E. Veitch, 1952 ; M. Karnaugh 1953

1
???? ???? ??????? ???????????? ???, ???
???????? ?????????? ?? ????

• ????? ?? ?????? ?????? ?????, ??????????? ??????
  ?"?

2
????? ???????? ?"? ???? ????
y

• E. Veitch, 1952 M. Karnaugh 1953
• ???? ?? ??? ??????

y
0 1
0 m0 m1
1 m2 m3
0 1
0 xy xy
1 xy xy
x
x
????? ?????
y
y

0 1
1 1
x
f m1m2m3
y
f xy
x
x
z

• ???? ?? ????? ??????

yz
00 01 11 10
0 xyz xyz xyz xyz
1 xyz xyz xyz xyz
00 01 11 10
0 m0 m1 m3 m2
1 m4 m5 m7 m6
x
x
y
?? ??? ??????? ?????? ???? ?????? ?????? ???
????. m2 m6 ? xyz xyz ? yz
3
y
00 01 11 10
0 1 0 0 1
1 1 0 1 1
x
z
f xyz xyz xyz xyz xyz
??? ???? ?? ???????? ???? ??????? "???????"
?????? ????? ?? ?"1"
??????? "?????"
??????? ??????
4
????? ?????
f(x,y,z) ?(0,1,5,6,7)
y
00 01 11 10
0 1 1
1 1 1 1
x
z
5
??? ?? ????? ??????
y
yz
00 01 11 10
00 1 1
01 1 1
11
10 1 1
wx
fxz wz
x
w
z
??? ?? ????? ??????
C
1 1 1
1
1 1 1 1 1 1
1 1 1 1 1 1
AB CDE
B
A
D
E
E
f AC ADE CDE BDE
6
??? ?? ????? ?????? ???? ??????
f ABDE ABDE
C

1 1
1 1

AB CDE
B
D
E
7
?????? / ??????? ??????
y
yz
00 01 11 10
00 ? 0 0 0
01 0 ? ? 0
11 1 1 ? ?
10 1 0 0 1
wx
x
w
z
? ? Dont Care ???? ????? ?"1" ?? "0
(??? ????? ???????)
f zw zx ???? ??????
8
???????
9
???????
10
????? ???????
?? ??????? ???????? ????? ?????? ????? ??????
xyzxyzxyzxyzxyzxyz
(xyz)
(xyz)
?????? ??????

??????
??????
????? ???? ????? ???? z y x
M0 xyz m0 xyz 0 0 0
M1 xyz m1 xyz 1 0 0
M2 xyz m2 xyz 0 1 0
M3 xyz m3 xyz 1 1 0
M4 xyz m4 xyz 0 0 1
M5 xyz m5 xyz 1 0 1
M6 xyz m6 xyz 0 1 1
M7 xyz m7 xyz 1 1 1

• ?????? ???? ??? ?? ??????? f
• ????? ?? f ?????? ?????? ?"? ????? Mi ????? f0.
• ??
• 2) ????? ?? f ????? ?????? ?"? ????? mi ????? f1.

minterm
(xyz)
(xyz)
xyzxyz
11
Product of sum design
12
Product of sum design
13
?????? / ??????? ??????
y
yz
00 01 11 10
00 ? 0 0 0
01 0 ? ? 0
11 1 1 ? ?
10 1 0 0 1
wx
x
w
z
? ? Dont Care ???? ????? ?"1" ?? "0
(??? ????? ???????)
f zw zx ???? ??????
14
?????? ?????? Combinatorial Logic
m ????? ?????
n ????? ?????

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

15
BCD gt Seven -Segment - Decoder
a
Seven Segment
b
f
g
c
e
d
??? ???? ?? 4 ????? ? BCD ??? 7 ????????
????????? ?? ??? ??????? ???? "1" ??"? ?-
Segment ?????? ???? ?????.

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

16
???? ??? ? BCD ? 7 Seg
n BCD IN BCD IN BCD IN BCD IN 7 Seg Out 7 Seg Out 7 Seg Out 7 Seg Out 7 Seg Out 7 Seg Out 7 Seg Out
n A B C D a b c d e f g
0 0 0 0 0 1 1 1 1 1 1 0
1 0 0 0 1 0 1 1 0 0 0 0
2 0 0 1 0 1 1 0 1 1 0 1
3 0 0 1 1 1 1 1 1 0 0 1
4 0 1 0 0 0 1 1 0 0 1 1
5 0 1 0 1 1 0 1 1 0 1 1
6 0 1 1 0 1 0 1 1 1 1 1
7 0 1 1 1 1 1 1 0 0 0 0
8 1 0 0 0 1 1 1 1 1 1 1
9 1 0 0 1 1 1 1 1 0 1 1
other ? ? ? ? ? ? ? ? ? ? ?
D
a
00
b
f
1 0 1 1
0 1 1 1
? ? ? ?
1 1 ? ?
g
(A,B,C,D)gta
c
01
e
d
B
11
A
10
AB
CD
C
17
a
b
f
g
c
e
d
(A,B,C,D) gte
18
??? ???? Half Adder
??? ???? ???? 2 ?????? ?????? ?? ????? (mod 2)
??? ????.
a b s c
0 0 0 0
1 0 1 0
0 1 1 0
1 1 0 1
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
19
???? ??? Full Adder
x y z c s
0 0 0 0 0
1 0 0 0 1
0 1 0 0 1
1 1 0 1 0
0 0 1 0 1
1 0 1 1 0
0 1 1 1 0
1 1 1 1 1
????????? s,c ???????? ? x,y,z
"??????" x,y,z ???? ????
S xyz xyz xyz xyz C xy yz xz
Y
Y
1
1 1 1
1 1
1 1
X
X
C
S
Z
Z
20
Ripple Carry Adder
21
4-Bit Adder
22
???? / ????
23
????? ????? ?????
b0
a0
b1
a1
b2
a2
b3
a3
C0
C1
C2
C3
Adder
Adder
Adder
Adder
"0"
??? ????
??? ???? ?????? ?????
S0
S1
S2
S3
???? ??? n4 ?????
Overflow Cn-1 ? Cn-2 1
?????? (57) 1 C3 ? C2

• ?? ? - ??????? ?? ???? ????? ??? ????
  ????? ?????? ??? ???? ????? ??? ????.
• ?? ? - ??????? ???? ????? ??? ???? ????
  ?????? ???? ????? ???? ??? ???? ????? ??????.

24
Full Adder with Overflow check
1
25
????? ???? - Comparator
1 Alt0 B?0
AgtB ??? overflow A-Bgt0 0MSB ? AltgtB
?? overflow MSB1 Agt0,Blt0 BgtA ???
overflow A-Blt0 MSB1 ?? overflow
MSB0 Alt0,Bgt0 c4 XOR c3 Overflow
No Overflow
1 A?0 Blt0
26
Decoders

• Multiplexor
• Connects one of many inputs to one output.
• n select lines for 2n inputs.

27
Decoders Multiplexer
28
41 Multiplexer
29
Multiplexer Binary function