combinational logic: decoders, encoders, and...

Digital Logic Lecture 10

Combinational Logic: Decoders, Encoders, and Multiplexers

By

Ghada Al-Mashaqbeh

The Hashemite University

Computer Engineering Department

The Hashemite University 2

Outline

Introduction.

Decoders.

Encoders.

Multiplexers.


Introduction

In this lecture we will complete viewing the standard combinational circuits.

Specifically, we will study:

Decoders.

Encoders.

Multiplexers.


Decoders I Decoder is a combinational circuit that

converts a binary value to a specific binary code one to one mapping.

Remember that an n-bit binary code can represents up to a 2n binary codes.

So, a decoder has the binary code as an input and the original binary value as the output.

For an n-bit binary code, the decoder has n inputs and a maximum 2n outputs (may have fewer outputs if you have unused codes as in BCD).


Decoders II We will study the n-to-m-line decoders where m <= 2n where it

simply generate the 2n minterms of the n input variables. Two types of decoders:

Decoders without enable lines. Decoders with enable lines.

Each type of these decoders can also be classified into: Active low decoders. Active high decoders.

We will take an example the 3-to-8 line decoder (see the next slide).

You can design any decoder (2-to-4, etc) just truncate the unwanted minterms.

Remember that decoder term can be used to denote any binary code converters in general.


Decoders III

Truth table of 3-to-8 line decoder X Y Z D0 D1 D2 D3 D4 D5 D6 D7 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1


Decoders II


Decoders IV

A particular application of 3-to-8 line decoder is binary-to-octal conversion.

That is based on the 3-bit binary value you get which octal digit represented by this value.

For example if the input is 111 convert it to octal you get digit 7, the output of the decoder will have a 1 (active) at D7 which represent digit 7 in octal.


Active Low Decoders

Constructed with NAND gates.

The selected minterm based on the input (i.e. the active one) will have a 0 on its output line and other unselected minterms will have 1 (the inactive state for active low output line).

How to differentiate between active low and active high decoders from its block diagram? Bubbles on the output lines active low decoder.

No bubbles on the output lines active high decoder.


Decoders with Enable Line I

The following is a general design or block diagram of binary decoder.

The “Enable” lines enables the decoder to perform mapping.

The “Enable” depends on whether it is active low (have a bubble in front of the line) or active high (have no bubble in front of the line or pin).

Active low enable must have an input of 0 to activate the decoder.

Active high enable must be asserted (has value of 1) to activate the decoder.


Decoders with Enable Line II

Some decoders may have more than 1 enable line where each line can be different from the other (i.e. whether it is active high or active low).

The state of the decoder itself (whether it is active low or active high) is independent of the enable line state.

So, you may have an active low decoder with active high enable line.


Decoders with Enable Line III

If a decoder is not enabled (not activated) this means that the minterm selected by the inputs is not activated and no specific output is selected.

The inactive state depends on the type of the decoder: Active low decoder all outputs will be high (at

level 1) when disabled (the enable line is not activated).

Active high decoder all outputs will be low (at level 0) when disabled (the enable line is not activated).


Active Low 2-to-4 Line Decoder With Enable Line


Building Bigger Decoders from Small Ones

We can combine two 3-to-8 decoders to build a 4-to-16 decoder (in this figure w is the MSB and z is the LSB).

Generates from

0000 to 0111

Generates from

1000 to 1111


Combinational Logic Implementation of Functions Using Line Decoders I

Decoders can be used to implement logic functions with the condition that these functions must be represented as a sum of minterms.

The minterms in the function are combined using an OR gate or NAND gate based on the decoder type.

Two cases: Active high decoders: combine these minterms with

OR gate. Active low decoders: combine these minterms with

NAND gate.

Note that NAND-NAND is equivalent to AND-OR circuit.


Combinational Logic Implementation of Functions Using Line Decoders II

What to do with functions that have a long list of minterms (larger than 2n/2)?

Active high decoder:

Simply get the minterms in F’ and combine them with a NOR gate.

The output of the NOR gate is F.

Active low decoder:

Simply get the minterms in F’ and combine them with a AND gate.

The output of the AND gate is F.


Example

Design a full adder using OR gates and 3x8 active high decoder.

Sol:

From the truth table of full adder, the sum-of-minterms representation of S and C are:

S(x,y,z) = Σ(1,2,4,7)

C(x,y,z) = Σ (3,5,6,7)


Example -- Solution


Notes

If you are given F(x, y, z) = xy’ + z, how you can implement it using an active high decoder and an OR gate?

First you must define the minterms in F. the simplest way is obtain the truth table of F and see at which minterms it has value of 1.

The previous note is applied for any representation of F other than sum of minterms.


Encoders

Encoders perform the inverse operation of a decoder. An encoder has 2n (or fewer) input lines and n output

lines. The output lines generate the binary code

corresponding to the input value. Two types of encoders:

Encoders with mutually exclusive inputs: only one input is active at a time.

Priority encoders: multiple inputs can be active at the same time.

Example of encoders with mutually exclusive inputs is octal to binary encoders (8-to-3 line encoder).


