LGICA COMBINACIONAL

0. In

troducci

n1. P

uerta

s lg

icas b

sica

s2. T

abla

de

Verd

ad y F

unci

n L

gica

3.

lgebra

de B

oo

le: le

yes, re

gla

s y teore

mas D

eM

org

an

4. R

eta

rdos

5. E

xpre

siones C

an

nica

s6. S

mplifica

cin

por K

arn

au

gh

7. L

gica

unive

rsal: N

AN

D, N

OR

8. M

ultip

lexo

res, D

em

ultip

lexo

res, C

od

ificad

ore

s, Deco

difica

dore

s9. S

um

ado

res y re

stadore

s. Carry/B

orro

wlo

gicL

L

H

En la

ele

ctrn

ica d

igita

l se u

san s

lo d

os n

ivele

s de te

nsi

n:

Nive

l lg

ico a

lto (h

igh

) HN

ivel l

gico

bajo

(low

) L

Dg

itos b

inario

s (bit):

Lg

ica p

ositiva

: 0=

L 1

=H

Lg

ica n

eg

ativa

: 1=

L 0

=H

L H

V

VH

min

VD

D

VL

ma

x

0

V

t

transito

rio

0

0

1

V

t1

1

0

V

t

INTRODUCCIN

Com

bin

acio

na

lin

put

outp

ut=

F(in

put)

nm

Secu

encia

lin

put

outp

ut=

F(in

put,sta

te)

nm

Circu

ito C

om

bin

acio

nal:

Es u

n circu

ito sin

mem

oria

Para

una d

ete

rmin

ada

entra

da, la

salid

a e

s la m

isma

Circu

ito S

ecu

encia

l: E

s un circu

ito co

n m

em

oria

(esta

do)

Para

una d

ete

rmin

ada

entra

da, la

salid

a d

ep

end

e d

el e

stado

(depen

de d

e e

ntra

da

s ante

riore

s)

state

INTRODUCCIN

INTRODUCCIN

Imple

menta

cion

es

Lg

ica d

iscreta

(TT

L 7

4X

X, C

MO

S 4

000.

)

Lg

ica p

rogra

mab

le(F

PG

A, C

PLD

, PA

L

)

INTRODUCCIN

Un circu

ito d

igita

l com

bin

acio

na

lpue

de d

escrib

irse co

mo:

abc

o

o=

a*(b

+c)

a b c o

0 X X 0

1 1 X 1

1 0 0 1

1 0 1 0

Circu

itoD

igita

lC

om

bin

acio

na

l

Sch

em

atic

Alg

ebra

Tru

eT

able

PUERTAS LGICAS BSICAS

ao=

aa a

0 1

1 0

NO

T

abo=

a*b

a b

a*b

0 0

00 1

01 0

01 1

1

a b

a*b

0 X

0X

0 0

1 1

1A

ND

abo=

a+

b

a b

a+b

0 0

00 1

11 0

11 1

1

a b

a+b

0 0

0X

1 1

1 X

1

OR

(inclu

sive O

R)

abo=

a

b

a b

a

b0 0

00 1

11 0

11 1

0

XO

R(e

xclusive

OR

)

a

ba=

b 0

a

b1

b a

b

0 a

1 a

b a

*b0 0

1 a

b a

+b

1 1

0 a

PUERTAS LGICAS BSICAS

abo=

a*b

a b

a*b

0 0

10 1

11 0

11 1

0

a b

a*b

0 X

1X

0 1

1 1

0N

AN

D

abo=

a+

b

a b

a+b

0 0

10 1

01 0

01 1

0

a b

a+b

0 0

1X

1 0

1 X

0N

OR

abo=

a

b

a b

a

b0 0

10 1

01 0

01 1

1

a

ba=

b 1

a

b0

XN

OR

NX

OR

b a

b

0 a

1 a

b a

*b0 1

1 a

b a

+b

1 0

0 a

PUERTAS LGICAS BSICAS

1

Sm

bolo

s Sta

ndard

y AN

SI/IE

EE

91-1

98

4:

