THESIS - DTICMaster's Thesis I From To March 1991 16 Supplementary Notation The views expressed in this thesis are those of the author and do not reflect the official policy or position

NAVAL POSTGRADUATE SCHOOLMonterey, California

00

THESIS

NONCOHERENT DETECTION OF BFSK SIGNALS WITH

LIN4EAR AND NONLINEAR DIVERSITY COMBINING

OVER RICIAN FADING CHANNELS WITH PARTIAL

BAND INTERFERENCE

* by

Ahmet Cern Karaagac

March 1991

Thesis Advisor: Clark RobertsonLCo-Advisor :Tri T. Ha

Approved for public release; distribution is unlimited.

9I 1121 0 21 91-16115

UnclassifiedSecurity Classification of this page

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6a Name of Performing Organization 6b Office Symbol 7a Name of Monitoring OrganizationNaval Postgraduate School (If Applicable) Naval Postgraduate School

EC6c Address (city, state, and ZIP code) 7 b Address (city, state, and ZIP code)Monterey, CA 93943-5000 Monterey, CA 93943-50008a Name of Funding/Sponsoring Organization 8b Office Symbol 9 Procurement Instrument Identification Number

(If Applicable)

8c Address (city, state, and ZIP code) 10 Source of Funding NumbersProgram Elaecait NumbcrI Project No I Task No I Work Unit Accession No

11 Title (Include Security Classification) NONCOHERENT DETECTION OF BFSK SIGNALS WITH LINEAR ANDNONLINEAR DIVERSITY COMBINING OVER RICIAN FADING CHANNELS WITH PARTIAL BANDINTERFERENCE (Unclassified)12 Personal Author(s) Karaagac, Ahmet C.13a Type of Report 13b Time Covered 14 Date of Report (year, morahday) 15 Pa amountMaster's Thesis I From To March 1991

16 Supplementary Notation The views expressed in this thesis are those of the author and do not reflect the officialpolicy or position of the Department of Defense or the U.S. Government17 Cosati Codes 18 Subject Terms (corinue on reverse if necessary and identify by block number)

Field Group Subgroup

19. Abstract (Continue on reverse if necessary and identify by block number) A performance analysis of Binary Orthogonal FrequencyShift Keying (BFSK) Fast Frequency Hopped (FFH) receivers implemented with both square-law and envelope detectors is performed.Bit error probabilities of the two types of receivers for linear combining , noise-normalization combining, and self-normalizationcombining under worst-case partial-band interference with nonselective Rician fading and thermal noise are compared. The analysis isrepeated for the case of no interference to point out the effect of fading. A study of nonlinear diversity combining receivers (self-normalization and noise-normalization) is also performed for a system model that is free from thermal noise. Envelope and square-lawdetectors for particular types of nonlinear combining investigated do not differ in performance, but this is not true for linear combiningdetectors. The visible superiority of envelope detectors for linear combining is noted. Nonlinear combining receivers achieve adiversity and performance improvement compared to linear combining receivers.

20 Distribution/Availability of Abstract 21 Abstract SecurityClassification

N unclassified/unlimited 11same as report [] DTIC uaeus Unclassified22a Name of Responsible Individual 22b Telephone (Include Area code) 22c Office SymbolRobertson. Clark. 408-646-2383 EC/RCDD FORM 1473, 84 MAR 83 APR edition may be used until exhausted security classification of this page

All other editions are obsolete Unclassified

Approved for public release; distribution is unlimited.

NONCOHERENT DETECTION OF BFSK SIGNALS WITH LINEAR AND NONLINEAR

DIVERSITY COMBINING OVER RICIAN FADING CHANNELS WITH PARTIAL-BAND

INTERFERENCE

by

Ahmet Cem Karaagac

Lieutenant JG, Turkish Navy

B.S., Turkish Naval Academy, 1985

Submitted in partial fulfillment

of the requirements for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

Maych 1991

Author.___________________________ Ahmet Cedrflaagac

Approved by: &

Clark Robertson, Thesis Advisor

Tri T. Ha, Co-Advisor

Michael A. Morgan, Chairman

Department of Electrical and Computer Engineering

ii

ABSTRACT

A performance analysis of Binary Orthogonal Frequency Shift

Keying (BFSK) Fast Frequency Hopped (FFH) receivers implemented

with both square-law and envelope detectors is performed. Bit

error probabilities of the two types of receivers for linear

combining, noise-normalization combining, and self-normalization

combining under worst-case partial-band interference with

nonselective Rician fading and thermal noise are compared. The

analysis is repeated for the case of no interference to point out

the effect of fading. A study of nonlinear diversity combining

receivers (self-normalization and noise-normalization) is also

performed for a system model that is free from thermal noise.

Envelope and square-law detectors for particular types of

nonlinear combining investigated do not differ in performance, but

this is not true for linear combining detectors. The visible

superiority of envelope detectors for linear combining is noted.

Nonlinear combining receivers achieve a diversity and performance

improvement compared to linear combining receivers.

Acvegs' ter

V4" TALL,

iii

TABLE OF CONTENTS

I.INTRODUCTION .............................................1

II. BACKGROUND INFORMATION ............................... .3

III. SYSTEM AND WAVEFORM ANALYSIS ......................... 6

A. DESCRIPTION OF THE WAVEFORMS ...................... 6

B. DESCRIPTION OF THE SYSTEMS, AND ANALYSIS OF THE

SYSTEM PERFORMANCES UNDER MULTIPATH FADING, PARTIAL-BAND

INTERFERENCE AND THERMAL NOISE ............................. 7

1. Linear Combining Receivers .................... 7

a. Envelope Detector ........................ 7

b. Square-Law Detector ....................... 13

2. Self-Normalization Receivers ................. 21


b. Square-Law Detector ....................... 26

3. Noise-Normalization Receivers ................ 26


b. Square-Law Detector ...................... 27

C. SYSTEM PERFORMANCES UNDER MULTIPATH FADING AND

THERMAL NOISE (NEGLECTING THE EFFECT OF THE PARTIAL-BAND

INTERFERENCE) .............................................. 30

1. Envelope Detector ............................. 30

2. Square-Law Detector ........................... 31

a. Self-Normalization Combining .............. 33

iv

b. Noise-Normalization Combining ............. 32

c. Linear Combining .......................... 33

D. PERFORMANCE ANALYSIS IN THE ABSENCE OF THERMAL

NOISE ...................................................... 34

1. Envelope Detector ............................. 35


b. Noise-Normalization Combining .............. 37

2. Square-Law Detector ........................... 39


b. Noise-Normalization Combining ............. 40

IV. NUMERICAL RESULTS ..................................... 43

A. PERFORMANCE ANALYSIS FOR WORST CASE PARTIAL-BAND

INTERFERENCE AND THERMAL NOISE ............................. 44

1. Linear Combining Detectors ..................... 44

2. Self-Normalization Detectors .................. 47

3. Noise-Normalization Detectors ................. 49

B. COMPARISION OF THE PERFORMANCES IN THE ABSENCE OF

PARTIAL-BANDINTERFERENCE .................................. 50

C. PERFORMANCES OF THE NONLINEAR DETECTORS UNDER NO

THERMAL NOISE ............................................. 50

V. CONCLUSION ............................................. 52

APPENDIX:FIGURES .......................................... 54

LIST OF REFERENCES ......................................... 136

INITIAL DISTRIBUTION LIST .................................. 138

v

I. INTRODUCTION

Previous studies have proven that Fast Frequency Hopped

(FFH) Spread-spectrum communication systems are alternatives

to conventional systems under the presence of multipath fading

and/or partial-band interference.

Difficulties in synchronous carrier recovery in a

multipath fadiiig environment enables noncoherent orthogonal

FFH Binary Frequency Shift Keying (FFH-BFSK) modulation to be

an attractive choice [Ref. 1]. At the receiver, demodulation

of the dehopped signal is performed by a circuit implemented

with bandpass filter and envelope detector arrays. Envelope

and square-law detectors are used interchangeably. Their

performances have been proven to be identical for some cases,

and have been accepted as identical for the others. An

envelope detector is easier to implement, while a square-law

detector is easier to obtain analytical result for. This

assumption of identical performance is examined for FFH-BFSK

orthogonal noncoherent modulation systems with L-fold

diversity and both linear (ordinary FFH-BFSK) and nonlinear

combining. For the latter case, two systems are analyzed:

1. Noise-normalization (Adaptive Gain Control (AGC) [Ref.

2]) ccmbining in which noise and interference statistics are

assumed to be known or predicted.

1

2. Self-normalization combining which does not require the

noise and interference statistics to be known.

In order to make the research applicable to satellite-to-

mobile applications a Rician fading channel is assumed.

The performances of the systems are also inspected under

the absence of interference to emphasize the effect of fading.

As a special case both of the nonlinear combining systems are

analyzed under the absence of the thermal noise to show the

effect of the thermal noise on the performances of the

systems.

Chapter II presents background information, and a

description of the models, and evaluation of the bit error

rates are given in Chapter III. In Chapter IV, numerical

results are presented. Conclusion are given in Chapter V.

2

II. BACKGROUND INFORMATION

The behavior of envelope and square-law detectors are

analyzed with a model similar to that presented in [Ref. 1]

and [Ref. 3].

FFH-BFSK communication systems employing a diversity level

of L, communicating over a channel of bandwidth W, are assumed

to be effected by an interference source. The interference is

assumed to be an additive narrow-band Gaussian process over an

equally probable portion y of the channel bandwidth W. The

BFSK modulator represents a binary input 1 with the frequency

fl and a binary input 0 with that of f2,in d binary symmetric

channel model scheme. The bit duration of Tb is equally

divided into L chips. The bit rate is Rb =l/Tb, and the

hopping rate is Rh=L/Tb =LRb. The binary signal is passed

through a baseband filter of bandwidth Rh+Rb. The output of

the baseband filter modulates the signal generated by a

frequency hop synthesizer. The frequency hop synthesizer is

driven by a pseudorandom code generator. The hopping

frequency fh is a discrete uniform process taking one of the

N possible levels where N=W/Rn. The modulated signal is

filtered by a baseband filter of bandwidth N, upconverted by

a RF oscillator, and transmitted.

3

If the cell bandwidth is small compared to the channel

coherence bandwidth, the fading process can be modelled as

frequency nonselective; furthermore, if the channel bandwidth

is large enough to assign a minimum spacing between two

consecutive hopping frequencies that is large compared to the

coherence bandwidth of the channel, each cell fades

independently. Under these assumptions the amplitude of the

dehopped signal is modelled as a Rician random variable. The

intensity of the fading is assumed to be constant for the

entire bandwidth, as a result, the statistics of the L Rician

random variables affecting the L hops of a bit are equal.

