Company
LOGO
DKT 122/3
DIGITAL SYSTEM 1
E d i t
y o u r
s l o g a n
h e r e
WEEK #9
FUNCTIONS OF COMBINATIONAL LOGIC
(DECODERS & MUX EXPANSION)
Topic Outlines
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



Encoder
Decoder
Multiplexers (MUX)
Demultiplexers (DEMUX)
Topic Outlines
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



Encoder
Decoder
Multiplexers (MUX)
Demultiplexers (DEMUX)
Decoders Expansion
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 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.
Decoder Expansion
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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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Example:
Implementing a Binary Adder Using a Decoder
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.
Decoder Expansion
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Implementing a Binary Adder Using a Decoder
- The logic circuit
Multiplexers (MUX)
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 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 datainput 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
n select
inputs
1 data
output
2n data
inputs
Logic symbol for a 4-input
multiplexer (4:1 MUX)
Multiplexers (MUX)
Company LOGO
2:1 MUX
Data selector
SELECT input code determines
which input is transmitted to
output Z.
Multiplexers (MUX)
Company LOGO
4:1 MUX
DATA-SELECT
INPUTS
2 data-select lines means that
any one of the 4 data-input lines
can be selected
D0
D1
D2
D0
D1
D3
D3 S
1
4-to-1 Z
D2 MUX
S0
Z
INPUT
SELECTED
S0
S1
0
0
D0
0
1
D1
1
0
D2
1
1
D3
S1 S0
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
Multiplexers (MUX)
Company LOGO
4:1 MUX
Logic diagram
for 4:1 MUX
Total expression for the data output is:
Y  D0 S1 S 0  D1 S1 S 0  D2 S1 S 0  D3 S1 S 0
Multiplexers (MUX)
Company LOGO
Question 3
Construct an 8:1 multiplexer using block diagram.
8 input lines means there must be 3 data select lines.
Multiplexers (MUX)
Company LOGO
Another design option for 8:1 mux
 Using construction of larger multiplexers from smaller
ones.
16-to-1 MUX: 74150
8-to-1-Line Multiplexer
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16-to-1-Line Multiplexer
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Multiplexers (MUX)
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Implementing a Boolean Function with a MUX
 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.
Multiplexers (MUX)
Company LOGO
Implementing a Boolean Function with a MUX
 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.
Multiplexers (MUX)
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Implementing a Boolean Function with a MUX
General procedure:
1.
2.
3.
4.
5.
Produce Truth Table for Boolean function.
The first n-1 variables are applied to the selection
inputs of the mux.
The remaining single variable of the function is used
for the data input.
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.
These values are then applied to the data inputs in
the proper order.
Multiplexers (MUX)
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Example
Implement F (X,Y,Z) = m (1, 2, 6, 7) using
4:1 MUX
Multiplexers (MUX)
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Example
Implement F (A, B, C, D) = m (1, 3, 4, 11,
12, 13, 14, 15) using 8:1 MUX
MUX Application Example
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74157- consists of four
separate 2-input multiplexers
Content-selector Display
Demultiplexers (DEMUX)
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 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 data
input
2n data
outputs
n select
inputs
1-line to 4-line DEMUX
Company LOGO
Data input is transmitted to
only one of the outputs as
determined by the select
input code.
1-line-to-8-line multiplexer
Demultiplexers (DEMUX)
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1:4 DEMUX
The expression of every output
Demultiplexers (DEMUX)
Company LOGO
Question 4:
Construct a 1:4 DEMUX using block diagram. Show the
equivalent Truth-Table.
I0
1-4
DEMUX
Q0
Q1
Q2
Q3
S 1 S 0 Block diagram
S1
S0
I0
Truth-table
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
Q0
Q1
Q2
Q3
Logic circuit
Mux-Demux Application: Example
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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.
Solve this..
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Design the following:
 16-line-to-4-line encoder using the 8line-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