a 'group table' .

5

Solution:

(i)

(ii)

(iii- )

(iv)

(v)

(vi )

¡4- 2xtLx5 = 11-0

2e < M < 27 ; therefore rj = 7

ëþ = 7'4L; therefore W = I

ó - (a+I)/2 = 4

M12 = 55

Construct Table 4.3

Table 4.3

Group Table for Example 3.

0

1

2

4

8

16

32

3

7

15

31

55

Numbers Group

0

1

2

3

4

5

6

Therefore multiplicative overflows can be detected by

(a) testing for incorrect sign of the procluct, and (b) com-

paring the sum of group numbers of multiplicand and multip-lier with the total numbers of groups A. If sum is greater

than U, overflow must occur; if it is l-ess than A, overflow

wil-l not occur:

50

4.2.2 !çç.:gq. J-ùe*_e.s-- f,çr Overflow Detec'Lion

Since the group number:s descrj.bed pr:ev-ì-ousIy are

assigned to the positive numbers and to M/2, and a number

may be positive or negatj-ve, the inf crmat-.ion of slgn aird

magnitude of the number is essent-Ía.l . The MRC (eppendix-5)

is the only way to get sign and magnitude inform.ation.

From the mixed radix digits (MRD) of a negatlve number it

is possible to form the MRDs of its positj-ve 'counterpart'

Iappendix-1, (apl (b)) ].

When number of MRDs gets larger, a par"t.ition between

them is essential. This partition divides the L"iRDs into

t\^¡o parts the most significant part (MSP) and the least

significant part (LSP). If the modulus Mn is chosen to be

even, and is placed at the most signifi-cant digit (t"ISn)

position drt, then ArL contains the information of sign and

falls in the MSP. With the sign informat-ion and MRDs of

the LSP, a look up table is formed for the group numbers

of positive/'counterpart' positive numbers j-n the LSP. ('fne

table 1s formcd from the decimal value (DV) of the MRDs.)

mation

i-tive

sign

the

the

and the MIìD of MSP

IviRDs of the LSP are

DV of the MRDs.)

When the number

whether: the LSP

'counterpart' .

is negati.ve the MSP

is zero to form the

From this information

a look up table is

zeros. (rne table

needs the rnfor-

MR.D of the poG-

(zero-LSP), the

f ormed, as--suming

is formed from

The

Ir{RDs (MSp

group number of a

+ LSP) . Frorn ihe

nr¡mkrer: depencis

observat-ion of

oTì the DV

the range

of all

of

51

numbers in a group of the MSP and the DV of the MSIJ, i L

will be clear that the group number of certain MRDs of the

MSP will transit to the next higher group number if certain

minimum numJ¡er from the LSP is adcled to the DV of those

particular MRDs. Those MRDs may be 't-ermed as 'transition'

MRDs and the min-imum number required for transition as

'transition' number. The transi-tion number is identifÍed

by observing whether the number, obtained as a resul.t of sub-

traction of the DV of the MRDs in the MSP from the lowest

number in the next higher group (wittr respect to the group

of the MRDs in the MSP), falls within the range of the LSP.

Therefore along with the group tables for the IvISP an<l LSP,

there must be one table contain-ing the transition numbers

and another table giving the negative binary values (2's

complement) of the LSP. (ttre transltion nurrtber of the .non-

transition MRDs is taken to be zero.) The transition num-

ber is added to the negative value of the binary number of

the LSP. The carry from the most sigrlificant bit (MSB)

position indicates that the group number of the MSP must

transit to the next higher group. Therefore the carry is

added to the group number of the MSP. The group number of

a number is given by the group number of the MSP, oE the

group number of the MSP plus one if transtion occurs, or the

group number cf the LSP if the group number of tlre I'ISP is

zero.

Overflow j.s detected by comparing t.he sum of the group

numbers of operands wj-th the total number of groups and det-'

ectillq the incorrect sign of the procìuct"

52

4"2.3 Des_i.gn E-:g-Up_lS. on Overflow nClCC!¿p:f

The moduli ffiq = 16, ffi,, = 15, ffiz = 13 and m, = lL

have been chosen for this design example. The MRDs are

designated by î*, &3, &, and ar. The range of the LSp is

13x11 = 143 (0 to I42) and thar of rhe MSp is 16x15x13x11

13xl-l- = 34L7 7 i.e. from 1,43 to 34t71 . The rrighest nurnber

in the LSP is 142 and the MSp is norì-zero for a number

grea'ter than I42.

Examining the

MSP, Table 4

'transition'

4 which

numbers,

Table

shows

can

4.2 and the DV of

the rtransiti-on'

be formed.

tl-ie MRDs in the

MRDs and

Transiti-onNumber

Table 4.4

Transition MRDs and transition numbers

TransitionMRDs

&\, &,

0

0

0

0

1

2

5

L

2

5

10

5

11

8

t

t

t

t

t

t

t

t43286

7IsL430

2660

5863

1 1869

B

9

10

1iL2

73

14

186

372

744

14BB

2976

5952

11904

186- L43

372- 286

744- 7Is

1488- 1430

2976- 2860

5952- 5863

11904-11869

=43=86=?9=58=775

=83=35

The DV of theTransition MRDs

= d\.15. 13 . 11* d2.13 . 11

CorrespondingGroup of the

MRD

The lowestnurnber inthe next

higher group

Figure 4

flow detector.

incorrect sign

sign of the X,

l- show.s the circuit for multiplicative over-

The circuit in the sol id l lnect box detects

of the product. If the Excl_us 1ve OR of the

/ and Z (SX,Sli, andSZ) is '0,, the sign of

53

r- litt,

lxt lx hr IX t,,3 3t

T, 1

þ9]g.lt.t¡¡logates ). NZ( X ):'l'meàns lXlt3 and lXl¡¡åfe r,.'n- zeÌos.

NZtX)DI D2l, l,

xtD3 t, Dl.

SXSX r(x)

i.NZ(X)

ISY

NZ(x )

I( tI(

c(MsP )

l(x )

overtlow Il 'l'

f From 'sign detectort of Z

F¡9.¿.1 Schemotic diogrom forirr the 16-15-13-11

(x)

+9(Y !

I (x) I 1

rF From tgroup generatort ol Y

muttiplicotive overf lc\.',/ det ectío nresidue system.

3

I

S SX

Tsz

NZ(Yf)

I

¿RoM - IT¡ble lor Dl .(Sec f i9.4.2(a))

PRo M- z

Tabta for 02 .(See fig.6.2(b)).

(See f ig' 1,2(d','),

PROM - 4Tabls ¡e¡ e¡ and 01,

P RO M - 3

l¿bte lor D3(See lig.1.2(crt ,

on-zcr05 e

nd I(X).04 are

btT¿ ¡aX m¿âI n

PROM - 5

r a1, sx

a

.)

;t

ber offig. L.

UM lrsec (l

¿B-O-l¡-:--9Tabtc lo r

group

PRo M -flrensitionnumbers.(See fig.

t,

PI

L sIs ee

inary vatue

PROM-8Negal¡ve

BAau:-sTable I o r

grâupnumber ofSP. (S

Futt Adde rs.

Fult Adders.

G(LS

Carry

I

t,

2:lt(x

r¡ e¡F rtecls

PR0M 0r Logic ckt.

Gener¿tes 0VERFL0W signat.

Function of PROM-1 ; (256x4)

It forns a table of D l' where

D, = lo". 1l- + d, lr.,

wlrere tL" amd cL, are the lr{RDs of the LSP.

a If Xr,,- [r^r,l*],,' [.#,,],.2

& lxl ,,I

54

Fig 4 .?-(a) : Function of PROlt{-1

Fig 4.2(b) : Function of PR0M-2

Function of PROM-2 t [256x4)

It forms a table of Dr, where

Dz=lo, !7+et1 5, where

a., and a", are same as in Fig 4.2(a).

Function of PROM-3 t (256x4)

t forrns a table of D 3' where

3 IP¡,. 1*r,, -D Il6 r6

Fig 4 .2(c) : Function of PRONI-S

Function of PlìOtr{-4 ; (256x4)

It forms a t¿rble of ctr, lvhere

tla lxl,u D It5 15

15

D4 lo, | = d.', since 15 < l-616

55

I.'ig 4.2(d) ; Function of PROM-4

Function of PIlOlvf-5 (256x8)

It forns a table of nu, SX, I(X), wherc

&

I6

t1t, if

ln, Dul4

16

SX = '1r if. nu > 7 ; I(X)

Da and Du are nonzero

Fig 4.2(e) : Function of PR0M-5

Function of PROIT{-6 ; 1512x4l

It folms a table of group number of the MSP. If N7(x)

= t0' or tl-r and SX = r0;, the decinal value, DV(O) =

44 . 15. 13. 11 * D+ . 13. 11. If SX = '1' , NZ (X) = r0r and

Du is nonzero, the decinal value, DV(1) = (LS-a.+)

.15.13.11 + (15-Dq).13.11. If SX = rir, NZ(X) =r0rand D,, is zero, the decinial value, DV(2) = (16-a.q)

.1-5.13.11 + (15-D,f)'13.11. If SX = '1', NZ(X) =r1r,the decjmal value, DV[3) = (15 -cL+) .15.13. 11 'r (14-D+)

. L3,1,1,. Tire grou¡t nuntbels arc as-signed ¿lccording to

rhe Table 4.2.

Fig 4.2G) : Func.tion of Pl{0M-6

56

Function of Prom-7 t (s12xB)

It forms a table of rt:cansitionr number

according to the table 4.4.

Fig 4.2(Ð : Function of PR0M-7

Function of PROlt{-8 (512x8)

It forms a table of negative binary value of the

LSP fron its decimal value. If SX = f 0', t-he

decimal value, DV(4) = 42.tt + 4t. If SX = I l- |

and a, is zero, the decrmal value, DV(s) =

(I3-a"r).LL. If SX = 11' and a-, is nonzero, the

decimal value DV(6) = (12-a).11 + (11-ar). e,

and a, are same as shown in Fig a.2@)

Fig 4.2(h) : Function of PROM-B

Function of PROI'{-9 (sL2x4)

It forns a table of group number of the

LSP fron its decinal value (shown in Fig

4.2(h) according to the Table 4.2.

Fig 4.2(i) : Function of PR0M-9

Function of IrROM-l-0 (256x4)

It forns a table of overflow

overfiow) accorcling to

(i) Overflow = '0', if g(X)

(ii) Overflow = rlt , íf g(X)

(not final

= g(V) < L6

= g(V) > 16

57

Fig 4 .2(j) : Function of PR0M-10

the product is correct; otherwise the sign is incorrect..

However, false indication of overflovi is given if one oper-

and is negative and another is zero. The cj-r:cujt rj-qht side

of the circuit in solid-lined box detects whether any of

the operands is zero.

Sj-nce lXlt. and lXltt contaj-n the informatjon on 0.2

and ar it is not necessary to generate 0,2 and a1. This

saves use of one PROM. The circuit in the dash lined box

generates the sign of X (SX), the MRD (ou) and the informat-

ion of nonzero c-+ and Da. Two additional clrcuits- are

required - one is for generation of the sign of y (Sy), the

group number of y (g(V)) and Nl(V) and the other one is for

the sign of the result Z (SZ) .

Performance:

Delay in overflow detectj-on is the delay through the

longest path v¡hich, for the circuit in Fig 4.I, is through

the circuit reqr.ri-red for: overf l-ow detect:ion by the 'group-

ing technique' . Therefore

Delay, D= (Dl =

+ (Dz=

+(Ds=

5B

delay in sign detection)

delay in getting Group Number of the LSP)

B bits adder delay + 4 bits adder delay

+ 2zL multiplexer delay)

delay in PROM-l-0)

delay in OR-gate at the last stage).

Again

Let

Therefore

+ (D+

+ (os

Dt =

m.Ll

J.2

TRoo

T"u"

To*

2Tt + Tz where

access time of

access time of

(?-56x4) pnou

(256x8) PRoM

= delay in: delay in

= dei-ay in

a 4-bit full adder.

2:1 multiplexer"

OR-gate.

D - 2Tt +

= 3Tr +

T2+T3+37

T2 * T¡ + 3T

T1 + Ton+

+

T +

+

where T3 = access time of (512x8) PROM

However, delay in generating group number (Og) is the

delay in generating g (X) . Therefore Dg can be expressed as

Dg Dr *Dz *Da

2Tt*Tz*Ts+37

ADD MUX

T TADD MUX OR

(4.1)

+TADD MUX

using TTL components (raure o. g (a) ),multi.pl-icative overflow detection is given

D = 3 40+60+60+3.7t-

(4.la)

the delay in

h'y

4 + 4

59

The delay in generating grcup number is given by

= 1.2O +L20 -F2I +B

= 269 ns.

= 2.40+ 60+60+3-

= B0+60+60+2L +4

= 225 ns.

(4.l_b)

(4.1c)

Dg 7+4

4.3

The method of division conslsts of two processes

(i) determination of first approximatj-on F, and then (ii)

iterations using successive approximation. (fne successive

approximation is the number which is equal to the previous

approximation divided by 2 ) , Since the group number con-

tains the information on the magnitude of the number, i! ispossible to use the same piece of hardwarerused for gener-

ation of group numbers, for determination of first approx-

imation.

The F determined from the group numbers of the oper-

ands is alwalrs positive, however, division is not limitedto positive operands on1y. The cjetail of the technique fordetermination of the first approxi.mation and successive

approximation has been givenin section 4.3.2.

4"3

The

chart in

Exclrrsi.ve:

(X is the

1 A Division Process

division process can be best descrj-bed with a'flow

Fig 4"3. The sign of the result of division is the

OR function of the signs of the operands X ;"lnd y.

dividend and / is the divisor.)

60

No

No

No

No

l=ig. ¿.9 Flcw cha rt cÍ res¡due integerd iv i sion P rocess

iN ( x, Y )

I

q.û

0

enera)

e vhardware

Is

?

F 0

0, <i- AVi, 4- Y'F

of tlre resutt

Is

+'?

the sígn0oes0verf

7

W Iow

Is

2

W>C hange q to

its negative

fornr .

E NDe sign o

resultthe +t

ls

,,

x <-x- iÂ/X+X+W

q, +pi+l

-+

Yes

s2(

hardware.)F + Fl2.

gene rated by

Yes

YesYes No

61

4.3.2 Hotg to_ f .r.:ê the f :!_r_q! êpp_Ë_o_ì(r]la!-t-A.q

-an-è SUCCCSS]-V e aÐprox-]' mations

T\n¡o look-up tables are required

successive approximation,' one is used

one to get success-ive approximation.

depends on the group numbers g (X) and

t.o obtain F and

to get F and other

The value of I-

g (V) as shown beIow.

(i)

(ri )

rf s(x) > g(x)

rf s(x) < g(Y)

schematic

2e(xl -s(YlF

F 0

(r or successive approximation is tabled irr residue form. )

Fi.rJ 4.4 is a

cuitry required to

The dotted line in

bus can be used to

multiplexer. The

iterations and to

diagram of

successive

the computer cir-

approxination.

eitherXor/data

get F

Fig 4.4

and

indicates that

issue 'previous approximation' to the

function of the computer is to perform

issue X and V to get F or successive

approximation from previous approxirrration.

If the address iines of an available are equal to or

less than the sum of bits of each of the modul-1, then the

cJ.rcuitry in Fig 4.4 Ís not feasible. An alternate approach

rr¡hich needs a smaller number of address .l-lnes to PROIvI is

sh.¡wn in Fig 4.5. Tn this case the PROM forms table of F

(which also gives successive approxinLation), according to

the number output from the subtractor. If the (r:+1)th bit

is 'L' (i.e. negative) , the PROM gives zero value; otherwise

it gives the residue number of 2k (where k is the positive

out--put from t--he subtractor) " The conìputer receives t\,{o sets

of information one j-s F or successive approximation and

the other is the value of c-. r'or lth iteration the

{9(x)re(Y)}Or

þrevious . ,APProrirna tt on'

62

f o r gene rot ionve APProxirnr:ticn'"

X

9r.X)

F or Successive

AÞprox imationt

6rouP

Gene rat o r

c0þ'l

PU

T

E

R

2:1 MuttiPt exer

@sThe Tabtes cl F and

Sqccessiv¿ APProx irnatio nt'

is '0', muttiPtexersetects 9(X) ¿nd

g(y). Atso itdivides the tocat¡ons

of PROM s into two

hatves. 0ne hatflor F ¡ other hatffor tSu c cessiv€.Approx irnatíon"

Prev i ous,dpp rox iÌn at ion'.

9(Y )

Y

C

tDiuid" by 2'slgnal. ¡f it

Fig. h.trI Scl-remGtic "dioçrGm.of F qrì d Succes sc

63

X

9(x )

C

n bits -r

tDiv¡d. b

signat.C.

,2

l' se Iects

(n+r ) bits.

(n+t ) bits

F orS u c c ess ive Appro xim a tio n

Fig.4.5 Schemqtic diogrorn of on olternotecircuit f o;' F ond Success ¡ve

Approximotion.

v¡

I

t

I

GroupGenerator.

2"1

Mutt iptexer2:l

Muttiptexer.

n bits. n bits

Subtracter

c0M

P

U

ïE

R

g(Y )

0ne

Y

c

l b its.

tD

2'l'

ivide byt s ignat.setec t s

0ne.

EIgUrTable for F and

Successive Appro ximation.

n bits.

computer

V) which

j-ssues the value of C (through

equal to C' obtained during

a new value of C' which is

64

the data bus X or

(i-1) th iteration(c-f), and the

l-s

approxintation.

Si-nce

and receives

group number

ulus, it is

For example,

in Fig 4.5

5 address

the

is

number of bri.ts required to represent the

far less than the sum of bits of each mod-

16, 7-5, 13 and LI, the sum cf bits of

Since a 64k PROM (f0 address lines) is

the circuit j-n Fig 4.4 is not

can be implemented

possible to use

1n the case of

lines.

PROM with less address 1ines.

the residue system of rnoduli

each modu]-us is 16.

not yet ar¡a j lab1e,

feasible. Whereas the circu:Lt

Performance

Delay in obtaining F by the circuit j-n Fig 4-4 is

given by

D(1) D

with a PROMs with S,/more than

+ Tuux * Tr (4.2)

= delay in forming group.

= delay in 2:1 multiplexer.

= access time of the PROMs.

the circuit in Fig 4.5, delay is given

D

g

where

T¡,tux

T,

According to

g

by

D (2) D +T +T +TI MUX SUB F

t-rtINV

delay in

(4.3a)

where truu =

Agtrin T, uu =

where Tr*u =

addit-i"on "

delay in subtraction"

ADD.

f nverter,' and TADD

de-laf in

65

Therefore equ.

D(2) = Dg

For 16-15-13-11

(ranre 6.3 (a) ) ,

D (2) 225

(4 " 3a) becomes

+T +T +T +T (4.3b)MUX INV ADD F

modul.i system, and using TTL components

Dg = 225 ns (from 4.Lc) and

+4+3+7+(Tn,=35)

{a- ([Alß] ) ß]ßo

cLz ßl + cLt ßo

274 ns (4"3c)

* for DM751'7, a (gZ B) PROM IEO] l4"4

Thj-s section proposes a new form of representat-ion

of floating polnt numbert it is represented by an integer

multiplied by a suitable base of exponent depending on the

moduli of the residue system. An advantage of this form

is that floating arithmetic can be performed within the

integer residue arithmetic environment. without the need for

any redundant modulus. The lmplementation is therefore

economical-.

4 .4 .1 T'heoretical concs!

If R is the highest absolute value of mantissa of a

computing system, and ß is a posj-tive integer such that

(ß-t)' < R < ß", then any number A within the range from 0

to R can be represented by

A {[Alß] ]ßI +

l-t = 35 nsLF

(4.4\

Where (12 and G, are the co-efficj.ent-s of ßl a,nd ß0

r.e &2 Ialg] and cLt (A--{[Alß] ]ß)"

66

The

SimilarIy

range of

R can be

the co-efficients is from 0 to (ß-1).

represented by

+ rr (4.5)R'- rz ß

Case ( i) :

Case (ii) :

values of R

be denoted

t-.e.

l_.e

For ß2 = R, fz =

For (ß-1)2 < R <

that can satisfy

bY o"o* ttd Rrror

*"o" = (ß-1)ß +

and r 0

the max|mrrm and

al¡ol'e condition

(say)

( say)

ß

2ß ml_n].mum

of ß can

(4.0¡

(4.r0¡

R"r*

rr

r

2 : (ß-1)

r = (ß-1)

(ß-1)2 +

2 = (g-2)

lz

the

respectively and expressecL as

(ß-1) (4.6)

(4.6a)

(4.6b)

1 - (ß-2)ß + 2 ..... (4.7)

(4.7a)

(4.7b)r

4 .4.2 ¡l-o_e!:L¡O point representat j-on of a number

A number X can be represented in floating point form

bv x = Aßt (4. g)

Vthere A j-s the mantissa, and lies within the range

from 0 t'o R. ß is tire base of the exponent which must sat-

isfy the conditions stateci previously, and n is the exponent.

4.4.3 .¡'loating P_o_i4! Arithmetic withzerq-exponent

4.4.3a Fl-oatinc¡ point additi-qn'subtract.ion :with zero -e>(ponent

x

V

= Aßo

= Bßo

A & CL+ß1

= B = b2 ß + It,

67

Therefore

7

7

2

Fl (Xty) (ez 1

Þ

(ct2

2 + (a

br) ß + (a, + 6r)

Where

I2c+c

34A+ßA

(4.11)

(4.11a)

(4.l_1b)

(4. L2)

1

+

c

c

br)

br)(a"

4.4.3a (i) Floatinq point normalisat-ron(def i-ni-t-i on )

A number is said to be normalised i-f overflow occurs

when the nurnber is multiplied by the base of the exponeni.

4.4.3a(ii) Normalisaiion r.q f!.g_?_!r¡q poi4t _è,idi!¿g2lsubtraction (zero-exponen.t)

The process of normalisatj-on in floating point

addition,/subtraction has been shown with the flow chart in

Fig 4.6. The exponent generated durj.ng normalisation may

be termed as 'generated exponent' and denoted by g. In

case of addition nonzero value of g is 1.

4.4.3b Floatinq point mul.tiplication(zer_o exponen!)

7 - Fl (X.V)

From (¿. g) and (4. fO)

z (a b2)ßb)s'

g

2bl)ß + (o,

(aI b+

AAß2

+2 + (4.13)

68

ßW <--c

0oesove

?

w rf I o,¡v

K <- W +c.l

0oesflve?

K rftow

r <-[rc,-[ßlz])/n] T <- [tc,+þ t2Jrlt]l

No {Ê

P +-c, + T

Z: Kßf¡

Ißz P

END

Yes

Yes

T

* These are the steps

for rounding.

Fig.4.6 Process of normotisotion in ftootingPoint res¡due ocldclition /subtroction c

Where bA=&

(4.13a)

69

(4.13b)

(4.l-3c)

(4 . 14)

(4.14a)

A

4

z

2,

2CL

A=a"

a.A

(4,*

bt

b2

br

Again Z can be written as

z c+D) g2+D+ +c2

ß

ß I

2

Where D

D

2c2

2c I

Finally

_ [A3 lß]

= ¡a, lßl

=[a DBIlJ

l-\t

+ IA, D, ß]

z cg ß' i cz ß + cr

Dl + D2)Where ca (Au +

c2

cl

4.4.3b(r) in floating Lo j-ntmultiplicalion (zero-exponent)

Figure 4.7 shows the flow chart of normalisation for

floating point multipJication- The generated exponent, 9,

in case of multiplication, ranges from 0 to 2.

4 .4.3c Fl-oati- p--ar-n! divi s ion ( zero-exponent )

Normali sation

z

rle= Fr(xlY) = ale

= Kß-x (say)

A (1lB) (4.15 )

Where K is the mantissa (wj-thin the range from 0 to R) ,

and x is the exponent. The val-ues of x (except for B = 0 or

l-) 1ie between 2 and 3. The conversj.on of f ls to its equiv-

alen't f loating point number can be done by a look-up tal¡le.

70

Yes

Yes

Fig.l,.7 Process of normolizotion in flooting pointresidue rnultipticcrtion.

¡r These o re the ste ps

for rounLJing.

ßW <-c

oesove rf low

?

P <- W+ c

?

ov erf Iowoes

Scz +

2ï <- P.ß

K +-[(cr- lß12)l/nj K <-[(cz.fß/2]¡7ßl

u+- +c

oes

)T ove rftow

7=UU<-T+c

No

YesDo es

?

U ove rtow

z0

UßT a-[(cr-lß12]l/ßl T <--[(cr*[ ßl2lllßl

E ND

U (- P*T

1L

Isee Appendix-2 for the eval-uation of tls.]

Therefore 7 = A' Kß x

l^ -Y\n K)ß ^ ".... (4.f0)

(a . K) can be evaluated by floating point multiplic-

ation. That is

FI (a K) - vß4 (=.y) (4 .17)

Therefore Z = Yß g-x = vßq-* --a

(4.18)

Where m = q-x

4.4.4 Floatinq point Arithmetic w-ith non-,zero

Vßff

.exponent+

and \/ - B3oLet X -Aß

x-t-

^

where ¿ and t are the exponents of X and y

respectivelY.

Also X and !/ can be written as

(ar\ + o"r) g

(b2g + br)ß

A

t(4.1e)

(4.19a)

V{ithout loss of generality it can be assumed that

4.4 .4a Alqorílhms for alisnment anQ floatinq poin.Ladditi-s¡n/subt-rÈction (non-zero exponent)

Let k - ^-L

Case (i) : if lç = 0, then

Lst step : compute D = Fl (etn¡

= cgg (say)

/5>Í.

72

2nd step Compute p = ^

+9.

lz = cßPl.

END

Case (ii) : If k - L, then

*l-st step : Compute D =

(rr b, is negatÍve,

othenvise [ +ø | Z] )

[tr,, + ¡telzlrle]

acld | -ßlzl;

*2nd step : Compute D' = &n + D; I The

aligned mantissa takes the form of

(0.ß + ¡-).]

3rd step : compute T - Fl (AlD') = cgg (say)

4th step : Compute p = ,s+t; lZ = CßP ].

END

Case (fii) : If k = 2, then

2nd. stcp

3rd step

*lst step : compute D = [tO, + Ltglz]] l àlL-t '-J-'I

Et b., is negative, add [-glz]tLIo'uherwise [+glZ]. The aliçrned mantissa

rakes the forrn of (0.ß + ",].

compute T - Fl (AtD) = cgg (say)

Computep=^+grU=CßP].

END

(these are the steps for making the aligned mantj-ssa a

roundecl f 1gur. e. )

Case (iv) : If k >- 3, then Z = Aß

*

^

73

4.4.4b Floatinq poing multiplicationalqorithms- (-4çp_ zefo e4pp-!S]]! )

z Fl (x. v) (a. s) ßL+t

g1st

2nd

step

step

: Compute D

: Compute p

END

Fl (A'B) = cß

t+L+g t lZ =

( say)

cßp l.

4. 4 . 4c Floating point -d-fyåq--i.a.n alqorithms( no n - z e r o çëpp_qç¡-ï=)

z p1 (xlV)

l-st step :

2nd step :

Compute I)

compute p

END

= Pr (aln) = cß

= ,s+g-t ; lZ =

g( saY)

cgP l.

4.4.5 How to tirìQ the co-efficients of a mantisea

The co-efficients of a mantissa A (in residue form)

be obtained by following steps:

(residue integerdivision)

2nd step : ComPute &, = A (residuesubtraction)

4-4 6 Errqr in the Arogore-è flqatinq pointarithmetic operations.

can

1st step : comnute 4, = [å]

CL2

Errors in floati-ng Point

tiplication are introduced

an.i Fig 4.7) in normalisation-

error may origj-nate from (i)1

f,Loating point number of fr and

multiplication "

addition,/subtraction, and' mul-

during rcunding process (fig 4 -6

Tn floating point division

error in approximation of

(j r) error 1n floating Point

14

4.4-6a Error j-n floatins pA-1.4._t_ ed.gitf-A-A,/sub tr:actlort

The maximum relative error

addition isIslz]

Itax (e) in f loating

Max (e)

(appentl.ix-4

(4.20)R

(e) )

4.4.6b Elqçr in f loatinq Lq¡¡1_t_ multip.liçation

The maxj-mum relative error in float-ing point multip-

"l.lcatj-on, when ß-ì-_s odd, is girzen by

(..1 2(lßlz)¡' +g[slzliU("o)l -----r-- tt'2r)l-'-''Jooo* R \'r

(appendix-4 (26) )

!{hen ß is eIe_n, the maximum relatÍve error is given by

2 (lß 2)t'U {."¡ (4.22)

RMAX

(appendix-4 ; (27))

4.5 Output Translation

The method proposed here requires Chat Ø . *i for at

least one of the moduli. @ is the base to which con-

version is to be done.) However, there is no restriction

whether gcd(w,mil = l or not. The method can be applied for

conversion to any base provided abcve condj-tion is satj-sfied.

4 .5 .1 Theoretical Asr>ects

Any Ínteger Io can be expressed in base ilJ as

N N_I ifo = ú¿NøJ + c{¡-. 1r.ü + .."".' -l- a,i,N +

+ cLrt'J + ao

anNl

n=o

n

75

The coefficients or digits arL (n=0,1, N) are

given by drr-l = f.r-, Øf' for n = I,2, ....t N

where ro

For

lst step : Find f¡

fïllì-- Jt,l-specj-al case a¡

'N+.1.' gives the number of digits requi:red to repres-

ent the maxj.mum range of a computing system.

4.5.2 Conversion Proced.ure

The co-efficient (digits) in the base fil can be found

by repeating the following 3 steps N times.

n=1 2

2nd, step : Find Inb = dr.,

3rd step : Find a In-I n- I I nb

Last step (special) aN

To convert a residue numL¡e¡: to its equi-valent decimal

number at least one of the moduli must be greater than or

equal to 10" The residue cf that particular modulus will

represent the co-efficient (digit) after third step of the

procedui'e. Therefore only one residue is required to be

extracted.

F;l

76

Example:

Let B, 7 and 11 be the moduli of a residue system.

The hi-ghest pos.itive and negative numbers that can be

handled are (307) r 0 and (308) I o respectì-vely which are

three digits figures.

Take for an example a resj-due number (0,5,10) which

represents 208 in decimal- system.

Tbe procedure to get decimal digits out of 1-he residue

number is

(rne residue representation of 10 is (2,3,10))

Forn=1

First steP :

Second step:

Thi rcl step :

For n

First step :

Second step:

Third step :

= (0,5,10) | (2,3,I0) =

= (4,6,9)x(2,3,10) =

= (0,5,10)-(O,4,2) =

(2. ,2 ,2)

(4,6,9)

(0,0,c)I

I N-rt^)

in each it-eration is

rb

rl

2

I

(4,6,9)

(0 ,4 ,2)

(0,1,9)I

= (20)

= (200)

= (B)

t0

t'0ï

&

=

I

I

&

t00

2

zb

f,ast step for

(1 2 (2,2,

[nivi-sion in t]re fj-::st step

e::al residue division. The arrow

tlre base lol

["-2

I

(4,6,g) (2 ,3 , LO)

(2 ,2 ,2) x(2 ,3 , t})i

(4,6,9)-(4,6,9)

3rd digit is

= (2^, to

= (20\' '10

= (0)' '10

The:refore the decimal number is 208 (=a zxro2 +a rxro I +a oxl-o

o )

a gen-

i.nindicates the digits

77

The arithmetic operatj-ons requi::ed for conversion

are multiplication, divj-sion and subt-raction. Therefore

arithmetic unit must have t.he provision for subtraction,

multiplication and division in the residue numloer system.

4.6 Conclusion

According to the proposed method, multiplicative

overflow det-ect-io': in 16-15-13-1.1- moduli system is consid-

erably fast (269 ns). fn conventional fast technique,

using PROMs of 40 ns access time, the deJ_ay would be 640 ns

(- 4-4"40) . T'he hardware requirement is not insignificant,

however, since sanÌe piece of hardware can be sharecl for

sign detection and determination of first approxj_mation,

the overall cost will not be much. The delay in obtaining

the first approximation, according to the proposed technique

is 274 ns (eq 4.3c); whereas in conventional fast division

technique, the delay would be 2.4.40 = 32O ns.

Floating point arithmetic which was difficult j_n

residue system, no\^/ can easily be implemented using proposed

method. The prelirninary requirement is the existence of

residue int.eger addition. subtraction, multiplicat-ive,

division, comparison and multiplicative overfl"L)\lr detector.

Provided circuits for those operations and a look-up ta.t¡le

for approximate floatlng poj-nt number "t å exist, floating

point arithmetic operat-ion can be performed in software

(residing permanently in PROMs). Therefore implementation

of floating point operatr-on rvill be cost effective. One

departure from the convent-ional f.Loa'tlng polnt format (wirere

2 or 10 -is chosen to be the base of the exponent) is the

7B

choice of the base of the exponent which depends on the

moduli of the residue system and will differ from systen-r

to system. The main advantage of. this choice of the base

is that addition/subtraction/multiplication among the

co-efficients does not produce overflow. Therefore the

cost of usi-ng redundant moduli is saved.

Since output translation is a less f;:equently used

operation, more time can be allotted for thi-s operation.

where there exists all- the basic resi.due j-nteger operat-ions,

the proposed method for output translation can be j-mplemented

with no extra cost of hardware. Moreover, i-n a situation

where base extension cannot be applied, the proposed method

is the only cho j-ce.

5. DESIGN OF A REAL TIIUE RESIDUE

ARITHMtrTIC COMPUTER

5.1 Introduction

The aim of the deslgn of the Residue Arj_thmetic Com-

puter (naC¡ is to verify the algorithms proposed 1n the

Chapter 4, and study the feasibil-ity of residue system in a

general purpose computer. The functional blocks of the RAC

(r'ig 5.1) are (i) Intel SDK-80, B bit microcomputer (pig

5.2(a)) and (if) Residue Arithmetic unit. The microcomputer

acquires and mani-puÌates data, while the RAU does all the

basic residue arithmetic operations. The design of the RÀU,

floatj-ng point arithmetic and residue to decimal conver.sion

are the main topi-cs of discussion in this chapter.

From the consideratj-on of bit efficiency, moduli 16

and 15 (total 8 bits) have been chosen for the test s)zstem.

This gives the range of the system frorn 0 to 239 (lvl=16x1-5=

2401 . The positive half starts from 0 to Lt9, and negative

half f¡:om I2O to 239 representing . -L20 to -1 respectivelli.

board

Thc.re

(ris

The

(rtg

are

5.3

RAU has been designed on (22.Scm x 17.Bcm) vero-

5 .2 (b) ) witb wire wrappinE technique (rig 5 " 2 (c) ) .

altogether fifty TC chips and seventy-six resistors

and Table 5.1) .

llhe microcomputer

prograrnmable peripheral-

conirnunicates with the RÀU through a

intel:face (PPI) of the m-icre;cornputer

79

BO

(fig 5.4). The A-port of PPI issues operation codes (fanle

5.2) and operands, while B-port receives the results from

the RAU. The lower three lines (C0, c1 and C2) of C-port

gj.ves the information of the sign of the resul-t (C0), the

overflow (C1) and the Exclusive OR of the sign of divisor

and dividend (c27, and the higher three lines (C4, C5 and C6)

address the latches (table 5.3 and Fig 5.5) assigned to the

operands and operation codes. The operation codes and oper-

ands are each eight bits long. Of the eight bits of each

operands, the most significant 4 bits represent the residue

of module L6 (mod-16) and the least significant 4 bits rep-

resent the residue of modul-e 15 (mod-15).

The f/0 device specifies operations with alphabetlc

symbols (table 5.4) and issues operands to the microcomputer

which makes decision on the type of operatlon (decode),

enters the operands and operatj_on code to the RAU and then

gets the result from t.he RAU. The final result is displayed

on the f/O device in both residue and decimal forms with

overflow character (if any by '?' ) and sign character ('+'

or'-'). In case of compariscn, the character ) or =, or <,

representing x is greater than, ot equal to or less than y,

is dì. spl ayed.

The programmes

from locations

requl-red for al1 operations

400H t'o 49F H and from 800

re-síde in

PROMs

of the SDK-80 microcomputer memory.

l,ater sect--ions discuss the RÄU -i-n detail-.

il to FEFH,

B1_

The R AC

J-

Fig.5.1 Functionol blocks of the RAC

I

J

D

Ë

V

IC

E

i/0 M

¡

crocomputer

R

A

U

o1

Fig. 5.2 (a) The S D K- B0 microcompu te r.

B3

Fig. 5.2 (b) Top v iew of the RAU boqrd

8'+

Fig. 5.2(c) Bock vÍew of the RAU boord

LOæ

L àtch2

L¿tcht

Latch6

Latch

IPROM

3

Latch

IPRO M

't 0

PRO M

5

Latch

llPR OM

l2PRO M

Ht8

PR OM

H

Ël29

Mutt¡pIe¡er

r$

ËH

tËi-i.

I?

5

7

I

r3

t5t719

zt2325

27293l33

353739L143L5

L9

æEsSøsü<10a...r . i!2

@.-,-.r?-oH"f-;izz

@.,-æ 3 2

F..Æil 3 6

æ48iÑ* 50

a

PROM

Ë"r

l3PRO M

1/.

PRO M

r5PROM

¡5PROM

t7PRO M

+5V

5v

30MuI t ip texe r

37Muttiplexer

43Inverter

49Inverter

;L

tro¡f

É.

OJl'

Co

UI

q,Coo_Eo(J

oC,oo(4

Cog)LO(V)

c,olLL

l9PR0l.l

tt

PRO M

2tVultipIexer

32Multiplexer

PR0 tl

25Muttiplexer

33Mul.t ¡plexer

26Muttiptexer

Muttipterer

39AND

15OR

Multiptex er

46OR

28Mutt ¡ptere r

36FuIt Addcr

PftO M

3ll.luitiptexer

JõMul,t¡ple xer

50

EX. OR

til40

ANOtl

ANO

t7ANO

r.2

Inve rte:

1AInverter

22PRO M

1

0EX R

B6

Table 5.1

The list of components and theirtype numb er/ spec if icat ion

FunctionalDescription of

Components

Type Number/Specificat ion

Loca,tion on theäAU BoarJ Quantity

Quad Latch DM 7475

3601

D[,I 74LS83

DM 741,53

9322

DI,t 74 15 1

DM 74H08

DM 74S11

DM 7432

DM 74586

DM 74504

1.8 K A (%lI)

(36)

(30) , (31) , (37), (38)

(54) , (3s)

(24) -

(2e),(32) , (33)

(39), (40),(47)

(41)

(4s) -

(47), (s0)

(43) , (44)

(42) , (48), (49)

R1 R76

6

Inte 1

PromBipol ar

[zg]t7

76

4 Bit Full ADDER

Dual 4: L

Multipl exe:r

Quad 2-inputMultiplexer

B ChannelMul tipl exer

Quad 2-input AND

Triple 3-input AND

Quad 2-input OR

Quad 2-inputExclusive 0R

Hex Inverter

Resistors

1

4

2

8

J

1

4

2

3

for, .n" ,...Sernicoirductor

components used, excopt PRONfs, are of National

rart 1

B7

SDK- 80Mic rocompute r

Boo rd . +sV

&tttr9

1

2

3

4

56

7

ì(u

ooF

EL(,()m3É.

r+-

oVIt-o*o(UCco(J

G)glEIJ

g

0

1

21J

l.5

6

7

BI

20

2\222321

5

2627

I0

Þ

7

333

lr

I0

11

4213

4t,5

tE

25-Pin-CannonPtug _J

Fig. 5.1 Interconn¿ctlans betweenthe PPI of nrierocomputerand the edge connectors of theRAU boa rd ,

7

I

I

Ir 15

3

6

1

12

5

5

18

5

I7

20

I21

I2

10

2

B7

B3B6

iBzr- B5

Ë81È BL.

BO

c3c2cl

i cor- ClEoLoÈcs

c4

I

t-ß()È

rnLO(\¡æ

A7

A3A5

A2A5

A1

A1

AO

Â-(L

+5V{t

Table 5.2

operation Codes and their definition

B8

0peration Codes

þ7, 06, 05, 04, 03, þ2, 01, 0o

Definition

1

0

0

1

1

0

0

0

1

1

0

0

0

0

0

0

1.

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

10

01

(i)

(ii )

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Add

Subtract

It{ulti p1y

Compare (Unsigned)

Cornpare (Signecl)

Get Fjrst Approxinate ofQuotie:rt (Integer Division)

Get a nunber divided by 2

Get the equivalent mantissaof the reciprocal of divisor(floating point division)

1

L

0

L

0

1

1

0

0

0

0

0

0

0

0

0

0L000000 (ix) Get the exponent of thereciprocal of the divisor

Table 5.3

Addresses of iatches

LatchesAddress

c7, c6, c5, c4, c3, c2, c1, c0

Latch of X

Latch of !/

Latch of Operation Code

100000

0

0

0

0

0

0

0

0

L

t

0

1

1

1

0

0

0

0

¿

3

3

_ 16d

r15e l'È

h 1l10

h) c

(de

5

a tgt15ct1o

(e)

2

c

hô

103

n' ''t 1556

235

at6

t t5

ito4)

s

($u

2

ch

10

t't5e 16

3(ds

e16

t15

ito

89

RAUBOA RD

18t716l5??212t9

c4

46c5

Irg'l

c6

ttig.5.5 Lotchirrg r¡f

operqnds -

r--

operatIon

co

d

e

Øz

ø4

6l

ø6

þr

x4

X5

4S

lr7

.lt

ctoÊ0

f

x

oØLo

-l-¿UoCCoc)oul'tf

LU

AO

5

QA

I(toE

A3

útF

I!o.z2

(ô

Iuoã

IJì

I1Jo

=

x7

0

Pe

rên

d

X

-l

oPefan

c

9)

X1

)oy1

y2

Y¡

y4

Ysy6'i7

Y

op*. ret i c; n e ocie oncl

90

Tab1e 5.4

Alphabetic symbols and operations

Alphabet Operat-i-on

A

B

c

D

E

F

G

H

fJ

Addition (lnteger)Subtraction (tnteger)Multiplication (Integer)Dj-visj-on (tnteger)Compar-ison (unsigned, Integer)Comparison (signed, Integer)

Addition (Floating Point)Subtraction (Ploating Point)Multiplication (r'loating Point)Division (Floating Point)

5,2 t Subtracti-on and Multi 1 ri n

Addition, subtraction and multiplication are performed

slmultaneously by means of six look up tables (fantes 5.5 (a) ,

5.5(b), 5.5(c), 5.5(d), 5.5(e) and 5.5(f)) stored in six

(256x4) lntel Bipolar PROM (3601), each having a 70 ns

access time. T\¡¡o PROMs are assigned to each operatj-on; one

is for mod-16 and the other is for ¡nod-15 operatioi:s (fiq

5.6 (a) , 5.6 (b) and 5.6 (c) ) .

