K-map Definitions. abc

Transcription

1 K-map efinitions b a bc Implicant ny single or any group of s is called an implicant of F. ny possible grouping of s is an implicant. b a Prime Implicant implicant that cannot be combined with some other implicant to eliminate a variable 3 Minimum Sum-Of-Products (SOP) The minimum SOP expression consists of some (but not necessarily all) of the prime implicants of a function. If a SOP expression contains a term which is NOT a prime implicant, then it NNOT be minimum. 2

2 Prime Implicants EH of these coverings is a PRIME IMPLINT (i.e. cannot be reduced) b, d, d Minimum SOP will have some or all of these prime implicants. The included prime implicants must cover all of the ONEs. F(,,,) = b + d (minimum set of PIs) = b + d + d (valid set of PIs, but not minimum) d + d (both PI s, but all s not included!) 3 Non-Essential vs. Essential Prime Implicants EH of the coverings is a PRIME IMPLINT. b, d, d F(,,,) = b + d (minimum # of PIs) NON-ESSENTIL prime implicant Prime Implicant d is non-essential because its s are covered by other PIs. PI is ESSENTIL if it covers a MINTERM that cannot be covered by any other PI. 4 2

3 n example with more than one solution EH of the coverings is a PRIME IMPLINT. acd bd bcd Recall that a covering is a Prime Implicant if it cannot be combined with another covering to eliminate a variable. 5 Two Solutions EH solution is equally valid. F(,,,) = + acd + bd Essential PIs 6 Non-Essential PIs F(,,,) = + acd + bcd 3

4 Minimal Solution minimal SOP will consist of prime implicants. minimal SOP equation will have all of the essential prime implicants on the map. y definition, these cover a minterm that may not be covered by some other prime implicant. The minimal SOP equation may or may not include nonessential prime implicants. It will include non-essential prime implicants if there are s remaining that have not been covered by an essential prime implicant. 7 Row F(,,,) x x 2 x 3 x 4 x 5 x on t ares on t ares are labeled as X s in truth table. an treat X s as either s or s F() Recognize numbers: 2,3,6 Non numbers are don t cares because they will never be applied as inputs! F(..) 8 4

5 on t ares treated as s or s X X X X X X Treat X s as s to make larger groupings. ll X s do not have to be covered. F(,,,) = c + c 9 Minimizing s Grouping s produces an equation for F. F(,,) = F(,,) = c 5

6 Minimize s, then omplement to get POS X X X X X X F (,,,) = + bd Take inverse of both sides F(,,,) = ( + bd) = c (bd) = c (+) Minimizing zeros, then applying inverse to both sides is a way to get to minimum POS form K-Map example esign Example: Two it omparator omparator F F 2 F 3 lock iagram = < > Truth Table F F 2 F 3 4-Variable K-map for each of the 3 output functions 2 6

7 K-Map example esign Example: Two it omparator K-map for F K-map for F 2 K-map for F 3 F = F2 = F3 = 3 6- Variable K-Maps EF = djacencies EF = EF = EF = f(,,,,e,f) = Σm(2,8,,8,24, 26,34,37,42,45,5, 53,58,6) = e F + a d E f + c F EF = EF = EF = EF = 4 7

8 EF = EF = Six-Variable K-Map F= d + bf + ace EF = EF = 5 More K-Map Method Examples, 3 Variables F(,,) = Σm(,4,5,7) F = F' simply replace 's with 's and vice versa F(,,) = Σm(,2,3,6) F = 6 8

9 K-map Method Examples: 4 variables F(,,,) = Σm(,2,3,5,6,7,8,,,4,5) F = 7 K-map Example: on't ares on't ares can be treated as 's or 's if it is advantageous to do so X X X minterms on t cares F(,,,) = Σm(,3,5,7,9) + Σd(6,2,3) without don't cares F = with don't cares F = 8 9

10 Variable-Entered Karnaugh Maps (VEM) K-maps are cumbersome for 5 or more variables The variable entered map method allows 8 to 6 variable maps to be represented as 3 to 5 variable maps Needed because typical oolean design problem involves 8 or more oolean variables onventional logic minimization: Time consuming Error-prone VEM Key idea: Represent values of function in terms of its variables (called mapentered variables) within Karnaugh map framework Group like variables in Karnaugh map cells 9 VEM example 5-variable karnaugh map,,, go, wait ells can now contain variables, as well as,,or X. on t care go go wait X wait F Minimisation approach Map all s onvert s to X s (i.e. don t cares) Map SIMILR components 2

11 Phase Map s with any don t cares VEM example go go wait X wait F f = (c + ab) +. 2 Phase 2 VEM example onvert s to X s and map like entries go go X wait X X wait F f = ( c + a b ) +. ( go + b go + a wait) 22

12 VEM example 2 5-variable karnaugh map,,, P, Q Q + p p + P p + P = X Q Q + p F X Q f = ( + ) +.. F 23 5-variable karnaugh map,,, P, Q VEM example 2 XX XX Q + p XX X Q F f = ( + ) + bq + p 24 2

13 Example using VEM Example function F(a,b,c) = + bc + a + ac rbitrarily choose as map entered variable or choose least used variable Insert map entries as follows F(a,b,c) = No minterm (i.e ) X (don t care) ondition --- = = = or --- Map entry c (c + ) or X 25 3