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

Part I Combinational Logic. Specification Implementation K-maps

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

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

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

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)

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

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

0 1 1 1 f(A,B) = A + B K-Map: Truth Table in 2 Dimensions A = 0 A = 1 0 2 0 1 1 1 B = 0 B = 1 1 3 A’B AB f(A,B) = A + B

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

On the K-map: A = 0 A= 1 0 2 B= 0 B = 1 0 0 1 1 1 3 A’B AB f(A,B)=B

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

Two Variable K-maps # possible 2-variable functions: iClicker 4 16 32 Id a b f (a, b) 0 0 0 f (0, 0) 1 0 1 f (0, 1) 2 1 0 f (1, 0) 3 1 1 f (1, 1) # possible 2-variable functions: iClicker 4 16 32 81 None of the above f(a,b) a b

Two Variable K-maps # possible 2-variable functions: Id a b f (a, b) 0 0 0 f (0, 0) 1 0 1 f (0, 1) 2 1 0 f (1, 0) 3 1 1 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)

Two-Input Logic Gates

More Two-Input Logic Gates

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)

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)

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 0 0 0 0 1 1 0 0 1 0 2 0 1 0 1 3 0 1 1 0 4 1 0 0 1 5 1 0 1 0 6 1 1 0 1 7 1 1 1 0

(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) 0 2 6 4 c = 0 1 1 1 1 1 3 7 5 c = 1 0 0 0 0 a = 1 f(a,b,c) = c’

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

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’

Another 3-Input example Id a b c f (a,b,c) 0 0 0 0 0 1 0 0 1 0 2 0 1 0 1 3 0 1 1 0 4 1 0 0 1 5 1 0 1 1 6 1 1 0 - 7 1 1 1 1

(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) 0 2 6 4 c = 0 0 1 - 1 1 3 7 5 c = 1 0 0 1 1 a = 1 f(a,b,c) = a + bc’

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 - 0 0 0 0 1 1 0 0 1 1 2 0 1 0 - 3 0 1 1 0 4 1 0 0 1 5 1 0 1 1 6 1 1 0 0 7 1 1 1 0

(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) 0 2 6 4 c = 0 1 - 0 1 1 3 7 5 c = 1 1 0 0 1 a = 1 f(a,b,c) = b’

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

4-input K-map

4-input K-map

4-input K-map

K-maps with Don’t Cares

K-maps with Don’t Cares

K-maps with Don’t Cares