&&

1 1

=1

=1

PUERTAS LGICAS BSICAS

Equ

ivale

ncia

s:

aba*b

aba*b

aba*b

aba*b

x y x+y

0 0

1X

1 0

1 X

0

x=a

y=b

x+y

a b

a*b

1 1

1X

0 0

0 X

0

x y x+y

0 0

0X

1 1

1 X

1

a b

a*b

1 1

0X

0 1

0 X

1

x=a

y=b

x+y

aba+

bab

a+

bx y x*y1 1

0X

0 1

0 X

1

a b

a+

b0 0

0X

1 1

1 X

1

x y x*y1 1

1X

0 0

0 X

0

a b

a+

b0 0

1X

1 0

1 X

0

x=a

y=b

x*y

aba+

bab

a+

bx=

ay=

b

x*y

PUERTAS LGICAS BSICAS

Puerta

s lg

icas d

e m

s d

e 2

entra

das

a*b

*c*d

a b

c d A

ND

1 1

1 1

10 X

X X

0X

0 X

X 0

X X

0 X

0X

X X

0 0

abcd

abcd

a*b

*c*d

a b

c d N

AN

D1 1

1 1

00 X

X X

1X

0 X

X 1

X X

0 X

1X

X X

0 1

abcd

a+

b+

c+d

a b

c d O

R0 0

0 0

01 X

X X

1X

1 X

X 1

X X

1 X

1X

X X

1 1

abcd

a+

b+

c+d

a b

c d N

OR

0 0

0 0

11 X

X X

0X

1 X

X 0

X X

1 X

0X

X X

1 0

abcd

a

b

c

da b

c d X

OR

Nim

par1

1N

par1

0

abcd

a

b

c

da b

c d X

NO

RN

par1

1N

impar1

0

TABLA DE VERDAD / F. LGICA

La ta

bla

de

verd

ad d

e u

na fu

nci

n l

gica

F se

obtie

ne e

valu

and

oF

para

todas la

s posib

les co

mbin

acio

nes d

e e

ntra

da

F(a

,b,c)=

a+

(b

c)bc

a

F(1

,b,c)=

1+

(b

c)=

1=

0

F(0

,b,c)=

0+

(b

c)=

b

c

F(0

,0,c)=

0

c=c

10

b

c0b

cb

c

00c

cc

1cc

c0

F(0

,1,c)=

1

c=c

a b c F

0 0 0 1

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 0

LGEBRA DE BOOLE

Co

nju

nto

de L

eye

s y Re

gla

s, que p

erm

iten a

na

lizar/sim

plifica

run circu

ito d

igita

l

a+

b=

b+

aa*b

=b*a

abba

==

abba

Ley a

socia

tiva (su

ma y m

ultip

licaci

n):

Ley co

nm

uta

tiva (su

ma y m

ultip

licaci

n):

a+

(b+

c)=(a+

b)+

ca*(b

*c)=(a

*b)*c

abbc

a

c=

bc

aab

c=

Ley d

istributiva

:

a*(b

+c)=

a*b

+a*c

bc

abc aa

=

LGEBRA DE BOOLE

Re

gla

s

a+

0=

aa+

1=

1

a*0

=0

a*1

=a

a+

a=

aa+

a=

1

a*a

=a

a*a

=0

a=

a

aa

0a

11

aa

1a

a0a1

aa

0a

10

aa

a+

a*b

=a

a+

a*b

=a+

b(a

+b)*(a

+c)=

a+

b*c

abab

ac b

bc

a

a

b

aa

LGEBRA DE BOOLE

Teore

mas D

eM

org

an

a*b

=a+

b

a+

b=

a*b

aba*b

aba+

b =

a*b

aba+

bab

a*b

= a

+b

LGEBRA DE BOOLE

Eje

mplo

: F

1 (a,b

,c)=a*b

+ a

*(b+

c) + b

*(b+

c)

a*b

+ a

*(b+

c) + b

*(b+

c)

a*b

+ a

*b +

a*c +

b*b

+ b

*c

a*b

+ a

*c + b

+ b

*c

a*b

+ a

*c + b

a*c +

b

F1 (a

,b,c)=

a*c +

b

abc

ac

b

x+x=

xx*x=

x

x+x*y=

x

x+x*y=

x

LGEBRA DE BOOLE

Eje

mplo

: F

2 (a,b

,c)=a*b

+ a

*c + a

*b*c

a bc

abc

F2 (a

,b,c)=

a+

(b+

c)

a*b

+ a

*c + a

*b*c

(a*b

) * (a*c) +

a*b

*c

(a+

b)*(a

+c) +

a*b

*c

a +

b*c +

a*b

*c

a +

b*c =

a+

(b+

c) = a

*(b+

c)

F2 (a

,b,c)=

a*(b

+c)

abc

F2 (a

,b,c)=

a+

(b*c)

Morg

an

Morg

an

(x+y)*(x+

z)=x+

y*z

x+x*y=

x

abc

RETARDOS

Reta

rdos: L

as p

uerta

s lg

icas n

ece

sitan u

n tie

mpo (td ) p

ara

re

accio

nar a

las e

ntra

das y p

ropagar su

salid

a a

las sig

uie

nte

s

abc

a b c a*(b+c)

0 X X 0

1 1 X 1

1 0 0 1

1 0 1 0

o=

F(a

,b,c)