The interference is assumed to be additive white Gaussian

noise. Whether the interference is a deliberate jammer or a

coincidental narrowband process, it is not always possible to

maximize the negative impact on the performance of the

communication link when the finite energy is spread over the

entire bandwidth. Reference 4 shows that for the linear

combining square-law detector, especially with relatively high

diversity levels (when the number of hops per bit is greater

than 2), it is not an effective jamming strategy to distribute

the total jamming power uniformly over the entire bandwidth

even when the signal and the interference energies are equal

at the receiver RF circuit. When the jamming power is not

distributed uniformly over the entire bandwidth, there exists

a certain portion (y) of bandwidth that maximize the Bit Error

Rate (BER) as a function of the variables:

4

1. Interference energy.

2. Thermal noise energy.

3. Detector type.

4. Hopping rate.

5. Severity of fading.

The average power spectral density (PSD) of the narrowband

interference is N,/2 when spread over the entire bandwidth W;

therefore, the conditional partial-band interference PSD is

N1/2y if it is present, zero otherwise.

5

III. SYSTEM AND WAVEFORM ANALYSIS

A. DESCRIPTION OF THE WAVEFORMS

The interference power is uniformly distributed over yW Hz

of the total system bandwidth W. The received signal after

the kth dehopping, where k is an integer taking a value from

1 to L, is represented as:

f sk(t) +nk(t) +ik(t) with probability ysk(t) +nk(t) with probability (l-y) (1)

(k-1) xh< n~k-zh

where sk(t) is the information carrying signal affected by

fading, nk(t) is the thermal noise component, and ik(t) is the

interference noise component. The information carrying signal

in the kth hop interval is:

{ akk 2 S cos(2 n ft +0 k) binary 1 is sent (2)sk(t) = akv2-i cos(2nf 2 t+k) binary 0 is sent

where 0 k and Ok are random phases uniformly distributed over

(0,2U). The average signal power is ak2S, and ak is a Rician

random variable. Channel fading is assumed to be slow

compared to the hoptime, but each hop is assumed to be

independent. The statistics of ak are assumed to be identical

for each chip of a bit. The probability density function of

ak is:

6

fAk (ak) _L e ~C+2/22 T-ak) ak' 0 (3)

where A2 is the signal strength of the nonfaded (direct)

component and 2a2 is the mean-squared value of the Rayleigh-

faded (diffuse) component. Io(.) Represents the modified

Bessel function of zero order.

B. DESCRIPTION OF THE SYSTEMS, AND ANALYSIS OF THE SYSTEM

PERFORMANCES UNDER MULTIPATH FADING, PARTIAL-BAND INTERFERENCE

AND THERMAL NOISE.

1. Linear Combining Receivers

a. Envelope Detector

A linear combining envelope detector receiver is

depicted in Fig. 1. Assuming that a binary 1 is sent, we

obtain the sampled detector outputs contaminated with only

wideband thermal noise as:

Xlk = I(ak V2 cosOk + nCk) 2 + (ak V2 sink + n,,2(4)

n 2 n2K Czk + C2k

and with narrowband interference added as:

Xlk = V (ak V/S COSEk + ncC + iC1 k )2 + (ak /2S sinEk + nSl + i(5)

X2 k = (nc, + i2k )2 + (ns 2 k + iS2k

7

where ncjk, nsJk, j=l,2 are independent thermal noise components

in the channels at the sampling instants t=krh (where Th=l/Rh) .

Both are assumed to be independent zero mean Gaussian random

variables with equal variances N 2=N0B, where B is the cell

bandwidth which is equal to the hopping rate Rh. The

interference components icjk, isik, are both narrowband zero

mean Gaussian random variables with a variance of o2 = NIB/y

Equation (4) and (5) can be represented as

Xlk = I (akIr2S cosek + Vlk) + (ak2 sinOk + V2k) (6)2 2

X 2 k =V3k + 4k

where viks (i=l,2 3,4) are independent zero mean Gaussian

random variables with equal variances ak2:

f = NoB with probability (l-y)Ok =2 = a + Cy = (No + N 1/y)B with probability yT 2

Narrowband interference, when present, is assumed to affect

both channels. The conditional probability density functions

for Xlk and X2 k are given in Ref. 5:

(xlk + 2SAk)

kl I° a xk xa kfXlk/A 2xk/ k0 k Xl o~a~k) Xlk : 0

k k/

(8)X2k

fX 2 k/Ak (X2k/ak) = f X ) X2 k 2 e It 2k 08 i

X~"'2'0

8

The unconditional probability density function of the

envelope of the output of channel one, fX1k (Xlk), is obtained

by integrating:

.fxk(Xlk) = fx,/Ak(X/ak) fA,(ak) dak (9)0

to get

IX1kj ' 2 - S 1

k1 2 02 ak 2 2O 22fxk(xlk) - e 2 ) e -k e

2 a 2Ok 0 a2

x [.r(. a) 10 =S xlkak Jjdak (0

(10)

Without loss of generality, S is normalized to unity, and

equation (10) is evaluated to obtain

fx,,(Xlk) Xlk ePk/( ,) 2 2

Ok ex+k[2a

x 2Xk)] Xlk ()

where Pk=A2 /ok 2 is the signal-to-noise ratio of the nonfaded

(direct) component of the kth hop of a bit and k = 2a 2/ak 2 is

the signal-to-noise ratio of the Rayleigh faded (diffuse)

component. It is possible to normalize A2 to unity and

equation (11) becomes

9

fx k(Xlk) - P1+ k Xlk e Pk'(1 (k) e-PkxKf /[2(1-tt)I

(12)

1 .1+Pk Xlk Xlk 0

The bit error probability for the receiver in Fig. 1 in

the presence of partial-band interference is

L

where P(e/l) is the conditional bit error probability when 1

of L hops of a bit have interference, and is given as

P(e/1) = Pr(x1 <x2/1) =fx (x1/l)[fx2(x2/1) dx2 dxY

0

(14)

where both xi and x2 are the sum of L independent random

variables, 1 of which are interfered. Thus,

4 (x 1 /) / f;) (xi, x) * (L1) (X(0))

() = 2-(l)) * (L (X*(0)XX (X11 (0) 2

where *m is m-fold linear convolution, and the superscript (1)

and (0) denote the random variables with and without

interference, respectively. In the following, the

superscripts are attached only to the names of the functions,

not to the variables and the constants.

Analytic solutions for fxl(xl/l) and fx2 (x2/1) include

nested infinite summations which make P(E) tedious to obtain

10

numerically. The following approach is preferred for numeric

results.

Define

fl (xlk(Xikk 1k) Xlk (16)

xlk 0 Xlk > M

and

F 0) (X) = fX (Xk) Xlk < M (17)k 0 Xlk > M

where M = max { M , M2) and

M. M2

f f( 1 (Xlk) "dx 1 and f fO (xl) 1 (18)

0 0

For the linear combining detector, M=M,. Define

_ k ={ f0x2* (x2k) X 2k < NT (X2 k) =(19)

0 2 k> N

and

S(x2k) f(X) X2k N (20)0 x2k> N

where N=max(N , N2 ),

JV N2

f fx,) (x2k) dx2k A 1 and f :X,(°) (x2k) dx2 A 1 (21)0 0

and M/N is chosen as integer. It is easier to explain the

method used when M=N. Define

ii

I( f -(1 M

.x - Xi) (xik)I where A-M1lanzl (22)

where A is the distance between samples that are taken from

the pdfs.

f (0) (n) = (O) (x

f 1 (0)kx,) (23

X20 (n) = KO)(X 2k)l (23)

f () = (x2k) IXk (n-l1) A

Probability density function for x1 and x can be

approximated as f x, (n) and f'X2 (n) where

x, (n) = IDFT ( EDFT(Pl, (n))] ix [DFT(f (') (n)) ] L-1 AL- (24)

DFT and IDFT are Discrete Fourier Transform and the Inverse

Discrete Fourier Transform respectively. Numeric values are

obtained using the Fast Fourier Transform (FFT) and the

Inverse Fast Fourier Transform (IFFT). In implementing the

analyses of a K point FFT (where K is an integer power of 2),

K is chosen such that K>ML/A+l, and remaining samples of the

functions between M/A+1 and K are padded with zeros. Redefine

f(() (X(, + _M ?-n;l1k(n) : Mk (25)xIJ, 0 K> M +ln> - +1

and the other functions can be redefined similarly. The

probability of bit error is obtained by integrating equation

12

(14) numerically and substituting into equation (13) for

L=2,4,6,8. For L= (slow FH), the exact result is obtained as

e _pk / (2_ +1k ) IM _ O , /(2+to))

P(E/L=) = y + (1-y) () (26)

b. Square-Law Detector

The linear combining square-law detector is

depicted in Figure 2. Assuming that a binary 1 is sent when

there is not interference, we obtain the sampled detector

outputs as

Xlk = (ak 2_Scos6k + nc.k) 2 + (akV2sinek + n.) 2

(27)

x 2 k = (nck + ns)

and when there is interference

Xlk = (akv/2ZcosOk + Cl + ) 2 + (akN/2Ssin~k + n... + ik ) 2

X2k = (n) 2 + =) + (n 8, + is8,)2

(28)

All the variables are as defined for the envelope detector.

Using equation (8) and equation (9), we obtain the probability

density functions for xlk and X2k via a transformation of x=y2,

where x represents Xlk or x2k and y stands for xlk and x2 k, in

equations (8) and (9), respectively. Hence

13

1 1/2(x ) 1 ( 2 )P~fxl (xlk) -0 2 (I+-k) 0 k l+k) Xlk 0

(29)

fx 2k(xk) e k X 2k 2 (3)2 k

The characteristic functions of the decision random

variables Xlk and X2k are

Cs(S) = f fX,,(Xlk) e-Xk dLXk-- (31)

Cs, 1 f 1 e-Pl(1.t) e-xk/(22

(1(k))

o 0 k 2()(1+

x o akF1 )) e-X Idxlk

Substituting Xlk=U 2 and dxlk =2udu into equation (31) and

integrating, one obtains

1 (S) e -Pk/(1 tk) u e I° P k u du0xkS)f o k ) (32)

The result yields [Ref. 6)

Cxk (S) = ePk/ (1 ' k) 1 el2°IC1,(" 2ok(1.4)

a (1 +&k) 2 s+- 142 1+ : Z k) (33)

14

Define xi (1) as the sum of 1 interfered random variables

and xi (0) as the sum of remaining L-1 random variables that are

not interfered, and c as an integer assuming a value of either

1 or L-1 for the superscripts (1) and (0) respectively. Then

X1 (x') : {Cxjs) IC} for i=0,1 (34)

which is

F2ait + 2

(i) j e-pc/(1+(,) e 2a 1+ k) (35)x ~ ~~ o (l+Ek) C 2 S+I 2 0 ( + )

2c(+ o(1+ k)

Using (8 Campbell-Foster Pairs 650.0] we obtain

e -CPk/(1k) xi (C-1) /2

2 G C+1 (1+9k) (Pk C) (23

(36)

