Digitalna Logika (Cveta Martinovska)

7/22/2019 Digitalna Logika (Cveta Martinovska)

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, 2011


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-

-

-

-

- -

CIP -

004.312(075.8)

-

ISBN 978-608-4504-40-5

COBISS.MK-ID 89142282

.

-.

.

; -

. .

-


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III

- ........................... 221 - ........................................................................ 222 - ................................. 224 - .......................................................................... 225

-............................... 227

........................................ 228 ......................................... 232 ............................................................ 234 ................................... 239 .................. 245 ............................................................. 251

................................................................................. 252 ................................................................................... 256

............................................................................... 262 .................................................................................. 269

............................................................................................. 272


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2

bmnnb PPPPPPPN ).......()( 21011 (1.1)

10 bPi .

m

m

n

n

n

nb bPbPbPbPbPbPN

......)( 22

1

1

0

0

1

1

n

mi

i

ib bPN)( (1.2)

nP

- 0P

- Least Significant Digit).

0123

10 104103102101)1234(


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9/142

1. 5

( ) ...N P b P b P bb mm

1

1

2

2

b P b P b P m

1

1

1

2

1( ( ... ( )...)) (1.4)

8

4:8 + 3 = 3,503,50:8 + 5 = 5,43755,4375:8 + 0 = 0,6796875

44:2 = 22 22:2 = 11 11:2 = 5

5:2 = 2 2:2 = 1 1:2 = 0

10=(101100)2.

10

6

10

0,6796875*8 = 5,43750,4375*8 = 3,50

0,50*8 = 4,00 10= (0,534)8.

( ) ( )16 1 7 6 7 1371 0

10

7

13 : 3 = 4 4 : 3 = 1

1 : 3 = 0

(13)10=(111)3

( , ) ( , ...) ( , )0 1 1 3 0 333 0 331

10 10

3

( , ) ( , ...) ( , )0 2 2 11 0 1818 0 18111

10 10

11


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1. 7

0+0=0 0

0+1=1 0

1+0=1 0

1+1=0 1

- - -

1001.011 (9.375)10+1101.101 (13.625)10

________ _________10111.000 (23)10

0-0=0 00-1=1 1

1-0=1 0

1-1=0 0

02 10

- 01___01

8

0112 11000 24

- 10001 -17______ ___00111 7

-

.

.101)2=(6.625)10 2 -2) (1.10101)2=2

-2(6.625)10=(1.65625)10 22 (11010.1)2=2

2(6.625)10=(26.5)10.

0x0=0

0x1=0

1x0=01x1=1

110.10x 10.1________11010

11010___________10000.010


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1. 9

Algori

- 110

2= (24)10 2= (6)10

11000 : 110 = 100

_____= = 00

- 111

2= (46)10 2= (7)10

101110 : 111 = 0110

_____1011

- 111_____=1001

- 111

______

(restoring and nonrestoring algorithm).

10

2:(1111)2 (12)10:(15)10.

1100

-1111 _____ q0=0 -0011restore +1111

_____

11000 -1111

______ q -1=1

-1111_______ q-2=1

-1111

_______ q-3=0 -1001restore +1111

_______01100

- 1111________ q-4=0

- 0011restore + 1111

________11000

- 1111_________ q-5=1

0.q-1q-2q-3q-4q-5


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1. 19

01100 (+12)+11011 (-5)

-------------- 1 00111

10011 (-13)+11001 (-7)--------------

1 01100

11000 (-8)

+00110 (+6)--------------

0 1110 (-

11101 (-3)+11010 (-6)--------------

10111 (-

20

2-(00001001)2 =2310-910

00010111 (+23)+11110111 (-9)--------------

1 00001110 (+14)

-

-

--8 1000 -

-7 1001 1111-6 1010 1110

-5 1011 1101

-4 1100 1100

-3 1101 1011

-2 1110 1010

-1 1111 1001

0 0000

1 0001 00012 0010 0010

3 0011 0011

4 0100 0100

5 0101 0101

6 0110 0110

7 0111 0111

(end around carry) 2-(11110110)2 =2310-910


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1 27 28


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1. 27

0 0 0 0 0

1 0 0 0 1

2 0 0 1 0

3 0 0 1 14 0 1 0 0

5 0 1 0 1

6 0 1 1 0

7 0 1 1 1

8 1 0 0 0

9 1 0 0 1

2

0010

1

0001

10= 0011 0110NBCD 10=1001 0010NBCD

28

0011 0110 36+0110 +0110 +6_________________

1001 1100+1001 0010 +92__________________1 0010 1110

-0110 -6__________________

0001 0010 1000 128

10=0100 0110NBCD 10=1001 0100NBCD

0100 0110 46+0110 +0110 +6

_________________1010 1100

+1001 0100 +94__________________1 0011 1 0000

+1__________________

0001 0100 0000 140

10=1001 0010NBCD 10=0011 0110NBCD

6

01120

1001 0010 92- 0011 0110 -36_________________0101 1100

-0110

__________________0101 10 6

1 29 30


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1. 29

-

a b c d e f g a b c d e f g

0 1 1 1 1 1 1 0 0 0 0 0 0 0 1

1 0 1 1 0 0 0 0 1 0 0 1 1 1 1

2 1 1 0 1 1 0 1 0 0 1 0 0 1 0

3 1 1 1 1 0 0 1 0 0 0 0 1 1 0

4 0 1 1 0 0 1 1 1 0 0 1 1 0 0

5 1 0 1 1 0 1 1 0 1 0 0 1 0 06 1 0 1 1 1 1 1 0 1 0 0 0 0 0

7 1 1 1 0 0 0 0 0 0 0 1 1 1 1

8 1 1 1 1 1 1 1 0 0 0 0 0 0 0

9 1 1 1 1 0 1 1 0 0 0 0 1 0 0

a

b

cd

e

fg

- -

30

3, t2, t1 0 3b2b1b0 3t3+b2t2+b1t1+b0t0.

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 1 0 0 0 1

2 0 0 1 0 0 0 1 0 0 0 1 0

3 0 1 0 0 0 1 1 0 0 1 14 0 1 0 1 0 1 0 0 1 0 0 0

5 0 1 0 1 0 1 1 16 0 1 1 0 1 1 0 07 1 0 1 0 1 0 0 0 1 1 0 1

8 1 0 0 1 1 1 1 09 1 1 0 1 1 0 1 0 1 1 1 1

-2,-

(8,4,-2,-

(8,4,-2,- 0 0 0 0 01 0 1 1 1

2 0 1 1 0

3 0 1 0 1

4 0 1 0 0

5 1 0 1 1

6 1 0 1 0

7 1 0 0 1

8 1 0 0 0

9 1 1 1 1


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1. 33 34


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3 0 0 1 1

9-3=6 1 1 0 0

5 0 1 0 1

9-5=4 0 1 0 0

-2,-

(8,4,-2,-

7 1 0 0 19-7=2 0 1 1 0

-2,-

1 0 1 0 09-1=8 1 0 1 1

b b b b3 2 1 0 0123 tttt .

00112233 tbtbtbtbd .

d

d

b b b b3 2 1 0 .

00112233 )1()1()1()1( tbtbtbtbd

)( 001122330123 tbtbtbtbtttt

dtttt 0123

90123 tttt .

-2,-1).

0 00001 0001

2 0011

3 0010

4 0110

5 0100

6 1100

7 1110

8 1010

9 1000

1. 35 36


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g b b i n

g b

i i i

n n

1 0 1

(1.7)

b i ni , 0 g i ni , 0

x y x y0 0 0

0 1 1

1 0 1

1 1 0

g b b i ni i i ( ) mod 1 2 0 1. (1.8)

1 1 1 0

b3 b2 b1 b0

1 0 0 1

2=(14)10

101

nibbg

gb

iii

nn(1.9)

i i i-1.

0 0000 0000

1 0001 0001

2 0011 0010

3 0010 0011

4 0110 0100

5 0111 0101

6 0101 0110

7 0100 01118 1100 1000

9 1101 1001

10 1111 1010

11 1110 1011

12 1010 1100

13 1011 1101

14 1001 1110

15 1000 1111

g3g2g1g00

b3=g3=1

g2=b3 b2 0=1b2 2=1g1=b2b1 1=1b1 1=0g0=b1b0 1=0b0 0=1

2=(13)10.


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1. 41 42


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0 1 1 0 0 0

1 0 0 0 1 1

2 0 0 1 0 1

3 0 0 1 1 0

4 0 1 0 0 15 0 1 0 1 0

6 0 1 1 0 0

7 1 0 0 0 1

8 1 0 0 1 0

9 1 0 1 0 0

( ) !

