Simplification of switching functions Simplify – why? –Switching functions map to switching circuits –Simpler function  simpler circuit –Reduce hardware complexity –Reduce size and increase speed by reducing number of gates Simplify – how? –Using the postulates –Ad-hoc

Simplification of switching functions Simplify – what? –SOP/POS form has products/sums and literals Literal: each appearance of a variable or its complement –Minimize number of sums/products Reduces total gate count –Minimize number of variables in each sum/product Reduces number of inputs to each gate PLDs have fixed # of inputs; only the number of terms need to be minimized there

Simplification of switching functions

Simplification using postulates

Simplification using Karnaugh maps

Karnaugh maps Karnaugh map (also K-map) is a graphic tool, pictorial representation of truth table –Extension of the concepts of truth table, Venn diagram, minterm –Transition from Venn diagram to minterm

Karnaugh maps –Adjacencies are preserved when going from c) to d) They are the same, only the areas are made equal in d), which preserves adjacencies Subscripts are dropped in e); realize that 2&3 is A; 1&3 is B In f) the labels change and become 0 and 1 –Each square of the K-map is 1 row of the TT

Karnaugh maps Might start with rectangles initially and get the same result  A B  –Each square of the K-map is 1 row of the TT

Karnaugh maps One to one correspondence between K-map squares and maxterms A A+B  M 0 = m 0 = AB B A A+B  M 3 = m 3 = AB B

Karnaugh maps One to one correspondence between K-map squares and maxterms A A+B  M 2 = m 2 = AB B A A+B  M 1 = m 1 = AB B

3-variable K-maps

Constructing 3-variable K-mapsA B B 0flip  01 C = 0C = 1 abutt CA B

3-variable K-maps Constructing 3-variable K-mapsA B 0 1 CB 1 0 0C = C = 0 11 A 10 B 0 1 1C = 1 0

4-variable K-maps

5-variable K-maps

6-variable K-maps

Plotting functions in canonical form