)× C .o EX., )e x/o(l(,

and

Cx(s) = 2 f e x dX2 k -

0 2s+k)(37)15

15

The characteristic function for the random variable x2 which

is the sum of 1 interfered and L-1 noninterfered random

variables, is

1x (SL-k) P)'Css =~2 ) 2 P / 1 0) L-)(38)Pkk

This can be separated as

Cx (s) - A11 , A 2 .. Ap(1) ! ^(1) \2 / "

S+Pk~l Pkj )2 S+ Pk j2 2) 2)

+ A 2 1 . A 2 2 2 +L-1(0) 1 (o) /2 (o) (L-)

2+ PkS+SPO

(39)where

1 d_ -j1d_(1 -J)

A(j 1 j) ! ds U-J) () (L-()

2 j -Pk2

which yields

AIj = (-1) (-j)( L-j-)) 2 L-j lj l 1 *L and 1*00)-(1))(-J)

(41)

16

1 d(L- - j -A2j = (L-j-1) ! ds(L-j-) (42)

2 Pk2

A is verified to be

A2 j = (-i)-' (L-J-! 2"(L- -1(3L-1-j) (,0)_ (1))L-j I j L.-1 (43)

The probability density function for the decision variable x2

is obtained by taking the inverse Laplace transform of the

characteristic function CX2(S)

S(p) (0) L)1 - L-j-I)

f~l, (2) j( k -1)x e2

(po)_p )) - (j e (PX2/2)]

+ L-1 2 (o))L-1 (_-1) 1 L- j- 1j.1 2j

,______________ xi-ix IP() P(1) ' j (j-l) e- (P,''°)x2) /

) -(j-1)

and when 1=L and 1=0

(Pk(') L-1 e-" p ',' X2)/f, (x 2 ) - 2 L (L-1) i e

(45)

fx 2(X,) 21 (L-1) x "I - e PkX2)/2=0

17

The probability of bit error is

L

P (E) = , (L) yj (1-y) (L-1 P(e/1) (46)

where

P(e/2) =f f,,.(x,) f f, (x,) dx2 dx1 (47)0 X2

and

.fX, (Xi) = f,(1) (xi) * f,() (XI) (48)

It is tedious to derive P(e/l) by using the exact solution for

equation (21) except for the cases 1=0 and 1=L;

hence,numerical analysis is preferred. As before,

.; j) = XDFT[(DFT(f (1 ) (n)) x ( DFT. (, (O)(n))] . (49)

is defined where

t i) - (nn).

( r,~' (X1 ) + -+l-fl 1 (50)

and

-xEi) (x 1 ) = fx (,!(x 1) Mkx 1 0 (51)0 otherwise

where M is the maximum of the reasonable limits of fx1(X1 ) or

fX1( 0)(xj) in order to have the areas under these probability

18

density functions as unity. The sampling distance is A. The

remaining part of the procedure is as explained previously for

the envelope detector. Also

P(E/L=I) - Y e-Pk /(2.k) + (l-y) e-p0l(2) tk" )

This result is the same as the one obtained for the envelope

linear combining detector.

When the signal is completely diffuse (Rayleigh

fading:pk(i)-O) the probability density function for a single

hop is

fxlk(xlk) = Pke -P where = 1

2 (k+202) (53)

and the characteristic function for the random variable x, is

CrI(S) =( (+0) where Ie4"O) P(l) (54)

which yields

fx -1/ ((Xi -1) L-j (j-l)

LX L-I - , (55 )fx (xiL) e (L-l) ! fx,(x 1 /1=O) = ( PL-lePx (

19

The probability density function for x2 is found in a similar

manner

1 - L L1 -m X2

2 x2 , e L-m a mx2 e

(X2 F 1 1) L-

iXX 2 /- (L-1) I iXIXV (L-1)

+ (xE L) = f (L2 1) X2 e

( M-.) (L -I) L

(56)

where

(0)2 a1 (1)2 (57)o 2ok°) 2ok 2 (7

After some algebra the conditional bit error probabilities

when 1=0 and when 1=L are found to be

L-I L-kP rP2c,'1=0 aokI3

k-o( L-1 (Do+a)2L-k-1

L-1 Lk L (58)P(e/L) (2L- k 2 al

L-1I (Pj+a1) 2L-k-1

In general

20

1 I-jpIpL- 1 L-

r (Po L-j -- 1 (ao- ) a -

m-1~ji-t-J

x j- t. ( +PI) (,.. -t-1

2. ~ ~ - aefNralzto Reeier

i _j) -jplnL-1 L.-1 L-1The selfnormaizat ion con -i . m R1

L-I I{ L- -_ .L-

j=1 (Po-Pi) I 1 i - M- ao-a)- in -m

[ -o~ ~ j- j- al),t-1 (P+a m ' j' Lj-

L- l- L ~ j - -1) -' 1 1

+ oo L - - m j t-["=_P (Lom I- M- - '"o1 (P3 (a)*-a-

a method of obtaining the predecision variables by normalizing

the outputs of the envelope detectors of Fig. 1 with the sum

21

of the detector outputs. The receiver is depicted in Fig. 3.

Equations (11) and (8) are revisited

2x ,( Xk A2S/(2o2S+o ) eXik/ 2(20o+0)] I0 2 S XlffS02+)(12 S02 +0

(60)x=2 k e-x,/20k' X 2k.0f,,,(= - e

We define Zlk and v1k as

Zlk - 0<Zlk!lXk+X2k (61)

Vi.k = Xk+X2k OVlk <

The Jacobian of the transformation is

aZlk 8zlk

aXlk (3X 2 k _ 1

(3_ V1 k()Vk = -(2

aXlk aX2k

Equation (61) can be inverted to yield xlk=vlkzlk and x2k=vlk(l -

Zik) . Since Xlk and x2 k are independent random variables, their

joint probability density function is

fx k.x2k(Xlk I X2k) = fx,,(Xlk) X ffx,,(X2k) (63)

fXlk X2 k ~A2S/p2 -a/ 2X2,2 2Sfxl, Xk, (Xlk I X2k) = 12c2 e it e- x/ic e-x/ o0 2k Ax

where kOk+ 2 Sa 2 . Hence

fzlk,k (Zlk, V1k) = I J I. fxlk,x"a(Vlk. Zik , Vlk. (l-zlk)) (64)

The probability density function for Zik is now obtained as

22

fZlk (zlk) = fz , V, (Zlki vlk) (65)0

Substituting equation (63) into equation (65), we get

- _k ( Zl) _eZ 2 22ok

x f Zlk(l-zlk) V~k e 2P Pk Ok ( I j vlJ dvlk

(66)

Making the necessary substitutions into the equation in Ref.

6 on page 394

f~~ ( (2m +AJo(at) em-p ' 2 2-) In = (67)J ed rm+) 2pt- 2 m

0 m-02p'+m

where r(.) is the Gamma function and Jo(.) is the Bessel

function of the first kind of order zero, we can evaluate

equation (66). Since F(m)=(m-l)! for m an integer

Zlk~1k~ -e A2S/02fzk (zk) -2ozj k (1 (l-Zlk)2P2~

- (SA2 m 2rn (8)

2(m1) Zlk

rn-a m 2p~ (:12)n2Zlk ('-zlk)

which simplifies into

23

-A 2SIO 2

Ok ~ ~ k 4 k 1 - zk) 2

(P2 2k Ok

m=O M !,, + (1 Zl, 1P2 2

Using

d~kye Y) M+1 yrn = e Y (Y+ ) (70)dy m-0 m.

and replacing for y

SA2 Z2~(Y 2 1k )21 (71)B4 Zlk + (1 ZlkJ

22 2

we obtain the pdf of z Ik as

fzl1k = 2 e SA2 101 Zlk lZk) 2

Ok J+ ( 1z k )2

( 2 2_ _

p 2 02

SAzk e~kP2 Ok lkZx, l+Z) x e kl~

2P2jJ

24

Substituting k=2a 2/ak 2 , Pk=A 2/ok 2 , and S=l in equation (72), we

get

2 e P I (1-k) (Zik'l-Zlk))

IZk(zlk) l 2(Zlk + ((-73) ()+))2

( 2 21 - PkZk1× +PkzIk x 1e ( k) (Z+k + (2 - Z( 2 ) 2 ( k) )

(1+ k) [Zlk + (1 -Zl) (l ]

Recalling equation (13) for probability of bit error

L

P(E) E ( )Yj 1 -l.y )L-1 P (e/ 1) (74)1-0

and using

, (Zlk Z2k) = Z1 Z2 L

1.0 (75)

P(e/l) = P( z2 >z1 /1)

we get the partial probability of bit error, when 1 of L chips

contain interference noise energy, as

P(e/1) = z 1 <.-/1) (76)

The results for probability bit error are obtained

numerically in a similar fashion to that previously explained.

The probability of bit error for L=1 is found to be the same

as in equation (26).

25


The square-law self-normalization detector is

depicted in Figure 4. P(E) is derived for L=2,4,6,8 using

the equation from reference 1

fzIk(zlk) = PkZlk + (l+k) [l+ k(l-Zlk) I e-P('z-k)/[1t~k(z1k)] (77)[l+k (1-zk) j3

in the method previously explained. For L=I, P(E) is derived

analytically, and the same result as in equation (26) is

obtained.

3. Noise-Normalization Combining Receivers

a. Envelope Detector

The noise-normalization combining scheme normalizes

the outputs of the envelope detectors of Figure 1 with the

noise power (square-rooted) obtained from the output of the

noise only channel (noise power prediction channel) at the

sampling instants to form the predecision variables Z1k and z2k

(depicted in Figure 5). The probability density functions for

zlk and z2k can be derived applying a linear transformation to

the pdfs of the linear combining detectors Xlk and X2k, such as

Zlk=xlk/ak z2k=x2k/ok. The results for fZlk(Zlk) and fZ2k(Z2k) yield

(Zl+k zz' I l+k

~Z: 1lk) Z1 . e I V Zlk) Zlk t0

fz 2k(z2k) = Z 2 k e - zk/2 Z2 k 0

26

The probability of bit error is obtained with a method

similar to that used in the linear combining envelope detector

case. An exact result is derived for the probability of bit

error of a single hop per bit (slow FH). The result is the

same as in the equation (26).


The square-law noise-normalization detector is depicted in

Figure 6. The decision is made by a comparison circuit which

accepts as inputs the sum of the random variables that are

obtained by normalizing the outputs of the detectors with the

output of the noise power measurement channel. Assuming a

binary one is sent, we get the pdfs for single hop random

variables [Ref. 3]

fzk(Zlk) -(z2p) / (2(1 )k Zlk >0

2 (1 + k) e I '{J ~kJ(79)

fZk (Z2k) e -ZkZ/22 Z~2> 2

The pdfs for the random variables zi (1) and z1(0), where (1) and

(0) represent the portion having interference and the portion

not having interference, are

(i) ( 1) /2 r.(lPk 2C! - ( ,fI (z ' )-e -[p zl )2cIP ') JJIi11"i)c (~ li

-l (2Ci )

(80)

f, (z 2 ) (z 2 / 2 ) L- 1 -(Z2/2)2(L-I) ! e

27

where ci is 1 if i=1 or L-1 if i=O, and

W(i = 1 (12 (1 + (1) (81)

The probability of bit error for L-=2,4,6,8 is obtained in

a similar fashion as for the previous cases, and P(E/L=l) is

found to be the same as in the equation (26).

