Cpr E 281 Digital Logic Instructor Alexander Stoytchev

Incompletely Specified Functions & Multiple-Output Circuits Cpr. E 281: Digital Logic Iowa State University,

Administrative Stuff • HW 4 is due today.

Administrative Stuff • HW 5 is out • It is due on Monday Feb

Administrative Stuff • Midterm Exam #1 • When: Monday Feb 24. • Where: This

The Combining Theorems of Boolean Algebra

Two-Variable K-map x 1 x 2 0 0 m 0 0 1 m 1

Two-Variable K-map 1 [ Figure 2. 50 from the textbook ]

These are also valid But try to use larger rectangles if possible.

Three-Variable K-map [ Figure 2. 51 from the textbook ]

Location of three-variable minterms Notice the placement of § Variables § Binary pair values

Three-Variable K-map [ Figure 2. 52 from the textbook ]

Two Different Ways to Draw the K-map x 2 x 3 00 01 11

Another Way to Draw 3 -variable K-map x 1 x 2 x 3 0

A four-variable Karnaugh map [ Figure 2. 53 from the textbook ]

A four-variable Karnaugh map x 1 x 2 x 3 x 4 0 0

Adjacency Rules adjacent rows adjacent columns

Example of a four-variable Karnaugh map [ Figure 2. 54 from the textbook ]

Other Four-Variable K-map Examples [ Figure 2. 54 from the textbook ]

Grouping Rules • Group “ 1”s with rectangles • Both sides a power of

Terminology Literal: a variable, complemented or uncomplemented Some Examples: _ § X 1 §

Terminology • Implicant: product term that indicates the input combinations for which the function

Terminology • Prime Implicant § Implicant that cannot be combined into another implicant with

Terminology • Essential Prime Implicant § Prime implicant that includes a minterm not covered

Terminology • Cover § Collection of implicants that account for all possible input valuations

Example • Give the Number of § Implicants? § Prime Implicants? § Essential Prime

Why concerned with minimization? • Simplified function • Reduce the cost of the circuit

Three-variable function f (x 1, x 2, x 3) = m(0, 1, 2, 3,

Example: Another Solution x 3 x 4 x 1 x 2 00 00 01

Example: Incompletely Specified Function

Three Ways to Specify the Function f(x 1, x 2, x 3, x 4)

SOP implementation x x x 3 x 4 1 2 00 01 11 10

POS implementation x 3 x 4 x 1 x 2 00 01 11 10

Example: A circuit with multiple outputs

Seven-Segment Indicator 1 1 1 0 0 1 1 0 1 0 1

Seven-Segment Indicator 1 1 1 0 0 1 1 0 1 0 1 d

Seven-Segment Indicator x 1 x 0 1 1 1 0 0 1 1 0

Seven-Segment Indicator 1 1 1 0 0 1 1 0 1 0 1 1

Individual vs Joint Optimization x 3 x 4 x 1 x 2 00 01

Slides: 93

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Cpr. E 281: Digital Logic Instructor: Alexander Stoytchev http: //www. ece. iastate. edu/~alexs/classes/

Administrative Stuff • HW 4 is due today.

Administrative Stuff • HW 5 is out • It is due on Monday Feb 17 @ 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 • Midterm Exam #1 • When: Monday Feb 24. • Where: This classroom • What: Chapter 1 and Chapter 2 plus number systems • The exam will be open book and open notes (you can bring up to 3 pages of handwritten notes). • More details to follow.

Quick Review

The Combining Theorems of Boolean Algebra

Two-Variable K-map x 1 x 2 0 0 m 0 0 1 m 1 1 0 m 2 1 1 m 3 (a) Truth table x 2 x 1 0 m 0 m 2 1 m 3 (b) Karnaugh map [ Figure 2. 49 from the textbook ]

Two-Variable K-map 1 [ Figure 2. 50 from the textbook ]

These are all valid groupings

These are also valid But try to use larger rectangles if possible.

Why are these two not valid?

