Min Max Term
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Transcript of Min Max Term
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Dr. D. J. Jackson Lecture 4-1Electrical & Computer Engineering
ECE380 Digital Logic
Introduction to Logic Circuits:
Synthesis using AND, OR, andNOT gates
Dr. D. J. Jackson Lecture 4-2Electrical & Computer Engineering
Example logic circuit design
Assume we want to design a logic circuit withthree inputs x, y, and z
The circuit output should be 1 only when x=1and either yor z (or both) is 1
Three possible combinations
x=1, y=0, z=1 => xyz x=1, y=1, z=0 => xyz
x=1, y=1, z=1 => xyz
The function could be written as
f(x,y,z)=xyz+xyz+xyz
sum of products form
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Dr. D. J. Jackson Lecture 4-3Electrical & Computer Engineering
Example logic circuit design
Dr. D. J. Jackson Lecture 4-4Electrical & Computer Engineering
Example logic circuit design
Implements f correctly,but is not the simplestsuch network
f(x,y,z)=xyz+xyz+xyz
xyz+xyz+xyz
xyz+xy from 14a
x(yz+y) from 12a
x(y+z) from 16a
1101
0110
0001
1011
1
00
0
x
111
001010
000
fzy
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Dr. D. J. Jackson Lecture 4-5Electrical & Computer Engineering
Example logic circuit design
Obviously, the cost (in terms of gates and
connections) of this network is much lessthan the initial network
The process of generating a circuit from astated desired functional behavior is calledsynthesis
Generation of AND-OR style networks from atruth table is one of many types of synthesis
techniques that we will cover
Dr. D. J. Jackson Lecture 4-6Electrical & Computer Engineering
Logic synthesis
If a function f is described in a truth table,then an expression that generates f can byobtained (synthesized) by
Considering all rows in the table where f=1, or
By considering all rows in the table where f=0
This will be an application of the principal ofduality
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Dr. D. J. Jackson Lecture 4-7Electrical & Computer Engineering
Minterms
For a function ofn variables f(a,b,c,n)
A minterm off is a product ofn literals (variables)in which each variable appears once in either trueor complemented form, but not both
f(a,b,c) -- minterm examples: abc, abc, abc
f(a,b,c) -- invalid examples: ab, c, ac
An n variable function has 2n valid minterms
Dr. D. J. Jackson Lecture 4-8Electrical & Computer Engineering
Minterms
Each row of a truthrow corresponds to asingle minterm
When a function iswritten as a sum ofminterms, the form is
called a standard (orcanonical) sum-of-products
7
6
5
4
32
1
0
Row
number
m5=xyz101
m3=xyz110
m4=xyz001
m6=xyz011
1
0
0
0
x
m7=xyz11
m2=xyz01
m1=xyz10
m0=xyz00
Mintermzy
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Dr. D. J. Jackson Lecture 4-9Electrical & Computer Engineering
Minterm notation
An equation may be written in terms of
m-notation
f(a,b,c)=m0+m1+m2+m4f(a,b,c)=abc+abc+abc+abc
f(a,b,c)=m(0,1,2,4)
000 001 010 100} } } }
0 1 2 4
0101
0110
1001
0011
1
0
0
0
a
011
101
110
100
fcb
Dr. D. J. Jackson Lecture 4-10Electrical & Computer Engineering
Minterm notation examples
What is the minterm notation for thefollowing function?
f(a,b,c)=abc+abc+abc+abc
What is the function (in terms of variables) ifthe minterm notation is the following?
f(a,b,c)= m(1,5,6,7)
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Dr. D. J. Jackson Lecture 4-11Electrical & Computer Engineering
Logic synthesis
Duality suggests that:
If it is possible to synthesize a function fbyconsidering the truth table rows where f=1, thenit should also be possible to synthesize fbyconsidering the rows for which f=0.
This approach uses the complement ofminterms, which are called maxterms
Dr. D. J. Jackson Lecture 4-12Electrical & Computer Engineering
Maxterms
Each row of a truthrow corresponds to asingle maxterm
When a function iswritten as a productof maxterms, the
form is called astandard (orcanonical) product-of-sums
7
6
5
4
32
1
0
Row
number
M5=x+y+z101
M3=x+y+z110
M4=x+y+z001
M6=x+y+z011
1
0
0
0
x
M7=x+y+z11
M2=x+y+z01
M1=x+y+z10
M0=x+y+z00
Maxtermzy
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Dr. D. J. Jackson Lecture 4-13Electrical & Computer Engineering
Maxterm notation
An equation may be written in terms of
M-notation
f(a,b,c)=M3 M5 M6 M7
f(a,b,c)=(a+b+c)(a+b+c)(a+b+c)(a+b+c)
f(a,b,c)=M(3,5,6,7)
0101
0110
1001
00111
0
0
0
a
011
101
110
100
fcb
0 1 1 1 0 1 1 1 0 1 1 1} } } }
3 5 6 7
Dr. D. J. Jackson Lecture 4-14Electrical & Computer Engineering
Maxterm notation examples
What is the maxterm notation for thefollowing function?
f(a,b,c)=(a+b+c)(a+b+c)(a+b+c)(a+b+c)
What is the function (in terms of variables) ifthe maxterm notation is the following?
f(a,b,c)= M(1,5,6,7)
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Dr. D. J. Jackson Lecture 4-15Electrical & Computer Engineering
Sum-of-products and minimality
A function expressed in standard sum-of-
products (or product-of-sums) form may notbe minimal
111
1
0
0
a
00
11
10
fb
f(a,b)=ab+ab+ab=a+b
Dr. D. J. Jackson Lecture 4-16Electrical & Computer Engineering
Form conversion
If a function f is given in m or M form, it is
easy to find for f in m or M form
Use the following form conversion table
Use numbersnot onmaxterm list(0,2,5,7)
Use numberson maxtermlist(1,3,4,6)
--Use numbersnot onmaxterm list
(0,2,5,7)
f= M(1,3,4,6)
Use numberson mintermlist(0,2,5,7)
Use numbersnot onminterm list(1,3,4,6)
Use numbersnot onminterm list(1,3,4,6)
--f=m(0,2,5,7)
f= Mf=mf= Mf=m
DESIRED FORMGIVENFORM