When the signal is completely diffuse (Pk(l-O), and the

characteristic function from reference 3, replacing f31=k3(1),

and .=3 k (2), is

cz, (s/.1 s) (82)

This can be inverted to yield

1I_, -jpI L- 1J- - ~fz,(zl/l) = --_ - L-j- z,

j.1 ( -1) j -j )(U-1)e

(83)

L-=1 -1 J-1) IP. I _P- )L-i L- -j (j-1)' L>

fzj (Zj/L) -(L l e-iz ' .z( J 1 0 0 z e- I (4

and

f2z (Z2 ) L e (85)

2 (L 1) I

Now

28

P(e/1) =fz (zl) dzl f_,, (z,) dz, (86)0 z2

Since

e ( 1 )k=ea N!

eaYy~dy = eaY - ) (N! for N is integer (87)fk-o ajC 1 (N- k)!

then

a[ L-1 L-1-k 1dz1 (88)P(e/ ) = fz (zl) e - z2/2 E 2- -__d_8

0 o 2 (L-k-I) !

Replacing fz1 (zl) in the equation above and solving for 1=L, we

get

1 (89)

P(e/L) - _ "____ e z -zil- dz(L-1) ! .k- 2 Lk1 (L-k-l) !

pL L-1 ~ ~2L-k22---(L-1) .k 2L-1(Lk~l) I (- 1 ) 1(2.-k-t-2)

2- 21 ZL- -O

The upper limit of the function yields 0, while the lower

limit has value only at t=2L-k-2; hence,

P(e/L) =L1j2L-k-2) (2L)L (90)k-0 L-1 ( 2 1 3 + 1 ) 2L-k-1

Similarly

L- 1

P(e/1=o) = L- (2L-k-2) (2p" 3)L (91)k-o L-1 (2 P 1 + 1 ) 2L-k-1

29

finally

P l 1 ((PePll L -j L-1j- 2 1k-0 2 L-k+j

2 (92)L1 ( -)-J L-_j-1 j-i 1 -~-

k02L-1 -k(pl+ ) ---

C. SYSTEM PERFORMANCES UNDER MULTIPATH FADING AND THERMAL

NOISE (NEGLECTING THE EFFECT OF PARTIAL-BAND INTERFERENCE)

1. Envelope Detector

An analysis for envelope detector for three kinds of

receivers previously discussed is similar to that used in the

partial-band interference analysis. In the absence of

partial-band interference, the performance of a noise-

normalization detector is easily proven equal to the

performance of the linear combining version of the same

detector. Hence

L L

x1 = , Xlk E X2 kk-1 k-1

(93)

Z,=L Zlk L -1k X

k-1 k-i 0k ak

30

Similarly z2=x2/Ok, and for linear combining

P (E) =fX ~(X1 ) *dx1 fX f(X 2) CZx 2 (94)o x

is equal to

P (E) =f fzjzi') .dz1 f fZ (Z 2 ) .dZ2 (95)0 z,

if the substitutions Y1=aCkZ, dxl=ak.dzl and X2=akZ 2, dx 2=ak.dz 2

are made, where - recall that fzl(zl)dzl=fxl(ak'Zl) "dxl and

fz 2 (z 2 )dz 2=fx2 (Q.' 2 ).dx2 for the linear transformation. This

proof can 1- applied to wideband (uniform) interference. The

probability of bit error for L=I can be extracted from that of

with-interference results, and it is

P(E/L=l) = ePk/(2k)96)2+ k (6

2. Square-Law Detector

a. Self-Normalization Combining

Results are obtained numerically except for L=1.

P(E/L=l) is found the same as for equation (96).

31

b. Noise-Normalization Combining

The probability density function for xi is derived

by replacing c=L in equation (80) to obtain

fx (xj) = k x L - ) P/2

X ) (2Lpk) (L-1)/2 e-1___ (2P 2 Lp-i ) x, ;0 (97)

The pdf for x2 is the same as in equation (38) where

)k=1/2(l+ k), and the probability of bit error is

P (E) = f f.1,(X 1 ) If4 ; 4(X 2 ) d2 I ] x (98)0 x,

Substituting equation (80) and equation (97) into equation

(98) and integrating the inner integral, we get

P(E) = Ok e-2 PkPk

2L (2Lpk) (L-1)/2

[L~1 -(Pk+-!9Xxx 2' f eL-2-l) !(L-1J/2 X(L-i-1) dLj24TUkxj

eutis oba id ! fro prvosnlsi9n9fudtob )h

The result of the integration is an infinite sum and numeric

integration is preferred instead. For slow FH (L-1) , the

result is obtained from previous analysis and found to be the

same as in equation (96).

32

c. Linear Combining

The probability density function for xi is obtained

from equation (36) by substituting c=L

fx, (xj) =

e-_____________1 x(L-1)/2

2 (L+1) /2 a (1+ ,) (pkL) (L-1)/2

( P(1 &l,) e(-x1/2o 2 (1s ) x1 0)

The pdf for x2 is derived using the characteristic function

method

fX, (X2 ) = - .ePk'2}) when A 2 =1 (101)

which yields

fX, (X2) Pk S+ { }) (102)

which reduces to

fX; (x 2 ) = ( Pk ) L-) 1 ePj'2/ 2 x 2 a0 (103)

Substituting equation (102) and equation (103) into the well

known equation for the probability of bit error for

noncoherent BFSK, we get

33

P (E) = fx ~(XI) [ fx.f (x 2 ) cix2] dx0 x,

Evaluation of the inner integral yields

P(E) e

(L-1) (1+ k) (pkL) (L-i)/2

(104)L- Pk

k-0 2k k! f Pk(i+ k)

Replacing x,=a k2z and dx=k 2dzi in equation (104), we obtain

equation (99). As a result, we see that noise-normalization

combining for the square-law detector detection procedure has

no performance improvement as compared to linear combining in

the absence of partial-band interference or jamming. P(E/L=l)

is found to be the same as in the previous cases.

D. PERFORMANCE ANALYSIS IN THE ABSENCE OF THERMAL NOISE

The neglect of thermal noise (No0-) is a basic

simplifying assumption in work regarding partial-band

interference and fading. This analysis is implemented for the

nonlinear combining detectors; it is found impractical to

obtain accurate results for the linear combining detectors

when thermal noise is neglected.

34

1. Envelope Detector


Recall equation (72) for the pdf of a single hop

2 e -P11l (1+Qk ( 1+ k) zlk (l-zlk)[Z2k+ (-zlk) 2 (1+ k) ]2

( Pk42 (105)

PZ2 Pkl

x+ k) [zk (1-zk) 2 (+)x e( (1+Qk (zlk)2(1V)

With no thermal noise, and for the sake of simplicity letting

B=1,we have

{0 when the hop is free of interferenceoy = when the hop has interference ()

If the random variable Zlk is not contaminated with

interference, the parameters Pk and tk in the probability

density function of the particular random variable go to

infinity in the limit. Define the condition set C as

Pk-0

C k- (107)C= (l-zlk) ('+Elk)

Zlk-1

where the third condition can take place if and only if the

order of Ek is greater than the order of i/(l-Z lk ) as zlk

approaches to 1. By a simple limit operation

35

= = | | !

limc fz, (Z(lk) oo (108)

We now redefine the pdf for Zik under the declared condition

as

fz (zlk) = 8 (1 -Zlk) (109)

where S(.) is delta function. Recall that the probability

density function of the sum of statistically independent

random variables is the convolution of the probability density

functions of the random variables included in the sum. The

convolution operation with a delta function is implemented

simply by shifting the function involved on the horizontal

axis as much as the distance of the delta function from the

origin. Every single cell in the combination that does not

have interference shifts the resultant pdf by one unit to the

right. The conditional probability of bit error when 1 of L

hops have interference power is thus

L/2

P(e1)= f [fz * 8(Z-(L-1))i . dz (3o)0

Hence

L/2P(e/l) = f; (zj) .dz, when >L/2 (M)

L-10 when 2 L/2

and the probability of bit error becomes

36

L()pf(ej1 Y(_Y)(L-1) L is even

P(E) = 2 (112)

L (L)Pce/1) y (1y)(L1) L is odd

2

Numerical results are obtained with the method previously

explained. For single hop per bit FH, the probability of bit

error is obtained as

e- kil(2 +t j )

P(E/L=:) = y 2(1) (113)

b. Noise Normalization Combining

Recall equation (78) for envelope detector noise-

normalization combining pdf for signal-containing random

variable zik

fz, ( Zlk ) _ e - P J,/ t ) -z, /i2k1j (k)] o10 _ Z 1 k) Zld> 0 (114)f~) P1+k)1t

Zlklk zlk eziI21 k)J. l+Pk

and z 2k from equation (78)

fz,,,(Z2k ) = Z 2 k ez ,/2 z 2kO (115)

If we replace B=1 for simplicity and if No approaches 0, then

tk and Pk approach to infinity, and

0 wi thou t interference

O= N.{ with interference

37

We separate equation (114) into 3 parts

( 2F2 Pk Zlk

f k(l e PII.j x Zik 1 + k 2 ( 1 +-k1 (1-17)1 +Ek e Z,,/ [2 (.1 +,) 1

(1) (2) (3)

Defining

2

!_k k P (118)2('+k) a + k

and recalling

I(2avk) a 2mx i xMM (!) 2 eM! ). (119)

we find that the first part assumes a finite value as ak2,0.

The order of 1o(2a,/x) is equal to the order of ex, so the third

part may also converge as well. The second part is infinite

as zlk-+. Defining the condition set C as

C{p240 } (120)

so lim C fzlk(Zlk) +, we have

f 0) (zlk) = lim.. 6 (M-zlk) (121)Z1k

which is an impulse at the infinity. The probability density

function for the random variable z2k is finite for all z2k.

38

The convolution operation of a function with an impulse at the

infinity yields zero for finite values of zlk. Therefore, for

every bit including even a single chip that is not

contaminated by the interference, the conditional probability

of bit error is zero. It is only necessary to evaluate

P(e/l=L) and

P(E) = yLp(e/1=L) (122)

The results are obtained numerically except for L=1, which is

found to be the same as in the self-normalization case.

2. Square-Law Detector


Recall equation (77). separating it into two

parts, we have

fzlk(Zk) = PkZlk+ (l+ k) [l+&k(lz1k) I x e-Pk(1

-z-k)/ [1 C

* ( 1 -z1k

)I

-[ (l-zlk) 33 (123)

(1) (2)

and the condition set C is defined as

P k.

