Lecture Digital logic design - Lecture 12: More about combinational analysis and design procedures

The main contents of the chapter consist of the following: Logic circuit analysis, logic circuit analysis, verification - circuit analysis, symbolic analysis, literal analysis, analysis versus design, digital design overview, design procedure (mano), combinational logic design,...

Lecture 12

More about Combinational Analysis and

Design Procedures

Logic Circuit Analysis

Analysis:

Determining the behavior of a system given its

description

The description of the system is often provided

in the form of a circuit diagram.

Logic Circuit Analysis

o For two-level circuits, the analysis process is simple.

o The Boolean expression representing the circuit can often be written by inspection.

 For multilevel circuits, the analysis process is much more complicated.

Cannot write a Boolean expression by inspection.

Must follow a procedure to implement the analysis.

Logic Circuit Analysis

1 Identify inputs and outputs

2 Track circuit behavior from input to output

3 Determine Boolean expression for output(s)

4 Determine Truth Table

5 Examine circuit timing, power dissipation, etc.

• Every circuit computes some function, which can be described with

Boolean expressions or truth tables

• So, the goal is to find an expression or truth table for the circuit

° The first thing to do is to figure out what the inputs and

outputs of the overall circuit are

Inputs: x, y,z Output: f

° We start with the circuit diagram

• We determine gate output expressions

• Intermediate expressions are combined in following gates to form

complex expressions

- It might help to do some algebraic simplification along the way

• We repeat until we have the output function and expression

° Symbolic analysis gives both the truth table and logic

expression

Literal Analysis

° Literal analysis is process of manually assigning a set

of values to the inputs, tracing the results, and

recording the output values

• For ‘n’ inputs there are 2 n possible input combinations

• From input values, gate outputs are evaluated to form next set of

gate inputs

• Evaluation continues until gate outputs are circuit outputs

° Literal analysis only gives us the truth table

° Once you know the number of inputs and outputs, list

all the possible input combinations in your truth table

• A circuit with n inputs should have a truth table with 2 n

° You can simulate the circuit by hand to find the output

for each possible combination of inputs

° Doing the same thing for all the other input combinations

yields the complete truth table

° This is simple, but tedious

° Remember that if you already have a Boolean

expression, you can use that to easily make a truth table

° For example, since we already found that the circuit

computes the function

f(x,y,z) = xz + y’z + x’yz’ , we can use that to fill in a table:

° The opposite is also true: it’s easy to come up with an

expression if you already have a truth table

° Convert a truth table into a sum of minterms expression

Truth Table → Equation

° Analyze the logic circuit shown below to determine

the circuit’s truth-table Using the truth table, derive the logic expression for the output F 1

Did you analyze the circuit BEFORE you turned the

power on?

Truth Table → Equation

a) Add test-points at the output of every gate.

c) Working from the inputs to the output,

complete the truth table for each

test-point, ultimately ending at the circuit’s

output.

d) From the completed truth table, identify

the Minterms from the truth table

anywhere the output is one

e) Using the extracted Minterms, write the

Steps (a)

Truth Table → Equation

b) Add a column to the truth table for every

test-point.

c) Working from the inputs to the output,

complete the truth table for each test-point,

ultimately ending at the circuit’s output.

c) Working from the inputs to the

output, complete the truth table for

each test-point, ultimately ending at

the circuit’s output.

d) From the completed truth table,

identify the Minterms from the truth

table anywhere the output is one

e) Using the extracted Minterms, write

the Sum-Of-Products logic

expression.

Truth Table → Equation

d) From the completed truth table, identify the

Minterms from the truth table anywhere the output is

one

e) Using the extracted Minterms, write the

Sum-Of-Products logic expression.

Steps (d) & (e)

Z Y X

Z Y X

Z

Y X

Z Y X Z

Y X Z

Y X F

Analyze the logic circuit shown below to

determine the circuit’s truth table Using the

truth table, derive the logic expression for the

output F 2

Y X

Z

Y X Z

Y X Z

Y X F

Y X

Z

Y X

d)

e)

The Process

a) Working from the inputs to the output, write the

cumulating logic expression at the output of each gate concluding with the expression for the circuit’s output

b) Using the circuit’s output logic expression,

derive the circuit’s truth table.

Equation → Truth Table Technique

Equation → Truth Table Technique

Analyze the logic circuit shown below to determine the logic expression for the output F 1. Using the logic

expression, derive the circuit’s truth table.