!( )!

5

2

5

2 5 210

1.19.

0 0 1 0 0 1

1 0 0 0 1 1

2 0 0 1 0 1

3 0 0 1 1 0

4 0 1 0 1 0

5 0 1 1 0 0

6 1 0 0 0 1

7 1 0 0 1 0

8 1 0 1 0 0

9 1 1 0 0 0

-

-

-

(5,0) (4,3,2,1,0)0 01 0 0 0 0 11 01 0 0 0 1 0

2 01 0 0 1 0 0

3 01 0 1 0 0 0

4 01 1 0 0 0 0

5 10 0 0 0 0 1

6 10 0 0 0 1 0

7 10 0 0 1 0 0

8 10 0 1 0 0 09 10 1 0 0 0 0

1. 43 44


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R(N)=N mod p.

2 3 5 70 0 0 0 0

1 1 1 1 1

2 0 2 2 2

3 1 0 3 3

4 0 1 4 4

5 1 2 0 5

6 0 0 1 6

7 1 1 2 0

8 0 2 3 19 1 0 4 2

2 3 5 715 1 0 0 1

5 1 2 0 5______________________

15+5=20 0 2 0 6

2 3 5 78 0 2 3 16 0 0 1 6

______________________8+6=14 0 2 4 0

2 3 5 7

2 0 2 2 28 0 2 3 1

______________________2*8=16 0 1 1 2

N1 321 321 mod 9 = 6N2 248 248 mod 9 = 5

----- ------------------N1+N2 569 569 mod 9 = 2 (6+5=11 mod 9=2)

-

R(N1)=(3+2+1) mod 9 = (6) mod 9 = 6R(N2)=(2+4+8) mod 9 = (14) mod 9 = 5

-checking).


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1. 53 54


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-

-

1111 0001 0010 0110 0101 1101

-

1. 55 56


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-

1.8

16-(256)8

-2), -3,-

(2AC)16=(0010 1010 1100)2=(684)10(256)8=(010 101 110)2=(174)10

10 1010 1100- 0 1010 1110

_________________________

1 1111 1110

(111111110)2=28+27+26+25+24+23+22+21=(510)10

0011.

(4,4,1,-

3 0 1 1 19-3=6 1 1 0 1

-

(8,4,-3,- 3 1 0 1 1

9-3=6 1 0 0 1

-3,-

3 0 0 1 09-3=6 0 1 0 1


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1. 59 60


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8 16 8

3 5 12.

(3EF)16-(1111)2

: 11010010.

-2-

- - -107.

-

-

-3, -2) -6)

14.

-

---n (m-out-of-

1. 61


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-out-of-

18. 7-

-

-

20.

64


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1. -

2. 3. 4. -

(Closure)


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67

68

(( ) )* (( ) )*( )


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x x x x ( ) *1

( ) *( )x x x x

x x x* x 0 x

(Absorption)

x x y x *

x x y x*( )

x x y x x y

* * *1 x y*( )1 x *1 x

(Simplification)

a) x x y x y *

b)x x y x y*( ) *

yxxxyxx **)(* 0 x y*

yx *

(Associative law)

a) x y z x y z x y z ( ) ( )

x y z x y z x y z*( * ) ( * ) * * *

A x y z x y z ( ( )) *(( ) )

( ( )) *(( ) )x y z x y z

( ( )) *( ) ( ( )) *x y z x y x y z z

( ( )) *( )x y z x y z

( ( )) *(( )x y z x y z

(( ) )* (( ) )*( )x y z x x y z y z

(( ) ) *( )x y z x y z

A x y z ( ) A x y z ( )

(Consensus)

a) x y x z y z x y x z* * * * *

b)( ) *( ) *( ) ( ) *( )x y x z y z x y x z

x y x z y z x y x z y z x x* * * * * * *( ) x y x z y z x y z x* * * * * * ( * * * ) ( * * * )x y x y z x z x y z x y z x z y* *( ) * *( )1 1 x y x z* *

(De Morgans law)

a) x y x y *

x y x y*

x y*

x y

. ( ) * ( * )x y x y x x y y

( )x y y x y y( ) x 1 1

yxyyxxyxyx ******)(

0* * *y x y y 0


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73

z z

74

2 5


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z z

z z

)()()()(5 xxyzyyxzzzyxzzyxE

xyzzyxyzxzyxzyxE 6

.

6

zxxyE

zxxyE

))(( zxyxxy

))()()(( zyzxxyxx

))()(( zyzxyx

))(( zyxzyxzzyxyx

))(( zyxzyxzyyxzx

))(( zyxzyxzyxxzy yxx y, zz

))()()(( zyxzyxzyxzyxE

.

2.5

2121 xxxx

2121 xxxx

321321 xxxxxx

321321 xxxxxx

x x1 2 x4 3434 xxxx

21214 xxxxx .

nn xxxxxx ...... 2121

nn xxxxxx ...... 2121 .

E x x x E x x xn n( , ,..., , ,*) ( , ,..., ,*, )1 2 1 2

yzxzyxzyxE

.

))()(( zyxzyxzyxE

75 76


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))(( zyxyzxzyzyyxzxyxE

))(( zyxyzxzx

zxzyxzyxy

E

))()()(( zxzyxzyyxE

))(( zzyzxzzxyxxxzyzxyyx

))(( yzxzzxyxzxy

yzxzyyzxxzyxzzxzyxyzxyyx

zyxzyxyzx

2.6

n

n

n

22n

.

22n

.

-

x y F

0

F

1

F

2

F

3

F

4

F

5

F

6

F

7

F

8

F

9

F1

0

F1

1

F1

2

F1

3

F1

4

F1

5

0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 11 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

F0 0

xyF 1

yxF 2x but not y

F x3

x

yxF 4y but not x

F y5 y

yxyxF 6

F x y7

F x y8

yxxyF 9Equil

F y10 NOT y

F x y11 if y then x (y->x)

F x12 NOT x

F x y13 if x then y (x->y)

xyF 14

F15 1


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81

2112 )1( xxxx

82

2.8


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2112 )(

)( 2212 xxxx

x x1 2

),,,( 4321 xxxxf 3x

4x

)1,0,,()0,0,,(),,,( 214321434321 xxfxxxxfxxxxxxf

)1,1,,()0,1,,( 21432143 xxfxxxxfxx

.

f x x x x f x x x f x xn n n( , ,..., ) [ ( , ,..., )][ ( , ,..., )]1 2 1 2 1 20 1

)......)(...(111210 nnn

xxxbxxxbf

)...)(...( 21122122 1 nn xxxbxxxb n

bi

).

x y x y xy x+y yxyx x y yx yxxy

0 0 1 1 0 0 0 1 1 1

0 1 1 0 0 1 1 1 0 01 0 0 1 0 1 1 1 0 0

1 1 0 0 1 1 0 0 0 1

yxxy

yxyx

xyyxyyxyyxxxyxyxyxyxyx ))((

yxyxyx

yyxxyxyx

)()( yxyyxx

))(( yxyx

83

84


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-

yxxyxy .

x y x y x y .

xxxxx

xyxyxyxy

yyxxyxyxyx .

x x x

x y x y x y x y

yyxxyxxyxy .

zyzxzyxzyxzyx )(

zxyxzyxzyx )( .

zxyzxyzxy

yzxyzx .

85

86

yxyxyx


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yxxy

yxyx

yxxyxy

yxyxyx

))(( yxyx

yxxy

))(( yxyx

yxyx

))(( yxyx :

87

yyxxyxyxyxyx

88

xyxyxyxy


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)()( yxyyxx

))(( yxyx .

yxyx

))(( yxyx :

yxyxyxyx

))(( yxyx

))(( yxyx .

))(( yxyx

yxxy :

))(())(( yxyxyxyx

yxyx

xyyx

yxxy .

xxxxx

yyxxyxyxyx

x x x

x y x y x y x y

yyxxyxxyxy


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91

92

2.10


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-Very Large

Scale Integrated) -

1.

(power dissipation)2.

3. (noise margin).

1. 2.

f =

)()()()(( 321321321321321 xxxxxxxxxxxxxxxf

xyzzyxzyxyzxzyxf

zyzyxxyf .

))()(( zyzyxyxzyzyxxyf

4

cddcabaf .

