Digital Electronics

Encoders and Decoders

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Decoders

A decoder has
 N inputs
 2N outputs

A decoder selects one of 2N outputs by decoding the binary value on
the N inputs.

The decoder generates all of the minterms of the N input variables.
 Exactly one output will be active for each combination of the inputs.

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Decoders

W = A'.B'
Out0 W
B I0 Out1 X X = A.B'
A I1 Out2 Y Y = A'.B
Out3 Z
msb Z = A.B
Active-high outputs

A B W X Y Z
0 0 1 0 0 0
0 1 0 1 0 0
1 0 0 0 1 0
1 1 0 0 0 1

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Decoders

W = (A'.B')'
Out0 W
B I0 Out1 X X = (A.B')'
A I1 Out2 Y Y = (A'.B)'
Out3 Z
msb Z = (A.B)'
Active-low outputs

A B W X Y Z
0 0 0 1 1 1
0 1 1 0 1 1
1 0 1 1 0 1
1 1 1 1 1 0

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Decoders
msb

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Decoder with Enable

Out0 W
B I0
high-level Out1 X
enable A I1
Out2 Y
Out3 Z
Enable En

En A B W X Y Z
1 0 0 1 0 0 0
1 0 1 0 1 0 0
enabled
1 1 0 0 0 1 0
1 1 1 0 0 0 1
disabled 0 x x 0 0 0 0

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Decoder with Enable

Out0 W
B I0
low-level Out1 X
enable A I1
Out2 Y
Out3 Z
Enable En

En A B W X Y Z
0 0 0 1 0 0 0
0 0 1 0 1 0 0
enabled
0 1 0 0 0 1 0
0 1 1 0 0 0 1
disabled 1 x x 0 0 0 0

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Decoders

Exercise:

Design a 4-to-16 decoder using

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Encoders

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Encoders

An encoder has
 2N inputs
 N outputs

An encoder outputs the binary value of the selected (or
active) input.

An encoder performs the inverse operation of a decoder.

Issues
 What if more than one input is active?
 What if no inputs are active?

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Encoders

D I0
C I1 Out0 Z
B I2
Out1 Y

A I3

A B C D Y Z
0 0 0 1 0 0
0 0 1 0 0 1
0 1 0 0 1 0
1 0 0 0 1 1

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Priority Encoders


If more than one input is active, the higher-order input
has priority over the lower-order input.
 The higher value is encoded on the output

A valid indicator, d, is included to indicate whether or not
the output is valid.
 Output is invalid when no inputs are active

d=0
 Output is valid when at least one input is active

d=1

Why is the valid indicator needed?

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Priority Encoders

msb

Valid bit

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Designing logic circuits using multiplexers

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Using an n-input Multiplexer

Use an n-input multiplexer to realize a logic circuit for a
function with n minterms.
 m = 2n, where m = # of variables in the function

Each minterm of the function can be mapped to an input
of the multiplexer.

For each row in the truth table, for the function, where
the output is 1, set the corresponding input of the
multiplexer to 1.
 That is, for each minterm in the minterm expansion of the
function, set the corresponding input of the multiplexer to 1.

Set the remaining inputs of the multiplexer to 0.

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Using an n-input Mux

Example:

Using an 8-to-1 multiplexer, design a logic circuit to realize the following

F(A,B,C) = m(2, 3, 5, 6, 7)

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Using an n-input Mux

Example:

Using an 8-to-1 multiplexer, design a logic circuit to realize the following

F(A,B,C) = m(1, 2, 4)

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Using an (n / 2)-input Multiplexer

Use an (n / 2)-input multiplexer to realize a logic circuit
for a function with n minterms.
 m = 2n, where m = # of variables in the function

Group the rows of the truth table, for the function, into
(n / 2) pairs of rows.
 Each pair of rows represents a product term of (m – 1)
variables.
 Each pair of rows can be mapped to a multiplexer input.

Determine the logical function of each pair of rows in
terms of the mth variable.
 If the mth variable, for example, is x, then the possible values
are x, x', 0, and 1.

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Using an (n / 2)-input Mux

Example: F(x,y,z) = m(1, 2, 6, 7)

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Using an (n / 2)-input Mux

Example: F(A,B,C,D) = m(1,3,4,11,12–15)

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 Using an (n / 4)-input Mux

The design of a logic circuit using an (n / 2)-input multiplexer can be

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Designing logic circuits using decoders

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Using an n-output Decoder

Use an n-output decoder to realize a logic circuit for a
function with n minterms.

Each minterm of the function can be mapped to an output
of the decoder.

For each row in the truth table, for the function, where the
output is 1, sum (or “OR”) the corresponding outputs of
the decoder.
 That is, for each minterm in the minterm expansion of the
function, OR the corresponding outputs of the decoder.

Leave remaining outputs of the decoder unconnected.

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Using an n-output Decoder

Example:

Using a 3-to-8 decoder, design a logic circuit to realize the following

F(A,B,C) = m(2, 3, 5, 6, 7)

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Using an n-output Decoder

Example:

Using two 2-to-4 decoders, design a logic circuit to realize the following
Boolean function

F(A,B,C) = m(0, 1, 4, 6, 7)

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Questions?

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