Equation → Truth Table Technique

a) Working from the inputs to the output, write the

cumulating logic expression at the output of

each gate concluding with the expression for

the circuit’s output

Step (a)

X

Y

Y X

Y X

Z

Y X

Y X Z

Y X

T

he Process

a) Add test-points at the output of every gate.

b) Add a column to the truth table for every

test-point.

c) Working from the inputs to the output,

complete the truth table for each test-point,

ultimately ending at the circuit’s output.

d) From the completed truth table, identify the

Minterms from the truth table anywhere the

output is one

e) Using the extracted Minterms, write the

Sum-Of-Products logic expression.

Equation → Truth Table Technique

b) Using the circuit’s output logic expression,

derive the circuit’s truth table.

Y X Z

b) Add a column to the truth table for every point.

test-c) Working from the inputs to the output, complete the truth table for each test-point, ultimately ending at the circuit’s output.

d) From the completed truth table, identify the Minterms from the truth table anywhere the output is one

e) Using the extracted Minterms, write the Of-Products logic expression.

Analyze the logic circuit shown below to

determine the logic expression for the output F 2.

Using the logic expression, derive the circuit’s

truth table.

Example #2: Circuit Analysis

B A

C B

A ABC BC

C B C B A

C B

° After finding the circuit inputs and outputs, you can

come up with either an expression or a truth table to describe what the circuit does

° You can easily convert between expressions and truth

tables

° The analysis and synthesis tools presented are

sometimes based on the fundamental concepts of

Boolean algebra

Find the circuit’s inputs and outputs

Find a Boolean expression for the circuit

Find a truth table for the circuit

° Design of a circuit starts with specification and

ends up with a logic diagram.

° Analysis for a combinational circuit consists of

determining the function that the circuit

implements with:

A set of Boolean functions or

A truth table, together with a possible

explanation of the operation of the circuit.

We can perform the analysis by manually

finding the Boolean equations or truth table.

oThe first step in the analysis is to make sure that

the given circuit is combinational and not

sequential (i.e no feedback or storage elements).

Digital Design Overview

° Design digital circuit from specification

° Digital inputs and outputs known

• Need to determine logic that can transform data

° Start in truth table form

° Create K-map for each output based on function of

inputs

° Determine minimized sum-of-product representation

° Draw circuit diagram

Design Procedure (Mano)

Design a circuit from a specification.

1 Determine number of required inputs and

outputs.

2 Derive truth table

3 Obtain simplified Boolean functions

4 Draw logic diagram and verify correctness

A 0 0 0 0 1 1 1 1

B 0 0 1 1 0 0 1 1

C 0 1 0 1 0 1 0 1

R 0 0 0 0 0 0 0 1

S 0 1 1 1 1 1 1 1

S = A + B + C

R = ABC

Combinational logic design

° Use multiple representations of logic functions

° Use graphical representation to assist in

simplification of function

° Use concept of “don’t care” conditions.

° Example - encoding BCD to seven segment display.

° Similar to approach used by designers in the field

BCD to Seven Segment Display

° Used to display binary coded decimal (BCD)

numbers using seven illuminated segments.

° BCD uses 0’s and 1’s to represent decimal digits 0 -

9 Need four bits to represent required 10 digits.

° Binary coded decimal (BCD) represents each

decimal digit with four bits

a

bc

ge

required inputs and outputs.

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

BCD to seven segment display

g

e

df

° List the segments that should be illuminated for

specification.

1 Determine number of

required inputs and outputs.

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

BCD to seven segment display

.01111

100

19

.111110

00

18

.001111

11

07

.110110

10

02

.001101

00

01

.111110

00

00

.edcba

zyx

wDec

° Derive the truth table for the circuit.

° Each output column in one circuit.

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

BCD to seven segment display

1 0

10

1 1

11

yzwx

10110100

1011

0100

For segment “a” :

Note: Have only filled in ten 

squares, corresponding to the ten numerical digits we wish to represent

° Find minimal sum-of-products representation for

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

Don’t care conditions (BCD display)

1 0

10

X X X X

1 1

11

yzwx

10110100

1011

0100

For segment “a” :

Put in “X” (don’t care), and interpret as either 1 or 0 as 

desired …

° Fill in don’t cares for undefined outputs.

• Note that these combinations of inputs should never happen

° Leads to a reduced implementation Design a circuit from a specification.

1 Determine number of

required inputs and outputs.

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

Don’t care conditions (BCD display)

1 1 X X

X X X X

1 1

11

yzwx

10110100

1011

0100

° Circle biggest group of 1’s and Don’t Cares.

° Leads to a reduced implementation

Design a circuit from a

specification.

1 Determine number of

required inputs and outputs.

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

1 1 X X

X X X X

1 1

11

yzwx

10110100

1011

0100

° Circle biggest group of 1’s and Don’t Cares.

° Leads to a reduced implementation

Design a circuit from a

specification.

1 Determine number of

required inputs and outputs.

