m

1
?????? ?????? Combinatorial Logic
m ????? ?????
n ????? ?????

• ???? ????? Design Principles
• ???? ?????.
• ????? ???? ????? ?????? ??????? ????? ?????
  ?????? ???????.
• ????? ????? ?????? ?????? ???????.
• ????? ???? ??? ??????? ?? ?????? ??????? ???
  ??????? ???????.
• ????? ???????? ????????? ???? ?? ?????.
• "?????" ?????? ?? ???????? ??????.
• ????? ?????? ???????? ??????.

2
BCD gt Seven -Segment - Decoder
a
Seven Segment
b
f
g
c
e
d
??? ???? ?? 4 ????? ? BCD ??? 7 ????????
????????? ?? ??? ??????? ???? "1" ??"? ?-
Segment ?????? ???? ?????.

• ???? ?? ???? ????.
• ???? ?? ag ?"? ???? ????.
• ?????? ?? ??????? ?"? ????? ????? ??????.

3
???? ??? ? BCD ? 7 Seg
D
00
(A,B,C,D)gta
01
B
11
A
10
AB
CD
C
4
(A,B,C,D) gte
5
??? ???? Half Adder
??? ???? ???? 2 ?????? ?????? ?? ????? (mod 2)
??? ????.
S X ? Y (a ? b)
C X Y (a b)
(ab c) (ab)(ab) (ab)(ab) aa
ab ba bb
(ab)ab
a
S
b
C
(ab)
ab
6
???? ??? Full Adder
????????? s,c ???????? ? x,y,z
"??????" x,y,z ???? ????
S xyz xyz xyz xyz C xy yz xz
Y
Y
X
X
C
S
Z
Z
7
Ripple Carry Adder
8
4-Bit Adder
9
???? / ????
10
???? ????? ????? ?????
Self complementary to 9s comp.
(X)BCD (X3)Excess-3
S3
S4
S1
S2
C5
C1
(?? ?????)
O
3
4
B1
2
3
4
A1
2
BCD
I
???? Excess-3 lt BCD
11
????? ???? - Comparator
1 Alt0 B?0

• ???? ????? "1"
• ???? A?0 ?- Blt0.

1 A?0 Blt0

• ???? ????? "1"
• ???? Alt0 ?- B?0.

• S31 ??? overflow ? A-Blt0

• S30 ??? overflow ? A-B?0

• AgtB A?B ??? A?B

12
????? ???? Carry Look Ahead

• ????? ???? ???? nC ????? ?? 2n ???? ?? ?????

c1
c1
cn-1
cn
2 ????? ?????
2 ????? ?????
??? ?????? ???? ????? ????? ?? ????? ????? ????
??????. ????? ????? ????? ??????? ?????? ????
????? ??????. ????? ????????? ????? ??????? ??
?????. ????? ?? ??? i ?. ?? ai1 ?-bi1 ???
???? ??? (ci11) ??? ???? ???? ci. ?. ?? ai1 ??
bi1 (?? ?? ?????) ???? ???????? ????? ??? ??
ci1 ??? ci11.
ai
bi
Ci
Ci1
Si
13
?????? ?????? ????
bi1, ai1 gt ??? ???? ??????? ????? ??? Gi
ai bi ai ? bi1 gt ??? ????? Pi ai ? bi
??????? ????? ???

• ????? ??? ?? Si, Ci1 ???????? ?? Ci,Gi,Pi
• Si Pi ? Ci
• Ci1 GiPiCi
• Pi, Gi ??????? ???? ??? ???? ???? ????? ?????.
  ???? ??? ?? ci ????? ????????.