Three-Variable K-map [ Figure 2. 51 from the textbook ]

Location of three-variable minterms Notice the placement of § Variables § Binary pair values § Minterms

Adjacency Rules adjacent columns

These are valid groupings

This is a valid grouping

Some invalid groupings

Three-Variable K-map [ Figure 2. 52 from the textbook ]

Two Different Ways to Draw the K-map x 2 x 3 00 01 11 10 0 m 1 m 3 m 2 1 m 4 m 5 m 7 m 6 x 1

Another Way to Draw 3 -variable K-map x 1 x 2 x 3 0 1 00 m 4 01 m 5 11 m 3 m 7 10 m 2 m 6

A four-variable Karnaugh map [ Figure 2. 53 from the textbook ]

A four-variable Karnaugh map x 1 x 2 x 3 x 4 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 1 m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 m 8 m 9 m 10 m 11 m 12 m 13 m 14 m 15

Adjacency Rules adjacent rows adjacent columns

Example of a four-variable Karnaugh map [ Figure 2. 54 from the textbook ]

Other Four-Variable K-map Examples [ Figure 2. 54 from the textbook ]

Strategy For Minimization

Grouping Rules • Group “ 1”s 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 the same minterm more than once • Can wrap around the edges of the map • Some rules in selecting groups: § Try to use as few groups as possible to cover all “ 1”s. § For each group, try to make it as large as you can (i. e. , if you can use a 2 x 2, don’t use a 2 x 1 even if that is enough).

Terminology Literal: a variable, complemented or uncomplemented Some Examples: _ § X 1 § X 2

Terminology • Implicant: product term that indicates the input combinations for which the function output is 1 • Example _ § x 1 __ _ - indicates that x 1 x 2 and x 1 x 2 yield output of 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 § Some Examples 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 § Some Examples 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 • Give the 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 the cost of the circuit § Cost: Gates + Inputs § Transistors

Three-variable function f (x 1, x 2, x 3) = m(0, 1, 2, 3, 7) [ Figure 2. 56 from the textbook ]

Example x 3 x 4 x 1 x 2 00 00 01 1 11 10 11 1 10 1 1

Example x 3 x 4 x 1 x 2 00 00 01 1 11 10 11 x 3 x 4 x 2 x 3 x 4 1 10 1 x 3 x 4 1 x 2 x 3 x 4

Example: Another Solution x 3 x 4 x 1 x 2 00 00 01 1 1 01 11 x 3 x 4 1 x 2 x 3 x 4 1 11 10 10 1 1 1 x 3 x 4 1 x 2 x 3 x 4 x 1 x 2 x 3 [ Figure 2. 59 from the textbook ]

Example: Incompletely Specified Function

Three Ways to Specify the Function f(x 1, x 2, x 3, x 4) = Σ m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)

Three Ways to Specify the Function f(x 1, x 2, x 3, x 4) = Σ m(2, 4, 5, 6, 10) + D(12, 13, 14, 15) x x x 3 x 4 1 2 00 01 11 10 00 0 1 d 0 01 0 1 d 0 11 0 0 d 0 10 1 1 d 1

SOP implementation x x x 3 x 4 1 2 00 01 11 10 00 0 1 d 0 01 0 1 d 0 11 0 0 d 0 10 1 1 d 1 x 2 x 3 x 4 (a) SOP implementation [ Figure 2. 62 from the textbook ]

POS implementation x 3 x 4 x 1 x 2 00 01 11 10 00 0 1 d 0 01 0 1 d 0 11 0 0 d 0 10 1 1 d 1 ( x 2 + x 3) ( x 3 + x 4) (b) POS implementation [ Figure 2. 62 from the textbook ]

Example: A circuit with multiple outputs

Seven-Segment Indicator

Seven-Segment Indicator 1 1 1 0 0 1 1 0 1 0 1

Seven-Segment Indicator 1 1 1 0 0 1 1 0 1 0 1 d d d d d d d d d d d

Seven-Segment Indicator x 1 x 0 1 1 1 0 0 1 1 0 1 0 1 d d d d d d d d d d d x 3 x 2 00 00 01 11 10

Seven-Segment Indicator x 1 x 0 1 1 1 0 0 1 1 0 1 0 1 d d d d d d d d d d d x 3 x 2 00 01 11 10 00 1 0 d 1 01 0 1 d 1 11 1 1 d d 10 1 1 d d

