1 Chapter 2: Combinational Logic Circuits EKT 121 / 4 DIGITAL ELECTRONICS 1

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Chapter 2: Chapter 2: Combinational Logic CircuitsCombinational Logic Circuits

EKT 121 / 4DIGITAL

ELECTRONICS 1

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2.1 Combinational Logic Analysis

Basic Combinational Logic Circuits

Implementing Combinational Logic

The Universal Property of NAND and NOR gates

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The AND-OR Logic

4

The AND-OR-Invert Logic

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The Exclusive-OR logic

How about Ex-NOR logic???

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X-Nor EquationX=ABAB= AB AB = AB AB = AB AB

= AB AB= AB AB =A AABABBB=ABAB

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Exclusive-OR Gate

A ⊕ B= A B A B

a) Gate Symbol, Boolean a) Gate Symbol, Boolean Equation Equation

& Truth Table & Truth Table

b) Timing Diagram b) Timing Diagram

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Ex-OR and Ex-NOR

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Implementing Combinational Logic

From Boolean Expression to Logic Circuit

From Truth Table to Logic Circuit

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The NAND and NOR gates as Universal Logic Elements

Both the NAND and NOR gates are universal gates because they can each represent the NOT, AND, OR and NOR functions.

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The Universal NAND gate

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The Universal NOR gate

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Equivalence Symbols

NAND

NORPlease sketch !!

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2.2 Functions of Combinational Logic Basic Adders

Parallel Binary Adder

Ripple Carry Adder

Comparators

Decoders

Encoders

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2.2 Functions of Combinational Logic Code Converters

Multiplexers (Data Selectors)

Demultiplexers

Parity Generators/ Checkers

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Decoders & Encoders

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Encoder

Encoder converts information such as decimal number or an alphabetical character into some binary coded form

Encoder is usually used for:•Data representation•Data security•Data compression

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EncoderExample: 8-to-3 Binary Encoder

Only one of the inputs can be 1.

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Design Exercise

Write the equations for Y2, Y1 and Y0.

Draw the logic circuit to implement this 8-to-3 encoder.

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The decimal-to-BCD Encoder

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DecoderA decoder is a circuit that creates an output based on the binary states of a given input

Selects 1 of several output lines, based on a coded input signal.

Do the opposite of encoder

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DecoderExample: 3 to 8 Binary Decoder

The inputs are treated as a binary number and the output selected is made active.

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A 4-line-to-16-line Decoder

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The BCD-to-decimal Decoder

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The BCD to 7-segment Decoder Accepts BCD

code on inputs and provide outputs to drive 7-segment display devices to produce a decimal readout.

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BCD to 7-Segment Decoder

A BCD to 7-segment decoder

has 4-bit BCD input and

the seven segment display

code as its output: In minimizing the circuits

for the segment outputs all

non-decimal input combinations

(1010, 1011, 1100,1101, 1110,

1111) are taken as don’t-cares

/Bl D C B A a b c d e f g 0 x x x x 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

-- d

on’t

car

e in

pu

ts -

-

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Figure 4–47 Seven-segment display format showing arrangement of segments.

Thomas L. FloydDigital Fundamentals, 9e

Copyright ©2006 by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458All rights reserved.

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Figure 4–48 Display of decimal digits with a 7-segment device.




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Figure 4–49 Arrangements of 7-segment LED displays.




Active-low Active-high

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Figure 4–50 Block diagram of 7-segment logic and display.




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Design Example

From the truth table, an SOP or POS can be written for each segment.

Derive an SOP for segment a.

Minimize the SOP using a K-map.

Draw the logic circuit to implement segment a of the 7-segment display.

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Figure 4–51 Karnaugh map minimization of the segment-a logic expression.




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Figure 4–52 The minimum logic implementation for segment a of the 7-segment display.




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BCD to 7-segment DecoderApplication Example:

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Lab 3

Use of 74S90 IC, a BCD counter – as inputs to the BCD to 7-segment decoder.

Part B Prepare the design before coming to lab! For unused input states, the output is either a ‘0’

or ‘X’. TT, K-map and circuit.

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Decoders & Encoders

- A Second Look -

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Convert numbers to be displayed

BCD to 7-segment display decoder

Hex to 7-segment display decoder

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7-Segment Display Configuration

Common Anode Requires logic 0 to

light up (ON) a segment.

