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Unformatted text preview: 1 ECE 15A Fundamentals of Logic Design Lecture 8 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Last time: Karnaugh Map Method to find minimum SOP Step 1: Choose an element of ON-set not already covered by an implicant Step 2: Find "maximal" groupings of 1's and X's adjacent to that element. Remember to consider top/bottom row, left/right column, and corner adjacencies. This forms prime implicants (always a power of 2 number of elements). Repeat Steps 1 and 2 to find all prime implicants Step 3: Revisit the 1's elements in the K-map. If covered by single prime implicant, it is essential , and participates in final cover. The 1's it covers do not need to be revisited Step 4: If there remain 1's not covered by essential prime implicants, then select the smallest number of prime implicants that cover the remaining 1's 3 Discussion Karnaugh maps are: Very effective for functions up to 6 variables Not useful for functions dependent on >6 variables K-maps depend on humans visual ability to identify prime implicants K-maps are not helpful in developing CAD tools 4 2-level simplification approaches Algebraic simplification Not a systematic procedure Difficult to tell when the minimum expression has been found Hand methods K-maps Quine-McCluskey method (aka Tabular Method) Computer-aided tools High quality solutions for large functions ( more than 10 variables) Heuristic methods applied Espresso Exact solutions for unate function Shannon expansion 5 Quine-McCluskey method Tabular method to systematically find all prime implicants Can be programmed 6 Outline of the Quine-McCluskey Method 1. Produce a minterm expansion (standard sum-of-products form) for a function F 2. Eliminate as many literals as possible by systematically applying XY + XY = X . 3. Use a prime implicant chart to select a minimum set of prime implicants that when ORed together produce F, and that contains a minimum number of literals. 2 7 Determining Prime Implicants ABCD + ABCD = ABC 1 0 1 0 + 1 0 1 1 = 1 0 1 - (The dash indicates a missing variable) ABCD + ABCD 0 1 0 1 + 0 1 1 0 We can combine the minterms above because they differ by a single bit. The minterms below wont combine 8 Quine-McCluskey Method Example 1. Find all the prime implicants + = ) 15 , 7 , ( ) 13 , 10 , 9 , 8 , 6 , 5 , 4 ( ) , , , ( d m d c b a f group 0 0 0000 4 0100 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 13 1101 Group the minterms according to the number of 1s in the minterm....
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- Winter '08