multiple output sop minimization
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Multiple Output Multiple Output SOP MinimizationSOP Minimization
Multiple-Output Minimization• Frequently, practical logic design problems require minimization
of multiple-output functions all of which are functions of the same input variables.
• This is such a tedious task that we relegate it to a computer program, eg, Espresso in the SIS package we see later in the course.
• Here, we will show what needs to be considered in multiple-output minimization, but advise that all such work be performed with the aid of a computer, ie, use a CAD tool.
Example of Multiple-output Minimization
• To illustrate multiple-output minimization, consider the following three output expressions, each of three variables:
f A B C m
f A B C m
f A B C m
1
2
3
0 3 4 5 6
1 2 4 6 7
13 4 5 6
( , , ) ( , , , , )
( , , ) ( , , , , )
( , , ) ( , , , , )
Minimizing f1
f1 = B’C’ + AB’ + AC’ + A’BC
1 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
1 0 11 1
Minimizing f2
f2 = A’B’C + BC’ + AB + AC’
0 0
BC
A 00 01
1
0 1
54
13 2
67
11 10
A
B
C
1
0 1 11
Minimizing f3
f3 = A’C + AB’ + B’C + AC’
0 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
1
1 0 11
Shared Shared Prime Prime
ImplicantsImplicants
Using Shared PIsUsing Shared PIs
• The object is to minimize each of the three functions in such a way as to retain as many shared terms between them as possible, thus optimizing the combinational logic of this system.
• Hence, we now need to look at the shared terms.
AND-ed functions: f1.f2
f1 . f2 = AC’
0 0
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
0 0 11
)6,4(. 21 mff
AND-ed functions: f2.f3
f2 . f3 = AC’ + A’B’C
0 0
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
1
0 0 11
f f m2 3 1 4 6. ( , , )
AND-ed functions: f3.f1
f3 . f1 = AC’ + AB’ + A’BC
0 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
1 0 11
f f m3 1 3 4 5 6. ( , , , )
AND-ed functions: f1.f2 .f3
f1 . f2 . f3 = AC’
0 0
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
0 0 11
f f f m1 2 3 4 6. . ( , )
Summarizing Product Terms The original functions are:
f1 = B’C’ + AB’ + AC’ + A’BC f2 = A’B’C + BC’ + AB + AC’ f3 = A’C + AB’ + B’C + AC’
The product terms, which must be included in the optimized expressions, are: f1 . f2 . f3 = AC’ - common to all three. f1 . f2 = AC’ f2 . f3 = AC’ + A’B’C f3 . f1 = AC’ + AB’ + A’BC
Including Shared PI: AC’
1 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
1 0 11
0 0
BC
A 00 01
1
0 1
54
13 2
67
11 10
A
B
C
1
0 1 11
0 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
1
1 0 11
f1 = AC’
f2 = AC’
f3 = AC’
Including Shared PI: A’B’C
1 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
1 0 11
0 0
BC
A 00 01
1
0 1
54
13 2
67
11 10
A
B
C
1
0 1 11
0 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
1
1 0 11
f1 = AC’
f2 = AC’ + A’B’C
f3 = AC’ + A’B’C
Including Shared PI: AB’
1 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
1 0 11
0 0
BC
A 00 01
1
0 1
54
13 2
67
11 10
A
B
C
1
0 1 11
0 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
1
1 0 11
f1 = AC’ + AB’
f2 = AC’ + A’B’C
f3 = AC’ + A’B’C + AB’
Including Shared PI: A’BC
1 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
1 0 11
0 0
BC
A 00 01
1
0 1
54
13 2
67
11 10
A
B
C
1
0 1 11
0 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
1
1 0 11
f1 = AC’ + AB’ + A’BC
f2 = AC’ + A’B’C
f3 = AC’ + A’B’C + AB’ + A’BC
Including Remaining PIs
1 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
0
1 0 11
0 0
BC
A 00 01
1
0 1
54
13 2
67
11 10
A
B
C
1
0 1 11
0 1
BC
A 00 01
1
0 1
54
03 2
67
11 10
A
B
C
1
1 0 11
f1 = AC’ + AB’ + A’BC + B’C’
f2 = AC’ + A’B’C + AB + BC’
f3 = AC’ + A’B’C + AB’ + A’BC
What have we learnt?• Multiple-output minimization is not for the faint hearted.• You should be able to find reasonably good solutions
from 5-variable Kmaps.• Good understanding of these principles will help you to
understand how software for SOP minimization works, coming very soon
• For any practical problem, use a suitable CAD package.• The principles illustrated above are used to create
efficient programs for multiple-output minimization.
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