simplification of switching functions simplify – why? –switching functions map to switching...
TRANSCRIPT
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