Karnaugh Maps (K Maps) K Maps with 3 and 4 Variables

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1 Karnaugh Maps (K Maps) Karnugh map is a graphical representation of a truth table The map contains one cell for each possible minterm adjacent cells differ in onl one literal, i.e., or Two variables, F =f(,) m m m2 m3 Function is plotted b placing in cells corresponding to minterms of function Eample, F = 7 of 92 K Maps with 3 and 4 Variables 3 variables, F = f(,,z); 4 variables, F = f(w,,,z) m m m3 m2 z z m4 m5 m7 m6 z z z z w m m m3 m2 m4 m5 m7 m6 w m2 m3 m5 m4 m8 m9 m m z 7 of 92

2 Eamples F = w = (+ )(+ )(z+z ) w F = w F = w z w 72 of 92 K Map oolean Funct. Simplification To write simplified function, find maimum size groups (minimum literals) that cover all s in map 8 cells --> single literal 4 cells --> two literals 2 cells --> three literals cell --> four literals Guidelines for logic snthesis Fewer groups: fewer ND gates and fewer input to the OR gate Fewer literals (larger group): fewer inputs to ND gate Snthesis (design) objectives Smallest number of logic gates Number of inputs to logic gate 73 of 92

3 Eample Consider the following K map Nothing must be a single cell Four groups of two cells each nothing left uncovered The group of 4 (z) term is not needed F = w + w + w + w z w 74 of 92 Product of Sum Epression Recall: Let F be the function F = (all minterms not in F) F = Π (all minterms not in F) (de morgan s theorem) Therefore, one can obtain F b grouping all s on K map, and then taking the complement to obtain product-of-sum form Hence, F = (w + )( + z )( + z) in sum-of-product form Should check both, sum of products, and product of sums One is often simpler than the other F = z + + w z w F = w + + z 75 of 92

4 Plotting Product of Sum Given, F = (w + )( + + z)( + z) F = w + + z w 76 of 92 Don t Care (Incompletel Specified) Conditions Some times, not all values of a function are defined Some input conditions will never occur We don t care what the output is for that input condition In these cases, we can choose the output to be either or, whichever simplifies the circuit Eample: a circuit is to have an output of if a binar coded decimal (CD) digit is a multiple of 3 digit w z F 77 of 92

5 don t care condition -,, -,, -,, -,, -,, F = (3,6,9) + d(,,2,3,4,5) 78 of 92 Don t Care: Plotting Don t cares are plotted as in the K map Sum of products: treat as if it allows a larger group Product of sums: Treat as of it allows a larger group F = wz + + (sum of products, (a)) F2 = z + w + z (recall F = (all minterms not in F) F2 = ( + z )(w + )( + z) w w (a) (b) 79 of 92

6 Observation: In general, F is not equal to F2 due to different values chosen for don t care cells 8 of 92 More Logic Gates NOT gate ND gate uffer gate NND gate OR gate OR gate NOR gate NOR gate 8 of 92

7 NND and NOR Implementation set of logic gates are functionall complete if an boolean function can be implemented b just these gates ND, OR, NOT ND, NOT ( ) = + ==> OR gate OR, NOT NND NOR NND and NOR gates are easier to implement (smaller area, less power consumption, faster) than ND and OR gates 82 of 92 Logic Implementation with NND/NOR F = () F = + = () Given F = z + w all implementations represent the same function Function can be implemented with NND gates onl Procedure from K map = ND-Invert Invert-OR gate F = z + w z w present the simplified function in sum of product form (ND-OR) use De morgan s theorem to represent the function in NND-NND form F = z + w z w Similar steps for NOR implementation starting from product of sums form F = z + w z w 83 of 92

8 Other Two-Level Implementations Wired Logic, Transistor-Transistor Logic (TTL) Wired logic: if outputs of two logic gates are shorted together TTL stle implementation allows wired connection + 5 V + 5 V + 5 V R R R Out Input Out Inputs ND logic Out NOT gate NND gate wired ND gate Other two level implementations are ND-OR-INVERT and OR-ND-INVERT 84 of 92 Simplest Two-Level Epression Some definitions Implicant: a grouping of one or more K map cells Prime implicant: an implicant that is not a subset of another implicant Essential prime implicant: a prime implicant that covers at least one minterm not covered b another prime implicant Eample, f(w,,,z) = (,,2,5,6,7,9,4) + d(3) w w essential prime implicants prime implicants 85 of 92

9 Essential prime implicants: z, Prime implicants: w, w z, w z, w, w z (,5,9,3) (6,4) w (,) w z (,2) w z (5,7) w (6,7) w (2,6) minterms covered * * C D E 86 of 92 ll minterms must be covered Essential prime imlicants must be included (*) Different combinations of prime implicants are: + C; or + D; or + C + E; or + D + E + C or + D are the simplest, hence the simplest function implementation is F = z + + w z + w z or z + + w z + w 87 of 92

10 Tabulation (Quine-McCluske) Method The map method of simplification is convenient if number of variables does not eceed beond 4 or 5 Tabulation method is preferred for a function with large number of variables for F = f(w,,,z) consider two adjacent minterms let a = m4 + m5 = w z + w z = w or = + = - similarl, let b = m2 + m3 = w z + w z = w or = + = - similarl, c = m4 + m5 + m2 + m3 = a + b = w + w = = = of 92 djacent minterms differ b a single bit in their binar representation Tabulation method consists of grouping minterms and sstematicall checking for single bit differences Eample, f(w,,,z) = (,3,4,6,7,8,,,5) + d(5,9) Group minterms according to number of s in binar representation Each element of each section is compared with each element of the section below it; all reductions are recorded in net column Mark terms that combine ll unmarked terms are prime implicants 89 of 92

11 w z of 92,4 (4) 4,5,6,7 (,2) ----,8 (8) 8,9,, (,2) ,5 () 3,7,,5 (4,8) ,6 (2) 3 8,9 () 5 8, (2) ,7 (4) 3, (8) ,7 (2) 7 6,7 () 9, (2) ---, () ,5 (8),5 (4) 9 of 92

12 Prime implicants 3 minterms covered ,4,8 4,5,6,7 8,9,, 3,7,,5 F(w,,,z) =,4 + 4,5,6,7 + 8,9,, + 3,7,, w z + w + w + or F(w,,,z) =,8 + 4,5,6,7 + 8,9,, + 3,7,, z + w + w + 92 of 92