The lower 4 address lines (pins 5,6,'7 and 4) are

connected to *;he 4 data lj-nes of operand x," while the higher

4 address lines (pins 3,2,1 and 15) are connected to the 4

data lines (corresponding residue) of operand y.

Irne proqram listings have been shown in Appendix-6,"

AP 0(a).1

9L

Table 5.5 (a)

Look up Table (Addition in lrlocl-16)

Each hexadecimal di git representsrloler 4 adclress bits t

0

O1234567B9ABCDEF

O1234567B9ABCDEF

234567B9ABCDEFO].

34567BgABCDEFO12

4 5 67 8 9AB CDE F O L23

56789A8C0EF01234

67B9ABCDEFO12345

1,

2

3

4

5

6

7

8

9

A

B

C

D

E

F

L23456789ABCDEFO

EactrhexadecinaldigitrepresentsIhigher4 addressbitsr .

9AB

ABC

BCD

CDE

DE F

EFO

F01

012

t23

F

0

1

2

5

4

8

9

A

B

C

D

E

F

0

7

8

9

A

B

C

D

E

F

cDEF0123456

DEF0t234567

8F012345678

0123456789

t234s6789A

23456789A8

3456789A8C

456789A8CD

56789AtsCDE

92

Table 5.5 (b)

Look up Table (Addition in lvlod-1.5)

Each hexadecimal digit TepresentsI lor'¡er 4 address bitst

0

01234s6789ABCDEF

0123456789ABCDE0

t23456789ABCDE01

23456789A8CDE012

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

34567B9ABCDEOT23

Eachhexadecimaldigitrepresentsrhigher4 addressbitsr .

4

5

6

7

B

9

A

B

C

D

E

0

5

6

7

I

9

A

B

c

D

E

0

1

67B9ABCDEOL234

78948C08012345

89ABCDEOT23456

9ABCDEOT234567

ABCDEOT2345678

BCDEOL23456789

cDE0723456789A

DE0L234s6789AB

E0123456789A8C

012 3 4 s 6 7 B 9 A B C n

123456789ABCDE

234s6789ABCDE0

93Table 5.5 (c)

Look up Table (Subtraction in Nfod-16)

Each hexadecimal rlj-git representsrlorver 4 address bitst

Eachhexadecinraldigitrepresentsrhigher4 addressbitsr .

0

1

)

3

4

5

6

8

9

A

B

C

D

E

F

0123456789ABCD8

0123456789ABCD8F

FO1234567B9ABCDE

EFO1234567B9ABCD

DEF0t23456789ABC

CDEFO1234567B9AB

BCDEFO1.234567B9A

ABCDEFO1234567B9

948CDEF012345678

8 9 A B C D E F O T 2 3 4 5 6 7

789ABCDEFO123456

6 7 8 9 A B C D E F 012 3 4 5

678gABCDEFO1234

56789ABCDEFO123

456789ABCDEFO12

34567B9ABCDEFO1

23456789ABCDEFO

5

4

3

2

L

94

Table s.5 (d)

Look up Table (Subtraction in Mod-15)

Each hexadecimal digit representstloler 4 address bitsr

0 01

EO

DE

CD

BC

AB

9A

B9

78

67

s64s34

23

t2

01

0123456789ABCDEF

EachhexadecinalcligitrepÏesentsrhigher4 addressbits' .

1

2

3

4

5

6

7

B

9

A

B

C

D

E

F

2

1

0

E

D

C

B

A

9

8

7

6

5

4

J

2

3

2

1

0

E

D

c

B

A

9

8

7

6

5

4

3

4

3

2

T

0

E

D

C

B

A

9

I

7

6

5

4

5

4

3

2

1

0

E

D

C

B

A

9

I

7

6

5

6

5

4

3

2

1

0

E

D

C

B

A

9

8

7

o

7

6

5

4

3

2

L

0

E

D

C

B

A

9

8

7

I

7

6

5

4

3

2

1

0

E

D

C

B

A

9

8

9

8

7

6

5

4

3

)

L

0

E

D

C

B

A

I

A

I

B

7

6

5

4

3

2

1

0

E

D

C

B

A

B

.1{

9

B

7

6

5

4

3

2

1

0

E

D

C

B

C

B

A

9

B

7

6

q

4

3

2

1

0

E

D

C

E

C

B

A

9

8

7

6

5

4

3

2

1

0

E

D

DO

DE

CD

BC

AB

9A

B9

78

67

56

45

34

23

L2

01

EO

95

Table 5.5(e)

Look up Table (Multiplication in Mod-16)

O1234567B9ABCDEF'

0000000000000000

Each hexadecinial. digit representst lower 4 address bits I

BCAB642OECAB642

F8DC8A987654327

Eachhexadecinaldigitrepresentsrhigher4 addressbitst .

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

123456789ABCDUI.'

2468ACEO2468ACE

3 6 9 C F 2 5 B B E T 4 7 A D

4BCO4BCO4BCO.lBC

5AF49E38D27CT68

6C2BE4AO6C28E4A

785C3A18F6D4829

808080808080808

9284D6F81A3C587

44E82C60A4882C6

B61C72D83E94FA5

c840c840c840c84DA741EBB52FC963

96

Tab1e 5. s (f)

Looh up Table (Ì,{ultiplication in lt{od-15)

Each hexadecimal- digit representsrlourer 4 address tlits t

O1234567B9ABCDE

0 00

01

02

03

04

05

06

07

08

09

OA

OB

OC

OD

OE

00

00

EO

DO

CO

BO

AO

90

80

70

60

50

40

30

20

10

00

0

2

4

6

B

A

C

E

1

3

5

7

9

B

D

0

0

3

6

9

C

0

3

6

9

C

0

J

6

9

C

0

0

4

oo

C

L

5

9

D

2

6

A

E

J

7

B

0

0

5

A

0

5

A

0

5

A

0

5

A

0

5

A

0

0

6

C

3

9

0

6

C

5

9

0

6

C

3

9

0

0

7

E

6

D

5

C

4

B

3

A

2

9

1

B

0

0

B

1

9

2

A

3

B

4

C

5

D

6

E

7

0

0

9

C

6

0

9

3

C

6

0

I

3

C

ó

0

0

^

5

0

A

5

0

A

5

0

A

5

0

A

q

0

0

B

7

3

E

A

6

)

D

9

5

L

C

B

4

0

0

C

9

6

3

0

C

9

6

3

0

C

9

6

5

0

0

D

B

9

7

5

3

1

E

C

A

B

6

4

2

0

Eachhexadecinaldigitrepresentsrhigher4 addressbitsr.

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

91

+ 5v rããxë^<ìñ5 '5ðxäðs:ss +5v

CJd\¡Þc)(t)

ÞC'(tr¡\

(1)

Fig.5. 6(sl

l"lod - 16 Mod - 15

Circuit for the 16-i5 moduti crddltion.

+5vYXXXX\<<q <XXXX\<q\<\ 6ì Ur I\ 1 Sr g¡ Õt (èi ¡\i j ¡;l Lú 6ä- r- Þ.,

+5v

UIc cdl Itr N)

Mod- I S Mod- 15

Fig.5.6(b) Circuit for the 16-15 rnodutî subtrqçtion.

5ãõXëts;X :rNããõtssts+5v +5v

t! G'(^, c

À,

Mod-16 Mod -t 5

Ø tJt tn ulc€,cc..\l.!tcn\¡

acc,a¡

CG^

ca¡,

It/)c

(-7

cc¡- (J!r Cn \¡

co

\ cocf,.

PR0t,t (7\

a,ct A ts)

r/l N\¡

PR OM (B}

'\)

\ll h,J(rr

(ô o

PROM (9}

àlôctr l\)

(tl \¡ (lì (¡1

PROM (ro)

,DdjË

UN\ t,T

7t

ul

PR0M (11)

(¡'ó=Ë

rtl À NlJt

PR0M (12)

¿að:ß

t, \¡ gl fn

Fig.5.6(c) Circuit for the 16-15 moduli rnuttptieation.

98

5.2 (a) De_ley in Addil-ion, Subtraction andMul tiplicat j-on

The delay in addition,/subtractl.on/multiplication

1 PROM delay

70 ns (access time of 3601 PROM)

5. 3 Co¡nparison

There are tl.'c types of comparÍsons compari-sons with

signed numbers and comparisons with unsigned numbers. Com-

parisons with s:Lgned numbers involve either no operation or

subtraction; whereas comparisons with unsigned numbers

inrzolve either addition or subt::action. Table 5.6 is a truth

table showing the comparisons, operations required and dec-

ision modes. It is clear from the Table 5.6 that subtraction

is requJ-red (in both types of comparisons) it the signs of X

and }/ agree; while addition is required (comparison with

gned number) if the signs of X ancl / do not agree.

99

Table 5.6

Truth table for decision modes in comparison

SignedNumbers

Decision

0

1

0

0

1

0

0

L

0

0

1

0

0

1

0

0

1

0

0

1

lt

0

0

0

l"

0

1

1

1

0

0

0

1

1

1

0

0

0

L

1

7

0

C

0

0

7

L

1

1

0

0

0

0

0

0

1

1

1

1

L

l_

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

I

1

1

1

1

L

t

1

1

L

1

1

0

0

0

0

0

0

0

0

0

0

0

0

SUB. O x> y

x<yX=V

x> y

x<vx>yx<vX=V

x> y

x<yX=V

x> y

x<yX=V

x<yx> v

X=V

x<yx> v

X=V

No 0p

No 0p

SUB

il

ll

lt

il

tr

ADD

tt

il

lt

lt

SUB

It

0

1.

0

0

1

0

0

1

0

0

1

0

0

1

0

[t'tote: SUB Subtraction; ADD Addition]

100

Comparisons require the sign of X, !z and the sign of the

resurt of addition or subtraction. However, the results ofaddition, subtraction and multiplicabion are generated sim-

ultaneously. Therefore, depending on signs of X ancl y, the

result of addition /subtraction is to be selected for sign

d.etection.

The sign is detected by a look up table (rabte 5.7)

implemented v¡ith a PROM. 'rhere a-re artogetherthree sj-gn

detectors; one is for X, one for V and the other one for the

result (one of the results of addition, subtracti-on and mul-

tiplication) .

Fig 5.7(a) and 5.7(b) show rhe clrcuits used tor sign

detection of X and f respectively. The data l-ines of the

residue of mod-16 are connected to the lower 4 address linesof the PRoMs and that of mod-l5 to the higher 4 address llnes.The most significant bit of the output indicates the sign

(' 1' for negative, '0' for positive) and the least signlfic-ant bit indicates zero/nonzero of the nurnber ('1' for zero-

number). Thus SX and ZX give the information sign and zero/

nonzero of X respectJ-ve1y. Similarly so for y.

The appropriate result j-s multipJ exed to the sign

detector by two bits (03, dîl of the respective operation

code (fig 5.S). However, þT is not connected di.rectly to the

selection pin (pin-i-4) of the mult:iplexers (two dual 4z1

multiplexers), since in case of comparison, the result ofeither addiLion or subtraction is 'fo be se-l.ected" Table 5.8

shov¡s the states of the selection pins and bhe sel-ected in-puts.

10L

oTable 5.7

Look up table for the sign and zero-nonzeroof a num.ber

Each hexaclecinal digit lepresentsIlorver 4 address bitsr

O1234567B9ABCDEF

Eachhexadecimaldigitrepresentsrhigher4 addressbitsr .

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

1B

00

00

00

00

00

00

00

BO

88

8B

88

8B

88

8B

i.8

00

00

00

00

00

00

BO

B8

88

88

88

88

88

8E

08

00

8

B

0

0

0

0

0

0

0

0

I

8

8

8

8

8

I

B

B

0

0

0

0

0

0

0

0

8

8

B

8

B

I

(t

Ò

B

0

0

0

0

0

0

0

0

B

8

8

I

B

8

B

B

8

0

0

0

0

0

0

0

0

8

8

8

B

B

8

B

8

oo

0

0

0

0

0

0

0

0

I

8

I

8

B

8

I

8

8

0

0

0

0

0

0

0

0

8

8

B

B

B

B

B

8

B

0

0

0

0

0

0

0

0

0

B

8

I

B

B

B

8

I

0

0

0

0

0

0

0

0

0

8

8

B

B

8

B

8

B

0

0

0

0

0

0

0

0

0

I

8

8

8

B

8

8

I

0

0

0

0

0

0

0

0

0

8

I

B

8

B

I

I

8

0

0

0

0

0

0

0

0

0

B

B

8

8

B

8

I

8

0

0

0

I02

xãär.äérã+5v

Fig. 5 .7(o)

+5v

SX

Circuitin the

sy

+5V

zx

for sign detection of

16-15 residue sYstem'

X

idt-¡-(^'Nro

+5v

f or sig n detection of

16-15 residue sYstemCircuitin the

PR OM (I g)-\ N) (,

FJ(O

5 --1 Or grl (tl

ni\)ro

7¡.

PR OM (15 )

N(o

¡-{Cnul G'N)(^)n(,\:

Fis. s.7 (b) Y

103

Table 5. B

The states of the select-ion pi-ns andcorresponding selected inputs

States ofSelected Pins

2, L4

Sel-ectedInputs

'03of

' and. '01' (inverted '01)operation codes (table 5 "2)' r03', '01'

Addition

Sukrt ract-i on

Mult.iplication

Not used

From the Table 5.8 it is clear that'03' and'01' can be

di;ectly connected to the selection pins 2 and 14 respec-

tively. However, since in comparisons the sign of the res-

ult of addj.tion (if the signs of X and l/ do not agree) /subtraction (if the signs of X and / agree) is required and

the '03' bit of addition, subtraction and comparison codes

(rab1e 5.2') 1s same (zero) , only '03' bit is directly conn:-

ected,' whereas 'S-i' is gated thqough the Excluslve OR (44,

t,2,3) to change conditionally the logic state of '01' to

select the result of addition,/subtraction for sign detection

during com;:arison.

For addition, subtraction and multiplication 'þ7' and

'06' of the operation codes (rabte 5.2) are zeros; therefore

the output of the AND gate (39 ,2, L,3) is ' O ' . The 'õ-1 '

passes out through the Exclusive OR (44,L,2,3) without inver-

sion. In case of comparÍson, subtractiotr and dlvision, t-Ìre

'þ-1 'which is t.he input to the Exclusive oRs (44,5,4,6) and

(44 ,I ,2,3) is ' 1 ' . The output f ::c-¡m the Exclusive OR (44 ,5 ,

0

1

0

0

0

1

0

1

0

l_

0

0

1

1

4,6) is the::efore

of the signs

ter (48, I,2) ,

ofX

Therefore the 'SC' is also

di.visor and dividend. In

gate (45,1,2,3) is '1' .

not agree 'St-' will be ' 1

will be ' 1' v¡hich inverl-s

104

the invertecl S/. The actual Exclusive OR

and y (SX,Sy) is the output from the inver-since 'SC' = 1 @ Sy o SX = Sy @ SX = S/ @ SX.

the Exclusive OR of the signs of

comparisons, the output of the OR

If the signs of the X and / does

and the output of the AND gate

Of ' (=' l-' in comparison) to '0' .

Theref ore the states of the selection pins 2 an<1 1,4 are 'O'

and '0' respectively and the result of addition will be

sel-ected. ITowever, if the s-igns of .{ and / agrees '3?, wil.lbe '0' and so the output from the AND. Therefore the output

of the Exclusive oR (44,I,2,3) will be 'õT' (='f in compar-

ison) ; the states of the sel-ection pines 2 and L4 are '0'and '1' respectively and the result of subtractlon will. be

selected for sign detection.

The outputs from the multiplexers are connected to the

sign detector (enou (fa¡¡. 'SZ' gives the sign of the result(addition/subtraction/multiplication) Z and ZZ gives the in-formation of zero/nonzero of the result

fUcte 'Lhat the Exclusive OR (44 ,5 ,4 , 6 ) and the inverter (48,

L,2) could be omitted by connecting S/ to the pin 10 of the

Exclusive OR (44,10,9,8) and the output from it ('SC') to the

pin 2 of the AND gate. However, it has not omitted, slnce

this circuit is also part of overflow detector (fiq 5.10)

v¡here the sign of the subtrahend is to be j.nverted by'01'(='f in subtraction) for subtractive overflow detection].

3

t2

{uiÀJ

oìä

2

ô.1

MultiPtexer{co

MuttiPte xe rl\)

2\¡ 'D \(.o

or6rrtN

MultiPtexe r€l o

2

¿,

Multiplexer\t .f,

(gr )

ëH€HQå

ßz)

<<ØV¡ÞÞCCccc]c'(nòr¡¡'tJr¡\

(¡¡Ð(¡,

(sa )

3.\¡øUn

ì9l¡J

Lo À)

PRO M (1/, )

=<(rL4ÞÞccGc,gËt¡N(^rtvl4rÀJ

B

09

3

t2

l_0s

gl .€

51.(t-4]l 6

SS

1st. Ex-0R)

44fiJü

6

O

6tr_ TfOr-()ctCJocçaco

C)

oc')Ð

UJ

19, \.o\ \¡

a

atlx

(Lt-l

2nd.Ëx-0R/

(lrl-l

sc (4 5)

øB)tö

19{(3s)

+-3rd. Ex-0R

+ 5v

SZ zz

Círcuit for sign detection of theresult of oddition/subtrnction /muttipticotio¡r in the 1ö*15 - resicJue

systerÍì.

33utØÞÞccccooâojoJo

s.lr)

(¡o)

+5v

Fig. 5 ,8

106

The finar decision in comparison is done with a lockup table (Table 5.9) formed in rhe pRoM (ZZ¡ (ri_g 5.9) .

Altogether 32 locations (locations DoH to DFH and FoH toFFH) are required. The contents of the odd locations give

the decision modes of comparison with sigr,ed numbers, where-

as the contents of the even locations give the decisj_on

modes of comparison with unsigned numbers. The 'þ7, bit ofthe operation code determines whether the l-ocations read are

even or odd. If 'þ7' is 'L', the odd locations are selected;otherwise the even locations.

The

C01 = '1'indicates

output C00

indicates X

X is equal

'f indicates X

is greater than Y

toV

is less

; whi-1e

than

Cþz =

yì

tl_'

Table 5.9

Look up table for decision modesin comparison

D

F

4 bits of each hexadecimal digit representtSVt, tSZt, tZZt and tþ7t

The secondbit of eachhexadecinaldigitrepresents

rSXt

01234567 sgABCDEF2244iI0022242L00

1 1 L 4 L 2.0 0 2 1 4 4 L 2 0 0

1,07

+5v

ø7

zzszsy

+5v

Fig. 5.9 Círcuit for decisíon rnoddsin comPorison in 16 -1 5- residu e

syste m .

5

50

R

l2

il

t0

l23lI

5

6 P

R

0M3

2

1

r5

108

5. 3 (a) Delay in Compar j-son

From Fig 5.8 and 5.9,

Delay in comparison = 1 PROM delay for operatlonof addition/subtractlon.+ 1 tutultiplexer delay

+ (¡ Exclusive OR + 1 inverter+ 1 AND) delay.

+ 1 PROM delay fortion

+ 1 PROM delay formodes.

sign detec-

decision

(70+ 2o+3x7+1"0+9+70+70) ns

= 270 ns (for the elements usedi-n the test facilities) .

5.4 Overflow Detection

Fig 5.10 shows the circuit for overflow detectÍon in

addition, subtraction and multiplication. The circuits in

the dash-lined box detect additive,/subtractj-ve overflow,'

while the rest of the circuit is needed to detect multiplic^

ative overflow.

5.4 (a) Multiplicative Overflow Detection

V'Ihen multiplicative overflow occurs

be of incorrect sign or may be outsj-de the

system.

the product may

range of theI

An overflow characterised by j-ncorrect sj gn is detec-

ted by the circuit shown in the solid-lined box. The signal

'SC' is the Exclusive oR of the signs of X and / (rig 5. B) .

In Fig 5.8, 'þ-f is 'O' for multiplication and addition.

Therefore the output 'SS' from the Exclusive OR (44,5,4,6)

is really S/ (sign of V) and the output'SC' from the Exclus-

1- 09

ive OR (44,10,9,8) is the Exclusive OR of 'Sy' and 'SX, (the

signs of x and V) . Tabre 5.10 is the truth table of overflow

characterised by incorrect sign.

Tab1e 5.10

Truth tabl-e of overflow characterised byincorrect sign

Sign of X, SX sign of V, SY sign of Z, SZ Overflow

0

1

1

0

1

0

0

1

0

1

0

0

0

1

0

1

0

0

1

1

0

0

1

1

0

0

0

0

I

1

1

1

From the Table 5.10 the overflow 6(þ') is given by

ó(þ'' :ï l;".,î'From Fig 5.8 SC = '0,1' e 'SY'

For: multi-plication and additj,on,

SC=S/@SX.

The expression (5.1) becomes

6(þ') = SC Ø SZ

'[T' is 'o' , rherefore

(5.3).

@ 'sx, (s.2)

a

110

In Fig 5.10, the output from the Exclusive OR (43,I,

2,3) is the Excl-usive OR of 'SC' and 'SZ'. However, a false

slgnal of overflow occurs when a negative number j_s multi-plied by zero" The zero operands are detected by the

inverters (49 ,3 , 4) and (49 ,5 , 6) and AND gate (39 ,6 ,7 , B) . Ifone of the operands is zero the output from the AND gate

(39,6,7,8) will be '0'; so is the output from the 3-input

AND gate (4I,5 ,3, L ,6) ( OT is , 1 , for multiplication) .

This means no overflow indication wj-Il be given.

The circuit outside the boxes (fi-g 5.1-0) detects over-

flow by 'group technique'. The PROMs (18) and (f0¡ form look

up tables (fable 5.L2) of group numbers of !¡ and X respec-

tlvely (where X and / are the numbers from 0 to +119 and -1

to -L2O) corresponding to the Table 5.11. (fne number of

groups is 8, since 27 < 240 < 28 and nt = 16.)

Table 5.11

Table of numbers and their correspondinggroup number

Numbers Group Number

0

1

2+3

4

I -+1516

32

64

-+ 31

->63+L20

0

1

2

3

4

5

6

7

111

Tab1e 5.12

Look up table of group number of X or V

Each hexadecimal digit represents tlie residueof N{od-16

0

Eachhexadecinaldigitrepresentstheresidueof ¡nod-15.

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0123456789A8CDEF

04s6677777776654

514 s 6 6 7 7 7 7 7 7 7 6 6 5

6524s66777777766

6 6 s 2 4 5 6 67 7 7 7 7 7 7 6

7 6 6 5 34 s 6 67 7 7 7 7 7 7

7 6 6 s 3 4 s 6 67 7 7 7 7 7

7766s3456677777

77766s34s667777

777766543s66777

7777766s4s56677

77777766s435667

777777766s43566

6777777766s4256

667777777665425

7 7 7 7 7 7 7 6 6 5 4 L

677777776654

6

6

6

5

6

4

7

7

7

7

7

7

7

6

6

5

0

TI2

+5v\i6r)ljls(^ttú-O

Të

(/.9) (49) (41)

+ 5v XXXXXXXX{('rr¡¡\(, l\)rC¡ +5v

I

o

(16t

ula¡

'oN'

g¡ l,ll

Ulxìgf\,

t¡ñFä

(1¡ )

(16)(39)

€¡¡\t I

Fig. 5 .1 0

(43 ) t11t

€tXrl

u.7l

(41) )t

u7t

(46 t (15 )

¡1t0verf Iow.

ctors Ot the R AU board -

Circuít f or overflow detection ìn the16-15 res id ue sYSte rn

'10 l1

a(^,@ .Dô

otF.

n¡\nõ'co

PROM (16)I'o o

¡.\¡ølt,l LN\¡ (tl L'ì ul

PR 0 M (18)JJ

.D o f\t

tf¡

l")rÆ

13 F.ADDER (36I

l.'l ¡., o, ll'

é-or^*

3

21

6

51

1l

213I

673

12

6

15

3

12

If the sum of the group numbers

than B, overflow must occur, if less

not occur, íf equal to 8 overflow may

if overflow occurs, the si-gn will be

113

of X and / is greater

than B overflow does

or may not occur, but

incorrect.

The group numbers are addecl- with the Full Adder. Ifthe sum exceeds B, the M.S.B. (at Pin 15) will be 'l-' and atleast one of other three bits wilt be nonzero. The firsttwo OR gates (46,2,L,3) and (46,5,4,6) detect whether the

least signifÍcant three bits is nonzero. rf it is nonzero

and the M.S.B. is '1' , the output from the AND gate (4'l ,5,6,6) will be 'f i.e. overflow has occurred..

In multiplication 'þ2' is '0' . This means that the

output from the additive/subtractive overflow detectors(dash-lined box) is '0'. However, 'F' opens the ÀND gate

(47,L,2,3). The output from the OR gate (46,13,L2,LI) gives

the information of multiplicative overflow ('f indicatesoverflow) .

5.4 (b) Additive and Subtractive Overflow Detection

After inverting the sign of the subtrahend, the sub-

tractive overflow is same as the adCi-tive overflow detection.The inve¡sion 1s done by the Exclusive OR (44,5,4,6) of Fig

5.8. In subtraction 'þT' is '1'; therefore the output 'SS'

= 37. In addj-tion ' pf is 'O'i therefore the output is 'SS'

= Sy. An additive overflow is saicl 1-o occur if the sign ofthe operands are same and the sign of result is incorrect.The Table 5. j.3 is a truth tabl-e of additive,/subtractive over-

flow decision modes.

LI4

Table 5. 13

Truth table of additive/subtractiveoverflow decision modes

'ss , Sy/Sf Sj-¡rn of X, SX Sign of Z, SZ overflow

0

I

0

0

0

0

1

0

0

1

0

1

0

1

0

1

0

0

1

1

0

0

1

1

0

0

0

0

1

1

1"

1

From the truth

expressed as

d (0)

table the additive overflow d (0) can be

='SS'. SX. SZ +'SS'. SX. SZ

This function is to be ANDed with 'þ2'the overflow is due to addition/subtraction.

actual expression becomes

d (ol = 'þ2' . d (O) = 'þ2' . ('3s '. sT. sz +

(s.4)

to signify that

Therefore the

,SS,. SX. 9l(5.s)

The implementation of the expression (5.5) needs three

inverters, tü¡o 3-input AND gates, one 2-input OR gate and One

2-inputANDgate. Assuming 'SS' exists, the total de1ay,

accordlng to the delay characteristics of the components used

in the test circuj-t, is One 2-input AND gate delay (g'S

ns) + One 3-input AND gate delay (4'75 ns) + One 2-input OR

gate delay (tZ ns) 25 ns.

The actual circuit of additive,/subtractj-ve

is shown in the dash-lined box of Fig 5.10. The

expression ú(A) of the circuit is

d (A) = (,þ2, . SX. 'SS') @ (,Q2,. g. SZ)

where SC 'ss' @ sx (rig 5.8)

115

overflow

logic

(5.6)

To form ('SS' @ SX) does not need any extra Exclusive

OR, since this signal already exists (for multipticativeoverflow detection). Only requì-rement is to invert to form

'ss' @ sx.

Therefore implementation of expression (5.6) needs

One fnverter, tÍro 3-input ÀND gates, and One 2-input Exclus-

ive OR. The gate counts are less than that woul-d be required

to implement the expression (5.5). Assuming 'SS' exists, the

deJ-ay is --- One fnverter delay (g ns) + One 3-input AND gate

delay (4'lS ns) + One Exclusive OR delay (l ns) = 15 ns. The

delay is less than that would be for the expressi_on (5.5).

However, the expression (5.6) can be reduced to the

expression (5.5)

(,þ2'. SX. 'SS') @ ('þ2'. SC. SZ)

,þ2'. [(SX. 'SS') @ (SC. SZ)]

' þ2' . ISX.

'þ2' - t tsx

'ss' . (sc. sz) + (sx. ,ss'). se. sz]

+ 'SS' ) ' (SX o-*S3l . SZ + SX. 'SS'.

(SX @ 'SS' + SZ) ]

116

'þ2',. Itsx *SZ + SX.

sx. 'ss,) .

sx. 'SS' + V) ]

Szl.

'0' . therefore the out-

detector is '0' . The

gives the information

indicates overfl-ow) .

'ss' )

'ss'

(sx.

(y.'SS' +

'SS' +

,þ2' . tsx. '33' . sz + sx. ss.

In addition,/subtraction 'þ2' -i s

put from the multiplicative overflow

output from the OR gate (46,13,13,11)

of additive/subtractive overflow ('1'

5.4 (c) Delay j-n Multiplicative Overflow Detection

Multiplicative overflow detectj-on needs trn/o processes

(i) detecticn of incorrect sign of the result and (ii) detec-

tÍon whether the product is outside the range of system

(using group technique)

(i) The detection of incorrect sign of the product

involves in detection of the sign of the result, 'SZ'.

De1a1r in finding 'SZ' is (from Fig 5.8)

D 2T +37 +T +T +T (5.7)SZ 3 xoR I AND MUX

(Ot 2 ,5, one T: is needed to form product term)

where t3 = the access tj-me of (256x4) pnotq.

TxoR = the delay in the ¡ixclusive oR'

Tt = the deJ-aY in the Inverter'

T¿,ut = the delay in 2-input AND gate '

TMux = the delay in 2 : i- multipl-exer -

For the components used in the circuit in Fig 5. B

tAND-IT: T0nstTXOR=7ns,T

= 20 ns. Thereforeïnu"

10ns i 8.8 ns

TT7

Dr, = L40+2L+10+B'8+20 =

From Fig 5.10 (circuit in the

detecting incorrect sign, DIS

Drr=DsztT3a"o*Txon

200 ns.

solid-lined Box), the delay 1n

is given by

* To* (5.7a)

where T

(since delay

in detecting

= the delay in

= the delay in

3-input AND gate.

OR gate.3AND

ORT

T L4 ns (for the OR-gate used

is less than the delay

'SC' has been omitted.)

in the circuit 5.10)

in forming 'SC'

'SZ' , delay in

(z T*o*)

forming

OR

Therefore

(ii)

nique' DGT

Dr, = 2oo+7+8. B+l-4 = 230 ns

Delay in overflow detectron

is given by (from Fig 5.10)

D"t=T3*T¿,oo+ATon+

(s.7b)

by 'grouping tech-

2TAND

(5.7c)

where Taot = the delay in adding 4 bits. For the

adder used in Fig 5.10, TADD = 25 ns. Tirerefore

D = 7O+25+4xI4+2x8.8

= L69 ns (5.7d)

From (5.7b) and (5.7d), it is seen that the overflow detec-

tion by 'group technique' j-s faster than by 'incorrect sJ-gn

detectj-on' technique. This is due to the fact that the sign

detection is not needed for the formation of group number.

The (5.7a) gives the actual expression for delay in multiplic-

ative overfl-ow in the present system.

5.4 (d)

Delay

is given by

Delav iltdetection

additive/subtractive overflow

l_ 18

ADDin additive/subtractj-ve overflow

(from Fig 5.10)

D =D +T +27ADD SZ XOR OR

=2OO+7+2x14

= 235 ns (for the components

detection D

(5.2e)

used inFig s.10)

5.5 Integer Division

fnteger division needs a first approximation of the

quotient and then proceeds by successive approximations each

of which is a number equal to the preceding approxlmation

divided by 2. The first approxÍmation is formed from the

information of the group number, g(X) of the dividend and

the group number , g (V) of the divisor according Lo the Table

5.14. (fne first approximation and the successive approxim-

ations are positj-ve numbers. )

Table 5.L4

Table of first approximatj-on of quotient

Condition First Approximation

rf g(x) > g(V) 2s (x) s (v)

rf g (x) s (v) Zero

(oivisj-on of zero or by zero j-s aborted.)

Tab1e 5.15 shows the possibl-e values of first

imations (in power of 2) and their residue forms in

and mod-15.

apprîox-

mod- l- 6

119

Tabl-e 5. 15

Possible values of first approximations

First ApproximationFirst Approximation in

residue formmod-16 mod-15

7

6

5

4

3

2

1

0

Since the successive approximation is the number equal

to the precedi-ng number divided by 2, Table 5. 15 also g j-ves

the possible values of successive approximation (except Z7 ) .

For the successive approximations, the number to be

divided by 2 is supplied by the X-data bus (ri-g 5.11). How-

ever, all eight bits need not be multiplexed. Frcm the T'able

5.15, it is clear that the resiclue in mod-L6 is either

zeÊo or equal to the residue in mod-l5. There.'ore only six

bits of information 4 bits of the residue in mod-15, 1 bit

of zero-nonzero information of the residue in mod-16 and one

extra bit signifying 'division by 2' , are required. The

requirement of 'division by 2' is commucicated by the 'Ô3' bitwhich iS connected to the input pin 13 of the multiplexer (34)

and to the selection pin 1 of the multi1:lexer (34) and (35).

B

4

2

1

I

4

2

1

0

0

0

0

0

8

4

2

1

0

2

2

2

2

2

2

2

2

0

T]Ccc)o=

oC

J.ocr

l.-¡--?

oJuO

+E

a I¿à

xã v,

o.-þ(uÈ

: =

o_tslJo'i

',n

OO

€l_l-Ø

o_¡- o. Ln;o-

I

-(OãE

-Ø

G)

tø-C

.';o+

¡¿

()oo.-

C)

l/l'{-

rf)ólr.

rn+

lnIÌt02rf,IÌto

Oc\¡!-t

ct¡rì

x4x5X6

x7

ûzgy0X

Ogy'X

Ix2

gy2X3ø3

gx0

gxl

AP

O

PI

AP

2

P3

AP

4

AP

5

AP

6

AP

7

AP

O

AP

'

AP

2

AP

3

AP

1

AP

5

AP

6

AP

7

l2t1

P'o

LI

321r5 567

rtt\'

Itl\'

ox2ø

3aolt

oì\'att

ocD

r¡,\'lfl

nl

tr1x9?Àr-9);¿

l?

235b34 0

I7I2

08.

IJxlrlJÀt--JÐ

2355t0t13I

1l:P

'oo-

9

567

12

321l55

R s4

R53

R 5659

R 58

R 57

121,

l{hen '03' goes high, four bits of the residue (X3,X2,

X1,X0) in mod-15, one bit of zero/nonzero informati-on of the

residue (X7 ,X6,X5,X4) i-n mod-l-6 and. '03' (- '1') are selected

to the output of the multiplexers (Z:t multiplexer). I¡Íhen

'03' goes low the group number of X, (gXI,gXI,gX2) and the

group number of y (SVj,gVL,gV2) are selected instead. (fne

group number of X and / are generated by the PROMs (f0¡ and

( 18 ) respectively of the circuit in Fig 5 . l- 0 . )

The multiplexed outputs are fed to the PROMs (19) and

(20¡ which form two l-ook up tables of the first approxi-mation

and successive approximations. one *-able (raUte S.16(a))

gives the residue in mod-16 and the other table (raUte 5.16

(b)) gives the residue in mod-15. Lo,cations (00H to 07H) ,

(10H to 17H), (20H to 27H) , (ZO H to 77 H) alrogerher

64 locations are allotted for first approximations and locat-

ions (80 H to 9F) altogether 32 locations are all-otted for

successive approximations .

The integer

can be shown with

division is an lterative process; and

the flow chart of Fig 5.L2.

ir

L22

0t23456789ABCDEF

Table 5.16 (a)

Look up table for first approximation andsuccessive approximation in mod-16

0000000000000000010000

Z E R O2L00042100842100842100000000000000200000000000

Z E R O

0

1

2

4

I

0

0

0

1

L

2

4

8

0

0

0

I

0

0

0

0

0

0

0

0

0

0

0

0

1

2

3

4

5

6

7

I

9

A

B

c

D

E

F

L23

0

Tabl-e 5;16 (b)

Look up table for first. approximation andsuccessive approximation in mod-15

000000000100000002100000042L0000 Z E R O

0842L00001842100021842L00421842108102000400000000010200040000000

Z E R O

L23456789ABCDEF

0

I

2

3

4

5

6

7

I

9

A

B

c

D

E

F

L24

Ycs

Yes No

No

Yes

No

No

No

Process ol integer division in the l6-15 residuesystem.

IN (X,Y )

l<-0; j+0; <-0

Input first approxima-iion: F.

I

Is

f =o ?

T<_Y Does

overftow?

t

7

the re sutt+t

Isth e sign s ign

r e sutt7

of+

Is

thethe

Q.ê 0- qq<- q

q, <- q,Ftag overf low.

ENDT ove rftow

2

Does

ls

?T X

he sign of the+

Is

resutt?

X <- X +Tx (--- x-T

Yes

Q¡*r* Q¡ + F,

i <--- i+1

Y¿s

+ /z;i+-i+t

No

Yes

Fig. 5 .12

125

Iterative processes continue such that the iterative

values of X continue to decrease. Therefore T (in Fig 5.12)

is added or subtracted from the iteratj-ve values of X depend-

ing on the sign of the result (Exclusive OR of the signs ofX and }/ i.e. SX O Sy). From Table 5.I7, it is obvious thatT is added if the sign of the result is negatj-ve; otherwise

T j-s subtracted. (The first approximations and the succes-

sive approximatioi:s are positive.)

Table 5.L7

Tab1e to decide whether T is to be' added or subtracted

SY SX Sign of TSign of

SXthe result@ sy Operation

0

0

1

1

0

1

0

1

0

0

1

1

0

1

1

0

Subtraction

Addition

Addition

Subtracti-on

(The sign of T follows SY. )

The quotient generated is in positJ-ve form. Therefore

it is ccnverted to negative forr.t (subtracting frorn zero), ifthe sign of the result is negati-ve.

The division of M/2 (fZO

a problem. M/2 is a negative

part' posj-tive M/2. The sign

is positive; whereas quotient

in this case) by

and there is

1 presents

no 'counter-number

of the

(M/2)

result of above divisionis negative.

L26

However,this problem can be solved by software. When

the positive approximations are added successively, the lastaddition will show overflow wÌ:,en M/2 is divÍded by 11. There-

fore when sign of the result is posj-tive and overflow occurs,

then overflow signal is to be flagged. If the sign of the

result is negative and overflow occurs, the quotient is M/2.

(ffre software listings have been shown in Appendix-6; Ap6(a).)

5.5(a) Delay i-n findinq approximations

Delay in finding the first approximation, DFA is given

by (ris 5.11)

DFA = Delay in finding the group number of X and V

+ I multiplexers delay (fig 5.11)

+ 1 PROM delay (rig 5.11)

= 1 PROM delay (rig 5.l_0)

+ 1 multiplexer delay + 1 PROM delay.

For the components used for the test circuit,

DFA = 70 ns + I7 ns + 70 ns = L57 ns (5.8)

Delay in findrng a successive approxlmation Oro is given by

(from Fig 5.11) '

DSA = 2 OR gate delay

+ 1 multiplexer ,1e1ay + 1 PRCM delay.

= 2xI4+L7+10 = 115 ns

(for the components used in the circuit).

5.6 Tloatinq Poiht Arithmetic

In this case the base of the exponent is chosen to beI' 12

S = @t¡2¡'' = (L20)'2 = 11. A number X j-s expressed as

I27

X A. 1ln (s. g)

where A is the mantissa and n is the exponent, the range

of which is from 0 to +119 and from -1 to -I20.

In floating point operations each mantissa is separateci

into two terms such that

A = az.Lt + o.y (5.9a)

0"2 and a1 êF€ the co-efficients of l-power and O-power of

11 respectively and the range is from 0 to 10. The steps to

find the co-efficients are

(i) o.z (Integer Division)

(ii) d.¡ A a.2.IL

During normalisation a co-efficient oí is needed to be

rounded. The steps that are requj-red for rounding are

11 l(rr a is negative (-5) is added; otherwise +5)

í- I

a +T

toí-, t 5

(i) Find T

(ii) Find the rounded a.1

Floating point arithmetj-c is done by sottware (appendix-6 (b) )

which resides in the PROMs of SDK-80 microcomputer.

5.6 (a) Floating Poip!. Addition

The floating point addition can be described with the

flow chart of Fig 5.13 (a) , 5.13 (b) and 5. 13 (c) .

L28

Yes ño

No Yes

No Yes

Addit¡on of Mantis:¿=.

Fig. 5.13(a) I Ftow chart of Floating Point Addit ion f n thel6-lS resídue system

E

IN( XJ }

X = A.11ntY = 8.11

n2

A:05

nl e0

n2e0

Coeff (A):) ql ,O2br bzCoeff (B)

Te-(nt-nz)

Negative ?

IsT

oiÈb1 ¡ o2+ b2

S(qt) --- 5(bl); Sloz)S S(bz)

Atignmeiìt of Mantissa.

NormaIisation.

Erponent & Sign Ot The

Resu I t

END

r29

Yes

Yes

No

Fig. 5 .13 (b, Proce s s

addition

* Process of rounding .

of atignment and ftoating point

in the l6- l5 residue system .

Yes

Yes

T

Is>t

?ls

?T 2

Is

?T 1

T(2) <- O2 + 2

T(r) <- o1 + þ1

T(2) <- O2

T(l)ê01

Normalisatíon 7

lss(bz) = 1'

*bt+ +

I

bz- 0Sign detect ionof the mant i s sa

of the re sul t n *bz<-0bt <-

END

Iss(b1)='r'

2

br+ bz+bze- 0

br <- bz+

bz <-0

130

No

Process

add it io n

Yes

of normati sat ion in

in the l6-15 residu e

No

floating Point

system.

T + T(2).1.|

Does

ove r7

T ftow

T+ T+Til)

YesDo

ov?

T c rftes

t+tT(tIs)

2

T<- lry,] Tê.t

T+-T(2)+TExpt X) <-Exp(X)+1

Yes

Mnt(Z | + T

Exp(Z¡+- ExP(X)

Find the sign of

Mnt (Z )

EIiD

Fi9. 5.13(c)

131

5.6(b) Floatinq Point Subtraction

After the subtraction of the mantissa of subtrahend

from zero, the floating point j-s same as floating point

addition. The flow chart of Fig 5.I4 shows the floatingpoint subtraction.

