Ch3_Logic_Minimization

Ch3_Logic_Minimizati - ECE 223 Digital Circuits and Systems Logic Minimization 1 Karnaugh Maps Introduction 2-Level Logic implementation using SOP

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1 Logic Minimization ECE 223 Digital Circuits and Systems
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2 Karnaugh Maps - Introduction ± 2-Level Logic implementation using SOP or POS is not the most economical in terms of #gates & #inputs ± A Karnaugh map is a graphical representation of a truth table ² The map contains one cell for each possible minterm ² Adjacent cells differ in only one literal; i.e. x (or x’) ² Function is plotted by placing 1 in cells corresponding to minterms ² Put 0 in rest of the cells
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3 K Map with 2 Variables m0 m1 1 0 m3 m2 0 1 y x m3 1 1 1 0 0 x m2 0 1 0 y m1 m0 F ± F =f(x,y) ± Example, F1 = x’y 1 0 0 1 y x
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4 K Map with 3 Variables ± 3 Variable, F = f(x,y,z); ± Given F2 = (2,3,4,5) ² Represent it on the K map ² minimize the function m0 m1 01 00 m5 m4 0 1 yz x m3 m7 m2 m6 11 10 x’y’z’ ’x’y’z 01 00 xy’z xy’z’ 0 1 yz x x’yz xyz x’yz’ xyz’ 11 10 01 00 0 1 yz x 11 10
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5 K Map with 3 Variables ± 3 Variable, F = f(x,y,z); ± Given F3 = (3,4,6,7) ² Minimize the function using K map Function minimization ² Find maximum size groups that cover all 1s in the map (Comment – a group should not be a subset of other group) 4 cell group Æ 2 literals can be removed 2 cell group Æ 1 literal can be removed ± Guidelines for logic synthesis (SOP) ² Fewer groups Æ fewer AND gates, and fewer inputs to the OR gate ² Fewer literals (larger group) Æ fewer inputs to an AND gate ± Synthesis Objective: Fewest # of gates and # of inputs
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6 K Map with 4 Variables ± 4 Variable, F = f(w,x,y,z) ± Given, F4 = (3,4,5,7,9,13,14,15) ² represent it on the map ² Minimize the logic ± Clues ² Make all possible groups ² Do we need “the group of 4”? F4 = w’xy’ +wxy +w’yz +wy’z m0 m8 01 00 m12 m4 00 01 yz wx m9 m13 m1 m5 11 10 11 10 m3 m2 m7 m6 m15 m14 m11 m10 0 0 01 00 1 0 00 01 yz wx 1 1 0 1 11 10 11 10 10 11 00
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7 Implicants & Prime Implicants, … ± Implicant: A group of one or more k map cell ± Prime implicant: an implicant that is not a subset of another implicant ± Essential Prime Implicant: a prime implicant that covers
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This note was uploaded on 12/07/2010 for the course EE ee012 taught by Professor Razarahim during the Winter '10 term at NUCES.

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Ch3_Logic_Minimizati - ECE 223 Digital Circuits and Systems Logic Minimization 1 Karnaugh Maps Introduction 2-Level Logic implementation using SOP

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