Cpr E 281 Digital Logic Guest Lecture by

Cpr. E 281: Digital Logic Guest Lecture by Kyle Tietz http: //www. ece. iastate. edu/~alexs/classes/

Minimization Cpr. E 281: Digital Logic Iowa State University, Ames, IA Copyright © 2013

Administrative Stuff • HW 4 is out • It is due on Monday Sep 23 @ 4 pm. • 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: 30 pm F 2: 00 -4: 00 pm

Recap

Four-variable K-map

Grouping • Group with rectangles • Both sides a power of 2: § 1 x 1, 1 x 2, 2 x 1, 2 x 2, 1 x 4, 4 x 1, 2 x 4, 4 x 2, 4 x 4 • 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 1 0 1 0 1 1 0

Terminology • Prime Implicant § Implicant that cannot be combined into another implicant with fewer literals § Ex. x 3 x 1 x 2 00 01 11 10 0 0 1 1 1 1 0 Not prime x 3 x 1 x 2 00 01 11 10 0 0 1 1 1 1 0 Prime

Terminology • Essential Prime Implicant § Prime implicant that includes a minterm not covered by any other prime implicant § Ex. x 3 x 1 x 2 00 01 11 10 0 0 1 1 1 0 0

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’ x 1’x 2 x 3 + x 1 x 3’ x 3 x 1 x 2 00 01 11 10 0 1 1 1 0 0

Example • Number of § Implicants? § Prime Implicants? § Essential Prime Implicants? x 3 x 1 x 2 00 01 11 10 0 1 1 1 1 0

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

Example: Minimization in SOP Form g= Z’YX’W’ +Z’YX’W +ZYX’W + Z’Y’XW +ZYXW + Z’Y’XW’ +Z’YXW’ +ZYXW’ Cpr. E 281 ZY XW 00 01 11 10 00 0 1 01 0 1 11 1 0 1 1 1 1 +ZY’X’W’ +ZY’X’W +ZY’XW’

Example: Minimization in POS Form g=(Z+Y+X+W). (Z’+Y’+X+W) (Z+Y+X+W’). (Z+Y’+X’+W’) Cpr. E 281 ZY XW 00 01 11 10 00 0 1 01 0 1 11 1 0 1 1 1 1

Minimization of both SOP and POS Forms ZY 00 01 XW 00 0 5 1 01 11 10 0 4 1 1 0 1 3 0 1 2 ZY 00 01 XW 1 00 0 1 01 0 1 11 1 0 10 1 1 Cpr. E 281 11 2 10 1 1 1 1 g=ZY’ +XW’ +ZW +Y’X +Z’YX’ 1 2 3 4 5 11 10 3 0 1 1 1 1 g=(Z+Y+X) 1. (Z+Y’+X’+W’) 2. (Z’+Y’+X+W) 3 Cost = 22 (5 AND gates, 1 OR gates 16 inputs) Assumption: Complemented forms of primary inputs are given at zero cost. Cost = 18 (3 OR gates, 1 AND gates 14 inputs)

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

K-map for 5 -variables functions F(A, B, C, D, E) = S 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’ Cpr. E 281

K-map for 6 -variable functions G(A, B, C, D, E, F) = S 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’ Cpr. E 281 Lec 15 23

Questions?

THE END