Seven-Segment Indicator x 1 x 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 d d d d d d d d d In this case all d's were treated as 1's. x 3 x 2 00 01 11 10 00 1 1 01 0 1 11 1 1 10 1 1

Seven-Segment Indicator 1 1 1 0 0 1 1 0 1 0 1 1 1 1 d d d d d d d d d

Seven-Segment Indicator x 1 x 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 d d d d d d d d d x 3 x 2 00 00 01 11 10

Seven-Segment Indicator x 1 x 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 d d d d d d d d d x 3 x 2 00 01 11 10 00 1 0 d 1 01 0 0 d 0 11 0 0 d d 10 1 1 d d

Seven-Segment Indicator x 1 x 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 d d d d d 1 0 0 0 1 0 d d d x 3 x 2 00 01 11 10 00 1 01 0 0 0 0 10 1 1 In this case some d's were treated as 1's, others as 0's.

Seven-Segment Indicator

Another Example

x 3 x 4 x 1 x 2 00 01 1 11 10 1 1 x 3 x 4 x 1 x 2 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 (a) Function f 1 1 (b) Function f 2 [ Figure 2. 64 from the textbook ]

x 3 x 4 x 1 x 2 00 01 1 11 10 1 1 x 3 x 4 x 1 x 2 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 (a) Function f 1 1 (b) Function f 2

x 3 x 4 x 1 x 2 00 01 1 11 10 1 1 x 1 x 2 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 (a) Function f 1 _ x 1 x 3 x 4 1 (b) Function f 2

x 3 x 4 x 1 x 2 00 01 1 11 10 1 1 x 1 x 2 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 (a) Function f 1 _ x 1 x 3 x 4 1 (b) Function f 2 _ x 1 x 3

x 3 x 4 x 1 x 2 00 01 1 11 10 1 1 x 3 x 4 x 1 x 2 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 (a) Function f 1 _ x 1 x 3 1 (b) Function f 2 _ x 1 x 3

x 3 x 4 00 01 1 11 1 1 10 1 1 1 1 (a) Function f 1 _ x 1 x 3 x 1 x 2 x 3 x 4 _ x 2 x 3 x 4 x 1 x 2 00 01 11 10 00 1 1 01 1 1 10 1 1 1 (b) Function f 2 _ x 1 x 3

x 3 x 4 _ x 1 x 3 x 1 x 2 00 01 1 11 1 1 10 1 1 1 1 _ x 2 x 3 x 4 (a) Function f 1 _ x 1 x 3 x 4 x 1 x 2 00 01 11 10 00 1 1 01 1 1 10 1 1 1 (b) Function f 2 x 3 x 4 f 1 x 3 x 1 x 3 f 2 x 3 x 4 (c) Combined circuit for f 1 and f 2 _ x 1 x 3

x 3 x 4 x 1 x 2 00 01 1 11 10 1 1 x 3 x 4 x 1 x 2 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 1 (b) Function f 2 (a) Function f 1 x 2 x 3 x 4 f 1 x 3 x 1 x 3 f 2 x 3 x 4 (c) Combined circuit for f 1 and f 2 [ Figure 2. 64 from the textbook ]

Yet Another Example

Individual vs Joint Optimization x 3 x 4 x 1 x 2 00 01 11 10 x 3 x 4 00 00 01 11 10 00 01 1 1 1 11 1 1 1 10 (a) Optimal realization of f 3 x 3 x 4 x 1 x 2 00 01 11 00 10 1 (b) Optimal realization of f 4 x 3 x 4 x 1 x 2 00 01 11 10 00 01 1 1 1 11 1 1 1 10 1 (c) Optimal realization of f 3 and f 4 together [ Figure 2. 65 from the textbook ]

Individual vs Joint Optimization x 3 x 4 x 1 x 2 00 01 11 00 10 x 3 x 4 x 1 x 2 00 01 11 10 00 01 1 1 1 11 1 1 1 10 1 (c) Optimal realization of f 3 and f 4 together x 1 x 4 x 1 f 3 x 2 x 4 x 1 x 2 x 3 x 4 f 4 x 2 x 4 (d) Combined circuit for f 3 and f 4 [ Figure 2. 65 from the textbook ]

Questions?

THE END