=a*(b

+c)

o abcc

cb+

c

b+

cRETARDOS

La fu

nci

n l

gica

F e

s vlid

a d

esp

us d

e u

n tie

mp

o tra

nsito

rioP

ue

de

n a

pare

cer g

litches

para

alg

un

as tra

nsicio

nes d

e e

ntra

das

abc

a b c a*(b+c)

0 X X 0

1 1 X 1

1 0 0 1

1 0 1 0

o=

F(a

,b,c)

=a*(b

+c)

o abcc

cb+

c

b+

c

tdA

ND

tdO

R +td

AN

Dtd

NO

T +td

OR +

tdA

ND

GLITCH

RETARDOS

El tie

mpo d

e re

tard

o (td ) d

epen

de

del n

m

ero

de n

ivele

s l

gico

s (logic-d

epth

)C

uan

do t td

(min

) la sa

lida n

o h

a ca

mbia

do

Cu

an

do td

(min

) t

td(m

ax) la

salid

a p

uede se

r corre

cta/in

corre

ctaC

uan

do t td

(max) la

salid

a e

s corre

ctaN

o tie

ne m

em

oria

: La sa

lida n

o d

epe

nd

e d

e e

ntra

das a

nte

riore

s

Com

bin

acio

na

lin

put

outp

ut=

F(in

put)

nm

td

i1i2

i3i1

i4i5

i2

F(i1 )

F(i1 )

F(i2 )

input

outp

ut

F(i2 )

F(i3 )

F(i4 )

F(i5 )

td(m

in)

td(m

ax)

EXPRESIONES CANNICAS

Todas la

s funcio

nes b

oo

lean

as

pued

en co

nve

rtirse a

las

exp

resio

nes ca

nn

icas:

Un

a S

um

a d

e P

roducto

s/Min

term

s(S

oP

)F

(a,b

) = [F

(0,0

)*a*b

] + [F

(0,1

)*a*b

] + [F

(1,0

)*a*b

] + [F

(1,1

)*a*b

]

F(a

,b) =

[F(0

,0)+

a+

b] * [F

(0,1

)+a+

b] * [F

(1,0

)+a+

b] * [F

(1,1

)+a+

b]

F(0,0) =

[F(0,0)*1] +

[F(0

,1)*0] +

[F(1,0)*0] +

[F(1,1)*0] =

F(0,0

)+0+

0+

0 =

F(0,0

)

F(0,1) =

[F(0,0)*0] +

[F(0

,1)*1] +

[F(1,0)*0] +

[F(1,1)*0] =

0+F

(0,1)+0+

0 = F

(0,1)

F(1,0) =

[F(0,0)*0] +

[F(0

,1)*0] +

[F(1,0)*1] +

[F(1,1)*0] =

0+0+

F(1,0)+

0 = F

(1,0)

F(1,1) =

[F(0,0)*0] +

[F(0

,1)*0] +

[F(1,0)*0] +

[F(1,1)*1] =

0+0+

0+F

(1,1

) = F

(1,1)

F(0,0) =

[F(0,0)+

0] * [F(0

,1)+

1] * [F(1,0)+

1] + [F

(1,1)+1] =

F(0,0)*1

*1*1 =

F(0,0

)

F(0,1) =

[F(0,0)+

1] * [F(0

,1)+

0] * [F(1,0)+

1] + [F

(1,1)+1] =

1*F

(0,1

)*1*1 =

F(0,1

)

F(1,0) =

[F(0,0)+

1] * [F(0

,1)+

1] * [F(1,0)+

0] + [F

(1,1)+1] =

1*1

*F(1,0)*1 =

F(1,0

)

F(1,1) =

[F(0,0)+

1] * [F(0

,1)+

1] * [F(1,0)+

1] + [F

(1,1)+0] =

1*1

*1*F

(1,0) =

F(1,0

)

F(a

,b)=

(0

F0 , 1

F1 , 2

F2 , 3

F3 )=

(0

F0 , 1

F1 , 2

F2 , 3

F3 )

Un P

rod

ucto

de S

um

as/M

axte

rms

(PoS

)

Tra

nsfo

rmaci

n e

ntre

SoP

y PoS

= [0

*a*b

] + [1

*a*b

] + [1

*a*b

] + [0

*a*b

] =

EXPRESIONES CANNICAS

Eje

mplo

:a b

a

b0 0

00 1

11 0

11 1

0

F(a

,b) =

[F(0

,0)*a

*b] +

[F(0

,1)*a

*b] +

[F(1

,0)*a

*b] +

[F(1

,1)*a

*b] =

a b

a

b0 0

00 1

11 0

11 1

0

F(a

,b) =

[F(0

,0)+

a+

b] * [F

(0,1

)+a+

b] * [F

(1,0

)+a+

b] * [F

(1,1

)+a+

b] =

= a

*b +

a*b

= [0

+a+

b] * [1

+a+

b] * [1

+a+

b] * [0

+a+

b] =

= 0

+ [a

*b] +

[a*b] +

0 =

= (a

+b) * (a

+b)