C = k (124)

&k (1 z1k) -A

where A stands for any finite real value. Under this

condition set the limit

39

Iiimc fZ,, (Zlk) "(125)

The second part is finite and nonzero, and the first part goes

to infinity (the denominator is finite, the numerator

approaches infinity in the limit); hence the probability

density function for a single chip which does not include

interference is

ff4' (Zlk) = (l-Zlk) (126)

The performance analysis is carried out numerically, and

results for equation (112) are obtained using the equation

(123).

b. Noise-Normalization combining

Equation (80)

f(z Zka0 (127)

gives a finite result for all finite values of Z2k. Separating

equation (81) into three parts and rewriting it for c

noninterfered hops, we have

)c-) /2

1L 2+ k. _0 C -) 2J (128)

cp(0) Zi1+t 2 )1eZ,/2 (1+ {('0

40

We derive a condition set C as

(kiO) ) 2-

(0 ) -'0

C k- (129)

Z, (0) 0

Defining

x __ a CP (130)

and recalling the series expansion for the modified Bessel

function with an integer order (c-i) and for the exponential

function

Ic-1 (2ax) = a x2mxc- eX2 (1-31)m-0 m! (m+c-1) . = -

we see that the limit of the first part of equation (129)

under the condition set C has a finite nonzero value. In

addition the third part, which has a form of Ic.1(2ax)/(ex2 ),

does not converge (the order of Ic.1(.) is equal to or greater

than the order of eX2; so if c>2 this may happen). Finally,

the second part may approach infinity depending on the order

of z i with respect to 1/ok2 . Hence, lim c fz (0) (z1)-w, and

fz, (zO) =M-. (M-z) for c=L-1:2 (132)

Rewriting this equation, we get

41

f (0) (z) ffO (0) * (0)Z1 "1k (Z* (z') when c=2 (133)

This is simply a two-fold convolution (auto-convolation) of

fZlk((zl),and equation (132) is valid if and only if

ff4i(z1k) = M - a zl) is true

Obviously, equation (132) is valid not only for c>2 but also

for c=l, and P(E)=yLp(e/l=L) can be derived from equation (99)

as

yLP3 (1 -2ft Lp~' L-1 Z' pP

P(E) - Y2e J: 2- fl2L(2Lpk()) (L-1)/2 i-0 0 (L-i-l) (135)

Results are obtained numerically by evaluating equation (134)

when L>1. For L=Il, numerical results can be found with

equation (113).

42

IV. NUMERICAL RESULTS

Bit error probabilities for worst case interference ratios

are obtained versus bit energy-to-interference density with

the following parameters: a) detector type, b) direct-to-

diffuse signal ratios (DD=A2/2a2 ), and c) bit energy-to-noise

power spectral density ratios (EIN o). These results are shown

in Figures 7 through 62. In the absence of interference, the

results shown in Figures 63 through 82 are obtained. They

illustrate BER as a function of EINo with the following

parameters: a) detector type, b) direct-to-diffuse signal

ratios. Figures 83-86 are an illustration of the performance

of the various nonlinear combining receivers versus bit

energy-to-interference power spectral density (Eb/N) ratios

when there is no thermal noise contamination (No-0) and DD=I0

(a moderate fading effect).

Worst case y values are obtained by inspection, and it can

be seen that for a particular detector (and normalization)

type the worst case y's are functions of the parameters: Eb/No,

EVNI, L and DD. For the linear combining analyses, we see

that all of the parameters mentioned effect the worst case y

(y0) in an inverse manner. For nonlinear combining, yo is

directly proportional to L, while the effect of the other

parameters investigated on yo is the same as for linear

combining. The relationship between yo and A2/2C 2 is very

43

loose. For strong fading, detector performance is not

sensitive to y [reference 1].

A. PERFORMANCE ANALYSIS FOR WORST CASE PARTIAL-BAND

INTERFERENCE

1. Linear Combining Detectors

(1) Et§N =13.35 dB and DD=0.01 (Figures 7 and 8)

There is not a visible difference between envelope and

square-law detectors. Both have a great amount of diversity

improvement for Eb/Ni 5 dB. The optimum value of L is greater

than 4 for the envelope detector with Eb/NI>I0 dB and the

square-law detector for Eb/NI>l 5 db. The square-law detector

has a slightly better performance for these optimum L values

when E./NI>3 0 Db.

(2) E/N-=13.35 Db and DD=1 (Figures 9 and 10)

Both detectors show a diversity improvement. The optimum

number of chips per bit is greater than 4 for the envelope

detector with Eb/NI12 dB and for the square-law detector with

Eb/N 1>I7 dB. The square-law detector has a slightly better

performance.

(3) E /N-=13.35 dB and DD=10 (Figures 11 and 12)

The region of diversity improvement begins for the

envelope detector with Eb/NI> 9 dB and for the square-law

detector with Eb/NI> 20 dB. The optimum L=4 for the envelope

44

detector with Eb/NI>l5 dB, while the optimum L never exceeds

2 for the square-law detector. Performance of the envelope

detector is better than the square-law detector for 25

dB>EbIN>7 dB and equal for EIN>25 dB.

(4) E IN dB and DD=I000 (Figures 13 and 14

The square-law detector does not exhibit diversity

improvement in this case, but the envelope detector does. The

performance of the envelope detector is much better than the

performance of the square-law detector. Simulations with

greater A2/2U 2 such as 106 for some values of EINI show that

there is a diversity improvement for the envelope detector not

only versus fading but also versus partial-band interference.

There is not any difference between the values obtained for

A2/2-2=1000 and 106, so A2/2 2'=i000 represents the no fading

condition (Similarly A2/2a2=0.01 represents Rayleigh fading).

(5) E/N, =16 dB and DD=0.0 (Figures 15 and 16)

Both detectors show a diversity improvement. For the

optimum values of L, the performance of the envelope detector

is better than that of the square-law detector up to EINI=35

dB. Comparing Figures 15 and 16 to Figures 63 and 64 (the

latter two figures illustrate performance when there is no

partial-band interference for the same values of EINO and DD),

we see that both detectors (linear combining) have almost the

same diversity improvement versus fading, but the envelope

detector is better versus partial-band interference.

45

(6) E,/N=l6 dB and DD=1 (Figures 17 and 18)

Both detectors show diversity improvement, but the

performance of envelope detector is superior.

(7) E /N-=16 dB and DD=l0 (Figures 19 and 20)

Diversity improvement is achieved by the envelope

detector, but not by the square-law detector up to Eb/N 1>27 dB.

The envelope detector performance is better.

(8) E_/N=I6 dB and DD=I000 (Figures 21 and 22)

The envelope detector show diversity improvement up to 3

dB, but no diversity improvement is reached by the square-law

detector. The envelope detector performance is better.

(9) EL/N,=1 8 dB and DD=I0 (Figures 23 and 24)

The envelope detector shows a diversity improvement, but

the square-law detector does not up to Eb/N1>33 dB. Note that

the diversity improvement region for this particular value of

A2/2a 2 is decreasing with increasing Eb/NO for the square-law

detector. If thermal noise is negligible, then partial-band

interference has a more significant impact on system

performance than fading, and no diversity improvement is

obtained with the square-law detector.

(10) E /N-=18 dB and DD=I00 (Figures 25 and 26)

A diversity improvement is obtained for the envelope

detector up to 3 dB but there is no diversity improvement for

the square-law detector. The envelope detector is 3 dB (or as

much as diversity improvement) better in performance. For a

46

particular value of L, it is much better than 3 db because of

the diversity degradation for the square-law detector.

2. Self-Normalization Combining Detectors

(I) EbN,=13.35 dB and DD=0.01 (Figures 27 and

28)

The performances of both of the detectors are almost the

same. Some diversity improvement is obtained, but performance

degradation is as much as 4 dB as compared to linear combining

detectors.

(2) Eb/N-=13.35 dB and DD=I (Figures 29 and 30)

The performances are the same for both of the self-

normalization detectors. These detectors show a diversity

improvement, but performance is degraded up to 3.5 dB as

compared to the linear combining detectors.

(3) Eb/N-=13.35 dB and DD=I0 (Figures 31 and 32)

The performances of both of detectors are much alike.

Some diversity improvement for Eb/N1>l0 db is obtained. Some

performance improvement is achieved by the square-law detector

for 12 db<Eb/NI<28 dB, but this improvement is because of the

deficiency in the performance of the square-law linear

combining detector. Both of the detectors suffer a

performance degradation for Et/NI>28 db, and this degradation

with respect to the linear combining envelope detector is 3 dB

at EINI=40 dB.

47

(4) E /N-=1 3 .35 dB DD=I000 (Figures - aiid 34)

The performances of both of the detectors are the same.

Diversity improvement is obtained for 10 L--/NI<38 dB.

Performance improvement for both of the detectors with respect

to the linear combining envelope detector is about 5 dB.

(5) Eb/N1 =16 dB and DD=0.01/I (Figures 35, 36 and

37, 38)

The performances of both of the detectors are the same,

with a diversity improvement but not a performance improvement

(with respect to the linear combining envelope detector).

(6) Eb/N I=I6 dB DD=I0 (Figures 39 and 40)

Both of the detectors have the same performance.

Diversity improvement and performance improvement are obtained

for Eb/NI>0 dB. The maximum performance improvement with

respect to the envelope linear combining detector is 7 dB.

(7) E /N=16 dB DD=1000 (Figures 41 and 42)

Performance of the both detectors is the same, a diversity

and a performance improvement is obtained up to 20 dB.

(8) EL,N=1 8 dB DD=I000 (Figures 43 and 44)

The detector performances are the same. Diversity

improvement and performance improvement (up to 15 dB) are

obtained.

48

3. Noise-Normalization Combining Detectors

(1) E /N-=13.35 dB DD=0.01/I (FiQures 45, 46 and

47 48)

The performances of the two detectors are the same.

Diversity improvement but no performance improvement is

obtained.

(2) E=bN=13.35 dB DD=10/1000 (Figures 49, 50 and

51, 52)

The detector performances are much alike. Diversity and

performance improvement relative to the linear combining and

the self-normalization combining detectors are obtained and

maximized for moderate values of Eb/NI.

(3) E/N,=16 dB DD=0.01/I (Figures 53, 54 and

55, 56)

The performances of the detectors are similar. A visible

performance improvement is not obtained.

(4) E=/N =16 dB DD=I0 and 1000 (Figures 57, 58

and 59,60)

Optimum L is 4 or at most 6. The performances of the

detectors are similar. Both have diversity and performance

improvement.

(5) E/N=18 dB DD=I0 (Figures 61 and 62)

Performance and diversity improvement are obtained.

