FUNCTIONS OF COMBINATIONAL LOGIC
(DECODER & MUX EXPANSION)
Lecture 8
Digital Design
Dr. PO Kimtho
Department of Computer Sciences
Norton University (NU)
Topic Outlines
Encoder
Decoder
Multiplexers (MUX)
Demultiplexers (DEMUX)
Topic Outlines
Encoder
Decoder
Multiplexers (MUX)
Demultiplexers (DEMUX)
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.
Example: A 3-to-8 Decoder Constructed with Two 2-to-4 Decoders
Decoder Expansion
Decoder Expansion
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
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
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
Implementing a Binary Adder Using a Decoder - The logic circuit
Decoder Expansion
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
2:1 MUX
Multiplexers (MUX)
Data selector SELECT input code determines which input is transmitted to output Z.
DATA-SELECT INPUTS INPUT
SELECTED S0 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 that
any one of the 4 data-input lines
can be selected
4-to-1
MUX
S0
Z D1
D0
S1
D3
D2
D0
D1
D2
D3 S1 S0
Z
4:1 MUX
Multiplexers (MUX)
Total expression for the data output is:
013012011010 SSDSSDSSDSSDY
Logic diagram for 4:1 MUX
4:1 MUX
Multiplexers (MUX)
Question 3
Construct an 8:1 multiplexer using block diagram.
8 input lines means there must be 3 data select lines.
Multiplexers (MUX)
Multiplexers (MUX)
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
16-to-1-Line Multiplexer
Multiplexers (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.
Implementing a Boolean Function with a MUX
Multiplexers (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.
Implementing a Boolean Function with a MUX
Multiplexers (MUX)
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
Example Implement F (X,Y,Z) = m (1, 2, 6, 7) using 4:1 MUX
Multiplexers (MUX)
Example Implement F (A, B, C, D) = m (1, 3, 4, 11, 12, 13, 14, 15) using 8:1 MUX
Multiplexers (MUX)
MUX Application Example
74157- consists of four
separate 2-input multiplexers
Content-selector Display
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
Data input is transmitted to only one of the outputs as determined by the select input code.
1-line-to-8-line multiplexer
1:4 DEMUX
Demultiplexers (DEMUX)
The expression of every output
Question 4:
Construct a 1:4 DEMUX using block diagram. Show the equivalent Truth-Table.
1 - 4
DEMUX
S 0
I 0 Q 1
Q 0
S 1
Q 3
Q 2
1 - 4
DEMUX
S 0
I 0 Q 1
Q 0
S 1
Q 3
Q 2
S0
S1 Q0
Q1
I 0
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)
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
Solve this..
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