= (a

+b) * 1

* 1 * (a

+b) =

a b

a

b0 0

00 1

11 0

11 1

0

a

bab ab

a

bab ab

==

=

(1,2)

=

(0,3

)

ab

a*b + a*b

12

ab

(a+

b) * (a

+b)

03

F(i1 ,i2 ,..,in

-1 ,in ) = [F

(0,0

,P,0

,0)*i1 *i2 *P

*in-1 *in ] +

[F(0

,0,P

,0,1

)*i1 *i2 *P*in

-1 *in ] + P

++

[F(1

,1,P

,1,0

)*i1 *i2 *P*in

-1 *in ] + [F

(1,1

,P,1

,1)*i1 *i2 *P

*in-1 *in ]

EXPRESIONES CANNICAS

Genera

lizaci

n p

ara

n e

ntra

das:

F(i1 ,i2 ,..,in

-1 ,in ) = [F

(0,0

,P,0

,0)+

i1 +i2 +

P+

in-1 +

in ] * [F(0

,0,P

,0,1

)+i1 +

i2 +P

+in

-1 +in ] * P

** [F

(1,1

,P,1

,0)+

i1 +i2 +

P+

in-1 +

in ] * [F(1

,1,P

,1,1

)+i1 +

i2 +P

+in

-1 +in ]

N=

2n-1

F(i1 ,..in )=

(0

F0 , 1

F1 , P

, (N-1

)FN

-1 , NF

N )=

(0

F0 , 1

F1 ,P

, (N-1

)FN

-1 , NF

N )

0F0

1F1

(N-1

)FN

-1

NF

N

S0

S1

SN

-1

SN

(S

k )

i1i2iN iN-1

0F0

1F1

(N-1

)FN

-1

NF

N

i1i2iN iN-1

P0

P1

PN

-1

PN

(P

k )

EXPRESIONES CANNICAS

abc

F=

a*(b

+c)

a b c F

0 X X 1

1 1 X 0

1 0 c c

a b c F

0 0 0 1

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 0

1 1 1 0

F(a

,b,c)=

(0

,1,2

,3,5) =

(4,6

,7)

(0

,1,2

,3,5) =

a*b*c +

a*b

*c + a*b*c +

a*b

*c + a*b

*c

(4

,6,7) = (a+

b+c) * (a+

b+c) * (a+

b+c)

01235

abc

abc

467

abc

==

SIMPLIFICACIN KARNAUGH

Mto

do siste

mtico

pa

ra la

simplifica

cin d

e S

oP

/PoS

:M

apa d

e K

arn

au

gh

(cod

ificaci

n G

ray)

Agru

par e

n m

nim

o n

m

ero

de la

zos to

dos lo

s 1 (S

oP

) 0 (P

oS

) L

azo

s de

l mxim

o (g

rupos 1

6,8

,4,2

,1) d

e ce

ldas

adya

cente

s (esq

uin

as y b

ord

es in

cluid

os)

a b c F

0 0 0 1

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 0

1 1 1 0

11

11

11

0

a

1/0

1/1

0/1

0/0

b/c

00

01 0

a

1/0

1/1

0/1

0/0

b/c

(0

,1,2

,3,5) =

a*b*c +

a*b

*c + a*b*c +

a*b

*c + a*b

*c = a

+ b*c

(4

,6,7) = (a+

b+c) * (a+

b+c) * (a+

b+c) =

(a+b) * (a+

c)

Eje

mplo

: Funci

n d

e 3

entra

das

SIMPLIFICACIN KARNAUGH

Para

aqu

ella

s entra

da

s cuya

salid

a e

s X (d

ont

care

), se p

ued

en

tom

ar o

no X

en la

agru

paci

n d

e 1

(SoP

) 0 (P

oS

)

Counte

rM

od. 1

04

Com

para

tor

2

a

6

a[3

:0]

o

a[3:0] o

3 2 1 0

0 0 0 0 0

0 0 0 1 0

0 0 1 0 1

... 1

0 1 1 0 1

0 1 1 1 0

... 0

1 0 0 1 0

1 0 1 0 X

... X

1 1 1 1 X

11

10/1

XX

XX

1/1

XX

1/0

11

0/0

1/0

1/1

0/1

0/0

a[1]/a[0]

a[3]/a[2]

00/1

XX

XX

1/1

XX

00

1/0

00

0/0

1/0

1/1

0/1

0/0

a[1]/a[0]

a[3]/a[2]

o =

a[1

]*a[0] +

a[2

]*a[1] +

a[2

]*a[1

]o = (a

[2]+

a[1]) * (a

[2]+

a[1

]+a[0

])