49

B. COMPARISION OF THE PERFORMANCES IN THE ABSENCE OF PARTIAL

-BAND INTERFERENCE

Linear and noise-normalization combining detectors are

analytically shown to have the same performance when the

effect of the partial-band interference is eliminated. Self-

normalization combining detectors have a very poor performance

in this case. Even though they show a diversity improvement

(gained versus fading), noncoherent combining losses are much

more than for the linear and the noise-normalization combining

detectors. All the detectors show a diversity imrrovement for

moderate values of direct-to-diffuse signal ratios, but for

the self-normalization combining detectors the improvement

region begins at greater values of Eb/N o. There is not a

visible performance difference between the envelope and the

square-law self-normalization combining detectors. By

comparing the performances of the envelope and the square-law

linear (also noise-normalization) combining detectors, one can

conclude that for the systems suffering from fading the

square-law detector performs slightly better. For no or

moderate amounts of fading, their performances are the same.

C. PERFORMANCES OF THE NONLINEAR DETECTORS UNDER NO THERMAL

NOISE

The nonlinear combining receivers are analyzed for

A 2/2a2=10. The performances of both detectors for self-

normalization combining (Figures 81 and 82) are found to be

50

the same. Diversity improvement is obtained for EIN>7 dB.

The self-normalization receivers show a visible performance

degradation regarding as compared to the noise-normalization

combining detectors (comparing Figures 81 and 82 with 79 and

80).

Performances of the two detectors for noise-normalization

combining are compared in Figures 81 and 82 . No visible

difference is found. Both have diversity improvement for

Eb/NI> 7 dB.

51

V. CONCLUSION

The linear combining receiver implemented with the

envelope detector is seen to have a diversity improvement both

versus fading and partial-band interference, while the linear

combining square-law detector has a diversity improvement

versus only fading. Performance differences are also

emphasized by the decreasing effect of thermal noise. Under

no interference or wideband (uniform) interference conditions,

the performances of the two linear combining detectors are

found to be the same versus fading.

Self-normalization combining receivers implemented with

envelope and square-law detectors are seen to have the same

performance. They have diversity and performance improvement

compared to the linear combining receivers versus only

partial-band interference. They are very sensitive to fading

and thermal noise. Self-normalization can be a good choice

for down-link communication under good weather conditions and

partial-band interference.

The square-law and envelope detectors implemented with the

noise-normalization combining scheme do not differ in

performance. They have the best performance. Their

performances approach the linear combining receivers under

wideband or no interference conditions. When the signal is

52

completely diffuse there is not any performance improvement

for the noise-normalization combining receivers with respect

to a linear combining receiver implemented with an envelope

detector.

For slow frequency hopping, all the possible detector and

combining type combinations that are inspected have the same

performance.

53

APENDIX: FIGURES

- -:

LL-JU1 0

LAIj

54

z

J-J+

------ /--- -

w3W DW I:

00

55

z

LL..z0

5

_d C

u )

w LL

0i0V

zI

CL.

57

zI00

uii

w w

R 58

00

c(i)

0

--- -- - --- -- - -- --- - ---9

100

EbfNo= 13.35 db, DD= 0,01

o

LL.0

0 102

....... .........

o. ...0 .. ....... .... ... ......

I= solid line

L--2 ... L--4 _._. L=_ _ =

1 030 5 10 15 20 25 30 35 40

EbINi db

Figure 7. Envelope Detector Linear Combining: worst case

performance of the linear combining envelope detector

receiver with diversity combining, partial-band

interference, and thermal noise in a fading channel for a

diffuse signal (A2/2o2-O.Ol) and EIN0 =13.35 dB.

60

100

L Eb/No= 13.35 db, DD= 0.01

oe 101

0

o 10L

L=1l solid line

L_-2 ... L--4 L- . U 6 L-- ~8

1030 5 10 15 20 25 30 35 40

Eb/Ni db

Figure 8. Square-Law Detector Linear Combining: Worst case

performance of the linear combining square-law detector



diffuse signal (A2/2a 2 -O.Ol) and EI/N0 =13.35 dB.

61

-. Eb/No= 13.35 dJb, DD= 1

o 10'

hL

LL.

O= I0 2

s o i c. ... - -- -

L L2.. L4 . = .L6 __ L=8

0 5 10 15 20 25 30 35 40

Eb/Ni db

Figure 9. Envelope Detector Linear combinling: Worst case




signal with equal direct and diffuse components (A/2c2=1)

and E1/N0 13.35 dB.

62

IIo=13.35 dh, DD= 1

S10'

~ 0 jL--1 solid line ..

L--2 ... L_-4 L_ . L6 L8~

10'0 5 10 15 2(0 25 30 35 40

Eb/'Ni db

Figure 10. Square-Law Detector Linear Combining: Worst

case performance of the linear combining square-law detector



signal with equal direct and diffuse components (A2/2 C2 =1)

and Eb,/N,=13.35 dB.

63

100

Eb/No=]13.35 db, DD= 10

o 102

0

L = 1 so lid line --------- .... ... ...

L_2... Li- -.4 . L=6 L=8"

10 5 1'0 15- 20 25s 3'0 3'5 40

Eb/Ni db

Figure 11. Envelope Detector Linear Combining: Worst case




relatively strong direct signal component (A2/2a2=10) and

E1No=I3.35 dB.

64

100

1 0 l Eb/No= 13.35 db, DD= 1000

L=- I solid line -----------

L_-2 ... L_- 4 _ L6 __ L= 810,

1050 5 10 15 20 25 30 35 40

Eb/Ni db





strong direct signal component (A2/2aC2=1lOOO) and EI/N0 =13.35

dB.

66

Eb/1No-13.35 db, DD= 1000

10'

0 2

101-

L-- 1 solid line

104 L--2.. L_-4 L_-6 - _ L 8

0 5 10 15 20 25 30 35 40

Eb/Ni dt)





strong direct signal component (A 2/2 C2=1000) and Eb,/N,=13.35

dB.

67

100~

K ~*Eh!No=]16 Ub, DD= 0.01

0 L-

1 0 1 ---- -----

L0-[ 3 solid line

[=-2 ... L= 4 L= .[6 L-8~ bId


performance of the linear co~mbining envelope detector



diffuse signal (A2/2o2-Q.Ol) and EI,,N 0 =16 dB.

68

Eb!No=16 db, DD= 0.01

10 <

LL "

0 1011

0

01 L--I solid line

L_2... L.4 ..8

0 5 10 15 20 25 30 35 40

Eb/Ni db

Figure 16. Square-Law Detector Linear combining: Worst




diffuse signal (A212 2 =0 01) and EN 0 =16 dB.

69

* Eb/No= 16 db, DD) I

0

LU

L_-1 solid line

2 .- ... L_-4 . _ .U6 __ L=8

5 10 15 20 25 30 35 40

Eh/'Ni db

Figure 17. Envelope Detector Linear combining: worst case




signal with equal direct and diffuse components (A2/2c2=:l)

arnd EjN0 =16 dB.

70

Lh,'No= 16 db, DD= 1

10

C 1O2L

L- I ol- m

-- - -----

101

Fiur 18 squae-La leetriierCobnngeos

case pefomac of th iercmiig qaelwdtco

and~LbN Eb/Nb6 B

Fiur 1. qureLa Dteto Lner ominng7Wrs

10,~ EhNo= 16 db, DLD= 10

0

]OIL

101

L--2 ... L--4 .. -L6 L__ L~8

0 5 10 15 20 25 30 J5 40

EbtNi dh





relatively strong direct signal component and (A 2/2c2=10

Et,/N,=lG dB.

72

100

Eb/N'o- 16 db, DD= 10

0

U 102

0

0 L_- I solid liznc

L=-2 ... L= 4 L,-6 - L 8

U 0 5202 3 354

1u~~u~ Eb/Ni db 54

Figure 20. Square-Law Detector Linear combining: Worst




relatively strong direct signal component (A2/2c 2=10) and

EIN,=l6 dB.

73

100

- -. Ebh'No= 16 dl,, DD= 1000

10,

WL 102

LL.

~ 0 '

0

L~I solid line

101-

L-- .. L= 4 L-- L6 _ _L8

1050 5 10 15 20 25 30 35 40

Eb/Ni Ob

Figure 21. Envelope Detector Linear Combining: worst case




strong direct signal component (A2 /2 C2=lOOO) and EtIN0 =16 dB.

74

100

E bNo= 16 d 1, DD= 1000

101-

0

10U'

L-- 1 sulid line

104 L-- 2 L_-4 _ _ -L6 L=8

1000 5 10 15 2 53 54

Eb/Ni db





strong direct signal component (A2/2a 2 -lOOO) and EbN 0 =16 dB.

75

Eb/No= 18 db, DD= 10

LJ 102L

L.

10'

0 L= I solid line

L=2 ... L=4

1050 5 10 15 20 25 30 35 40

Eb/Ni db





relatively strong direct signal component (A2/2a2=10) and

E,,/No=18 dB.

76

100

101 "\ " Eb/No= 18 db, DD= 10

o N

10z ',

0LI 1 solid line

104 L_,2 ... L=4 ..

100 50 15 25 40

Eb/Ni db





relatively strong direct signal component (A2/2C2=10) and

EINo=18 dB.

77

10)f b.No 18 d b, DD= 100

0

102

0

10~~~hN db-Isli i~

Fiue2.Evlp0eetrLna obnn: Wrtcs

pefrac ftelna obnn neoedtco

receiver.. wihdvrst obiig arilLn

10'.frece an-hra4oiei aig hne o

strog drec sigal.comonet (A2 /c2=00 and b,/,=1 dB

10>- 78

100

101 Eb/No=18 db, DD=100

0

S 101"

0 L= 1 solid line

L=-2 ... L 4 __

0 5 10 15 20 25 30 35 40

Eb/Ni db





strong direct signal component (A2/2C2=100) and Et/No=18 dB.

79

109s

Eb/No= 13.35 db, DD= 0.01

L L= I solid line ---

L~2. L~4_ L=_ L=S8

10 2 .. _____________________________ _

0) 5 101 15 20 25 30 35 40

Eh/Ni db

Figure 27. Envelope Detector Self-Normalization combining:

Worst case performance of the self-normalization combining

envelope detector receiver with diversity combining,

partial-band interference, and thermal noise in a fading

channel for a diffuse signal (A2/2oC2=0.01) and EIN,=13.35

dB.

80

K EtO/No= 13.35 db, DD= 0.01

L11

101I

0

L--I solid line r

L-- 2... L= 4 L_-6 _ b 8

10 2L520 5 10 12025 30 35 40

Eb/Ni Jb

Figure 28. Square-LaW Detector Self-Normalization

Combining: Worst case performance of the self-normalization

combining square-law detector receiver with diversity

combining, partial-band interfeirence, and thermal noise in a

fading channel for a diffuse signal (A2/2a2=0.Ol) and

EIN,=13.35 dB.

81

F Eb/No= 13.35 db, DD= I

C 0 Nj N

L-I SOUf 11rc

L=-2.. L_-4 L- L 6 ---- -- .... .

