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					Â
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
Â	Â	Â	Â	Â	Â

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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