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DKT 122/3DIGITAL SYSTEM 1

WEEK #9 FUNCTIONS OF COMBINATIONAL LOGIC

(DECODERS & MUX EXPANSION)

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Topic Outlines

Encoder Decoder Multiplexers (MUX) Demultiplexers (DEMUX)

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Topic Outlines

Encoder Decoder Multiplexers (MUX) Demultiplexers (DEMUX)

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Decoders 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

Decoder Expansion

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The Operation

The MSB input, A2, functions:– As enable, EN, of one decoder – As its complement, 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.

Decoder Expansion

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The function of EN 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.

Decoder Expansion

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For an addition of X, Y, and Z (as Cin), the S and Cout expression are as follows:

S(X,Y,Z) = m (1, 2, 4, 7)C(X,Y,Z) = m (3, 5, 6, 7)

So, there are 3 inputs and 8 minterms Use a 3-to-8 decoder.

Example: Implementing a Binary Adder Using a Decoder

Decoder Expansion

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Implementing a Binary Adder Using a Decoder - The logic circuit

Decoder Expansion

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Multiplexers (MUX)

MUX is a device that allows digital information from several sources to be routed onto a single line for transmission

It is made up of several data-input lines and a single output line. It also has data-select inputs which permits digital data on any one of the inputs to be switched to the output line.

MUX is also known as data selectors

Logic symbol for a 4-input multiplexer (4:1 MUX)

n select inputs

1 data output

2n data inputs

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2:1 MUX

Multiplexers (MUX)

Data selectorSELECT input code determines which input is transmitted to output Z.

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DATA-SELECT INPUTS INPUT

SELECTEDS0 S1

0 0 D0

0 1 D1

1 0 D2

1 1 D3

If a binary 0 (S0=0 and S1=0) is applied to the data-select lines, the data on input D0 appear on the data-output line

2 data-select lines means thatany one of the 4 data-input linescan be selected

4-to-1MUX

S0

ZD1

D0

S1

D3

D2

D0D1

D2

D3 S1 S0

Z

4:1 MUX

Multiplexers (MUX)

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Total expression for the data output is:

013012011010 SSDSSDSSDSSDY 013012011010 SSDSSDSSDSSDY

Logic diagram for 4:1 MUX

4:1 MUX

Multiplexers (MUX)

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Question 3Construct an 8:1 multiplexer using block diagram.

8 input lines means there must be 3 data select lines.

Multiplexers (MUX)

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Another design option for 8:1 mux

Using construction of larger multiplexers from smaller ones.

16-to-1 MUX: 74150

Multiplexers (MUX)

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8-to-1-Line Multiplexer

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16-to-1-Line Multiplexer

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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.

Implementing a Boolean Function with a MUX

Multiplexers (MUX)

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Another method (more efficient way) Implementing a Boolean function of n

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

Implementing a Boolean Function with a MUX

Multiplexers (MUX)

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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. 4. 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.

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

Implementing a Boolean Function with a MUX

Multiplexers (MUX)

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ExampleImplement F (X,Y,Z) = m (1, 2, 6, 7) using 4:1 MUX

Multiplexers (MUX)

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ExampleImplement F (A, B, C, D) = m (1, 3, 4, 11, 12, 13, 14, 15) using 8:1 MUX

Multiplexers (MUX)

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74157- consists of four separate 2-input multiplexers

MUX Application Example

Content-selector Display

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Demultiplexers (DEMUX)

DEMUX reverse the multiplexing functions It takes digital information from one line and distributes it

to a given number of output lines DEMUX is also known as data distributor

1-line to 4-line DEMUX

1 data input

n select inputs

2n data outputs

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

1-line-to-8-line multiplexer

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1:4 DEMUX

Demultiplexers (DEMUX)

The expression of every output

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Question 4:Construct a 1:4 DEMUX using block diagram. Show the equivalent Truth-Table.

1-4DEMUX

S 0

I0Q1

Q0

S 1

Q3

Q2

1-4DEMUX

S 0

I0Q1

Q0

S 1

Q3

Q2

S0

S1 Q0

Q1

I0

Q2

Q3

S1 S0 I1 Q3 Q2 Q1 Q0

0 0 1 0 0 0 1

0 1 1 0 0 1 0

1 0 1 0 1 0 0

1 1 1 1 0 0 0

Truth-table

Block diagram

Logic circuit

Demultiplexers (DEMUX)

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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.

Mux-Demux Application: Example

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Design the following:

16-line-to-4-line encoder using the 8-line-to-3-line encoder in cascade

A 4:1 MUX using 2:1 MUXes A 8:1 MUX using 4:1 MUXes A 1:4 DeMUX using 1:2 DeMUX A 1:8 DeMUX using 1:4 DeMUX

Solve this..

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