0 5 10 U20 25 30 35 40

Eb/Ni d b

Figure 29. Envelope Detector Self-Normlalizationl Combining:


envelo~pe detector receiver with diversity combining,


channel for a signal with equal direct and diffuse

components (A2/2c2=l) and Eb/N.=13 .35 dB.

82

100

Eb/No= 13.35 d b, DD= I

0

o 101

9L L

L- I olid line r'

10 2~ L ~ 4. I~ ~..0 5 10 15 20 25 30 35 40

EbfNi db

Figure 30. square-Law Detector Self-Normalization

Combining: worst case performance of the self-normalization


combining, partial-band interference, and thermal noise in a

fading channel for a signal with equal direct and diffuse

components (A 2/2 02=1l) and Eb,/N 0=13.35 dB.

83

101'

Eb;Nou' 13.35 dOb, DD= 10

F01

C 10

I S o i 1 1CU -- --- --

1 1 . I Ind ln

0 5~~ 10 15 20 25 30 35 40

Eb/N dob

Figure 31. Envelope Detector Self-Normalization combining:




channel for a relatively strong direct signal component

(A2/2o2-lO) and EI/N0 =13.35 dB.

84

100.

Eb,/No=13.35 db, DD=K10 -

o 10 L

.r

0..........

L=2 L_-4 . -. L 6 L-8---------- - -

10j0 5 10 15 20 25 30 35 40

Eb/Ni db

Figure 32. Squ~are-Law Detector Self-Normalization


combining square-law detector receiver with divel'sity


fading channel for a relatively strong direct signal

component (A2/2c'=10) and E1,N0 =13.35 dB.

85

EhfNo= 13-35 db, DD= 1000

101

0-

1 -- ~ 2 .. L--4 __ L=6 L= 8

0 5 1015 20 25 * 30 35 40

E 1,N dOI

Figure 33. Envelope Detector Self-Normalization Combining:




channel for a strong direct signal component (A 2/2aC2=1OOO)

and Eb/N 0O'l3.35 dB.

86

10 0,- --1

C44

102

10~

10' L-- 1 solid line ~-~

L-- 2... L=4-.-. L -6 __ L=8..

1050 5 10 15 20 25 30 35 40

Eb/Ni db

Figure 34. Sqluare-Law Detector Self-NorMalization




fading channel for a strong direct signal component

(A'/ 2 a2 =-1000) and EINO=13.35 dB.

87

100

EbINo= 16 d b, DD= 0.01

0 iu0-

LL

0 1 2~

- L= I sulid kne ----

1) L _ L2 ... L= 4 . L _ L8~

(51015 20 25 30 35 40

Eb/Nj dOb





channel for a diffuse signal (A2/2cT 2=O.Ol) and EI/N.=16 dB.

88

Ell No= 16 d b, DD= 0.01

o 102

L-- I solid line-----

L--2 ... L--4 L--6 L= 8

(1510 15 20 25 30 35 40

Eb/Ni db

Figure 36. Square-Law Detector Self-Normalization




fading channel for a diffuse signal (A2/12a2=O.01) and

Eb,/N 0=l6 dB.

89

10o

Eb/No= 16 db, DD= 1

0 101~

o \,

0 102 "

------- ---- .......

L = 1 solid line ------ ------1.,=2 ... L -- 4 . _ . 1, 6 L -8 "

10 5 1 15 20 25 30 35 40

Eb/Ni db






components (A2/2a2=1) and Eb/NO=I6 dB.

90

Eb/No= 16 db, DD= I

..... ....

0 I soi5ln

5 101 2 5 0354

Eb/Ni 7

Fiur 38 SqolidLa Deetiel-oraizto

Foire 3.square-law detector SeeeNrmiaionrit

combining, partial-band interterence, and thermal noise in a

fading channel for a signal with equal direct and diffuse

components (A 2/2 C2=1l) and Eb/NO=l6 dB.

91

100

EbINo= 16 db, DD= 107

'± 102 -

L.0

0

10' -1., I solid line

L--2 ... L-4 _.. L=6__ L.-8"

10'1105 10 15 20 25 30 35 40

Eb/Ni db






(A2/2a2=l0) and Eb/N0=16 dB.

92

1as

10*'

EbIN'o=6 db, DD= 10

0

103

0

S101

LI solid line *: .: ...

L--2 ... L--4-.-. L--6 L=

0 5 10 15 20 25 30 35 40

Eb/Ni db


Combining: Worst case performance of the self-normalikatibn




component (A 2/2 C2=1O) and Eb/110=16 dB.

93

100)

101-. Eb/No= 16 d b, DD= 1000

10g2

0

LL.

j 1()-4

0

105

L--I solid line106-- ---- -

L_-2 ... L--4 L_-6 1 8

10110 5 10 15 20 25 30 35 40

Eb/Ni db





channel for a strong direct signal component (A2/20r'=1000)

and E1,N0 =16 dB.

94

. Eb'No= 16 ib, DD= 1000

0

S10 3

Ui.

0

0

L_- I solid line

L--2 ... L_-4 L_-6 __ L 8

10' 20 5 10 15 2025 30 35 40

Eb/Ni db





fading channel for a strong direct signal component

(A'/ 2oC2-lOOO) and Eb/NO=1E dB.

95

EbINo=12 db, DD= 10

0 1012

o 101z

L_ I solid line

L--2 ... L_-4

106 ---0 5 10 15 20 25 30 3;40

Eb/Ni db


Worst case performance of the self-normpijzation combining



channel for a relatively stronff direct signal component

(A2/2a2=±o) and EIN0 =18 dB.

96

1011

Eb/No= 18 db, DD=10

0

0 I

L= I solid line

10 _ L=2 ... L_,=4 .....

0 5 10 1'5 20 25 30 35 40

Eb/Ni db






component (A2/2a 2=10) and EINo=18 dB.

97

100

- . EbfNo=13.35 db, DD= 0.01

o 10'

LL.

0 10 , - ---------

L= I solid line

L_-2 ... L_-4 .. L=6- L=83

103,0 5 10 15 20 25 30 35 40)

Eb/Ni db

Figure 45. Envelope Detector Noise-Normalization Combining:

Worst case performance of the noise-normalization combining



channel for a diffuse signal (A 2/2C2=O.01) and Eb,/N=13.35

dB.

98

100

Eb/No= 13.35 db, DD= 0.01

C) 10'

LUJ

0

C)102~

L=1solid line

L--2 ... L--4 L-- L6 L=8

1030 5 10 15 20 25 30 35 40

Eb/Ni db

Figure 46. Square-Law Detector Noise-Normalization

Combining: Worst case performance of the noise-

normalization combining square-law detector receiver with

diversity combining, partial-band interference, and thermal

noise in a fading channel for a diffuse signal (A 2/2 C2=-.Ol)1

and EIN,=13.35 dB.

99

ELV'u= 13.35 dt,, DD= 1

0 102.Fr

0 1 31 L

Figur 47li lnveop DeetrNieNraiainCmiig

10~~ L ~ 100

I UO

Eb/No= 13.35 db, DD= I

jo11

0 1K

-------------- --------------- -----

L--2 ... L--4 L- . L 6 _ L 8'

10 , 1 L-0 0 15 20 25 30 35 40

Eb/Ni db





noise in a fading channel for a signal with equal direct and

dif fuse components (A 2/2a 2=1l) and EIN0 =13.35 dB.

101

100

Eb,'= 13.35 db, DD= 10

ol

i_- I...................................

10)0 5 10 15 20 25 30 35 40

Eb/Ni db

Figure 49. Envelope Detector Noise-Normalization combining:





(A'/2ac2=10 ) and Eb/NO=l 3 . 3 5 dB.

102

10L:

EO No!N= 13.3 5 (lb, DD= 10

IG1

------- -------- -------- -------10-

10,1

0 5 0 1520 2 30 5 4EON db

Fiue5.Sur-a eeco os-omlzto

0 5 10 15 ~02 0 54

100

Eb/No= 13.35 db, DD=100010'

, 102 '

LLo

L=1I solid line ---------- d10'-

L-- .. L 4 . _ . L 6 I_-8 "

0 5 1 15 20 25 30 35 40

Eb,/N'i db





channel for a strong direct signal component (A2/2C2=1000)

and EINo=13.35 dB.

104

100 -_____________________________________

10' Eb/No= 13.35 db, DD= 1000

LL

0

I1 solid line

L_-2.. L-1 L-- L6 __

105tL _L

Eb/Ni db

Figure 52. Square-LaW Detector NOiSe-Normalization

Combining: Worst case performance Of the noise-

normalization combining square-laW detector receiver with


noise in a fading channel for a strong direct signal

component (A2/2a 2 -1000) and Eb/N 0=l3.35 dB.

105

0Eb/No= 16 db, DD= 0.01

Q102

L

0

10. -L0*

0 5 1-0 15 20 5 30 35 40

Eb/Ni db





channel for a diffuse signal (A2/22=O.O1) and Eb/No=16 dB.

106

100

10' Eb,Tho= 16 Ob, DD= 0.01

o 102

0

................................

C 1051 52 53 54

L-orma liatin combining----- squar-lawdetecor-rceive-wit

and 2I0 1 .. dB.4L- -

1107

100

- . : Eb/lNo= 16 db, DD= I

101

*I-

o 102

0

10 3 L,- 1 so lid line ----------------------------------

---------------------- -I---

L_=2 ... L_=4 . . L=6 L=8"

10-1.0 5 10 15 20 25 30 35 40

Eb/Ni db






components (A2/2a 2=l) and EIN.=16 dB.

108

i~4: -. ~Eb,o~= 16 db, DD= I

LL.

10: L-I solid line

-- -- -- - -- - - - -

0 = 5. ~ 10 15 20T~id 25 30 35 40

Figure 56. Square-LaW Detector Noise-Normalization

combining: Worst case performance of the noise-



noise in a fading channel for a signal with equal direct and

diffuse components (A 2/2C2=1 ) and Et/N,=16 dB.

109

10o~

' ]02 ,N= 16 db, DD= 10

10 Ft

C44

0 10 '. ...

< oo iu''.

L= - I solid l c"': ' -" :< ""

L= ... L=4 _ 6 8:.

106!0 5 10 15 20 25 30 35 40

EbNi db






(A2/2C 2=O) and EWNo=16 dB.

110

I V)

101Eb/No 16 db, DD= 10

0 10 "

0 10-

o) 10 4

L_. 1 solid line

10' 5

L= 2 ... L= 4 L_ -6 .L=8=

1060 5 I0 15 20 25 30 3 5 40

Eb/Ni db

Figure 58. Square-Law Detector Noise-normalization

Combining: Worst case perfor ,a.nce of the noise-



noise in a fading channel for a relatively strong direct

signal component (A2/2a2=I0) and Eb/NO=I6 dB.