Eje

mplo

: Funci

n d

e 4

entra

das, co

n sa

lida

s X

3*A

ND

-2 +

1*O

R-3

1 O

R-2

+ 1

OR

-3 +

1 A

ND

-2

LGICA UNIVERSAL

Se p

ue

de im

ple

me

nta

r cualq

uie

r exp

resi

n b

oo

lea

na

usa

nd

oexclu

sivam

ente

las p

uerta

s NA

ND

o la

s NO

R

aa

+a

=a

ab

ab

a+

b=

a*b

ab

a+

ba

+b

ab

a*b

a*b

aa

*a=

a

ab

a*b

a*b

ab

ab

a*b

=a

+b

ab

a+

ba

+b

LGICA UNIVERSAL

abc

def

a*b

a*b

*c=a

*b+

c(a

*b+

c)*d=

a*b*c+

d

e*f

(a*b

*c+d

)*e*f=

(a*b

+c)*d

+e

*f

ab

def

c

a+

b(a

+b

)+c=

(a+

b)*c

e+

f

(a+

b)*c+

d=

((a+

b)+

c)*d

((a+

b)+

c)*d+

e+

f=((a

+b)*c+

d)*(e

+f)

Eje

mplo

: An

lisis de la

funci

n l

gica

LGICA UNIVERSAL

Eje

mplo

: Imple

me

nta

cin N

OR

de o

=c*(a

+b)

cba

c*(a+

b)

c

b

a+

b

cba

cb

a+

b

c*(a+

b)

a+

b

a+

b

cc+

(a+

b)

cba

b

a+

b

c+(a

+b

)

c*(a+

b)

LGICA UNIVERSAL

Imple

menta

cin N

AN

D a

partir d

e S

oP

Imple

menta

cin N

OR

a p

artir d

e P

oS

F=

Pa +

Pb +

P+

Pj

F=

Pa *P

b *P*P

j

F=

Sa *S

b *P*S

j

F=

Sa +

Sb +

P+

Sj

F=

Pa +

Pb +

P+

Pj

Pa

Pb

Pj

(P

k )

Pa

Pb

Pj

(P

k )=

(Pk )

Sj

Sb

Sa

(S

k )

Sj

Sb

Sa

(S

k )=

(Sk )

F=

Sa *S

b *P*S

j

LGICA UNIVERSAL

Eje

mplo

: Imple

me

nta

cin N

AN

D y N

OR

de o

=c*(a

+b)

o =

(a*c)+

(b*c) =

pa

+ p

b=

pa

* pb=

(a*c) * (b*c)

o =

(c)*(a+

b) =

sa

* sb =

sa

+ s

b=

(c) + (a

+b) =

(c) + (a

+b)

cba

b

a+

b

c+(a

+b)

c*(a+

b)

bca

bb

*c

c

a*c

b*c * a

*c

b*c +

a*c

d0

d1

01s

ys y

0 d0

1 d1

y = s*d

0+

s*d1

d0

d1

00

01s

y

2

s[1:0] y

0 0 d0

0 1 d1

1 0 d2

1 1 d3

y = s[1

]*s[0]*d

0+

s[1]*s[0

]*d1 +

+ s[1

]*s[0]*d

2+

s[1]*s[0

]*d3

d2

d3

10

11

MULTIPLEXOR (MUX)

MU

X2

MU

X4

La sa

lida (y) e

s sele

cciona

da (s) a

una d

e la

s entra

da

s (di )

MULTIPLEXOR (MUX)

d0

d1

01

d2

d3

01

01s 2s[1:0

]

y

0 1

Un M

UX

de n

entra

das, se

pued

e co

nstru

ir a p

artir d

e M

UX

ms

simp

les

d0

d1

00

01s

y

2

d2

d3

1011 M

UX

4

Un M

UX

de m

-bits se co

nstru

ye co

n m

MU

X

d0

d1

01

444

y

s

01010101

4y

y[3:0]

s

44

d0

d1

d0[3:0]

d1[3:0]

3210

3210

3210

DEMULTIPLEXOR (DEMUX)

La e

ntra

da (d

) es se

leccio

na

da (s) a

una d

e la

s salid

as (y

i )

y0

y1

01

s

ds y0y1

0 d 0

1 0 d

DE

MU

X2

y0

= s*d

y1

= s*d

d0

d1

00

01

s

d

2

s[1:0] y0 y

1 y

2 y

3

0 0 d

00

0

0 1 0

d0

0

1 0 0

0d

0

1 1 0

00

d

d2

d3

10

11

DE

MU

X4

y0

= s[1

]*s[0]*d

y1

= s[1

]*s[0]*d

y2

= s[1

]*s[0]*d

y3

= s[1

]*s[0]*d

Tam

bi

n e

s posib

le co

nstru

ir DE

MU

X d

e n

salid

as a

partir d

eD

EM

UX

ms sim

ple

s, y exte

nd

er e

l nm

ero

de b

its m en

las

entra

das y sa

lidas

CODIFICADORES (COD)

