1. CSE 140 Lecture 3 Combinational Logic: Implementation
Professor CK Cheng
CSE Dept.
UC San Diego

2. Part I Combinational Logic.
Specification
Implementation
K-maps

3. Definitions Literals xi or xi’ Product Term x2x1’x0
Sum Term x2 + x1’ + x0
Minterm of n variables: A product of n literals
in which every variable appears exactly once.
Maxterm of n variables: A sum of n literals

4. Implementation Specification  Schematic Diagram Net list,
Switching expression
Obj min cost  Search in solution space
(max performance)
Cost: wires, gates  Literals, product terms,
sum terms
We want to minimize # of terms, # of literals

5. Implementation Specification  Schematic Diagram Net list, Flow:
Truth Table
Karnaugh Map (truth table in two dimensional space)
Sum of Products or Product of Sums
Schematic Diagram of Two Level Logic

6. Truth Table vs. Karnaugh Map
2-variable function, f(A,B) ID A B
f(A,B)
f(0,0) 1
f(0,1) 2
f(1,0) 3
f(1,1)
A=0
A=1
B=0
f(0,0)
f(1,0)
B=1
f(0,1)
f(1,1)

7. Truth Table An example of 2-variable function, f(A,B) ID A B f(A,B)
minterm 1
A’B 2
AB’ 3 AB

8. Function can be represented by sum of minterms:
f(A,B) = A’B+AB’+AB
This is not optimal however!
We want to minimize the number of literals and terms.
We factor out common terms –
A’B+AB’+AB= A’B+AB’+AB+AB
=(A’+A)B+A(B’+B)=B+A
Hence, we have
f(A,B) = A+B

9. 0 1 1 1 f(A,B) = A + B K-Map: Truth Table in 2 Dimensions A = 0 A = 1
B = 0
B = 1
A’B AB
f(A,B) = A + B

10. Another Example ID A B f(A,B) minterm 1 A’B 2 3 AB1
A’B 2 3 AB
f(A,B)=A’B+AB=(A’+A)B=B

11. On the K-map:
A = A= 1
B= 0
B = 1
A’B AB
f(A,B)=B

12. Using Maxterms ID A B f(A,B) Maxterm A+B 1 2 A’+B 3
A+B 1 2
A’+B 3
f(A,B)=(A+B)(A’+B)=(AA’)+B=0+B=B

13. Two Variable K-maps # possible 2-variable functions: iClicker 4 16 32
Id a b f (a, b)
f (0, 0)
f (0, 1)
f (1, 0)
f (1, 1)
# possible 2-variable functions: iClicker 4 16 32 81
None of the above
f(a,b) a b

14. Two Variable K-maps # possible 2-variable functions:
Id a b f (a, b)
f (0, 0)
f (0, 1)
f (1, 0)
f (1, 1)
# possible 2-variable functions:
For 2 variables as inputs, we have 4=22 entries. Each entry can be 0 or 1. Thus we have 16=24 possible functions. a b
f(a,b)

15. Two-Input Logic Gates

16. More Two-Input Logic Gates

17. Representation of k-Variable Func.
Boolean Expression
Truth Table
Cube
K Map
Binary Decision Diagram
(0,1,1,0)
(0,1,1,1)
(1,1,1,0)
(1,1,1,1) B
(0,0,1,0)
(0,0,1,1)
(1,0,1,0)
(1,0,1,1) C
(0,1,0,1)
(1,1,0,1) D
(0,0,0,0)
(0,0,0,1)
(1,0,0,0)
(1,0,0,1) A
A cube of 4 variables: (A,B,C,D)

18. Truth Table vs. Karnaught Map
3-variable function, f(A,B,C) ID A B C
f(A,B,C)
f(0,0,0) 1
f(0,0,1) 2
f(0,1,0) 3
f(0,1,1) 4
f(1,0,0) 5
f(1,0,1) 6
f(1,1,0) 7
f(1,1,1)
(A,B)
(0,0)
(0,1)
(1,1)
(1,0)
C=0
f(0,0,0)
f(0,1,0)
f(1,1,0)
f(1,0,0)
C=1
f(0,0,1)
f(0,1,1)
f(1,1,1)
f(1,0,1)

19. Three-Variable K-Map Id a b c f (a,b,c) 0 0 0 0 1 1 0 0 1 0 2 0 1 0 1

20. (0,0) (0,1) (1,1) (1,0) 1 1 1 1 0 0 0 0 Corresponding K-map b = 1
Gray code
(0,0) (0,1) (1,1) (1,0)
c = 0
c = 1
a = 1
f(a,b,c) = c’

21. Karnaugh Maps (K-Maps)
Boolean expressions can be minimized by combining terms
K-maps minimize equations graphically

22. K-map Circle 1’s in adjacent squares
Find rectangles which correspond to product terms in Boolean expression
y(A,B)=A’B’C’+A’B’C= A’B’(C’+C)=A’B’

23. Another 3-Input example
Id a b c f (a,b,c)

24. (0,0) (0,1) (1,1) (1,0) 0 1 - 1 0 0 1 1 Corresponding K-map b = 1
(0,0) (0,1) (1,1) (1,0)
c = 0
c = 1
a = 1
f(a,b,c) = a + bc’

25. Yet another example Id a b c f (a,b,c,d) 0 0 0 0 1 1 0 0 1 1 2 0 1 0 -

26. (0,0) (0,1) (1,1) (1,0) 1 - 0 1 1 0 0 1 Corresponding K-map b = 1
(0,0) (0,1) (1,1) (1,0)
c = 0
c = 1
a = 1
f(a,b,c) = b’

27. Karnaugh Maps (K-Maps)
Consensus Theorem:
A’B+AC+BC=A’B+AC

28. 4-input K-map

29. 4-input K-map

30. 4-input K-map

31. K-maps with Don’t Cares

32. K-maps with Don’t Cares

33. K-maps with Don’t Cares