Lecture (09) Karnaugh Maps 2 By: Dr. Ahmed ElShafee Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I ١ Determination of Minimum Expansions Using Essential Prime Implicants • Cover: A switching function f(x1,x2,…,xn) is said to cover another function g(x1,x2,…,xn), if f assumes the value 1 whenever g does. • Implicant : Given a function F of n variables, a product term P is an implicant of F iff for every combination of values of the n variables for which P=1 , F is also equal 1.That is, P=1 implies F=1. Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I ٢
29
Embed
Lecture (09) Karnaugh Maps 2 - Dr. Ahmed ElShafee - …draelshafee.net/Spring2017/logic-design-1---lecture-09.pdf · · 2017-05-01Lecture (09) Karnaugh Maps 2 By: ... Given a function
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Lecture (09)Karnaugh Maps 2
By:
Dr. Ahmed ElShafee
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١
Determination of Minimum Expansions Using Essential Prime Implicants
• Cover: A switching function f(x1,x2,…,xn) is said to
cover another function g(x1,x2,…,xn), if f assumes the
value 1 whenever g does.
• Implicant : Given a function F of n variables, a
product term P is an implicant of F iff for every
combination of values of the n variables for which
P=1 , F is also equal 1.That is, P=1 implies F=1.
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢
• Prime Implicant: A prime implicant of a
function F is a product term implicart which is no
longer an implicant if any literal is deleted from it.
• Essential Prime Implicant: If a minterm is
covered by only one prime implicant, then that prime
implicant is called an essential prime implicant.
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٣
• On a Karnaugh Map
Any single 1 or any group of 1’s (2k 1’s, k=0,1,2,…)
which can be combined together on a map of the
function F represents a product term which is
called an implicant of F.
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٤
• A product term implicant is called a prime implicant if it cannot be combined with another term to eliminate a variable.
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٥
• If a minterm is covered by only one prime implicant,
then that prime implicant is called an essential
prime implicant.
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٦
Examples
• f=wx+yz,
• g=wxy’
• g=1 (w=1,x=1,y=0)
• implies
• f=1.1+0.z=1,
• f covers g.
• g is a product term, g is an implicant of f.
• g is not a prime implicant.
• The literal y’ is deleted from wxy’, the resulting term wx is also an implicant of f.
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٧
• f=wx+yz,
• h=wx is a prime implicant.
• The deletion of any literal (w or x) results a new product (x or w) which is not covered by f.
• [w=1 does not imply f=1 (w=1,x=0,y=0,z=0 imply f=0)]
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٨
Example:
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٩
111
11
1
11
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٠
00 01 11 1000
01
11
10
abcd
111
11
1
11
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١١
00 01 11 1000
01
11
10
abcd
111
11
1
11
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٢
00 01 11 1000
01
11
10
abcd
• The minimum sum‐of‐products expression for a
function consists of some (but not necessarily all) of
the prime implicants of a function.
• A sum‐of‐products expression consisting a term which
is not a prime implicant cannot be minimum.
• The essential prime implicant must be included in the
minimum sum‐of‐products.
• In order to find the minimum sum‐of‐products from a
map, we must find a minimum number of prime
implicants which cover all of the 1’s on the map.
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٣
Example:
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٤
11
111
111
11
00 01 11 1000
01
11
10
abcd
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٥
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٦
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٧
Example
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٨
11
1111
111
00 01 11 1000
01
11
10
abcd
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I١٩
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٠
Essential prime implicates: BD,B′’C, AC
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢١
Example
• Simplify the function
• f (A, B,C,D) = Σm(0,1,2,4,5,7,11,15).
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٢
• f (A, B,C,D) = Σm(0,1,2,4,5,7,11,15).
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٣
11
11
111
1
00 01 11 1000
01
11
10
abcd
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٤
Example
• Simplify the function
• f (A, B,C,D) = Σm(4,5,6,8,9,10,13) + Σd(0,7,15)
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٥
• f (A, B,C,D) = Σm(4,5,6,8,9,10,13) + Σd(0,7,15)
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٦
11x
111
xx
11
00 01 11 1000
01
11
10
abcd
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٧
Example
• Find min min‐terms expansion
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٨
1
111
111
1
00 01 11 1000
01
11
10
abcd
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٢٩
1
111
111
1
00 01 11 1000
01
11
10
abcd
•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٣٠
Other forms of 5‐Variable Karnaugh Maps•
Dr. Ahmed ElShafee, ACU : Spring 2017, Logic Design I٣١