ECE 124 – Assignment #2: Algebric simplification and applications 05/15/18 Presented by: Mahmoud KhalafAlla Supervised by: Prof. Cathy Gebotys ECE 124 – Assignment #2
Outline § Brief Introduction to simplification theorems § Assignment #2 solutions § Questions PAGE ECE 124 – Assignment #2 2
Brief Introduction to Simplification theoreoms § Consensus theorem XY + X’Z + YZ = XY + X’Z XY + X’Z + YZ(X + X’) = XY + X’Z + XYZ + X’YZ = XY(Z + 1) + X’Z (Y + 1) § Dual form: (X+Y)(X’ + Z)(Y + Z) = (X+Y)(X’ + Z) § Minterm expression: Sum of products where F = 1 § Maxterm expression Product of sum where F = 0 PAGE ECE 124 – Assignment #2 3
Problem #1 § Find the consensus term in each expression and delete it: § abc’d + a’be + bc'de Let X = a , Y = bc‘d, Z = be , then can be rewritten as : XY + X‘Z + YZ bc‘de to be deleted à abc‘d + a‘be § (x'+y+z)(x+w)(y+z+w) Dual form of consensus form is (X+Y)(X'+Z)(Y+Z) à (y+z+w) to be deleted à (x’ + y + z)(x + w) PAGE ECE 124 – Assignment #2 4
Problem #2 § Simplify each expression by only using consensus theorem: § bc'd'+abc'+ac'd+ab'd+a'bd’ bc’d’ can be deleted and keep abc’ + a’bd’ à abc’ + ac’d + ab’d + a’bd’ ac’d can be deleted and keep abc’ + ab’d à abc’ + ab’d + a’bd’ § (b'+c'+d')(b+c+d)(a+b+c)(a'+c+d) (b + c + d) can be deleted and keep (a+b+c)(a’+c+d) à (b'+c'+d') (a+b+c)(a’+c+d) § a'b’c + abd + a’cde + bcde + a'bde bcde can be deleted and keep abd + a’cde à a’b’c + abd + a’cde + a’bde à a’cde can be deleted and keep a’b’c + a’bde à a’b’c + abd + a’bde PAGE ECE 124 – Assignment #2 5
Problem #3 § Simplify each expression to a sum of three terms : § a'bcd+a'bc'd+b'ef+cde'g+a'def+a'b'ef a'bd[c+c'] + ef[b'(1+a')] +efa'd + cde'g =a'[b]d + ef[b'] + [efa'd] (consensus of 1st two terms) + cde'g = a'bd + efb'+cde'g ; a'bd+b'ef+cde'g § w'x'y'+w'xz'+((x+y+w'z)(x'+z'+wy'))’ w’x’y’ + w’xz’ + (x+y+w’z)’ + (x’ + z’ + wy’)’ à w’x’y’ + w’xz’ + x’y’(w’z)’ + xz(wy’)’ à w’x’y’ + w’xz’ + x’y’(w + z’) + xz(w’ + y) à x’y’(w’ + w + z’) + w’xz’ + w’xz + xyz à x’y’ + w’x(z+z’) + xyz à x'y’ + w’x + xyz PAGE ECE 124 – Assignment #2 6
Problem #4 § Simplify F = a’b ⊕ ⊕ bc bc ⊕ ab ⊕ ⊕ ab ⊕ b’c b’c’ b(a' ⊕ a) ⊕ (bc ⊕ b'c’) à (b ⊕ bc) ⊕ b’c’ à b(1 ⊕ c) ⊕ b'c’ à bc' ⊕ b'c’ à c'(b ⊕ b') = c' PAGE ECE 124 – Assignment #2 7
Problem #5 § Factor Z = abc+de+acf+ad'+ab'e' and simplify it to form (x+x)(x+x)(x+x+x+x) where each x represents a literal. Then express Z as a minimum sum of products in the form xx + xx+ xx+ xx a(bc+cf+b'e'+d')+de à (a+d)(a+e)(bc + cf +b'e'+d' +d)(bc +cf +b'e'+d' +e) à (a+d)(a+e)(bc + cf + b'e' + d' + e) à (e + e’b’ = e + b’) (a+d)(a+e)(bc+cf+b'+d’+e) à (bc + b’ = c + b’) (a+d)(a+e)(c+cf +b' +d’+e) à (a+d)(a+e)(c+d'+b'+e) (a + de)(c + d’ + b’ + e) à ac + ad’ + ab’ + ae + de + dec + deb’ à (ae consensus term of ad’ + de) ac+ad'+ab’+de(1 + b’) à ac+ad'+ab'+de PAGE ECE 124 – Assignment #2 8
Problem #6 § statement always true? if a + b = c then ad' + bd' = cd’ YES PAGE ECE 124 – Assignment #2 9
Problem #7 § A network has three 1-bit inputs and two 1-bit outputs. The output signals b0, b1 represent a binary number which is equal to the number of input signals which are zero x y z b1 b0 Minterm b0 = m(0,3,5,6) 0 0 0 1 1 Minterm b1 = m(0,1,2,4) 0 0 1 1 0 Maxterm b0 = M(1,2,4,7) Maxterm b1 = M(3,5,6,7) 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 0 PAGE ECE 124 – Assignment #2 10
Problem #9 § Find the simplest expression for F(a,b,c) and specify the values for the don't cares (x) that lead to this expression. a b c F 0 0 0 0 0 0 1 1 0 1 0 x 0 1 1 1 1 0 0 0 1 0 1 x 1 1 0 1 1 1 1 0 PAGE ECE 124 – Assignment #2 11
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