5.6 (c) Floatlnq Point Multiplication

The theory behind the flóating point multiplicarionhas been given in sections 4.4. 3 (b) and 4.4.4 (b) . The

floating point multipl-icatj-on in 16-15 moduli system can

be describe<l with rhe f low chart of Fig 5.15 (a) , 5.15 (b) .

To find the exponent of the result, Exp(Z) is a bitcomplicated process. The Exp(Z) can be written as

Exp(Z) = Exp(X) + Exp(Y) + g

where g is the 'generated exponent' (section 4.4.4 and

Fis 5. 1-5 (b) )

Vthen Exp(X) and Exp(/) are positive, the overflow in

E*p (X) + Pxp (!/) j-ndicates that nxp (Z) is outside the range.

(g is always positive.). When Exp(X) and Exp(l¡) are negacive,

Exp(X) + Exp(/) may overflow, however, Exp(Z) may or may not

remain wi-thin +.he range due to the addition of g.

In later case it has beenobservedthat when Exp(Z) 1s

really within the range, and if Exp(X) + Exp(y) overflows,

addition g makes a second overflow such that Exp(Z) comes

within the range.

t32

X = A.11

l= 8.1 1

IN (XY)

n1

n2

B O- B

Ftooting oddition

E ND

Fig.5.1t, Flow chortsubtroction insystem.

of ftootingthe 16- 1 5

pointresidue

133

rN(xy)¡= A-ilnl; Y= 8.tfl2

Coeff(Al+a1ra2Coef f (B) :+ br, bz

T(3)<_a2.b2 ) T(2lea2.bl

T(l) <- at - b2 ; T(0, <- al. bt

t {P.| ] 7 TU,t<-T(2)-T-tl ; T(3)eT(3)+T

t-[Hl ; T(5) eTfii- T.lr ¡ T(3)<- T(3)+ T

T(2) <- T(r.) +T(5) ¡ t - [.,I!E] ; T(t) <- T(ol- T

T(2t + T(2) +T .

Norrnatisation

Fi nd Exp (Z )

Find sig n of Mnt (Z

ENt)

Fig.5.15(o) Ftow chort of flooting poínt

muttipticqtion in the 16-15 res¡cue

system.

134

No Yes

No

Nc

No

No

Fig. 5.15(b) Ftow chart of normatisation in the l6-15 moduti

f loating point muttiptícat ion .

T<- T(3)'11

DocS

ove ra,

T ftow

T(5)+T+l(3)ls

T(2) +, 2

T(4'+ 2

Does

ove?

T(5) rf tow

T(6)<- T( 5). r IT(4'<- il

2

t(z)+T(31+T(4)T (6, overflow

oes

7

T(4) <- T(6) + T(l) g<- 2

0oes

ov7

T(4 ) erf tow

IsT(1) '+' ?

Mnt(Z) ê- T(¿)

g<-0 T(4)<- T(4 ) <- fi(1)+5

Mnt(Z¡ <-T(5) +T(4)Exp (Z) <-Exp(X) + Exp(y).rg

g<- 1

Yes

END

135

Therefore if overflow occurs twice or overflow does

not occur at all, Exp(Z) is v¡ithin the range, otherwise

EXP(Z) is outside the range. The flow chart of Fig 5.15(c)

explains how to determine whether Exp(Z) overflows or not.

5.5(d) Floatinq Point Division

The flow chart of Fig 5.16(a) shows the floating pointdivj-sion. (The division of zero or by zero is aborted.)

The mantissa and exponent "t * are Eenerated with two

look up tables,' one is for the mantissa "t å and the otherfor the exponent

"t *

The mantissa of the result, Mnt (7) is found by float-ing point murtiplication of the mantlssa of x and that "t å.This multiplication results in a 'generated exponent' , g.

The calculation of the exponent of floating pointdivision is complicated too.

Exp(Z) = [exp(X) nxp(V)] + [g + EE] (rts s.1o(a))

Exp(X)-Exp(}/) may overflow,- however, Exp(Z) may remain

within the range due to addition of the te.rm I g+EE ] which may

be positive or negative and does not overflow.

(g =0or1or2 andEE=0or-1or-2 or-3)

Applying the same reasonings as given in determining the over-

fl-ow conclition of Exp(Z) in case of floating multiplicatlon,the overflow condition of Exp(Z) in case of floating pointdivision can be shown with the flow chart of Fig 5.16 (b) .

136

T <- Exp(X) * Exp(Y)

T overf tow

D oes

2

T+- T + gT <-T + g

Yes NoT overflow

Does

7

T ove rf tow

Does

?

No ove rf low

0verf Iow0verflow.

Yes No

No Ye s

Fíg.5.15(c) Overftow detectíon of Exp(Z) in

flooting point muttipticotion in the

16-15 resid u e syste m

t37

rN(xy)X = 4.11

n1

)(B

E x p

EM + Mnt

EE

Mnt( Z) e- Ft.Pt. (A'EM )

Erp(Zl+Erp(X)- Exp(Y)+ 9+ EE

END

Fig.5.16(ol Flow chortdivision insyste m .

of f lootíng point

the 16-15 residue

No Yes

Does

T ove rflow?

Yes

Overf low detect io n offtooting point divisionres¡d ue syste m .

No

tn16-15

T+Exp(X)-Exp{Y)T0)+ 9 + EE

TI

Doesov¿ rf

T ê- T + T(l)T- T + T(l)

Yas NoT ov e rftow

0oe s

7

0v e rf[ ow No ove rf lowNo ove rf [ow

Fis. 5 .16 (b) Exp(Zlín the

the

-2

of

138

5.6(e) GgnerjrFion of mantissa and exponent "f +

For a base of exponent ß, a general method to calculate

mantissa and exponent of f n"r been described in Appendix

Applying that method the values of mantissa and. exponent

for the base of exponent ß=11 (eleven) have been tabul-

in decimal and residue forms shown in Appendix-3.

1B

ated

The mantissa and exponent of * in resj-due form areIJ

stored in PROMs (fiq 5.17). The value B is fed to the PROM

(I7), (21) and (231 through the X-data bus and PROM (17) and

(Zt¡ output the mantissa in mod-16 and mod-15 respectively,

while the PROM (23) gives the value of exponent.

The exponent "t * may be of any value of -L, -2 and

The residue form of those values are shown in Table 5.19

3

Table 5.19

Resj-due form of the exponent of

Exponent in Residue form

1

B

Exponentin

Decirnalform

-1

-2

-J

In Hex In Binarymod-16 noJ-15 nod- 16 ¡nod- 15

1i 10

1 101

1100

E

D

c

F

E

D

1111

1110

1 101

From Table 5.1-9 it is seen that two most significant bits of

the residues (in binary) in mod-16 and mod-l5 are same and

constant (11). Therefore in the formation of a look up table,

storÍng of these two bits is not necessary; only two least

significant bits of each residue are required.

only one (256x4) PRotl is needed.

139

Therefore

Two M.S.Bs are generated at the output cj-rcuit of the

RAU and this will be discussed in section \.7 .

The PROM(23) (rig 5.17) forms the look up rable forthe generation of the two least significant bits of each

residue of the exponent "t å

The Tables 5.20(a), 5.20(b) and 5.20(c) show the con-

tents of the pRoMs (L7) , (2I) and (23) respectively.

5.6(f) Delav in findinq the mantissa andexponent of 1

B

Delay 1 PROM delay (from FÍg 5.17)

70 ns (for PRoM, rntel 3601).

140

xt,x5X6

x7

x0xt

X2

X3

EEO

EEI

EE1

EE5

EMO

EMI

EM2

EM3

EM4

EM5

ËM6

EM7

ïw0EXP.

L. S.

IN

IW0 L. S.

EXP. IN

EEO

EEI

EE4

EE

EMO

EMI

EM4

EM5

BITS OF

MoD-15 ,

BITS OF

MOD - 16

rn

IÌto2

G)oaoao)o

3o)=.tnU¡oo

(p

xt'X5

X6

x7x0

X1

x2x3

E

E

x4

X5

X6

x7XO

X1

x2

X3

(o

I

!oE

E

E

+5v

Fig. 5.17 C ircu it f or o pProx imote f tootingnumber of 1/ B in the 16 -1 5

system -

pointresidue

5

6

7

1

3

2

ll

10

( zglI

15

P

R

0M

5

6

7

t,

3

2

P12Rt0M 10

l21l s

5

6

7

64

6

56

R 67

R

R41

5

6

7

I3

2

1

12

1t

10

$719

t5

P

R

oM

R ¿5

141

Table 5.20(a)

Look up table for the mantissa of Iin nod-16 B

0

Each hex digit represents the residue of B innod-16. (Lower 4 address bits)

t 4 B 6 3 8 4 B F I D 0 5 4 C E

01234s6789ABCDE

0 7 4 2 A E 13 s D F 2 6 E C 9

3Bt229E135DF26DB

AE D A 019 E 13 5 C E 15 C

c8A81F098135CE15

5 B 7 6 E 5 6 0 8 D t 3 5 4 2F

Eachhex digitrepresentsthe residueofVinnod-15.(Higher4 addressbits).

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

E

C

4

2

F

B

4

6

D

0

04

EO

CE

4C

24

F2

BF

4B

53

74

A

4

0

D

c

4

2

F

A

2

5

A

4

0

D

B

4

2

E

A

F

4

9

3

0

D

B

4

1

E

4

D

5

9

3

F

D

B

3

1

3

1

A

2

I

5

F

D

B

j

9

1

F

7

1

8

2

F

D

5

E

6

F

D

5

0

7

2

F

D

7

D

3

c

c

A

0

7

2

F

D

7

c

1

8

B

1

F

7

2

0

c

6

B

D

2

F

0

E

6

3

0

c

6

A

A

B

6

E

E

4C

24

02

c05C

95

68

32

F5

c9

L42

Table s .20 (b)

Look up table for the mantissa of Iin mod-15 '^ B

0

Each hex digit represents the residue of B innod-16. (Lorver 4 address bits)

8BAE07C024D057ED

0123456789A8CD8

0110 8 C 0 2 4 D 0 3 7 0 E E

Eachhex digitrepresentsthe residueofVinnod- 15 .(Higher4 addressbits) .

1

2

3

4

5

6

7

8

9

A

B

c

D

E

F

c

D

6

2

E

C

5

1

D

9

2

313C87C024C826D

AEA9BD7CO24CE26

c9AO44D6802431D

5C87937C68E23CE

15B73525858E23C

E 15 B 612 0 2 ^

5 A E 13

C E 15 A 5 D O D E 9 4 A E 1

3CD74A4ADAC84AE

tscD1.4938C6873A

Dl5BD()492880563

9 D 13 B D 0 312 4 6 51s

2L8C028D0380154

1ic8c024Ð0370EE

329D158D03815CEC

7

0

L43

Table 5.20(c)

Look up table for two least significant bits of each

residue (in nod-15 and nod-16) "f *

Each hex digit represents the residue of B innod-16. (Lower 4 address bits)

0

1

2

3

4

01234567894BCDEF0444444444444444

4844444444444444

4494444444444444

4449444444444444

4444994444444444

4444499444444444Eachhex digitrepresentsthe residueof/innod-15.(Higher4 addressbits) .

5

6

7

I

9

A

B

C

D

E

F

44444499 4

9

9

4

4

4

4

4

4

4

4444444

4444444

4444444

44

44

44

44

44

04

4

4

4

4

4

4

4

4

4

44449

44444

44444

44444

9444444

9944444

4994444

4 4 4 4 4 4 E

44 44444

44 44444

4499444

4444944

44 44444 4444494

44444

44444

r44

For ß = 11, t glr) = 5. The highest positive number

for the system under consideration is +119; and the highest

negative number is -I2O.

5.6.1(a) Error in Floatinq Point Addition/Subtracti-on

From (4.20) ,

numbers is

the maximum relative error for positive

| ø¡z)

5.6.1 Error in Floatinq Point AdditionSubtraction and Multiplication

(i) Max (¿) 5 .o42R 119

The maximum relatj-ve error for negative numbers is

.o42

5.6.1.(b) Error in Floatinq Point Multiplication

From (4.2t¡ ,

rs is

the maximum relative error for positive

(ii) Max (¿)

U (zo)ß/, ß/t

2 25+3 5

65L20 .54

5L20

n l,j + 3'max R

65119 .55119

The maximum relative error for neqatlve numbers i-s

U (¿olmax

5 .7 Outr¡ut Circu t of the RAU

The outputs from the RAU are the results of addition,

subtractl-on, multiptication, comparison, approximate quotient

for integer division, and mantissa and exponent of $ tot

floating division.

145

The sign of the result of addition, subtraction and

multiplication, overflow signal and the Exclusive OR of the

sJ-gn of the divisor and divj-dend are connected to the pins

4, 5 and 6 respectively of the RAU board (fig 5.8 and 5.10)

and are eventually" connected to the C-port (lower part

C0, Cl, C2) of the SDK-80 microcomputer (fi-g 5.4) .

The result cf addition, subtraction, multiplication,

comparison, approxi-mate quotient, and the mantissa and1exponent "f Ë are multiplexed by three least significant bits

of the operation codes (fig 5.18).

The pin 2

are connected to

most significant

of the multiplexers (24) , (25) , (28) and

+5V through 1.8 K reslstors to generate

bits for each residue of exponent of

(2e)

two

_lB

The pin 4 of the

and (28) are connected

5 MSB positj-ons of the

least significant bits

multiplexers (24) , (25) ,

to ground which generates

comparison sj-gnals; only

give the information of

(26) ,' (27)

zeros in the

the three

comparr_scn.

Table 5.2L shows the states of the selectj-on pins and

the selected inputs.

L46

87I

+5V

+5Y

|,1V7

7

EM7AD7SU7

MU5

EE5AP5

EM5AD5su5

Øzþtûo

U3

AP3

EM3AD3su3

+5vMU

AP6

EM6AD6su6

MU4

EE4AP4

EM4AD4SU4

BS13

2

QzþtØ

ôz$r6o

(2t,'

( 26 )

(2 8'

(32 )

(25)

B5 t

3Cr{1'mxm7

2

3

5

B4

Q7)

B3

oÉ.

C)co

3

É.

tro

tJ)É.oF(-)LUzzO(J

lrl(9êLTJ

I

2

9 101121

0

øþø 40

+5vAP2cþ2Ern2AD2su2

( 29)

B1

BO

39

ØzQtøo

ØzØtØo

MUIEElAPlc0lEMIADlsul

2

3

1

=c--{P5rrl

¡ìn

MUOEEOAPO

2

3

I

(33)

=ct-{'t->mxmvr0 1l

2

3

II

-!

cÉ0EMO

ADOsu0

72

314

9Øz

þtØo

ØzötØo

R6 2

3

2

3

1

I

3cJ.

}5m)<mv

10 11I 1011

1

2

3

L2

13

=cr-{T5-mxmÐ

1tI

2

3

1

2

3

t,

.SC

-tT'->mxrftn

61

3ct-1¡].mxmn

I2

5

I 1011

2

3l.

R60ct-

-o

mxm7l0 I

2

3

2

3

I9

5

Fig.5.18 Output circuit of the RAU

L47

Table 5.2L

States of selection pins and selected inputs

Statesþ2, þ1, 00

selection pinsbits of op. code) Selected inputs

of(¡

0

1

1

1

0

1

0

1

0

1

0

0

1

1

1

1

0

0

0

1

0

Addition

Subtraction

Multiplication

Approximate quotient

Compar j-son

Ma-ntissa of

Exponent of

The outputs from the multiplexers are connected to the

pins 37 through 44 of edge connectors of the RAU board. The

outputs are eventually connected to the B-port of SDK-80

microcomputer (fiq 5.4). The most significant 4 bits of the

outputs represent the residue in mod-16 and the least signif-

icant 4 bits represent the residue in rnod-l5. In case of

comparison, three least significant bits gives the information

of the results of comparison.

5.8 Conversion of Residue Numbers to Decimal Numbers

The 1-6-15 residue system is a three-digit <iecimal sys-

tem (ttre highest positive number is +119). The theory behind

the conversion has been described in secti-on 4.5.

1B1

B

Fig 5.19 shows the residue

one decimal diqit (residue form)

operations required to

of a positive number X.

find

1.48

IN (X}

i<-0

Is

23

t-fExtract O

lrom T . T+

END b <- T. 10

c<- x-b

fromExtract O

b

Yes

No

x<-i<-

T

i+t

Process of re sid u e to decimq t

conversion in the 16-15 res¡duesystem .

Fig . 5 .19

L49

Since 16 and 15 are greater than 10, the residue of the

decimal digit in mod-16 is equal to the residue in mod-15

and each residue is equal to the decimal digit. Only one

residue is needed to be extracted. Decimal numbers are

represented in signed magnitude form. Thcrefore if X is

negative, it is converted to its positive form by residue

subtraction from zero or by residue multiplication with -1.

Successive digits are found by taking T (ln Fig 5.19)

as a new value of X.

The first decimal digit is extracted by residue

multiplication with 01 and second digit by residue multip-

Ii-cation with 10. The residue addition of first and second

extracted digits gives two consecutive decimal digits which

are stored at an 8-bit-byte location. The third digit(t'l.S.O) is extracted by resj-due multiplication with 0l- and

is stored at another location. The contents of those locat-

ions, in proper order, give the decimal number of X.

(The program listings have been given in Appendix

-6 (c).)

5.9. How the Residue Arithmetic Compute r Works

The RAU together with the SDK-80 microcomputer forms

a real time residue arithmetic computer (nac). The communic-

ation between the RAC and outside is thus made through a

teletype or VDU using SDK-80 programmes.

With the starting command - G

return), the RAC undergoes the phases

the flow chart of Fig. 5.20.

Boo) ( ù= carriage

of operations shown by

150

START

G 80oJ

tnitialise the p0RTs

Give signat to get thetype of operation required.

Wait

operationavaitable

Is

,)

Give signat to getope rand s .

Wa it

Areoperandsaveitabte

7

Pe rf orm

disptay

opera t ion and

the resutt .

Yes

Yes

Fig . 5. 20 Prog rom e loops of the R AC

l_5 1

5.10 Ho$¡ to operate the RAC

with the teletype or vDU connected to sDK-80 micro-

computer and power on to the SDK-80 and RAU, the

'reset'. Then the procedures that are followed

(i) Key in c Boo )

The computer responds with

SDK-80 is

are

0

(ii) Key in any

operati-on.

= (for type of operation).

alphabet A through ,f for the requj-red

(see Table 5.4)

(iii )

( iv)

If the operation is an integer operation the

computer responds with

M = (for fi-rst operand X).

Key in two hex numbers (resj-due number) .

The computer responds with

M = (for second operand Y).

Key j-n two hex numbers (residue number) .

Thc computer displays Che result, first in

residue form and second in decimal form with

overflow (if anlr) and sign characters. (tt¡e

overf low character j-s '?' . ) If operat j-on is

a comparison one, the display is ) or = or <

to indicate X is greater than or équal to or

less than Y. The computer signals

0 = (for further operatlon).

If

responds

operation is a floating point one, the computer

with

M = (for the mantissa of X).

Key in two hex numbers (residue number).

The conìputer responds with

E = (for the exponent of X).

L52

form)

f orm).

(iiÍ )

(iv) Key in two hex numbers (residue number).

The computer responds with

M (for the mantissa of V).

(v) Key in two hex numbers.

The computer responds with

E (for the exponent of V).

(vi) Key in two hex numbers.

The computer displays the result as

EM (in residue

(in decimalM

(fne dash-line indicates the digits. The results are

displavs with proper sign and overflow (i-r any) characters.)

Then the computer responds with

0 = (for further operation).

153

5.11 Conclusi-ons

The delay in multiplicative overflow detection, acc-

ording to the proposed technique, is 230 ns; whereas deIay,

accordj-ng to conventj-onal fast technique [-.rsing PROMs of

70 ns access tlme (tntel Bipolar PROM, 3601)] would be

4.2.70 = 560 ns. Therefore multiplicative overflow detec-

tion is considerably faster, accordlng to Lhe proposed tech-

nique. In determination of the first approximation for

residue integer division, the proposed technique introduces

a delay of I57 nsi whereas conventional fast division tech-

nique would need 2.2.70 = 28O ns. Therefore determination

of first approximation is also faster in the proposed tech-

nique.

Until now there has been no residue computer which

performs floating point operations. The introduction of

floating point arithmetic in 16-15 moduli system reveals

the fact that floating point arithmetic operations are feas-

ible in residue system, and that these operations are cost-

effective. The maximum relative errors that are introduced

in addition and multiplJ-cation [section 5.6.1] are not really

significant and quite normal for a single precision macl-^1ne.

Therefore the proposed algoriuhms on floating point arith-

metic have great impli-caticn in the residue system.

In the residue-to-decimal conversion process, where

the moduli are 16 and 15,the conversion radix is 10 and

neither A(X) methodnor base

fore the proposed technique

there is no reason for not

extensj-on is applicable. There-

is the only choice. Moreover

using the proposed technique where

all- the integer arithmetic

can be allotted for output

154

operations exist and more time

translation.

CHAPTER VItttrr

CONCLUSIONS

This concluding chapter presents a comparative study

between residue and binary number systems, and puts forward

a suggestion on further study on residue number systems.

6

The bases of compari-son

consumption in performing the

cation, addition,/subtraction

are the speed, cost

basic operations

including overflow

and power

multiplic-

detection.

For the purpose of comparison'16-15 and 16-15-13-11

moduli systems have been considered and the fastest avail-able J-ogic components of both TTL and ECL technology have

been chosen when dealing with implementations.

The range of 16-15 moduli system (16x1-5=24O) is lëss

than 28, however, greater than 27; likewise the range

16-15-13-l-1 system (16x15x13x11 = 34320) 1s less than 2" ,

however, greater than 2rs . In this cornparison, however, a

16-15 residue system has been compared wj-th an B-bit binary

system,: while the 16-15-13-11 residue system has been com-

pared with a 16-bit binary system.

In multiplj-catj-on the 16-15-l-3-11- system needs four(256x4) pRO¡ls. The component requirement for multiplicativeoverflow detection (on the basis of Fig 4.L) has been shown

in Table 6. 1 (a) .

155

1s6

In the circuit of Fig 4.L the generation of the group

number of the operand X, g(X) has been shown. The generat-

ion of g (V) needs similar sorts of components as required

for S (X). Again for the generation of g (X) the information

of NZ(X) is required; and ñZ(X) can be formed with one B

input NAND gate and one inverter with two levels of gate

delay. Similar type of components are needed for NZ(V).

The sign detectioi: (by mixed radix conversion) of the product

(SZ) needs five (256x4) pRottts. The circuit configuration is

sj-mi1ar to the circuit used for SX, except that (256x8) pnOU

is replaced by a (256x4) Pnot"l.

The requirement when using ECL components is same as

that of TTL components, however, since (256x8), (512x8) and

(5L2x4) ECL PROMs are not available, those PROMs are to be

built from available (256x4) enot"ts. The (256x8) is a PRoM

with 8-bit address lines and 8-bit data li-nes. The function

of that PROM can be confì-gured by using two (256x4) enOus.

One is used for four J-east signi-ficant data outputs, and

other one is used for four most significant data outputs.

The (512x8) pno¡,t can be built from two (256x8) pRotqs (con-

structed by above method). The B-bit address lj-nes are used

to select any L:cation of (256x8) pnotq and the remaining

address bit selects one of two (256x8) pRO¡,1s. The outputs

from (256x8) pROt"ts are wire-ORed. Therefore four (256x4)

PROMs are needed to construct an equivalent (512x8) enot'ts.

construction of a (5I2x4) pno¡¿ needs tu/o (256x4) pRot"ts. The

8-bit address lines select any locatj-on of (256x4) PRO¡,1,

whilst the remaining address line selects one of two (256x4)

PROMs. The outputs of (256x4) pROt'ls are wire-ORed. Table

L57

6. 1 (b) shows ECL required f or multipS-ication and its over-

flow detection in 16-15-13-L1 moduli system. In place of a

4-bit ful1 adder (nCf,) which Ís not available, a 4-bitArithmetic Logic Unit (ef,U) has been chosen. Since an 8 in-put ECL NAND is not avaj-Iabl-e, 2 j-nput AND gates are to be

used for the generation of Nl(X) and liJZ (y) . A total of

fourteen 2-input AND gates are needed and the arrangement

has three level-s of gate delay using currently available

components.

In a 16-15 moduli system, multiplication needs two

(256x4) pRotvls. The components f or multiplicative overf low

detectj-on can be calculated f rom the cj-rcuits in Fig 5.l-0

anC 5.8. The circuit in Fig 5.8 has been used for sign SZ

of the result Z. Tabte 6.2(a) shows the TTL components

required for multi-plication and its overflow detection in

a 16-15 moduli system.

The ECL components needed are the same as the TTL com-

ponents, however, since ECL 3-input AND gates are not avail-able, 2-input AND gates are to be used. Therefore for three

3-input AND gates, six 2-input AND gates are substituted.

Tabl-e 6.2(b) shows the ECL components needed for nultiplic-

ation and overflow detectj-on j-n a L6-15 moduli system.

Of the TTL family Schottky TTL ISZ ] is rhe fasresr;therefore Schottky TTL components have been chosen. Of the

ECL family [Se ], EcL-Iff is the fastest,- however, ECL-IIT

has a limited number of functions, therefore ECL-10000 has

been chosen for the ECL components. Tables 6.9(.) and 6.3(b)

show the characteristics regarding delay, cost and power

158

consumption of TTL and ECL components respectively.

Table 6.1(a)

TTL cornponents required for nultiplication andits overflorv detection in a L6-15-15-1L modu1isystem. (Yaking into account of generation ofg(V) and SZ.)

Function ConponentRequired

Numberof

Conponents

Number ofPackages

TrpeNo.

Multiplication (256x4) PROM 82527

82527

825714

82S130

82S115

sN74S283

SN74S15B

SN74S86

SN74SO8

SN74S32

44

OverflowDetection:(According tothe circuit inFig 4.1,requirement forgeneration ofS(n and signdetection of7 i.e. SZ.)

(256x4) PROM

(25óx8) PROM

(512x4) PROM

(512x8) PROM

4-bit Ful1 Adder

2:1 Multiplexer

2-input Exclusive OR

2-input Al,{D

2-input 0R

2

4

4

t4

2

4

4

t4

6

2

1

1

1

6

2

2

2

3

I B-input ttANlInverter for

SN74S3OSN74SO4

2

2

a

IgeneraNZ (x) ,

ticn of

^/z (y) . l

42

(Totalpackage

count)

159

Table 6. 1(b)

ECL conponents required for multiplicationand its overflow detection in a 16-15-13-11noduli system.

Function Coml.ronentReqrrired

Nunberof

Conponents

Nunber ofPackages

TypeNo.

Multiplication (256^4) PROM 1014 9

10149

l_ 0181

t0L73

10L07

10104

IvlC 10103

4

42

6

2

1

3

(Hex AND)

1

4

6

OverflowDetection:(According tothe circuitin Fig 4.1,the require-nent forgenerationof g(V) andthe signdetectionof Z i. e. SZ)

(2s6xa) PRoM

4-bit Arithneti" MCLogic Unit

2:1 Multiplexer MC

2-input Exclusive OR lvfc

2-input AND MC

2-input 0R

42

2

2

16

3

59

(Totalpackage

count)

160

Table 6,2(a)

TTL conponents required for nultiplicationand its overflow detection in a 16-15 nodulisysten.

Functinn ConponentRequíred

Nunberof

Components

Number ofPackagesof gates

TypeNo.

Muttiplication(rig s.6 (c) )

OverflowDetection:(According tothe circuitsin Fig 5.8and 5. 10.)

(2s6x4) PROM

(2s6x4) PROM

4-bit Ful1 Adder

2-input Exclusive 0R

2-input 0R

2-input AND

Inverter4:1 MultiplexerS-input AND

82527

82527

sN74S285

SN74S86

SN74S32

SNT4SOB

SN74S04sN74S153SN74S1 1

22

3

1

2

2

1

3

1

5

5

4

L

41

343

L7

(Tota1package

count)

161

Table 6.2(b)

ECL conponents required for multiplicationand its overflow detection in a 16-15 modulisysten.

Function ComponentRequired

Nunberof

Components

Number ofPackagesof gates

TypeNo.

l4uttiplication (256x ) PROM 10149

1 0149

MC 10181

MC 10107

MC 10103

MC 10104

IvlC 10195MC 10174

22

OverflowDetection:(According tothe circuitsin Fig 5.8and 5.10.)

(2s6xa) PROM

4-bit ALU

2-input ExclusiveOR

2-input 0R

2-input AIttrD

Inverter4:1 Multiplexer

3

1

2

3

1

5

5 2

2

(Hex AND)

10

L

434

L7

(Totalpackage

count)

Table 6. 3 (a)

Characteristics of TTL components required forResidue Multiplication and its overflow detection

Components

(256x4) pno¡,t

(256x8) Pnou(51,2x4) pnou

(5i-2x8) pnou

4-Bit Adder3-input AND

Hex Inverter2-input AND

2-input OR

Quad Exclusive CR

2:1 Multiplexer8-input NAND

4: l- Multi-plexer

TypeNumber

82527

I2S 11_4

B25 1 30

825115

sN74s283SN74S1 1

SN74SO4

SN7 4S O8

SN7 4S3 2

SN74S86

sN74s158

sN74s30

sN74S15 3

Delayin ns Power

mwr/Bitpwlnitmw/eitlw/sitmw

mw/gatemw/gatemw/gatemw/gatemw

mlf

mw/gatem14r

Price*

25.2519.006.50

17.002.O0

.571 .08

.57

.57

.852.39

.862.39

Manufacturers

sisnetics [Ze ](i)(ii )

(iii)(iv)(v)

(vi )

(vii )

(viii )

(rx)(x)

(xi )

(xii )

(xiii )

40

60

BO

60

7

4.753

4.754

7

4

3

6

.6165

.3165

510

31

19

32

35

250

195

L9

225

il

I

ll

Texas Instrument ISZ ]il

il

il

il

ll

il

il

I

il

il

ll

il

il

il

It

il

79 -03POìN* All prices are i-n Australian dollars and sma1l quantity price.

Table 6. 3 (b)

Characterj-stics of ECL components required forResidue Multiplication and its overflow detection

Function

(256x4 ) pnopt

4-bir ALU

Hex fnverter

Hex 2-input AND

2:1 Multiplexer

Triple Exclusive OR

Quad 2-input OR

Dual 4zI Multiplexer

TypeNumber

LOL49

MC 10181

MC 10195

MC 10104

MC 1017 3

MC 1_01_07

MC 1010 3

MC 10174

DeIayin ns

20

7

2

2.7

2.5

2.5

2

3.5

Power

mwr/Bit

mw

mwlekg

mw/vkg

mw

mw,/ekg

mwlPkg

mw

Pri-ce*

22.32

35.24

1.31

.51

7.10

.55

.5 i-

7.06

7 4.56

Signetics

Motorola

Manufacturers

(i)

(ii)

(iii- )

(iv)

(v)

(vi )

(vii )

(viii)

.66

600

100

140

270

110

100

305

[ 78]

Ise]il

ll

* A1I prices are in Australian dollars and small quantiti price.POr(,

L64

Presently the MPY/HJ family [56] of binary multi-p1j-ers are the fastest. Table 6.4(a) shows comparj-sons

of a 16 x 16 multiplier Vs. 16-15-13-11 moduli multiplier

and an B x B bits multiplier Vs. 16-15 moduli multiplier

using TTL technology. The delay in overflow detection ina 16-15-13-11 moduli multiplj-cation has been calculated

using the expression (4.1)r whilst the expression (5.7)

and (5.7a) have been used for calculation of delay in over-

flow detection in the 16-15 moduli multiplication. Table

6.4(b) shows similar comparisons using ECL components.

From the examination of Table 6.¿ (.) and 6.4 (b) it

is clear that residue multlplj-cation is much faster and

more cost-effective, specially using ECL technology. How-

ever, when residue multiplication is associated with over-

flow detection, the system gets slower and more costly.

However, since the same piece of hardware, used for over-

flow detection, can be used for sign d.etection and generat-

ion of first approxi-mation for resj-due j-nteger dÍvision,

the cost is not as much of a concerning factor as would at

first appear.

Table 6.4 (a)

Comparison of (i) 16x16 binary multiplier Vs 16-15-13-11 mod muttiplierincluding its overflow detection

(ii) 8x8 binary multiplier Vs 16-15 mod multj-its overflow detection (TTL components.

lier includingp)

Function

(1) 16 x 16 bits multiplier

(2) Residue Multiplication(16-15-13-1l-mod)

(3) Overflow Detection

(4) 8 x 8 bits multj-plie::

(5) Residue Multiplication(16-15 mod)

ComponentRequired

rRw/, MPY-16 HJ

See Table6. 1 (a)

See Table6. 1 (a)

TRw/MPY-8 HJ-]-

See Table6 "2 (a)

Number ofComponents

See Table6. 1 (a)

See Tab1e6. 1 (a)

1

See Table6.2 (a)

See Table6. 2 (a)

Delay inNS

100

40

269

45

40

L29

Price*

$265.00

$101.00

$s06.21

$120

$ 5o.so

$ e3.00

Power

3w

2.56 W

18w

1W

1.78 vü

4W

1

(6) Overflow Detectlon See Table6.2 (a)

* All prices are in Australian dollars. POlC¡

Com,oari-son of (i)

(ii)

Table 6.4 (b)

16x16 binary multiplier Vs 16-L5-13-11 mod multipl-icationincluding overflot¡ detection8xB binary multiplication Vs 16-15 mod multiplicationincluding overflow detection. (Using ECL.)

Functi-on

(1) L6 x 16 binarymultiplication

(2) Residue Multiplication(16-1s-13-11)

(3) Overflow Detection

(4) SxSbinaryMultiplier

(5) Residue Multiplication( 16-15-mod)

ComponentRequired

Motorola /L0L83(2x4 multiplier)

See Table6. 1 (b)

See Tab1e6. 1- (b)

Motcrrola /I0Ig3

See Table6 .2 (b)

Number ofComponents

32

See Tab1e6. 1 (b)

See Table6. 1 (b)

See Table6 .2 (b)

See Table6 .2 (b)

Delay inNS

100

20

L26

50

Price*

@ 30.84ea986.88

89.28

1165.39

@ 30.84ea246.72

44.64

320.00

8

Power

25.6 w

2.7 W

33.16 W

6.4 w

1. 35 V{

5vù

20

65(6) Overflow Detec;ion See Table

6.2 (b)

* All prices are in Australian dollars. HOrOì

t67

A comparison of a 4-bit binary adder (fff, and ECL)

with a 4-bit residue PROM adder has been shown in Table 6.5.

From the table it is apparent that in TTL technology a

4-bit binary adder is six times faster than a 4-bit residue

PROM adder. Therefore in addition of equivalent 24 (=6x4)

bits or higher, residue PROM wil1 be faster, provided the

access time of the PRoMs used is 40 ns. However, since the

cost of one (256x4) pnO¡¿ (TTL) is twelve times higher than

that of a 4-bit binary adder (TTL), the residue adder isnot cost effective.

In ECL technology, a 4-bit binary adder is three times

faster than a -bj-t residue PROM adder. with ECL technology

addition of equi-valent L2 (=3x4) ¡its or higher will be

faster in a residue system assuming that the access time ofthe PROMs used is 20 ns. However, sign detection in 3

moduli (each module having 4 bits) system wilt need 40 ns

[ = (3-I)'2O; see Appendix-S]; whereas the sign detection ina 12 bì-t binary system will need 3x7 = 2L ns.

From what has been discussed above, it is clear thatpresent day technology still is not favourable towards the

impiementation of resÍdue j-n general purpose computers. The

binary system wil-1 maintain its p::edomillance. However, inspecial applications wÌrere there is no use of sion and over-

flow detection, residue multiplication will be the firstchoice and may also be a contender 1n very fast and large

systems where the carry free property is of great importance

in minimising the chip connections and inter chip delays.

Table 6.5

Comparison of a 4 bit binary adder(TTL and ECL) Vs. a 4 bit residuePR0M adder (TTL and ECL).

Function Technology ConponentsNunber ofConponents

Delay inns.

168

Price in $'.4.

A4bitbinaryadder

TTL

ECL

TTL

sN74S283

MC 10181

82527(256x4) PROM

1 7

7

2.00

35.20

25.25

1

1 40A4bitresiduePROM

adder 10149(2s6x4) PROM

ECL 1 20 22.32

As it has been discussed earli-er that the residue

floating point arithmetic can be performed in software prov-

ided all the residue integer arithmetic operations are avaj-1-

ab1e. In fast processors the residue floating point

operations will be fast. However, in comparison with binary

floatj-ng point arithmetic the residue floating point ¿rrith-

metic is sl-ower since (f ) the mantissa.s of the residue

floating point operands are to be broken down into two

co-efficients and (ii) in normalisation process the mul.ti-

plicative overflow information is required. However, at

least it has been shown that the floating point arithmetic

can be performed in the residue system, and the operations

are cost-effective, since the only requirements are PROMS

for generation of floating point numbers of the reciprocal

of the mantissa of divisor and memories to store the prog-

ramme required for resj-due floating point operations.

169

It is clear from the discussion above that the multi-plicative overflow detection is the speed limiting factor

in the residue system. Also sign detection is a problem.

Thêrefore more efficient algorithms for sign detection and

overflow detection should be investigated and also the

possibility of using logic gates or FPLAs (instead of PROMs)

should be looked at. In the near future VLSI circuit may

open the way to j-mplementation of overflow detectors with

combinational logic and a dj-fferent comparison between the

systems may then result.

The resj-due PROM adder,/subtractor is slow. The com-

binational logic -i s another possible \^/ay for implementing

resj-due addition,/subtraction and should be a matter of

f urther investigat j-on.

1

BIBLI.OGRAPHY

Pawlak, Z. "An Electronic digital computer based on'-2' system", BuJ-1. Acad. Pol-onaise Sci.Serie des sciences techniques. Vol-7, No.12,Dec. 1959, pp.713-721.

Pawl-ak, Z. and Wakuliz A. "Use of expansions wj-tha negative basis in arithmometer of a digitalcomputerr'. 8uI1. Acad. Polonaise Sci. C1-3,Vol-5, No. 3, March 1,957, pp .233-236.

Brusenzov, N.P., Zhogol-ev, Y..A,., Verigin, S.P.,Maslov, S.P. and Tishulina, A.M. "The SETUN smal1

automatic digital computer" (in Russian),Moscow University Vestnik, no.4, 1962, pp.3-72.

Israel, Halpern and Yoeli, M. "Ternary arlthmeticunit", Proc. IEE, Vol-115, No.10, October 1968,pp.13B5-1388.

Vranesic Z.G. and Hamachar, V.C. "Ternary Logic inParallel Multipliers", The Computer Journal,VoJ--1-5, No.3, Nov. L972, pp.254-258.

6

2.

3

4-

5

7.

8.

9

10.

Vranesic Z.G. and Smith, K.C. "Engineeringof Multivalued Logic Systems", Computer,L974, pp.34-4I.

Aspectssep -

Mine, H- and Hasegawa, T.Operations", System.No.1, L97I, pp.46-54.

"Four Ternary ArithmeticComputers. Controls. YoI-2,

Ataka, H. "C)n the Definition of Ternary ThresholdFunctions", System. Computer. Controls . YoI-2,No.1, t97L, pp.92-93.

Merril, R.D. Jr- rrA Tabular Mirtimization Procedurefor Ternary Switching Functions", IEEE trans onElectronic Computers. Vol-Eó-15, No.4, AugustL966, pp.573-585.

Mukhapadhyay, A. "Summetrlc Ternary SwitchingFunctions", IEEE trans. on Electronic Computers,Vol-E6-15, No.5, .Oct. L966, pp.73I-739.

Santos J. and Arango H. "On the Anal.ysis and Syn-thesis of Three Valued Digital Systems", Proceed-ings Spring Joint Computer Conference, 7964,pp.463-475.

1t_ .

L70

L2.

13.

t4.

15.

16.

L7.

18.

19.

20.

2t;'

22.

23.

171

Porat, D.I. "Three Valued Digital Systems", Proc.IEE, Vol-1L6, No.6, June 1969, Pp.947-950.

Mouftah H.T. and Jordan, I.B. "Design of TernaryCOS,/MOS Memory and Sequential Circuits", IEEEtrans. on computers, YoI-C-26, March L977, pp28L-288.

Ivlouftah, H.T. "Three valued C.M.O.S. Cycling Gates"Electronic Letters, 19th Jan. L978, Vo1-14, No.2,pp. 36-37.

Huertas, J.L., Acha, J.I. and Carmona, J.M."Implementation of some ternary operations withcMos integrated circuits", Electronic Letters,Vol-12, July I976, pp.3B5-386.

Carmona, J.M., Huertas, J.L. and Acha, J.I."Rea1j-zat j-on of three valued C.M. O. S. CyclingGates", Electronic Letters, Vol-14, No.9, 24LhApril 1978, pp .2BB-290.

Kaniel, A. "Trilogic, a three leveIs logic systemprovides greater memory density", EDN, L973,pp.80-83.

Mouftah, H.T. and Jordan, I.B. "Integrated Circuitsfor Ternary Logic", Proceedings of the Internat-ional Symposium on Multivalued Logj-c, May L974,pp . 285- 302 .

Etiemble, D. and Israel, M. "Implementation ofTernary Circuits with Linear Integrated Circuits",IEEE trans. on Computer, 7977, C-26, pp.L222-1'223.

Higuchi, T. and Kamayama, M. "Static Hazard FreeT-gate for Ternary Memory Element and its applic-ation to Ternary Computer". IEEE trans. onComputers, YoI-26, No.12, Dec. L977, pp.L2L2-L22L.

Tichdao, T., McC1usk.y, E.J. and Russel, L.K."tn1¿irzalued Integrated Injection Logic", TEEEtrans. on Computer, Vol-26, No.12, Dec. 1977,pp. L233-I24L.

Anderson, D.J. and Dietmeyer, D.l. "A Magnetic Tern-ary Device". IEEE trans. on Electronic Computers,Dec. 1963, pp.9II-9I4.

Pugh, A. "Application of binary device and BooleanAlgebra to the realization of 3-valued Logiccircuits", Proc. IEE L967, lI4, pp.335-338.

Rath, S. S. "A ternary f lip-f lop circuitrr , Tnt. J.Electron j-cs, 1,975, Vol-38, No. 1, pp.41,-47 .

