simplification of switching functions

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 M0 = m0 = AB

B

A

A+B M3 = m3 = AB

B

Karnaugh maps

• One to one correspondence between K-map squares and maxterms

A

A+B M2 = m2 = AB

B

A

A+B M1 = m1 = AB

B

3-variable K-maps

3-variable K-maps

• Constructing 3-variable K-maps

A A

B 0 1 1 0 B

0 flip 0

1 1

C = 0 C = 1

abutt

CA

B 00 01 11 10

0

1

3-variable K-maps

• Constructing 3-variable K-maps

A A

B 0 1 CB 1 0

0 C = 0 00

1 01

C = 0 11

A 10

B 0 1

1 C = 1

0

4-variable K-maps

5-variable K-maps

5-variable K-maps

6-variable K-maps

6-variable K-maps

Plotting functions in canonical form

Plotting functions in canonical form