ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS

DESIGN

Lecture 7

Dr. Shi

Dept. of Electrical and Computer Engineering

BINARY ADDERS

Overview

Addition of binary data is fundamental Need to determine hardware implementation

Represent inputs and outputs Inputs: single bit values, carry in Outputs: Sum, Carry

Hardware features Create a single-bit adder and chain together

Dealing with overflow What happens if numbers are too big?

4-Bit Adder

Think top-down: How to break a problem into pieces?

Multiple-bit Addition

A3 A2 A1 A0

0 1 0 1A 0 1 1 1B3 B2 B1 B0

B

0 1 0 10 1 1 1

A

B

0

1

0

1

1

1

1

Ai

+Bi

Ci

Si

Ci+1

Consider single-bit adder for each bit position.

Each bit position creates a sum and carry

Full Adder

Block Diagram

Single-bit full adder, building block of large adders

Common piece of computer hardware

Full Adder

0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

Ci Ai Bi Si Ci+1

1 1

1 1

Ci

AiBi

00 01 11 10

0

1

Si

Full adder includes carry in Ci

Notice interesting pattern in Karnaugh map.

Full Adder

0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

Ci Ai Bi Si Ci+1

1

1 11

Ci

AiBi

00 01 11 10

0

1

Ci+1

Now consider implementation of carry outTwo outputs per full adder bit (Ci+1, Si)

Note: 3 inputs

Full Adder Implementation

A

B

S

C

C i+1

i

i

i

i

Si = Ci (Ai Bi)

Ci+1 = Ai · Bi + Ci · (Ai Bi)

Full Adder

A full adder can be made fromtwo half adders (plus an OR gate).

Hardware repetition simplifies hardware design

4-Bit Adder

C 1 1 1 0A 0 1 0 1B 0 1 1 1S 1 1 0 0

Chain single-bit adders together.What does this do to delay?

Half Adder (not very useful)

C A B S 0 0 0 1 A 0 B 0

S 0

C 1

0 0 0 00 1 1 01 0 1 01 1 0 1

Dec Binary

1 1+1 +1 2 10

Add two binary numbers

• A0 , B0 -> single bit inputs

• S0 -> single bit sum

• C1 -> carry out

Multiplexers

Multiplexers

Select an input value with one or more select bits

Use for transmitting data Allows for conditional transfer of data Sometimes called a mux

4– to– 1- Line Multiplexer

Quadruple 2–to–1-Line Multiplexer

Notice enable bitNotice select bit4 bit inputs

Multiplexer as combinational modules

Connect input variables to select inputs of multiplexer (n-1 for n variables)

Set data inputs to multiplexer equal to values of function for corresponding assignment of select variables

Using a variable at data inputs reduces the size of the multiplexer

Implementing a Four- Input Function with a Multiplexer

Multiplexers & Shannon Expansion

Shannon Expansion Theorem

The Shannon expansion develops the idea that Boolean functions can be reduced by means of the identity:

where F is any function and Fx and Fx' are Shannon cofactors of F.

Example

f = yz + xyz' + x'y'z

= yz(x' + x) + xyz' + x'y'z

= x'yz + xyz + xyz' + x'y'z

= x'(yz + y'z) + x(yz + yz')

= x'gx' + xgx

Expand about 2 variables

Multiplexer Implementation

DECODERS

Overview Binary decoders

Converts an n-bit code to a single active output Can be developed using AND/OR gates Can be used to implement logic circuits.

Binary encoders Converts one of 2n inputs to an n-bit output Useful for compressing data Can be developed using AND/OR gates

Both encoders and decoders are extensively used in digital systems

Binary Decoder

Black box with n input lines and 2n output lines

Only one output is a 1 for any given input

BinaryDecoder

ninputs 2n outputs

Enable

2-to-4 Binary Decoder2-to-4 Binary Decoder

From truth table, circuit for 2x4 decoder is:

Note: Each output is a 2-variable minterm (X'Y', X'Y, XY' or XY)

F0 = X'Y'E

F1 = X'YE

F2 = XY'E

F3 = XYE

X Y

Truth Table:

2-to-4Decoder

X

Y

F0

F1

F2

F3

Enable

Enable

Enable = 1

3-to-8 Binary Decoder3-to-8 Binary Decoder

x y z F0 F1 F2 F3 F4 F5 F6 F7

0 0 0 1 0 0 0 0 0 0 00 0 1 0 1 0 0 0 0 0 00 1 0 0 0 1 0 0 0 0 00 1 1 0 0 0 1 0 0 0 01 0 0 0 0 0 0 1 0 0 01 0 1 0 0 0 0 0 1 0 01 1 0 0 0 0 0 0 0 1 01 1 1 0 0 0 0 0 0 0 1

F1 = x'y'zE

x zy

F0 = x'y'z’E

F2 = x'yz'E

F3 = x'yzE

F5 = xy'zE

F4 = xy'z'E

F6 = xyz'EF7 = xyzE

Truth Table:

3-to-8Decoder

X

Y

F0

F1

F2

F3

F4

F5

F6

F7

Z

Enable

Enable

Enable = 1

Implementing Functions Using Implementing Functions Using DecodersDecoders

Any n-variable logic function can be implemented using a single n-to-2n decoder to generate the minterms OR gate forms the sum. The output lines of the decoder corresponding to

the minterms of the function are used as inputs to the or gate.