24.

172

25.

26.

27.

28.

29.

Vùatson, R.W. andcomputationPrOc. IEEE,

Hastings, C.I47. "Self-checkedusing residue number arithmetic",Vol-54, Dec. 1966, pp.1,920-1931.

Mandelbau¡n, D. "Error correction in residue arj-th-metic", IEEE trans. on Comp. Vol-C-21, Junet972, pp.53B-545.

Barsi, F. and Maestrj-ni, P. "Error correcting prop-erties of redundant rcsidue number systems",IEEE trans. Computer, YoI-C-22, March I973,pp.307-315.

Szabo, N.S. and Tanaka, R-I. "Resj-Cue Arithmeticand its application to computer Technology",McGraw Hill Book Company, 1967.

Garner, H.L. "The Residue Number System", IREtrans. on Electronic Computers, Vo1-EC-8, June1959, pp.140-L47 .

30. Keir, Y.4., Cheney, P.!{. and Tannenbaum, M."Division and Overflow Detection in ResidueNumber Systems", IRE Trans. on Electronic Com-puters, Vo1-EC-11, August 1962, pP.501-507.

31. Kinoshita, E., Kosako H. and Kojima, Y. "FloatingPoint Arithmetic Algorithms in the SymmetricResidue Number System", IEEE Trans. on Computers,Vol-C-23, No. L . , January 1,97 4, pp .9-20.

32. Sasaki, A. "The Basis of Implementation of AdditiveOperations in the Residue Number System", IEEETrans. on Computers, Vo1-C-17, Nov. l-968, pp.l-066-1078.

33. Svoboda, A. "Digitale Informationswandler",Braunschwelg Germany : Vieweg and Sohn, 1960.

34. Flores, Ivan. "The Logic of Computer Arithmetic",Prentice Hall Inc., 7963.

35. Banerjee, D.K. and Brzozowski, J.A. "On TranslacionAlgorithms in Residue Number Systems", fEEETrans. on Computer:s, YoI-C-2L, Dec. 1972, PP.L28i -L285 .

36. Chu, Y. "Dj"gj-tai Computer Design Fund.amentals",McGraw-Hill Book Company Inc., L962.

37. Lewin, D. "TheoryNelson, L972.

and Design of Digital Computers",

Deverall, J. "The design of cellular arrays forarithmetic", The Radio and Electronic Engineer,YoI-44, January L974, pp.2L-26.

38.

39.

40-

4L.

42.

43.

44.

45.

46.

47.

48.

49.

50.

L73

Jayashri, T. and Basu, D. "Ce1lular arrays formultiplication of signed binary numbers",The Radio and ElectronÍc Engineer, YoL-44, No.1,January I974, pp. 18-20.

V'Ia11ace, C.S. "A suggestion for a Fast Multiplier",IEEE Trans. on Electronic Computers, Vo1-EC-13,Feb. 1964, pp.14-17.

Guild, H.H. "Fully Iteraiive Fast Array for BinaryMultiplication and Addition", ElectronicsLetters , L21u}:¡ June 1969, Vol-5, No. 1.2, pp.263.

Majithia, J.C. and Kitai, R. "An It.erative Àrrayfor Multiplication of Signed Biirary Numbers",IEEE Trans. on Computers, Vo1-C-20, Feb. I97L,pp .21.4-2L6 .

Fenwick, P.M. "Binary MuJ-tiplication with OverlappedAddition Cycles", fEEE Trans. on Computers,Vol-C- l- 9 , January 1969, pp .71-7 4 .

Motorola Semiconductor Products Inc., "High SpeedRipple-Through Arithmetic Processor:, AN4B7.(Application Note).

De Mori, R. "Suggestion for an f.C.Multipliêr", Electronics Letters,L969, Vo1-5, No.3, pp.50-51-.

Fast6rh

ParallelFebruary

Thomson, P.M. and Belanger, A. "Digital arithmeticunits for a high data rate", The Radio andElectronic Engineer, Vol-45, No.3, March L975,pp.1L6-L2o.

tsandyopadhyay, S., Basu, S. and Choudhury, A.K."An lterative Array for multiplication of signedBinary Numbers", IEEE Trans. on Computers,Yol-C-2I, August L972, PP.92L-922.

Ha1lin, T.G. and Flynn, M.J. "Pipelj-ning of Arith-metic Functions", IEEE Trans. on Computers,YoL-2L, August 1'972, PP - BB0-886.

Habibi, A. and Wintz, F.A. "Fast Ì{ultipliers " ,

IEEB Trans. on Computers, Vo1-C-19, Feb. L970,pp.153-1957.

Toma, C.I., "Ce1lu1ar Loglc Array for High-SpeedSigned Binary Number Multiplication", IEEETrans on Computers, Vol-C-23, 7975, pp.932-935.

Motorola Semiconductor Products Inc., "High SpeedBinary Multiplication using the MC L0181",AN-566, I972. (Application Note).

51.

52.

53.

54.

55-

56.

57.

58.

59.

60.

61.

62.

63-

64.

65.

I74

Pezaris, S.D. "A 40-ns 17-Bit by 17-Bit ArrayMuItipJ-ier", IEEE Trans. on Computers, Vol-C-20,April L972, pp .442-447 .

Chung, Tl.J-. and Bedrosian, S.D. "Iterative DigitalMultiplier Based on Cellular Arrays of ROMs",Electronics Letters, VoI-11-, No.1B, Sept. I975,pp. 4 26-428 .

Knigh, R.B.D. and Stockton, J.R. "Development ofChung and Bedrosian Iterative Cellu1ar Multip-lier", Electronics Letters, VoI-72, No.B, April1.976, pp .LB2-l-83.

"MPY-Series Multipliers", TRW, Redondo Beach,Cali-f ornia, I97 6 .

"MPY/HJ family of Multipliersrr, TRW, Redondo Beach,California, L978.

The TTL DATA Book for Design Engineer, Texas Instrum-ent Inc., Dallas, 1976.

"MECL Data Book", Motorola, Phoenix, Arj-zona, I976.

Jacobsohn, D.H. "A Combinatoric Division Algorithmfor fixed Integer Divisors", IEEE Trans. onComputers, pp. 608-61-0.

Dean, K.J. "Binary Division Using a Data-Dependentfterative Array", Electronj-cs Letters , 721|-]:,JuIy 1968, Vol-4, No.14, pp.2B3-284.

Guild, H.H. "Some Cellular Logic Arrays for Non-Restoring Binary Division", The Radj-o and EIec-tronic Englneer, Vo1-39, No.6, June I970. pp.345-348.

Dean, K.J. "Ce1lular Arrays for Bj-nary Division",Proc. IEE. Vol-11-7, No.5, May I970, pp.9I7-920.

Majetha, J.C. "Nonrestoríng Divisionular Array" , Electro¡rics Letters,Vol-5, No. 1û, pp. 303-304.

Using a CeIl--14th May 1970,

Cappa, M. and Hamachar, V.C. "An Augmented Iterat-ive Array for High-Speed Binary Division", IEEETrans. on Computers, VoI-C-22, No.2, Feb. 7973.

V'Iadel-, L.B. "Negative Base Number Systems", IRETrans. on Conìputers (corresp. ) Vol--EC-6, June7957, p.I23.

"Conversion from Conventional Binary to Negative-base Number Representation", IRE Trans. onElectronic Computer, (Corresp. ) Vol-EC-10, Dec.1961 , p.779 .

66.

67.

68.

69.

t75

Zohar, S. "Negative Radix Conversioh", IEEE Trans.on Computers, Vol--C-19, March 1,970, pp .222-226.

Krishnamurthy, E.V. "Complementary Two-way Algor-ithm for negative radix conversionsrr, IEEETrans. on Computers, Vo1-C-20, No.5, May I97I,pp.543-550.

Yen, C.K. "A noteIEEE Trans. on325-329.

on Base -2Computcrs,

Arithiretic Logic",YoI-C-24, L975, pp.

70.

7r.

72. Rao,

73.

74.

't5.

76.

77.

78.

79.

80.

Sankar, P.V., Chakrabarti, S. and Krlshnamurthy, E-V"Arithmetic Algorithms in a negative base",IEEE Trans. on Computers, YoI-C.22, Feb. 7973,pp. 1 2o-L25 .

"Deterministic Division Algorithm in a negativebase", fEEE Trans. on Computers, YoI-C-22, Feb.I973, pp.I25-L28.

G.S., Rao, M.N. and Krishnamurthy, E.V."Logical Design of a negative binary adder-sub-tracLor", Int. J. Electronics, Vo1-36, No.4,L974, pp.537-542.

"A variable shift negabinary multiplier", Int.J. Electronics, I974, Vo1-36, No.6, pp.749-757.

Agrawal, D.P. "Arithmetic algorithms in a negativebase", IEEE Trans. onComputers, Yo!-C-24, OctoberL975, pp. 998-1000.

"Negabinary Division and Square-rooting",Digital Process, Vol-l-,Autumn L975, pp.267-274.

"Negabinary Carry-look-ahead adder and fastmultipli€r", Electronics Letters, Vol-10, JulyL975, pp.312-31-3.

Waser, S. "State-of-the-Art in High Speed Arithmeticfntegrated Circuits", Computer Design, Juì-y L978,pp.67-75.

"Signetics Bipolar and MOS Memory Data lulanual",L97i. 811 East Arques Avenue, Sunny Vale,California 94086.

"fnte1 Data Catalog", L975, fntel Corporation,3065, Bower Ave., Santa Clara, CA - 95051--

"Memory Data Book", L976, National SemiconductorCorp., 290O Semiconductor Drive, Santa CIara,cA - 95051.

"TTL Data Book", I976, National29OO Semiconductor Drive, Santa

9505 1 .

Semiconductor,Clara, California

81.

A1

APPENDIX - 1

Some Important Proofs

Apl (a) :

If 2rt <M< 2r^*', U= n+1, andg(X) andg(/l arethe

group numbers of X and V respectively, prove that

(i) if glxl+g(V) > U, overflow will occur-

(ii) if gl(l+glVl < U, overflow will not occur'

(iii) if g(xl+glvl = u, overflow may or may not occur.

If it occurs, the sign of the product will be

incorrect.

Proof:

The multiplication of the lor,/est numbers j-n two

groups is gj-ven by (from Table 4.1)

7=W

Z=W 2

z

2-^*glXl . v{ .2-^+glVl

2-Z^+glXl+slVl

(1)

( 1a)

Let glxl+TlV) = A+L- If A is an even number, W=(M

and ¿ = U/2+L. Therefore (1a) can be written as

z = M . 2 z(q/!+ll+¡tr+l = M/2 .""(2a)

If A is an odd number, W=

fore (1a) can be written as

Y!l¿2t and ¿ = lq+1)/2. There-

(M/2) . 2-2{(q+1 | I 2}+U+1 = M/2 (2b)

it

The equarions (2a) and (2b) indicate that overflow

will occur when the lowest positive numbers in two groups,r

sum of which is greater than U, is multiplied- | The

L

A2

equations (2a) and (2b) do not indicate overflow if one

of the lowest numbers is negati-ve, sínce M/2 is within

the range of negative half. However, in most cases @þ

or ëÌt is not an integer number. Therefore error will be

introduced in determination of W, and this error will make

7 ín equdtions (2a) and (2b) greater than M/2, and there-

fore the product of the lowest numbers one of whÍch is

negative will overflow" The method of determination of W

and construction of group tabre when @þ or ëlt is not

an integer number has been discussed in sectj-on 4.2.1)_t

Let g lxl +g lV I

can be written as

q. If. A is an even number, then (1a)

Z=M 2-2lql2+11+a = |11

o

2-¿ ( 3a)

If A is an odd number, (1a) becomes

(M/2) . 2 2{la+l1/2}+a = M.2 -2z ( 3b)

The equatj-ons (3a) and (3b) indicate overflow will not

occur.

The multiplicati-on of the highest numbers of two

groups glxl and g(/) is given by

z(

tw 2-^+gt,Xl+i -1 2-^*glVl+t l.

ì

[''

Z=w2 z-2^+g(Xl+SlV)+z _ hr . 2-^+glXl+1

-w.2-^+glVl+1 +1 ..... (4)

_1W

1-w

,'s-9(vl

z^ g (xl= !i[

2 2-2^+slxl+slvl+1 l,ìI

)+1

=w z-2o*g(Xl+g(V)+t

r.+r-fr.2^-s(vl

1+1

z^-s(xl

z^-s(xl

A3

(4b)

1!v

+1

(4a)

Since group 1 contains number L, overflow will

never occur on multiplication of number in group 1 with

the number in any other groups. Therefore overflow for

the groups greater than 1 will be consj-dered;

Let S lKl +S lY I A. The equation ( a) can be written

asZ=vtrZ.2-2d+ti+1 1

W

1

UIz^-slvl +1

( 4c)

ì

The minimum value of Z depends on the minimum value of

^-s(xl

The expression ( c)

to 2. Let glxl = 2.

slvl

From the value of M,

,nl z

2^

1-+ -2 1 .^-9 lV I-Ïl"

If tJ is an even number, then from (4c) and (4e)

is mj-nimised when SlXl or S(Yl is equal

Therefore

= r1-2 (4d)

W can be expressed as

n+l<w<2-v- ( e)

2s-a+2 < I _ 1

,nlz2Ul2+1-2

2U/2+l-ti+2,n/ z

-2 1vü

1-+

A4

2(n+1) /2+1-Z 2ln*1ll/2+1-ln+1l¡+2

n/2 2n/2

l-2n+1 | I 2+3

mtn

L-

= L-+-22

z

orZ >Mm1n

2

2-2lU/2+11+q+7

However, for n s 4, 2l-2n+f) l2+3 t t

Therefore the minimum value of ('ic) is a positive number

for n > 4.

In other words for n > 4

Z-2o* U* 12vl (4f)

1=Pt.2

The expression (5) indicates that when the sum of group

numbers is equal to {, multiplication of the highest numbers

in two groups will produce overflow-

In similar way the expressed (5) can be proved to be

valid for odd value of q. Again from (4)

2 -2^+g(Xl+SlVl+Z

(5)

Z<w 2 (6)

Therefore for glx)+g(yl : A an<.l evnn value of ti, (frcm (6))

Z<M ( 6a)

If glxl+glVl = A-!, and A is an even number then from (6)

z<w 2-2lU/2+ll+q+l+l

orZ<M.2 -1 (6b)

A5

The expressì-on (6) indicates, t^rhen the sum of two

groups is less than /, the product of the highest number

of two groups does not overflow. Similarly it can be

proved that expression (0U¡ is also valid for the odd

value of U.

if

Combining

slxl+s(Vl

Combining

(2a) ,

> U,

(3a),

(2b) and (5), it cêrr b€ written that

overflow must occur.

(3n¡ and (6b), it can be concluded

that

if glxl +glyl < U, overflow will not occur-

combining (3a), (3b), (5) and (6a), it can be

concluded that

if glXl+$lVl =

If overflow occurs, the

U, overflow may

product will be

or may not occur.

of incorrect sign.

4n-t, ....., &zt At Afe mixed radix digits

APl (b)

If d-nt

of a number

ffinr ffin-r t

dÍgits bnt

(negative

¡ lfl2 ¡ III ¡

bn-rt ....t

or positive

{ (mi 1)

(positive or negative) in system of moduli

the mixed radixrespectively,

br., br of the 'counterpart' n.mber

number) are given by

oi\ for oil 0 (i=n, nrl, k+l )

the

the

then

b.1

bn {m.kon] and

pfor l0andþ - 0 nPok

(p:k-l- , k-2, 2, 1).

0

A6

Proof:

The 'counterpart' number is formed by subtraction of

the number from zero i.e.

(or, orr-, ok*, ou ok-, o" or)

Since o"

by (*r o*.) .

propagates up

formed by

bk*, =

b\*, =

Example:

Ifm 2

(or, 4rr- i on*, ou ou-, o, o ,)

, to 4k-r are zeros, the kth digit is formed

This subtracti-on produces a borrow which

to nth digit. Therefore other digits are

(^k* t 4k* r -1)

(*k*. &k*, -1)

{ (m¡1 r -1) -a¡a , }

{ (mt+ z -L) -at*, }aIt

{(m -L)-a }.nnb (m a _1)n n

b 0 forp k-1, k-2,

n

2, 1.P

= 0,

given

,fr2

then

by

6s =

b-2

ô-I

= 7, R,

the MRDs

(2-1)-1

(7-5)

0.

and 4 1 5a33 3 2

O.t

is

of the 'counterpart' number

0

2

A7

APPENDIX - 2

AP2z Floating Point va.Iue "t å

For B > l- and positive,

(1)

Let (2)

where OB and rU are the quotient and remainder of the left

hand side division in (2).

+a2

å=ou

Case (i ) rf O

divided by B.

Let

g-2L=g'.BB

tBB

ßl-B

1 ß3BB

r p

B+

P

33.B

ß R then 7",) is added to r and thenBB

r + vt (s)

(3a)

( 3b)

(s)

(6)

t-.e.

r rBBB =O +r

The rounded Qu j-s given bY K where

K=Q oB

Hence K

Case (if) If QB ß < R, then

ß-3

+

2

r

o

is added to rp and then divided bY B

ttB

+r+

D

vlB

v^=Q t

(6a)

The rounded is given by H, where

+ t

AB

( 6b)

( 6c)

is the negative

The exponents

OP

0oHP

Hence

ff H > R, then H is taken to be equal to R

For B 1 B¡

ß-3

The floating point value of

form of mantissa of positive

å-H

*=ß

negative

1

Bof both positive and. negatj-ve are same.

1B

1B

Table 5.18: Table of nantissa and exponent offor B = 7 to I20. (11 is the base of exponen

DecinalNumbers

1

2

3

4

5

6

7

I9

10

11

T2

13

14

15

16

Residue NumbersMod-15 Mod-16

Mantissa inDecinal Fonn

11

61

40

30

24

20

L7

15

13

t2

11

111

102

95

89

83

Exponent inDecimal Forn

-1

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-3

-3

-3

-3

-3

Mantissa in ResidueForm

Mod-15 Mod-16

Exponent inResidue Form

Mod-15 Mod-16

F

E

E

E

E

E

E

E

E

E

E

D

D

D

D

D

E

D

D

D

D

D

D

D

D

D

D

C

c

C

C

c

B

1

A

E

8

+

1

F

D

C

B

F

6

F

9

3

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

1

B

D

8

0

9

5

2

0

D

c

B

6

72

5

E

8

'ÚtDzH

I

(r¡

Þ\c

c)or+

coo-

DecinalNunbers

L7

1B

19

20

2L

22

23

24

25

26

27

28

29

30

3T

32

33

Residue NunbersMod-15 Mod-16

lvlantissa inDecinal Fonn

Exponent inDecinal Form

Mantissa in ResidueForn

It{od- 15 Mcd- 16

Exponent inResidue Fonn

Mod-15 Mod-16

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

c

C

c

c

C

c

C

c

c

c

C

c

c

C

C

C

c

E

A

6

3

F

D

A

7

5

3

1

0

E

C

B

A

8

3

E

A

7

3

1

D

A

8

6

4

3

1

E

D

C

A

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0

1

2

3

4

5

6

7

8

9

A

B

c

D

E

0

1

2

3

78

74

70

67

63

61

58

55

53

51

49

48

46

44

43

42

40

-3

-3

-3

-3

-3

-3

-5

-3

-3

-3

-3

-3

-5

-3

-3

-3

-3

ÞtsO

t)or+H.

oo.

DecinalNumbers

34

35

36

37

38

39

40

4t42

43

44

45

46

47

48

49

s0

Residue Nunberst{od-15 I'fod-16

Itfantissa inDecinal Forn

39

38

37

36

55

34

33

32

32

3L

30

50

29

28

2B

27

27

Exponent inDecinal Forn

I,.{antissa in ResidueForm

lvlo<i-lS Mod-16

Exponent inResidue Form

It{od-15 Mod-16

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

C

C

c

c

C

c

c

C

C

c

C

c

c

C

C

C

C

7

6

5

4

3

2

1

0

0

F

E

E

L.

c

C

B

B

9

8

7

6

5

4

3

2

2

1

0

0

E

D

D

C

C

2

3

4

5

6

7

8

9

A

L}

C

D

E

F

0

1

2

4

5

6

7

8

IA

B

c

D

E

0

1

2

3

4

5

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

tsH

a)oal

oÞ,

DecirnalNuinbers

51

52

53

s4

55

56

57

58

59

60

61

62

63

64

65

66

67

6B

Residue NunbersItfod- 15 Mod- 16

l4antissa inDecinal Forn

26

26

25

25

24

24

23

23

¿5

22

22

2t

27

2t

20

20

20

20

Exponent inDecinal Forn

-3

-5

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

Mantissa in ResidueForm

Mod- 15 Nlod- 16

Exponent inResidue Form

Mod-15 lvlod-16

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

C

c

C

c

c

C

C

C

C

c

C

c

c

c

C

C

C

C

A

A

9

9

8

B

7

7

7

6

6

5

5

5

4

4

4

4

B

B

A

A

9

9

8

8

8

7

7

6

6

6

5

5

5

5

3

4

5

6

7

8

9

A

B

C

D

E

F

0

1

2

3

4

6

7

8

9

A

B

c

D

E

0

1

2

J

4

rJ

6

7

8

tst\)

t)o(-lF.

oÊ,

DecinalNumbers

69

70

7l72

t5

74

t5

76

77

7B

79

80

81

82

83

84

B5

Residue Nt¡mbersMod-15 Mod-16

Mantissa inDecinal Forn

19

19

19

18

18

18

18

1B

T7

t7

t7

t7

16

76

16

16

16

Exponent inDecimal Forn

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

Nfantissa in ResidueForrn

Mod-15 N,cd-16

Exponent inResidue Fonn

l,lod-15 lvlod-16

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

C

c

c

c

c

c

C

c

c

c

c

c

C

C

C

c

c

ó

3

3

2

2

2

2

2

1

1

1

1

0

0

0

0

0

4

4

4

3

3

3

3

J

2

2

2

2

1

1

1

1

1

5

6

7

8

9

A

B

C

D

E

F

0

1

2

3

4

5

9

A

B

C

D

E

0

1

2

3

4

5

ó

7

8

9

A

ts(^

(-)oclts.

oÊ.

DecinalNumbers

B6

87

B8

B9

90

91

92

93

94

95

96

97

98

99

100

101

Residue NumbersMod-15 Mod-16

Mantissa inDecinal Forn

15

15

15

15

15

15

14

L4

t4

L4

74

T4

t4

L3

15

L3

Exponent inDecinal Form

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3z-.,

-3

-3

-3

Mantissa in P.esidueForn

Mod-16 Mod-16

Exponent inResidue Forn

l"lod-15 l"lod-16

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

C

C

c

c

C

C

c

c

C

c

c

c

C

c

C

c

F

F

F

F

F

F

E

E

E

E

E

E

E

D

D

D

0

0

0

0

0

0

E

E

E

E

E

E

E

D

D

D

6

7

B

9

A

B

C

D

E

F

0

1

2

3

4

5

B

C

D

E

0

1

2

3

4

5

6

7

8

9

A

B

Þè

cl

rlP.

oÈ

DecinalNu:nbers

t02

103

104

105

106

L07

108

109

110

111

Lt2

L73

rt411s

LL6

777

118

119

720

Residue NunbersMod- 15 Ilod- 16

Mantissa inDecinal Form

Exponent inDecinal Forn

-3

-3

-j

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

Mantissa in ResidueFonn

Mod-15 Mod-16

Exponent inResidue Form

Mod-15 lvlod-16

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

c

c

c

c

c

c

c

c

c

c

C

C

c

c

c

c

C

C

C

D

D

D

D

D

C

c

C

C

C

c

c

c

C

B

B

B

B

5

D

D

D

D

D

c

c

c

C

C

C

C

C

C

B

B

B

B

4

6

7

8

9

A

B

C

D

E

F

0

1

2

3

4

5

ó

7

8

C

D

E

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

0

T3

73

L3

73

T3

72

L2

L2

l2t2

l2L2

72

T2

11

11

11

11

-11

ts(/r

A16

APPENDIX - 4

AP4: Error Analysis of Floating PointAddition and Multiplication Operations

In case of floating point additj-on/subtraction, and

multiplication, the errors are introduced during rounding

process in normal-ì.sation. In floating point division

error is introduced from two sources (i) error from

approximation of floating point, vafue of ] and (ii) theIJ

error from floating point multiplj-cation of FI(A.K).

However, errors only due to floating addition and multip-

lication will be evaluated, assuming the exponents of the

operands are zero.

aP4 (a) : Error in floatinq point additon

The actual value of Z is given bY

ß + cl

The positive error occurs if

z cz (from (4.11) ) .

(cz ß + "r)>R or c

2a Rß

and

if ß is odd.

if ß is even.

Under these condi-ti-ons the normalised rounded Z (Fig 4-6)

becomes

Z. = (cz + 1)ß

", ''V4

A17

Therefore positive error ø+ -- Z' z

ß -c (1)

(1a)

(2)

(3)

I

Case (i) If ß is odd, then

v^ ß-12

z .þl*tor

(c,). ='m1n

(c,) . j¡ mÌn

Ê

Under error condition since cr > vr, and cl is an

j-nteger, therefore the minimum value of cl can be given by

( 1b)

¿+ in equation (1), Ís maximj-sed,

9-l*'For cr =

therefore

(ø+¡m ax

Case (ii) If ß is even, then

V, = B/z

= ,.|9^orß

Under error condition since cl >

therefore the minimum value of ct,

(", )*i' v^For c. = (c.) . ,r ¡ ml-lr

therefore

vl and ct is an integer,

in this case, is gj-ven by

(4)

e+ in equation (1), is maxlmised,'

AlB

Therefore in both cases, positive maximum error is same.

The relative positive error U(ø+¡ is given by

?+crB+c,

(5)

)*r* and cz ß + cr

is given by

(6)

if ß is odd.

U (¿+)

and

lu- |

If ß is odd,

negative error, U(e

The U(ø+¡ is maximj-sed when Ø* = (¿+

Therefore the maximum value of U (¿+)

R

fs ^l( ø+¡U ( u*)*"* max

R

The negatj-ve error occurs if

(cz 3 + cr) > R or c2

R

Rß

c v^

cr < if ß is even.

Under those conditi-ons the normalised Z becomes

., =czß

Therefore the absolute value of negative error is given by

or

lz' -zl

V,'

is given by

v^

c

lu-l-)max

U ( ¿- )*r*R

(7)

A19

If ß iq even, lu-l v^

{ (cs

and maximum I u- | is siven by

(since lu-l is an integer.)

v^-'

v^

The maxi-mum relative negative error, when ß is even,

is given by

U { ø-)*.*

From (6), (7) and (B),

¡lax(¿) is given by (6)

Max(¿) = U(ø+) .*

ZR=c2ß+cr

R(B)

it is clear that the maximum error,

or (7) , therefore

(e)R

AP4 (b) : Error in .f_-I-e-ati_ng. point multip lication

Actua1 value of Z is given by

I - cg 92 + cz ß + cr (from 4.I4) .

Case (i) If c3 = 0, Z is reduced to Z whereR

(10)

In this case, maximum errgr is Same aS the maximum errgr

in floating point addition.

Case (ii) If ca and cr are nonzero, then positive error

will occur if

ß + "r) or ce

vl if ß is odd.

ß]>R

c2

or

and

c wl if ß is even.

A20

In this case the normalised rounded Z (Ffg 4.7) becomes

z (cs + 1)ß2

Therefore positive error 8-+ -- Z' Z

= g'-(cr.ß+cr) ( 11)

or e+ < g2 ""'9

Case (iii) If ca j-s nonzero and c, i" zeto, then positive

error will occur if

ca ' ß + cz > R or ca ' ß > R

c2>and

If ß is even t c2

if ß is od.d.

if ß is even.

(L2)

in this case

2c

v^ú,1

Under these conditions, the expresslon for positive error

can be derived from (11) as

ø'+ = 92 "r. ß ..... (13)

From (13) and (tZ), it is seen that ø' > ø- Therefore to

evaluate maximum positive error, the expression (13) will

be used.

Q.+ j-s maxj-mised if cz = ("rLir, .

If ß is odd cz , Vl (under the error conditions

Therefore the minimum value of c2 can begiven above).

written as

(", Li,, vl., (14)

V^, theref ore ( c2

is (under error condition)

(", ),ni' v4

m 1n

(rs¡

A2I

Therefore if ß is odd, the maximum positive error is

given by

(¿'+Lr* = ß2 (cz)*in' ß from (13)

s2 l\rl*rJ o rr.-om (ra¡

= lr.ú,^*'] ' IFzrl.'j l, Wù.')f ,:om ( 1a)

Therefore when ß is odd, (¿'*) "*

( ro¡

When ß is even, the maximum positive error is given by

{ (ce

and

Wr' [, 'V^)

ß from (13)

from (fS¡ß

from (3)

(12¡

The negative error occurs if

2c

c

ß + cz) or ca . ß ) > R

v^ if 3 is odd.

e,, if ß is even.2

(ror c and c are nonzero)3 t

Under these conditions, the normalised rounded Z becomes

c3

z g2

The absolute vafue of negative error I u- | is given

A22

(ro¡

by

lu-l = 17' zl

)max

3 + c, .....(18)

v^

c2

The maximum value of õ r , (", )*"* = ( 3-1)

If ß is odd, under the above condition,

cz <

Therefore the maxi-mum value of c can be !.'ritten as2

Therefore the maximum absolute value of negative error,

when ß is odd, is given by

c2

c2

vr (19a)

ß + (c,)¡ max

ß + (ß-1) from (19a) and (19)

. Frr].'] . [, .V^+1-1]

(ø-)max

(c" )- max

v^w^.v + 3.

I'

l'If ß is even , under the above condition

Therefore the maximum value of c can be wri-tten as2

w^

I (20)

(2L)c w^ -12 max

Theref ore the maximum abso.'l-ute value of negative error,

when ß is even, is given by

lu-L.* = l""l*.* . ß + l.rl*r*

vl-,wt-,

(ß-1) from (19) and (2L)] ß+

]. 2- W^ + 2'Wl-,from (3)

A:23

= ,[E ù)' 1 (22)

Comparing (16) and (2O') , the maximum error, when ß is odd

is given by (20), therefore

(ø (o) )

([ (øo) )*"* =

(U (¿¿) )o'"* =

v^ þ^max

(ø(ø) )*r* = 2

=2 +3

ì

2

2

2

(23)

Comparing (ZZ¡ and (tl¡, the maximum error, when ß is even,

is given by (]-7) , therefore

w^ (24)

by

V'lhen c is nonzero, the minimum value of Z is3

(z') =ft (zs¡m¡-n

The maximum relative error, when ß is odd, is given

(e(o))max

(z)m1n

2 2 +3. 2(26)

by

R

from (23) and (ZS¡

The maximum relative error, when ß is even-, is given

(ø(ø) )maxt) man

vr)(

(

t2

2

Rfrom (24) ; ,r::'.'

A24

APPENDIX - 5

AP5 Mixed Radix Conversion

For a set of moduli *i (i=L,2,3,

X can be expressed j-n mixed radix form

n) a number

Izs ] as

{=a n+a3m2mr+a2mr*4r..... (1)

where a. (i=L,2,3, .... Ì n) are the mixed radix digits. The1

digits can be found by

lxl*,

+n- I.'fi1= I

a

a

X-e.,n

Rr '

2

&3 m m2 3

In this h/ay, by successi-ve subtraction and divislon,

all the digÍts can be obtained. Conversion of a number X

(residue form) to mixed radi-x fcrm (shown in (1) ) is called

mixed radix conversion (MRC) -

Example:

For m, = 16, m2 = 15, mr = 13, find mixed radix

digits of X >

A25

Solution:

Residue of X

Subtract a"r

Moduli: 16 15 13

7 I 4r B

8 B

15 L2

denotes multiplicatj-ve inverse of 13

and 16

5

* Multiply by 5 7

11 9 &2=9

Subtract a2

* Multiply by t+1,, l-5L4 0.. g

From example, it can be shown that total paralleI

residue operations required are

(i) (n-1) subtractions and (if) (n-1) multiplication.

If these operations are carried out with PROMs (by look up

table) then subtractions and multiplication can be over-

lapped (since the multipliers are constant numbers). In

this case total operations are (n-1). In other words, delay

is (n-1) PRoM-delay (access time).

The decimal value (DV) of X = 14x15x13 + 9x13+8

= 2855

lrl| 13 lmt

92

* !1

151

L3 m. '1 16

and l-5 with respective to mí (rs and. 16)

respectirr"rv.J

A26

APPENDIX 6

PROGRAM LISTÏNGS

A-i

Appcndix-6 consists of the program listings for the

RAC and is divided into three parts AP6 (a) , AP6 (b) and

AP6 (c) . The AP6 (a) contains the program listings for

integer arj-thmetic operations together with initialisation

of the ports of the microcomputer, decoding of the operation

codes etc.; the AP6 (b) contains the program listings for

floating point arithmetic operations and the AP6 (c) contains

the program listings for the conversion of residue numbers

to deci.mal numbers.

The programs are the subroutines nested together. The

labe1s of the subroutines and routines, in alphabetical order,

are given belor^¡.

Labels Subroutines/Routines Paqe

AD

ADD

AP

AI2A30

BLN

S

R

R

R

R

s

S

R

S

R

S

S

S

S

R

R

A30

A33

A32

A33

A33

A29

A38

A54

A2B

A39

A62

A3l_

*

A30

A33

A33

CAD

CAP

CAR

CAT

CBCD

CCHR

CNVBN

CM

COMS

coMu

Labels

coR

COSI

COUS

COVER

CV

DAD

DIDIVDSV

Subrout

ECHO

EEQ

EPT

EQ

EXP

EXPO

EXT

FAD

FA

FBCD

FD

FDI

FEND

FFC

FINTS

FTXE

FM

FMT

FMU

FS

FSU

GAT

GET

GETCH

GGT

S ut Paqe

A31

A32

A32

A54

A30

A-ii

A38

A30

A34

A32

*

A32

A2B

A28

A54

A43

A57

A45

A32

A62

A32

A57

A41

A57

A40

A40

A32

A59

A49

A32

A48

A39

A29*

A31

Labels

GT

IBCD

TNI

JPM

KAT

KATCH

KATH

KATR

KKK

LLTLOAD

LT

MAD

MM

MNT

MU

MUL

NCHRT

NEG

NEGA

Nl"lOUT

NORL

NORM

oPc

OSIGN

OUÎPU

OVER

OVFL

ut

PASS

PEG

PEND

S utines

A-iii

Paqe

A2B

A6l_

A2B

A30

A36

A46

A36

A36

A58

A31

A29

A2B

A47

A32

À28

A30

A33

A65

A28

A41*

A52

A42

A2B

A31

A45

A44

A59

A36

A50

A40

Labels

PElPOS

POVER

PSÏ

QZERO

RRET

RSÏ

RZERO

Subroutines /Routines

R

s

R

R

A-iv

Paqe

A34

A28

A53

A31

A56

A60

A63

A56

A40

A32

A64

A34

A57

A39

A28

A37

A39

A65

A30

A33

A64

A34

A35

A35

A47

A31

A30

A38

A32

A64

A28

A56

SAT

SB

SCHRT

SDIV

SDK

SELE

SET

sscSSEP

SSSP

SU

SUB

SUM

TAB

TAC

TAD

TAD

TAG

TAKE

TAP

TEST

TOR

TTT

TZERO

R

R

R

R

R

R

s

S

R

s

R

R

S

S

S

R

S

R

R

R

s

R

S

s

R

R

R

S

SXTAR A29

Labels

ZRET

ZERO

Is

Subroutine s,/Routine s

A-v

Paqe

A60

A36

R

R

Subroutine : R Routine l

* Resides in the Monitor Prom of the SDK microcomputer.

A21

AP-6(a) : Li-stings .of fnteger Arithmetic operations

i Í1Íiili.tfi'ql'lH lrilFt l:t'{'f Ë:ü[:fi rqül] T'f T illì r flLJ$TfirìC:T'Ï üNi M t.J t,. T T F' L. T C Ä l' T il tJ Ër i'l tr f' ( l l'l l::',.ì [( ]: 5 n Ì{,t fTË'$l:Iit.J[:. rìlt].T'l"li'1[::'f ÏC ( t'filI]t.Jl...l' i 1ó'âl'll1 lli )

ti t{;j{}+iiir;{¿*rt¡\ii,;)fii|{)fi,$}hiK**)|i}i{.$}i{)+i}ir)ft*I(}i(tcil{i{ri{r.',{;$f{hk*{r'H}i'$¡c)ll}ii;{il{}f,$Ï(4¡¡a;'it¡t/,t*

í [1ftl1lìft'tt'lF' fil'Alil'!ì l':'ftt]l'l l.-00. 'll'OE]ü0 üFi 'l l'l[: $Irli-'Í:jt)Í t4 ï ill:t nc01.1l::'lJ T' L. lt

"i *)l.i,i{y,{*iii*:fr}iiIr¡t$ii*i)$t*>h>k.r\il>lq>hri<ff)i{rttr{,\c}kÑ*,$il{\cÏ(s,**}k¡qr*;ü}i(;ì!*r'|()i{¡?hk\cr{1:'Idv

t *}|ilitiliirtii¡{;i(r(,ir*¡k,{d¡\,,¡\rft,*I(¡k}l(fr;l\rü,**)i(¡t}t{td*)Elkl.:h\;'krití;'k:ftrür'f;'Fll¡k;J(ht¡\Ï(,\\:*¡t:;liÏr:;'1fill:(f.ì 'll'f;ì00

I' Fftili'l $ËTs;

üNVI'.tl.l I $HT ,[,ü J. [¡AE:Cl-lU i tìË:'f '['0 J. Ë'4

G[:T'f:l-l f lSfiI' ,[,03:1. [tl.lt'l0l.l'ì'l $Ë'f 'ftü:l[:3

rrl:ul sili:T {,0(trJÙFrllll $Ë'f '[.0ülillFfit.l ! sËT ,ll0üEltFÌ'f t.J I $E:T ll,0[¡10F'I1T I Sìtr1' 't'040ü

L rJ{:[t î $ËT' ,1,ÛtiIrÛ

i ht*{)Í{:)rt)H*rl:Ii,fi;i(/ù*)H}i(Í{t{$}i(h\¡th\ilihq*;.kÏi*ìk'$*(}c;tI(hk)&;i(;i{/,()ii}hI{ifis.v}k)lc)i(¡tri{}i(if{)iisI

i F:'{lt.ji:t t}f" ÏþlH $UfitiUU'l INI:ijj tf-¡[.'l'.'fjfr Êrlìii:: ST0[t[:Ilí l:t{ 1't-¡H I'f ill'll:T't}ri ' F'ft0Ì"f ' Ll¡:i Tl"lH fi[rlt'-$0 .

i lÊ:t'll.JIt $l.JIifi{ltJI'l:Ì'JE.$ ËrltË: 1 ) 'ÇNVFl.l ' -üÜl{UlIFt'IÍìirllîÇTI f{[:[¡[(Ë:fiìli:t'iI'Ë¡'I].0Ì,1 [Jl:: r'l i"iLJl'1FË:i:t .iN'fü :t]'Í]i IlT i{¡*¡RY Uôrl...tJ[:: ( I'lËX I'lt.Ji'JIït:ft ) 'i:¿)'Hüi"ltl' -ï'tlitî{:ì Sil:Nt'il...L' ill"lrâliñtlTliiR ÉrS l:l'lË't.Jl'

i'qNIlv Vl.l¡ Ì"iUlli:'IUlt Sii:i{lliii'll"lrâ'1" tlþl'}[tr*rn'fHH T0i T'l-ll;:: l.J$L:H 'If:l:t 11 l:l'lËi1.. "í 3 ) ' (il-'f CH ' *i:i[::'T't.Jl'(i''ll1 ]'l"ll::: üþLâ[lûüTlIli T l'l 'll-ll:

i l: i'l F't,J l' Íì'1R H. rl I'l I'C] 1'þlfi L: A t..1.. :i N tì [:' lli ('J [ì l:i ttì 14 Ë: +

i "T )' l,li'íO t.J'I' -' ü0 i'l UH Ft'f fl [l'-Ï:l :t ]' U äÍì :l" Gi'll:::Il Ï t'l l'l: tìHIiiIl.l rì*.Íil-ffiI$]'Hli INT'ü I'tdfi fåEìüT.t []l'.iÊl:t¡âu1'l:rltiiì rtl',11.¡

û [iË:N[r{} ]'0 T'l-lË: çuÌ'.1Íì01-F- ,t'Ï Ïl-lti clJlIIlliNl' f¡l:t l'Ì'Jl'i F'tllj i:T r ni.l 0F' T'l.lË üüÌ'J|:ì01.-l;::

"

A2B

ä 1'l-lI lì l:tüi.J'f l: t'li:r :l N T't l:'tl.- T Í:ìl:i$ l'l'lH f¡0ftï'ì nF l:1F'[

ifi-'Fr0i:l'f íìÌ',lI:r (::-Fill:iT' (tJl'tFl::l:i) 11ìlî[:: f:ìll::1' ]fl üt'l"l11t'Jl'

i l"ltlllË:" B'-l:rültT ¡îlltr ü-'Ë'tlRl' ( l.-üt'J[:l:t ) f'rl:ilii EìË:]' Tlli Tl,lFLll' Ì'iflllt:: '

:1. tl I ; 1...)( 1. Í:i[:' v 'ft J' 3FËl'1V I É¡ v '['$ '3

i l:ì'f ,ttili [tt]l:l*lÏË:R 'i'.ll tjiï' Tliì 1'þlH ü(lN'flitll.-û u t.l I {l l..l $ H: f ii l' l-l H l:' cl l:t I' si

i fìV0Ffi ìf i.ll:rli$.i f:ìtil'ì[t ]:T Ttl lr'[t Ï ç

trrq T 'tïtl

üU'T-J 14 [t

i ¡t * iH il{ h\;.ii }iilfir+{ )+l ilt li :i'Jli;li

,l'17T'Ë:$T

*:+:)X ¡s;t * hc;{ ti il( ii{ ii( * l( il'.li }t ri }i4;t ri( S ¡\ ¡\ iii ¡c il( \c r'[ $ $ ].k;''l \c¡c ]ici1

v

'].þlË$[: ¡âl.iH T.|^lla $LJlt[tütJ.T.:i'NIiÏì t,Jþl]:Ë1.{ nUÏFtjT. .Il'.lli::

IrilL.l'.Ütd:l.i'lË

íül-lrlltr:rü'T'[::l:ilil'-'L'f{trtL-[:''v'O'v'Yt'v'l:i"'v'li[:'r*¡üli'i'v'-'v''l-'v'yt.,,'yt'.lr'y'Î' ov

Iv

t.ll: Ii[:l;:'ütiË Ëì lltJtll:rËfi I t¡trT ÜÉr'tlii{ì l-lËX I'lLJì'rltliiFi '

srïT I

0F'ü !

