4.3 Combinational-Circuit Synthesis Most algebraic methods are based on ageneralization of the combiningtheorems, T10 and T10':given product term-y+given productterm:y = given product term(givensumterm+y)·(givensumterm+y = given sum termWe can apply this algebraic methodrepeatedly to combine minterms1,3,5,7 of the prime-number detector.BackNextReturn
We can apply this algebraic method repeatedly to combine minterms 1,3,5,7 of the prime-number detector. ◼ Most algebraic methods are based on a generalization of the combining theorems, T10 and T10’: ➢ given product term·y+ given product term·y = given product term ➢ (given sum term+y) ·(given sum term+y ) = given sum term 4.3 Combinational-Circuit Synthesis Return Back Next
4.3 Combinational-Circuit SynthesisF=Zng,Na,M.n.N,.NN.N(1,2,3,5,7,11,13) =→N, .N, .N,. No+ N, .N.N.N.+N, N2 .N.N.-N,.N2.N,.No+N, N,.Ni.No +N,.N2 Ni-NN,·N2·N+N,:N,·N,.N.+N,:N,N+ N, .N,.N, . N。+N.N,·Ni.N.= N,N+N, .N,.N,.N。+N,-N2N,.N。+N.N,.Ni.NBackNextReturn
4.3 Combinational-Circuit Synthesis Return Back Next 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 , , , 3 2 1 0 (1,2,3,5,7,11,13) N N N N N N N N N N N N N N N N N N N N N N N N F N N N N N N N N + + + + + = = + 3 2 1 0 3 2 1 0 3 2 0 3 2 1 0 3 2 0 N N N N N N N N N N N N N N N N N N + + = + + N3 N0 N3 N2 N1 N0 N3 N2 N1 N0 N3 N2 N1 N0 = + + +
4.3 Combinational-CircuitSynthesis5.Karnaugh MapsAKarnaugh Map is a graphical representationof a logicfunction's truth tableyyzWX10000111XGray0Codex1yxyzX10001101W01ZzNextBackReturn
4.3 Combinational-Circuit Synthesis Return Back Next 5. Karnaugh Maps A Karnaugh Map is a graphical representation of a logic function’s truth table. y x 0 1 0 1 y x x yz x 00 01 11 10 0 1 z y yz wx 00 01 11 10 y x z w Gray Code
4.3 Combinational-Circuit Synthesis6.MinimizingSums of ProductsExamples.Simplify the following logic function:(1) F=Zxy,z(1,2,5,7)(2)F=x·y·z+x·y·z+x·y·z+x·yz+x··zvlyzyzX00011110X00011011001F=y+x·zF=y·z+x·z+xy·zBackNextReturn
4.3 Combinational-Circuit Synthesis Return Back Next 6. Minimizing Sums of Products Examples. yz x 00 01 11 10 0 1 Simplify the following logic function: (1) F=∑x,y,z(1,2,5,7) (2)F = x y z + x y z + x y z + x y z + x y z F = y z + x z + x y z yz x 00 01 11 10 0 1 F = y + x z