Co

nve

rsin

de u

n d

ato

de e

ntra

da a

otro

tipo d

e d

ato

de m

enor

tam

ao

Dife

rente

s tipos d

e C

OD

, depen

die

nd

o d

e lo

s tipos d

e d

ato

de

entra

da y sa

lida3

Enco

der

8 to

3

d1

d2

d3

d4

d5

d6

d7

a

d1 d

2 d

3d4 d

5 d

6 d

7 a[2:0]

00

00

00

0000

10

00

00

0001

01

00

00

0010

00

10

00

0011

00

01

00

0100

00

00

10

0101

00

00

01

0110

00

00

00

1111

Resto

XXX

a[0

]= d

1 +d

3 +d

5 +d

7

a[1

]= d

2 +d

3 +d

6 +d

7

a[2

]= d

4 +d

5 +d

6 +d

7

Eje

mplo

: Cod

ificador 8

a 3

CODIFICADORES (COD)

Eje

mplo

: Cod

ificador co

n p

riorid

ad 8

a 3

3

d1

d2

d3

d4

d5

d6

d7

a

d1 d

2 d

3d4 d

5 d

6 d

7 a[2:0]

00

00

00

0000

10

00

00

0001

X1

00

00

0010

XX

10

00

0011

XX

X1

00

0100

XX

XX

10

0101

XX

XX

X1

0110

XX

XX

XX

1111

a[2

]= d

7 +d

6 +d

5 +d

4

a[1

]=d

7 +d

6 +d

5 *d4 *(d

3 +d

2 )

a[0

]=d

7 +d

6 *[d5 +

d4 *(d

3 +d

2 *d1 )]

Prio

rityE

nco

der

8 to

3

DECODIFICADORES (DECOD)

Co

nve

rsin

de u

n d

ato

de e

ntra

da a

otro

tipo d

e d

ato

de m

ayo

r ta

mao

Dife

rente

s tipos d

e D

EC

OD

, depe

nd

iend

o d

e lo

s tipos d

e d

ato

de e

ntra

da y sa

lida

Eje

mplo

: Deco

difica

do

r com

ple

to d

e 3

-bits

3D

eco

der

3 to

8

y1

y2

y3

y4

y5

y6

y7

a

y0

a[2:0] y0 y

1 y

2 y

3y4 y

5 y

6 y

7

000 1

00

00

00

0

001 0

10

00

00

0

010 0

01

00

00

0

011 0

00

10

00

0

100 0

00

01

00

0

101 0

00

00

10

0

100 0

00

00

01

0

111 0

00

00

00

1

y0 =

a[2]*a

[1]*a

[0]

y1 =

a[2]*a

[1]*a

[0]

y7 =

a[2]*a

[1]*a

[0]

P

Eje

mplo

: Deco

difica

do

r BC

D a

7 se

gm

ento

s

DECODIFICADORES (DECOD)

47-se

gm

ents

Deco

der

bcdefg

d

ad[2:0] a b c de f g

0000 1 1 1 1 1 1 0

0001 0 1 1 0 0 0 0

0010 1 1 0 1 1 0 1

0011 1 1 1 1 0 0 1

0100 0 1 1 0 0 1 1

0101 1 0 1 1 0 1 1

0110 0 0 1 1 1 1 1

0111 1 1 1 0 0 0 0

1000 1 1 1 1 1 1 1

1001 1 1 1 0 0 1 1

101X X X X X X X X

11XX X X X X X X X

a

bcd

e fg

SUMADOR

Re

aliza

la o

pera

cin a

ritmtica

sum

a

Sem

i-sum

ador (h

alf-a

dder). L

a sa

lida ca

rry(c

o ) perm

ite e

xten

der

la su

ma a

ms d

e 1

-bit

aboc

o

a b

coo

0 0

00

0 1

01

1 0

01

1 1

10

Sum

ad

or co

mple

to (full-a

dder). U

na e

ntra

da d

e ca

rry(c

i ) que se

conecta

a la

co

de la

sum

a d

el b

itante

rior

aboc

oc

i

cia b

coo

0 0 0

00

0 0 1

01

0 1 0

01

0 1 1

10

1 0 0

01

1 0 1

10

1 1 0

10

1 1 1

11

o=

a

bc

o =a*b

aboc

o

o=

a

b

ci

co =

a*b

+(a

+b)*c

i

co =

a*b

+(a

b)*c

i

aboc

o

ci

aboc

o

ci

SUMADOR

El su

mad

or co

mple

to p

ue

de im

ple

menta

rse ta

mbi

n co

n d

os

sem

i-sum

adore

s en se

rie

Un se

mi-su

mador p

ue

de o

bte

nerse

con u

n su

mador co

mp

leto

,co

necta

nd

o 0

al ca

rryde e

ntra

da

o1

o2

co2

co1

ab

ci

o1 =

a

b

co1 =

a*b

co2 =

ci *o

1 ==

ci *(a

b

)