Common Cathode Requires logic 1 to

light up (ON) a segment.

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3-5 DECODERS

DECODER

n m

m ≤ 2n

Converts binary information, from n coded inputs to a maximum of 2n outputs

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3-5 DECODERS

Functional Specs:

To generate the 2n (or less) minterms of input variables.

For each input, 7 outputs are equal to 0 and only 1 equals to 1. The output that is 1 represents the

minterm equivalent of the input number.

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Truth Table for 3-to-8-Line Decoder

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3-to-8-Line Decoder

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A 2-to-4-Line Decoder

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Decoder Expansion

When a certain decoder size is needed, but only smaller number of sizes is available.

Combine 2 or more decoders in a hierarchy, i.e. cascade the smaller decoders to form a larger decoder size.

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Example: A 3-to-8 Decoder Constructed with Two 2-to-4 Decoders

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… Its Operation The MSB input, A2, functions:

As enable, EN, of one decoder, and As its complement, N(EN) to the other decoder. When A2=0,

Top decoder enabled Generates minterms D0 to D3.

Lower decoder disabled Outputs equal to 0.

When A2=1, Top decoder disabled Outputs equal to 0. Lower decoder enabled Generates

minterms D4 to D7.

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… Enable Input

Very useful and convenient way to interconnect 2 or more functional blocks.

For the purpose of expanding digital functions into: Similar functions with more inputs

and outputs.

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Exercise

Construct a 6-to-64-line decoder, using:

Four 4-to-16-line decoders and one 2-to-4-line decoder.

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Combinational Circuit Implementation of Decoders

Binary Adder

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Implementing a Binary Adder Using a Decoder

S(X,Y,Z) = m (1, 2, 4, 7)

C(X,Y,Z) = m (3, 5, 6, 7)

3 inputs and 8 minterms

Use a 3-to-8 decoder.

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Implementing a Binary Adder Using a 3-to-8 Decoder

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3-6 ENCODERS

ENCODER

m n

m ≤ 2n

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3-6 ENCODERS

Inverse operation of decoder. The output lines generate the binary code

corresponding to the input value. Assume only 1 input has the value of 1 at

any given time. Example:

Octal-to-binary encoder.

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Truth Table for Octal-to-Binary Encoder

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Priority Encoder

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What about when 2 inputs are 1 at the same time?

Use “Priority Encoder” To ensure only one input is encoded.

e.g.

D3 = D6 = 1

Output = 110

(D6 has a higher priority than D3)

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Priority Encoder …

If 2 or more inputs are equal to 1 at the same time, the input having the highest priority is the one encoded.

Example: 4-input Priority Encoder Use condensed TT (5 rows to represent 16 rows). V (valid) =1 when 1 or more input is equal to 1.

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Condensed Truth Table of Priority Encoder

X @ output => don’t careX @ input => product term that is not minterm

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Maps for Priority Encoder

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Logic Diagram of a 4-Input Priority Encoder

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Functions of Combinational Logic

Adder Encoder Comparator DecoderMultiplexer DemultiplexerParity Generator Parity CheckerCode Converters

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ADDERS

Half Adder

Full Adder

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Half Adder Adding two single-bit binary values, X, Y

produces a sum S bit

and a carry out C-out bit. This operation is called half addition and the circuit to realize it is called a half adder.

X0011

Y0101

S0110

C-out 0 0 0 1

Half Adder Truth Table:

Inputs Outputs

S(X,Y) = (1,2)S = X’Y + XY’S = X Y

C-out(x, y, C-in) = (3)C-out = XY

HalfAdder

X

Y

SC-OUT

Block diagram/ Symbol/ Black-box view

X

YSum S

C-out

Circuit

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Full Adder Adding two single-bit binary values,

X, Y with a carry input bit C-in

produces a sum bit S

and a carry out C-out bit. Full Adder Truth Table

S = X’Y’(C-in) + X’Y(C-in)’ + XY’(C-in)’ + XY(C-in)S = X Y (C-in)

C-out = XY + X(C-in) + Y(C-in)

X00001111

Y00110011

S01101001

C-out 0 0 0 1 0 1 1 1

C-in 0 1 0 1 0 1 0 1

S(X,Y, C-in) = (1,2,4,7)C-out(x, y, C-in) = (3,5,6,7)