))()(( dcdcabacddcabaf

))()(( zyzyyx

)()()()()()(

)()()())()((

zyzyxyzyzyxy

zyzyxyzyzyyxzyzyyx

)()()(

()()())()(())()((

zzyzyyyyxx

zyzyyxzyzyyxzyzyyx

))(( zxzyyx

93

94

cz1a

b


53/142

))((*))((*

))(())(())((

zzxxzyyyyxzxzyyx

zxzyyxzxzyyxzxzyyx

)()()()()()(

)()()())(())((

zxzzyyxxzxzzyyxx

zxzyyxzxzyyxzxzyyx

zxyx - - - -

))(( zxyxzxyxzxyx

)()(())(())(( yxzyxzyxzyyxzzyxzxyx

xzxzyyx )( - - - -

zyxxzyxxxzxyxzyxxzxzyyx )()(

zyxzxyxyzzyxzyxzyxyzx

yyzxyyxzzzyxzzxyzzyxzzyx

zxxzyxxyyxyxxxzxxyyyx

)()()()()()(

)()()(

1 2

z2

yxyx *c

z1a

b

z2

cbacbaabccba

cbacbacbaba

cbabacbaba

cbabacbaz

))((

)()(

)()(1

abcbabcaabcbaz )(2

10. 1 2

cz1a

b

z2

95

a) xyyx

96

2.1


54/142

cz1a

b

z2

cbacbaabccba

cbacbacbaba

cbabacbaba

cbabacbaz

))((

)()(

)()(1

bcbaacbacbababacbabaz ))((]))[((2

11. ->,-

qpqp .

qpqppqpq

qpqp

ab c a b c ( ) .

( )( )( )ab ac a b a c

x y w yz x y wz z( ) ( ) a b c uv ( ) .

x y x y

x y x y

xy x y x y .

abc a bc b c a c ( )( )( ) .

E ab ac bc1

E x y y z y z2 ( )( )( )

E x y y z x z3 ( )( )

f x x x( , , )1 2 3 f ( , , , )2 3 5 6

97


55/142

f x x x x x x x x x x( , , ) ( )1 2 3 1 2 2 3 1 1 3 .

( )( )x yz y xz

100


56/142

3.

- - -

1. xy xy y

2. x x y x y

3. xz xy yz xz xy


57/142


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3. 105

00 01 11 10 00 01 11 10 00 01 11 10

00 00 00

01 01 01

106


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11 11 11

10 10 10

00 01 11 10 00 01 11 10 00 01 11 1000 00 00

01 01 01

11 11 11

10 10 10

00 01 11 10 00 01 11 10 00 01 11 10

00 00 00

01 01 01

11 11 11

10 10 10

00 01 11 10 00 01 11 10 00 01 11 10

00 00 00

01 01 01

11 11 11

10 10 10

00 01 11 10 00 01 11 10 00 01 11 10

00 00 00

01 01 0111 11 11

10 10 10

00 01 11 10 00 01 11 10 00 01 11 10

00 00 00

01 01 01

11 11 11

10 10 10

00 01 11 10 00 01 11 10 00 01 11 1000 00 00

01 01 01

11 11 11

10 10 10

00 01 11 10 00 01 11 10 00 01 11 10

00 00 00

01 01 01

11 11 11

10 10 10

- -

- - -


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61/142

3. 111

3.10.

112

f f ( )( )


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yzyz

x

00 01 11 100 1

1 1 1 1

x z

3.4

f yz xz .

f x y z w( , , , ) ( , , , , ) 0 1 4 5 9

.

zw

xy

00 01 11 10

00 0 0

01 0 0

1110 0

f xz yzw .

f x z y z w ( )( ).

f

f f x z y zw

( )( )x z y z w

xy yz xz z xw zw( )

xy yz z xw zw( ) ( )z zw xy xw

z xy xw

z

zw

xy

00 01 11 10

00 1 1

01 1 1

11 1 1 1 1

10 1 1 1

xy x w

f z xy xw .


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3. 117

118

x3x2x3

00 01 11 10

0 0 0


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x1

x2 x3 f

0 0 0 00 0 1 10 1 0 00 1 1 1

1 0 0 01 0 1 11 1 0 01 1 1 0

f x x x x x x x x x

1 2 3 1 2 3 1 2 3, f x x x x x x x x x x x x x x x ( )( )( )( )( )1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 .

f x x x( , , ) ( , , , , )1 2 3 0 2 4 6 7 .

x1 0 0 0

1 0 0 0

x x1 2

3 x x1 2 f x x x ( )1 2 3.

f x x x 3 1 2

f f x x x x x x 3 1 2 3 1 2( ) .

f x y z w( , , , ) ( , , , , , , , , ) 0 1 2 4 5 6 8 9 10

.

f x y z w( , , , ) ( , , , , , , ) 3 7 11 12 13 14 15 .

f x y z w ( )( ) .

3. 119

zw00 01 11 10

00 0

120

f x x x x x


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xy00 0

01 0

11 0 0 0 010 0

4 =16

NBCD 0 0000 0011

1 0001 0100

2 0010 0101

3 0011 0110

4 0100 0111

5 0101 1000

6 0110 1001

7 0111 1010

8 1000 10119 1001 1100

f x x x x x1 1 2 4 2 3

f x x x x x x x2 2 3 4 2 4 2 3

f x x x x3 3 4 3 4

f x4 4 .

x1x2x3x4 f1f2f3f40 0 0 0 0 0 0 1 1

1 0 0 0 1 0 1 0 02 0 0 1 0 0 1 0 1

3 0 0 1 1 0 1 1 0

4 0 1 0 0 0 1 1 1

5 0 1 0 1 1 0 0 0

6 0 1 1 0 1 0 0 1

7 0 1 1 1 1 0 1 0

8 1 0 0 0 1 0 1 1

9 1 0 0 1 1 1 0 010 1 0 1 0 b b b b

11 1 0 1 1 b b b b

12 1 1 0 0 b b b b

13 1 1 0 1 b b b b

14 1 1 1 0 b b b b

15 1 1 1 1 b b b b

x1x2

f1 x3x4

x1x2

f2 x3x400 01 11 10 00 01 11 10

00 00 1 1 1

01 1 1 1 01 1

11 b b b b 11 b b b b

10 1 1 b b 10 1 b b

1 2

3. 121

x x

f3 x3x4

x x

f4 x3x400 01 11 10 00 01 11 10

00 1 1 00 1 1

01 1 1 01 1 1

122


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x1x2 x1x201 1 1 01 1 1

11 b b b b 11 b b b b

10 1 b b 10 1 b b

f3 4

1x0 1y0

x1 x0 y1 y0 z1 z20 0 0 0 0 0

0 0 0 1 0 0

0 0 1 0 0 0

0 0 1 1 0 0

0 1 0 0 0 0

0 1 0 1 0 1

0 1 1 0 0 1

0 1 1 1 0 11 0 0 0 0 0

1 0 0 1 0 1

1 0 1 0 1 0

1 0 1 1 1 0

1 1 0 0 0 0

1 1 0 1 0 1

1 1 1 0 1 0

1 1 1 1 1 1

1z2

x1x0

z1 y1y0

x1x0

z2 y1y000 01 11 10 00 01 11 10

00 00

01 01 1 1 1

11 1 1 11 1 1

10 1 1 10 1

3.11

1 2

z x y1 1 1z x y x y y x x y2 0 0 1 1 0 1 0 1 .

3. 123

124

f1 zw f2 zw

00 01 11 10 00 01 11 10

00 00


68/142

xyzwwzxywzxyzwyxwzyxf ),,,(1

xyzwwzxywzxyzwyxwzyxwzyxwzyxf ),,,(2

xy xy

00 00

01 01

11 1 1 1 11 1 1 110 1 10 1 1 1

f xy z xzw1

f xz xw2

3.24.

xy

f1 zw

xy

f2 zw

00 01 11 10 00 01 11 1000 00

01 01

11 1 1 1 11 1 1 1

10 1 10 1 1 1

f xy z xzw1 f xz xzw2


69/142

3. 127

x1x2x3x40 0 0 0 0 1 0 0 0 1 2 0 0 1 0

128

00-- 00-- 6,7.