ïTl'l

NËü i

Cr*'L.LHHl'}.1V Ï

ECI{t}

ü r {'4F

CAftl i"lVI Cv'll'0I.¡

Cr4l.-t.. ËCl-luMVI C v il0Ér

i/:[;0ft' Igi Tl"lH ¡qi:ìüI'[ rf'ì[lt: r:0r{i üAËlF( I ÉrGË. ft[::T'Ll[tl'] .

i fi I tìi'lôL .î ' {'0rì ' I Íl l'þlË: ñlìti T I [:Üil8 [:0f(il...l:NE Fl::[iI1.i 5I ül'{'âL .

i'{,,1F' l.li'Il"lE rll;iCl:1 (]OtlE: Ffilti TþlH 1,..h:T'1'Hlt ' (l' ,í Il l:fìFL-'qY .

í,,li,ljlt' I$ I'H[it 'fH[:: $ I Cil'J ¡ '''' +

'1¡$(:TI üütrE: F'llti

i',1!!l[t' ]:ti I'l'{Ë 'à$CIT ü0tll:i ¡::Üi:i

i Tl-l[: ÊiJ:Cìt'l ''ìL' o

i,.x,4[¡, ]:S Tl-lE riiitlt ¡¡'¡ttiii F0lti T'l"lir t- E 'I I'Ë: R ' i'1 ' .

i r :þs|!i ' T ii 1'l'lE rìfÏÇ:t I üilf¡[i FÜl:lî T'l-lH l.-[iT'f f::Íi 'lii' o

¡ z :[;$t.l ' I Eì T'HH râliü Ï T {]ütl[: F:Üft

û l't'lH f:il:tiN r:::r.

C:Éìt-1.- Hül-lt]

"il.il-' H:R

l"iU I ¡" :!:flfl

JI'IFF0$ i l'1u ï

-J Ì'f r¡r,f NT ; l.íu l:

$h:Tt y,ll,lF

EìF:T

Ë v,li,4ll

-Ji.lFHF'I I ¡'ÍVÏ

Jt"íFL_Cr r l.1U ï

T'Tt v,il 4lT

't''r l'f v ,11,.5il

J¡'ÍF.üT Î i'IUÏ

Í:ì E: 'ì-

ü v,t'.311 î'.113Ë' ïÍ;i l't'.t Ë: lì r Li l'ì

û',U,äC' Illî ïl"llï $ì T L'ii'l

-,Mrtt_T 3 Ì.ru ï

$L:l'[)v,ll,3ü

T'l iH ¡â$C T I CCltrE l:r(J[t.r+

'f þiH ô'flü l: .l C{lt¡E f:f}ft"-a

A29

JMTI $ETtJL-NI l''lt",ï üv{'f0 i',ll:lü' I$ T'l'lH: 'qfiiCII CilnË F0R

í T l'{E '$[:'¡l{]Ë' .-ll,l13 sE'r

X'f AR * l'f V.l. C ¡ 'ü,.ìF' í',1[:3F' ]:Íi Tþ¡E: râSfll.T tUIrË ]îül:ti'rHH ÍiïffiN '"!",

í )'kil()Í{ñIi*tcï{rd;t¡cï-*ll;of f lol,*l'*n,$¡[t¡:ltfr*>|<acißil(]i(i,chtï(]ftif rft¡qii(x{r,í).¡i)i(ï{)¡r*ï(}fiI{ii(rd

1

i \c¡T)'l:iliI{ili*}f,hqí()h}i{}h}kt{Ii*{fr/,(l\)$:¡çil{$}r;'F.Y:r}F}lf .tñ}ir)i()ft}.k;ì\Ï{Íi*}liht;ii;'t}$:)l()i(*)fIi}i{}irii{i{ûT'þlIfi !il.Jlirlltll.JT'l:NH GË'I$ Tl,,J0 ,t$üïï ül"irii:t¡tt'ï[:l'tiì Fl:ttìl'1íCfll'l$tll.-Ë t¡Ë:Ul:CE: Al'JIr fi0i.lU[:l:iT$ 'Il"lEl'l ]:l,lïCl ûNH IrYTËi ( $-.ß T T'ti þlË.x ) [¡ÉrI'rl .

fiHT' t CAl.-l- GIIT'Cl-l i {.ì[iT' ûl"lË Â$ü I T CH'tlt'ìC'l'[iR .cfìl-.1.- Hcþlü iÏ:rÏ$l:¡1."râY"CALL CNUttN t fiüNV[:ltï' -l T' ï l'{ï0 ]:I'lì .

i tJ.L l'IARY Vril.-tJH,RLCf'{LtRLI:t(L-i: iAlÏütiËI'l'l[:l:t l:OtJl:t LEr[:'T $l{ïF'fü'STrâ '[,J. 360 $ S'f 0ft[: ËtT' l...üC , ,ll,][.3t]ö ,Ctrì1.-l- G[iTCl'1 í tu*l::T' Êl.lt]l'l"ltrfI r*r$Ç ï l. t:HARÉ'Cl't{Í(CÉìl-l- ËCþ10 inï$frl.-rqY,CÊl...1.- CNUE{N iC0tlVEliT l:1' INTt} I'r$

i fi l:NÉrltY U'r¡1.-t.lË .¡'ftlV C¡A il40V[: ïT T'[] ü'-fth:Gl:ãl'ER.L"IlÉr ,[,].3åO iLCl¡âI¡ ¡l- [i[iGïiiT¡:ilt Fl:tLìÞl

;l-CIC.'l'13rå0Alltr C iËrll[r tl].Tl'J C-ftlIUI$'fHR.F{IiT

í )Í{hcs*;*;ri{ilr¡H;ftrliil{r+{**rt*.*il(*}k*I(a+{ì\Í(}iill:il{il{tt,r{iy,r}|(r.l(¡tif )F)Kil(,v}|(ì\:)rt)i{¡rr,t)|t}rtÏi/,c,*

I

i **l*¡fi***}[.*rfi]f{)+{,ü¡cili,iili<*:lilir{<¡lcti/,ii,çrfi,$;i,{hc¡thcï{;,r\¡cI{rf,ht}ir*,t)i{*;ili}iit{ilr:fl¡t}i{Ï{il(*t'fHl:Sì lilJIrltlJLJl'Ti.lli l-CIÉ¡I¡iì üI¡Ë[tËrl'J[rS ñN[r üF'ËRËrTTUl{inütrli T'ü filrf:ìïtrLJH 11¡ltITl.'ll.íHI':1.ü lJl'lIT([tÉrLJ)'ql.]ilt üET$ 'I¡-ll:i ftE:$LlL..T' t¡UT ü1" T T'"v

L..0,lnÍ þlt'rL Ay,[,s0

0LJ'f ,t.1tå

Lü'â ,lt13É J.

cllJT' {,14i''lVl: tâ r {.50

0UT,[,J.f.¡LIrÄ ,[,]l..5C¡3

i ' .ll {i0 ' l:lrì 'l"l-lH Êl1nftfi:$f:ií {lF' T}lt: 1...râT't:l'J r.1fì!ìT[ìi'lHI]Ët50Ët FïH$1', ilFtiltÄ¡.tü {x)

"i $Hl-ltr Tn fi*[]0l:t'f ( uFfrFfi ) .i [,f]AIr [:Tfliì1' ilftEl:(rqNI] [j'l:iûl'fi ll-0t:: o {t,l.3d 1. ,t $Hlllü L T' l'ü

'â-'F 0f(T .

i',fi.lio' l:5 l'l.lH ATlllFil::Íi$t 0F 1'l{H l.-rq'f Cl'l r:rÍlS L Ltl.lËili F0ft sËc0Nn üF frRÉlNïl ( \') .í flËiN[¡ TCI U'-FUF:T ( LIF ÊËlt ) .i [- 0Ërtt $[;:fi ü]i'.lt¡ 0 [tliRrql{IJ fTl:t 014

i l*t}C.,il,J.3/r.3 'i $[::]itr l:T 'f 0 A'-tiüitT .üLJ'f ,[,J.4

A30

MUI: ô¡v{'i50 i''ll':T0' fS Tt'l[:' rîItI][iESfr'û 0f: T'þllii l-'âTnl'{ ËrSfi.t ül'll:'Ilr^ f:fl1:ì tìF HItril'l:i.ìl'l [)t]lll: -

ÛLJT {'.[/r i$[:i{il T0 C'-F'i]ft'f (LJF'l']l::fi)'

t-trfl ,[,:t:Iófi i L.C]ôll üFllil:tAI'1. üN C{lllla l:rlTilì'Ii L'0C o 'ft J.3óIi .

0U'f 't, 14 í $li:l'lÏr .T T l'ü) r'¡-lsUltT 'I N 'R'1.:i $ Ë[i'T' I'l-l[: R[::$ìt,I¡'-T' F[iÜi'l [t*FC]R'1" '$TA '['J.:T70 i$T0[i IT Ë]l' l"-0(:.'t'L.T70o¡'lnu F v ¡t i ¡-ltluH I T' 'f ü I'r- l:il:i{l . .l:N {,1.ú itjl:iT ËìIGtl Al.ltr C}U[::filî[-(]kl

v .tl'lF[l[it'l¡â'ITt]i,l f:ftLll'f il-f¡filtl' ( t-fit'r|f::[t ) '$TA '['1lT7J. iST'üËË IT rq'l" 1..0Ü.'['137J."RË

i fr¡cï1¡k;g;¡c*)fi)i(hc)+írkts*¡\il(*)i.:¡\:)f:¡\$$)+{iE,\tI(rír'F:;*r$;Srl\:lrChtlit}F*11.{(il(ì(il{y'{**}s)i()i(}i(*il()i{}$:r

t Tittàr{rkil(}i4ili:t.üH¡ir¡t;rrihc\r:ìc),k¡ttfihkil1¡t.\c$;'|{¡k¡ch'(}|l:,vht;$ikh\)i{:fi}iiriiÏ{;i{$:rfi}i(rf.lLii:$¡g;¡ç.1k{t>lokiT'l-ll::$E 'lRli: T'l-lH: tìtJ[itií]tJ1'Ii'JlI$ Frilli rtIlIr:t1'l:ili,,lrlitJll'Ï'Riì[)'fÏ0i''{; l'lLll...T'l:Fl-..t fiÉrT'T tll'J v üÜl'íl::'ràR 1:$0¡ll, Itl-.l[r f'0R GË.TI I i"lGiË Tli$T AFF[(nX].i'irìTTflÌ'l F'{lR :thlT[itìHfi ilTUTi:ìfül'l'

l'11.J I I'f UT A r'['Oti

JF i'j I $T'Él 't'1rTóliüAt-t- t..üAIlRET

A[l I MV ]: rl y ,ll r]ó

JMI.. JF'i,I$1.,î l"luI É¡v{lû5

û,,ftü[r' I$'il.|]: üü[tE FCIlT

i i'1t.JLT I ItL T fltl'I Ï {lÌ'l'iSi'fütiE:'âT'll'J.3C¡5*

i ' .11 0ó ' T $ T'lJl;i L:Clrr[: r:0¡:tâAIrttT'fIül'1.

i '{,OTi' I Tl'{t: Cütl[: fi0ftí SìU[I'TFiAC]T.I OI.J '

c¡'j;JF'i'lËr v {,4ü

.JMF,

I.fVÏ í'{,40' I$ Tl'lH cûI1H f:o[tt Ëüt'tF Êtt I $ül{ ( t.Ji,¡5:I til'{ËÏli I'lUl'lI{E:Í( ) .

Jr"iF JF l.lCU! ¡fUT A¡{'S0 í ',ft{30' I$ T'tji: ¡''ç¡ttH F'{llt

i C0t'll:''îiltI$01'l ( $IGN[:t¡i NLJI.IHË:R ) ,

nïr ;iiji li:iltr. t',rr,0:r' ïsi î'r{r, f'ürrri Ftrrîr

i GE:T"I Ï i'¡Ci A¡:rf:'ËilX T l'lÊ''f ü.

i GLJil'II [iN1' il:]:tilìÏ; Af-'F'ËnXIt'lrt'i.tili'l ) .

-Jl'fF ,,1-'t'1

î $ft,|(¡\rk¡I:fi,s¡\;¡(;lci[;{<.$Ii:t{*,i{).1.:il()fi)[${ìt)ft*}kyí;+:}Eikil<)ftt{>ii¡l,i*ii(:l:¡l'rfi.¡$./,c}iiIciltdc}i'ri(ht{*

îí )li#t!:¡t{){ih'(}í.:¡thtI,#}f,.Sl(,il{il{)t{}Hfrri{}H*$$*1,(ñ;(ïr}F.Èh?}i.:r*{il1)i(tifi}ftrfi*}Í(iii.tililttiI(ii{)fifrt T14l:$ lilJI:rIt0tJl'.l.NlI ÍìT|ti'IALS l'ü Tl-lE CÜl{Í$nt..E IrüiVTt:[iî 'f ü fi E:

'f 0 Ë'lii. nt ¡l i',1 I.¡ fì .v

T'ÅliH t üËìL-1.- t'1NT'

ülìLL {lHl', $ I tìl.,i,fl.-, / ivl;:;:,- {

; ri l:: T' 0 N Ë I{'í T'fi: IJ rq'r rq *

A31_

$ì'fô'[,13Cr1. iST{]I:(H: ôT .lll.î5C¡1.C'qL.L nlN iGTV[: $FAü[:.fiÉ¡1...t- l'fi.lT í$Ifìh{'âl- '14=" o

CAt..L. CìE:T î GliT' 'ôhlü'll.lË[t ËY'f ï: 0[:' [rrtTô.

$iïÉ' {,1.:16.T i $'f ilftH AT ,ll 1;Itl.5 .RIiT'

i )$l;si{)ii;ü}E*)l(i}i}i()$(iíiT{)'qil(}l(,{(I(S{:io$í*l{tk;{ilf¡k}kh\¡ftcï1ri{Í(,vï{s}kfr{{*\trìc;.&l(ï(ï(ï{i1)fìqhk¡r.t}i{v

û ìf,ili¡\ii(iiir,(*il(fi*)i{Ìi.*ht}8.*¡k*:lrìthtï{,t;r!,{:i<¡t¡a¡¡q4ilili,\\..ti¡trfi$il(;t¡\*}i(g:il¡c*t{t(,1{il<:.*ñil{tT'þl:LÍi iiLJIjfttll.Jl'l:Nh $iItìNr:ìt..Í:ì 0Uf:[{Fl...0['l ('!') rl::'0$Iï"tUË (.1) y

Êl'lH:LìAT:tU[: (-') r/.1Ì.ltt $[rl'lIr$ T'l-lH l:tEÍiLJt-T T0 1'll[:: üüt-l$ilLEÊ nËu:i:cLt .I

0$ItìNi L-trÉr,ll'J.371 il.-0É¡tr lìl:c'jN rlNïr üV[ittl:iLOUi Il'{F'0[iÌ.1'q'ïIfiÌ.1 f:Rü1.1 ,il,].:T7:l .

I'IIJU I y A i SÉrVË IT' It,l [r-'Illiü. "

'âNl: {,ûíl ;'f[:liìï Ïfr RHSUL'Ïi üutrttFt...fJ[,J{i

"JU ü01( iIf: l.luTy ,JUI{F 'Iil 'C0tt'.CALL. X'f'êR í0'fHl:Ftt^JIii[: -ci:LGl,lÊrl- r¿it/ r

C0ttÎ l'TüU ñ¡f¡ ttit{T' li6rtli Il'llîtftl'f'q'fI0Ni f:tt0M Il*ft[:tì. .

ANï {'0J. ËTHSi'f TF ftEi$tJLT T$i I'lËCìåT J:V[:: . o

JZ. F $I i lti l-l0T ¡ JtJtll:¡ 'f ü 't:,$ï , .C¡+it-L NEG r^0TþlËRLJTSE SIGl.l¡tl.. '-' r

Fsï ; ;ill-r o..Fåå= i srrìr.¡,âr.- ,.t., ,

t_[rft d,]..I70 ; t._ortIt T].tH R î5ut.,1' FltclM ,þ 1370 .Ci*¡l-L Ni.f 0LJT i CtJNUHI:iT' Tl-lFi ttliiiUt..T'

t INTCI rqC$:t l: f:01:(1.1.RH]'

í T<*ttl¡lrÍt)f<*il(Ì(i$trc¡1í$:;)t¡tìil<ti>Eg¡q*;stq*:,í(il(*,v**;s)¡iirs¡qx;fi*$)kli(¡c*.àt¡qli:y,(;{í,**I{.ìt}.1r,{<ilçti *:)liï{/dfi#fi)ftr{.,Í;ôl:;t',Ë;'i(*iliiÍ{rft;'l;).iot(Tisi(¡ctk/,{*I{rfhqï{.,ir}i{¡kH(ikï(}K}ki(:.foi{;.lrfs::f*ioi{t{ïih\,tirþlr$ iiuFJlifiLJl'l:N[: EìTt]i',tÊ'L.S ü0MFl\ttriiüi,t fl{ÊrftÉrfi-t'[:.[tstT'0 I'l.l[: C0i'lïìt]t-H IrEVIüË,,

cüt'¡tt I t,.I'rí\ '[,J.370 t[-ûÊtI] ïþlH Ítlif:iUL-'l- 0Fi flill'lFÊ'ñ:I Í:ì f 0l{ .i S¡1VE l:N Il-l:il::G:t f:ìTl:::ñ .i T[:$T tJl'llT:'f l']El:t l.-lIfîiïËft.i :[ f: YE$i y ..JtJl'f [] 'ì"û '' l-1.-'f ' .Ê tìHT' 'Il"lH L l'lF 0[il',fr*¡'l'T 0i']Ë Fft0l.t IJ..rì8ffi o

"t TESì'f ülH[:]'l'l[:ft Cifi[::rt¡T'li:It .il:F YË.fiy .Jl.Jl'fIl 'fü 'Ëfr''ì"' o

t0'fl{l::ltt{IÍìË JLi¡'iF¡'Iü'ElLil' .í S:tffi1,¡'.ì1._ t .'ti., oL_LT't

[iGI'i

t4üvËìtl I"lþ¡r¡f üu

Ët l,,l l: ,t'01?

",1'lu tü'r-,i'1F ËËQürqt-L LT'ÍtE'rn'*rl...L- GT'

Rl:iT'

fi¡f\u.tL,LTt.t y .Ll

isTGl,lfll.- ,.i,'t

A32

t;:Htlt ü'å1.-1. Hll tSìïtìl'l'tl.- ",'.ËË'r$,T{Ï(,r\}i(}k;$i)i(*)li)lr#*i)i{$ttlçK¡lrilr,ft:Ícs¡crfi}.¡ilfi;üí¡t}ir}¡{}h*ï{)kil(:'kil(¡crfi)i(}fi}i{rfii¡(}ioi{;fi}i(f;}ftIy ¿\lf.4\.¡rr.,'11,ì\ ð. rf..Í.41 /tl d!.f..ï1 4\ + tô ¡F,¿tl.Þ 4\ /ii ¿ì\ /r\,i\ 4\+ + ,,l\ ô,I'.1'41.t.4\,} ¿$/¡\ 4\ 41.+./ß,Þ++ ¡r1ò¿r1rT'J'ô.

iTl-lTti F'ltüf.iR'1rl'iË: ô!ìlih- Ë'(llï ill:ifiËûl'T{li'l tlt)fi[:$ '*'l.lI1

'fH5]'t Tl'ltr 'fYFlI 0F:' UFËRËr'I'l't]l'J liE:[¡tJÏ[tË.I¡'i {:ül.l$üLl;i nHU:1.[[: :tËl;itJlïfiì Tl"li:i Fill.-l-.ü|,il]:i\iüi ËtlIll:fi i -' ôr ( ÉrIrll.t l'.1.üi''l ) v Il ( 5ìLl[r'ï'[trâ0'lï0l.{ )

iü (ì'1U1.-T'ïf:'l...1:{:r'':r1'l;tlN) vt¡ (trTVTfï[il,l] vli: ({1ül'f[]i1rl:tÏEiÏClÌ'lt üF ul{$ I {.ìt,l[::[r l.,ltJl,iIr[iii ) v lî ( fi01.11]rì[t ]:Í:ìilN nF fï:t ü;Nl;:Ilil.lt.Ji.ll;{[:r()rtì (F'1...üË¡T']:Ì"1$ Ët]ïÌ{'1" râI1ïll:'il.c)N)vþl (Ë1...ÛÍ\'Il:l¡ti,^Fnïl'll'$t.J$T'fiÉ\tll'.tüN)vl. (Ë'1.-üÉ¡T'l:i'J{ì Fr(lIl.ll'l'il.ll.-'i'IFl.-ï"iCAI'Tllt'l ) r J ( l'r'[-ü'lTIi'{tì Füïl.lï' n:iVï$:tOl'J } "r

TË:$1'I ClÁ[-L üâlt t C¡âltIt I rqü[i Il[:'f Ul:ti,{ 'tl,lüt L- ï NE ËË:tilr .

CAL-L tË¡C í$I(ìi'lAl.- fJ;;; Tn G[:'f; 0F Ëlt¡ì'f I0N üOIrL. .

CALI- üETUI-l ißËl' t')Fr[:FfÊrTïC]l'l ü0llË'C/\LL [rüHü i I]:t$Fl-rt'¡,.l40u üyc ;s,qu[: I]. Ë¡T. [r._.t:iÉ.cì. .côL.t- E{[-l.l tGIVli Ë[¡¡lcHþlClU Ë't r Lt

AF I üF I ,t,4lL $ T[:$T Tf- 'tt' ..17. Ë¡Ittr i IF 'fl:$ -It_li'lf¡ T'u 'Ërltfr' o

$Ill CFI '4'4? it]ïl'lHlttJT$lî,v 'tFfl'I ïlr '$'.

Ì,i,.,i dË'ull*' i['lËfi'Íliii"'J,i''Tl;l'0,;, .-lz t1t.JL. irf: Yti$ ,.JtJt'tF TCI 't.1t.J1...'.

Ir$V! CF'r {t,44 íCt'rHt:ifttJl:5Ë: T'ËSjT' ïtl,fr',.

c0rr$' dir''lÏ*'Il;"1iiil'iljlln'''ll,''ÏËulü', .J¿ {:0MU iïF \',[igi -lt.li'1Fi Ïu 'cnÌ,il.J'o

c{lçiïi cFï,['4d i0Tt.'¡iit(til:t$[:ì-[i$'I-rF'tî"

F,âî ,jË'*iii'i['lË,'irf,I|]"u18,'fl;I{Ti:.JZ FË¡tl iïF Y[:S -lui'lF ]'U r¡::f¡fl/

+

FS 3 {:F I '1140 î UTt{l::ltlrll:$li T'[i$ì]' ïF 'l"l' *

FÈ'r,lfr"lTu'i['l[ü,ìili'nll,,,lT'l;uT':.-lL f:i'it.J ¡IF Yf:r$ Jt.Jl'fi:¡ ïü 'Fl'ftj'.

FII ; üF I ,[,44 i OT'þlF:fit,J:t $l:i 'IEírì'I T Ë' ' ul' ,

:if,."'Ër,' I[,'lllfi'íli';"''J;l],;l;li''''Ê 0F Ë:l:i¡ôl' T []i{ ü{iIl[:.

i ?i{ili¡\)+(}K;fil\:;\t,i(Tr}hT{rc..fiÍ(Tr**)[{ìk$f{Ti,$:firËiFhkrlr:'l{ii{ïir.l:ii(;K;*il{;Ì1il(iir;fi)ii;*&yíÍr,ïift*tctc

A33

i {i ïr )i: )h hr{ }i{ "{i:,ii

il1 h"; tk il( }ii ;Jr il( }i{ }H )i( i( x{ s :# I{ r+{ Ii }F tk * /,( f ik )l{ }h ){( It }i( \k }f }h It ,\c ri{ hc )k ¡tí yc ik I( lk Ii ìi( ii )ii

iTþtH t.,l,âl:t.J [¡Rt)[ìfirtÌ.1[::$ [rL']li Il{I'L:.Ë[:lT Érl'(T]'fii'18'fItl 0llHfiAT'Ïüi'J5'I

i t.iÊ-T Tl,Jü 0FiË:ttÉrl.ltrÍi .;ttfJ ËrllttT'fIül.l'í C'q¡ÌË( l:Aüü. ÏtH'ILJI:ii'l ñN[¡û t._ fi'JE r:[ih.l]

"i$TËl''l'â1."Í; ilVE:RFt..Utl ( l:f:

"\l'JY)t $ I Ght ,qhlÌ:r [r.l'g;¡:r¡_6ti'rH[: trH$tJi-Tû I N ItHÍi l:[rt.J[: lrüfil'l ,i l.tI í:iF l,-râY Tl"lL: ftli:$l.J1..1'i IN IIE{:T¡'1ñ1.- l::üf(l'f .; GET 0[r[:RÊrl.lÏ:rli .i nü sLJË'ïft AcT'l:üi'ì.

Atrn I

rl L:l i

ü'âL.1.. Tfr^ I I

^L, f-r t... t- t'.1

ü'ât,.t- C

r\lil:IrHT\

Cr.1L..t- Cl$ I riN

-JMl:r ï ErUIr

$lJIi; CÊ'1..t- TÉ¡ltHilÅ1...1- $LJ

Jl'lF ôl.rl'f t.Jt,.l üÊrl-.1.- T¡tltl:

ürlL-1.- i'fU-Jt.f F' A 1i

00l'f$ ! CÊ'1.-L- TÉrliF:{:'âLL. CV

t 0[:T üF[:Fi'qNIr5ì.i ttU t'lUl.-l T [1[.. T Ërq'I ï 0Ì'l .

i tìET 0lllirltËrlv'I¡fl .i C0l"fli¡'tl:tl:t'IþlH $jl:[ìlltiili l.ltJ14fiER$.

ü0r'ru, illl-,-nT,Ï,*,,' tGËî' 0rtiiFiÉ.'t'lt¡$.CÊ1.-t- CÌ"1 ;Cüt'fF'ôRl:1 Tl'lE UN$f üi''ll:iÏl

i l,llJi'TIll-:ft$ 'Ër30; U'â1.-l- CrqF{

üÉrl.-t- ccl"lR tSTGt'{flL- ct}¡1r¡A[iT$Ïni-lÊ Ct'lrtFtrì{l'ì-[i Ft

".,1'lF TËST idìSlt Ëüft Cll-'[:RÊ'T'].f:]N

i çtlI¡ti.i lT**)FrSï{;ii;tilili;i{}¡tÍíili)i.)+{fr:}¡(àCfrï(;tr{i,$ilXrfi)i{*}'i(*r'l1rir}ii}ii}f ii{,$}ii}i{:.ü}'lt}ii;ìt;fitr;Sil(

í${is{i1þ1þtþtl;lli$.+:Þ${iiliiþ:F$lli:ltfli:Flli${;iliiþÍÞiË't's:þiltiF:Fss{tÍlis#l;iFl,ilifilFf'ENn

A34

i ^JlIrilr:FiFliiiii;ili1;:rr$:lilillrlriE.${'fi1;$ilr:litliiliTi{ilifitß$il;tlis'titb.sl;?[|ßiFlitir':litl;.F!ßllili

i F It [) Il Ér iÌT lr: f:' {] lt l: Ì'l T'i:: tì ¡::: [t [i i:: $ :t I:r i.i l;:: ll I V T Ïì T t] l'J *

il:ti:$:i:[rt.Jti ¡âft]:'Il-lÌ'l[::nT':tt] (l'l(1ilt.J1...ï$ tó li:Ll:l ]"û ;i( H hc lÌ{ r{i;i{ ri( }.h ht{ ;i{ li( ii; ?l{ * $:i(;it r'F. rf( ;li }ü: ht ¡t:.\t:rq }f rrq:.'li{i:'k ili:i< ¡t:.lq :K ;lc ¡t;l\ ¡t Ï1if )i:)i{;i{ $ ¡\ }i( * >ii *:i<

Ël'lìl:$ l:1[ç(][ìl:tiìi'jl:i fiT',tl"tT'5i Fltt)ä 1.-ilC. {lÖ$fi0 {iI:i T' t-l H f3 I1 li * li û i"i T ül:i n Ü 0 Ì4 i::' tJ T'[ilt

"

I

ül:itì ,llütifiü

" frl:tot.j sH'rÍlt ;¡ ;;; ; :;;; .¡; ..;; ;

l'f t.J i $lîT' 'R'0f:i!'0Êü å $trT' ,[,0ii]?T!ìt.J I IiFiT ,11,0$l'flii

cl.1î siHT',11,û{lËr:5nIi $El''ftû{:}'àf

'âJ.;: ¡'SËT'il'ü1;'iJ*l

ïHliì'f i $[:T ,11,0$'i$

t..U'til î !;ET n'0{;iC}'tTrIliE. M[:T'll'0Í:iFî

i ti(¡cili$;Jrr{iht:*iilit\}i(Íi:fifr}.lr*{¡c)Fr.i.il{}k..i{rd}iiiltht)i<ì!)rt:ii;i<;{iiii}fr{iiki¡ih(h?}i{ii{rf;fìii:flir(il:*iKtl.

t )'trii¡!{rir)Kr'|i**¡rNi:i(ï( t'1ËrTN [¡1.(ut]Iiflùlli: r{iÍ(;ft,1T$d(;fi}i{?iir,}:i|l*,'i(*rÍ(;'li;'i{}ft$:;|ifril{¡i)i{

y Ë r--;;;:;r::;:;:r ;;;;; ;:;; ;; -. .

Irl:ul ürâ1...1- l'¡âliH âtì[i]' 'rHü ßYTli$ 0F [rATËr'cô1...1.- $trTu îIltl t¡IuÏsÏ0i'l"

f'Ë'fi Jl'lF Ë'J.;i âIrI5l"'Lii'f Tl{Ë 0UH¡:(l:r1...(ll¡J

;(IF Al'l\')v TþlFi iiI[ìN tâi'ltl; l' I'lË:: fi [i fii ti t.. 1' ]. I'l [t Ü T l-i

i tt fr Sì l: t¡ t.J l;:: rt i''l i.t Il Ëi {: :t I'l rì l.- [: 0 Ft i'f r

;l'l-ll:$ $iUfrñilt.Jl'ïl.,lE IrIUI:IrËi:i I'Ut0 ïi{1'11üL.It Nti¡iIll:ti$.

$ilïUî [-IrA ,['1iióJ. t L-0ôil ït.tE: Irr u T rri:l-lI F [ic]l'ft L-t-1C r ,8 1.3ú J. 'i $T0i:t[: T I' Í\T d'J.340 r$TA

ì)ttlMUïSf Éì

{,L 34ü'[;1;$,{lrH'UV,& J.3[ì0

:iTfì,[,J.:i$4

LII¡â ,1 1Só3

$Tñ ,[,1343'fÉ¡lt I CËìL-L- trl:

i rr.l:i:l'rAL-l:$iE 'ft{H Liln. c}Ft S ï {:ìl.l v ,[ l3$ö Tü ?:H¡:{l] .t I ill:'r r É1.l:iiri l'l-lE l-[]u . ÛFi fiULl'f T [ii,tT' v it ]..i8"I 'f ü ;|HÍt0

"t t..0Ë'il Tl"l[: I] I U I ËlffFt FlttJl'lí L-üil ,[.:l.3óiJ .t Í;1'0ttH ñl' ,$ "[ ii,Ill r

i [ì [: 'f r'i Ir t] [t í] X I i'l Ér 'l'Ë

G ü i.J'ì' l: Ë. i'l l' 'i ( F l. Fl$T A[]l:il:i0y l:l.lÉiT T nl.,l ) .i¡:::[i.lI¡ ]'l-lE !:iTtìN ü[:' Tl"lll IiHÍilJL'f'i $'Ifil:{f: I'i Ér'i i.'-[][:] . ,t,J.'.JS0 ¡

Al.l ïsT/l

04{,J..ill0

'fôni HnU Avß

A35

i t'lnuli 'rHË AliFt:tüxl:l{A'r[:û üt..t0T' t: Ët'J'I riËill'Ji Il.*fiËG, 'T'[J r]r.'-lii:.[ì. .igìT'l)[t[: râ'ì' ]i':l.,iTÍ31 -

i l'H Íi'f I F il f:q f] ft f l )i i: l'1 rl:r l'[:í nLJnl'T H:ÌlT r í+ ^rli[ìfi .i IF iliHU v .Jt.Jl'ï[:' T'U lHl:i0.î üT'l..lrilti,,.l1.liìH Íi'It.¡i:il;i :t 1'

!'fì¡ ;¡l;l.*)r),) +

t t-.0AI1 Tt.'tE Ilïv:tïìtJl:t rr(ilÌ4í ,[,:l..34.'l .

i MUl... ï I ft L. Y Ërtilrf( f) )( I l'1¡â f H ['l Uü'f .t ôi,tfi l't.{H ïl:[uïfiilR

"t'f HST :t Ff :t T tìiUiii.i:tl::'[-{]t+S .i l:F YE.$ r -JtJl'lF '10 ' $$ü ' .

t l-.tJåIt tr I uI [¡ËNì] r:[i0¡l't, 1 3"1'] o

i Ctlt'lFrt¡Fll:i Tl'{E li[1St.Jl.-]' nF'; l"lLJ l.- ï l: tt l.- l: (:l Ér T I [] þ¡ llJ I 'i l"l T' l"l Ë:

t n ï V:t t¡ül'lïl .

í ï[iSìT' I 15 ( Fltül]tJü'f ) .:'' ( ïl:L U:t I¡ü:NI1 ) .i l:F YËfi v JLJI'IF 'IU '$Íì(l' *

iüH TH[i IiTËi.¡ OF TþIE:

i RFi$ilJl-T FIt{ll.f ,[,:[.5{Jü o

t I EllT' t-Jl'lËT'HËl:i Fûl)'1.'f I UF: "

t l:l;: Nü'Ir.JLJl'1t-' l't]'F'¡,rÍ:iÍi' "

i $itJIiT'[tr*rü'I 'f H[: t']RüIllJËT 'f Ë:It14

t f- f'{ül'f Tl'lH I.¡T U:[ I]i::l,lll .il.Jûv[i 'rH[i FIË$Lil.-'i t]F $tJII*i'f HAü'l'.L Ul,l 'f {l rì'*i:il:iG . .t$Tnft[::IT ÊrS dì l.lEt I1]:UÏtrL.i'JIlt É¡'f ;ft I Jrf Q

"il.-UÂIr Tl-lH f¡Êrlt'fI'iL flLJ0T'.i Fli0tf ,ü,13êi4 0

iLíl'til 1'l-lEr ô[¡FlttlXïl'l'âTH nLJflT.tFRUI'1 ,¡r,J.3fiì1.

í '4

tr fl T U F n Fi l'í il I'l E:trl [1Ër [t T' I r.¡ L.

i tluü'T'T [ii'.lT .i ílTufii: S:L Gl.l íÈUUËii:if:'1..ütrl.

îliìTCIRE Tl' ÊrT' 't'.[.1i34.i Li [: I' T'l-lH tl U il'f I L. l'l T i:r I V I ilrl:: t¡î fJY T'tJü ( I'T' I b* 'f Hl:: tltJl'lÏi[:fiû 0 lii t'i H. Iliì T l!: I¡'îË'f [i r( I] I u l. Ï1 l: l'¡ tìi T'1.{Ë t{Ui'1f'rË:Ft IìT'üfifitr AT ,lt,l. ij$ L

i ftV I'[,JCI ) .

SìTôANT

ï'AIt; t{fiu

$ÏA

Lttô

{, 1. .10 t.[: l::'l:r

"t7. ?Ë:Fil):} I É{ :ll:.1. r5c).}

l.-[tr] .[,J..T4î

$'f É{ '[, J.3S ].

UËi1...1... I'f t.J

ANT O?-,t'tr sì$üI'10U A v [tì] I fì 1l;J.1C)JLIIÊ' ,il l.:J40$'rA .!,.l..äó LCôLL ËI.I

H0uHI,,I IJNUL.tlËr

h-t t rr01:r ì:} t.,,[,J.3Ê]0

ÉrNl: {1,04

-JNf FÊ'tìSicAl_.1. fitJ

Al,B

{r.134 0

{,111$-tr

SiTË¡

LlrÊ+13ó$'il J. :{ f:} J.

$TA,['J.3í¡1CÉ¡l-.L Ë¡['

$T'.ti''lüuif I t'l.,l.lF

:l; J. .J _1. ôAvff,l! L 3g4

;iHIiü I L-tlrâ ,[,13[]0

A36

i t-0tìrr 'rH[: l:i r $l-l nr- 'rHH

í R[i$ìuL-ï [rRt]Ì.1 ,ll:[:3uü *

i d\t-T0tiliTþlE:tt 'rt,l0 R r tìH'r $l.l:t F"r$ .IIRT:ItF(üÌ'iilvt_ttËr

ANÏÉìtrll$I'ËrCFÏ

ft r f.l,ll,l.3 I Cr

0^1:

Ë{,1.37 L(J .5 i'fE{:ì'f ïïi Ér-FiHG.

i Ë0NTñ:L Ìliì 0:it I F Y[i$ v ..JtJ]'lF 'f 0 '' l{râ'T'l"l ' .t'IESìT r:F 01.i :t f: YE$ ¡ ..ltJl'j[:' T{1 'l{É\Tlt ' 'tT[i$ï TF T'l{l;: RHStJt-'ÏtI$ FnÍìIT'lUE.í IF Ylilìi v .JtJi'lF 'ffl 'l(rô'f '' .i 01.'IIthtI t]ti t..0¡ritr T.þtE

i [IUüT'. fjITLJi'f 'll'1:i{:}*'tr.5ìT'râ d,1363l'1Uï A t 00$iTÊr ,4,J.3óL

CÉit-L St, iflUtlT'ËACT T'l'l[: fttJ0'f .i F[i0l.f zHltu 'rû r0iii.f; NtiG¡tT'r uE [:ilF(l'f []t' Qt.J0r' .

¡'f usìTf( [:

¡i'â'ï'l{ I l'lV tTl{I5 TS T'FiË SIG}..l ÉrNI¡i OV¡:fiF LütÀl I i{f:fl|:il'f rq'l I CIll .

$'l''4t ,['J. 371.Jt4F KrrìT

ItôI'Ftl l'lUl. Ar03 iTl-lI$ ISì T'l'll:: ÍiItìÌ'{ ËrNt¡

í tlvElt[:'LüH I i'{Ëfl Frl.lriT Ï Ui,l "

sìl'¡*l ,8,1371ItôrT'l l.-il'l ,il'13$4 i LUFìIl Tþlfi nLJn'f .

â r:[ì0t4 {,J.:Ilt4$T'lt -$. 1370RHT

i hcli{il(}Ír1i.:ìtl+ilF,fiir)'iiilI{il1¡tri{),i<;fi}i{ik)k,i(fi¡(*il(;fiil':ii(ìch\'/}i,'r,(firi{}i(}i(,ÏI(I{,t$)í(i\}'F4{h\rfi*I

i Í{ {i }ii $ il{ hs jli hq jii .1q ¡!: Ti ri{ }ii il: ¡t :lr ¡fi ;ir ¡jç ¡t ¡l: ¡ft lli )|( T{ # '/'{ )?i )fi iÍ i{ }Ï r'I 't }i( ri( }( i$ :ft ;i{ ì\ )ii i{ rft ri( i't lt h\

t Tl-lï$ l:.linilltrìi.Ín. Ërlltrfiì T't'li::: F'ItclrrLJc't T'[il:tl'f tJ]:Ïl-lí 1'þtfi [r]:u T Il[iNL¡ v T l: ]'l-lF- 5 r fil.l ÜF' 'rl-iH l:iF-$t.Jl..'I

t I$ NËttËr'IIVli v'lü ltrillit'l .Ér Ì''{l:tl F'¡'ltiTl.ill- f{l:l'lrqIl'{[¡l:iÏt.,

F Aii$ i t)ill|..L- r*rlr

,Jlllt 'IÂIri )f{ /r * ÉJ{i )ii ¡\ ñ il{ ht 4t àc Í( iii:ii àq )li }ii :,i( ti h\ fr ii( )i{ ¡\ rt{ il{;f{ )$( T{ }i:}H }+( Ï( il( hk ¡t /'( ¡t t( }k;i{ il( iF '$ N ri( f{

-f;:, liËrTl"lTF,T O:I

-lL ltrâI'[(Ai'l ï 01

"t7. trAT1..I1Ër ,[,1384

rEtJ.3/U

1U

VÉrl.'r .I.

T'

IÉì !

A37

í tklFh(.ì|iil([{h\}h*is¡q)ii*)h)i4*$)|(,*àcIi;ìt#il{*¡\I{}i<h\àc},t)iiil(¡tEt{$"*s¡kri(*,,l(;1t,$¡k*,\ksil(û T'l'l:t sì FitÍlü[tÉ¡i'f ll; üET$ Ér l.lt.Jl.iIrHri IrT v.t ilHn I{'r TL.JÜ .,

$$U t l-l¡É¡ ,n,J.3t J.