Inputs Outputs

C-in

X

0 1

00 01 11 10

Y

C-in

XY

0

1

2

3

6

7

4

5

1

11 1

Carry C-out

Sum S

C-in

X

0 1

00 01 11 10

Y

C-in

XY

0

1

2

3

6

7

4

5

1

1 1

1

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Full Adder

Y Y’Y

X X’X

C-in C-in’C-in

Full Adder

X Y

S

C-inC-out

Full Adder Circuit Using AND-OR:

Block diagram

XY

YC-in

C-outXC-in

X

X

Y

C-in

Y

C-in

X’Y’C-in

XY’C-in’

Sum SX’YC-in’

XYC-in

X’

X’

X

X

Y’

Y

Y

C-in

Y

C-in’

C-in’

C-in’Circuit

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Full Adder

Full Adder

X Y

S

C-inC-out XY

YC-in

C-outXC-in

X

X

Y

C-in

Y

C-in

Sum S

X

Y

C-in

Full Adder Circuit Using XOR:

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HA`and `FA symbols

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Arrangement of 2 half-adders to form a full-adder

CASCADE

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Exercise: Determine the output values

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2-Bit Parallel Adder

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Determine the sum of this 3-bit parallel Adder

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4-bit Parallel Adder

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Bigger Adders

8-bit using two 4-bit Adders

16-bit using four4-bit Adders

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Larger Adders Example: 16-bit adder using 4, 4-bit adders Adds two 16-bit inputs X (bits X0 to X15), Y (bits Y0 to Y15)

producing a 16-bit Sum S (bits S0 to S15)

and a carry out C16 from most significant position.

Propagation delay for 16-bit adder = 4 x propagation delay of 4-bit adder = 4 x 2 n x8or 32 gate delays

4-bit Adder

C-inC-out 4-bit Adder

C-inC-out C0 =0 4-bit Adder

C-inC-out 4-bit Adder

C-inC-outC4C8C12C16

Data inputs to be added X (X0 to X15) , Y (Y0-Y15)

Sum output S (S0 to S15)

Y3Y2Y1Y0X3X2X1X0Y3Y2Y1Y0X3X2X1X0Y3Y2Y1Y0X3X2X1X0Y3Y2Y1Y0X3X2X1X0

S3 S2 S1 S0S3 S2 S1 S0S3 S2 S1 S0S3 S2 S1 S0

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A 4-bit parallel ripple carry adder




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n-bit Carry Ripple Adders An n-bit adder used to add two n-bit binary numbers can built by connecting

in series n full adders. Each full adder represents a bit position j (from 0 to n-1). Each carry out C-out from a full adder at position j is connected to

the carry in C-in of the full adder at the higher position j+1. The output of a full adder at position j is given by:

Sj = Xj Yj Cj

Cj+1 = Xj . Yj + Xj . Cj + Y . Cj

In the expression of the sum Cj must be generated by the full adder at the lower position j-1.

The propagation delay in each full adder to produce the carry is equal to two gate delays = 2

Since the generation of the sum requires the propagation of the carry from the lowest position to the highest position , the total propagation delay of the adder is approximately:

Total Propagation delay = 2 n

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4-bit Ripple Carry AdderAdds two 4-bit numbers: X = X3 X2 X1 X0 Y = Y3 Y2 Y1 Y0 producing the sum S = S3 S2 S1 S0 , C-out = C4 from the most significant position j=3

Total Propagation delay = 2 n8

or 8 gate delays

Full Adder

X1 Y1

S1

C-inC-out Full Adder

X0 Y0

S0

C-inC-out C0 =0 Full Adder

X2 Y2

S2

C-inC-out Full Adder

X3 Y3

S3

C-inC-outC1C2C3C4

Data inputs to be added

Sum output

4-bit Adder

X3X2X1X0

S3 S2 S1 S0

C-inC-outC4

Y3Y2Y1Y0

C0 =0

Inputs to be added

Sum Output

Cascade 4 full adders to get a 4-bit full adder

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Other types of Parallel Adders Carry Look-Ahead

Carry-Select Adder

Carry-Save Adder

Etc.

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1’s Complement and 2’s Complement Hardware

The 2’s complement of a binary number is formed by adding 1 to the 1’s complement.

2’s complement = (1’s complement) + 1

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How to use an adder as a Subtractor?

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Building a Subtractor using an Adder and 2’s Complement (for positive numbers)

A – B = A + (-B) = A + B + 1

To perform subtraction using an adder, we invert B, and use Cin =1.

