Menu. X + /X=1 and XY+X /Y = X(Y + /Y) = X
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1 Menu K-Maps and Boolean Algera >Don t ares >5 Variale Look into my... 1 Karnaugh Maps - Boolean Algera We have disovered that simplifiation/minimization is an art. If you see it, GREAT! Else, work at it, work at it,, A method to exhaustively minimize is due to Quine/MCluskey and is programmale. As the numer of variales inrease, it gets quite tedious and is not appropriate for our purposes Reminder X + /X=1 and XY+X /Y = X(Y + /Y) = X Two terms whih differ in only one literal an e minimized/ redued y one literal. A K-Map is a graphial aid (a pattern reognition tehnique) for humans to implement the aove two equations. 2 1
2 K-Maps Consider the following arrangement for f. A BC Positions of Truth Tale Rows in K-Map A BC R 0 R 4 01 R 1 R 5 11 R 3 R 7 10 R 2 R f SOP = /A /B /C + /A /B C + /A B C + A /B C + A B /C But the minterms of funtion f are now rearranged so they differ in 1 literal. Look at row 1 (BC=01) of the K-map. The terms of funtion f differ only in one literal, e.g., f ontains /A /B C + A /B C, the seond and fourth terms of the equation, = (/A + A) /B C = /B C. In fat, any two adjaent rows or olumns of 1 s differ in one literal, and thus, an e simplified. 3 K-Maps A BC R 0 R 4 f SOP = /A /B /C + /A /B C + /A B C + A /B C + A B /C Now, onsider the following arrangement for f. A BC m 0 But the minterms of f are now arranged so they differ in 1 literal. m 5 Look at the seond row of the K-Map. The terms of f differ only in one literal, e.g., f ontains A B C + AB C = (/A + A)B C = B C. In fat, if any of two rows or olumns of 1 s differ in one literal, they an e simplified. m 6 01 R 1 R 5 11 R 3 R 7 10 R 2 R 6 ABC f
3 Definitions K-Maps Impliants: terms in a K-Map that are 1 iff the funtion is 1. Prime impliant: a term in a K-Map that is as ig as it an e. If we irle in the K-Map those impliants that are logially and spatially adjaent and if we make sure that all the 1 s of the funtion are overed, we an synthesize the funtion diretly. When a 1 is overed y one and only one (prime) impliant, it is alled an essential (prime) impliant. Goal: > Step 1. Cirle all the (prime) impliants > Step 2. Selet a suset that overs the funtion starting with the prime impliants 5 K-Map: MSOP Example <Example 1> Simplify a given funtion g in SOP form. g = /a / + /a / + /a + a + a / Prime Impliant (eause of m 5 ) (a + /a) / = / Prime Impliant (a + /a) = (eause of m 6 ) g = + / + 6 3
4 K-Map: MSOP Example <Example 1> Simplify a given funtion g in SOP form. g = /a / + /a / + /a + a + a / /a /( + ) = /a / g MSOP1 = + / + /a / = /a / + / + /a ( + /) = /a g MSOP2 = + / + /a = /a + / + 7 Another way to look at it K-Map: Another MSOP Viewpoint g = /a / + /a / + /a + a + a / /a / g MSOP1 = + / + /a / = /a / + / + g MSOP2 = + / + /a = /a + / + /a a / / 8 4
5 K-Map: MPOS Example <Example 2> Will it work for POS if we use 0 s & maxterms? NOTE: Reversed laels from SOP g MPOS /a + + / + = (/a + + )(/ + ) a /a / 9 K-Map: MSOP vs MPOS Let s ompare MSOP with MPOS. g MSOP = /a + / + g MPOS Q: Whih is heaper? Ans: Depends on ost funtion; proaly MPOS Q: Why? Ans: Beause MPOS requires less # of gates and inputs. /a g MSOP requires 4 gates and 9 inputs / g MPOS requires 3 gates and 7 inputs / /a = (/a + + )(/ + ) 10 5
6 K-Map Heuristi Heuristi: (not a guarantee) We note that the POS was minimum (ompared to the SOP) perhaps eause we had only three 0 s vs five 1 s. In general, we synthesize the funtion ased on the least numer of either 1 s or 0 s. But to e sure we must do oth and hek! Oservation: We got 2-level logi with these, i.e., the maximum delay is the delay of two gates. It is sometimes possile to otain a heaper (in gates) ost funtion using more than 2-level logi. If your appliation an tolerate larger delays, this is OK. /a / NOTE: 2-Level logi is the / fastest logi possile and K-Maps give it to you. /a 11 Examples of 4-input K-Map Example: Groups of 2 / /d prime a / /d prime a /d prime a / prime g 1MSOP = a / + a / /d + a /d + / /d 12 6
7 4-input K-Map Example: Groups of 2 and Redundany Examples of (ontinued) Redundant! What is eqn? a prime /a d prime /a prime a d prime g 2MSOP = /a + /a d + a + a d 13 Examples of (ontinued) 4-input K-Map Example: Groups of 4 /a prime d prime g 3MSOP = /a + d 14 7
8 Examples of (ontinued) 4-input K-Map Example: More Groups of 4 / d prime /d prime g 4MSOP = / d + /d = xor d / + /d prime + d prime g 4MPOS = (/ + /d) ( + d) = xor d Same funtion! 15 Don t Cares Don t Cares: Suppose we have the following system a C i f( ) d What if some ominations of the inputs a,,,d are never produed y C i? >Q: What should we write in the Truth Tale, 0 or 1? >A: We write a don t are (X). We are free to hoose X to e 0 or 1 at our onveniene. However, one you pik a partiular value for X, it must e the same throughout the design! 16 8
9 Examples of with don t ares Don t Care Examples Is g 1MSOP = g 2MPOS? Why or why not? 17 5-variale K-Maps Five Variale : Break the funtion into two 4-variale K- maps, e.g., for a funtion f (a,,,d,e) of 5 variales, pik one of the variales, say e, and do a K-map when e=0 and when e=1. Then f = e f e=0 + ef e=1. Can simplify to a d Not in lexial f = e f e=0 +ef e=1 = e ( d+d ) + order e(a +a d+d ) = a d e + d + e ( d) + e(a ) 18 9
10 5-variale K-Maps Or use: > f = f e&e + e f e=0 (not f e=1 )+ ef e=1 (not f e=0 ), i.e., > f = (shared overs) + e (overs in f e=0 only) + e (overs in f e=1 only) f = a d + d + e ( d) + e(a ) = f e&e + e /f e=1 f e=0 + e /f e=0 f e=1 Not in lexial order 19 The End! 20 10
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