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

Don’t care conditions (BCD display)

For segment “a” :

z x F

1 0

10

1 1 X X

X X X X

1 1

11

yzwx

0100

xz F

1 0

10

1 1 X X

X X X X

1 1

11

yzwx

10110100

1011

0100

° Circle biggest group of 1’s and Don’t Cares.

° All 1’s should be covered by at least one

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

Don’t care conditions (BCD display)

For segment “a” :

xz z

x w

y

F

1 0

10

1 1 X X

X X X X

1 1

11

yzwx

10110100

1011

0100

° Put all the terms together

° Generate the circuit

Design a circuit from a

specification.

1 Determine number of

required inputs and outputs.

2 Derive truth table

3 Obtain simplified Boolean

functions

4 Draw logic diagram and

verify correctness

BCD to seven segment display

.01111

100

19

.111110

00

18

.001111

11

07

.110110

10

02

.001101

00

01

.111110

00

00

.edc

ba

zyx

wDec

° Derive the truth table for the circuit.

° Each output column in one circuit.

Inputs Outputs

BCD to seven segment display

1 1

01

1 0

11

yzwx

10110100

1011

0100

° Find minimal sum-of-products representation for

each output

1 1

01

1 0

11

yzwx

10110100

1011

0100

X X X X

X X

Fb1 = W

1 1

01

1 0

11

yzwx

10110100

1011

0100

X X X X

X X

Fb2 =Y Z

1 1

01

1 0

11

yzwx

10110100

1011

0100

X X X X

X X

Fb3 =W X

1 1

01

1 0

11

yzwx

10110100

1011

0100

X X X X

X X

Fb4 =YZ

1 1

01

1 0

11

yzwx

10110100

1011

0100

° Design of a circuit starts with specification and

ends up with a logic diagram.

° Analysis for a combinational circuit consists of

determining the function that the circuit

implements with:

A set of Boolean functions or

A truth table, together with a possible

explanation of the operation of the circuit.

We can perform the analysis by manually

finding the Boolean equations or truth table.

oThe first step in the analysis is to make sure that

the given circuit is combinational and not

sequential (i.e no feedback or storage elements).

Multilevel Logic Circuits

Boolean expressions of a few variables.

two-level logic circuit may result in fan-in

problems.

Fan-in refers to the number of inputs to a logic gate

technology used to implement the logic circuit.

Standard TTL and CMOS chips Field Programmable Gate Array (FPGA) Complex Programmable Logic Device (CPLD)

Multilevel Logic Circuits

Example:

Realize the following Boolean expression using only

2-input AND gates and 2-2-input OR gates.

F(A,B,C) = m(0, 5, 6)

Multilevel Logic Circuits

equivalent two-level logic circuit.

Reduced (silicon) area Decreased cost

Fewer literals results in fewer interconnecting wires

equivalent two-level logic circuit.

Each additional level adds to the propagation delay

Decreased speed

Multilevel Logic Circuits

Objectives:

1 Design logic circuits that meet the fan-in

requirements of the chosen technology.

2 Design a minimum-cost logic circuit.

Multilevel Logic Circuits

Two techniques that can be used to realize multilevel

logic circuits:

1 Factoring

2 Functional Decomposition

Factoring

Example:

Realize a logic circuit that has a maximum fan-in of two

for the following Boolean expression.

F(A G) = ACF' + ADEF' + BCG + BDEG

Example:

Design the minimum-cost logic circuit that implements

the following Boolean expressions.

F 1 (A,B,C,D) = m(1,2,3,7,11,15)

F 2 (A,B,C,D) = M(0,1,2,3,4,8,12)

Functional Decomposition

Example:

Design a minimum-cost logic circuit to implement the

following Boolean expression.

F(A,B,C,D) = A'BC + AB'C + ABD + A'B'D

NAND and NOR Circuits

As with two-level circuits, multilevel circuits can be

realized using NAND or NOR gates only.

NAND and NOR Circuits

NAND and NOR Circuits

NAND and NOR Circuits

NAND and NOR Circuits

Create a truth table or equations, whichever is

most natural for the given problem, to describe

the desired behavior of the combinational logic.

as a

gate-based

circuit

For each output, create a circuit corresponding

to the output’s equation (Sharing gates among multiple outputs is OK optionally.)

° Analysis and Design Procedures (Combinational)

° Important concept – analyze digital circuits

• Given a circuit

- Create a truth table

- Create a minimized circuit

° Approaches

• Boolean expression approach

• Truth table approach

° Both results can then be minimized using K-maps (Leads to minimized

hardware) -° Need to formulate circuits from problem descriptions

1.Determine number of inputs and outputs

2.Determine truth table format

3.Determine K-map

4.Determine minimal SOP

o There may be multiple outputs per design