o2 =

ci

o1 =

=c

i (a

b)

oco

co =

co1 +

co2 =

=a

*b+

ci *(a

b

)

o=

o2 =

=a

b

ci

aboc

o0

aboc

o=

SUMADOR

Un su

mad

or co

n a

carre

o e

n se

rie (rip

ple

-carry) d

e n

-bits se

im

ple

me

nta

con n

sum

adore

s, pro

pag

an

do e

l carry

entre

eta

pas

aboco

ci

aboco

ci

ab[3:0]

aboco

ci

aboco

ci

aboco

ci

aboco

ci

a[3:0]

b

44

4

co

o

0

0

01

23

1

1

2

2

3

3

o[3:0]

aboco

ci

aboco

ci

0

Usa

nd

o m

sum

adore

s de n

-bits, se

obtie

ne u

n su

mad

or d

e

(m*n

)-bits

aboc

oc

i

444

aboc

oc

i

888

444

abci

co o

444

abci

co o

88

8

a[7:0]

b

o coc

i

[3:0

]

[3:0

]

[7:4

]

[7:4

]

[7:4

][3

:0]

ab

[7:0]

o[7:0]

co =

co [n

-1]

ci [n

]=c

o [n-1] (n

1)

ci [0

]=0

co =

co [m

-1]

ci [m

]=c

o [m-1

] (m1

)c

i [0]=

0

SUMADOR

El p

rob

lem

a d

el su

ma

dor rip

ple

-carrye

s el tie

mpo d

e re

tard

odeb

ido a

la se

al d

e ca

rry

aboco

ci

aboco

ci

ab[3:0]

aboco

ci

aboco

ci

aboco

ci

aboco

ci

a[3:0]

b

44

4

co

o

0

0

01

23

1

1

2

2

3

3

o[3:0]

aboco

ci

aboco

ci

0

aba1b1

b0

a2

b2

a0

o co

co 1

co 2

o1

o2

4*td

co 0

o0

4*td

aboc

oc

i

td

ab0111

0101

0000

0000

1111

1000

o[0

]c

i [1]=

co [0

]

0000

o1100

0111

0111

+0101

011

00

0x7

+0x5

0x0

C

co

o[3

:0]

a[3

:0]

b[3

:0]

0 1

1 1

0c

i

SUMADOR

1111

+1000

101

11

0xF

+0x8

0x1

7

co

o[3

:0]

a[3

:0]

b[3

:0]

1 0

0 0

0c

iE

jem

plo

:

00101000

10011011

1111

o[1

]c

i [2]=

co [1

]o[2

]c

i [3]=

co [2

]o[3

]c

o =c

o [3]

SUMADOR

El su

mad

or p

roduce

la su

ma d

e n

m

ero

s codifica

dos e

n b

inario

natu

ral (u

nsig

ne

d) o

en co

mple

mento

a 2

(signe

d)

Sum

a u

nsig

ne

d=

> C

arry

si co [n

-1]=

1 (c

o =c

o [n-1

])

Sum

a sig

ned

=>

Ove

rflow

si co [n

-1]

ci [n

-1]=

1 (c

i [n-1

]=c

o [n-2

])

Neg

ative

si o[n

-1]=

1E

jem

plo

:

-7+

5-2

1001

+0101

011

10

0x9

+0x5

0x0

E

co

o[3

:0]

a[3

:0]

b[3

:0]

0 0

0 1

0c

i

9+

514

1011

+1110

110

01

0xB

+0xE

0x1

9

co

o[3

:0]

1 1

1 0

0c

i

11

+149

-5+

-2-7

5+

5-6

0101

+0101

010

10

0x5

+0x5

0x0

A

co

o[3

:0]

a[3

:0]

b[3

:0]

0 1

0 1

0c

i

5+

510

1011

+1001

101

00

0xB

+0x9

0x1

4

co

o[3

:0]

1 0

1 1

0c

i

11

+ 94

-5+

-7+

4

Ca

rry!!!11

+14

=9

Ca

rry!!!11

+9

=4

Ove

rflow

!!!(-5

)+(-7

)=+

4O

verflo

w!!!