2’s complement

Try drawing this diagram…

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Other types of Parallel Adders Carry Look-Ahead

Carry-Select Adder

Carry-Save Adder

Etc.

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ADDERS

– A Second Look -

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Adders and its Applications

Subtractor

Multipliers

Dividers

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ARITHMETIC CIRCUITS

Is a combinational circuit that performs arithmetic operations, e.g.

Addition Subtraction Multiplication Division

with numbers in binary form.

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Half Adder

Generates the sum of 2 binary digits.

HALFADDER

X

Y

Sum = X Y

Cy = X.Y

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Truth Table of Half Adder

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Logic Diagram of Half Adder

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Full Adder

Forms the arithmetic sum of 3 input bits.

FULL ADDER

X

Y

Sum = X Y Z

Cout = X.Y + Z (X Y)Cin

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Truth Table of Full Adder

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K-Maps for Full Adder

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Full Adder

A Full Adder can also be implemented using 2 HALF ADDERS and one OR gate.

Cascade two half adders

(Array method) Design “Smart”

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Full Adder (Array Method)

Design “Smart”

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1’s Complement and 2’s Complement Hardware

The 2’s complement of a binary number is formed by adding 1 to the 1’s complement.

2’s complement = (1’s complement) + 1

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How to use an adder as a Subtractor?

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Building a Subtractor using an Adder and 2’s Complement (for positive numbers)

A – B = A + (-B) = A + B + 1

To perform subtraction using an adder, we invert B, and use Cin =1.

2’s complement

Try drawing this diagram…

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Design Exercise

2-bits x 2-bits Multiplier

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Terms

2

X 3

--------------

6

Mutiplicand

Multiplier

Product

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Multiplication in binary form?1. Rewrite the multiplication in binary

form.

1. Sketch the black box view.

1. The multiplier multiplies two __?__ bits numbers.

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2-bits x 2-bits Multiplier Design

Two techniques:

Using the standard K-Map

Using Arrays (cascaded approach)

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Method A: Using the K-Map Technique Sketch the bLack box. Sketch the Truth Table for a 2-bit “multiplier” and 2-

bit “multiplicand”. Input (Multiplier) = A1 and A0 Input (Multiplicand) = B1 and B0 Output (4-bits) = S3, S2, S1 and S0 or S[3..0]

Using K-Maps, obtain the boolean expression for each output.

Sketch the schematic diagram.

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Method B: Using the Array (Cascaded) Technique

Create the 2x2 multiplier using Full ADDERS.

Design “Smart”

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… tHE cONcEpt

A1 A0B1 B0x

A1B0 A0B0A1B1 A0B1+

S0S1S2S3

CC

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A 2-Bit by 2-Bit Binary Multiplier

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AND computes A0 B0

Half adder computes sum. Will need FA for larger multiplier.

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The 4-bits x 4-bits Multiplier

Using Array 2x2

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Basic Idea of a Larger Multiplier (4-bits by 3-bits)

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Multiplier Product

The product of m-bit x n-bit numbers is an (m+n)-bit number.

=> The product of two 4-bit numbers is an 8-bit number.

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How about this one?

13

X 11

--------------

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Mutiplicand

Multiplier

Product

110

1

1

1

1

(143) Product1110001

1011

0000

1011

011

(11) multiplier101X

(13) multiplicand011

Partial products

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S0S1S2S3S4S5S6S7

A0B3A1B3A2B3A3B3

A0B2A1B2A2B2A3B2

A0B1A1B1A2B1A3B1

A0B0A1B0A2B0A3B0

B0B1B2B3

A0A1A2A3

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From the previous slide:

1. The multiplier multiplies two __?__ bits numbers.

1. Sketch the black box view.

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Design a 4-bits x 4-bits multiplier using the Array (cascaded) technique, by utilizing:

The 2-bits x 2-bits Multiplier and full adder designed earlier.

Hints : Look back at the concept of 2x2 multiplier. Take the same step.

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4x4 Combinational Multiplier

Note use of parallel carry-outs to form higher order sums

12 Adders, if full adders, this is 6 gates each = 72 gates

16 gates form the partial products

total = 88 gates!