70/142

3 0 0 1 1

6 0 1 1 0 9 1 0 0 112 1 1 0 0 7 0 1 1 1 11 1 0 1 113 1 1 0 1 14 1 1 1 0

x1x2x3x4

0,1 0 0 0 -0,2 0 0 - 0 1,3 0 0 - 1 1,9 - 0 0 12,3 0 0 1 - 2,6 0 - 1 0

3,7 0 - 1 13,11 - 0 1 1 6,7 0 1 1 -6,14 - 1 1 0 a

9,11 1 0 - 1 9,13 1 - 0 1 b12,13 1 1 0 - c12,14 1 1 - 0 d

x1x2x3x40,1,2,3 0 0 - - e

1,3,9,11 - 0 - 1 f

2,3,6,7 0 - 1 - g

x x x2 3 4 , x x x1 3 4, x x x1 2 3 , x x x1 2 4 , x x1 2 , x x2 4 x x1 3. -

0 1 2 3 6 7 9 11 12 13 14a x x

b x x

c x x

d x x

e x x x x

f x x x x

g x x x x

3.13


71/142


72/142


73/142

3. 135

x1x2x3x49,13 1 - 0 1 b12,13 1 1 0 - c

136


74/142

7,15 - 1 1 1 d

13,15 1 1 - 1 e

0 9 13 15

a* x

b* x x

c x

d x

e x x

f x x x x x x x x x x 1 2 3 4 1 3 4 2 3 4f x x x x x x x x x x 1 2 3 4 1 3 4 1 2 4 .

-

3.10:

0 4 16 12 24 28

a x x

b x x

c x x

d x x

e x x

f x x


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3. 139

x1x2x3x4

140

-


76/142

x1x2x3x4

4,5,12,13 - 1 0 - d

3.17

4 5 10 11 12 13 14a* x x

b x x

c x x

d* x x x x

f x x x x x x x x ( )( )( )1 2 3 2 3 1 3 4 f x x x x x x x x ( )( )( )1 2 3 2 3 1 2 4

.

f x x x x b( , , , ) ( , , , , , ) ( , )1 2 3 4 3 4 6 9 10 15 8 11

.

41.

x1x2x3x44 0 1 0 0 8 1 0 0 0

3 0 0 1 16 0 1 1 0 9 1 0 0 110 1 0 1 0

11 1 0 1 1 15 1 1 1 1

3 4 6 9 10 15

a x x

b x

c x

d x x

x1x2x3x4

4,6 0 1 - 0 a8,9 1 0 0 -8,10 1 0 - 0 3,11 - 0 1 1 b9,11 1 0 - 110,11 1 0 1 - 11,15 1 - 1 1 c

x1x2x3x48,9,10,11 1 0 - - d


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3. 143

144


78/142

2 11 12 13 14 15

a xb

c

d x x

e

f x x x x

3 7 11 12 13 14 15a

b x x

c x x

d x x

e x x x x

f x x x x

3 7 12 13 14 15

a

b x x

c x x

d

e

f x x x x

1,

2 3

p adf d f adf1 ( )p b e b c e d e f c d e f e bd f2 ( )( )( ) ( ) ( )

p b b c f c f bf3 ( ) ( )

p p p p adf e bd fbf abdf e bd abdf 1 2 3 ( ) ( )

2 11 12 13 14 15

a x

b

d x x

f x x x x

1

1.

2142143211 xxxxxxxxxfdaf .

3 7 11 12 13 14 15

a

b x x

d x x

f x x x x

2

3. 145

2

214314312 xxxxxxxxfdbf .

3 7 12 13 14 15

146

1x0 1y0


79/142

3 7 12 13 14 15

ab x x

d

f x x x x

3

214313 xxxxxfbf .

x1x0y1y0 f1f2f3f40 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0

2 0 0 1 0 0 0 0 0

3 0 0 1 1 0 0 0 0

4 0 1 0 0 0 0 0 0

5 0 1 0 1 0 0 0 1

6 0 1 1 0 0 0 1 0

7 0 1 1 1 0 0 1 1

8 1 0 0 0 0 0 0 0

9 1 0 0 1 0 0 1 0

10 1 0 1 0 0 1 0 0

11 1 0 1 1 0 1 1 0

12 1 1 0 0 0 0 0 0

13 1 1 0 1 0 0 1 1

14 1 1 1 0 0 1 1 0

15 1 1 1 1 1 0 0 1

x1x0

f1 y1y0

x1x0

f2 y1y000 01 11 10 00 01 11 10

00 00

01 01

11 1 11 1

10 10 1 1

x1x0

f3 y1y0

x1x0

f4 y1y000 01 11 10 00 01 11 10

00 00

01 1 1 01 1 1

11 1 1 11 1 1

10 1 1 10

0111012

01011

yyxyxxfyyxxf

004

0110010101013

yxfyyxyxxyyxyxxf

3. 147

1, x0, y1 0 1,x0 y=(y1,y0

148

3.6

f x x x x( , , , ) ( , , , , , , )1 2 3 4 0 2 5 6 7 8 10 .


80/142

x1x0y1y0 f1f2f30 0 0 0 0 0 1 0

1 0 0 0 1 1 0 0

2 0 0 1 0 1 0 0

3 0 0 1 1 1 0 0

4 0 1 0 0 0 0 1

5 0 1 0 1 0 1 0

6 0 1 1 0 1 0 0

7 0 1 1 1 1 0 0

8 1 0 0 0 0 0 1

9 1 0 0 1 0 0 1

10 1 0 1 0 0 1 0

11 1 0 1 1 1 0 0

12 1 1 0 0 0 0 1

13 1 1 0 1 0 0 1

14 1 1 1 0 0 0 1

15 1 1 1 1 0 1 0

x1x0

f1 y1y0

x1x0

f2 y1y000 01 11 10 00 01 11 10

00 1 1 1 00

01 1 1 01

11 11 1

10 1 10 1 1

f3 y1y0

x1x0

00 01 11 10

00

01 1

11 1 1 1

10 1 1

01010101010101012

010001111

yyxxyyxxyyxxyyxxf

yyxyxxyxf

010001113 yyxyxxyxf

f x x x x( , , , ) ( , , , , , )1 2 3 4 1 3 5 7 13 15 .

f x x x x x b( , , , , ) ( , , , , , , , , , , , , , ) ( , , , )1 2 3 4 5 0 1 5 6 8 9 12 15 17 20 21 22 25 28 3 14 27 30

)18,10,8()31,30,28,27,25,22,20,19,17,15,12,11,9,6,4,3,1( bf

-

f x x x x( , , , ) ( , , , , , , , )1 2 3 4 1 4 5 6 7 9 11 12 .

-

f x x x x b( , , , ) ( , , , , , , , ) ( , )1 2 3 4 0 1 2 5 7 12 13 14 8 11 .

-

f x x x x( , , , ) ( , , , , , )1 2 3 4 1 3 5 7 13 15 .

-

f x x x x b( , , , ) ( , , , ) ( , , , , )1 2 3 4 2 4 8 15 0 7 10 12 13

9.

f

f

f

1

2

3

0 2 4 5 9 10 11 13 15

2 5 10 11 12 13 14 15

0 2 3 4 9 11 13 14 15

( , , , , , , , , )

( , , , , , , , )

( , , , , , , , , )

3. 149

-

f

f

1 2 3 5 7 8 9 10 11 13 15

2 3 5 6 7 10 11 14 15

( , , , , , , , , , )

( )


81/142

f

f

2

3

2 3 5 6 7 10 11 14 15

6 7 8 9 13 14 15

( , , , , , , , , )

( , , , , , , )

f a b c d b( , , , ) ( , , , , ) ( , , , , , ) 2 6 8 10 11 3 4 5 9 12 15 .

152


82/142

)

-

- -

2n

4. 153

-

154

i

iiiiiiiiiiiiiiiiiiiiiii CyxyxCyxyxCyxCyxCyxCyxS )()(

iiiiiiiiiiiii CyxCyxyxCyxyx )()()( (4.3)


83/142

xi

i i Ci i i+1-

yi Ci Si Ci+1 Si: Ci+1:

0 0 0 0 0 xi xi0 0 1 1 0 0 1 0 1

0 1 0 1 0 00 1 00

0 1 1 0 1 yiCi 01 1 yiCi 01 1

1 0 0 1 0 11 1 11 1 1

1 0 1 0 1 10 1 10 11 1 0 0 11 1 1 1 1

a) )

i i+1

iiiiiiiiiiiii CyxCyxCyxCyxS

(4.1)

)(1 iiiiiiiiiiii yxCyxCyCxyxC (4.2)

i i

p

- p.

)()(1 iiiiiiiiiiiii yxCyxyxyxCyxC . (4.4)

i i+1 --

- -

- -


84/142

4. 157

i i+1 Ci

158

i i+1 i+2, Bi, Bi+1, Bi+2 i.


85/142

Carry-

iiiiii

iiiii

iiii

CPPGPG

CPGPG

CPGC

111

11

1112

)(

(4.7)

i+1 =1 i Pi+1=1

i i=Pi+1=1.

