**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

- 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.

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.

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