Any combinational circuit with n inputs and m outputs can be implemented with an n-to-2n decoder with m OR gates.

Suitable when a circuit has many outputs, and each output function is expressed with few minterms.

Example: Full adderS(x, y, z) = (1,2,4,7)

C(x, y, z) = (3,5,6,7)

3-to-8Decode

rS2

S1

S0

x

y

z

01234567

S

C

x y z C S0 0 0 0 00 0 1 0 10 1 0 0 10 1 1 1 01 0 0 0 11 0 1 1 01 1 0 1 01 1 1 1 1

Implementing Functions Using Implementing Functions Using DecodersDecoders

Enable

Enable = 1

Use two 3 to 8 decoders to make 4 to 16 decoder In this example, only one decoder

can be active at a time. x, y, z effectively select output line

for w

Encoders

Encoders

If the a decoder's output code has fewer bits than the input code, the device is usually called an encoder. e.g. 2n-to-n

The simplest encoder is a 2n-to-n binary encoder One of 2n inputs = 1 Output is an n-bit binary number

.

.

.

.

.

.

2n

inputsn outputs

Binaryencoder

8-to-3 Binary Encoder8-to-3 Binary Encoder

At any one time, only one input line has a value of 1.

Inputs Outputs

I 0 I 1 I 2 I 3 I 4 I 5 I 6 I 7 y2 y1 y0

1 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 1 00 0 0 1 0 0 0 0 0 1 10 0 0 0 1 0 0 0 1 0 00 0 0 0 0 1 0 0 1 0 10 0 0 0 0 0 1 0 1 1 00 0 0 0 0 0 0 1 1 1 1

I0

I1

I2

I3

I4

I5

I6

I7

y0 = I1 + I3 + I5 + I7

y1 = I2 + I3 + I6 + I7

y2 = I4 + I5 + I6 + I7

8-to-3 Priority Encoder8-to-3 Priority Encoder

• What if more than one input line has a value of 1?• Ignore “lower priority” inputs.• Idle indicates that no input is a 1.

Inputs Outputs

I 0 I 1 I 2 I 3 I 4 I 5 I 6 I 7 y2 y1 y0 Idle

0 0 0 0 0 0 0 0 x x x 01 0 0 0 0 0 0 0 0 0 0 1X 1 0 0 0 0 0 0 0 0 1 1X X 1 0 0 0 0 0 0 1 0 1X X X 1 0 0 0 0 0 1 1 1X X X X 1 0 0 0 1 0 0 1X X X X X 1 0 0 1 0 1 1X X X X X X 1 0 1 1 0 1X X X X X X X 1 1 1 1 1

Priority Encoder (8 to 3 encoder)

Priority Encoder : H7=I7 (Highest Priority) H6=I6·I7’ H5=I5·I6’ ·I7’ H4=I4·I5’ ·I6’·I7’ H3=I3.I4’·I5’ ·I6’·I7’ H2=I2.I3’·I4’ ·I5’·I6’ ·I7’ H1=I1· I2’ ·I3’·I4’ · I5’·I6’ ·I7’ H0=I0·I1’ ·I2’·I3’ ·I4’ ·I5’ ·I6’ ·I7’ IDLE= I0’ ·I1’ · I2’ ·I3’ ·I4’ ·I5’ ·I6’ ·I7’

Inputs Outputs

I 0 I 1 I 2 I 3 I 4 I 5 I 6 I 7 y2 y1 y0 Idle0 0 0 0 0 0 0 0 x x x 01 0 0 0 0 0 0 0 0 0 0 1X 1 0 0 0 0 0 0 0 0 1 1X X 1 0 0 0 0 0 0 1 0 1X X X 1 0 0 0 0 0 1 1 1X X X X 1 0 0 0 1 0 0 1X X X X X 1 0 0 1 0 1 1X X X X X X 1 0 1 1 0 1X X X X X X X 1 1 1 1 1

I1

I2

I3

I4

I5

I6

I0

I7

H1

H2

H3

H4

H5

H6

H0

H7

IDLE

Priority Encoder (8 to 3 encoder)

Assign priorities to the inputs When more than one input are asserted, the output generates

the code of the input with the highest priority Priority Encoder :

H7=I7 (Highest Priority) H6=I6·I7’ H5=I5·I6’ ·I7’ H4=I4·I5’ ·I6’·I7’ H3=I3.I4’·I5’ ·I6’·I7’ H2=I2.I3’·I4’ ·I5’·I6’ ·I7’ H1=I1· I2’ ·I3’·I4’ · I5’·I6’ ·I7’ H0=I0·I1’ ·I2’·I3’ ·I4’ ·I5’ ·I6’ ·I7’ IDLE= I0’ ·I1’ · I2’ ·I3’ ·I4’ ·I5’ ·I6’ ·I7’

Encoder Y0 = I1 + I3 + I5 + I7 Y1 = I2 + I3 + I6 + I7 Y2 = I4 + I5 + I6 + I7

Y1

Y2

Y0

IDLE

I1

I2

I3 Y1

Y2I4

I5

I6

I0

Y0

I7

Binary encoder Priority Circuit

I1

I2

I3

I4

I5

I6

I0

I7

H1

H2

H3

H4

H5

H6

H0

H7

IDLE

I1

I2

I3

I4

I5

I6

I0

I7

Priority encoder