SYEN 3330 Digital Systems Chapter 2 – Part 6 SYEN 3330 Digital Systems

Table Methods for PI Generation SYEN 3330 Digital Systems

An Example: F(x,y,z)= m(2,3,6,7) SYEN 3330 Digital Systems

Results - Step 2 SYEN 3330 Digital Systems

Step 3 SYEN 3330 Digital Systems

The Results of Step 3 SYEN 3330 Digital Systems

Computational Complexity Issues SYEN 3330 Digital Systems

Q-M on F(x,y,z)= m(2,3,6,7) SYEN 3330 Digital Systems

The Result of Step 3 SYEN 3330 Digital Systems

Result of Step 4 SYEN 3330 Digital Systems

Review of Boolean Logic SYEN 3330 Digital Systems

Canonical Forms SYEN 3330 Digital Systems

Minimum Literal SOP Form SYEN 3330 Digital Systems

Tabular Method to Find a Cover SYEN 3330 Digital Systems

Table Method Example SYEN 3330 Digital Systems

Select Essential Prime Implicants SYEN 3330 Digital Systems

Select Essential Prime Implicants SYEN 3330 Digital Systems

Less Than Prime Implicants SYEN 3330 Digital Systems

Secondary Essential PIs SYEN 3330 Digital Systems

Cyclic Structures SYEN 3330 Digital Systems

Cyclic Structure: Pick One SYEN 3330 Digital Systems

Less Thans SYEN 3330 Digital Systems

Secondary Essential PIs SYEN 3330 Digital Systems

Now Go Back and Try Again SYEN 3330 Digital Systems

Finish Up SYEN 3330 Digital Systems

Quine-McCluskey (tabular) method 1. Arrange all minterms in group such that all terms in the same group have the same # of 1’s in their binary representation. 2. Compare every term of the lowest-index group with each term in the successive group. Whenever possible, combine two terms being compared by means of gxi+gxi´=g(xi+xi´)=g. Two terms from adjacent groups are combinable if their binary representation differ by just a single digit in the same position. 3. The process continues until no further combinations are possible. The remaining unchecked terms constitute the set of PI. SYEN 3330 Digital Systems

Using prime implicant chart, we can find essential PI Ex) f(x1,x2,x3,x4) = (0,1,2,5,6,7,8,9,10,13,15) 1 1 1 1 15 (5,7) (5,13) (6,7) (9,13) 0 1 - 1 - 1 0 1 0 1 1 - 1 - 0 1 0 1 1 1 1 1 0 1 7 13 (1,5) (1,9) (2,6) (2,10) (8,9) (8,10) 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 5 6 9 10 0 - 0 1 - 0 0 1 0 - 1 0 - 0 1 0 1 0 0 - 1 0 - 0 - 1 1 1 1 1 - 1 (13,15) (7,15)  1 2 8 (0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15) - 0 0 - - 0 - 0 - - 0 1 - 1 - 1 (0,1) (0,2) (0,8) 0 0 0 - 0 0 - 0 - 0 0 0 0 0 0 0 x1,x2,x3,x4 # 0 0 0 1 1 0 0 0 0 0 1 0 Using prime implicant chart, we can find essential PI 0 1 2 5 6 7 8 9 10 13 15 (2,6) (6,7) (0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15)                     SYEN 3330 Digital Systems

If we choose p1 first, then p3, p5 are next. The essential PI’s are (0,2,8,10) and (5,7,13,15) . So, f(x1,x2,x3,x4) = (0,2,7,8) + (5,7,13,15) + PI’s Here are 4 different choices (2,6) + (0,1,8,9), (2,6) + (1,5,9,13) (6,7) + (0,1,8,9), or (6,7) + (1,5,9,13) The reduced PI chart A PI pj dominates PI pk iff every minterm covered by pk is also covered by pj. pj pk m1 m2 m3 m4  (can remove) Branching method p1 p2 p3 p4 p5 m1 m2 m3 m4 m5 If we choose p1 first, then p3, p5 are next. p1 p4 p3 p5 p2 Quine – McCluskey method (no limitation of the # of variables) (2,6) (6,7) (0,1,8,9) (1,5,9,13) 1 6 9 SYEN 3330 Digital Systems

Quine-McCluskey example F(A,B,C,D) =  (3,9,11,12,13,14,15) + d (1,4,6) SYEN 3330 Digital Systems

Ex) f(A,B,C,D) = (3,9,11,12,13,14,15) + d (1,4,6) PI chart: 3 9 11 12 13 14 15 (1,3, 9, 11) (4, 6,12,14) (9,13,11,15) (12,13,14,15)              Reduced PI chart: 12 13 14 15 (4, 6,12,14) (9,13,11,15) (12,13,14,15)         Result: (1,3,9,11) + (12,13,14,15) SYEN 3330 Digital Systems

Minimum SOP to Minimum POS SYEN 3330 Digital Systems

Minimum POS Example Given g(w,x,y,z): Form the Complement (Circle Zeros): SYEN 3330 Digital Systems

Table Method Minimum SOP SYEN 3330 Digital Systems