A 0 B 0 A 1 B 0 A 0 B 1 A 0 B 2 A 1 B 1 A 2 B 0 A 0 B 3 A 1 B 2 A 2 B 1 A 3 B 0 A 1 B 3 A 2 B 2 A 3 B 1 A 2 B 3 A 3 B 2 A 3 B 3

HA

S 0 S 1

HA

F A

F A

S 3

F A

F A

S 4

HA

F A

S 2

F A

F A

S 5

F A

S 6

HA

S 7

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Array Multiplier

4 x 4 array of building blocks

1 building block

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4x4 Multiplier

(your design)

Tenth & UnitSegmentDecoder

7447BCD to

7Seg

7447BCD to

7Seg a - g

a - g

A0A1A2A3

B0B1B2B3

Pin Configuration to input(use flex switch 1-8)

Pin Configuration to output(use flex digit 1 & 2 )

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118

Comparator

Equality Comparator

Inequality Comparator

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Equality Comparator

A B A=B

0 0 1

0 1 0

1 0 0

1 1 1

• To compare two binary strings (or binary words) to determine if they are exactly equal.

•Truth table for a comparator:

What is the Boolean expression for this truth table?

Can you draw the circuit for basic comparator?

120

Equality Comparator

A B A=B

0 0 1

0 1 0

1 0 0

1 1 1

• Comparing two binary strings (or binary words) to determine if they are exactly equal.

•Truth table for a comparator:

A=B is same as output for Ex-NOR gate

1-bit comparator

121

Basic Equality Comparator Operation

122

1-bit Comparator with Inequality outputs To compare 2 binary numbers, each 1-bit

wide. The result of the comparison are single bit

outputs that indicate whether the numbers are equal to each other, or if one number is greater than or less than another. A=B ? A<B? A>B?

123

Design Exercise

For the comparator explained in the previous slide ,

Derive its truth table. Minimize the output equations using K-maps. Draw the logic circuit.

124

2-bit Comparator

Sketch the black box view/ symbol

Produce the truth table

Use K-maps

Sketch the circuit

125

Design Exercise

Repeat the previous design exercise for a 4-bit comparator with inequality outputs.

126

4-bit Comparator with Inequality Indication

127

8- bit Comparator To design a comparator to evaluate two 4 bit numbers, we need 4

Ex-NORs and a 4-input AND gate 8 bit magnitude comparator (cascade two 4-bit comparators):

128

Code Converters

129

Code Converters A device that converts data in one type of

code and produces an output expressed in another type.

Example, BCD to binary, BCD to 7-segment display, Gray-to-binary code and binary-to-Gray code.

Still remember…………………………….? Binary code vs Gray Code.

To convert binary to Gray code or Gray code to binary, we use X-OR gates. How???

130

Four-bit binary-to-Gray conversion logic




131

Four-bit Gray-to-binary conversion logic




132

Multiplexers& Demultiplexers

MUX & DEMUX

133

Mutiplexer (MUX)

A circuit that selects binary information from one of many input lines, and directs the information to a single output line.

The selection of a particular input line is controlled by a set of input variables, called “selection inputs”.

Normally, there are 2n input lines and n selection inputs.

Output lines = ?

134

2-to-1 MultiplexerData selectorSELECT input code determines which input is transmitted to output Z.

A 2 input multiplexer

135

4-to-1 Multiplexer

A 4 input multiplexer

Write the output equation for Z

Verify the circuit above

136

8-to-1 Multiplexer Larger multiplexers can be constructed from smaller ones.

An 8-to-1 multiplexer can be constructed by cascading smaller multiplexers as shown:

16-to-1 MUX: 74150

137

“Smart” Design Exercise

Design a 16:1 MUX by cascading 4:1 MUXes

138

MultiplexerApplication Example:

74157- consists of four separate 2-input multiplexers.

139

Demutiplexer (DeMUX)

Opposite function of the multiplexer.

Information received from a single line is transmitted to one of 2n possible output lines.

The output line chosen is controlled by the n selection inputs.

Black box view?

140

DemultiplexerData input is transmitted to only one of the outputs as determined by the select input code.

1-line-to-8-line multiplexer

141

1-to-2 DeMUX

0D1

D00

Y1Y0Select

Black box view

Output equations

Logic circuit

142

1-to-4 DeMUX

143

Mux-Demux Application: Example

•This enables sharing a single communication line among a number of devices.•At any time, only one source and one destination can use the communication line.

