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ECE 331 – Digital System Design

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Logic Circuit Design

(Lecture #7)

ECE 331 - Digital System Design

Combinational Logic Circuits

Outputs are functions of (present) inputs

No memory

Can be described using Boolean expressions

Hierarchical design

Used to solve large design problems

Break problem into smaller (sub-)problems

Solve each sub-problem (i.e. realize design)

Combine individual solutions

ECE 331 - Digital System Design

Specification

Describes the problem to be solved.

Describes what needs to be done,

not how to do it.

Implementation

Describes how the problem is solved.

ECE 331 - Digital System Design

Issues

Most solutions are not unique.

More than one solution may meet the specifications.

Cannot always satisfy all of the requirements.

Must identify (and study) design tradeoffs.

Cost

Speed

Power consumption

etc.

ECE 331 - Digital System Design

Identify requirements (i.e. circuit specifications)

Determine the inputs and outputs.

Derive the Truth Table

Determine simplified Boolean expression(s)

Implement solution

Verify solution

ECE 331 - Digital System Design

Exercise:

Design a combinational logic circuit to meet the following specifications:

1. 3-bit input (A = a2a1a0)

2. 1-bit output (z)

3. Output is high (logic 1) when 2 < A <= 5.

ECE 331 - Digital System Design

Example:

Design a 7-Segment Decoder

ECE 331 - Digital System Design

1. Circuit Specification

The 7-Segment Decoder must decode Binary Coded Decimal (BCD) digits so that they can be displayed on a 7-Segment Display.

ECE 331 - Digital System Design

2. Determine Inputs and Outputs

Input: Binary Coded Decimal digits

ECE 331 - Digital System Design

Assign a 4-bit code to each decimal digit.

A 4-bit code can represent 16 values.

There are only 10 digits in the decimal number system.

Unassigned codes are not used.

How do we interpret these unused codes?

Hint: think about K-maps.

ECE 331 - Digital System Design

ECE 331 - Digital System Design

2. Determine Inputs and Outputs

Output: 7-Segment Display

ECE 331 - Digital System Design

ECE 331 - Digital System Design

ECE 331 - Digital System Design

3. Derive Truth Table

ECE 331 - Digital System Design

ECE 331 - Digital System Design

4. Determine simplified Boolean expression(s)

ECE 331 - Digital System Design

A

y

z

w

x

00

01

11

10

00

1

0

1

1

01

0

1

1

1

11

d

d

d

d

10

1

1

d

d

A = ?

ECE 331 - Digital System Design

B

y

z

w

x

00

01

11

10

00

1

1

1

1

01

1

0

1

0

11

d

d

d

d

10

1

1

d

d

B = ?

ECE 331 - Digital System Design

Student Exercise:

Determine the minimized Boolean expression for each of the segments of the 7-Segment Display.

ECE 331 - Digital System Design

5. Implement Solution

6. Verify Solution

ECE 331 - Digital System Design

Multiple-Output Logic Circuits

ECE 331 - Digital System Design

Example:

Given two functions, F1 and F2, of the same input variables x1.. x4, design the minimum-cost implementation.

ECE 331 - Digital System Design

x

x

x

x

1

2

1

2

x

x

x

x

3

4

3

4

00

01

11

10

00

01

11

10

00

1

1

00

1

1

01

1

1

1

01

1

1

11

1

1

11

1

1

1

10

1

1

10

1

1

(a) Function

f

(b) Function

f

1

2

F1 = X1'.X3 + X1.X3' + X2.X3'.X4

F1 = X1'.X3 + X1.X3' + X2.X3.X4

ECE 331 - Digital System Design

x

2

x

3

x

4

f

1

x

1

x

3

x

1

x

3

f

2

x

2

x

3

x

4

f

f

(c) Combined circuit for

and

1

2

ECE 331 - Digital System Design

Example:

Given two functions, F3 and F4, of the same input variables x1.. x4, design the minimum-cost implementation for the combined circuit.

Note: the minimum-cost implementation for the combined circuit may not

be the same as the minimum-cost implementations for the individual circuits.

ECE 331 - Digital System Design

x

x

x

x

1

2

1

2

x

x

x

x

3

4

3

4

00

01

11

10

00

01

11

10

00

00

01

1

1

1

01

1

1

1

11

1

1

1

11

1

1

1

10

1

10

1

(a) Optimal realization of

(b) Optimal realization of

f

f

3

4

F3 = X1'.X4 + X2.X4 + X1'.X2.X3

F3 = X2'.X4 + X1.X4 + X1'.X2.X3.X4'

Logic Gates required:

2 2-input AND

1 3-input AND

1 3-input OR

Logic Gates required:

2 2-input AND

1 4-input AND

1 3-input OR

Total Gates and Inputs required:

8 Logic Gates

21 Inputs

ECE 331 - Digital System Design

x

x

x

x

1

2

1

2

x

x

x

x

3

4

3

4

00

01

11

10

00

01

11

10

00

00

01

1

1

1

01

1

1

1

11

1

1

1

11

1

1

1

10

1

10

1

(c) Optimal realization of

f

and

f

together

3

4

F3 = X1'.X4 + X1.X2.X4 + X1'.X2.X3.X4'

F3 = X2'.X4 + X1.X2.X4 + X1'.X2.X3.X4'

Logic Gates required:

1 2-input AND

1 3-input AND

1 4-input AND

1 3-input OR

Logic Gates required:

1 2-input AND

1 3-input AND

1 4-input AND

1 3-input OR

shared logic gates

Total Gates and Inputs required:

6 Logic Gates

17 Inputs

ECE 331 - Digital System Design

x

1

x

4

x

f

1

3

x

2

x

4

x

1

x

2

x

3

x

f

4

4

x

2

x

4

f

f

(d) Combined circuit for

and

3

4