-

4- -

- i -

-


86/142


87/142

4. 163 164

2n


88/142


89/142

4. 167

- i.

168

1 0 i.


90/142

f x y z x yz xy z x y z xyz( , , ) ( , , , ) 1 2 4 7

. (4.9)

3- f s s D s s D s s D s s D 1 0 0 1 0 1 1 0 2 1 0 3. (4.10)

s x1 s y0 . f x yD xyD x yD xyD 0 1 2 3. (4.11)

D D z D D z0 3 1 2 . (4.12)

f x y z x yz xy z x y z xyz( , , ) ( , , , ) 1 2 4 7

f xzD xzD xzD xzD 0 1 2 3

1=z 0

. (4.13)

3

D y D y D y y0 1 2 D3 .

wx

yz

00 01 11 10

00 1

01 1

11 1 1

10 1

f w x y z( , , , ) ( , , , , ) 0 1 6 7 11

-

1 0

f s s D s s D s s D s s D 1 0 0 1 0 1 1 0 2 1 0 3. (4.14)

4. 169

1s0 0 yz yz y z z y ( ) . D y0 .

yz yz y s

170

D D x D xw x w0 1 2 w D3 .

-


91/142

yz yz y 1s0

f D 1. D y1 .

D3 0 D yz2 .

f w x y z( , , , ) ( , , , , ) 0 1 6 7 11

f w x y z( , , , ) ( , , , , ) 0 1 6 7 11

1 0

.

3

C2 C1 C0 D0 D1 D2 D3 D4 D5 D6 D70 0 0 x 0 0 0 0 0 0 0

0 0 1 0 x 0 0 0 0 0 0

0 1 0 0 0 x 0 0 0 0 0

0 1 1 0 0 0 x 0 0 0 0

1 0 0 0 0 0 0 x 0 0 0

1 0 1 0 0 0 0 0 x 0 0

1 1 0 0 0 0 0 0 0 x 0

1 1 1 0 0 0 0 0 0 0 x


92/142

4. 173

- - -

174


93/142

6 3

12

-

- AB AB AB, , AB

ABC ABC ABC ABC, , ,

-

- - i -

-

- w x y z s1 s2 s3 s4 s5 s6 s7

0 0 0 1 1 1 1 1 1 1 1 0

1 0 1 0 0 0 0 0 1 1 0 0

2 0 1 0 1 1 0 1 1 0 1 1

3 0 1 1 0 0 0 1 1 1 1 1

4 0 1 1 1 0 1 0 1 1 0 1

5 1 0 0 0 0 1 1 0 1 1 1

6 1 0 0 1 1 1 0 0 1 1 1

7 1 0 1 0 0 0 1 1 1 0 0

8 1 0 1 1 1 1 1 1 1 1 1

9 1 1 0 0 0 1 1 1 1 0 1

- 6.

4. 175

6 s6 3 5 6 8 9 11 ( , , , , , )

176

s5 3 4 6 7 8 9 10 11 12 ( , , , , , , , , )s6 3 5 6 8 9 11 ( , , , , , )s7 5 6 7 8 9 11 12 ( , , , , , , ) .

n


94/142

s6 ( , , , , , ) .

-

-

-

s1 3 5 9 11 ( , , , )s2 3 5 6 8 10 11 12 ( , , , , , , )s3 3 5 6 8 10 11 12 ( , , , , , , )s4 3 4 5 6 7 10 11 12 ( , , , , , , , )

n

n -

4.5 ROM (Read Only Memory)

n n

n


95/142

4. 179

(PLA-Programmab (PAL-

180

4.


96/142

n n

Array).

n

f xz xz1

-

f yz xz2

f xz yz3

f x y yz4

xz xz yz, , x y.

1, f2, f3 4

f1 x

yz

0 1

00

01 1

11 1

10 1

f2 x

yz

0 1

00

01 1 1

11 1

10

f3 x

yz

0 1

00 1

01 1 1

11

10 1

f4 x

yz

0 1

00 1

01 1 1

11

10


97/142

4. 183

184

-


98/142

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 61 1

2 1

3 1

4 1

5 1

6 1

7x x x x x x1 2 5 7 11 12 1 0 - - 1 - 1 - - - 0 11

8 1

9 1

10 1

11 112 1

13 1

14 1

15 1

16 1

A

-

B S P

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

-

S: A P: A0 1 0 1

B0 1

B0

1 1 1 1

S A B AB AB

P AB -

S AB AB

AB AB AA BB

( ) ( )A B A A B B

AB A ABB

( )( )AB A AB B

4. 185 186

( ) ( )A B A A B B

( )( )A B A B

( )( )A B A B

A B A B

A A B B A B


99/142

-

P AB

AB

AB AB .

-

-

S AB AB AB AB AA BB

P AB

AB

A B

A A B B .

-

-

S ABC ABC ABC ABC

-

C AB AB C AB AB( ) ( )

4. 187

C AB A B C A B AB( ) ( )

C AB A B C A B AB( ) ( )

C AB AB C AB AB( ) ( )

CS CS 1 1

1 -

188


100/142

P ABC ABC AB C ABC

C AB AB AB( )

CS P1 1 1 -

B C S P0 0 0 0 0

0 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

-

S ABC ABC ABC ABC

3

P ABC ABC AB C ABC .

4. 189

x0

i i k=1.

x1 x2 x3 x4 x5 x6 x7 z0 z1 z2

190


101/142

0 1 2 3 4 5 6 7 0 1 2

1 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 1 00 0 0 1 0 0 0 0 0 1 10 0 0 0 1 0 0 0 1 0 00 0 0 0 0 1 0 0 1 0 1

0 0 0 0 0 0 1 0 1 1 00 0 0 0 0 0 0 1 1 1 1

z x x x x0 4 5 6 7

z x x x x1 2 3 6 7

z x x x x2 1 3 5 7

y x x I x x I x x I x x I 1 2 0 1 2 1 1 2 2 1 2 3

y x x x x x x1 1 2 1 2 1 2

y x x x x x x2 3 4 3 4 3 4 .

4.35.

C0 C1 C2 C3 v1 v2 w

0 0 0 0 0 0 01 0 0 0 0 0 10 1 0 0 0 1 10 0 1 0 1 0 10 0 0 1 1 1 1

4. 191

v C C1 2 3 v C C2 1 3 w C C C C 0 1 2 3.

C x x0 2 4 C y x x1 1 1 2 C x x2 1 3 C y x x3 2 3 4 .

192

z x x x x x x x x1 1 2 1 2 3 4 3 4 z2

z x x x x x x2 1 2 1 3 3 4 ( )

x x x x x x x x1 2 1 3 3 4 3 4( )


102/142

v C C x x x x1 2 3 1 3 3 4 v C C x x x x2 1 3 1 2 3 4 .

F x x I 0 1 2 F x x I 1 1 2 F x x I 2 1 2 F x x I 3 1 2 .

1 z2 3 z F x x v x x x x x x2 1 1 2 1 1 2 1 3 3 4 ( )z F x x v x x x x x x3 3 1 2 1 1 2 1 3 3 4 ( ) .

1, z2 3

z v x x x x1 2 1 2 3 4

x x x x x x x x1 2 1 2 3 4 3 4.

z XX XX XX XX11 10 01 11 00( ) { , , , }

x3x4

x1x2

00 01 11 10

00 1 1

01 1 1 1 1

11 1 1

10 1 1 1 1

x x x x x x x x1 2 3 4 1 2 1 3

3

z x x x x x x3 1 2 1 3 3 4 ( )

x x x x x x x x1 2 3 4 3 4 1 3( )

x x x x x x x x x x x1 2 3 4 1 2 3 4 1 2 3.

1z X3 1 1100 1111 111( ) { , , }

x3x4

x1x2

00 01 11 10

00

01

11 1 1 1

10

z x x x x x x3 1 2 4 1 2 3 .

-

-

4. 193

f1 f2 f3A B AB A=B

0 0 0 0 10 1 1 0 0

1 0 0 1 01 1 0 0 1

194


103/142

-

1, f2 3

f AB1f AB2

f A B AB A B AB A B AB3 .

-

-

-

f yz xy x yz

f s s D s s D s s D s s D 1 0 0 1 0 1 1 0 2 1 0 3.

s x1 s y0 .

f x y z x y f z xyf z x y f z xyf z( , , ) ( , , ) ( , , ) ( , , ) ( , , ) 0 0 0 1 1 0 1 1 .

f i i z( , , )1 2

ij { , }0 1

z z 3-1

f i i z( , , )1 2 j 1i2.

