Adders

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Adders

• Used to perform addition, subtraction, multiplication, and division (sometimes)

• Half-adder adds rightmost (least significant) bit

• Full-adder adds all other bits, since a 1 may be carried into it. Use carry-out from one adder as the carry-in for the next adder

• Combinational circuit (no memory)

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• Half adder– Inputs two 1-bit values, X and Y – Outputs their 2-bit sum as bits C and S

• C is carry• S is the sum

• A Half Adder (HA) is a 2-input, 2-output combinational circuit that adds the inputs and produces a Sum and a Carry

• The Boolean expressions and the circuit for the Sum S and the carry C are given below:

Half-Adder

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Half-Adder

• Only this circuit is a Half-Adder – Other circuits may be equivalent, however

Result

Carry

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Half-Adder

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How to know a circuit is equivalent to half-adder ?• Circuit is equivalent to a half-adder if both truth tables have identical output

• Cannot be used if there could be a carry-in bitSo used to add least significant (rightmost) bit.

Very important circuit in computers.• Addition is very common • Half-adder is found in every computer, calculator, digital watch…

Half-Adder

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Binary Full Adder

• A Full Adder (FA) is a 3-input, 2-output combinational circuit that adds the inputs and produces a Sum and a Carry• The third input can be perceived as the carry from a previous addition• The Boolean expressions and for the Sum S andthe carry C, obtained from their K-Maps, are given below:

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MultiplexerA multiplexer (MUX) is a device that accepts data from one of many input sources for transmission over a common shared line. To achieve this the MUX has several data lines and a single output along with data-select inputs, which permit digital data on any one of the inputs to be switched to the output line. The logic symbol for a 1-of-4 data selector/multiplexer is shown below, along with its associated table.

Logic symbol for 1-of-4 multiplexer

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Active high MUX

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Active Low MUX

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Data Select Inputs Input Selected

S1 S0

0 0 D0

0 1 D1

1 0 D2

1 1 D3Table of Operation

Note that if a binary zero appears on the data-select lines then data on input line D0 will appear on the output. Thus, data

output Y is equal to D0 if and only if S1=0 and S0=0

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Similarly, the data output is equal to D1, D2 and D3 for             ,               and             , respectively. Thus the

total multiplexer logic expression, formed from ORing terms is                                                                                         

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Larger MUX

• Technique: use hierarchies of smaller components

• Example: creating 4x1 mux from 2x1 mux– Will create a 2-level mux tree– First level takes the initial inputs– The results from the first level are fed into the

second level

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Hierarchy Approach

• Technique– Divide the truth table into equal sections

• Number of sections given by type of second-level MUX

• If the second-level MUX is 2x1 then need 2 sections in the TT

S1 S0 F

0 0 Input 0

0 1 Input 1

1 0 Input 2

1 1 Input 3

Section 1

Section 2

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Hierarchy Approach

• Technique– Connect the outputs corresponding to individual

sections of the TT to the data lines of the individual first-level MUXs

2x1Mux

2x1Mux

Input 0

Input 1

Input 2

Input 3

Outputs from Section 1 ofTruth Table

Outputs from Section 2 ofTruth Table

S1 S0 F0 0 Input 0

0 1 Input 1

1 0 Input 2

1 1 Input 3

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Hierarchy Approach

• Technique– Connect the outputs of the first-level MUXs to the

data lines of the second-level MUX following the order of the sections

2x1Mux

2x1Mux

Input 0

Input 1

Input 2

Input 3

2x1Mux

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Hierarchy Approach

• Technique– Connect the outputs of the first-level MUXs to the

data lines of the second-level MUX following the order of the sections

2x1Mux

2x1Mux

Input 0

Input 1

Input 2

Input 3

2x1Mux

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4-to-1 MUX made from 2-to-1 MUXs

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Decoders

• A decoder:– Accepts a Boolean value (number) and activates the

corresponding output line• All other lines are deactivated

– For n inputs there are 2n output lines• Each possible input value corresponds to an output line

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Decoders

Motivation (one example)

CPU needs to select channel => assign address to each device

Need a way to activate device.

CPU

COMMUNICATION CHANNEL

MEMORY

PRINTER

MOUSE

DISPLAY

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Decoders

Motivation

CPU needs to select channel => assign address to each device

Need a way to activate device.

CPU

ADDRESS

CODE

MEMORY

PRINTER

MOUSE

DISPLAY

ADDRESS

DECODER

ACTIVATION

LINE

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Decoder

–An n input decoder has 2n

outputs.

–Outputi is 1 iff the binary

value of the n-bit input is i.

–At any time, exactly one output is 1, all others are 0.

1, iff A,B is 00AB

1, iff A,B is 01

1, iff A,B is 10

1, iff A,B is 11

i = 0

i = 1

i = 2

i = 3

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• In general, attach m-enabling circuits to the outputs• See truth table below for function

– Note use of X’s to denote both 0 and 1– Combination containing two X’s represent four binary

combinations

• Alternatively, can be viewed as distributing value of signal EN to 1 of 4 outputs

• In this case, called ademultiplexer

EN

A 1

A 0

D0

D1

D2

D3

(b)

EN A1 A0 D0 D1 D2 D3

01111

X0011

X0101

01000

00100

00010

0000

Decoder with Enable

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2-to-4 decoder with active high enable

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2-to-4 decoder with active low enable

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Encoders

• An encoder:– For 2n inputs there are n output lines

• Outputs the Boolean value corresponding to the input line number

• There is a special output line V that indicates whether any input lines are active.

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Encoder

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Priority Encoder