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Combinational Logic Design

Digital Computer Logic

Kashif Bashirhttp://www.taleem.greatnow.com

Email: kashif@pafkiet.edu.pk

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Module Outline Combinational Circuits Design Topics Analysis Procedure Design Procedure Decoders Encoders Multiplexers Binary Adders Binary Subtraction Binary Adder-subtractors Binary Multipliers Decimal Arithmetic

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Combinational Circuits The outputs are a function of the present set of

inputs only The inside of a combinational circuit is made of

logic gates Combinational logic circuits are important

components of digital systems Each output can be thought of as a function of

all the inputs – if there are m outputs and n inputs then there are m boolean functions, one describing each output

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Combinational Circuits

Combinational Logic Circuit

m Outputsn Inputs

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Design Topics - Design Hierarchy

Uses divide and conquer approach Break the circuit into smaller pieces called blocks If a block is too large break it up into still smaller

blocks At each level or hierarchy we have a representation of

the blocks such as block diagram, gates, circuits Benefits

Reusability of blocks Quicker design completion Easier to verify and isolate problems

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Figure 3-2

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Diagrams Representing the Hierarchy for Figure 3-2

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Design Topics - Computer Aided Design

Digital Design of real world systems has become too complex to be done by hand alone

Digital design tools or software helps in creating design and verify it

Uses Hardware Description languages such as VHDL, Verilog

Computer Aided Design Tools exist for: Schematic Capture Simulation of Design Verification Synthesis Etc.

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Design Topics - Top Down Design

Ideally deign is top down which means the design is specified at very high levels of abstraction as text along with constraints on cost, performance size, reliability, etc.

Then the design is repeatedly divided into smaller and smaller blocks and implemented with these blocks

In order to obtain reusability and to make maximum reuse of predefined modules, often portions of the design are performed bottom-up

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Analysis Procedure Derivation of Boolean Functions Derivation of the Truth Table Logic Simulation

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Design Procedure1. From the specifications of the circuit,

determine the required number of inputs and outputs and assigned a letter symbol to each

2. Derive the truth table that defined the relationship between inputs and putouts

3. Obtain the simplified Boolean functions for each output as a function f the input variables

4. Draw the logic diagram

5. Verify the correctness of the design

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Some Design Examples BCD to Excess-3 code converter BCD to Seven Segment Decoder

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BCD to Excess-3 code converter

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BCD to Excess-3

code converter K-Maps

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BCD to Excess-3 code converter Logic Diagram

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Decoders n to 2n decoder - a

combinational logic circuit that converts binary information from the n coded inputs to a maximum of 2n unique outputs

Also called the n-to-m line deciders for example: 2-to-4 line decoder 3-to-8 line decoder

DecoderCombinational Logic Circuit

n Inputs 2n Outputs

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Decoder Diagrams

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Decoder with Enable Input

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Decoder Expansion Can use smaller

decoders to build bigger decoders.

e.g. using two 2-4 decoders we can build a 3-8 decoder or using two 3-8 decoders we can build a 4-16 decoder

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Combinational Circuit Implementation using Decoder A n-input decoder generates the minterms

corresponding to a boolean function of n-variables

Can use the outputs of a decoder along with an OR gate to implement a Boolean function in SOP form

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Combinational Circuit Implementation using Decoder

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Encoders Performs the

inverse operation of a decoder

Has 2n or fewer input lines and n output lines

The output generates the binary code corresponding to the input value

EncoderCombinational Logic Circuit

n Outputs2n Inputs

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Examples of Encoders Octal to Binary Encoder Priority Encoder

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Priority Encoder

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Multiplexers A multiplexer is

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

MULTIPLEXERCombinational Logic Circuit

1 Output2n Inputs

n Selection Lines

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4-to-1-Line Multiplexer

MULTIPLEXERCombinational Logic Circuit

Output4 Inputs

S0 S1

D0

D1

D2

D3

S1 S0

Function table

Output

0 00 11 01 1

D0

D1

D2

D3

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Building MUXes

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Building MUXes

A MUX can be built using transmission gates

Mutiplexer blocks can be combined in parallel with common selection and enable lines to perform selection on multi-bit quantities

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Implementing Boolean Functions Using MUX

Any Boolean function of n-variables can be implemented using a MUX with n selection lines

A more efficient method is using a MUX with n-1 selection lines.

Let us take a look at each method

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Example of Implementing Boolean Functions Using MUX

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Example of Implementing Boolean Functions Using MUX

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Demultiplexers Performs the inverse operation of a multiplexer A combinational circuit that receives input

from a single line and transmits it to one of 2n possible output lines

The selection of the specific output is controlled by the bit combination of n selection lines

Same as a Decoder with an enable – how?