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![Page 1: Company LOGO Edit your slogan here DKT 122/3 DIGITAL SYSTEM 1 WEEK #9 FUNCTIONS OF COMBINATIONAL LOGIC (DECODERS & MUX EXPANSION)](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649eab5503460f94bb123d/html5/thumbnails/1.jpg)
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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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