111

100

Eb/No= 16 db, DD= 100010' -

102

LL.0 104-

< 105

10 6,"' '-L=- I solid line

101-~" -------

109

05 10 15 20 25 30 35 40

Eb[Ni db





channel for a strong direct signal component (A2/2a2=1000)

and E/No=16 dB.

112

100

10, Eb/No= 16 db, DD= 1000

102 -4

S10 3

0 10-' ,

< 105

106

L=-I solid line

107 L=2... L=4 _ .6 L -

1080 5 10 15 20 25 30 35 40

Eb/Ni db






component (A2/20 2=1000) and E,/N,=I 6 dB.

113

100

101 Eb/No= 18 db, DD= 10

1ok -102

103'

LU

105 -.

106 L 1=1 solid line

L=2... L=410'

108 L_

0 5 10 15 20 25 30 35 40

Eb/Ni db






(A2/2c2=I0) and EINo=18 dB.

114

100

10 1 ~Eb/No=18 db, DD= 10

102~

o03r

0 104

< 10.5 V10-6

L-- I solid line

1 7 - L_-2 ... L-4 ------ -----

]OR80 5 1 0 15 20 25 30 3'5 40

Eb/Ni db

Figure 62. square-Law Detector Noise-Normalization





signal component (A2/2C 2-lO) and Eb/N.=18 dB.

115

DD=0,01

o102-

I-Io 103

o 10 -

L--2 ... L--4 L--6 __ L= S

1010 2 4 6 8 10 12 14 16 18 20

Eb/Na db

Figure 63. Envelope Detector Linear and Noise-Normalization

Combining: Performance of the linear and noise-

* normalization combining envelope detector receiver with

diversity combining, and thermal noise in a fading channel

f or a dif fuse signal (A2/2c'=O.01).

116

10,

FDD 0.01101i

0 102 -

II

10 5- L=2.. L-4 L-6 Li

10-1

0 I6046 21)

o io'o d

n L 1 solid cinc10sk L=2.. L..4 __. L.6 __ L8"

1117

0 2 4 6 8 1012 14 16 1'8 21)

Eb/No db

Figure 64. Square-Law Detector Linear and Noise-

Normalization Combining: Performance of the linear and

noise-normalization combining square-law detector receiver

with diversity combining, and thermal noise in a fading

channel for a diffuse signal (A2/2a 2 =0.01).

117

l00

10' DD=I

L0,

S101

L= 1 solid line

1 0 6

l0U 0 = 2 .. 24 6 8 16 1 48 20

Eb/No db



normalization combining envelope detector receiver with


for a signal with equal direct and diffuse components

(A2/2a2=1)

118

100

l------ DD=1

102

S 102~

0

10'"

0

0 "-

- iOs ,.,

L--1 solid line

06[ -=2 ... L_-=4 __ L=6 __ L=8

10 7,

0 2 4 6 8 10 12 14 16 18 20

EbINo db





channel for a signal with equal diffuse and direct

components (A2/2 2=l).

119

102

10

oa I

C) 10 4IH I

o 106 '< 10 1

lo l

10 b L= i soi d line "

L_,2 ... L_,4 __. L=6 _ _ "' 8"" '10'

0 2 4 68 10 12 14 16 18

Eb/No db





for a relatively strong direct signal (A /2a 2 =10).

120

100

D 1

102

I01-

l 0--.. .. .

i'd

o 10 4

< 10 iO

10k- L=I solid line

101 L-=2 ... L=4 .. L=6 __ L=8

1081

02 4 6 8 10 12 14 16 18

Eb/No db





channel for a relatively strong direct signal (A2/2a 2=10) •

121

]0o

DD= 1000

0 10 3 .

LU

r ~ 104

0

0 106 ,.,

L= 1 solid line

10 L--2 .. L--4 L_ L=

1010 2 6 8 10 12 14 16

Eb/No db





for a st-_:ng direct signal (A2/2o2=1000).

122

IU -.-- -- . . DD= 1000

S10< 2

10 L

L~L10"

1 \I1 L

L--2 ... L_-4 . = L 6 L8

1091

10




channel for a strong direct signal (A,/2,,,lOOO)

123

---------------- D D = 0.01

S 101-

o N

101"

I L=2 .L=4 L= 6 I 8

0 4 6 8 0 12 14 16 18 20

Eb/No db


Performance of the self-normalization combining envelope

detector receiver with diversity combining, and thermal

noise in a fading channel for a diffuse signal

(A /2o2=0.01)

124

100

DD= 0.01

10'2

0o L2. L4 _L6 8 10 12 14 16 js 2

Eb/No db


Combining: Performance of the self-normalization combining

square-law detector receiver with diversity combining, and

thermal noise in a fading channel for a diffuse signal

(A2,'2a2-O.O01)

125

10 2E"':' ~DD I0 . -

I.. i solid line '

I_- 2 ... L--4 L.-. _. 6 _ L=8 "".

05

0 46 10( 1'2 1'4 16 1L8 20

Eb/No db


Performance of the linear and self-normalization combining

envelope detector receiver with diversity combining, and

thermal noise in a fading channel for a signal with equal

direct and diffuse components (A2/2o 2=I).

126

DD= I

•01 "

0 2:

1I-

o

LLL

10) L= I socid iiic ,

- ,= .. L.=4 .__ L=(6 _ =8'*

0 2 4 6 L 10 124 1 6 1'8 20

EbiNo db

Figure 74. Square-Law Detector Self-normalization



thermal noise in a fading channel for a signal with equal

diffuse and direct components (A2/2ao=l).

127

10 .. DD= 10

102.

0

o -

D 101-

<"x

010 5"',

L= I solid line

106 L-2 ... L -4 _. --

1070 2 6 112 I4 I6 18

Eb/No db





signal (A2/2o 2=lO)

128

101 DD= 10

102~

L.

0

01

L=I solid honc

U 2 46 8 10 12 14 16 is

EbINo dJb




thermal noise in a fading channel for a relatively strong

direct signal (A2/2a2-lO) .

129

101

. . - -DD= 1000

o:'!

C 103 ,

O 2

cz I

LL

Sio0Li.. =. . L 6 = "

0

o 106 d

I solid line101(

L 2 ... L-- 4 L-- =6 -3

10,

0 24 6 8 10 12 14 16

f Figure 77. Envelope Detector Self-Normalizationl combining:




(A2/2a2-l000)

130

10 0 , - .. . ... . . . ,

DD= 1000

IU I

S10%

L. I2 _ -

10%. -.-

<N"

S 1061 -\

L=J solid linc107

L 2... L=4 ._ L=6 L S

U 2 4 6 8 10 12 14 16

EbNo db




thermal noise in a fading channel for a strong direct signal

(A 2/2a 2=I000)

131

,on

10'

0 10,

< 1050

r," L_- I solid line "

L=-2 ... L _ _. "4,

107

10

0 5 10 15 20 25

Eb/Ni db


Performance of the noise-normalization combining envelope

detector receiver with diversity combining, and partial-band

interference in the absence of thermal noise, and in a


(A/2O2=I0)

132

100'

102

0

U..

1050 I',

0

L= I solid line106 -

_ L=2... L=4

10 1.

0 5 10 15 20 25

Eb!Ni db


Combining: Performance of the noise-normalization combining


partial-band interference, in the absence of thermal noise,

and in a fading channel for a relatively strong direct

signal (A2/2-2=10) .

133

,, .

101"

0",

C

LLU

0

0

: I_ 1 solid line-10~

L=2... L=4

100 5 10 15 21 25

Eb/Ni db



detector receiver with diversity combining, and partial-band

interference in the absence of thermal noise, and in a


(A2/2a 2=10).

134

I0r10 'i . --._.2 2

L_= I solid line.- 105

.= 2 L= 4

0 5 10 15 20 25

Eb/Ni db




partial-band interference, in the absence of thermal noise,

and in a fading channel for a relatively strong direct

signal (A.2 /2C 2=1).

135

REFERENCES

1. R. C. Robertson, and T. T. Ha, 'Error Probabilities ofFrequency-Hopped FSK with Self-Normalization Combining ina Fading Channel with Partial-Band Interference, I IEEETrans. Commun., forthcoming.

2. J. S. Lee, L. E. Miller, and Y. K. Kim, 'Probabilityof Error Analyses of a BFSK Frequency-Hopping System withDiversity Under Partial-Band Jamming Interference Part II:Performance of a Square-Law Nonlinear Combining SoftDecision Receivers, ' IEEE Trans. Commun., Vol. COM-32,no. 12, pp. 1245-1250, Dec. 1984.

3. R. C. Robertson, T. M. Clemons III, and T. T. Ha,'Error Probabilities of Frequency Hopped MFSK with Noise-Normalization Combining in a Fading Channel with Partial-Band Interference, I IEEE Trans. Commun., forthcoming.

4. J. S. Lee, R. H. French, and L. E. Miller,'Probability of Error Analyses of a BFSK Frequency-HoppingSystem with Diversity Under Partial-Band JammingInterference-Part I: Performance of Square-Law LinearCombining Soft Decision Receiver,' IEEE Trans. Commun.,Vol. COM-32, no. 6, pp. 645-653, June 1984.

5. A. D. Whalen, Detection of Signals in Noise. New York:Academic Press, 1971.

6. G. N. Watson, 'A Treatise on the Theory of BesselFunctions, I Macmillan, New York, 1945.

7. L. E. Miller, J. S. Lee, and A. P. Kadrichu,'Probability of error analysis of a BFSK frequency-hoppingsystem with diversity under partial-band jamminginterference Part III: Performance of a square-law self-normalizing soft decision receivers, ' IEEE Trans.Commun., vol. COM-34, no. 7, pp. 669-675, July 1986.

8. G. A. Campbell, and R. M. Foster, 'Fourier Integralsfor Practical Applications, Van Nostrad, Princeton, NewJersey, 1954.

9. J. S. Lee, L. E. MILLER, and R. H. French, 'TheAnalyses of Uncoded Performances for Certain ECCM ReceiverDesign Strategies for Multihops/Symbol FH/MFSK Waveforms,I IEEE J. Sel. Areas Commun., vol. SAC-3, no. 2, pp. 611-621, Sep. 1985.

136

10. W. C. Lindsey, 'Error Probabilities for Rician FadingMultichannel Reception of Binary and N-ary Signals, ' IEEETrans. on Infor. Theory, Vol. IT-10, pp. 339-350, Oct.1964.

11. B. Solaiman,a. Glavieux, and A. Hillion, 'ErrorProbability of Fast Frequency Hopping Spread Spectrum withBFSK Modulation in Selective Rayleigh and Selective RicianFading Channels,' IEEE Trans. Commun., Vol. 38, No. 2, pp.223-240 Feb. 1990.

137

Documents

THESIS - DTICMaster's Thesis I From To March 1991 16 Supplementary Notation The views expressed in this thesis are those of the author and do not reflect the official policy or position