Lecture 8 More Karnaugh Maps and Don’t Cares. The main contents of the chapter consist of the following: Karnaugh maps with four inputs, same basic rules as three input k-maps, understanding prime implicants, related to minterms, covering all implicants, using don’t cares to simplify functions, don’t care outputs are undefined, summarizing Karnaugh maps.
Lecture More Karnaugh Maps and Don’t Cares Overvie w ° Karnaugh maps with four inputs • Same basic rules as three input K-maps ° Understanding prime implicants • Related to minterms ° Covering all implicants ° Using Don’t Cares to simplify functions • Don’t care outputs are undefined ° Summarizing Karnaugh maps Karnaugh Maps for Four Input Functions ° Represent functions of inputs with 16 minterms ° Use same rules developed for 3-input functions ° Note bracketed sections shown in example Kar nau gh °ma F(A,B,C,D) = m(0,2,3,5,6,7,8,10,11,14,15) p: F= 4vari C + A’BD + B’D’ able A exa 0111 1 mpl e 0 D C 1 1 1 C 1 0000 D A B 1111 1000 B Solution set can be considered as a coordinate System! Des ign exa mpl es C A A A 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 D C D C B B B Kmap for LT Kmap for EQ Kmap for GT LT = A' B' D + A' C + B' C D EQ = A'B'C'D' + A'BC'D + ABCD + AB'CD’ GT = B C' D' + A C' + A B D' Can you draw the truth table for these examples? D Can you draw the truth table for these examples? A C 0 0 0 1 1 0 B D Physical Implementation ° Step 1: Truth table A B C D ° Step 2: K-map ° Step 3: Minimized sum-ofproducts EQ ° Step 4: Physical implementation with gates A C 0 0 0 0 0 0 B Kmap for EQ D Kar nau ° Fourgh variable maps Map CD s 00 01 11 10 AB 00 0 F=A BC +A CD +ABC 01 1 +AB C D +ABC +AB C F=BC +CD + AC+ AD 11 1 1 10 1 ° Need to make sure all 1’s are covered ° Try to minimize total product terms ° Design could be implemented using NANDs and NORs Don’t cares ° In digital systems it often happens that certain input conditions can never occur For example, suppose that x1 and x2 control two interlocked switches such that both switches cannot be closed at the same time Thus the only three possible states of the switches are that both switches are open or that one switch is open and the other switch is closed Namely, the input valuations (x1, x2) = 00, 01, and 10 are possible, but 11 is guaranteed not to occur Then we say that (x1, x2) = 11 is a don’t-care condition , meaning that a circuit with x1 and x2 as inputs can be designed by ignoring this condition ° A function that has don’t-care condition(s) is said to be incompletely specified Karnaugh maps: Don’t cares ° In some cases, outputs are undefined ° We “don’t care” if the logic produces a or a ° This knowledge can be used to simplify functions A CD C AB 00 01 11 10 00 0 X 01 1 X 11 1 0 10 X 0 B D - Treat X’s like either 1’s or 0’s - Very useful - OK to leave some X’s uncovered Defi niti ° Implicant on •of Single product term of the ON-set (terms that create a logic 1) ter ° Prime implicant ms • Implicant that can't be combined with another to form an implicant with for fewer literals two ° Essential prime implicant •leve Prime implicant is essential if it alone covers a minterm in the K-map •l Remember that all squares marked with must be covered sim ° Objective: plifi •cati Grow implicant into prime implicants (minimize literals per term) •on Cover the K-map with as few prime implicants as possible (minimize number of product terms) Exa mpl es to illus trat e ter ms C A X 1 0 1 1 6 prime implicants: A'B'D, BC', AC, A'C'D, AB, B'CD D essential minimum cover: AC + BC' + A'B'D B A 5 prime implicants: BD, ABC', ACD, A'BC, A'C'D essential minimum cover: 4 essential implicants C 0 1 1 0 1 1 0 B D Prime Implicants Any single 1 or group of 1s in the Karnaugh map of a function F is an implicant of F A product term is called a prime implicant of F if it cannot be combined with another term to eliminate a variable. If a function F is represented by A Example: C 1 B 1 1 this Karnaugh Map Which of the following terms are implicants of F, and which ones are prime implicants of F? Implicants: D (a) AC’D’ (a),(c),(d),(e) (b) BD (c) A’B’C’D’ Prime Implicants: (d) AC’ (d),(e) (e) B’C’D’ Essential Prime Implicants • A product term is an essential prime implicant if there is a minterm that is only covered by that prime implicant • The minimal sum-of-products form of F must include all the essential prime implicants of F Example of Prime Implicants ° Find ALL Prime Implicants CD C BD 1 BD A AB 1 1 1 D AD ESSENTIAL Prime C Implicants BD 1 B BD A 1 1 1 1 B D BC Minterms covered by single prime implicant Prime Implicant Practice ° Find all prime implicants for: F(A, B, C, D) (0,2,3,8,9,10,11,12,13,14,15) m C B’D’ Prime implicants are: A, B'C, and B'D’ A All prime implicants are essential A 1 12 1 15 1 D 11 13 B’C 1 14 10 B Don’t Care Conditions Ex: Ex: Don’t Care Conditions After labeling and transferring the truth table data into the K-Map, write the simplified sum-ofproducts (SOP) logic expression for the logic function F4 Be sure to take advantage of the don’t care conditions V R S T U F4 0 0 X 0 0 1 0 1 X 0 0 1 X 1 X 1 1 0 1 0 1 1 1 1 X 1 0 X 1 1 1 0 1 1 Ex: Don’t Care Conditions After labeling and transferring the truth table data into the K-Map, write the simplified sum-ofproducts (SOP) logic expression for the logic function F4 Be sure to take advantage of the don’t care conditions Solution: R S T U F4 0 0 X 0 0 1 0 1 X 0 0 1 X 1 X 1 1 0 1 0 1 1 1 1 X 1 0 X 1 1 1 0 1 1 RT TU V TU TU TU RS X X RS X X RS X 0 RS 1 X F4 RT RS RS Ex: ° Simplify F(A, B, C, D) given on the K-map Selected Essential C 1 A 1 1 C 1 B A D 1 1 1 B D Minterms covered by essential prime implicants F(A, B, C, D) = A’B + A’CD + AC’D + B’C’D’ Example with Don't Cares ° Simplify F(A, B, C, D) given on the K-map Selected Essential C C A x x x x 1 D x B A x x x x 1 D B x Minterms covered by essential prime implicants F(A, B, C, D) = A’B + AB’D + B’C Don’ t-care Conditions POS implementation SOP implementation Ex: Ex: Summary ° K-maps of four literals considered • Larger examples exist ° Don’t care conditions help minimize functions • Output for don’t cares are undefined ° Result of minimization is minimal sum-of-products ° Result contains prime implicants ° Essential prime implicants are required in the implementation ... circuit with x1 and x2 as inputs can be designed by ignoring this condition ° A function that has don’t- care condition(s) is said to be incompletely specified Karnaugh maps: Don’t cares ° In some... terms ° Design could be implemented using NANDs and NORs Don’t cares ° In digital systems it often happens that certain input conditions can never occur For example, suppose that x1 and x2 control...Overvie w ° Karnaugh maps with four inputs • Same basic rules as three input K -maps ° Understanding prime implicants • Related to minterms ° Covering all implicants ° Using Don’t Cares to simplify