$TA,['J.3ó].MUI A r,[,0?

i l-tJËrt¡ T'l'lH llt.Jl"ífinÍt,TU FE IlrUïrrË:rr FËül.ltl...0ür '['J.S[3J. '

í ''[,09' ].

i c{lIrE: [:c)l'JE: Ul:'Ëlt'âTIüNTUl.trË $T:ll.

clTJI

ftrrSi'f A 'Í'13C¡5Clll-L LÜ'1fiJMI-' TÂC

i **:h\)i(,$ï{,ñ*,\c;|{I{il{}i(}fiT(*¡9¡k;t¡qlt:¡tl4l.$IÌiliil(âlï(},\hth\,$,$tr)i(hkil{}i(¡ic¡ir¡\¡k:ft¡\¡l¿¡ft*)fixhc¡ 61¡1$rFrFli{il|i1,l|irßfli{;il;lr'#S$$S#rFlirF:l;:Fli{;:Þ{iiFllilli'|i$l;rlillilFiþfþ+fitÞf rb{}lirþ$

liNIt

A3B

AP-6 (b) : Listings of Floating Point Operations

i )|{ )|( l|i .* fi y,( Í{ tr o nq r¡a ¡1 t¡ tk ;.k )k :fi ht * ¡s ¡\ )ií ¡t }F * ì\: ¡\ )F )ìt rft r$ ,* ¡c ìc ti ¡t il( ¡t ;lt ,$ ,'l( il( $ hc ili i|( ht ,\t }ii ft ß ¡c )lk ){(

i[:'L0ô]'Tl.l(ì IrtlÏl.l'I l:iI.rtrT'ITüN t¡Ï'l't'l H1...Iu[::i"l '*rli Tþlli:

t'fHft ËA$ìË: 0F [i,'(F't]l.lËi.l'l (',ll'lrlr' :t{iì T'l"ll;i l:t[::l:rf([:$[ii!'l''t1'IC]l'l$ 0F' E:1._[::U[:i.t ï¡'{ l:tl:ifìTtrt.Jlî: lî'í15'ì'l::i4 üIr i"'l(iI]t.J1....[ :[* ä1i:i

i F¡F{{lti[tí.:l¡ll::: $iI'rr.ì[i'Iiï Fttt]i'l l,..L]i:]. ,f,0rl*0 [J[: ]'HIi 1ìillt'-il]0i l',lr tHtiüÜl.lFLJ'T'[:'l:l ]

t *il(¡k)$iil{à'{hq*r{iiHtírfi,t¡\¡tfr¡cìtil(hc*¡crfi)|(*Iif(¡cil{,t)rc*h\¡\ht*)i4I(*hk}'¡(hcÏ(}t;**;s{{)i(;'!(hsr''cI{ii(C)Ä$ 'ü'0ArâÖ

i il(**t(ilr\kr'thq$}l{rß}i{.\c}ìk.*rfi¡t,$)l(*)|t:+:)'¡()F¡l:,SIià\rß.lt¡kil<¡c.$}tlrfi,$,$)i(Ñfr**il(I(}fi*tßh(hkil(hki'tiÏ{' FftÜÌ'f $[:T'$

l'f Nl't $[rT 'Þ0817H:[t]' I $:ìHl''[,0{3:ll;fJLtlI {ilÏl''llü{:1"I0LìHT' I f:ìli:'f 'il'0fiì4rà

l'11,1 f:ìËT''ll'0t390ôil Í f:ìËl' 'ft0til9'9$t.l I $ü:T' 'X'0{llHCM 3 riHT 'll'0{:irâ:3

0UËft I $H'I 'll'0t]0Û$$i$F i $H'f 'il'0l:ilrt.:I$llTUl $i[::T''ll'Û$tfJf

i il(*,$il(I{}firfi,t*I(h\$r(h\il(ilr},c)t(*Irirhtï()l(Iltk}t(¡k*hk$tk}F}|(}|(rt¡c]l{}l(*s;ü;(rfi4(ht,1(}it,l\$il{¡t:{;{ti * fi hrí )i{ I,i }i( tñ 11 $( ¡t il( \k ,* ¡k t\ t4 ¡r{ ¡q tc .ï il( ;{{ I( * r.i( ¡c .ï )fi rti il{ \k t{í r{i }ü }}t ,* )ft ,Ï ,{,{ il(;* ht }¡( ik tr Ñ }ìc ;r\ ;{i ¡t ;{:i< * h\

t'fHh. $LJFfiill.ll':ti'lË' []ôll' f:iTt'ii'lô1.-$ ÏU G[:'Ïí Ì'lAtlT I lllrìA$ ''ìi'{il HXt¡üNliNÏ'$ .,

CAIIl l"lvI Itr01lL.X:t l'lv{'l.illiÖ

t'ôI1 å üAl.-L ]'TN'r

CË¡t-l-. flHTI'IOU i'f t A

INIï l-

CAt-L IrL..N

cr\t.-t_ HF'îCÊ'1.-L GETMtlU i'l r fl

TNlr l...

üAt..L ßt_NI]ün [r

t [-il1âI] I1*ltliü * td T l'l-l 'ft0f .Ê l.-tlrâIl H 'âl'JIl

t- RlIti I l:iT[::ltf]t t l T T' þl ,!, ll. .{ l1T') .i$IüNñ[.. /fil:::'.t (iì[:T i''lËrl{'f I $$Ê.ì .i $T'üFt[:: T'l-lË f"l'âN'l-:ii:ifi'îl A'fâl'1.{E [-Ür]. F0[t14[;:il ITY Il-lF-

i C(lÌ'.1'l'[ii.lïli f][: l'l Ai'll] l.-

; ft [i fi ï f:ì ]'t:. [( li .í I l'lÇt'tl.{rôliìlii Tþl[: CC]Ì'{TIINTIìi {lF 'f H[: lti::Ë.L f:ì'f Ii[t l- .t ti T UH ' lìl::'É'tCil. ' .t li T fìl'lÉtt.. ' Ít::: r +

i li[:'t 'il-lL: E:X[¡0N[:NT'.t $'f n[([: T 1' l\1' ]'þf l:r l.-fl(1.iF0[tìiHI] ITY I't'lË tt)i{T'e ¡{S

i {lF 1'l.lË: [tlÏtu'r 5iT'lil:t F'A Ï rtâ ([Jct..).i I Nü[tlii'qi:iE 'f l{H Ct}ï'11'ËN1'Íi 0l'rt 'f l{ [: [t H: tì :t f:ì ]' l¡i l:t l'- 'ifiIVH 'Ëìf:'rtüH'.û Illiiü[tË.iìSË IJ--RH[i. .

A39

JNZ Il¡âI1 il:f:' NOi'l ¿lxlT0e Cì[:T ÉìN0Tl{Hftt {:ìlr.T' illï i'lrtÌ,1'l'l:$$l\ rlNItt [: )(l;t(] l'¡ E:: l{'f '[tH]' t 0'f l'l[::ttl,l I {iH l:tE'f LJliÌ,l '

i filk\kìkI()ft)iofiI()í{il(ì(hk)f,h\$)|(ñilcli¡kil(;lk¡lc¡c;.Fri(.-.fiI()|(hkil{il(iF:rq;.i(i()k;ff\k*}f,v}.kàc¡\)|(*ikhkûï¡'lIfì $l.JIìrltClt.lTIi.l[:. filiFAI('âT'Ël$ 'q R[;:frì]:Tlt.lI NtJÌ'lIfER ]:Nl'ut l'tlfi T'Ël:tÌ4$. ,í A=' A ( lll ) ¡c ('li'llrj )I .{. l\ ( J. ),$ ('llÏrrÏl )0,i :t N $'qM[: tJÉtY T T fi'tl',| i.rH tlR ]:T"IHl'l T'HA'Îi I.:{::: I.J ( 3 ) rt ( 'll'Iin lr .Slr ( l. );.k (:lLlIIr )0+

fi (:::;:(] ( ll ) ¡k (,11,[r]ir ll.t-C i l. );ft (,ll'[rli¡ )0.iþJl"lH:11E: A,'" F"l'fT'ïT i'l'1l.lTl.{iìfiì'q" B-'SL'C(ll.l[¡ I'IANTT{if]'ô.$Ç:::: Tl"lH l'l¡tl',lT'1.ÍillA tlF Tl"l[: tII:$l.Jl.-T"$T'þlï$ fit.JIi{ItiltJ'ï'TN[: ilÉrLL..qì T'l"lË: f:ìlJ]IR0t.JT'Täl:: '$$$Ë''i tdl'll:il1'l (j l:çrli::I:ì ï'l-lH ütJ0'r ï [iNT' 'âNI]

nt[:i'1'â ï NI;rt,l;tihll-lEN l\ Nl.,i"iËEi:t ¡t'[ l.-lJ[-'.,[.]..;i?1 l:$ ïlIVl:ïrliIl IrYíTl'lH: Ì'ltJi''lIJ[:li f]'T' 1...0ü. ,11,1.390"T'l"lH l.-llC. {lFtClUüïl:ËhlT' l:$ ,lll.:5?? llNIr ü[-- l:tti14r:tIl.lÌ1Hft I$ ,ll,.13?3.t

$ll¡:ï1 I i.1Uï A ¡,[']l[t î/:l¡IJË' I$ ïl'tL: tï[i$rrrl.J[iâ ñ IIF lt [::$l:r i'.1'f ô ï ]: tltl [J F E l-. Ë U Ë l''l .

Sl'A'['1390Crt\Ll.- S$fï["

LnA ,['L:{4,S

ANI 01JZ TATsTA {'1341:

t trl:V:tïlH 'f HH C)01{'f [:hll] râT'

i l...CIil . ,ft J..J91 $Y ¡aLI:U[rN ,t At'l[r U[:'f tILIU]'. rqNIlt ÍtHÞlô ï l'ltr[:Ii .t t-O'q[t 'ï'l-lE

Sì ï t]tl 0F 'tþlË

t'rHt: üür:FFïüï[iNT 0t;i 0--F{ll¡lt;:t:t {lf: ( {trIÍIJ ) ( [:L.f:uËN ) .

iïË:5'f ïf: N[iLì¡tT'ïUH.t ï F NCI'r .JLJHtl T'ü ', CAT ', .i OT'þlHfttl:til[: 5ìI'0RH 'fHË $]:tiN nF'â ïl-lH i'1'ât',lT ï $ll'à A1' {,1.:i"I3i .tL0rlIl TþlE c{Jl::'l::l:il:t::LHNT 0rri 0-l:.0tJËlt CIF' {'[,lf B ) ( El.-ËU[:l.l ) .

CrlI' 3 l-IrA {Þ 1:.{93

l'f 0U Il y l\LnA {,139:t it-ü'âïl'IH[i finFI:l:rIüIliNT' t]F

i L-l:¡0tJË:tt ilF ( {'l]fi ) (Ë1.-HU[iN) .RE'T

i f({(s*\k)ü)l(?shcI{tkÍ<àk*.il()|(*)ftI(I(il(*)t(ht,f(¡cìkI(il(Í(;fi¡|(*.$¡ft*¡c¡k.{vr:fi{qtcN(*,$**I{*hkIi *Í(il(:i{)k)k)rk¡lí¡k.ttkiß)k,$¡ir*;fc*¡cE*;,t.4(tkhliloßtctttt/d}k¡qht)ii¡t¡r,:¡tlt.}Kil()F¡k,V*.$.ïX()k*It

'l lTr¡r ftuIf ltill.JT'ïi',1È. '{:i[:L-Ë' Â[-Tl]Ì,{$ 'rHË t'rfli'r'ì'TÍìr-ìfi Ai.rrr

, iCüNV[:ftT'Íì 'fHH AL-]:{.il''¡HIl l'fAN'Il:il$'â l\ I'iül.Ji'JIl[;:ïr filfitJltlii"t -'""-'-'-'- *

$H:l-.li::i ANT 0L ifrINII 'fH[: $ItìN 0lï 'fHE

i [rXF' . *IJ I [:F . fil:l0M 'l'Hli li ï Gl'li Il{f:'0Hi'l'lTl:{lÌ'l TN ¡q*Rli(3. .

Cl.lZ ïA$ i I f: l.l[:Lìrì'l'T Ulii r .llJi'lF ' Tílü ' .üAïl L.llA ,[,1.:IliJ. il*0'qT] l'þlH [;,,(Füi.lHi,ll' 0F'

i $i{'qLLH[ì t]11[:tt14Ì.ln "

A40

fi'l'd\ ,[':[ 37i:ifiAl'I l.-llA,ll'.1,:$;70

[iT.¡q ,[, J..T"iFf:iT'rt\ '['J..3$.$l'4UI ôr'['.[].ti T'ô ,ll, I .í ó LftA|,..L Cl"t

t4üV l\ v FANI 01.

..JNZtïnul\l.l ï

i f:il'0fi[:: T'f Aï ,ß:l..J7l1i .t l._ürt[r'lþl!: Hxtt0l'll:ii,l'Ii I't'[ f;FF'l:il:::Ì.lCH [:[tüM ,[']..J70 .i iSrtV[: râ'f ;[:'[ iJ

¡'il:' .

i ü0tt[¡Âltli 'IH[l HX['. -'nÏF tr.ô tJ I1'l{ 0i.lH ( 'fl'1 :l ) .

r' 1'[r$T' T I-' ]'Hf:: HXI¡ . --il l:[:F .

trïAv04

t,,v

,,

t I$ (!ttË'Éì1'[:[i "

Ì.lIf:i ûïF Y[i.$ìv "JlJi'{P '[rTÌrl.[Íi"$

i0Tl..ltil:(tl.l.l:ìliiv T'[:Ëì1' lF T]{[1

i HX[l + '*I.t ] ¡:'¡::' . :l' fj [r{ll.lrâl- .

-,1{f $[:hll] â T [: Yliil] p .ltJM¡:¡ I ¡::'ç'[rl]'r / o

l([::'f$HNIl l t:¡11.-L- t][iNlt

t'1UT AvüÖfìI''â 'll'J.:5l:iCltH"f

* il{,\t Il tk )F ht il{¡c Í1ht¡k,t il( ).k ht },1,*,\c }fi t{ * y,( }i( * il( 4íÍ( rß il( }i{¡c ht }H lt * ¡c rfi ilt }i( u, o ¡, ¡;¡ç ¡¡< ¡ t|( rrk * il;

( i,lü'IË: I Ë::xfl . -l.rI lrF . $Trlt'llli:ì [it]II [i){l:i{]Nl:ii'{'r [tI r:F'H[t[::l-lcE: ) .

f( )t( )ft ]ft ht ilrtk )k t<,1ç * * ¡c )ft ¡k ï( ¡k il(¡q¡\ ¡'c )|\ +i hk¡k Iiil( l( ltil( )$:rc hc rfi lc.1( $,It Ñ \k lt ¡\t$ * *¡c ilt,$ )k hY,$

FII'lII:il MUI Av{'??$T¡ô '['l[ .ió LüAt-L Cl'f û Cill"lFôfi[:. 'IFIE Ë.Xf:'. -lJIFli.

i l,J I T l-l '[ í,1 í,1 ( ]'td t] ) .t4uuAt{ ïJNZ

¡00J.35nL 35tl

FTXE:I tl-u'àIl T'l'lH f[J[r[:frTüI[iN'fi {-l F' :[ - F' 0 tl l:: l:t t] F' ". l'l'll' Il ÏJ )

t ( $l"lAl.-t-Hft ütjt--ltAl'l[r)i $i T'0 [t E: rà ]' 'tr' J. .I ii I.¡$Td¡:[:135[l

Ì'1VT Ér ¡ O0$'f íâ ,ll'J.:I5ü t $'f 0RE. :{Hfiü rlT' {'L.JIit .

rl0FÉt

.[,

;[:

,[:

}.1V Ï$TA$TíIft[:Tl.- Il r.\

vll4I X[:

i'rH$l' TF HC.IUôL't I l- Yti$ v Jt.Jl'lF 'F:'Ï XII ' .

û CITþlHlil¡J:t !ì[: $T'0¡:tlli ZE:ft0 rtT t..0Ë . {'J.3liC & 'll'i[ 35ft .

t 35c;

tiT'þlIEì {:it.lËFtC)l.J1'l.Nlii r' FEl'lIJ' T$ At-ÍìU ü¡Il.-LEIr FCIR

tltOtJi{IlTtl$ rq NLJÌ'.1HË:l't trt.IItIl.lLì t'lüIti'J¡q[-:tl:ìl\TI0i'l rlFI'ËR,Fl.-0Ë'ïl:t'lti Ir(lI1'-l'f MLJL.TT[1[-Ï1:r4'fIOl't 'ât'lil ''ìI¡l]ÏTIüN.í -*--.*..-'

FËNIll L-trfì,[,J.3.'ill iLcl¡âI] fïTüi'l 0[: Tl-l[:t14ANT':I{5I:ìÉì tlþlTCl-l l.Sì T0 [tEi R0LJi'JIlli[r.

A41,

lìNï 0:L

",NZ AlllCì'tMUï rt v,['iiS

F[:i'ltl: '[,J.:5,f, J.

:ft 1:TIifl

sTll ,[,]L.Id.itAt-1, ¡t\11

/ìvFJ,llt:3IL

t- $i$ËIr

fìïÉr {.:1,$C.¡ I

L['l\ '['J. iJllC

S'f 'q {,:L.ió3CAl|-L AT]

Èl0V ll y litS1'rl {,J.:J:iIl

R[:'fNIIGI\l MU¡ 6':[;]irfl

.JMF TI[iNI]TAtì I l-Ilrâ '[' :L :Tli J.

I'10

L-Ir

SìTÉ¡ '8,.l. iiii:[MtlU Éì y ll$ï'â 'tr 1:ilt:T

L.IlÉÌ ,[,:t.IlT{":}

MOL-tl

û 'f [i ïi ï T't'l[: {:i T fì N .

i l:l:: Ì,,lliitìA'I T U[i y .Jt.ll'lF 'NliGill ' .Í",ûl5li' (l::IVH) Ifri l'l{Ëi .I.RI.JNO/Ii'HI]

NI.JiJIIH¡t OFÍ l"lrì1.-F {lt Hl..HVliN.

û l.-lJAIt 1'l.lH lillJi'lIrliIt tll"lT ül"l I $ 'f üô ltla ttUl.Ji.l[r[:I] [:'R0¡'T {, J. :iliI].

û Atrtt 1'0 T'l'lli:i üON'Ilil.lT'$ üË' 1...üC .,t'J.3C¡3 .

tltl:VTIIH 'fþlË l:tH$tJL..T ü[:'t ÉrIlIlITI0N nY HLL:Vf:tl.t Ahltl [ì[:T fiLJüT'. Íti l:tl:illrqï Nil[ilt.i |iìl'ult[: T'1.{H fitJü'r.û l\T {,:L:.{ó J. .í1.-$/ln (ltl[:F'l-'IüIHl{'f nF

i :l *[]0tJER uË' ,['F]J ( E:t-HU[:].1 ) .

t 'ânï:r

Tl'lH nLJflT'. ( {llINli:F{AT'EIlí il{ 'slilrfr, ) .

â$T'0rr[: ô$ T'þtE: c{]HF}:.i [JF' Z[:H0 []UtdHR 0F ,['H]Jí (ËL.HU[i].t).

í',ll,F¡A' I$ Tl"lH NE:Ë'.¡I'IUt:i r:0fti'f nF F r uH It'li ütH$:tIitJ[:: iìTSi'I[:i"1 ( M0l1i L16 ¡ J.li )

t LnAII Tl{[i HXI:]UN[:ll'I (]F'

i FIft$'f 0FËR'{hllr.

â L.,t']Ë¡Ir 'Tl-lË: HXIr0i{Ê.NT' 0F'i f:iÍ:#üNIr {'lt"F'Ët ËìN[r .iSiT'üIt[: IT''âT',*:[.ilï1.

i $''f {ll:tH T'f rq'f ,S I 33î13 .i ( tiXCþl'ti'l{:Ë: llf-' I'tJfii C iltl'f t: i,J $ IJ [:'l-t J[:[: i'.1,t, 1,3 i:i :L

i ô¡'JIl ,ll,J..Iljil ) ,ût-üAIl ïl{H fiIUN tl[: 1'HI;â F ïl:ttìl' tJ[1titt,âl'ln.

û1..,(.ll\tr 'fHË SiTËN 0l:: Tl..l[:i f:ì[:{:0NIr tl[1[if(AN11 Fl{üi'f ,[,l.5lil1t .

fj; 't'í\

Lilô

MfJU,i:, I t-lCrâl...

V F¡yl\

'â ,l,J.l.l5liT

v/â1.;JÎJIJ

Vßrì :1,

A42

S'f 'â¡ 'ft:L:lli$

I'f tlU ¡â v [t$T'q '[,:l..ii:ilt

i $'f 0ft[: I ]' A'1" '[,1ill;ï$ .

Ln'{,1l:L::lli9

i $ïtlltË I I' r^tl' ,[,135[¡ .t ( HXüll/ìÌ'{(.ìH ülî Tl.lH üÜi''¡l'Ë:}'lllt Il[iT'tJH.[iN 'll,:l..llif:] rtNIl ,F:[:$5Ï'l ) .t t..(l'qIr Ër ( "? ) tlF [i ï H$ïi ¡'îr¡l'lT I fri$l\ Flitll'1 ,[,J. "{]:i$ '

Pl t]L.ï:r t l."t.l'qil IJ ( Í,1) tlli' SìË:t0l'{il

i 1.1/\N1'IliJ:ìrî frli0l'f ,ll,l[3liü.i13'I0ttH IT ôl' +J.;{5ï.STI\ 'U,13li9

Ì{llu ¡ì v }l{¡T'A {,1..T5u i l¡T'0ItH I T r\T' 't,ll.iLÎ0 .

û ( HXül-l'âNtl[: 0F 1']'lË CüNT[:N'l-f:ìt ItHI'l,J[iHN ,[ ].3119 Âl'.1il ,ll'J. iiliü ) .t 1..0¡âI1 É1( 1) ü[j [:ïftll'tt t4râN't I I3f:ìfi t"ft0l'l {l' J..3liA

LItr.t,&L.i5l\

MOU [t I rrì

t-ilÉ\ {'J.35Ir í t-0AIr ¡T ( l. ) t}F fiL.c0NIliMANI'rliì$A lîËü¡'f ;!:J.3Ii[¡ ]

SìT''ô {,1;iliÉl i$'f0ft[: IT A'f 'ß1.5lil\.t{0u A v ll$Trl,t,:l..ilîIl t$'l-tlF(f: T'r l\l' +1.:T5Ir.

t ( ËXCþhâl''ltiH 0F ütli'tl'Hi''lT'$t fn'fUlH[:t'l'll']..iliÉ' ÉrNIt'[' J.i511ÎI.l ) .

RHi ***)kil(*,$¡i()f )Ftkh\*il(Stt*àt,V{(il(*hc>t<*;*rftrlt;ft**I(}kil(hct(ñ,*ilt¡kì\*t(*)k¡clk)k*I(hc}l()|(v

i s,*il(x{¡q)kI(s},ktctkil(*íil<**ht¡'(il(il()'k¡t,ÏÏ{,fi$¡iI(}ftx(*lY;ft1\i(*,t;g;fiI(*4{:*,'ft;thkltil{tftÑ,$*Ïßt T't'lI {ì $Unltt}l.JT'l:i.lli: Nü[tt'f Al- ]:I:il:$ 'f HE FtH$tJt-l' Ê[r'f H[t; 6trnrT'Iül''l / {}tJIlTt(ôc'T'T 0l.l.

NüR¡"T i L-llA ,['J.3É¡0 il...clAn c{?) üF Tl-lEt [tEf$LJl..'f FHüi'1 ,ft L 3ô0 .

s'IÉ¡,[,]l'36J.Ì'lUI /..t v,ll$fJ$I'rl ,F J.3$:Tcl\t_1. i,lLJ

U Ilvrtrà ,[ L:illi0

Érl.lI 0fJN¿ ClViliÞlOV Ér v ï:t

il'1t.,LTTFl...Y ü(2) 0FtTþrË: ÍrË:litJt-T' ItY HL-EU[iN.

' TÊ:$T T F' T'T OUF[I[:LülA|$ .

i lF 'lli:fl v ..ltJl'1F '[JVEI[{' .i ü'l'þl[;ltLJ.t $[i v l'l0UH T'l'{H

i FIt0tttJü'I F:'ttCll'f IJ-'R[:C, .i râ¡*[tH[ì. .

A43

ST'à,[,:L3C¡lL- tl'â ,ll':1i57'7 tt'0AIl n( I ) 0[j' Tþl[:

t [tt:$tJl.-T' F'fï[]i4 {'1.177' .$T/l ,[,:L.ió:lCrìt-L ôtl i Ëìtrtr c ( :t ) 'âNI]

'rF{[r

i FÍtunuüT'.t'rH$'r TF 0utiltr:'L.CIuli T F' YË:$ t .Jt.Jl'f F' '0UIiþt ' .

0:rüuElt¡â¡Il,[' ll .5 7 C¡

i **<rt<.$¡qlF¡krii:*frhC;1k¡\Í(tk¡SX(*I(**)kil(1(fiI(I(lt)*)i()k¡kil<¡C$)l(S*4(¡tc*tfc*¡C;lt,ÏI{il(I(I{hC$h\Iü,$)F)|(*,\t$),q.\t:\c}{($*11tk.t{,'t()i(}ì'()i{vril(il(il("ùìr,$I(;strlkrkil(Ñ*;*}'k;'k.{*)+1)k*hcil(il(*il(**h\hq*tftt T'l-lI {;i {it.JIlliOt.JT:I i'lH: [:'Lìfil'1$ l::){F'ÜN[::t'lT' lJ'[ [:FFFIHNt-:[::

i¡âNn firqL.L.{i 'f:ì[:1.-Ë' t:il[t fll.-It]i"l¡\'lHi'l'l' Ofi 'fHli

t MÉ¡NT'Tf:ìll'â{ì.t *..-*".*-..

HXIrCll Lftrâ ,[,:L3IîL tL0râtt 1'HH [:Xf¡il1'.lt:i.lT ü[:i FI[(l]'f {lF[.1RÊrt'J[t [rfiü]'lí ,[,J,,55 ][ .

$'rA {'1.361LIil\ ,ll:t:ii:i:T t t-04I] ]'1.{H [:xF0l'lHi!]' 0l::'

t $[:üt]NIl t]F Ë:n'tNil FH0t'lí ,[,:l .Jl':].í .

ST'É¡,[,J.Só3cALr... $tJ

Crq[-L- SË1.-tir{ET

i N(*¡ß**t<t\triloßtcÍük¡kà\:fifi*¡\¡[t¡cÍ<I{\k)líiE,tls¡k*h\}i(h\,t*¡c;lcI{;*I{\crfiÏ{}l(]sI(',*fr;fi,**)fii $$$$$:þ{;$$S{i$$$$-J;{;{i{r${,$S$$${,$S$'li$$$llt{t:li{i$${t$$$$tlt'b$$:lj$$

I:NTI

fìN IJNZMNU$'f râ

[IHT

t $t.tH1'Rl\ÇT Tl'lLr EXIltlNlIÌ'11'i üf: F I ttf$T 0FE:R'qÌ'-lÏr Í:ftnÌ'1t Tl-ltt EX[:'0l.ll:iÌ'.1'f t]f fiË:UtlN[li UF EÍl'ql'ln.

ç

A44

ûs:lirÞ{i:ü1,:li.ilitlilr$,liili{;',li{,',litli:lr:1,*l;$$|liil,rlitþtlr{i9,$$lF$fl;':l;||;:þtb$'$$$$'iþtli',lillitl,{¡:li$llj

iüÚNl.l:Ì.lt.Jr*¡'Il:ilN 0[i f:'|...t]¡ql'Ti.lt] l|rilIÌ.,1T ,lïrIl:['I]:t]Ì'l ,lÌ..¡Ir

i T N T' tt üT:rt.J C'ï l: fl i'l U F fÏ t.JIr T' [t f\ ü'f ]. 0 i.l .í tt[::{ì ]:lrtJl::: È[t I'f HÌ"1H'l'T ü . ( 14t]llt.JL- l: I I /r li J.li )i )t{}i1rNIls;l{Ír$r.¡{*;r+íHl{y}kilc+r)rk}.1(ik}þ:11}fi¡kìcìt}i{rl\¡(,'k}k)ttlt)ikil(t(Ii)t(.$*It,*,*lc¡\hc,*,$lk,$,$rìt

v

t **¡cs;liloir/,{iF)i(f1;t(ttxhc,*,+:il{},ctllc¡c;{,Ì(il(]{(¡\)iqlt,$,t;$i\,ì\àc,i{l{¡c\k¡kI(,{rr*¡c:{{}¡(;**.v)$,$lti F'ltflüÍi¡ìtf lii i:il'AliT'lì F liüt'J l.-ilü ' '['0ü00 f][: 'Ï']'lli

$ f:ì tr li * l3 ç1 Ì.Í :t t) l:ï {l ü tl i,.¡ f¡ [.J'I l:: [t +

i ìiíI{tiil{$)íiri{}rt)l{)ii}i(àcfrì1,fi)i()tt}ft)i{)i()¡\r'ftI(Ii;lc:\khk:+:il(,$}l{hkrk\cr1(:{$;t}ft;{<;it;lt:f¡kE.'fc:f,ìs*i{¡t¡k

It *I'(tC*X(àk*ri()|{)fiìkS)Í{:lí¡kI{),q}i($:fi:S,\k.¡ß:tililrkìChk2FEhk,$:iñ}.k,$,l(;ìtr;*ikht;J{à(}kil(hC)l()k¡tt(,1ilk*

0ItË ,ll'0[]û0i t(*htil(,T()k,*¡c*f ${ií{$tû{*hl)l{)¡\t{}rkTil{r\ìkri{r.i(*}i(*h\lk,fihk]|ofiil('$rì\H,$tft*,ßfiIrlkilt*vdlk\clk,vi rrft{ltf liH'r$j

fiAlt 3 l:il:'f '[.0tl0râMl'.lT' I $[iT,ll,ü{31?7E:FT'å $[::'I 'ß0{3lCI'lt-hl i lìË'f '['Û{:}4 Ö

f:Hüll i fï[iT '['0[::[::liI'7-[:.ftn i fiH'f 'F0[:ti'n

NüRH Ì fIH'T ;[:ÜT{?Ii{

h,X[rf] å |iiliT. ,[,0Irr.]5

lï'lT:1,1il I $ì¡;T 'l]üÏil^ïI:{:flIl I f:ì[:.T' '[,0/-trti)

SSiHt¡ I $[:T ,fl'OÉ¡üo

AI:rÎ ÍiEl' ,ft0[:]99$l.J I SiË'f '['0f:]91ï

t)$ T üN 3 f;[i]' ,[01]ü{;}$nl:!,I I:;ËiT' ,['0?F?

; ¡¡ags*:$*ri(ÏiìtÏrhk*)ik,\t,$,**il(h\t(¡tlt*il(,\crii¡c)fi)i(il()h.$,$,\t{lchc¡k,Ï¡\}¡{)l(y'()ft)l(,*)|(¡c*(*},l(*tT'þlI$ fltJfil:tl¡tJ1-Ti'lH TÄliHli CÊrliti üF' fi0LJNlrTNË Ër

i N tJ t"t t{ t: R I1 t.J [t 1: l.l ü I'l ü ñ þi ñ1.- r $ A'r Ï ü Ì,,1 "

ICIuËR 3 Ltt'â '8,J.37li

sT'A ,[,J..J$ Il"lvT râ v'll'll llSTll +1.;{C'3f:l\1..L !qil

'îl'lT 03

sìTì'fCI

$'r

L-ttA :g;1:J;r[:

sT'A {,J.3lifil.-trlf 'I'J.J';t'7

iLnÂfr'l'lJ[: E),il]tlN[:]'1T' 0f:i l-lIfil'lH[t 0FË[IËrNI] [:R0i'l'[L.T7li.

û AIIII ,il'I :l ( l)ll[:: ) T0 1'þlH HXf:'oi li: I i'll:i Tl{[i fì:l tii'l Ërl-ltl {]uHRl':1.-0tdÊ 1 P¡::'i'¡¡:,t.înl'T 0l'l .t !ìT'tlli[i I T rl'l ,[,1.37/4 'i !iTnll[: 'f þlti [ttr$tJL.'f 0Ft r4Itll I I' T ll i.l

'l T' ,ll, J. i5 7 5 .

tl.-0rqn 'IH[i $]:lii'{ üFî I't{Ë R[:liL,t-T' riñ{]i.l ,['J..J7F .i gìTOft[i ï T rqT' ,fl']. illli]J

"tl-0'ôn l'l'lE Cilliil::f:I{lIL:NT' OFi l Hl:tü F0tJli::R C)[:' 'll'Il]"Jt ( [it-[uHi'l ) l::llil14 '¡l¡:1.3'7'7 .í Íi'ï'0 1:l [: ï ]' lÏT' ,ll' 1. 3 Li lJ .

+

:1374r$:L.l7li

A :ll,

VÉrA '[,

sl'A .il,J. "{li[l

A45

Ln'â'ü'tl.i5'7$ iL-[]Érll'l"l'lH U(.¡E[:f- IüI[1]'lï Ul-

i[iItt$'r Fil|JJ[:'R 0[: ,fl,[r[r

i ( [:1...[:U[:l{ ) F'fi{]i{ '[':l;17ó .$TA +:1.::ï5Cl í f:i'l'tl[tH T'l' ËrI' 'll'J.3l'iü o

f:ôt..L []Hl{n íüf\1...L'ttËÌ'lü' F0lt ltUt'lÌ'JIlTl'lU't- tlir ,U, :L :l liÏl t t.. tl

'1 Il l'l"l Ë: l': ü L,f'J Itli Il Ì'l l.l l'1 H H It

ô r:Rili"l ,ll :[:lliil.$TA ,[,137óR[:'T

t $fr)ftil(.*sfi¡til(*yd¡khkàkil(4}|(*t(.'ft)fihqì\;ç¡q¡q¡k¡t¡k;kil.il(il{il(*shtfr**t(¡\;+|lt{,$sil(hk,'$.hk)tkIt )ß${il:il{hc;{il{il{)'t.)[í}li]l(]{(;i;{{¡t¡k$¡lrs¡cilis}ìc;icil(lc¡kÏ(\c,k)i{li(*i|(I($,v,$,{<¡tlk;*;ftfr$Eìk,$,$,ìktT't.lTsì fiit.JIJR{lt.J1':lii¡:: Ïr:t$Itl.-14Y(:ì 1't'lË Fllilìl.Jl..l' llt' Ël-.. l::'Ï.û Atl l.l'þlÌ'lHl'I ü 0f]l':l:i'ìT'I Ui'l$ .t "- -. *.- -.*. .

0t.,T'F tJ I üAL.L- ilAË ô üôlt¡:t I At3[: ttE:]'LJl:Il'l râNtt$ t..l.l'¡H F HHII .

CrqLL. ]'¡N'f û$ÏtiNrî|.- tll:::t,t-tr/ì 'ft I.il7C¡ $ l.-0Ë¡ll ïl"lH: l'1r1N'I I $$Ä 0F:'

íl'l'lË ft[::l:ìt.ll.-1' FRt]M'F.[.5,7ó.s'râ {'J.370l-I¡¡â '!'J.37[r í 1".(]'âtt fi I üN ôl'ltr {lUË.ntF'l'-tlt'J

i :t NFüt'il'1ôl'T 0l,l\ülT.IHË$ M'1NT'T fSÏirì il[:' ]']'lH nH$LJL-'Ii fin{l¡f ,[,][ li7[: .

$'f A ,F:1.371

ül\LL- n!:ìItiN ûl:il:tii''l'â1., nUHRliLOtl ( Il::' lìÌ{Y)i 'âNIl

$ I üÌ.1 üf: 'f HE ltE:5ìl.ll.-'f .iANIr TH[: ltH$tJl.-]' tMôi',11'ISj$'r\)i IN RËSil.trl.Jü: [:0[t14.tüTUH 'Í:ìFrâ[][:.' .

t $:t üÌ'lAl* ' lä=.' ,i |...ü¡ttr 'f HË h.X[:'. 0F Tl{Ht'rHE ftE$LJl.-T'.

$Trì ,[ J.370L-trlq ,l'lli5'7 4 ûl...0Atr TH[: $]GN ANt¡

i üuËl:tFt-0il l:Nl::üRMAl'T 0l'l

' CIF Tþ{T:

; HXIX. üF TI.¡fi: NH$UL.T.STrl '['1.57 ].

CALt- O$TüN¡tL:-ï'

i fr¡¡t¡kr$¡firF¡\)Kfitc}l{}|(flydÍr*}k;f ilflìcil(hk,$il(r(;{<hkri(il(il(,$il(¡v;{lt¡tï(lc*lt;*il(il(àt;{)}(;***

,í il<>k:,Ë*¡k*¡k)i{f{*}ft,'ft***Í(¡c,*t{¡c¡k.$v,{:ft¡fi¡k¡k¡k¡*il<*h'(;tk¡k,gI(.ÍÍt*)tt,T(h\hc)*hk**l¡cÌ(iT'H:tS l.$ l'þlE: MlìTN Fl't0tìft'1i"iH üf' FLn F'Ï'oi A[t[rIï'ïr]i.I.

CÉìLl- I:t..NCÉìt-1.- HF:'T

L.trí\ 't,J.37fi

,FA[I3 CALL üTITI

ü'qL.L TZtiRC)t $E't 'f kl0 F1... o [:'T. l'lt.JÌ'lll[:Itf].t ïHgìT F'ül:t ;:Ë:[(c)t MrâNl'.lSì{:i'tlì .

A46

cALt- liA'r'fiFl ûl::.tNIl 1'l-lË: f'ì(lrrl::FrüÏ[iN]'f:ii ill::' (:ftlitfr ) & (,1¡,FIl )i l:rfl(li'1 MrôN1'l:$ì$ÊlÎ .

Cf\L"L HXltl0 il:'Ïi,,1il [iXl::'o-'I]TI"F' É¡N[lirll.-.ttìN'Il'lE {\[]ErFf:Iül:Ii'lT'$ r

cALt- M'ìJ] i'ân[r 'Il"lH 00[::F']:üÏE:N'r$ì.CAt-l* 'l-'îil i F :L NIl Tl-lH h'T Gl'l 0F [:XF . tfirâLL- Nf)ttÌ"f û l,Jtllti'lôl-. Ï $ì[:: .{: rå L. L T' r*r lil i l::' :l: Ì''l n I' Fl [i Ëi ]: f.ì hl 0 l': 'f

H [ii [tH$LJ[-'l'.

ür1l...1... 0t.JTFU íT1l.IiFtt..¡lY []U[:RF't-ttJi(TF 'âNY) vfii.l.tll'l r*¡l'ltti ftHl¡t.Jl.-'f l:N H[::Íi I IiLJII F0IIM .

*Jl,{F [iIlnI] íIr:[|:if:'L.AY L]U[:tiÍ:l-.fltJi ( l:Ë Ëri'lY ) v !1ì.1. fiN r1Ì'{tt

i l:tl:;:SìtJt-T Tl',1 Il[:tiTMlì1.. Ëüftl"¡.i )|(**il{)ts}l{r'l{}i4ìcil(*ilrfl($:|(¡\à\I(}r,(il()|qrft.vìtr$ili\c**$}i{¡t¡lr¡t$c¡ft;i!*httkil(tt:l(àktt¡t}Ñì(¡k}ft,$v

û fr*>Ft,$t<,Jchtlk)|{).ñ){{il(,tll{hkl|tifiìctc,*¡k)rk,*,{ik,\c,\k}s¡k).k,$*il(,$;l\;{}i(,*ilt,t,'fc,*,*:¡k¡lc,tr*s*:*t T'l'l T $ I:ì t.lIi ft U l.J T' I I'l E {iìlr. F'fi ItÉ¡'IliiiS l:i ñL: þl {lF l'þll:íM'âN'rT$f:ìÉt$ Il.l'I0 T'L,lu'IHlti.l$. -Ï'l"lÏsì

¡'1H¡ìÌ{Í:ìiA=, f.t(2)¡t (,ll,l¡ll )t .+. Í\ ( 1 ) }lc {;[:]ir[¡ )0.

ig::;;fi (;1 ) ]i( (,ll,IJIl )l +.Lr ( :1. ) $: (,[,Ill¡ )0.

t I N tll'þl lii: It[,] f] ft Ili:ì 1']'l I.$ $tJlirlt0 t.J'l I ¡'iH F T t'lÏ1$ûÉ¡(:l) vrì(l) v[.t(lil) li $tl) +

!li'tT{ll-l I Ln'{¡ ,[,:1. i35ü

$TA,[,]:r9].CrlL.L- $$HIlSTI\,11,:l..5liË

MüST

L.n'â 'S J.,54Ii

t l.-0Ë¡n l'l'âN"T'T Íì$A 0F F l tt$T'i 0F[:[t'â1.1il.

î G[::]' l'tJU 'f Hl:{l'1$ 'i$1 üfiH l'Hli CüEFf:TII[rÌ''lTt( râ(:l) ) 0f:'i F l:tt!ìT' I¡ütJË:fi 0F 'FFIIt ( tit..tiUtii{ ) r.rT '[ 1359 .

t $TflIìl;:'rH[: Cc]fi:FË'ICItil'{'fi( A(1) )

i[J[: ;:H[til F0tJH¡i 0F {'I]$í ( HL..HVIii"J ) A'f ,!'J.3litI .i L-il'1il l'l{H f:ì l:ül''l üf TFIH:

t F l:RfiI' 0FliRfiÌ'll.r ( Ì'frlilTI$f]¡t ),tfj'rclHË: l:l' flT .tr,:l.358].

i l.-il'âl.l T'l"lE Ì'fiâtl'T'l.liii¡rt 0f:'t llHü0NIl 0Ir[::ltÉ¡Ì'lIi .

s'rL. tl

13:iüJ. .) -J ^.:

V ArIlA ,fl,1.:iliA

ñ+f\ ,ft

i f.ì[iï 'ItÀJ0 Ttrltl'l$¡.Ê $T',nRti'rH[: u0[:J:i::T{]IEN'ri( ït(l) ) 0F f:IHÍl'f ÍrühltrftiOF,ll,IirIir ([it..l;iUtiN) AT,[,].;ï58.

MnU A Y lit

$'l'4,ßl..li:i11 ílil'ilRli T'þlH f:nlif:FICIË:l'l'rt( 11(1) ) 0l:: fE:ftÜ [i0tjtiFti 0fi 'ü'llH ( [:.1-.8:U[:i'l) .

Ln¡q ,ß,J.345 í L.ilirìl T'l"lË: fl T üÌ.1 0F l'H[:t $[::n0NI] llFL.fiAhlÏ1 ( I'f ANTIf:ìSrq ) .