144

Multiplexer & Demultiplexer

- A Second Look -

145

3-7 Multiplexers (Pemultipleks)

Selects binary information from one of many input lines and directs the information to a single output line.

The selection of an input line is controlled by a set of variables, i.e. the “selection” inputs.

Also called “Data Selector”.

146

Mutiplexer (MUX)

147

4-to-1-Line Multiplexer

148


149


150

… Multiplexer

Also called “MUX”. Resembles a decoder circuit. 2n input lines. n selection inputs (SEL0, …, SELn). Can be constructed from:

Decoders. Transmission gates.

151

… Multiplexer

May have an Enable, EN, input, to control its operation. EN = Inactive Outputs Disabled. EN = Active Normal Operation.

EN is useful when using 2 or more MUXes to obtain a bigger MUX (i.e. more inputs). Combine MUXes in parallel with the same SEL

and EN lines.

152

Quadruple 2-to-1-Line Multiplexer…

153

…Quadruple 2-to-1-Line Multiplexer Has ____ number of MUXes. YO can be selected from either A0 or B0.

Y1- from either A1 or B1, etc.

EN=1, S=0, all A inputs are passsed to the outputs. S=1, all B inputs are passed to the outputs.

EN=0, all outputs are 0.

154

Implementing a Boolean Function with a Multiplexer … A multiplexer is basically a decoder that

includes the OR gate within the block. To implement a Boolean function of n

variables with a mux having n selection inputs and 2n data inputs, one for each minterm. The minterms are generated in a mux by the

circuit associated with the selection inputs. Individual minterms can be selected by the data

inputs.

155

… Implementing a Boolean Function with a Multiplexer

A more efficient way To implement a Boolean function of n

variables with a mux having only n-1 selection inputs and 2n-1 data inputs.

Example 1:

F (X,Y,Z) = m (1, 2, 6, 7)

156

Example 1

157


General procedure:

1. Produce Truth Table for Boolean function.

2. The first n-1 variables are applied to the selection inputs of the mux.

3. The remaining single variable of the function is used for the data input.

158


1. For each combination of the selection variables, we evaluate the output as a function of the last variable, i.e. a 0, 1, the variable or its complement.

2. These values are then applied to the data inputs in the proper order.

159

Example 2: Implementing a 4-Input Function with a Multiplexer

F (A, B, C, D) = m (1, 3, 4, 11, 12, 13, 14, 15)

Implement the function with a mux having 3-selection inputs.

160

… Example 2

161

Demultiplexer Performs the inverse function of a

multiplexer. Receives information from a single line and

transmits it to one of possible 2n possible output lines.

The selection is is controlled by the bit combination of n selection lines.

A demultiplexer is identical to a 2-to-4-line decoder with enable input. A decoder with enable input =

A Decoder/Demultiplexer.

162

1-to-4-Line Demultiplexer

163

Design Preparation for Lab 4

Extra Exercise

A 16:1 MUX using cascaded 4:1 MUXes

1. 4:1 MUX

2. 1:5 DeMUX

Truth Table K-maps & minimized

equations Circuit diagram

164



165

Error Detection & Correction Codes

PARITY METHOD:

1. Even Parity

2. Odd Parity

166

Parity Bit

Is an extra bit included with the data bits.

To detect errors in data communication & processing.

167

Error-Detection Codes

Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors.

A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.

A code word has even parity if the number of 1’s in the code word is even.

A code word has odd parity if the number of 1’s in the code word is odd.

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4-Bit Parity Code Example Fill in the even and odd parity bits:

The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3-bit data.

Even Parity Odd Parity Message - Parity Message - Parity

000 - 000 - 001 - 001 - 010 - 010 - 011 - 011 - 100 - 100 - 101 - 101 - 110 - 110 - 111 - 111 -

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Parity Generator & Parity Checker

Odd Parity

Even Parity

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Error-Detection

A parity bit is a scheme for detecting errors during transmission of binary info.

A parity bit is an extra bit included with the binary message to make the number of 1’s either odd or even.

The message, including the parity bit, is transmitted and then checked at the receiving end for errors.

An error is detected if the checked parity does not correspond to the one transmitted.

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Error-Detection

The circuit that generates the parity bit in the transmitter is a parity generator.

The circuit that checks the parity bit in the receiver is a parity checker.

