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ece15_14_12_6

ece15_14_12_6 - Today Test#2 Material review QM and...

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1 ECE 15A Fundamentals of Logic Design Lecture 14 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Today: Test #2 Material review QM and Petrick’s methods for functions with don’t cares. PI chart reduction rules Mux and decoder; expansion MUX as circuit element Modular design Propagation delays/circuit hazards Generating test patterns 2-level circuits with NOR and NAND gates Quine-McCluskey method for functions with don’t cares. Find all prime implicants for the function with all don’t cares temporarily set to 1. Build the implicant chart (table) for the minterms in care set. Find essential PIs and reduce the table. Apply Petrick’s algorithm on the reduced table. The selected solution determines don’t care assignment. Some don’t cares may be set to 0. 3 Implicant chart simplification The fewer rows and columns in the chart, the simpler the Petrick’s expression is. Simplification #1 – remove columns and rows related to essential PIs – we know this! Can we simplify more? Yes. But 2 cases must be considered: Case 1: Only 1 minimal solution is sought Case 2: All minimal solutions are to be determined What are minimal solutions? 4 Implicant’s Cost Solution in a SOP form 5 a b d c a’ f e F=e + ab + a’cd Implementation with 3 gates and 8 gate inputs. We will seek the solution with the fewest # of gates; if tie – minimum sum of inputs. Cost(PI)=#literals +1 if #literals > 1, otherwise Cost(PI) = 1 P1 P2 P3 Cost(P1) = 4 Cost(P2) = 3 Cost(P3) = 1 Column dominance A column i in a prime implicant table dominates a column j if i has x’s in all the rows j does, and I has at least 1 x in a row that j does not. Two columns of a prime implicant table are equal if they have x’s in exactly the same rows. 6 m a m b m c m d m e m f P1 x x P2 x x x P3 x x x P4 x x x x x m c dominates m b m d and m e are equal m c can be removed and one of m d or m e
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2 Row dominance A row i in a prime implicant table dominates a row j if i has x’s in all the columns j does, and i has at least 1 x in a column that j does not. Two rows of a prime implicant table are equal if they have x’s in exactly the same columns. 7 m a m b m c m d m e m f P1 x x P2 x x x P3 x x x P4 x x x x x P4 dominates P3 P3 can be eliminated If Cost(P4)<Cost(P3) Example F(v,w,x,y,z) = m(1,9,10,11) + d(0,3,14,25,27) Pis: A=v’x’z; B=wx’z; C=v’w’x’y’; D=v’wx’y; E=v’wyz’ 8 m1 m9 m10 m11 cost A x x x 4 B x
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