14
????? ????????
0
C1 G0 P0C0 G0 C2 G1 P1C1 G1
P1G0 C3 G2 P2C2 G2 P2(G1 P1G0)
G2 P2G1 P2P1G0 C4 G3 P3C3 G3P3(G2
P2G1 P2P1G0) G3 P3G2 P3P2G1
P3P2P1G0 Ci1 Gi PiGi-1 PiPi-1Gi-2
PiPi-1 P1G0
????? C4
G3
C4
15
????? ????? ?????? ?"? ????? ?? ???? ????? ?? 2
G0
G0
P1
P1
P2
P2
To C4
To C4
P3
P3
????? ?????? "??????" ????? ?? ??? ?????? ?? ????
(???? ??? ?????? ?????) ? - O(n) ? O(log2n).
???? ???? ????? ?? ???? ????? ????? ???? ?"?
????? ??????? ?????
16
????? ????? ?????? ?"? ????? ???????
17
????? ????? ?????? ?"? ????? ???????
18
Adder w/ Carry Look Ahead


19
????? ????BCD ?"? FAs

• ???????? 4-bit FAs ????? AND, OR

????? ?. ??? AB?T ?"? 4-bit FA. ?. ???? ?'
0?T ?9. ?????? gt ?????? (S1,S0)(0,T) . ?. ????
?' 10 ?T ?18. I. 1? S1. II. ?????
?????? 10 ? T ??? ???? ?? ?????? ??????.
20
??? ?'
10?T?18
over flow
lt 10?T Cout 1 OR S3 1 AND (S21 OR S11)
Cout S3(S2 S1) ???? ??????
21
??? ?'
?? T?10 ????? ?????? 10
)Cout S3 S2 S1 S0)2
-
(10 (10
( X3 X2 X1 X)2
???? ? 9
???? ??? ? 2sCo

-10 2s Comp

• ??????? 4 ????? ????? ?? ??????.
• ??? ???? MSB ???? ??????? ???? ??????? ?? ??????
  ? (X3X2X1X0).
• ???? ???? ?????? ?"? ????? (S3S2S1S0)
  (0110)!

22
???? ???? BCD ?"? FAs
???? BCD
A B (S,C)
?0,1,,9
?0,1
23
???? ??? ?"? ??????
B1 B0
?
A1 A0
0 A0B1 A0B0

A1B1 A1B0 0
S3 S2 S1 S0
S0 A0B0 S1 Least Significant Bit
ADD(A0B1,A1B0) S2 2nd bit of (0,A0B1)
(A1B1,A1B0) S3
add
24
??? ?????? ????????
? ? ? ? ?

• ????? ?? ????? ????? ???? ???????

(b2 b1 b0)
?????
(1011)2 11 1000 8 0010 2 0001 1

• ???? ????? ?"? n ??????? ?? ??????

25
????
????
?. ?? bi0 ?. an-1an-2a1a0000 ???? bi 1.
i ?????
????? ?. ???? ?' ???? ????? ???? ?????? ????
???? 0. ?. ???? ???? ?' ????? ?????????? ??
i. ???? i0 ??? ?????? ?? ???
???? ??? ???????. i1 ??? ????? ???? ????
20010 . ?????? ??? ?????? ? 2. ????? ?? ????
???? ?????? ????? ?????.
26
????? ?? ???? ? least
???? a00 ??? ????? 0 ????? 0. a01 ??? ?????
???? 0 ???? 1. ? S0, Ca0 ???? ??? ??????
???? c2 a1 ?? ????? ???? 0 a0 ?????? ????
???? ?? ?. ci ai-1 ?. si ai-1 ????"?
27
??? ??? ?????? ???? ???? ?"? ?????
b0an-1b0an-2b0a0

b1an-1b1an-2b1a00

bn-1an-1bn-1a00



??"? ????? 2n ????? ??????. ???? ? - 22n 2n1
1 (2n 1)(2n 1) ?????? 2n ????? ??????.
28
??? ??????? ???? ?????? ????? ???? ???? ????
??????. ?????
?
0 0 0


0 0
1 4 8 32 45
????? ???? ?? ????? ???? ??? n ????? ????? ??? m
????? ????? ? - n M ????? ?????? ??????.
29
????? ??? ???? 3B ????? 4B
A0
B0
B1
B2
B3
B1
B2
B2
B0
A1
0
a2
a3
a0
a1
b3
b2
b0
b1
4 bit FA
C4
C0
O
S2
S3
S0
S1
A2
B0
B1
B2
B3
4 bit FA
C4
C0
O
S2
S3
S0
S1
O2
O1
O0
O3
O5
O6
O4