CS2100 Computer Organisation

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Presentation transcript:

CS2100 Computer Organisation http://www.comp.nus.edu.sg/~cs2100/ Quine-McCluskey (AY2015/6 Semester 1)

Quine-McCluskey A tabulation method similar in concept to K-map Applicable for functions with any number of variables K-map is useful for functions with at most 5 or 6 variables Tedious on paper, but can be automated (programmed) Non-examinable But knowing it may enhance your understanding of K-maps CS2100 Quine-McCluskey

PIs AND EPIs To find the simplest (minimal) SOP expression from a K-map, you need to obtain: Minimum number of literals per product term; and Minimum number of product terms. Achieved through K-map using Bigger groupings of minterms (prime implicants) where possible; and No redundant groupings (look for essential prime implicants) CS2100 Quine-McCluskey

EXAMPLE: F(A,B,C,D) = Sm(2,3,4,5,7,8,10,13,15) Step 1: List out all minterms in groups with same number of 1s in their binary codes. 1 C A 00 01 11 10 00 01 11 10 B CD AB D 1st column 2: 0010 4: 0100 8: 1000 ------------ 3: 0011 5: 0101 10: 1010 7: 0111 13: 1101 15: 1111 Codes with one 1 Codes with two 1s Codes with three 1s Codes with four 1s CS2100 Quine-McCluskey

EXAMPLE: F(A,B,C,D) = Sm(2,3,4,5,7,8,10,13,15) Step 2: Combine codes that differ by 1 bit into bigger group, write the combined code in next column. 1 C A 00 01 11 10 00 01 11 10 B CD AB D 1st column 2: 0010 4: 0100 8: 1000 ------------ 3: 0011 5: 0101 10: 1010 7: 0111 13: 1101 15: 1111 2nd column   2,3: 001- Codes with one 1  2,10: -010  4,5: 010- 8,10: 10-0 ------------------   3,7: 0-11    Codes with two 1s 5,7: 01-1   5,13: -101 ------------------    7,15: -111   Codes with three 1s 13,15: 11-1   CS2100 Quine-McCluskey

EXAMPLE: F(A,B,C,D) = Sm(2,3,4,5,7,8,10,13,15) Step 3: Repeat step 2 – Combine codes that differ by 1 bit into bigger group, write the combined code in next column. 1 C A 00 01 11 10 00 01 11 10 B CD AB D 1st column 2: 0010 4: 0100 8: 1000 ------------ 3: 0011 5: 0101 10: 1010 7: 0111 13: 1101 15: 1111 2nd column 2,3: 001- 2,10: -010 4,5: 010- 8,10: 10-0 ---------------- 3,7: 0-11 5,7: 01-1 5,13: -101 7,15: -111 13,15: 11-1 3rd column  5,7,13,15: -1-1  5,7,13,15: -1-1       We have completed Phase 1: Identifying all the Prime Implicants (PIs)!      CS2100 Quine-McCluskey

EXAMPLE: F(A,B,C,D) = Sm(2,3,4,5,7,8,10,13,15) Phase 2: Identify the Essential Prime Implicants (EPIs) Draw the PI chart 1 C A 00 01 11 10 00 01 11 10 B CD AB D 2 3 4 5 7 8 10 13 15 2,3: 001- (A'.B'.C)  2,10: -101 (B.C'.D) 4,5: 010- (A'.B.C') 8,10: 10-0 (A.B'.D') 3,7: 0-11 (A'.C.D) 5,7,13,15: -1-1 (B.D) EPI EPI EPI Where are the EPIs? Look for columns containing a single tick. EPIs are: A'.B.C', A.B'.D', and B.D But we are not done yet. There are still minterms not covered by the EPIs! CS2100 Quine-McCluskey

EXAMPLE: F(A,B,C,D) = Sm(2,3,4,5,7,8,10,13,15) Phase 2: After identifying the EPIs Draw the reduced PI chart if there are minterms not covered 1 C A 00 01 11 10 00 01 11 10 B CD AB D 2 3 4 5 7 8 10 13 15 2,3: 001- (A'.B'.C)  2,10: -101 (B.C'.D) 4,5: 010- (A'.B.C') 8,10: 10-0 (A.B'.D') 3,7: 0-11 (A'.C.D) 5,7,13,15: -1-1 (B.D) EPI EPI EPI Answer: B.D + A'.B.C' + A.B'.D' + A'.B'.C 1. Find out what are the minterms covered by the EPIs. 2. Remove the EPIs and minterms they cover from the chart  reduced PI chart. 3. Find the minimum number of remaining PIs to cover the remaining minterms. CS2100 Quine-McCluskey

END CS2100 Quine-McCluskey