$T'â ,ll'1.5liH t $l'UltË: l:l' É¡l' ,t,1..3i:i[1 .l:IHT'

í t<f<¡t*¡k¡lr.*Íiik4ï(;ì{4{r'ft)¡i}kI(x*)Í{;s$à(¡r(.T(;r\¡\)i(*¡k,fif(il(*il(.{r*rfiI(,**il(Ï(il(ì\il{tk;ftiril(f{)k

STA'[,11]?;LC"\l-L I:;SìHIrS'f A ,[,:t:3lit:

A47

i,*)k,$il1htlh)H;ttI()i(¡t)rCl(il(¡C)|(,$Í(¡C,\CtkE,$Ï{;+(¡CIthk¡\hkilCrk)'Cil(S)rkr'F)Et{ì(il(I(*il(N(Ithl(.\k,*,*il(i Tll.ffi $t.lIl¡:(tltJT':tl-lH r*rIltr$ I l'lE (lilli:Í:FÏüÏË:l'l'Ii:ìt'fl{Ëtï I'lË:rIÌ{Iì ( A(ïJ)'{'fi(f) )

'tNIlt( 'ì(J.)+'F(.[) ).

M'qIll t [..0rqn TþlE ü0E:FF'. f\ ( ? ) .

í t-0Ät¡ 'rH[: c0ËFF o Ii ( I ) .

t"\ilil râ(I) l\l.lï1 fi(l).â$Tilfttï T't-rH $l:ül{ ANIl 0UlïltFL.ütJt INFü[tt'l'q]'T{lN'lT'['1."{7|i.

I'10V A v [.t

$'rrì '$:137¿, ;$T'0Iil¡i Tl.lli Rr:$LJL-T fiFi

'â (.1,1 )'{'I:t ( ? ) A't ,ft;137C¡

$Trl ,ll,L 3¡â,O i Ëìl.ltt ËrL$ü ,qT' ,ll,tl. JrâO .¡-[trt '&:|.:iTIi'â t l.-n'*rI] TtlH f0ü:f:F'. /l ( l. ) .!iTÉt '['].3ó 1

t-Ilrn¡ ,[,].3i.jll il..O,ân THË t0trFl.'. E{(J.).$T''â '[,:[;i$:TcÉrt_t_ AII ¡ 6trll

'â (:L ) ANII B ( :[ ) .

i'l0u râ v FSìTdì 'ùL,I'7'7 i ST'Clft[: 'f H[i tt[:$lJl..T 0f:

t ll ( J. )..1.$ ( J. ) AT )1137'7 'ST'A +13'âJ. tANIi 'âl-fltl É¡T ,[134.[.RËÏ

' ****)|(*)f,hk*)ft*xl(*8tcÍ(*tc)&,$.$***lofthc,tloft*)kt()ftir.*)f,s*,\k¡k¡t,$Í(*)ft)ft,*

Ii,J<)kil(il(*Í(il(*)i1**ì\*$,$*t()Ffr,\t,*)ß,\til(/,i)ß)F$iiloH*,$*l*¡!ç*s,ftak,$*)fi;Jrt(;ìt;*;**siTHIlS ÍìUTIR0UT'Il.{H [:Il''lIr$ ]'l"l[r {lTt]Ni0fi TltH Ë:XFüNHl'l'ï ül:: 'fl'lH: Ft[:lìUl..T.

1...1'â ,['J.3ïi?Ëì'f Ë¡ ,[ ]L 3C¡:[L-tJA .ü,1315u

$iT'A +J.3ó3üô1...t.. Ë.tI.t

gìTÄ {'.lrT7F

,'TÉ¡trl L.nA'[,]lll75 il.-0'qil TþlE: EXF'. Of; Tþlüi

i R[ist.lL'r.S.'fA *J.3CrLl{VJ: f\ v 00$T''â ,9,:1.:5ó.5

côLl.- A[r i 'tnn THË: [:xF' . tJ Ï TFI u [:RfJ "

$T'rl ,8,1374 t 5T'ünfi 'f l{H fl Ï CìN l\T ,ll' L -{74 .RE]'

i )ktkI(l(il(,$)&¡try,*t\r|(*;rkÍ(il(**/ß)ftÍ(ï(fihkI{Í(¡\;{,fihkilrÏ(f(,tìk1(,$tt)|(*sÏ()ß*Í{il(ìtx*.

A48

i il(*ìkil(tc.l1il(¡t*/r)fi,t,T()[;rftSX()$:¡\lrlkili,fid<;fi**¡k>l1¡k¡shklk.1(il(il(]ft1{X*;ß;ft;ßS:ft,r\,,{c¡$:¡kt T'l'l I Íi {ìtJIrl:t iltJ l' T N I gi l.JB I'It'ô{:'Iii ]'l'l H i'1 T i.l tJtiÌ.J[li ( i"l¡tlNl'T$ì$iÉ¡ ) F ftll14 ;{[:ftU. 'Il"lEi.l i]'tL-t-S;í lrL0AT'It{Lì Fnl:N'T' í\ïrI:r:tT'IUi.l.iï'l-lH LJþlü1.-H: Írft0tiìftrqi'jh I$ t.J$[:ï] ï:Utiî FL0ôl'I Nü 11il r i,lT StJIrl'fl,ùü'r ï clN .f

t ffiË'f 'f t,Jü tJIt[il:tÉ.ii'lIr{i .i l-û'til l"f I Nt-lËl'JIl ( 14ÉrN'Í l:$$ÍlÞi FltuÌ't {':l:515:t .

STô ,ll,;1..5ó.:f

MUï ûv00$TA,[,J.lici.[{:d\t_L l:ìl.J Ë $l.JIl'tft l-¡CT 1'l'lË M ï l.lU[:]'¡tr Fti0Ì"1

t ¿ [;[tü.MOU A v Ii{

$TA +J.3liíl r$T'ü[rË 'r]{H RH$rJr.-'r 0r:i $lJ[r T' ltn t:]' I 0¡,1,

.J¡'IF. FAI].I.:5i **)¡r\c*)¡c*)fi$)ftÍ(rkil(fr,'fi¡tE,*)'kil(*hktrï(Ii,$,*,$f(l()k**,|(t(àcr(+)fi,*åc*Í(ï(***Í(X)lrJ

i )ft,$tk*,,l,.hc,tà()ft$,tr*t(rkifiifiißhcïiiliàkhk,fixil(r(*)ft].F*rft*tt)ftIr)$:,$I(hc:rk>lç*.Y¡l*il<¡t.{,<.,|qñdr

î T' l{ f fl {i l.l Ir ft ¡;¡ ¡1 1' l r'l tt t¡ ti T H tt Ì.1 T i.l H l;i

tTþlE $Itil'l nF H¡âN'tï$SA ülrt T't.il!: ttL:${Jt_'r

TAFI L.IlA'4'L376 il-üí\tr TþlE t'lÉ¡l-lïI$$É\ 0Fû l'Htr F{E{:ìl.Jt..'f .

$'frâ,t'13cil.MVï rt v 0OsTA,$J.3C,:rCltLt- lìïl t¡âI]n Tþ|E MÉ¡i'{ÏTS$l\

ô UïT1.{ 7-Ënn.$T'1 ,FLä7F igìTU[tti Tl-lH l]IGi'l 'âT '[,1.57[r.Rriï

$ ttil(t\il(il(,&*)fi,.ftil(,fi)i(x(*,*xil{ìt?'cï(il(Í(*ï(t(tkI(.$*.$***}lí¡c}F*}l{tc*tcà\h\àtil(I{,{r,$)&,ti$$r|;fF.J,{r$fÞ{,:lrt|jl|i$$$r|,$:li,lr:Þíj4i1,+'S${i,l;,lisss'sl;l|rrj$,ljsslj$g1ili$'#s$$$

ti t't ïl

tif)t.J î CAI-L üÉ¡11

l-Ilrâ .$ l. ï:i3

A49

í ,:|j$:b$rþrF{i!|i$||r"ß$$Sflj.${;${i1;if':li$Í,r1,$$Sibrli:biþS$iþ{'+$'J;:F{,$$Ìl'SlitF',[$|lt$lß'S

i ¡t¡k¡t>fc¡k¡ft}|.l}l,¡\trhk*\k¡\,{}t,vht:ì\x{iKàk$t(ii(,,'trftili:i(tktt¡k¡crkÏ(il(,t)l{,}F,'hIklk*I(sli{¡lii(*lkl\rf I(

, F'l...tlr*rl'T Nlj [if] T i.l'ì- i'jUL'f l. l;11.- I [:lì'Ï'Ï ni'líltÊ,1ìl:l1tJür ÉrHT'fHi"iE'ì'Iil (l"lilnLJ[-]: i Ió &lli)v

i il(,\c*tr{$I(Ir:,ftï(il{*il()k,$,v,*¡t*î{}i{*,\\ìkIlii(h\.*$hcÍ(}i(,*H.-,fi)ir*¡t}kfrhk*Í(il(hc*I{¡c*itrI($*tfti FItüliltrll'1I:i Si'l f\[tT'îì t-'[iili'J l.-l]Ë. '[,01.r;]Ö 0F TþlE

t $rrti-f3û t{ l:(:lT0{:iii'lF't,JT'[::Ë .Ê il(il(fi)tk¡k.$ttril(ñI{hs.$iftÀ}fttk*}fi$rli:,li}k:{il(¡cilçtI(}ftÍ;hk}'ftr¡{il{¡kli{I{rfi,'fi¡k¡k¡k)l<Tr$¡fi>lr}i<,*hts¡\)f,

i il($)F¡EïoB,\\,Sr+.tt.*¡k,Vil{)K,\k¡t#tttkrfi*.il($il(,kìc}i(iliil(*I($*)NÏ{ìcI(ik,*tk.1(;.fiI(;#hktt¡t¡\¡klk,rt$0titì :[:0Il:10

î *il(,$il()kil()k¡lc*>ft¡lr,J<,rttc**I(}l(il{,*i|\;lk**il(t\]fiil(v,1*,t,*,$Ï(*iß)it}fivr)lihk,$tk;)k¡k,\Ììq$¡\¡k¡lc*¡[i tlRili'f fi[:I'$'

[:].f l'l $ET ,[,04]:i/rC¡âI1l {:ilîT'tr'0rt¡rl0I'fË¡ll I $tiT 'll'0Çtl9l'rqF'l $l;:l''Þ0I.rO.$

MIJ I $HT ,!'ö{390Éìl1i f:ì[:T 'll'0f;]ï9

S$l:rl:'l f:ì[iT'['0rì[]O11[i.Nlri lìH'I'llÜIi:lF'FFJCI.T i SìHt',fl,0[il:Ii

otJT'FlJ I 5ìËT' ,t,01:33Itl\l'Ël"ll $i[:'t'['üü7fi

í s)$:.tïo$.*I{,T(*ìq;'kï(tkx)ß¡k>lç>lr*¡i<.*,{r,$s,*tkï(ï(il($il¿,'{<;{)Kil(*,\til(*.t,$*x(,*fr}fi,**il($iloßr(Ii *)F¡g¡lt.*,*.**)k*)k,t¡sx,$,YàcI(s:\t)$¡$.il{)|r;fi¡c)t1fiI(¡k;#il{ìcrfi)'kÍ{ì\àt,$)'cÏ{,*,t,\k;'fiìc,'fi¡lç;firt¡Íc*iT'þlË: MÉìTi{ [1Rn[ìitl\þ18. [:0fi l:il.-. F''l'' ¡'llJL..TIl'rl-Iü]qTÏ0N.

v

,I- l'Tl.l I üAl.-1. CrtIl i tì[:T T't]tJ {l[t[:tirqNI.rfi

tAL.l,- [-t'l'Ï' íi',lt.JL-T':Il'¡l-Y'âi'¡[r I'lüRi"lñl-Ï$HCf¡L-L- ¡::XI] ûFTi'Jfi 'fl{[i HXF' 0f: T¡'{l:

i rtHfllJ[-T'.cril...t- tlt.l'rFU il:r:t$l:tl-AY Tl-lt{ 0uHRril-c}tJ

i (:llî fll{Y) v Tl-l[i. $IËl'lt Al'lr T'FlH tt[:$tJt-]' r I'l; ft[il;tr ilt.JË: [iürii"î.

JMF ti¡liln íl:rr$FL-fqY ïFlH riÊ,silJl.-T' ÏNi Il[iilrl',1Êtl.- [:rJl:tl'l

"t **sÍ(l()F**)ft)H,ïI(,,fi88X,kï{tt{¡r,$rlr¡ßhkil(hC}i(il(r(¡\fi)'kI(4{iK¡vl()|(fr;{\k¡t*)ft,*}¡t,$)'1,$;s)$

A50

i *,vlkil{}rqN':,$}]kt(il(,*il(}ii)&il(¡\)*:¡krF¡fi¡|(>k¡c¡f:lt¡lr¡t:rt¡F,'fc¡k}fà\y,,i¡":,tjf4(,*¡kIri($:)f,\c),iiy,i,rk¡ir¡f,*i T'þlïf! l:it.JIJItill.JTTtlH r\t.Jl...l'Il:rl.-l:[::i:i T'tilfi i'fÉrti]'ïlllrir*¡i:ii f\ ANlt lJ , tll.lHltti ¡'.\;::¡N (.1:l ) )|{ (,A,llli )f 'f' rrt ( :1, ) ¡k (,[,Il]J )0,i f\NI1 l.:¡:*fi (:,1) hq (,fl'fiB )l .l fJ (:t ) t¡: (,11,[it]il)0.

iôil($:::;('l::: A(ÍJ)iktt(il)ï((,[,Iirß)2.f.râ(;,1)Ìill(:1.)*(,&nT.])1t +A( J. );.kft(;:)il{('il,[rti)l +. Ëì{ J. )ìcH( I )¡r(,þFF)0rí0R c ;::; c(3))fi (,[,I1|FJ)2 + {:;(:L).t('['[ilJll +.f:( :t )s(,|l'ËH)0.í tJþlli:ttiÍ U (:5 ) ;:i ¡t (

^? ) il(It (:li ) ( l'l-lÉ: f'¡'l¡:¡::'¡::'. Oti'

i .?*t¡{llÀllil:t üF ,il,Iirl1r)y [')(i!)* rt(:iÌ)y,{F(1) .{.Ët(1)),kÏl(11)t ('f H[r ü[]l::'F'[r . (Jf: l -'frnLJ[::[t C]Ë' :fi:gg )it)(1) ;:: rì(1.)il{IJ(L) (l'l-lH {\0FlïF. üF O.-l:r[lt.lfilt fil-'í:[:]1[t ) +

f

Flii{.i Í [-tr¡q '[,J..JLi9

$T'A'[':1330$Trl ,1113,$1.

l.-n'l ,['J.lTliC

fåTÉr ,['L:i3f$Trâ,['.[3Cr3cf\1...t,- t.ît.J

M0U ll y Il

$TA,[,1:5t0Ln'q {,.1-TliIl

s'rA ,& L:3:i.3$'f A + 13C¡3Crq[-L MLJ

MO$l'

$TÉ¡ ,!,:[.iIi0LItrl +J.3::i"\

sïA ,þt.il31$T'A '& 1.:ICr lüAl"t_ i'1t.J

M0U É.¡ r Ii¡{rìTl\,ll.L.Jl3l-Itrl ,[':1..{$ü

STrq,[1.5ó.3cr\t-L Mr.J

Ë L-t]¡ttt 'f HIi üü[:Ë'F. 'ô

(:t )i liltili'î 'll :l:llïç .r' $ÂU[r I'l- ràT' ,ll ;[.T.Tö .

i 1..0'f¡tI T'l-l[: Cün:[rF'. i]( I )t I:'Itfll'f ,ll L ::TSü .t $'qVH I T'

'q ï ,ll J. 33? .

i ¡'It.Jt_Tr[rt..Y A (;J ) l\Nü n (:i )í'rc) Fütil'1 T't-il;: r0FF rr o [Jt:ii:;Ì*ü¡OtJ[:lt üf:',[,$IJ (l;il.-t:UliN) .i i'1UU[: T'l-l[:: tlfr$UL..T' üt:i i{t.,1-T II¡1. T ü'âT'T 0N l'ü rl--Ftlirti.t f:ì'l'0ftH I'f AT' ,[,J.310 .t l-0rlIl Tl{Ë ['[J[:'F'l::' . [r ( 1 )i FIiilM ,[:[.3lIr *t $AUli T T 'qT

,t. 13i53 .

t l'ltJt"Tl[tL-Y ,.¡ ( ít ) d\NIr [r ( I )í T'ü F'0tr14 'l'llEi n0[iF'tr. [Jl::i 1.'"FUhlËlt üF' {,$Ii{ ( [:L..HU[:N ) .

i $T'clrrti T't-tË ft[:!irJt-T 0r-t i'lt.ll.-'f l: Fl- I C r.\ I' T fl N ÊrT',ll. J. 3l ü .t.qtïilftË: I T fiT ,ü,1.31î0 ô1..f;ì(l .í L_tlr:lÏl 'f H[: [][Jfî[:l::. A (.l )i FI(ili'1 ,ll J..lliA .iÍìr*tUË IT AT' ,&13.it|..

i¡'lt.ll.'fIFLY A(1 ) rlNll I:r( J. )t'rfl [:'0RM TþtH C0tïF'rr. üt:i 0.-ttf)l¡jF It 0F 't,ltir ( f:L.HV[îN ) .

i$T'tll:tË IT rq'f ,[.13Í1.5.i 1...(l'ôTl 1'l.Jfi ['(lF[:'fi. .t] (:ll )¡ ¡::'¡:¡('li"f ,U,:[.Tlj[; ,

tl'jt.ll.-TIFI-Y r*ì ( J. ) /lNIr Il(:l)tT'n [r0fti"l 'rHtï ü0Hrf:. [J[:t J. ""F'0141[iR 0[i ,[,FIl ( EL-HVü:tl ) .

U Ayllfì '[,:L 3il:L

MOV AvËSi'f rô ,ll' 1 :T:i .?

$ïô ,ü,J.,Tlï:l

CrlL.t- ltˡT'fiH

üAL.l- 14Ërtt

t-11A ,1,13'Ì $

STf\,[,J..5,åJ.l-Ilrl ,lt L l51ll0

$ l' rl {, 1 :3 C¡;5

Cf\t-t- r.lll

Þt0vgi'f rt

LIr'q {,11lil.T

ST'¡â '!'1.3TlCrlL-t- $lì[::F'

STrq {,J..3ó I

l-t'l\ ,[,J. S4l:

5'f Éì ,ll'137[ll40U rà v ï:t

SI'râ {'l.i7f:l-llrâ,ÞJ..iI77

A5L

û ST'il|:iH I'f rtT '['J..I?? .

i lì[:: I'tôltrt'Il::: [rË¡ül"lû (.)il[: Ij'l-' :t fi ï lïi'¡ 1'$ì

i :[ -'ij'ül.i[:lt l]i::,ü']iií TNt',n I'ht0 'rHRH

t râ ïr[r 'I þt ti c] {l H:Ë'F' ll {l T li: ll'l' v

i Ci[iNlr:I{'â'l l:;.1] l:N ' l{ÂJ'tl'l ' .î 't l-l T Ti

'à Il n ï l' ï t] i'l {l [: N l;: l:t

'l ]'l;r Il

t 'I tllt) ['f l H [:'Í:' . ( :[ ) 't '-' F ['l UJ F'¡tå ilË' 'ü'[i]l r ( :? ) 0*F'ÜlÀlHR Üliâ,11 fiI,J ( [:1...[::Lrli:Ì'{ ) -i ( T'l'l[:: ["flf:'l::F T t::L l:::ì{'Iïì üFî l. *f!tltlH:l't r4rl'lIl 0-"!t(lLJ[:Iì 0F'í,[,I.J]ii Tl'l 'i.iûll' :f$ì lt[:rTl.-Yi 'f Hli t:LlliiIil'i T (: T l;:l{l'$ ilFt Í1.."¡:1{)l,,lHtt rtNIr :L -'FnU[:.ft flFt :U:$l;{ Itlilììlþ[:(.:]' ïUHt"Y y .li''ltl'þlË: FrìR'II'il.- FR0[rlJf:]'$ *'F .[:'', ) .i t-U'âIt l'þlE: [:t][:[jF. 0Fi 1"-f¡ilLJ[:'Ë (]F :llï:{}l

í fiHN[:l:t'à'f [:11 f N 'Mrtl.t ' .

i l.-il)'til 'f H[: Cüt;F'lî . 0Fí?'-Ftll,lHlt ( Ti{ [¡Ë¡lt]'IÉ¡[-i FtttllrLJËl') .

i í\ilil 1'þlH c0EFIi, {)Fi ;.1-'FtltJf:l:t ( I i.l llrtfiT ï É1t..

t Fliu[rLJüT') Ai',1il 1-Frl]LJ[:Ri ( G[:l'l[:ft'q'fHl] TN' l'lËtÏ¡' ) .

t $'f 0ËÊ: Tl'{H: Ë[:Í:ìUt-Ti ilF ílIln I T I0i'l r\T ,tr,1374 'Í [-fl¡ll] Tl-lti ü0HFÍ:'. 01"'

i O-F'cltJlil:t ( ït{ F'.1-')i H0Ì'1 ,t'rl .{ü3 .

í lr.tNI¡'ttJ0 {*0F rrf:'ï üIH:l.lT'{:iiIJF' Tl"lti l,lt.Jl"l$[il:t 'l]' '[].39l[oi $'l'ilft[i T't{H ("0F F tr , ill-i 1*FilblEl:t ( Lìf:Ì,.l[::ftÊïE:I1 Ï f'li 'ÍlÍilll¡ ' ) rq'I ,[,:l:Tí]:t 'i l._1.1"\[r 'r]{H iì l. Lìiil {lFt T't.tH f:ilHFF'. [JF'

t 0'-f¡ü[,Jf:Ft (]F '[,[tF ( I N [:' . [:') .û Fltn¡\'¡ ,ll:L:34:i.i $ï0fìH T'f AT ,S:[ 37ft .t l"l0Vlir l'l-l[: il0[iFF . 0Fi û.-llut,ll..-H nr: ,ü,I1fi

(l

iltifJ

r.l I

F I't'l Ë:

( Hi..!:Uiri.¡)

lì',${'J.37Él

i ( ütrNlîl:tA'It:i t:'[iLli-1 n--nfr $l'ilft[i ï T'

t l.-ü'1n TH[ri ilF ö.,-F0ldt;.i ( G[:ì'lË:ttrâ'f t:

, fsðr-t1 / \it;)l: r' ¿ .t\-.R[iu.

:.J7Ë.t

;1, IJ lir

'MrâïJ' )

ilTNü. T'u

'â'f ,[,].

fl{ltïFfrft 0t:[r :tN

A52

$T''â 'll1"iS3üÊùL.L At¡ t/ìnn 'il{H: c0rir:'F. []F'

i 0"'Ftlhl[:It üF ,&l¡Ët

i ( $[iNlirltÄ'l'HI] :t l'{ 'l'f fllr ' )

trqÌ{n 'l'l"l[: çl]F[:[:. 0Fi J.-t'nüdIR t]F :[:[Jt{

t (tilrN[:.rtrl'l'tr[r :tN'f:ìsir;F' )

t SiT{ltt[: T'l{H lì I tìN 0F T'}'lH

í Allttl: I'TCli.l Ë¡'I ;ft 'l .I7I:r r$T'â {,:L:{7Il

l4üU A r Ì1ST'A 'I':LiT7ü t$'füRH THH trTNllL üüËFt.

i {lF 1-Irr:)Lltil:t ilF' :lll1fit A'f ,il J..57f.: ' ('fHIi:ì f$i'fHË Ël.NAL- tltJfrF[:" 0F:i J.-F1tlHl:t üÍ: :[:]Jli{ I N f:' .l-' ) .

RH

í Ì(*)ktd)Ë)&ht4¡a¡a¡¡r¡,$:krfi)fiï(il(hkhc*¡kñ.T(¡c*il{$il(¡k:,ftili}ft*il(¡\il{¡c)ft*)k*}KNil(ht}'chkhcl{.$,ti T' þl T $ Íì t.JI'JIt U tJ'l' I i'.l l¡r N Cl lt H rq L. T $i L. f:;

,N0Ët... I [-rrA {,1:Ï7Ë¡ itCI'ôn I'FlH {l0liiFF. 0F'

i .?-rrüuJ[::l:t c]F ,[,FJß ( :t N It . F )t F[tc]l'4 ,& J..{7/ì .

$TÉt,[':L,TS:lMUT Ar00sìT/l {,L 3il4 t$T0n[: ¿l:iF(u 'E]' '¡'1334

t (L.UC. IJF tì[iNË[t'â'l'[:l] ËXF. )

Ì{UT /¡¡;l!:fi$SìTrq {,13ó3C'qLL i'fU i i'ltJl.-T.t f:'L-Y 'f H[: üÜ[::F F . 0[i

i"1.t-F0[ü[it:l cJF {,H}1 ( rt'l F'. F )i hJT'fH ,U,IllJ ( [:L-11UEt{ ) .

$Td¡ +13.Jf:' i$Tü[iË: ]'l{E sÏGNi I Nl:0Ri'J'1'I I tJi'l Ë¡T

i'Ini$'f TIi üVHlii[:l-.0i TF YH$ v JUi'lF '

'âilln ü'['J.33[:Ll.FCIIJFFI

VËRlT[.üUJo

+

ANÏ-,NZI'f CIuS'f Ër

LÏ14

o?F¡0uliftArIl,[. L ,l C.¡ 1.

,[,:[.T7C

S'f ô {,1. -1TdilCfìLL rlll

$TA'[,].;S.{F

t l-th4¡it iþlH Ct'lL:FF. nFÊl*Ftlu[:l:t 0fr l|itr$ (IN F.F)

i'âfrI'fH[: FRHU:inU{3iFltülttJü'r T'[:Hi'l Tü TH[: ü0HtiFi.ô üF 1-'r¡übJEl.t IJF ;[:fifir ( ]. ¡!F * Fr ) .i$'f0RH 'l'l'¡H {iÏGtl ÉrÌ'lI1 0UËItf:LütJí ItlFüftÌ"f'âT'Tüi'l ÜF râIlnITÏüNt AI' ,ft.[,$.TF' .t Tlill'T I F tlUHftfrL0tJ .i ïF Y[i.liì r *,lJMIt 'f:'0Ulllit' .

t Sì'f nR[: ]'l-lH ltHStJl.'l' 0t:âl\fiIlIl'Iül{ AT',t,:l.li7í\.

AN..,NMO(t'ï'ilt

TO¿F'VrlA :[:

?0Uliiltv I'l{

1:37Ë¡

A53

$l'rî¡ ,I,J.3Cr L

l'fUI A/,11,$Ii!:ìTrq ,[':1.:TC].3

Cr\1.-1.- i'ft.l t i'f LJl.-'f T F[-Ì' Tl-lH ]t[::f:tJl..1' üFi"\n[r:L]'Tül'l F'",H'¡]l ( [i1...[::U[:i.l)

"i flT'(liit::'fli[i fä I U;',1 rtri{I.r üU[::l:tFl...(]lti T tl F' il Í( l'l ¡l'I I U ì'-l 0 !:: i'j t.J l- T' I f] [- T ü'l'I T 0 N

t râ'f ,['J. il.5f: .û 1'E:f:iT' I F tlU[::ttËf l.-0tJ .t IF YË.5ri v .Jt.J¡'lF'llÜVE.l:i' .

$TA ;[:l.i("iJ::'

Al.lï 0:lJNT ËOUEIIi'lClU û r H

$ìTA.ü,1.3C)I i $T0l:tl;: 'l'l'l¡:: [iË$tJt..T' Ülr; MUL'f T[]l-Tü¡*¡'Il:{lN rlT''ll'1.:idl .í t,.0Í\[r 'f H[: [:0HFF. {JI:

i 0-'F'ilU[::l:t üF' 'U,nf:t ( I N F:'F' ) .l.-llrl ,[,t.T7t:

$Tfl ,[,1.T6:3

Cô'L-1.- '1il

¡6tril T'l'¡H IlFtliUTilt.l$ RH$l.Jl.-1'â 0f:' ¡'1tJl...'f I1"1...I t)râl'T 0l'l i'üt ZHtiU F llLlFR il[i ;llrlll:{ ( I l'¡ [:' o [:' ) .î tìTül:tti'T'1.18: ËiTffil,t ril.tI.r üUE:R[fl.-ütJí INF0fü4r.1'il:0i{ tilî Êtr[rT'rÏ0Nt rl"l' ,ll,J..5,Tl:: .t Tli{il' :t F .0uEFlFl-utJ 'iIf '/[ifì¡ ,.JUM[" 'CUU[:[I'.

$TA {,1.$3[:

fIN T

..tNuþtclvSi'f É,¡ t $T0R[r 'I¡"1[: N0Ri'l'tl.- T $tiIt

i l'lt.,MIl[::fi rlT '[':[.J,7$ .t LtlËr[r 'f Hh. $ I GN rqNIli 0UH[(fïL.0tl I NF 0Ri'f

"\'f I0N

í F'lt{ll'f ,[,.1..315l:f .

t-ïtA ,¡,1.3:5li

Atl T 0 L i f:;t i'l[r ]'1"¡lî ùri I LìN C]l-'

t THti F'l:N'll.- R[:-ÍìlJ[-T' .STA,&J.37F í$T0Ft[i IT AI' Jl'].*17[:'.RTiT

t s*)ftÍ1il(il(I(il#ft,*'fi)i('.i(ï(,\()Hlk*àc)H¡'()h)l(il<*,¡cÌ(,fiil()k¡\,$h\,T(,\cx(il(,Ïsrft¡\¡lr.k,Yils.*¡ltf(hci+{htrijil{]û ¡\>&¡lç41*¡(,$),k*¡c$¡c,*il(ift)'k¡cil(il(t(il{tc$(t(;Íil(,\kx¡k$tt,$rc$I{)f,yril{h(hk¡c¡Yil()¡cil(s¡\xhcil()f,il(trt Tl"lT $ l:tütJ'I T t''lH 1'Al{tr$ Ü'tRH üF' [t0l'JNI] I ]'lü

'àtl'lt.lMIflilt. ( ÊF:l'HR F'TltIì'l' 0ft $L:üül'1fi üU[:Fii:iL-0tJi rrËïËüT'I llN r N Tl.l[: siu[r[iiltJT'r NH ' l{0lll* ' ) .!

F()UE,I.i3 t'lUT É¡ r'['llü â'{,iìl' (T'tjfl) IS l'l"lË; l'ifil'lHl:t'âTlÏll llX[tCIN[rN'I .í $l'tll:iË: l:T',rt'r :[, I :3:]¡l r

í L0Arr Tl-lH $ r lìN t)[: ]'þll:i r[J[:'Frr o üF ]. ".r¡ü[,Jlilii (ll:i,[,$l'] (TN F'F ) .Fftill"l .[,13îrlt.i ÍìïilRt: :t T 'â'f 'þ 13li[t .t L-0¡qIl 'IþlH C0f:F ti . (JF

i J..-.F0uHrt CIl:r ;l!li{$ t TN F'.F )

t lrtrül.f ,['1.T7{.] .

$iTfi ,t':L:534L-n'.¡ '[,J.37I]

o?ü0Vl::[trl v lit

'[..1.37ó

Íì T'

LNJ.:{5fi1:J7u

A :l'

A ,9,

A54

$ l"A '['].:Tf;ll1..[trl ,ll ].37Ê¡

$'fA,[,J.3:iü$ALt F'EiNTI

$1'l\ ,b:l'.:l7Cr

CALL J'¡qfl

i "?-F'$hl[;R [JÍri FlttlM 'll'1.:r;7¡t .tfri'fillt I'l' rt"f

,[,TIIJ ( IN Í:'o F')

,t,l,.i SliC .

î $'rilftH I'r AT' '['J. -ïlïr¡ +

i l.-ürôIl Tl-ll;:. fi(l[::t'Ë. []F'

CÉrFt I tilr\ '['J.:Ilill

SiTdì {,137[rR[:T

i ,R***hk)|<*)fi)kilo,c}fi}fi¡(.,{*;*àkh\,fi1(}!(¡\)Kil(il(Ï(,$rfi)kI(r,k¡ft¡lt¡k¡k$¡lqlkil{ltiil{)t{,t}'Kh\#'$t(}k

"t )$*ñ

' TþIÏ

îô N

i Irti:l'

UOUH[t Í MU.l. l\ Y'E'l[ J. i'{,:[].' (cli.lË) T$ T't"ll:

i ü[:N[:t:t't'Ititr trxttüN[:i{l' .

i Ii'I0l:tË: 1. ï AT '['l[.ì1.$4 'iLü¡qIl '1'l'lH EìItìN Cl[:' Tl-lH

iüC)[:.fiËo ü[i 0-'f¡üHl:t 0Fi{,$B (IN [:'.F') FR{]¡'lí'&L37Il.;SìT0Iiti T1' r'1T :g:13liFr

i t..0AI1 l'þlti ü0E.t"t' . [][ji 0-F0t\tlil:t nF '['ß$ ( I I'lÍ¡ ' F )

; r- ftni{ ,l} L 37H .i$Tü[ili:: I]' AT :[:1.51iÏ'r+

t L-tlA[r T'l-l[:: C0HfrË. 0Fi l, *F'0td[if{ nti '['FIl ( I l-{ [:'* [:' )i r:R0M ,ll l37A .t $iT0RE: I T AT '[':[.55ü .t Cjr:1' ff üLlNiiHIl NIJ'r'1lJ[:'Ft

i 015 'rl{H nil[:FF. [lF'

i 1-F{ltJl¡::R 0F :!:$ft o

$ï'q ,E'L '1..{4

Ltlrq {,137I.t

$TA 413lit3l.-IlA,t'137[i

$TA +L:515Ï:lt-l.rA'F1374

sTA {,:l.:i15ncAL.L Ft:N[l

t LìtîT R0tJi''¡[tË:Ii Ntii'f Ill::rti 0F T'llE ütlH[:['I r]:t [iN]'i ilti :;l-'Ftll¿Jl::[t {:i::''ü'}1IJ't l*0rlll T l-lH l:(üt'JNITH:ll

í l',lU¡'lt+[::lt F R011 'll''L::ilil] 't$T0ltH Il''âT'[']..:l7C¡'i F I Nl.r l'l-lË $:1. üìl'l ilF:

¡l't'l[:: 1'1ñN'II$lì'ô 0F T'HË

i ltHf:ìt.,1.-T.i $ìTül:tH: I1' A'f 'il'J..ï7[r .

)l{,k il{ * h\lsil{ $\khk fi)ft,**¡c}l{lkil()ll¡cìc.{ It*,$il(,v,\l{ Ñ* il(}t(;ia¡;,1¡¡t r$ $:ii,*¡cil{}fi l{*:fi*

lì tlt(tltiltrqi'Jü''frâl(H{ii ü¡âfiH üF fi0UNIl:tNtìrlulirnAF'Ih.[t1't{IRI]o[tFtlUF(T'l'{0UH[tFLüt''Ëii'rri:ru TÌ'J l'H[: lìt.li]ftilt.Jl'rNl¡i' i"l[)ft["''

i ,$)s,t{il(il(ñ*,t *,$,,r,fi*lïoli'tloooo}i(il(*il(il(*'tht)$*,,k\k*ht**sr{¡kï(*il{hk'$"fi}ft'*tI'HI$ l:ìuIJIt0t.JT'Tti FÏ1.1il$ 'tl-lE [ixttt]N[:t'l'r 0F T'l'{[:

t MË.ìNl'I fi$'â ilF 'f HE till{ïl'l['I''v

[:XF I l...tlA 'n'13151 i l.-0'ln Tl'lE [iXF . 0fi; Tl{[: l:r T ItÍ]T' üF'Ë.FiAi{l.r

i frtiüÞ1 ,['J..lli:[ .

sTA.[,1.ióJ.

Lttrl ,ll :[;5i5;$

SI'4,Í,1.3ó.1c/\r..1.ANÏ O

MOU C

MOV Avll

$TÊt ,['l3Cr3LIt'q {,.[ 3.54

ST/\ '!,1..5{¡:lÇAL.l- tlI.tl'lOU l-lt ô

ANÏMC}V

STA ,t'l..J7ii

MOU HvC

l40U A v [i

Xttr.¡ [t

MtU fi r /.1

l'f CIV É¡ v 11

É¡NT 01rllltr tJ

AI],:l

v É.)

A55

t t-0'i\il Tþl[i E:X[1. (]Fi'f !{H ïìlïüilNIl 0Fli[tAi'.ltt.í Ëfttll'{ ,[ 13li.J .

it\t1I¡'ruü Ë){F¡üNliNT$.t'rHïìl' IF iluliRË1.-0hJ.i l-10V[:'Iþlli: tivrliHl;:Lf]l,Ji T NË:'(ll:il.114'l'l:t.iN Ë fiCìi'1 rl-'ft[:G .û'rü #-[tL.ci., MilU[i Tl..tH [1i$LJt-T. üt, f.¡trTt.t..f ï0Ni Ftttlþ1 t]*ItHLì. J'0 râ"-l:tfi{.i o

í [-nfitr ] l{E liH.Nlä[t¡â'l'fr.Ir liixl:' .t FftCIM ,þ 1334 .

t AItIl Tl{[i fi[:N[:Ë/l'ì [:It [::X[1.í MCIUE 'Il"l[: Eì I ttÌ'l ÄNIrt 0U[::f(f:L.0lJ T irllr(]ltMÉrT T 0Ni lîRil¡'1

'q*RË:L'i. 'I0 H*RHti "

i l'ti$'r ï F tluHllË'1...0tJ .it{üuH'rH[: {]vIifiFl.-0ti I NfiüKi'f

'q'f T ilN F:'ttÜM rq-'fl[::(ì .

t Tü [i*ltËG "

î MüV[i T'l'lË n[:!:ìtJl.-T 0lrí filitll: T'IüN Í:li0l'..t IJ-'rïË.ü.t T'0

'ô-ltHü -

i $'f t]HH T'f ( F i NrâL HXF . );A'f ,[,1375.ítf0v[i'l'l"lH 0vH:ttF'LÜtli Il.lF Uft l'Jô'I l: Ë)l'l ('qf:T[:ftiËIft$'f 'î¡[rllITïilN) Ftt0i"¡ {]-'l([:fi .i T'0 H-.R[iG.i l{{lUH 'l'l-l[: 0U[:ftf [-nLÚ

i T NF flftMÊl'T 0i'l ( rIl':'f [:Ft

i $Ë.(:üt{ü 'î¡llIl

T T I ül',1 )

tF[tfil'1 ti*ftËt"i. T'0 É¡*ltËË.ÍËXüI.-UIJTUH üIT I,I:ITHi râ-[t[:ffi ANI] H"-[t[:fi .i MtlU[: f:'T NË¡[.. 0V[:FIËl-.0td

iTN[:üttMrtT'I0N 0lî llXF.iFlitli{ É¡--RH.Lt. Tü B-ft[::ti.

tl'jilUH 'ft{[: fiI{ìN nlNI] üUËft1:rL.ütli I NI:i(l[tMÉr'f T üN ( AF'f [iËi $F.ütlNIl 'ôlltrI

l'I flN ) lÏRill'¡ l'l-fl[:ü .t Tü Ér-liEiG 'iFIi'ln I'Fltr. "ciI{Tl'l 0ti [i]':i:'.i I:'0ltl'1 friTüN Ai'lil 0UEtt[rl'-tlt'Ji I tlF(lÍtMrìl'I Cli'l {)Fi [XF '

n'?

[ivA

t'lCIU rl v Fl

fiT'â {':[374RET

i ¡c*)Í(il(*s*r&:ftfc;ft;s¡k*;fiÍ()|ok*:.,t(hkl()$htfl(]fiÏ(hq)fi*t{x8sx(,Y¡tsI(*hkEÏ()k**Ï(}k¡\I(il(;fi

À56

i stktc)¡ttc)l(;stcil(¡\,*4{ii{¡qì\il{}c.*ïr,$\qs¡t.$il(il(,\t(h\/,(h\*ilc*¡t,Jcltrk,*t¡;\t*ilorttkht(rkil()|ilÑx¡lt T'H 1fì liì1.,[tfi{:llJ'ï'.r i.¡Ë 'f tr$iT'lii tlFl[:T'l-lËR 'lNY üi{H 0Ii Tl-l[:i l'1'tl't I I $ì$ì'ô$ l:fì i:l[iR0 . ï Ë ?[::ftn 'f ]'l[; [iXIlOi'J[:l'l'l üF Tþl[:iCüft[tl::$iftf:]¡'ltrïl.li:ì i'lANT'IÍì$ir.l Tl;ì fitiT''t0 ïË:Rt].i T't{ I {ii $UIJ[(L]t.JT'T N[r T $ ilñl'-l...titr I]LJtì L l'1fi [:'l- . F 'f .; 6¡tll I T' I 0N¡'fiì t.J ÌJ'l'ft rt[]'I T U N ¡::' (l fi'f Ë!]T' T N ti Z f::fi Ü-i lt1'âN f ï $ìÍ:ì't$ 'I

T'f [::[t0 I l-ïr'à ,ll 1:Tlj0

üFr 00

í Lüd\tr 'f H¡:r MAN'f ï Sï$ì/ì UlriFI[t!]T l.)l:lti.ftANI¡ l.ift(]M *1.31i0.t'i'E:$T It' ZtTRU I$ 0Íi l'HHt F'ilRM ',00' .

C,'T. FI2:H:ftCI

ËF I '1,öF

cr nzHrinL-trô {,1lli::

CFÏ ÛO

J7. nZElrC)ct¡ï {,0F

Ë ïf:ilrt:$;ô fiür(i TF'

i LOAi $Eüi 1'HSí ri0R

¿ trltTTMIZ[:Ft[¡ I'ut,tût'ïMI

0 r t'qLL 'lt7-[iFtt].F Z[ifro r$ 0F Tr-rE0l:' ' ,(.l v {l'ttl- 'R;l[:ftO ' .þllï I',l'1i'{'l ï$flrl üf:

üIrH.[tÉrNI¡ t"ltü].f '[,L Sli? .F ;t[iftü rf] 0[: THlrUU' ¡

Jr. QZUftC)trHl'

RZHR0 î l',lU.t f\ v 00$'rA,[.13liJ.

tIF ZE.fi0v JtJi'lf' 'n;:Hlt{l'.