D D D z0 1 20 1 , , D z3 .

4. 195

f yz xy x yz

f

D00MUX

4x11 D1

196

f x x x x x x x x x x( , , , )1 2 3 4 1 2 3 4 3 4

.

3 4 1 2

f x x f x x x x f x x x x f x x x x f x x 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 20 0 0 1 1 0 1( , , , ) ( , , , ) ( , , , ) ( , ,


104/142

f

X y

4x1z

z

D2

D3

f yz xy x yz

f x y z yz f x yzf x yz f x yzf x( , , ) ( , , ) ( , , ) ( , , ) ( , , ) 0 0 0 1 1 0 1 1 .

D x D D x0 1 20 , , D3 1 .

f yz xy x yz

f

D0x

y z

MUX

4x10

x

1

D1

D2

D3

f yz xy x yz

D D x x0 3 1 2

x x x x x x1 2 2 2 1 1( ) ( )

x x x x x x1 2 1 2 1 2

D D x x1 2 1 2 1 1 1 2

x x x x x x1 2 1 2 1 2

f x x x x x x x x x x( , , , )1 2 3 4 1 2 3 4 3 4

6:

4. 197

3

0

1

2

MP1

x3

x1 x3

x3

x1

x1

2/4

DC f

198

21212211 )( xxxxxxxx 1x

21212111222121 )()( xxxxxxxxxxxxxx

21 xx

E0

1

1

1MP1

X1


105/142

3

0

1

2

MP2

x1 x2

x1

x2

x3

x4

MP1:

31131131331331 xxxxxxxxxxxxxxa

MP2:

4213212121421321221121 xxxxxxxxxxxxxxxxxxxxxxb

baI 0 baI 1 baI 2 abI 3

bababbaaababbabaIIIf )(321

421321212131 xxxxxxxxxxxxf

421131 xxxxxx

42131 xxxxx

4231 xxxx

) 4231 xxxxf x3 4

3

2143

214321432143

)1,1,,(

)0,1,,()1,0,,()0,0,,(

210

D

DDD

xxfxx

xxfxxxxfxxxxfxxf

10 xD 211 xxD 132 DD

2

3

0

0

X1 X2

MP1

E0

1

2

3

1

1

0

1

X3 X4MP2

ED0D1D2D3

CD1

1

f

X1 X2

X1 + X2

7:

3

0

1

2

DCx3

x4

3

0

1

2

MP1 z1

x2 x3

3

0

1

2

MP2 z2

x2 x3

3

0

1

2

DPx3

0

x1 x2

1 2 1 2

4. 199

a) DC:430 xxD 431 xxD 432 xxD 433 xxD

DP:3210 xxxF 3211 xxxF 3212 xxxF 3213 xxxF

MP1:2320320321 0 DxxFxxDxxz

4332321324332 xxxxxxxxxxxxx

432432 xxxxxx

MP2:3323321321322 DxxFxxFxxDxxz

200

1x0y1y0

x1x0y1y0 z1z2z30 0 0 0 0 0 0 0

1 0 0 0 1 0 0 1


106/142

3323321321322z

433232132321324332 xxxxxxxxxxxxxxxxxx

432432 xxxxxx

30

4324321

II

xxxxxxz

0101 41414141430 xxxxxxxxxII

1

x1 x4

MUX

x4

1

0

0

x2 x3

MUX

z1

0

30

4324322

II

xxxxxxz

1010 41414141430 xxxxxxxxxII

1

x1 x4

MUX

x41

0

0

x2 x3

MUX

z2

0

1 0 0 0 1 0 0 1

2 0 0 1 0 0 0 1

3 0 0 1 1 0 1 0

4 0 1 0 0 0 0 1

5 0 1 0 1 0 1 0

6 0 1 1 0 0 1 0

7 0 1 1 1 0 1 18 1 0 0 0 0 0 1

9 1 0 0 1 0 1 0

10 1 0 1 0 0 1 0

11 1 0 1 1 0 1 1

12 1 1 0 0 0 1 0

13 1 1 0 1 0 1 1

14 1 1 1 0 0 1 1

15 1 1 1 1 1 0 0

0

x0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

x1

y0

y1

z2z1 z3

4. 201

x1x0

z1 y1y0

x1x0

z2 y1y000 01 11 10 00 01 11 10

00 00 1

01 01 1 1 1

11 1 11 1 1 1

10 10 1 1 1

z3 y1y000 01 11 10

202

01011 yyxxz

1011010010110010102 yxxyxxyxxyyxyxxyyxz

0101010101010101

01010101010101013

yyxxyyxxyyxxyyxx

yyxxyyxxyyxxyyxxz

x y


107/142

x1x0

00 01 11 10

00 1 1

01 1 1

11 1 1

10 1 1

x0x1 y0y1z1 z2 z3

x1x0y1y0

x0y1y0

x1y1y0

x1x0y0

x1x0y0

x1x0y1

x1x0y1

x1x0y1y0

x1x0y1y0

x1x0y1y0

x1x0y1y0

x1x0y1y0

x1x0y1y0

x1x0y1y0

x1x0y1y0

1x0 1y0

x1x0y1y0 z2z1z0

0 0 0 0 0 0 0 01 0 0 0 1 0 0 1

2 0 0 1 0 0 1 0

3 0 0 1 1 0 1 1

4 0 1 0 0 0 0 1

5 0 1 0 1 0 1 0

6 0 1 1 0 0 1 1

7 0 1 1 1 1 0 0

8 1 0 0 0 0 1 0

9 1 0 0 1 0 1 1

10 1 0 1 0 1 0 0

11 1 0 1 1 1 0 1

12 1 1 0 0 0 1 1

13 1 1 0 1 1 0 0

14 1 1 1 0 1 0 1

15 1 1 1 1 1 1 0

2, z1 0

4. 203

0

x0

1

2

3

4

5

6

7

8

9

x1

y1

204

x0y0

x0y0


108/142

10

11

12

13

14

15

y0

z2 z1 z0

x1x0

z2 y1y0

x1x0

z1 y1y000 01 11 10 00 01 11 10

00 00 1 1

01 1 01 1 1

11 1 1 1 11 1 1

10 1 1 10 1 1

010001112 yyxyxxyxz

010101011010110111011 yyxxyyxxxyyxxyxxyxyyz

x1x0

z0 y1y000 01 11 10

00 1 101 1 1

11 1 1

10 1 1

00000 xyyxz

x0x1 y0y1z2 z1 z0

x1y1

x1x0y0

x0y1y0

y1y0x1

y1x1x0

y1x1x0

y1y0x1

y1y0x1x0

y1y0x1x0

4. 205

0 0000 0000 0

1 0001 0001 1

206

f x x x( , , ) ( , , , )1 2 3 0 4 6 7


109/142

1 0001 0001 1

2 0010 0011 3

3 0011 0010 2

4 0100 0110 6

5 0101 0111 7

6 0110 0101 5

7 0111 0100 48 1000 1100 12

9 1001 1101 13

10 1010 1111 15

11 1011 1110 14

12 1100 1010 10

13 1101 1011 11

14 1110 1001 9

15 1111 1000 8

0

b

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

a

d

c

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0

y

x

w

z

f x x x x x x x x x x x x x( , , )1 2 3 1 3 1 2 1 2 3 1 2 3

f x x x x b( , , , ) ( , , , , , ) ( , )1 2 3 4 3 4 7 8 11 13 0 5

4. f x x x x b( , , , ) ( , , , , ) ( , , )1 2 3 4 3 4 6 8 9 10 11 15

-

x x1 0 y y1 0 4

-

x x1 0 y y1 0

- -

4. 207

8

208

f1 2

f1 3 5 6 7 ( , , , )f2 1 2 4 7 ( , , , ) .


110/142

f w x y z wx yz wxyz wxyz wx yz1( , , , )

f w x y z wx yz wxyz wx yz2 ( , , , )

f w x y z wxyz wx yz3( , , , ) .

x y z w f 1 f20 0 0 0 0 00 0 0 1 0 0

0 0 1 0 0 00 0 1 1 0 1

0 1 0 0 0 00 1 0 1 0 1

0 1 1 0 0 10 1 1 1 1 01 0 0 0 0 0

1 0 0 1 0 11 0 1 0 0 11 0 1 1 1 01 1 0 0 0 11 1 0 1 1 01 1 1 0 1 0

1 1 1 1 1 0

-

-

f1 f2 f3 4

210


111/142

n

5. 211

m

212


112/142

m (finite state machine).