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Error-DetectionParity generator truth table*For odd parity, the bit P is generated so as to make the number

of 1’s odd (including P)

X Y Z P0 0 0 10 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0

P

X

Y

Z

Odd Parity Generator

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* The three-bit message (X, Y, Z) and parity bit (P) are transmitted to their destination, where they are applied to a parity checker circuit. An error occurs during transmission if the parity of the four bits is even, since the binary info transmitted was originally odd. The output C of the parity checker should be a 1 when an error occurs, i.e. when the number of 1’s in the four inputs is even.

X Y Z P C 0 0 0 0 10 0 0 1 00 0 1 0 00 0 1 1 10 1 0 0 00 1 0 1 10 1 1 0 10 1 1 1 01 0 0 0 01 0 0 1 11 0 1 0 11 0 1 1 01 1 0 0 11 1 0 1 01 1 1 0 0 1 1 1 1 1

Odd Parity Checker

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• Parity is used in digital circuits to check for errors in transmission.

• In Four bit transmission a parity bit is added to make the fifth bit.

• In a eight bit transmission a parity bit is added to made a ninth bit.

• And so on.

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ODD Parity

• In ODD parity when we add the bits together disregarding weight we get or want to get an odd number.

0000 is a four bit message add a parity bit to make it odd

10000 Odd parity is satisfied

00011001 is an eight bit message add a parity bit to make it odd

000011001 Odd parity is satisfied

Parity bit

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EVEN parity

• In EVEN parity when we add the bits together disregarding weight we get or want to get an even number.

0000 is a four bit message add a parity bit to make it even

00000 Even parity is satisfied

00011001 is an eight bit message add a parity bit to make it even

100011001 Even parity is satisfied

Parity bit

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How to generate a parity bit

Use exclusive ORs and Exclusive NORs

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How to make a parity checker

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Odd or even parity~ 0 out of a parity checker means the parity checks and all is ok

~ 1 out of a parity checker means there is an error

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Parity Generator/Checker

- A Second Look -

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Tip to generate/check “parity”

“The sum (disregarding carries) of an even number of 1’s is always 0, and the sum of an odd number of 1’s is always 1.”

To determine if a given code has even or odd parity, all the bits in the code must be summed. Summation can be done using Ex-OR gate

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Parity Checker Circuit

When the no. of inputs is even Output, X, is 0

When the no. of inputs is odd Output, X, is 1

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The 9-bit parity generator/checker.

Input is 8-bits of data and 1 parity bit

When there is an even no. of 1’s at the inputs, the Σ Even is high while the Σ Odd is low.

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Parity Checker Function

When used as an EVEN Parity Checker,

If a parity error occurs, the Σ Even is low while the Σ Odd is high.

When used as an ODD Parity Checker,

If a parity error occurs, the Σ Odd is low while the Σ Even is high.

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Parity Generator Function

When used as an EVEN Parity Generator,

The parity bit is taken at the Σ Odd output. It is 0 if there is an even no. of 1’s, and is 1 if there

is an odd no.

When used as an ODD Parity Generator,

The parity bit is taken at the Σ Even output. It is 0 if there is an odd no. of 1’s, and is 1 if there

is an even no.

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Design Exercise

Derive the gate-level schematic circuit for the 9-bit Parity Generator/Checker explained above.

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Extra Design Exercise - Decoder

1. The 4-line-to-16-line Decoder (Active low outputs)

1. The BCD to Decimal Decoder

1. The BCD to 7-segment Decoder

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Extra Design Exercise - Encoder1. 16-line-to-4-line Encoder using the 8-line-to-

3-line encoder in cascade

1. The Decimal to BCD Encoder

1. The Keypad Encoder, as in the next slide

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Keypad Encoder The keypad consists

of the keys 0 to 10. When one of the keys

is pressed, the decimal digit is encoded to the corresponding BCD code.

When a key is depressed, the line is connected to ground, i.e. low input.

The “0” key is not connected as the BCD output is 0 when none of the other keys is depressed.

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Design Exercise

Use truth table and K-map to design:

1. A 2-to-1 MUX

1. A 1-to-2 DeMUX

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Extra Design Exercise – MUX and DeMUXUsing the circuits from the previous slide,

design by using the cascade/array/”smart design” technique:

1. A 4:1 MUX using 2:1 MUXes



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1. A 1:4 DeMUX using 1:2 DeMUX



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1 Chapter 2: Combinational Logic Circuits EKT 121 / 4 DIGITAL ELECTRONICS 1