Guest Lecture by Kyle Tietz alexs/classes/ CprE 281: Digital Logic.
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Transcript of Guest Lecture by Kyle Tietz alexs/classes/ CprE 281: Digital Logic.
![Page 1: Guest Lecture by Kyle Tietz alexs/classes/ CprE 281: Digital Logic.](https://reader033.fdocuments.net/reader033/viewer/2022051401/56649f1e5503460f94c351e0/html5/thumbnails/1.jpg)
Guest Lecture by Kyle Tietz
http://www.ece.iastate.edu/~alexs/classes/
CprE 281: Digital Logic
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Minimization
CprE 281: Digital LogicIowa State University, Ames, IACopyright © 2013
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Administrative Stuff
• HW4 is out
• It is due on Monday Sep 23 @ 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
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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
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Recap
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Four-variable K-map
x 1 x 2 x 3 x 4 00 01 11 10
00
01
11
10
x 2
x 4
x 1
x 3
m 0
m 1 m 5
m 4 m 12
m 13
m 8
m 9
m 3
m 2 m 6
m 7 m 15
m 14
m 11
m 10
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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
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Recap Example
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Terminology
• Literal
A variable, complemented or uncomplemented
Ex. X1
Ex. X2
_
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Terminology
• Implicant Product term that indicates the input combinations for
which the function output is 1
Ex. x1 - indicates that x1x2 and x1x2 yield output of 1
Ex. x1x2
x 2
0
1
0 1
1 0
01
x 1
_ _ _ __ _
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Terminology
• Prime Implicant Implicant that cannot be combined into another implicant
with fewer literals
Ex.
x1x2x3
0 1
1 1
1 1
1 0
00 01 11 10
0
1
x1x2x3
0 1
1 1
1 1
1 0
00 01 11 10
0
1
Not prime Prime
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Terminology
• Essential Prime Implicant Prime implicant that includes a minterm not covered by
any other prime implicant
Ex.
x1x2x3
0 1
1 1
1 1
0 0
00 01 11 10
0
1
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Terminology
• Cover Collection of implicants that account for all possible
input valuations where output is 1
Ex. x1’x2x3 + x1x2x3’ + x1x2’x3’
Ex. x1’x2x3 + x1x3’
x1x2x3
0 0
0 1
1 1
0 0
00 01 11 10
0
1
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Example
• Number of Implicants? Prime Implicants? Essential Prime Implicants?
x1x2x3
1 1
1 1
0 0
1 0
00 01 11 10
0
1
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Why concerned with minimization?
• Simplified function
• Reduce cost of circuit Cost: Gates + Inputs Transistors
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CprE 281
0 1
0 1
1 1
0 1
1 1
1 0
1 1
1 1
Example: Minimization in SOP Form
00 01 11 10
00
01
11
10
ZYXW
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’
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CprE 281
0 1
0 1
1 1
0 1
1 1
1 0
1 1
1 1
00 01 11 10
00
01
11
10
ZYXW
g=(Z+Y+X+W). (Z’+Y’+X+W) (Z+Y+X+W’). (Z+Y’+X’+W’)
Example: Minimization in POS Form
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CprE 281
Minimization of both SOP and POS Forms
0 1
0 1
1 1
0 1
1 1
1 0
1 1
1 1
00 01 11 10
00
01
11
10
ZYXW
1
2
34
5 1
2
3
4
5
g=ZY’ +XW’ +ZW +Y’X +Z’YX’
0 1
0 1
1 1
0 1
1 1
1 0
1 1
1 1
00 01 11 10
00
01
11
10
ZYXW
1
2
3
g=(Z+Y+X) .(Z+Y’+X’+W’) .(Z’+Y’+X+W)
1
2
3
Cost = 22(5 AND gates, 1 OR gates 16 inputs)
Cost = 18(3 OR gates, 1 AND gates 14 inputs)
Assumption: Complemented formsof primary inputs aregiven at zero cost.
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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
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Examples
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x 1 x 2 x 3 x 4 00 01 11 10
1 1
1 1
1 1
00
01
11
10
x 1 x 2 x 3 x 4 00 01 11 10
1
1 1
1 1
1 1
00
01
11
10
f 1 x 1 x 3 x 1 x 3 x 4 x 1 x 2 x 3 x 5 + + =
x 5 1 = x 5 0 =
Five-variable K-map
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CprE 281
K-map for 5-variables functionsF(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’
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CprE 281 Lec 15 23
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’
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Questions?
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THE END