-

SR (Set-

0

1


113/142

5. 215

Q

Q

Q

Q Q Q

Q =1

216


114/142

Q

S R Q Q

1 0 0 1

1 1 0 1 0 1 1 0

1 1 1 0 0 0 - -

t+1=Qt

Q

Q

Q S RQt t 1 .

5. 217

Qt S R Qt+1 S R Qt+10 0 0 0 0 0 Qt0 0 1 0 0 1 0

0 1 0 1 1 0 1

0 1 1 - 1 1 -

1 0 0 11 0 1 0

1 1 0 1

1 1 1

218


115/142

1 1 1 -

8:

SR

Qt

00 01 11 10

0 b 1

1 1 b 1

Q S RQt t 1

S

R


116/142

5. 221

-

t1 t2 t1

222

- - - -

Q Q


117/142

-

-

-

- -s - - -

-

0

1

t1 t2

t4

t1

Q

D

C

Q

t3

0

1

t4

Q

D

C

Q

t3

Q1 Q1

SR master- -

1 1 Q1 Q1

Q1 Q1

Q2 Q2 .

-

- - - - - - -

5. 223

JK master- -

224


118/142

-

Qt J K Qt+1 J K Qt+10 0 0 0 0 0 Qt

0 0 1 0 0 1 00 1 0 1 1 0 1

0 1 1 1 1 1 Qt1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

-

-

- -

-

- Q J Q KQt t t 1 .

JK

Qt

00 01 11 10

0 1 1

1 1 1

Q J Q KQt t t 1

-

-

- -

-

5. 225

226

- - -


119/142

-

-

-

- - -

- -

- -

- -

-

Qt Qt+1 Qt+10 0 0 0 Qt0 1 1 1 Qt1 0 1

1 1 0

-

- TQQTQ ttt 1

TQQTQttt 1

-

-

T

Qt

0 1

0 11 1

5. 227

-

-

228

-


120/142

-

Qt Qt+1 S R Qt Qt+1 J K

0 0 0 b 0 0 0 b

0 1 1 0 0 1 1 b

1 0 0 1 1 0 b 1

1 1 b 0 1 1 b 0

Qt Qt+1 D Qt Qt+1 T

0 0 0 0 0 0

0 1 1 0 1 1

1 0 0 1 0 11 1 1 1 1 0

-

-

-

1 2 1 2. - n

-

S x Q1 2 2 , R x Q1 1 1 - S x Q x Q2 2 1 1 1 ,R x x2 2 1 -

z x Q Q 2 1 2 .


121/142


122/142

5. 233

-

234


123/142

1

x =0 x =1

A A,0 B,0B A,0 C,0

C D,0 C,0

D A,0 B,1

1101


124/142

5. 237

-

x =0 x =1

238

x =0 x =1

a

A b,0 a,0

D b,0 a,0

G c,0 a,0

H c,0 a,0

b B b,1 a,1

cC d,1 a,0

E d,1 a,0

d F d,0 c,1


125/142

x 0 x 1A B,0 G,0

B B,1 H,1

C F,1 D,0

D B,0 H,0

E F,1 D,0

F F,0 C,1

G E,0 A,0

H E,0 A,0

-

1={ADGH, B, CE, F}:

x =0 x =1

a

A B,0 G,0

D B,0 H,0

G E,0 A,0

H E,0 A,0

b B B,1 H,1

cC F,1 D,0

E F,1 D,0

d F F,0 C,1

2={AD,GH, B, CE, F}.

x =0 x =1

aA b,0 D b,0

G c,0 a,0

H c,0 a,0

b B b,1

cC d,1 a,0

E d,1 a,0

d F d,0 c,1

5. 239

x =0 x =1

a b,0 e c,0 a,0

b b,1

c d,1 a,0d d,0 c,1

240

.40: 0110


126/142

-

- .

-

- -

0110

x =0 x =1

A B,0B C,0C D,0D D,0

0110

Q1Q2

x =0 x =1Q1

+Q2+,z Q1

+Q2+,z

0 0 0 0,0 01,0

0 1 0 0,0 10,0

1 1 0 0,0 11,0

1 0 0 0,1 11,0

5. 241

- - -

Qt Qt+1 J K

0 0 0 b0 1 1 b

1 0 b 1

1 1 b 0

242

K1 x

Q1Q2

0 1

00 b b

01 b b

11 1

10 1

J1 x

Q1Q2

0 1

00

01 1

11 b b

10 b b

J2 x

0 1

00 1

K2 x

0 1


127/142

- 1, K1, J2 2 1Q2.

Q1Q2

x =0 J1K1 J2K2

x =1 J1K1 J2K2

0 0 0 b 0 b 0 b 1 b

0 1 0 b b 1 1 b b 1

1 1 b 1 b 1 b 0 b 0

1 0 b 1 0 b b 0 1 b

.43: -

1, K1, J2K2

J xQ1 2 , K x1 , J x2 , K x Q2 1 z xQ Q 1 2 .

-

-

Q1Q2

00 1

01 b b

11 b b

10 1

Q1Q2

00 b b

01 1 1

11 1

10 b b

z x

Q1Q2

0 1

00

01

11

10 1

5. 243

- - -

Qt Qt+1 D

0 0 00 1 1

1 0 0

1 1 1

244

D1 x

Q1Q2

0 1

00

01 1

11 1

10 1

D2 x

Q1Q2

0 1

00 1

01

11 1

10 1

z x

0 1


128/142

-

Q1Q2

x =0 x =1

D1D2 D1D20 0 00,0 01,0

0 1 00,0 10,0

1 1 00,0 11,0

1 0 00,1 11,0

-

-

D xQ xQ1 1 2

, D xQ xQ

2 1 2

z xQ Q

1 2 .

-

-

Q1Q2

0 1

00

01

11

10 1


129/142

5. 247

x =0 x =1

A B,0 C,0B D,0 E,0

C F,0 G,0

D H,0 H,1

E H,0 H,1

1a DE,DF,DGEF,EG,FG

2 FG,BC,DE

3 DE FG

248

-

Q1Q2Q3

x =0 x =1Q1

+Q2+Q3

+,z Q1+Q2

+Q3+,z

A 0 0 0 1 1 1 1 1 1 1 0


130/142

F H,1 H,0

G H,1 H,0

H A,0 A,0

3 DE,FG

Q1

Q2Q3

0 1

00 A H

01 B C

11 G E

10 F D

A 000

B 001

G 011

F 010

H 100

C 101

E 111

D 110

A 0 0 0 1 1 1,1 1 1 1,0

G 0 0 1 1 1 1,0 1 1 1,0

- 0 1 1 - -

C 0 1 0 1 0 0,1 1 0 1,0

E 1 0 0 0 0 1,0 0 0 0,0F 1 0 1 0 0 0,0 0 0 1,0

B 1 1 1 0 1 0,0 1 1 0,1

D 1 1 0 1 0 1,0 1 0 0,1

x=0 x=1

Q1Q2Q3 Q1Q1+ Q2Q2

+ Q3Q3+ Q1Q1

+ Q2Q2+ Q3Q3

+

0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

0 0 1 0 1 0 1 1 1 0 1 0 1 1 1

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

0 1 0 0 1 1 0 0 0 0 1 1 0 0 1

1 0 0 1 0 0 0 0 1 1 0 0 0 0 0

1 0 1 1 0 0 0 1 0 1 0 0 0 1 1

1 1 1 1 0 1 1 1 0 1 1 1 1 1 0

1 1 0 1 1 1 0 0 1 1 1 1 0 0 0

5. 249

X=0 x=1

Q1Q2Q3 J1K1 J2K2 J3K3 J1K1 J2K2 J3K30 0 0 1 1 b 1 b 1 b 4 1 b 1 b 1 b

0 0 1 1 b 1 b b 0 1 b 1 b b 0

0 1 1 b b b b b b b b b b b b

0 1 0 1 b b 1 0 b 1 b b 1 1 b

1 0 0 2 b 1 0 b 1 b 3 b 1 0 b 0 b1 0 1 b 1 0 b b 1 b 1 0 b b 0

1 1 1 b 1 b 0 b 1 b 0 b 0 b 1

1 1 0 b 0 b 1 1 b b 0 b 1 0 b

250

J2 xQ1

Q2Q3

00 01 11 10

00 1 1

01 1 1

11 b b b b

10 b b b b

K2 xQ1

Q2Q3

00 01 11 10

00 b b b b

01 b b b b

11 b b


131/142

1

J1 1 , K Q xQ1 2 3 , J Q2 1 , K Q2 3 , J x Q Q x Q3 1 2 1 ( )( ),

K x Q Q3 2 1 ( ) z xQ Q xQ Q 1 3 1 2 .