Octal-to-Binary Encoder Design I

Truth Table D0 D1 D2 D3 D4 D5 D6 D7 X Y Z 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1

Note that we need not to complete the whole table where we

have listed only 8 entries equal to the number of inputs (not 2^8 entries) since only 1 input is allowed to be active at a time.


Octal-to-Binary Encoder Design II

From the truth table we can see that the outputs of the encoders are based on ORing the minterms found in the inputs:

z=D1+D3+D5+D7

y=D2+D3+D6+D7

x=D4+D5+D6+D7

So, the internal implementation of octal-to-binary encoder is composed of 3 OR gates.


2n-to-n Encoder Design Limitations

This encoder has many limitations: Only one input can be active at a time, otherwise

it will produce undefined combination which do not correspond to either of the activated inputs (based on the OR expressions). Solution: give higher priority to inputs with higher

subscripts, e.g. D5 has higher priority than D2 and so on. So, when both D5 and D1 for example are active the outputs will be 101 since D5 has higher priority.

Note that when all inputs are 0 the output will be 000 which is the same output when only D0 is active. Solution: provide an additional output to indicate

whether at least there is one input is 1.


Priority Encoder

Priority encoder is an encoder that solves the limitations of 2n-to-n encoder discussed previously (include priority and an additional output which is called the valid bit (V) to indicate that at least one input is 1).

We will design a 4-to-2 priority encoder.


Priority Encoder Design I

Truth table: Inputs Outputs D0 D1 D2 D3 x y v 0 0 0 0 X X 0 1 0 0 0 0 0 1 X 1 0 0 0 1 1 X X 1 0 1 0 1 X X X 1 1 1 1

Note that we have inserted the don’t care conditions in

the input side just to avoid the need to list all 16 entries in the truth table (not to be used in simplification as the don’t care found in the output).


Priority Encoder Design II


Priority Encoder Design III


Multiplexers I

Multiplexer is a combinational circuit that selects one of the inputs and directs it to a single output line.

It is abbreviated as a MUX.

It is also called data selector and a digital switch.

The operation performed by a MUX is called election.

The selection of the inputs is controlled by a selection lines or inputs.

In general a MUX has 2n inputs, n selection lines (implied from the inputs so they are not counted with the inputs), and exactly 1 output.


Multiplexers II

Each input of the multiplexer is 1-bit only. The internal implementation of the MUX is an

AND-OR circuit. Each input is fed into one AND gate in

addition to one input from each selection line. The selection lines are very similar to the

enable line in the decoder. If their values are 1 then the AND gate is enabled, otherwise the AND gate is disabled so the input has no effect on the output.


Multiplexers III

A multiplexer is named as follows: (number of inputs)-to-(number of outputs) MUX or simply (number of inputs)X(number of outputs) MUX.

For example: you may have a 2X1 MUX, 4X1 MUX, 8X1 MUX and so on.


2X1 MUX Design I


2X1 MUX Design I

Boolean expression of the MUX output is:

Y = I0S’ + I1S

Note that which input that will be passed to the output (selected) depends on the value of S which is the selection line here.


4X1 MUX Design I


4X1 MUX Design II

Boolean expression of the MUX output is:

Y = I0S1’S0’ + I0S1’S0 + I0S1S0’ + I0S1S0

Note that which input that will be passed to the output (selected) depends on the product value of S0 and S1 which are the selection lines here.


Note

MUXs can have also enable line(s).

Similar to the decoder the enable line will be fed into all AND gates in the MUX.

The enable line can be active high or active low.


Boolean Functions Implementation Using Multiplexers I

We have used a decoder with OR gate to implement Boolean functions.

Active high decoder contains all the minterms represented by AND gates inside it.

Have a deep look at the internal design of a MUX you will find that a 2nX1 MUX is simply a nX2n decoder with OR gate.

So, you can use a MUX to implement any Boolean function since all the needed minterms are available without the need to OR gate since it is already found.

The only difference that now you apply the inputs of the Boolean function to the selection lines of the MUX in the proper way.


Boolean Functions Implementation Using Multiplexers II

To implement a Boolean function with n inputs you have two options:

Either to use a 2nX1 MUX where you have n selection lines.

Or to use a 2n-kX1 MUX where you have n-k selection lines.


Boolean Functions Implementation Using Multiplexers III

General rules for implementing any Boolean function with n variables using 2nX1 MUX: Use a multiplexer with n selection lines and 2n

data inputs.

List the truth table of F.

Apply the n variables to the selection inputs of multiplexer with the same significance order.

For each data input (which is a minterm) place a 1 if F has an output of 1 for this minterm and 0 if F has an output of 0 for this minterm.


Boolean Functions Implementation Using Multiplexers IV

General rules for implementing any Boolean function with n variables using 2n-kX1 MUX: Use a multiplexer with n-k selection inputs and 2n-1

data inputs List the truth table of F. Apply the first n-k variables to the selection inputs of

multiplexer with the same significance order. For each combination of the selection variables

evaluate the output as a function of the last k variable.


Example I


Example II


Additional Notes

This lecture covers the following material from the textbook:

Chapter 4: Sections 4.9, 4.10 and 4.11

combinational logic: decoders, encoders, and...

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