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Simplification Of Boolean Functions Part-IV

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QUINE-McCLUSKEY MINIMIZATION
Quine-McCluskey minimization method uses the same theorem to produce the solution as the K-map method, namely X(Y+Y')=X

Minimization Technique

  • The expression is represented in the canonical SOP form if not already in that form.
  • The function is converted into numeric notation.
  • The numbers are converted into binary form.
  • The minterms are arranged in a column divided into groups.
  • Begin with the minimization procedure.
    • Each minterm of one group is compared with each minterm in the group immediately below.
    • Each time a number is found in one group which is the same as a number in the group below except for one digit, the numbers pair is ticked and a new composite is created.
    • This composite number has the same number of digits as the numbers in the pair except the digit different which is replaced by an "x".
  • The above procedure is repeated on the second column to generate a third column.
  • The next step is to identify the essential prime implicants, which can be done using a prime implicant chart.
    • Where a prime implicant covers a minterm, the intersection of the corresponding row and column is marked with a cross.
    • Those columns with only one cross identify the essential prime implicants. -> These prime implicants must be in the final answer.
    • The single crosses on a column are circled and all the crosses on the same row are also circled, indicating that these crosses are covered by the prime implicants selected.
    • Once one cross on a column is circled, all the crosses on that column can be circled since the minterm is now covered.
    • If any non-essential prime implicant has all its crosses circled, the prime implicant is redundant and need not be considered further.
  • Next, a selection must be made from the remaining nonessential prime implicants, by considering how the non-circled crosses can be covered best.
    • One generally would take those prime implicants which cover the greatest number of crosses on their row.
    • If all the crosses in one row also occur on another row which includes further crosses, then the latter is said to dominate the former and can be selected.
    • The dominated prime implicant can then be deleted.

Example
Find the minimal sum of products for the Boolean expression, f= (1,2,3,7,8,9,10,11,14,15), using Quine-McCluskey method.

Firstly these minterms are represented in the binary form as shown in the table below. The above binary representations are grouped into a number of sections in terms of the number of 1's as shown in the table below.

 

Binary representation of minterms

 

Minterms

U

V

W

X

1

0

0

0

1

2

0

0

1

0

3

0

0

1

1

7

0

1

1

1

8

1

0

0

0

9

1

0

0

1

10

1

0

1

0

11

1

0

1

1

14

1

1

1

0

15

1

1

1

1

. . . .   .

Group of minterms for different number of 1's

No of 1's

Minterms

U

V

W

X

1

1

0

0

0

1

1

2

0

0

1

0

1

8

1

0

0

0

2

3

0

0

1

1

2

9

1

0

0

1

2

10

1

0

1

0

3

7

0

1

1

1

3

11

1

0

1

1

3

14

1

1

1

0

4

15

1

1

1

1

Any two numbers in these groups which differ from each other by only one variable can be chosen and combined, to get 2-cell combination, as shown in the table below.

.

..... . . . .

2-Cell combinations

 

Combinations

U

V

W

X

(1,3)

0

0

-

1

(1,9)

-

0

0

1

(2,3)

0

0

1

-

(2,10)

-

0

1

0

(8,9)

1

0

0

-

(8,10)

1

0

-

0

(3,7)

0

-

1

1

(3,11)

-

0

1

1

(9,11)

1

0

-

1

(10,11)

1

0

1

-

(10,14)

1

-

1

0

(7,15)

-

1

1

1

(11,15)

1

-

1

1

(14,15)

1

1

1

-

From the 2-cell combinations, one variable and dash in the same position can be combined to form 4-cell combinations as shown in the figure below.

4-Cell combinations

 

Combinations

U

V

W

X

(1,3,9,11)

-

0

-

1

(2,3,10,11)

-

0

1

-

(8,9,10,11)

1

0

-

-

(3,7,11,15)

-

-

1

1

(10,11,14,15)

1

-

1

-

The cells (1,3) and (9,11) form the same 4-cell combination as the cells (1,9) and (3,11). The order in which the cells are placed in a combination does not have any effect. Thus the (1,3,9,11) combination could be written as (1,9,3,11).

From above 4-cell combination table, the prime implicants table can be plotted as shown in table below.

Prime Implicants Table

Prime Implicants

1

2

3

7

8

9

10

11

14

15

(1,3,9,11)

X

-

X

-

-

X

-

X

-

-

(2,3,10,11)

-

X

X

-

-

-

X

X

-

-

(8,9,10,11)

-

-

-

-

X

X

X

X

-

-

(3,7,11,15)

-

-

-

-

-

-

X

X

X

X

-

X

X

-

X

X

-

-

-

X

-

The columns having only one cross mark correspond to essential prime implicants. A yellow cross is used against every essential prime implicant. The prime implicants sum gives the function in its minimal SOP form.
Y = V'X + V'W + UV' + WX + UW

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