C.S.Choy39 TERMINOLOGY Minterm –product term containing all input variables of a function in either true or complementary form Maxterm – sum term containing all input variables of a function in either true or complementary form Canonical Form – a function expressed in either fully minterms or fully maxterms Literal – each occurrence of a variable of a function in either true or complementary form

C.S.Choy40 DESIGN MINIMIZATION Reduce Hardware Reduce Number of Inputs In Boolean Expression –Minimum number of terms –Minimum number of literals

C.S.Choy41 DESIGN MINIMIZATION USING BOOLEAN ALGEBRA Example F = (A+B+C)(A+B+C)(A+B+C)(A+B+C) (A+B+C)(A+B+C) no. of terms = 6 no. of literals = 18 F = ABC + ABC

C.S.Choy42 DESIGN MINIMIZATION USING KARNAUGH MAP K-Map A graphical representation of a truth table organized in such a way to facilitate minimization based on the following Boolean property AB + AB = A Each square corresponds to a minterm A 1 is place in the K-map if the corresponding minterm produce a 1 in the truth table. Otherwise, a 0 is placed

C.S.Choy43 3-VARIABLE (INPUT) K-MAP –Adjacent squares differ only in the value of one variable (remember gray code) –Top and bottom, and extreme left and right are joined pairs, quads, or octets –Minimization is performed by looping adjacent 1s in pairs, quads, or octets. Any complementary variables (true and complement) in the loop are redundant –Since variables A and B have complementary occurrence in the loop, A and B are redundant in the expression F = C101 F=ABC+ABC+ABC+ABC

C.S.Choy44 4-INPUT K-MAP There are alternatives in looping that may give rise to equivalent minimal solution

C.S.Choy45 K-MAP MINIMIZATION GUIDELINE 1.Loop all isolated 1s 2.Consider each remaining 1 separately. If it can be looped in more than one way, try another 1 instead. Otherwise, include it in the largest possible loop 3.A minimal solution is derived as soon as all 1s are covered. In the process of making the largest loop, it is permissible to use previously covered 1s

C.S.Choy46 HANDLING OF DON’T CARE CONDITIONS IN K-MAP There are conditions in digital systems that are immaterial to a certain task. In these conditions, output is irrelevant F = ABC + ABC + ABC + ABC Normal expression does not indicate any don’t care conditions. F is simplified to F = AC + BC + ABC With K-map, 1 can be assigned to any don’t care to –Form largest possible loop –Combine isolated 1 to form a loop F = C + AB X – don’t care input condition

C.S.Choy47 5-INPUT K-MAP Alternative

C.S.Choy48 DESIGN EXAMPLE Design a 4-input circuit that will function as a majority detector. The circuit should output high when a majority of the inputs are high