1 2 3 4J1 xQ1

Q2Q3

00 01 11 10

00 1 b b 1

01 1 b b 1

11 b b b b

10 1 b b 1

K1 xQ1

Q2Q3

00 01 11 10

00 b 1 1 b

01 b 1 1 b

11 b 1 b

10 b b

11 b b

10 1 1 1 1

J3 xQ1

Q2Q3

00 01 11 1000 1 1 0 1

01 b b b b

11 b b b b

10 0 1 0 1

K3 xQ1

Q2Q3

00 01 11 10

00 b b b b

01 0 0 011 b b

10 b b b b

z xQ1

Q2Q3

00 01 11 10

00 1

01

11 b 1 b10 1 1

5. 251

252


132/142

- -

-

0, D1, D2 3 -

- -

5. 253

-

i

i

- i(i=0,1, 3).

254

- -


133/142

-

- -

-

Shift - - - -


134/142

5. 257

- 64.

-

258

0 1 0 2 1 Q3 2


135/142

000 000 0000

001 001 1000

010 011 1100

011 010 1110

100 110 1111

101 111 0111

110 101 0011

111 100 0001

- - -

- - - -

4-

Q3Q2Q1Q00 0 0 0

0 0 0 1

0 0 1 0

0 0 1 1

0 1 0 0

0 1 0 1

0 1 1 0

0 1 1 1

1 0 0 0

1 0 0 1

1 0 1 0

1 0 1 1

1 1 0 0

1 1 0 1

1 1 1 01 1 1 1

-

-

-

5. 259

- -

-

260

Q1Q2

x =0 x =1D1D2 D1D2

0 0 10 01

0 1 00 11

1 1 01 10

1 0 11 00


136/142

4-

0: -

5.68.

D Q x Q x Q x1 2 2 2 D xQ xQ Q x2 1 1 1 .

-

D2 x

Q1Q2

0 1

00 101 1

11 1

10 1

D1 x

Q1Q2

0 1

00 101 1

11 1

10 1

5. 261

-

262

5.8

- + Q.

-

L G Q Q+

0 0 0 00 0 1 10 1 0 0


137/142

0 1 0 00 1 1 01 0 0 0

1 0 1 11 1 0 11 1 1 1

L G Q+

0 0 Q

0 1 0

1 0 Q

1 1 1

GQ

L

00 01 11 10

0 1

1 1 1 1

LGQGQ

L G Q Q+ T

0 0 0 0 00 0 1 1 00 1 0 0 00 1 1 0 11 0 0 0 01 0 1 1 01 1 0 1 1

1 1 1 1 0


138/142

5. 265

-

Q1Q2Q3Q4 Q1+Q2

+Q3+Q4

+

0 0 0 0 0 0 0 1 0

1 0 0 0 1 b b b b

2 0 0 1 0 0 1 0 03 0 0 1 1 b b b b

4 0 1 0 0 0 1 1 0

5 0 1 0 1 b b b b

6 0 1 1 0 1 0 0 0

7 0 1 1 1 b b b b

266

5. -

Q3Q2Q1Q0 Q3+Q2

+Q1+Q0

+

0 0 0 0 0 0 0 0 1

1 0 0 0 1 0 0 1 0

2 0 0 1 0 1 0 0 13 0 0 1 1 b b b b

4 0 1 0 0 b b b b

5 0 1 0 1 b b b b

6 0 1 1 0 b b b b

7 0 1 1 1 b b b b


139/142

8 1 0 0 0 1 0 1 0

9 1 0 0 1 b b b b

10 1 0 1 0 1 1 0 0

11 1 0 1 1 b b b b12 1 1 0 0 1 1 1 0

13 1 1 0 1 b b b b

14 1 1 1 0 0 0 0 0

15 1 1 1 1 b b b b

Q1Q2

D1 Q3Q4

Q1Q2

D2 Q3Q400 01 11 10 00 01 11 10

00 b b 00 b b 101 b b 1 01 1 b b

11 1 b b 11 1 b b

10 1 b b 1 10 b b 1

Q1Q2

D3 Q3Q400 01 11 10

00 1 b b

01 1 b b

11 1 b b

10 1 B b

21313211 QQQQQQQD 32322 QQQQD

33 QD 04 D

8 1 0 0 0 b b b b

9 1 0 0 1 1 0 1 0

10 1 0 1 0 1 0 1 1

11 1 0 1 1 1 1 0 012 1 1 0 0 1 1 0 1

13 1 1 0 1 1 1 1 0

14 1 1 1 0 1 1 1 1

15 1 1 1 1 0 0 0 0

Q1Q2

D3 Q3Q4

Q1Q2

D2 Q3Q400 01 11 10 00 01 11 10

00 b 1 00 b01 b b b b 01 b b b b

11 1 1 1 11 1 1 1

10 b 1 1 1 10 b 1

0123133 QQQQQQD 012023122 QQQQQQQQD

Q1Q2

D1 Q3Q4

Q1Q2

D0 Q3Q4

00 01 11 10 00 01 11 1000 1 b 00 1 b 1

01 b b b b 01 b b b b

11 1 1 11 1 1

10 b 1 1 10 b 1

013011 QQQQQD 00 QD

5. 267

a b c d x y z w

0 0 0 1 1 3 0 0 0 0 0

1 0 1 0 0 4 1 0 0 0 8

2 0 1 0 1 5 0 1 0 0 4

3 0 1 1 0 6 1 1 0 0 12

4 0 1 1 1 7 0 0 1 0 2

5 1 0 0 0 8 1 0 1 0 10

6 1 0 0 1 9 0 0 0 1 1

7 1 0 1 0 10 1 0 0 1 9

268

Q1

LSB

Q1

Q2

Q2

D1 Q3

Q3

D2 D3

x

C

MSB

Q4

Q4

D4

0123456789101112131415


140/142

7 1 0 1 0 10 1 0 0 1 9

8 1 0 1 1 11 0 1 0 1 5

9 1 1 0 0 12 1 1 0 1 13

0

b

1

2

3

4

5

6

78

9

10

11

12

13

14

15

a

d

c

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0

y

x

w

z

-

z

5. 269

270

x =0 x =1

A A,0 C,0

B D,1 A,0

C F,0 F,0

D E,1 B,0

E G,1 G,0

F C,0 C,0

G B,1 H,0

H H,0 C,0


141/142

1 2

-

-

-

- J xQ1 2 , K xQ1 2 , J x Q x Q2 1 1 ( )( ) , K x Q x Q2 1 1 ( )( )

z Q1 2 z Q2 1 .

Q1Q2x1x2,z

00 01 10 11

0 0 1 0,0 1 1,0 0 0,1 1 1,0

0 1 1 0,1 1 1,0 0 0,1 1 1,1

1 0 0 0,0 0 1,1 0 0,1 0 1,0

1 1 0 0,1 0 1,0 0 0,1 0 1,1

5. 271

L M Q Q+

0 0 0 00 0 1 0

0 1 0 00 1 1 01 0 0 1

1 0 1 11 1 0 1

1 1 1 0

- 1 2 1 z2

1. Balabanian N., Carlson B.: Digital logic design principles, John Wiley& Sons, New York, 2001

2. Balch M.: Complete digital design: A comprehensive guide to digital

electronics and computer system architecture, McGraw-Hill, New York,2003

3. Mano M., Ciletti D.: Digital design, Prentice Hall, New York, 2007


142/142

1 2 1 2

Q1Q2x1x2,z1z2

00 01 10 11

0 0 00,00 01,00 11,00 00,00

0 1 01,00 01,00 11,00 01,00

1 0 10,11 01,11 11,10 10,10

1 1 11,10 01,10 11,10 11,10

Beograd, 1994

projektovanja digitalnih sistema, Nauka - Beograd, 1995

6. cCluskey E.: Logic design principles: With emphasis on testablesemicustom circuits, Prentice Hall Englewood Clifs, New Jersey, 1986

7. ull L., Lobur J.: The essentials of computer organization and

architecture, Jones and Bartlett, Boston, 2003

digitalne elektronike, II deo- knjiga, Beograd, 1985

9. anenbaum A.: Structured computer organization, 3rd edition, Prentice

Hall Englewood Clifs, NJ, 1990

Documents

Digitalna Logika (Cveta Martinovska)