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EE101Lecture8

EE101Lecture8 - Introduction to Digital Logic Lecture 8...

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© Mark Redekopp, All rights reserved Introduction to Digital Logic Lecture 8: Karnaugh Maps
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© Mark Redekopp, All rights reserved Gray Code Different than normal binary ordering Reflective code When you add the (n+1) th bit, reflect all the previous n-bit combinations Consecutive code words differ by only 1-bit 0 0 0 1 1 1 1 0 when you move to the next bit, reflect the previous combinations 2-bit Gray code differ by only 1-bit
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© Mark Redekopp, All rights reserved Gray Code Different than normal binary ordering Reflective code When you add the (n+1) th bit, reflect all the previous n-bit combinations Consecutive code words differ by only 1-bit 0 0 0 1 1 1 1 0 when you move to the next bit, reflect the previous combinations 2-bit Gray code 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 0 3-bit Gray code differ by only 1-bit differ by only 1-bit differ by only 1-bit
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© Mark Redekopp, All rights reserved Karnaugh Maps 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 WX YZ 00 01 11 10 00 01 11 10 0 1 3 2 4 5 7 6 12 13 14 15 8 9 11 10 0 0 1 1 1 0 0 1 XY Z 00 01 11 10 0 1 0 1 2 3 6 7 4 5 3 Variable Karnaugh Map 4 Variable Karnaugh Map Every square represents 1 input combination Must label axes in Gray code order Fill in squares with given function values F= Σ XYZ (1,4,5,6) G= Σ WXYZ (1,2,3,5,6,7,9,10,11,14,15)
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© Mark Redekopp, All rights reserved Karnaugh Maps W X Y Z F 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 WX YZ 00 01 11 10 00 01 11 10 0 1 3 2 4 5 7 6 12 13 14 15 8 9 11 10
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© Mark Redekopp, All rights reserved Karnaugh Maps 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 WX YZ 00 01 11 10 00 01 11 10 0 1 3 2 4 5 7 6 12 13 14 15 8 9 11 10 We can derive minterms from squares with 1 in them We can derive maxterms from squares with 0 in them Maxterm: w’ + x + y + z Maxterm: w’ + x’ + y + z Minterm: w•x’•y•z Minterm: w•x’•y•z’
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© Mark Redekopp, All rights reserved Karnaugh Maps 1 1 1 1 0 0 0 1 XY Z 00 01 11 10 0 1 0 1 2 3 6 7 4 5 3 Variable Karnaugh Map Groups of adjacent 1‟s will always simplify to smaller product terms (compared to the minterm representation). F= Σ XYZ (0,2,4,5,6) = m0 + m2 + m6 + m4 = x’y’z’ + x’yz’ + xyz’ + xy’z’ = z’(x’y’ + x’y + xy + xy’) = z’(x’(y’+y) + x(y+y’)) = z’(x’+x) = z’ = m4 + m5 = xy’z’ + xy’z = xy’(z’+z) = xy’
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© Mark Redekopp, All rights reserved Karnaugh Maps WX YZ 00 01 11 10 00 01 11 10 0 1 3 2 4 5 7 6 12 13 14 15 8 9 11 10 XY Z 00 01 11 10 0 1 0 1 2 3 6 7 4 5 3 Variable Karnaugh Map 4 Variable Karnaugh Map Adjacent squares differ by 1-variable This will allow us to use T10 = AB + AB‟= A or T10‟ = (A+B‟)(A+B) = A Difference in X: 010 & 110 Difference in Z: 010 & 011 Difference in Y: 010 & 000 1 = 0 0 01 4 = 010 0 5 = 0101 7 = 01 1 1 13 = 1 101 Adjacent squares differ by 1-bit 0 = 0 0 0 2 = 010 3 = 01 1 6 = 1 10 Adjacent squares differ by 1-bit x’yz’ + xyz’ = yz’ x’yz’ + x’yz = x’y x’yz’ + x’y’z’ = x’z’
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© Mark Redekopp, All rights reserved Karnaugh Maps 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 WX YZ 00 01 11 10 00 01 11 10 0 1 3 2 4 5 7 6 12 13 14 15 8 9 11 10 2 adjacent 1‟s (or 0‟s) differ by only one variable 4 adjacent 1‟s (or 0‟s) differ by two variables
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