EECE 320 Lecture 6, Fall 2011

EECE 320 L6: Combinational Logic design Principles 1 Chehab, AUB, 2003 EECE 320 Digital Systems Design Lecture 6: Combinational Logic Design Principles Ali Chehab EECE 320 L6: Combinational Logic design Principles 2 Chehab, AUB, 2003 Minimizing Sum-of-Product  Adjacent cells differ only in 1 variables  Cells 5 and 7: X.Y’.Z + X.Y.Z = (X.Z).Y’ + (X.Z).Y = X.Z  Cells 1 and 5: X’.Y’.Z + X.Y’.Z = Y’.Z  We can simplify a logic function by combining pairs of adjacent 1-cells (minterms) and by writing a SOP expression covering all 1-cells.  EX: F = X.Z + Y’.Z + X’.Y.Z’ EECE 320 L6: Combinational Logic design Principles 3 Chehab, AUB, 2003 Minimizing Sum-of-Product  In general, 2i 1-cells may be combined to form a product term containing (n-i) literals (n = number of variables)  EX: F = X,Y,Z(0,1,4,5,6)  F = X.Z’ + Y’ EECE 320 L6: Combinational Logic design Principles 4 Chehab, AUB, 2003 Minimizing Sum-of-Product  EX: Prime number detector. F = N3,N2,N1,N0(1,2,3,5,7,11,13) EECE 320 L6: Combinational Logic design Principles 5 Chehab, AUB, 2003 Minimizing Sum-of-Product  A minimal sum of a function F(X1,X2, …, Xn) is a SOP expression of F such that no SOP of F has fewer terms and for the same number of terms it can’t have less number of literals.  Function P(X1,X2, …, Xn) implies F(X1,X2, …, Xn) if for every combination where P = 1, then F = 1. P  F. F includes/covers P  A prime implicant (PI) of F(X1,X2, …, Xn) is a normal product term P(X1,X2, …, Xn) that implies F, such that if any variable is removed from P, the resulting term does not imply F.  Prime Implicant Theorem: A minimal sum is a sum of prime implicants. EECE 320 L6: Combinational Logic design Principles 6 Chehab, AUB, 2003 Minimizing Sum-of-Product  The sum of all prime implicants is the complete sum which is not always minimal  A distinguished 1-cell is covered by only 1 prime implicant  An essential PI covers 1 or more distinguished 1-cells  To obtain a minimal sum, we first include essential PI’s and then we try to cover the remaining 1’s. EECE 320 L6: Combinational Logic design Principles 7 Chehab, AUB, 2003 Minimizing Sum-of-Product  All prime implicants are essential, and hence they are all included in the above example, but it is not always the case. Cheaper EECE 320 L6: Combinational Logic design Principles 8 Chehab, AUB, 2003 Next Lecture and Reminders  Next lecture  Reminders