ECE2030 Introduction to Computer Engineering Lecture 7: Simplification using K-map Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech

Hamming Distance The count of bits different in two binary patterns Examples: Dh(1001, 0101) = 2 Dh(0xADF4, 0x9FE3) = ?? Unit-Distance Codes Reduce errors during transmission such as rotary positional sensor E.g. Gray Code

Gray Code Construction 000 001 011 010 110 111 101 100 1 1 00 01 11 10 1 1 1 10 11 01 00 1 100 101 111 110 010 011 001 000

Gray Code Binary Encoding Gray Code Encoding Decimal b3 b2 b1 b0 g3 g2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Rotary Position Sensor

Karnaugh Map (K-Map) A graphical map method to simplify Boolean function up to 6 variables A diagram made up of squares Each square represents one minterm (or maxterm) of a given Boolean function

Karnaugh Map Examples Note that the Hamming Distance between adjacent columns or adjacent rows (including cyclic ones) must be 1 for simplification purposes

Karnaugh Map Adjacent columns or rows allow grouping of minterms (maxterms) for simplification

Implicant Definition In K-map, an Implicant is A product term is an Implicant of a Boolean function if the function has an output 1 for all minterms of the product term. In K-map, an Implicant is bubble covers only 1 (bubble size must be a power of 2) CD 00 01 11 10 1 AB

Prime Implicant Definition In K-map, a Prime Implicant (PI) is If the removal of any literal from an implicant I results in a product term that is not an implicant of the Boolean function, then I is an Prime Implicant. Examples BCD is an implicant, but CD or BD or BC do not imply a 1 in this function; BCD is a PI B’C’D is an implicant, but B’C’ is not an implicant, thus B’C’D is not a PI In K-map, a Prime Implicant (PI) is bubble that is expanded as big as possible (bubble size must be a power of 2) CD 00 01 11 10 1 AB

Essential Prime Implicant Definition If a minterm of a Boolean function is included in only one PI, then this PI is an Essential Prime Implicant. In K-map, an Essential Prime Implicant is Bubble that contains a 1 covered only by itself and no other PI bubbles CD 00 01 11 10 1 AB

Non-Essential Prime Implicant Definition A Non-Essential Prime Implicant is a PI that is not an Essential PI. In K-map, an Non-Essential Prime Implicant is A 1 covered by more than one PI bubble CD 00 01 11 10 1 AB

Simplification for SOP Form K-Map for the given Boolean function Identify all Essential Prime Implicants for 1’s in the K-map Identify non-Essential Prime Implicants in the K-map for the 1’s which are not covered by the Essential Prime Implicants Form a sum-of-products (SOP) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 1’s

Example for SOP Identify all the essential PIs for 1’s Identify the non-essential PIs to cover 1’s Form an SOP based on the selected PIs BC 00 01 11 10 1 A

Example for SOP Identify all the essential PIs for 1’s Identify the non-essential PIs to cover 1’s Form an SOP based on the selected PIs CD 00 01 11 10 1 AB

Example for SOP Identify all the essential PIs for 1’s Identify the non-essential PIs to cover 1’s Form an SOP based on the selected PIs CD 00 01 11 10 1 AB

Prime Implicants All the prior definitions apply to ‘0’ (or maxterm) as well Consider these implicants imply a ‘0’ output

Simplification for POS Form K-Map for the given Boolean function Identify all Essential Prime Implicants for 0’s in the K-map Identify non-Essential Prime Implicants in the K-map for the 0’s which are not covered by the Essential Prime Implicants Form a product-of-sums (POS) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 0’s

Example for POS Identify all the essential PIs for 0’s Identify the non-essential PIs to cover 0’s Form an POS based on the selected PIs BC 00 01 11 10 1 A

Example for POS Identify all the essential PIs for 0’s Identify the non-essential PIs to cover 0’s Form an POS based on the selected PIs CD 00 01 11 10 1 AB

Don’t Care Condition  X Don’t care (X) Those input combinations which are irrelevant to the target function (i.e. If the input combination signals can be guaranteed never occur) Can be used to simplify Boolean equations, thus simply logic design In K-map Use X to express Don’t Care in the map Don’t care can be bubbled as 1 or 0 depending on SOP or POS simplification to result into bigger bubble

Don’t Care Example  BCD to Gray Code BCD stands for Binary Coded Decimal Each digit of a decimal number is represented by one code BCD only encodes from 0 to 9 Require 4 bits Only use 10 out of 16 encoding space BCD-coded input Gray Code Output Decimal b3 b2 b1 b0 g3 g2 g1 g0 1 2 3 4 5 6 7 8 9 10 X 11 12 13 14 15

BCD to Gray Code (Do not use Don’ Care) BCD stands for Binary Coded Decimal Each digit of a decimal number is represented by one code BCD only encodes from 0 to 9 Require 4 bits Only use 10 out of 16 encoding space BCD-coded input Gray Code Output Decimal b3 b2 b1 b0 g3 g2 g1 g0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

What g2 is? g2 = F(b3, b2, b1, b0)

Example: g2 of BCD-to-Gray Without using Don’t Care Take Don’t Care into account b1b0 b1b0 00 01 11 10 1 b3b2 00 01 11 10 1 X b3b2

Another Example of Don’t Care (SOP) CD 00 01 11 10 1 X AB

Another Example of Don’t Care (POS) CD 00 01 11 10 1 X AB

Use Karnaugh Map in B5 or B6 2 K-maps are to be constructed (2 submaps) Consider one submap is on top of the other Cells that occupy the same relative position in 2 maps are considered adjacent Bubble can be constructed in vertical dimension when stacking up 2 maps In B6 4 K-maps are to be constructed (4 submaps) Similar to B5 , yet another dimension needs to be considered

Karnaugh Map Example in B5 CD CD 00 01 11 10 1 00 01 11 10 1 AB AB E = 0 E = 1

Karnaugh Map Example in B6 YZ YZ 00 01 11 10 1 00 01 11 10 1 WX WX U,V=0,0 U,V=0,1 YZ YZ 00 01 11 10 1 X 00 01 11 10 1 WX WX U,V=1,0 U,V=1,1

Minimal SOP and POS w/ Don’t care Could lead to different function if the same x is treated as 1 in SOP but as 0 in POS. Even though the final Boolean functions could be different, but for those ones with non-don’t care square, correct functionality was maintained. 1 x B A F = Ā (SOP) F = B (POS) F = Ā (POS)

Use Karnaugh Map in B5 or B6 It is getting hard and complicated Eye-ball simplification on several submaps is error-prone, leading to sub-optimal results Do we have a better algorithm? Quine-McCluskey Method