Guest Lecture by Kyle Tietz CprE 281: Digital Logic

Minimization CprE 281: Digital Logic Iowa State University, Ames, IA Copyright © 2013

Administrative Stuff HW4 is out It is due on Monday Sep 4pm. Please write clearly on the first page (in block capital letters) the following three things:  Your First and Last Name  Your Student ID Number  Your Lab Section Letter

Administrative Stuff Exam 1 on Monday Sep 30. Details to follow. Homework Office Hours  Pratik Mishra  TLA  M 5:30-7:30pm  F 2:00-4:00pm

Recap

Four-variable K-map

Grouping Group with rectangles Both sides a power of 2:  1x1, 1x2, 2x1, 2x2, 1x4, 4x1, 2x4, 4x2, 4x4 Can use same minterm more than once Can wrap around edges of map

Recap Example

Terminology Literal  A variable, complemented or uncomplemented  Ex. X 1  Ex. X 2 _

Terminology Implicant  Product term that indicates the input combinations for which the function output is 1  Ex. x 1 - indicates that x 1 x 2 and x 1 x 2 yield output of 1  Ex. x 1 x 2 x x 1 ___ _ __

Terminology Prime Implicant  Implicant that cannot be combined into another implicant with fewer literals  Ex. x 1 x 2 x x 1 x 2 x Not prime Prime

Terminology Essential Prime Implicant  Prime implicant that includes a minterm not covered by any other prime implicant  Ex. x 1 x 2 x

Terminology Cover  Collection of implicants that account for all possible input valuations where output is 1  Ex. x 1 ’x 2 x 3 + x 1 x 2 x 3 ’ + x 1 x 2 ’x 3 ’  Ex. x 1 ’x 2 x 3 + x 1 x 3 ’ x 1 x 2 x

Example Number of  Implicants?  Prime Implicants?  Essential Prime Implicants? x 1 x 2 x

Why concerned with minimization? Simplified function Reduce cost of circuit  Cost: Gates + Inputs  Transistors

CprE Example: Minimization in SOP Form ZY XW g= Z’YX’W’ +ZY’X’W’ +Z’YX’W +ZYX’W +ZY’X’W + Z’Y’XW +ZYXW +ZY’XW + Z’Y’XW’ +Z’YXW’ +ZYXW’ +ZY’XW’

CprE ZY XW g=(Z+Y+X+W). (Z’+Y’+X+W) (Z+Y+X+W’). (Z+Y’+X’+W’) Example: Minimization in POS Form

CprE 281 Minimization of both SOP and POS Forms ZY XW g=ZY’ +XW’ +ZW +Y’X +Z’YX’ ZY XW g=(Z+Y+X).(Z+Y’+X’+W’).(Z’+Y’+X+W) Cost = 22 (5 AND gates, 1 OR gates 16 inputs) Cost = 18 (3 OR gates, 1 AND gates 14 inputs) Assumption: Complemented forms of primary inputs are given at zero cost.

Strategy 1.Generate all prime implicants 2.Find the set of essential prime implicants 3.If set of essential prime implicants covers function, Done! 4.Else, decide which non-essential prime implicants to add to complete minimum-cost cover

Examples

Five-variable K-map

CprE 281 K-map for 5-variables functions F(A,B,C,D,E) =  m(2,5,7,8,10,13,15,17,19,21,23,24,29,31) F(A,B,C,D,E) = CE + AB’E + BC’D’E’ + A’C’DE’

CprE 281Lec 1523 K-map for 6-variable functions G(A,B,C,D,E,F) =  m(2,8,10,18,24,26,34, 37,42,45,50,53,58,61) G(A,B,C,D,E,F) = D’EF’ + ADE’F + A’CD’F’

Questions?

THE END