8 - ECSE-2610 Computer Components & Operations (COCO)...

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ECSE-2610 Computer Components & Operations (COCO) Fall 2006 Part 6b : Synthesis of Combinational Circuits II
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Don’t-Cares in K-maps 00X0 11X1 1100 0X00 AB CD 00 01 11 10 00 01 11 10 In some applications, there are certain inputs that cannot occur (e.g. 11-15 in BCD). We call these don’t-care inputs . Since a don’t-care input can’t occur, it doesn’t matter what output we assign for that input. We represent these inputs in the K-Map by an X rather than a 0 or 1. We can include the Xs or not include them in our cubes to make the simplest functional form.
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Don’t-Cares in K-Maps 00X0 11X1 1100 0X00 AB CD 00 01 11 10 00 01 11 10 00 1/0 0 11 1/0 1 0 1/0 AB CD 00 01 11 10 00 01 11 10 A don’t-care that is not in any covering is assigned a 0 A don A don ’t-care that is care that is not in any not in any covering is assigned a covering is assigned a 0 A don’t-care that is included in any covering is assigned 1 A don A don ’t-care that is care that is included included in any covering is assigned in any covering is assigned 1
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Don’t-Care Example 00X0 11X1 1100 0X00 1 of 2 First, let’s minimize F without using the don't-cares [Assign 0 values to all don't-cares] AB CD 00 01 11 10 00 01 11 10 F = A' D + B' C' D , without don't-care set
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Don’t-Care Example 00X0 11X1 1100 0X00 AB CD 00 01 11 10 00 01 11 10 2 of 2 Now, let's use the don't-cares to further simplify the Boolean equation for F For don’t-cares used, assigned 1 assigned 1 , the others are assigned 0 assigned 0 0 1 0 F = A' D + C' D , with Don't-Care set F = A' D + B' C' D, without Don't-Care set
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What About Product of Sums (POS) Forms? Problem: Truth tables are inherently in SOP form — so are K-maps! Solution: Minimize the SOP form of the complement function Apply Apply DeMorgan DeMorgan s Theorem s Theorem to get to get POS form POS form of original function! of original function! Must consider the minimum POS form as well as the minimum SOP form Must consider the minimum POS form as well as the minimum SOP form
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A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 SOP POS SOP and POS forms are related by De Morgan’s Law SOP and POS forms are related by De Morgan SOP and POS forms are related by De Morgan ’s Law s Law SOP form : 1s of truth table F = A'BC + AB'C' + AB'C + ABC' + ABC But, F = (F')' = (A'B'C' + A'B'C + A'BC')' = (A+B+C)(A+B+C')(A+B'+C) = POS form ! {Using De Morgan’s Law}
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Minimal POS Forms AB CD 00 01 11 10 00 01 11 10 0 1 1 1 1 1 1 1 1 1 1 1 0 00 0 1 of 2 Complement function is 1 when the original is 0 Complement function is 1 when the original is 0 So, group 0s instead of 1s So, group 0s instead of 1s F' = B C' D' + B' C' D + { A B C' or A C' D}
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Minimal POS Forms 2 of 2 F' = B C' D' + B' C' D + A B C' F = (F')' = (B' + C + D) (B + C + D') (A' + B' +C) Apply DeMorgan’s Law Apply DeMorgan’s Law We want the POS form of F, not the SOP form of F' We want the POS form of F, not the SOP form of F'
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K-Map Summary Minimization Combining Theorem Karnaugh Maps aka “K-Maps” Fill out the K-Map Find the largest covering cubes Find the essential cubes Find minimal form(s) Don’t-cares POS minimal forms using DeMorgan
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ECSE-2610 Computer Components & Operations (COCO) Fall 2006 Quine-McCluskey: Tabular Minimization Method
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This homework help was uploaded on 04/22/2008 for the course ECSE 2610 taught by Professor Ji during the Spring '08 term at Rensselaer Polytechnic Institute.

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8 - ECSE-2610 Computer Components & Operations (COCO)...

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