(+5

)+(+

5)=

-6

RESTADOR

Re

aliza

la o

pera

cin a

ritmtica

resta

, en b

inario

-natu

ral

Resta

dor co

mple

to (su

bstra

cter). L

a se

al b

orro

w(b

i , bo ) p

erm

ite

exte

nder la

resta a

ms d

e 1

-bit

Substra

ctor

abob

ob

i

bia b

boo

0 0 0

00

0 0 1

11

0 1 0

01

0 1 1

00

1 0 0

11

1 0 1

10

1 1 0

00

1 1 1

11

1011

0101

001

10

0xB

0x5

0x0

6

bo

o[3

:0]

a[3

:0]

b[3

:0]

0 1

0 0

0b

i

1000

1011

111

01

0x8

0xB

0x1

D

bo

o[3

:0]

a[3

:0]

b[3

:0]

1 1

1 1

0b

i

Substra

ctor

abob

ob

i

444

Sub

stractor

abobo

bi S

ubstractor

abobo

bi

ab[3:0]

Su

bstractor

abobo

bi S

ubstractor

abobo

bi

Su

bstractor

abobo

bi S

ubstra

ctor

abobo

bi

a[3:0]

b

44

4

bo

o

0

0

01

23

1

1

2

2

3

3

o[3:0]

Sub

stractor

abobo

bi S

ubstractor

abobo

bi

0

RESTADOR

Si lo

s nm

ero

s van co

difica

dos e

n co

mple

mento

a 2

, la re

sta a

-bpue

de h

ace

rse co

n u

na su

ma a

+(-b

)U

n n

m

ero

b e

n a

2 cam

bia

de sig

no p

as

ndo

lo a

com

ple

mento

a 1

y sum

and

o +

1

0100

1011

b[3

:0]

b[3

:0]

0 1

1 1

ci

a[3

:0]

b[3

:0]

o[3

:0]

0000

+1011

0110

0

b

000

0-bco

1

4

44

4

+4=

010

0=

0x4

-4=

110

0=

0xC

-7=

100

1=

0x9

+7=

011

1=

0x7

1001

0110

b[3

:0]

b[3

:0]

0 0

0 1

ci

a[3

:0]

b[3

:0]

o[3

:0]

0000

+0110

0011

1

abci

oco

RESTADOR

b a

a-b

bo

bi

n

nn

nabc

i

oco

b a

ab

co /b

os/a

nn

nabc

i

oco

Substra

ctor

abo=

a-b

bo

bi

nnn

Un re

stado

r de n

-bits se

pued

e co

nstru

ir a p

artir d

e u

n su

mador,

neg

an

do u

na e

ntra

da e

invirtie

nd

o e

l borro

wde e

ntra

da y sa

lida

Un su

mad

or/re

stador d

e n

-bits se

pued

e co

nstru

ir a p

artir d

e u

n

n

ico su

mador, y u

na se

al s/a

para

imple

menta

r resta/su

ma

n

Adder

Substra

ctor

abc

o /bo

s/a

nnn

o=

ab

Adder

Substractor

abc

o0

444

o=a

+b

Adder

Substractor

abb

o1

444

o=a

-b

s/a

b

01n-1

01

n-1

nn

b[n-1:0]

c[n-1:0]

RESTADOR

El re

stador p

roduce

la re

sta d

e n

m

ero

s codifica

dos e

n b

inario

natu

ral (u

nsig

ne

d) o

en co

mple

mento

a 2

(signe

d)

Resta

unsig

ne

d=

> C

arry

si co [n

-1]=

0 (b

o =c

o [n-1

])

Resta

sign

ed

=>

Ove

rflow

si co [n

-1]

ci [n

-1]=

1 (c

i [n-1

]=c

o [n-2

])

Neg

ative

si o[n

-1]=

1E

jem

plo

:

-7-6-1

1001

+0101

011

11

0x9

+0x6

0x0

F

co

o[3

:0]

a[3

:0]

b[3

:0]

0 0

0 1

1c

i

91

015

1111

+0101

101

01

0xF

+0x6

0x1

5

co

o[3

:0]

1 1

1 1

1c

i

15

105

-1-6+

5-7+

5+

4

1001

+1010

101

00

0x9

+0xB

0x1

4

co

o[3

:0]

a[3

:0]

b[3

:0]

1 0

1 1

1c

i

9

54

0010

+0101

010

00

0x2

+0x6

0x0

8

co

o[3

:0]

0 1

1 1

1c

i

21

08

+2

-6-8C

arry!!!

2-10

=8

Ove

rflow

!!!(+

2)-(-6

)=-8

Ca

rry!!!9

-10=

15

Ove

rflow

!!!(-7

)-(+5

)=+

4