'T'E:ST TF' ZHTtü TS OF TI.IE

i [r{)til"î 'oF:'' .i Itr ZËRCl r .JLJi'f F:' 'f 0'üZ[:R0' .

t $TütiË Zf:f{0 AT [-0f: ' C]F' Tl'lEi HXF' . O[i TT{Ë f: T RÍiT'i 0F]ËlTAl.lll ( {l':1" 35 J. ) .

tt[:TCiZËFtü 3 IlV.t Ér r 00

$Tê' ,I':1.35:l tllÏ0R[i i:tiftn AT 1.0{:. tlF Tl{F-i HX. 0F 'rH[: $F:CtNIl

i Ul¡ERrâNI.t ( ,¡,13li,5 ) .ft[:'f

í ***rtyr¡ir$hc8)fr{t()${il(t{¡t)Hil{**il()Íoßhq¡krvI(¡k*)lr\t¡c¡l)+r¡s\k)'\tk,t)ft.$,T(,*il(*)ft*,T(}l(il()l(lkt N0l'ti:l* ¿ :'.1:1' $T'ANï:lli Flltt'[]rqftl'TAl.- lR0IllJC'f'.; )ft*(ï(fiil{**àk,*)+f )|(};Iril(il(il(¡\*i*,.fthcil(à\ìc*,$}r.*tfi*}k*<¡k*:lc,rc.$)fis**h\,tx,'ft¡tàt*)|(hk)Fil(

ËNTI

A57

tF'L-üAT'T¡{tì F'il1:i'¡l' Ir.li\J.tlìTüi.l-Ïl-ll;: E:Xl:¡[t[:i:ìÍìIt)i{ tJÍiliT] ÏI:ìi ttËI:ì f. ilt.J[:. rqft:t T'l-ti'll;:]':L rl . ( i'rt)[rt.Jl... ]: I :[ ci fì, ], lj )

i;.fi**.fi)ii;ìths:lrfi:lc.fifi;fi*¡,I{¡\h'{ifi,*ìk,$rli}{i;ftrlilk}fi¡q¡\)$.Ï*:*)i4¡c¡\)ttht*)fìc,*;+rh\**/llkslt)frilihkh\Ï{It ItlTUt][(Ërl'Jli:: liìI'¡rrltT$ì f:'[tf:]M l.-{l(.: ' 'll'040Ö ilF' I'l'1Ë: $[rl(-'t30$ l,t :t üli ü [;t] l.1F't.J'l' [:: ti .

v

i tc[{g(,\kg}*.tc}h}i(hsil{*)fir'l{,i:ri<¡k,{¡ri(¡l/¡l¡t,tttt)liri{)i{}k,t)¡(\kht¡f)t(ltls¡\*l(i(*hc4r)i{8.v,'|(*)l{,lt*f{)l(,{¡t.ìkflllü ,11,04ü0

' F[tü¡'f $i[::]'lìv

$t.J I Eì[:'r ,ft0$i?t:

AII I frìE:T' '["i{:}??Irll(} I $Hl. ,ll,0I.rilt:C'TIl å Í:ìl:rT' 'll'{}l\ÉlÖ

r:rtcIl I $lÏ1' ,llüHHIÏT't::$I'3 fïtil' '['0f Ö{3

NU[1... i $[:T. ,[,0[rfjFJ

It'1Ìl'Cl'l I ÍlHl' 'fl'0ü7C0t.JT'FlJ t f:ìF:l' 'þ0U.3;$

t il(}l{*îrhc}F}¡qrsr.Íri(;l(}}l¡|rlt/,r}ií¡\,$i(l\,'kÍ<;\ti(ãc)ü*s¡crfrfhsl(rgìt¡ci($)l(i(Ñ)f,\q:*ìc}fii}k*}'K*,*Ï(I{,JdÏ{tT'l{[:: ÈlAIi'l FItüfittôi'lË: F(][t F[-. [:'T. I]TUIriIlJi'J.,

r:rrTI 5fi1;"[Ëi iTTl,',]f'.J'-i:i.iil^lili,ìi,,,''*,'o*

i A[(ti THlil].FF'fl fl'qL.L tlUF'1.- tü[r]'Tl-lH Uñl..lJti 0fi 'J-/lJ"

. $Hi I C'111.-1... f:t'f 'f i Í1[:'ft[:üfii''l I:'[-. [:'"Ï'.; i{Ut-1' I ttl.. T ÜÊrl' I t)i.l .

tr'1t-1.- ËXÏ ; F I ¡'l[r ]'þl[: [Xlri]1.{[:l'l]' .flAl-L tll.lT'F'lJ iÏrl:SìF'L.ttY T'l-l[: nti.lil.JL'.T' 1'0

i 1.t.lH 0t.tTt:rt.Jl. [rË:V T {::[:: .t ( ri{ ftf:f:iTJlt.lti [:{]Fii"i ).

JMIt liHür i [rT l:iF'1...,\T 1'þlËi Ft[:$Llt-]'t T i''l nHil T ¡'l't1... F[][ti'1 "

i )&il()Hli(¡cs,*'i(¡kil(ifirilI{i(,tï1},!f;j(#tt{I(;{riol,{,,f;'l(;k,$;Írfi{;shsh\;fH$hcy,(ìt*i}'c}H;fik;'f,tf(f}|(11i{)FiflkI

iT't.|.ïl:ì $lJIll:(0lJl'Ii''lH Ë:'Ii'{[r$ T'l'l[:: [:xl:r(]l"ltii'{'r üÍ: 'IHL.

û [t[::ïìLJt..T.v

L-rtA,ll,:135:L$I A 'll':t:ló:LLnA ,ü,:l..:lli.i

sìTA ,ll,J..ió.iürqli..1... $LJ

í t..{lrâ11 Tl'{E [i)tF'. []f:' ilIUIIl[:NI] -

t1...üf\ll T'l"l[: [iXF'. 0f I]IUI$0li'

i $t.JËI'nAü'l' l'l'lË EXÍ:'o üFíüIVIli0[i F'lïil1'l Tl'{¡l'r 0Fû ü I V:t [r[::l'{Ï:r '

[i.xT I

A5B

ÉìN I 0?l"tüV E ¡ ô

l(l{ltl ¡"fnU AvÍl

rîN l: 0;ll4üU tlv ê¡

l4ClU [r p [t

t"llA ,[,L il:T4

I3l'¡ì '[,]..TC"¡ ll.

l.- tr É¡ + L :T (t

{:l

$ï'q {,1.5C¡.7fil\1...1.- Éìn

l'JüU ô r Il

siïÌ.r {)

STÉr,!,J.3óLüAl-1... flll¡"f 0U l-lv rä

$1'rl d,137T;l'lütJ B v f.)

l'10U A v [i

Xftrâ [t

Mnt'ril

1::!ó;:lrIl

fll ,¡,

V 'ît

û'l-lxÍìT' I r:' 0tJHFtl::1...01.*J *

í ¡'liluË T þf [i iluHl:tF:1.-t]l'dt I NFüftt'jr\l'T tll',1 Fl:ii'li"l rI-'R[iG .i T'0 {::-ft|;iLi * .i l"lUtJlr T'tl[:: l'ttiSìtJL-]' 0lr

i l:ì t.lF l' [t rì t: T ]. {l I'l F Fi tl i'l I1t -' Il [: [ì .

û"1'0 I.r-.[iH{i. .î 1..$r¿rIl T'l"lti HX[]' t]lirl'llii.Rfil'[i.Ï1í Ilt.JR:t Nü l::'L o F "l o i'Ít.J|...'I I Itl- ÏÊ --ËA'r l. ilN l::'tttl¡'l ,Û l. iTiT4

"

â 1...{.}rqll Tþllï: HXF' . 0F' ' 'I /FJ' o

â F f(tli4 ,ll J.il9[].

û AII[r Tl'l[:: [::XF'. tì[::NH.[t/à'I[::11

i trtJl.t.l Nrì [:'1.- . F"T' .t t't l.t 1..l' T Í1 1.. I {:: A'I Ï ü Ì'{ l',Ü T'þlH

âHXF. ü[i ''.|.¡'fi'oi i'10U[:: 'Iþlli:: nttiÍìlJl,-'i 0l:í l\trIl ï ï T nÌ'l I:ftÜÌ',¡ IJ'-[tHCi .t Ttl rt'*ltliiü. .

t I'f ilU[. ]']'ll:: ttH$lJL'f 0F'i SiUIlTItrôCl':t tll.l Irlittll'l n-'It[:.tì .t'f 0 r:¡"Itl..G. .

V [¡vl\U Érpl'l

i AÏII1t$râVE Tf'¡l;i Si:ttìhlAl'¡It üUH[t[:'L-0tJi I NIïüFtMr:rT I ilN f. i'{ l{*'RlI{i ., TH$iT :[[:' üUHliË1.ütl.tMilU[:'f]'l[: ÜVF'ÍtFl-f]t¡li TNFüFtl.Jrâ'IIUi'{ F [tUi''¡ nl-'l:tËü.i T'0 Ë-.R[i{i. .t t{üuH l'l-lH [;xF ' 0[:' 1'l'lHt fi[iSl.Jt-]' Ftttl14 H'-[t[:ü.; Tfl A-[iliiü. .i lîTClÍtfi: I 1' rlT '['J.3,71Ì .; i'1(lt'Ë'fl"l[:: üUHÍlfL.{]14i T Nþ tlliM'4ìl':t üN ( rql:r'IHIii $u$'ì'R/tt)l'I{ll.l ) tiftÜ¡".|

û f:*.|:tËffi " l'(l $-[tE[.i. .i MilUH: 'f HH rlUË:ttrr[-fit^,i I Nf:ilft i'î'1'l' l: {.lN ('qF'l'[:ltíAtrItTl'Tni{ ) riË0Mi Ë*ftfi:ü. 'I0 râ'-fl[r{i . .t HXü1,-l.li:ìT UI: 0F( r4r*ltlifi .í t^l ï I'l"l IJ "'fl Í:: (-.ì . .i l',t0Vl¡r I I' T'(.) IJ*[t[:ü . .i t.ttJur:: T'l.ilii si T fiN fIi'l[r tluH[ilï't-{]þJi I N [r il t:ï ].J rq l' I il N ( íì F' l' [:: Ii,ì tr [r :t T' .t t]i'.I )

i frttüÌ'.{ l.¡*rt[:G T'u A"tlH['i. .

A59

$T¡â'['13)74

iliTi.l:(l l'l"lH ll.l'{iiN 0[: TllÏ: fi[:llLll.-Ï'.ül::URi'i Tt'll:i $f Lìi''l

'âi'll1 flV[:.IiFr[-ilt'J

i T N[:i]ri14É\'I I (-l i'l .i $T'ilftti .L I' í\'ì' ,[,1,J74

ft[il'i sg¡1í)k¡qlli)i{,gtk,ï)lrilr¡F*¡khctchci(ï(¡c)i(i|(*¡t*r.K,$¡cÍ(hc)l1ï\hcÏrI(i{I(ìSI(t{;'i(*hCh\il(SI(,\thtT(h\/dr'k)'li)üI

t]'þtIlì Íjt.JTlttilt.jl':ll.ll:r [rF:'Itl.:'lltil'l I:1.,.. [:'T'. i'lU1...]':LFL-:t{:¡àT':ttll'1.

l::Ì41'lilAL..l.-lirt\l'üt'li[;']:t'lü1']'lH:Cü[iF[:'ICIli¡{1'tii ilF ('llllIJ ) & ('il'u$ )

i f:'tt 0 i4 T't'l [:: t'1 Éì N 1' T f:ì11ì 'â

!ì 'ütqL..t.'F]ElffiîÌ.1l.Jt.T.Tl]t.'Y(r'L0f\]'Ii'l0l.1c}IN].).(:'1¡1...1.- I'lüft1.- i NtlÍti'lrlt- T $H: 'RIi'T

i *tr{Íitchsï.:1(l(}i4)fih\,giEri{ii{*il(il()¡(i(tc;\\l(¡l:.h:f¡kfr:rt¡k}i{hti{¡khc¡c,\c)fR;f{ihclthl*}l(l(\ti<*f*i,\¡qÏ(¡ti{v

ü )Kil{rï.$tCil(Ti¡rí¡khkil(¡kftrqil(tr{tc¡q,'l(ttI(il{*sñ.'fi*ii(¡ch\r¡qhtx)'\rtt:)i{ifi¡\*h\**il{*il{I(*,$)ßhCti}I:Ï(¡\diil(tT'l'lIfi $i,JIift0l.Jl':tNÏ:. G[:'fi] Tl-lH U¡ât'U[: Uf: L'/FJ

$ f:l:tüt'l ïl"l[:: T'''¡U[..];r f:ì'l'üft[:I] IN F'l'tfli'l'

ClUF.l._l MUl. rây{,ó0 i,{,rá0, I$ Tl-lti râttfrR[:.i$$ Ofii TþlF- l,-rlT'üt'l AsiEì T Gl,lHIl l::$ftínrur$üÍ{.

0tTJ

ËìN:!'

ô[tÏl

0r.J

LNT :0'

't {;1óa *rf-:4J. .i) *J *1 il.-0'àn TFI[: Mr4rÌ'{'f T$$É¡ Üf:

t Il T U:t $tltt [ill0¡Í 'Í':l ..ili2 .

í',[,.i0' TÍì ]'þl[: l\l]ilf([:f:ìfiìifl[i 'ì-t{H t-¡trTCt{ bJl'l:tüH

iL-ATüþll;i$ l'þlË: Ût)11H: F0li G[:I'Ï*í rNË fiixF:'. üF' ''.1/ß"

i'0?' TÍì 'rt{H ctlllti Fill:titìHÏT'TNtì'rH[: li'-xl:'. üF' r/ß"

î GHT' Tþlti [::Xl] . 0fj ' 1" /li'i' ,t $iT'UR[r T ]' II'f 'il'l[ 3?$

"ï ''4,O'7 ', l:Si 'l'þlH tÛlll¡: FtltTi ü[:T"r I l{ri'Ï'tlfr l'1Éìl'lT' ]: fìÍìAûFARI' ilF ''.1.¡'Ir:.t' ,

0LlT' '['.l. 4l'lUT Étv,ü,30

0LJ't {'J. ú¡fUI Av,[,0liÌ

0tJT '[,1;iI I'l 'Í'L li$r'A {,139ÛMUI rl¡,ft07

üLJ'f '[,].4I N ,ü,1.Ii t ti[:T 'f !'¡f:: ¡''ll\t'l'f I $5ì'q FÊrË'f

i 0r- 't/ß' ,f:i'f Ä 'ft J.:{l5lft[:T

û )&fr**àC.$)klk)Ëil(*I()k¡C)fi)¡\S)kilortX*)K¡\ÍiI{,*il<11:{¡c)¡t)t{il(I(il(*;il(l\il1,*Ï(il{lCXf(*ltâk)k)i{iliICt(,thk

A60

ir tcxl(x,$4ytlßx;1(lKx¡()K\(S)K)El(,Ytr1()Fhc,Ï,'t()Kli(>l($I(#l{H,Yll}r(,$:t(.lH)t(h\ilcÍ<¡ltf{ìc)N\r¡\w}fixxhtx'Ï}:tTl-ll:{ì l:tË(ltiltfìtl[:: T'[r$T'Í:i If:' Al'lY 0Nf;. ül::' T'Hr:: i'1rìN'IT|:ìl]tqÍì

í l:$i lHl:iOo .[[:' I lï.ftt] vNf.] ilFtiRÄl'l:il1'{"t ".**.***-

ir-RhT'Î [-tr¡à 'll'.L3i.iÛ tl.-il'i\il 'lt'llä ¡'1åN'l'1.$ìliiËr 0Ft ü Ï v:t trHt'ln FR['ìÌ",| 'ü'1.3liÖ .

cF':I 00 tl'[i$i]':tFr UHRül:!i clf:'Tt'l[:'$ F0[iM ',00' .

JZ Rltliï' t I F' Tti'lii I 'lLJl"lF Til " ItlT[''l' ' 'CFI :l[fif iTlif3]' IlÏ ?-[::lTü Ï$ üF'I'þl[r

t früRl,l / 0l:i / ,

;1fi0n*liTI,,,, i ll,nil"'jï;'0,''l;lI]Ii ,l$*,,on'ä'i' ' '

r å I.rT U l:!:ìilR tîftCll'l 'll' J..$lil .cFrr o0 i T'[iÍìT rI' ;:HIt0 ].$ 0l- Tl-lL:

i r:'üËi'f '00' .J'¿ ftRH:T' t T [:' YHÍì r .JIJI'4I] T'0 ' [tFt[i'J' ' 'CF f :[:Q[:' t f H$'I IF' U [rftU :tÍ:ì ilF 'I]'lH

i [rl]Ë¡'1 / 0F'¿

-l'¿ RIiE:'f tIF Y[i.lit -JtJÌ'fF T'U'Ftl-'([iT".JM[¡ F'r:'il

[tFt[iTl ..JMl:t .ì l:::{:ìT. i$ìÏ1.ì}'lÉ.)1..

.rü üË'Ï ü[1Ë:ft'âi',ltl$.

t *hq¡chkil{il{*¡t¡k)$:T¿trtk*il(,til{il(¡cilifi)f,)kiF¡\ìti,*ii{)l<i{\c},k)i{,i(¡\)|(d{}i{ì\s/,iiiliÏiÌ{hc¡ci(\c;t}i{'\ti(Í('$*#Í(i g{i,1,{i{i:üg.:liili$t¡i,{;s$${,{,ts{i$$$$$$'l;{it{;{;{;'li$$s${i{iiF{igi$:[$lFtlb:liiþili':lt:þ${i$$stli

Ei'{n

A61

AP-6 (c) : Listj-ngs of Residue to Decimal- Conversion

i {ilË,$ss'litli',li$,l;:|;$$:liil;¡:¡;:È,|':l'{,{r${iûr|;"1;#ilrf':liü:[:l!{ifr*l,f;:l;iÞs.#$,J'st:l¡rliiþili',l)'i;${jt¡r

iTl-ll:$ l::'litltìi:ií\l'j[:: ü;illJVl::.|:t]'$ì r'l fifi!ì:tïrl.J[:: lrL]i'i$[::[tt 'I 0 rl [r [:: [) :t i"l ¡â 1... i{ t.i Ì.J if [r lt ,i) (ttl:::$ìl:T:rt-l[:: r.llt.t'l'l"li"lliiT'].(.). Ì'lnnllt-:1. í J.$ Ë. lli )

v

iFl:tü(iltrt¡lF' fiiT'r+rl:iï$ F[{ili.1 l.-t]ü. ,llû[::[10 (JIi 'fl{[:: 5il.ll{.-'$0i M ï üt:iü il flt'iF:' LJ I'Ë: R .I

ilftLi 'll'ö[::i:r0

i rrllüÌ'l .ÍìHl'$ìv

$IlïU I $ì[::T''[,Oïï:tij¡f:itll liìH'I,['ûfi]ï[rMl.I I $[::ï' ,il,0ü]90

'qI t I:ìlïT' 'll'0t:]9ïC'âft I f:ì[::]' ,[0[;]ür:\

Xl'É¡[( I l:ilir'f ,il 0{}45F[]$ I $[::'f {'0$?:ÌN[:$ I fiì[:]' 'ftûf] L n

I'lMCtt.JÏ i läHr' 'tr,0Íìn3 \

T H{iI'l lìH'r ,ft01ül}I'll'l l' I $i [:: 1' '[. 0 [:] ll 7$1.-i.l I f:i[iT' '['0{:i4üIiF'T'l fl[:1',1'üil;:ü

i tksyrlktcï(tc*,$*¡lc;fii'\¡irrrt*il(*,\t:rfiil(I{*il1*rcÍ{Ï(il(.$)f,hlx*Í{,\tI()ftli{)ìc$:rc*lk;'lrh\:ì\;srfi\c¡cil(lc,$;{,'${*t(J

I ),Cà\,*tt)khk¡ct\,\t,t$il{,tt\,.+{)+rtc}k,*;lcil(Erik.*¡kil(,*à\t'(*tfiI(àc)h*s,.|($h\¡ir>i(¡c¡k}|C.f{#.$il<rk¡c:fi,\c*¡c},c¡q**

i'ïnüil' l:15 TþlH: l'rttilüFtËìil[;; t,ll"lIül'l I]Tfi[¡l...rlYIi 'Il"li;r [tH$l.Ji...]'i üf:' l:NT'[::[ìHËt Aft T Tl-li"J[::I'ï ü üF'[illô]'l:il1'l$ì Ï N IrËi{: l. þf

'tL-i F'0[ti"f .I

IItütr I üAl.-l- C'âIl t üÉrttl:t I ÊrtrìF- [tt]]'LJt(Nâ Ë¡NI¡ 1..ïN[:: I- E.lllt

"L.nA +I:.{70 iL..tJllll T'l-lH $Tt.ìl'l rtNIt 0U[::Rf:L.Utl

v -[l{[:'0ttÌ'lËr f :lüN F'Rni'f 'll,].ä70$'fA ,il,J.:37ó i lj'f[il:(Ë: I "f rô]' 'll':[:57C]C"\1..t- lìül'lFtT' i|:iTGNñ1.- t'lVlittFl...ÜtJ ( ÏF

t l\NY ) 'qhlll l:ì l:{lN .CÉ¡1..1... ü$fi[r i f:Tþlli tr[:U:ti"f¡âl.- I'lLJi'Ífj[:.R

t ilË Itti.fiì I ItLJE. NtJi'jïrË:l't .C/\t..1.. NCI-l[iT íl¡[([::$iHi'l'I ]'þlE: Rt;:$it.JL.T

i I I'l I:rti:Cl:l ¡.141.. Füftl'l '..J Þ1[i 1'[::{:i I' Ë lì ]. Lì itl Ll 1... 'T'ü ti Ë. T' Ü [1H tl rt ]' l. 0 N .

t )ft)ft)ftil(il(,*tctks{rtcili)l()l(}K*¡k}{o'c¡cttrrtil(il(¡t,ß)titkì*\\¡cilr¡c)t(tk)k,$.v,YìT)f,}tràl(l(*il{;*,tìkt($I(rkt(I(

A62

i ),(lt)t{l\X}tol(fi$Ðí11N)fo'(X+:¡tX)tt¡<X;t<#:¡Ki¡(ì\:ly}S:f,)lo$-ä(T1$ïi}lr;$,y,Xï(I(}i(.r-,1()N)irT(Xl{)K

i'FIlCf ' Ifi ll'lH l:rl(t'ìtu*fi'ôM[: tJl..l]:ü1..1 frTIì[t1.../..rYfjì 'Il'lH l:i[::$l.Jl.-Tr^f:'[-fiAT']:N{:ì FüLþlT' ôRT1'l-li.'lHTl:C t}l:11:lftr'1'Il:üi',t$i IN tr[:ü:ti'tr'\Li F'lJ [t i'l .

F'IlLitrl üAL-1. ü:Ati iCAltItIrqüË ItË'l'LJItN rjrt,¡tlt 1...I hl[: F[::HI] 'C'tt..|.. l{NI' Ë f:i l:{}l'l'll.- t i!::: t ,

t-IJr.1 ,t'tl..T7Ii t 1...üA 'tl.'lH ÍìI GN 0Ër

i'l-l-'lli: I'l'âÌ,1 l'Ti:ìiìñ {lF' T'l-lt:i ftlïIìlJL..'l' F fiütf

'tt, J. 37[i.ST''1ì ,[';l.T7J. tfì]'fll:tF_ T'f LìT +1;i7J..ti't1...l... {:ì{:l'lË'l i liì:t ül'{Al.- 0UIil:tFL..0tJ ( I I:

i ,4¡NY ) 'lNIl f; I {:ìN ,Cl'â1.-t.. Cfiütr iFIi.tn 'ì't-,tI tr[::t]l:t"tôt.. üf:'

i T'lllir [tHçì]:ilt.lH Nt.J],lIl[:f{,Cri¡L-1... l.lül.-1ft f' t F'ItHIl[:.i',,1T' T'1.{lì: tt[ifit.Jt.-'T' IN

t [rl;: Ë ]: i'f Ër l'.. rr ü R ¡1 ,Crâl..L Hl.-N i li l: UH " $F:'AllF- ' y

CÂll.L- ti[l'l í $ìI[ìi{'â1,- / lii*; / .l-Ir:1 'il'J.374 it-ilÉrtr I'l.l[: f5I[iN nF: [:xF.

í flfr Tl'lË fihÍ:ìtJL-l''S1'ô + 1:57 J. t $ìTtllltr .t T' ¡ô'l' ,l! 1::17 J. .L-Ilrrt ,[,:L:3;'i;i t t-fi'trl] l'HË: [:;([] . üti

tl'l.tH ttË$ut_T' FRü¡'l,ü'tt3Tli.$T''â ,ftL37$ ûSìT'0liH I't r'.tT' ,tr,:L¡T7C¡,

CAL..t- $Ct'lttï ä liìTüi,l¡â[- UUË:ftf:L.nLJ ( ïF:: i ll¡l''lY ) 'âNIl 5ì l: fil.l üF' [::)(f¡ . .Cfì1.t.. ilÌ1ilü iF:rÌ.ln TH[: IJ[iüIi.,i,ôt* NUi"rFtirt

i Tl.lË ËixF. r

. Cfll-|... l.{Cllf('f û fr[ttil:ìE:i'.lï' 'I].t[: HXF, , I Ní I1[i:ü;r t1ñt_ Ëc]fttj.

J14Fr l'[::$T ifil[ìi,lr'lL- ]'tl $[:T 0l-'HliÍ\T']:üi''lS.i *¡c*hk*)¡tïiÍ{I{,*,t*,'Id,*tt,$ïrIihc}li)$:*$frI{¡tfi:$ìtrk¡t}t*ïi)fi¡t{,¡ktt)l()Í{;*.*ìilfiï()r\,$,ì(ìt;tkï(il(ìi *il{¡\¡\*;,$.il1il{,*ü}lr}¡k,$il{xÏ{,Yhtil(il()ftÏ(il{I()Í{)Hl()|{+*¡Cr|{tqI(tt,$h\iÍ*}+:I(il{/,(,,',(il{}};¡\il(yt*}$).Ft 1'l-lE gìl.JIlltilt.JT'IÌ'.JH 'ütJil[r' il0i'.ltJ[:ftT'$ A tt[:$iIfiUli t{l.J¡iti{[:ttt 1'tl Éì Il[:ü:t ¡'l¡*rL. Nl.itlü[itï 'I

{:ncn 3 [..I.r/¡ ,[,J..37ó

$TA '[,1.íry' LÌ'1U ï d\ r ,ll É.rl\

$TA,Í,tliTg0CÉ¡t-t.. $I:ì!ift

i l-0ôtr 1'l'{li: [tHgi I [rU[r NUi'f fJtiRitll.llül-l IÍì 'Ifl nli t:0NVHË't'Hïli l:N1'0 ttti.CIl.Tô1.- I'lLJl.,lgh:ïti tl R fli.l ,[, :l .T 7 /r .û $ì'ïüftti TT' /ì't',ü1.T9J. .$',ll'Érl\' (T'[ilJ) IÍi ]'þlH Il.TVIii[]It.iriT'ClItË Tl''tT'[':l,.i90.i üriT T'l-lH ftLJtlI'l:HNT' rlNIli T l'lH t(til'1rl:[ l.JtrËlìi ,í L-0rll.¡ 'Il"l[:. $ I üN üFí ïl-lH nIUl:It;:¡'.t[r (]'t.t[:t tt[:Íì]: ilt.,ti l.ilJi..1ItL:rt HFt l: üt-lt l:f; 'f 0 fi[i [:0i'lU[iIi't[:tlí T'0 lil:ÇTÌ"f At... Nl.Jt'lttHrr ) .

t-IlA,!,1.ïli¡â

A63

¡î¡N ï 01Cl{f f:;Ui',|

L"IlA ,ll:l..i93

Si l'1t ,l' L .il C.¡ .3

MUï Érv,ll,0J.

tr l"A ,ll,1.TC¡ll

CÉ¡1.t.. Mt.J

M0U [i v nj

l.-I]'â ,ll,J.3tt:l

$'f 'îì {, :L ;5 ç' ILlÉrLl- $iliì$[¡

"$Tl\ {':L:Tó3

l'luï A v,ll L0

$ïô,['L3C¡LüfìLL. ¡'f l.l-

MCIU Av[J

STA'8,.l..:TC¡J.I'lUU A v Ii

s't"A {,J.363ü'qL.L f-¡ll¡'10U Ë r Ii

þIH TIF{Ë:U T üIJË; Fftü*Ë:[iM F'[t0M E*RItì.HË. n

û'fHf:ìl' l: 13 ¡',,!H$rì1':tVH .û l: F' Tfiflì p ..]LJi'f[¡ 1'{] '{ìl.Ji',l ' .û L-t)ílll T'l-lH [tHi'j4 ï Nï][:[t

; früt0i'f '[,]1...T93 .

i ' ,F0:1. ' ï $ì 'fH¡:: fiÊ.i:il: nLJ[ri ri[rF'ft[:$ìr:t'l ì''ô'I ]:rli'J ill::' $i l:x'IIHi'l .

i MLIL..1'I I:'LY ' {,01. ' rôNIl ïl"lË.tIt[ri'J"\TÌ'lnË:ñ'fü Ëf'lfiÌ'l ïl'lË:i ItHSì:t IrtJ[:: [Jli ]'Hlii FCllll'1 ' ûX 'ül¡Jt'llÍ.[tH X ii 0 'fü 9ot tf uuË. T HIi [rt:tüuLlt]'ï' 'rfi:Fil.f

û F'ltül'1 ü-'It[:(.ì 1't] ti'"RE:t.i ' .i 1...0'àil 'Il-lti ftLJül'ï [ri.l]'i f:l:l{li'f ,ll ;1.:i?:l.i f:ì'f{lHË ï'f ô'ï ,ft J. ii9 J. .í tì H: l' 'T'l"l[:. ü l.J il T' ï H:l'l T' Ër ¡'l ûi[tH¡'1Ér.tI1n[;ft 0l:r ]'Flti ïïUïnË.Ì,,ltli 'ql' [-f]ü . ,ll ;l.3t/:i .i Íl'r 0l:t Ë: 1't{ H: t:t H ¡'1rt T N Il H lti ôï {'L;5C}::l .i'{,rl.0' T$ I'l.lË: ftH$ll]t.J[ii R[rF[tÉ.$Hi'lT'¡tt'ì'].tll',1 L'lf: ( *Lli ) 't l'1 l.J 1.. T :t [1 l.- Y ]' þlË: ft ti l'f r.ì ï Ì.1 Il [r Iti [tY ' ,t,1 0 ' ]'ü f:0[ii"¡ 'f Hliift[iÍì:tI]LJIi tlF I'I'lH Füfi¡'li'X0' t^,l{[iË(Ë l:* 0 T'U 9.tMc)ut: T't.tH [iR0nLJü1' ïËttMi f:ftilùl Ïir-lttitì. 1'0 A*IiËtì.

i t"ttlví Il r.J f:)

t'tt)

HT'Tl'râ'-It

,'É\l1ni¡'i0UI T'l-lt: RËf:ìl.Jt-I t]Ë' 1âtrI:rTTT{lNí r:t;iul.{ H-.[tHLi. i'ü F:--ttHtì r

i l.-t.lr*rIl I'þlH {ll.JüI'T [1¡{T' F[tili'l ,ü ].3?il .å$Ttll:tE IT' É'T',t'J..ïóJ.,

t..llÄ ,['J.39?STËì '[,:L.iI$ J.

þlU:t A v ü:1.

$I'A ,ll'J.::lCr3

cAt.t.- MU i Mt.,t._ï:tF Li" oL' ( $ìIX"lË:H:N ); É.¡l.ll:¡ Tl-l[: ültJil1'TIiN'r T'0i F:'UF(tf T'1.{H t([iI:ì:t IrUH üf:' ï't{Hi tiüItl"1 'OX' l¡Jl'lËllf: X;::0 'l'ü 9.

l'¡0U l{ ¡ rlRE:'T

í )k)Fttli(às)fril(hcàcii*)fiil(*{il(x*ïr{(,$){{,ï;'Ff(il()+(iF:ìc{{}¡(àk*;.l(8,t;.!()ii;ft*¡kï(,$I(,$,$)i(,$,*¡\Ï(is*hkÍ{

A64

:ì {,\t/ù',r.'*}k\l')V\k\t,\k\þ*.ü{,*Vr\k*'rk*'*\þ*,V,,\k."çr.1,',klÅ,.l:rk*\k\ti:k\k*!þ\k\k\k1t iq.$,{,V*\v*{.,*y 4! 4r ,Þ ,lr ,F ¡F ,Ê ,I¡ rF ¿t\ rt\ ,¡\ ,F ,t ¡lr ¡t ,tr ,tr rI¡ ,l\ rtvll r¡! jÞ ,rr ,¡r rÞ ¿F ,F ,F ¿Þ ,tvF ,Ir 4ì .t¡ {r ,t\ 4! ¿t\ 4\ ,tr 4r rF ¿lr .ì\ ,} ¿F ,t\ 4r ¿F ¿1.

û T'l"lË f:it.JHItüt.J'fINli' $ìtJÌ4' {:0Ì'lUli;I(l'Íi'â l''¡Ë:tìÄTIUlii I'lLJ¡'.f IJ[:ri 'T'0 ï ]'{: FÜïì ï'r ï u[: FilRÌ'1 .v

ÍtUl"l ¿ t..llÉì ,ll,:L,l9il

STA {,J.,5/r.ïl''f UI f-r z {,[:'H.

$iT''t 'll L 3ó Icftt-t. l"lLJ

tt_1.)AIt'rH[i (.ìtJü'rr[:NT' [:'[tüi.li,11 13??.

í',¡'ç:'¡;'' ISi l'l'lË: ftH$TIll.J[ii t:'0tt ¡'l il r:' i'f T l¡ tJIi ü i'l H: .

i 'ïlJt-I'I

FL.Y Tl{L: ilLJ0T I trNT't¡lNIl -1. ({,FH.) 'f0i ütlllVIilTl' ¡ilË:tÌA'rï VH FUtlt4ínË l:ttrlìIIltJ[: ]'ü l]flSilT'l:U[:i f:'lJfii"l.

Mr)$Tt-n

v lìt

J.::T?rL;:lï3

UrlA :¡:

ô :[; Í [_fJûtr THH IiHl.lfl I I'lIrËrti f:'tiui"l ,[,] *{9.]{ .

S'f't,ftJ..Jó3Çt\t-r- MrJ t MtJ[-TTFLY'fHE [(fi¡"fuì ]:l'l-'

i IlHfi rqtlil -'J. (:[:[:L') T0, CtINUEftT' T't'lH I'lEü'lTïU[:i Ftllti\î ü[i ÍtHù1Ér ï l.lil[:fi I't)i rl'$ F0firT'l:vH Fütti'i,

t'lCIU râ r HllT'rl ,t,J.3î.1FTHT'

i *¡sy¡,1r¡fc¡lr:$ìc¡t)rtft)i(rftil{*t¡<hkfrf{Ï(Ï{fix}l(,**)rt/,{ili)|(.*¡cil(hc;fiIit(*,r\Ñ)rk}|(¡liìft;fit\,ft,\q)+r:ì\r+ofi)rij

i *ï(Í{*Ê*il{¡càt¡c*)|;),(tk}F)fiì\;{(àkìcht,{11$ï{hq$¡\**(t(tt*,1{il(,t*;1rÍ(yí}.lrt{*iir¡\}fi)kr*¡\$I()|(st'rHr: $u¡iti0l.JI I ¡lri ' siüll[{T' ' si r [ìNAl.-$ T'].1Ë: li ï ül'l râl''1il

i 0UtiRF l-.fll,,t| ( I F fìl.lY ) T'0 üLJ'lItLIl' Il[:V T t]l;:: .

SCþlf(ïl L-trA 'il,J.i5'7J. tL.tlAIl $TüN Al{Ir C}UËftË[..iltJi T ùl [:' tilii"l'â ]' l: C] l.l f:' f( 0 l'l,l' l[,i 7 J. .

$Tô,11,:1.;5$'q tiì'f0F([i IT AT',ll1-31iÉ¡.¡Y0U [r v l\ i ¡'10UÉ: T T '10 [r'-t{[rtì. .rlNI 0:ì il'[:$T' Ïti (]uHIfFLüLl.-lZ T'nR i J:f: N[] r ..,[J¡'JF ''f ilË' 'ü'â1.-l... XT''âft t0'll'{ËfttJI$[iv fiItìl'lÊ1L 'Ît r

T0[t i l'lClU rl ¡ ï.¡

'tN I 01. t 'f ËS't I F l'Jtrtlf{T ï V[r .

-,?: RliìT i:[F'N0v JtJl'lF'fü 'ül$ï'.CÉìLl.- Nfi[ì tilTl-lHlitlIlitir $IüNÉ¡L ¡*' +

ËHI'R$I l []¡:r1...1. F 0Íì t $I0l''l'ât- ''l'' +

R[:Tí ¡|(,$/,<fi)|l:lr)+llr)|{,$tcrß***il(¡Í(il()k**ïofiïÇS}|(ii{.*il(tc]l,,$il(*hrirkik*St\)i:I(,$ilofi¡q{iil(,1\il(S}ftS

A65

i *{tkIixilot{)iiil{I(.*)ll;fi}i{il<}ft#tkii(ii:+i¡qì\h(¡\,t¡t)'\i(Ï()i{¡k$}|(i{Ï(;f',t)i(,ü¡k¡(}l(t(,*Ï(rfÑ}+{;f{tcrì\;rq,$,*,,1.t T'1.'l[: Í:ìLJl]Fiilt.J'I'l:i''lH 'i!ü1.'lf(T' ÏrTl:ìl:'1..¡âY$ I.rlì'f"'l'-r^ [.. i'll.Jl'lüliiftiì.v

NüHftTî l.fCIV rr¡vl.l i¡fnvt: Ïþltr i'jüÍìÏ {:ìÏüi.lIrÏ(lfll'l'Iitrl::üTif'lt- ill:tiïl' F'Ít0i'4 l"l^'[tU:Li.û T'0 'ä'-ft[:{]" .

U'qL.L Nt'ftll.JT' iilIf;F'L.AY l'Ilti Sl:tìl'lïFïü'âl.lï'i Il l: ti ï 'l'!i .

MOU rô¡[:. il'lUVL. T'tl0 t-HÁSiI' [ìTC'il'l]:F:l"Crti'lT'$ Ir Ë:{:: l: l'1'â L- [r ï f] ï 'ï'l:i F'R ü i'l [: - R li: f.ì .i l'tl râ¡'-ltË.Li ' .

Cíll.-L Nt'10tJ'l' !i [rï ïiFL-ËrY 'l'tli] L..HAfiT' fi ï Lìl'l l. Ë ï (ì¡ìl,lT'í Itli.fi.l l'JrtL- Il I ü ï'l'$ 'f 0 fiLJ1'ltlJl'i ltfiiv ï {::t;i .

liFr l't,*àk¡k}líil(:'ftht**àcy,iht)+i)fiht,$r+it(,\\)li)i{ì\ht}l(¡t,li¡!:i(I{àcI(s*y,{T(Tr¡t)i(;\\.{fc¡(,{r¡k,rt¡fr:f}i(hq,'ríI{i(i(¡t¡!.J

ÊT'l''lH l:it.JIl[ttjU]':il.IË 'f:ìlìf:ìl''' Ff i'.lïr!ì T']'lË (lt.J0l':tl:i,lT'rn¡i.lI.¡

ift[::¡1r*r]:i'.lfr[::lt fiI:'T'þ¡É. trTUl.illìi¡'l[r (r*rT'l.-ilC. .ft].:i9J.)t Âþ{fi Tt.'tE: It:t u l:$nft ( åT' 1...{)(: . .[':[ 3?0 ) .v

li$!it¡i L.It'â,[,L.ä90

$TA {'il3/r.3t_l1A .lt'1iT9:L

$ï'A {,J..tó ICrqL.L liïlïUl-[t¡tr ,[, J..T7 L

il...0AIr Tl{[: ilïUIliüR f:ltCIMí {t,1390 .

i l.-0AIl 'l-l.lË tl I U I IrË.l.lLt [iRCIi4i {,1.39l. .

$TL- tr

1.J4tt1.3'70

inrut[r[i.; L.CÌAï 'ftrEt ttLlU't ï [:N'f; sil'rltrË ï'rí L-uôtr l'l-ltii ,!,I3 i/û 'i ij'r0[rE; r I'

tì I tì Ì-l il t-' 1'þllI[ilifil"l {,1 .i7 ]. ,

'âT .[,;iJi4$.

cluclT' l: HÌ'{T' [:rtnÌ'l

^ 'f .d. { -t ¡t ¿'t

tJ ¡ .[.t *) 7.: r{, J. .{ I i.l,[, :l ;'5 C] :L

,!,1:i?û

$ì T'il .Í, L 3 l¡ 3cAL_t_ tfu

t'l[lV ¡â r H

ÍiI'A ,1 .[ 3ú.5L.Il¡q ,&:[,3?'l

$TÍ\ ,[,J..Jt5ll

ËË¡L-L $U

Ë.r ,[,

¡q .$;

fì T'rlfi'f É¡

l- [l¡l í l.-CIâ11 TþlH IlïUïlìtlli Frl'i0l'li,[,13'10.

; MUI-TTIrL,.Y IIUï$t]lt Ér].1ü

i [¡uoT'ï tit't'í .i HuuE: r't.tF_ t-'ftflItLJüT l'ERMi FFiUI''l IJ"'RË:ü. 'l'[J râ-lt[:ti.

i L-lJí\tr 'f l.{fi il I U T Ii[il{n f:R0¡.,|ô,&J.3?1.

i $t.JIlT'[i/lilT' I] l: U :t Ilhi'Iil Ff{0Ì'fÊ FItCItrtJtl'f T'[ili]4,

+

A66

$rA'r'1:54ó ;il;g"li*'l',iï",'iul,li**,1ì1,",*o*r*o*i A'f 'ü'.l.346.

M0V Él r [tfiT''â '['1.3?.3 i S]i'f 0ËH: ]']lË. ftË14A I NIlliR

t ( l'l-lti lt[::l]t.JL'r fl[:' EitJIJI'RÉ¡{:TÏ01'.1 )t A'[ ,s 139.J .

M{lv r¡ v dì t M0v[i 'r]{H rrËMA l. t{ilH[t F flnPlt Ê¡-ltË{i o Til Il""ltHü. .

F(HT

i )|(]gs,*tqhc)fi*àks*¡r¡r¡fiÍ{I(fitk¡lr,$il(,'ftï(/,(¡kì\il(ìkfril(,$*,$,$}t(,$Í(*}i(}k*}hhl,v,$lk,$.**}fifrti(I(fri$-.F$,.l,rl,rlr:F:litß16l,tliiFfþ's's$$ili{iililE:lr,Jr:li$'s{illt':F$iüs"1,:lifþ'$${,{,{i{i',1¡$$$$l$li'slt{iil;'s

t¡.Ntl