Multiplexer, Decoder and

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Multiplexer, Decoder and Circuit Designing
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• Decoders
• A Decoder is a combinational circuit that
  converts binary information from n input lines to
  a maximum of 2n unique output lines.
• As an example consider the 3-to-8 line decoder
  circuit below

D0 xyz
D1 xyz
z
D2 xyz
y
D3 xyz
D4 xyz
x
D5 xyz
D6 xyz
D7 xyz
3

• The three inputs are decoded into eight outputs,
  each representing one of the minterms of the
  three input variables
• The three inverters provide the complement of
  inputs,
• Each one of eight AND gates represent one of the
  minterms.

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• Multiplexer
• A multiplexer is a combinational circuit that
  selects binary information from one of many input
  lines and directs it to a single output lines.
• The selection of particular input line is
  controlled by a set of selection lines.
• Normally, there are 2n input line and n selection
  lines whose combinations determine which input is
  selected.
• A 2-to-1 line multiplexer connects one of two
  1-bit sources to a common destination as shown in
  the next slide.
• The block diagram of the circuit is also shown
  below.

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• The circuit has two data input lines, and one
  selection line S.
• When S0, the upper AND gate is enabled and the
  I0 has a path to the output
• When S1, the lower AND gate is enabled and I1
  has path a to the output.
• The multiplexer acts like an electric switch that
  selects one of two sources.

I0
Y
I1
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• A 4-to-1 line multiplexer is shown below
• Each of the four inputs, I0 through I3, is
  applied to one input of an AND gate.
• Selection lines S1 and S0 are decoded to select
  a particular AND gate

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• By looking at the truth table below we can see
  that
• When S00 and S10 then YI0
• When S01 and S10 then YI1
• When S00 and S11 then YI2
• When S01 and S11 then YI3
• A multiplexer is also called a data selector,
  Since it selects one of many inputs and steers
  the binary information to output line.
• The AND gates and inverters in the multiplexer
  are Decoder circuit, and indeed they decode the
  selection input lines.

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• Example
• Design a combinational circuit with three inputs
  and one output. The output is 1 when the binary
  value of the inputs is less than 3. The input is
  0 otherwise.
• Answer
• From the specifications of the circuit, determine
  the required number of inputs and outputs and
  assign a symbol to each. In this question, we
  have three input (lets call them x,y,z) and one
  output (lets call it F)
• Derive the truth table that defines the required
  relationship between inputs and outputs. (from
  question, it says that the output is 1 if the
  binary value of input is less than 3

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• Obtain the simplified Boolean function for each
  output as a function of the input variables
• F xy xz
• Draw the logic diagram and verify the correctness
  of the design

y
yz
11
01
00
10
x
0
1
x
z
x
y
F
x
Z
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• Example
• A majority circuit is a combinational circuit
  whose output is equal to 1 if the input variables
  have more 1s than 0s. the output is 0
  otherwise. Design a 3-input majority circuit.
• Answer
• 1) From the specifications of the circuit,
  determine the required number of inputs and
  outputs and assign a symbol to each. In this
  question, we have three inputs (lets call them
  x,y,z) and one output (lets call it F)

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• Obtain the simplified Boolean function for each
  output as a function of the input variables
• F xy xzyz
• Draw the logic diagram and verify the correctness
  of the design

y
yz
11
01
00
10
x
0
1
x
z
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• Example
• Design a combinational circuit that converts a 4
  bit Gray code to a 4-bit binary number. Implement
  the circuit with exclusive-OR gates. A Gray Code
  is a type of code where any two successive codes
  differ by only one variable, like Karnaugh map
  cells )
• Answer
• 1) From the specifications of the circuit,
  determine the required number of inputs and
  outputs and assign a symbol to each. In this
  question we have four input (lets call them
  A,B,C,D) and four outputs (lets call them
  W,X,Y,Z)

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• Derive the truth table that defines the required
  relationship between inputs and outputs.
  (convert a 4 bit Gray code (a type of binary
  where any two successive codes differ by only one
  variable, like Karnaugh map cells ) to a 4-bit
  binary number)

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• 3) Obtain the simplified Boolean function for
  each output as a function of the input variables
• WA XABABAB

C
C
CD AB
CD AB
00
01
10
11
00
01
10
11
00
00
01
01
B
B
11
11
A
A
10
10
D
D
C
C
CD AB
CD AB
00
01
10
11
00
01
10
11
00
00
01
01
B
B
11
11
A
A
10
10
D
D
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4) Draw the logic diagram and verify the
correctness of the design
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• Example
• Design a 4-bit combinational circuit incremented
  (the circuit that adds one to a 4-bit binary
  number). The circuit can be designed using four
  half adders.