# lecture #16 - multiplexers, decoders and encoders

TRANSCRIPT

ECE 301 – Digital Electronics

Multiplexers, Decoders and Encoders

(Lecture #16)

The slides included herein were taken from the materials accompanying

Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,

and were used with permission from Cengage Learning.

Spring 2011 ECE 301 - Digital Electronics 2

Multiplexers

Spring 2011 ECE 301 - Digital Electronics 3

Multiplexers

● A multiplexer has

– 2n data inputs

– n control inputs

– 1 output

● A multiplexer routes (or connects) the selected data input to the output.

– The value of the control inputs determines the data input that is selected.

Spring 2011 ECE 301 - Digital Electronics 4

Multiplexers

Z = A′.I0 + A.I

1

Datainputs

Controlinput

Spring 2011 ECE 301 - Digital Electronics 5

Multiplexers

Z = A′.B'.I0 + A'.B.I

1 + A.B'.I

2 + A.B.I

3

MSB LSB

A B Z

0 0 I0

0 1 I1

1 0 I2

1 1 I321 20

m0 = A'.B'

m1 = A'.B

m2 = A.B'

m3 = A.B

Spring 2011 ECE 301 - Digital Electronics 6

Multiplexers

Z = A′.B'.C'.I0 + A'.B'.C.I

1 + A'.B.C'.I

2 + A'.B.C.I

3 +

A.B'.C'.I0 + A.B'.C.I

1 + A'.B.C'.I

2 + A.B.C.I

3

MSB LSB

A B C Z

0 0 0 I0

0 0 1 I1

0 1 0 I2

0 1 1 I3

1 0 0 I4

1 0 1 I5

1 1 0 I6

1 1 1 I72220

m0

m1

m2

m3

m4

m5

m6

m7

Spring 2011 ECE 301 - Digital Electronics 7

Multiplexers

2n-120

Z = ΣΣΣΣ mi.Ii

Spring 2011 ECE 301 - Digital Electronics 8

Decoders

Spring 2011 ECE 301 - Digital Electronics 9

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.

What does “active” mean?

Spring 2011 ECE 301 - Digital Electronics 10

Decoders

A B Z0 Z1 Z2 Z3

0 0 1 0 0 0

0 1 0 1 0 0

1 0 0 0 1 0

1 1 0 0 0 1

msb

2-to-4Decoder

A

B

Z0

Z1

Z2

Z3

active-high output

Zi = mi

m0

m1

m2

m3

Spring 2011 ECE 301 - Digital Electronics 11

Decoders

A B Z0 Z1 Z2 Z3

0 0 0 1 1 1

0 1 1 0 1 1

1 0 1 1 0 1

1 1 1 1 1 0

msb

active-low output

Zi = (mi)' = Mi

M0

M1

M2

M3

A

B

Z0

Z1

Z2

Z3

2-to-4Decoder

Spring 2011 ECE 301 - Digital Electronics 12

Decodersmsb

3-to-8Decoder

Spring 2011 ECE 301 - Digital Electronics 13

Decoder with Enable

En A B Z0 Z1 Z2 Z3

1 0 0 1 0 0 0

1 0 1 0 1 0 0

1 1 0 0 0 1 0

1 1 1 0 0 0 1

0 x x 0 0 0 0

enabled

disabled

active-high enable

En

2-to-4Decoder

withEnable

A Z0

Z1

Z2

Z3

B

Spring 2011 ECE 301 - Digital Electronics 14

Decoder with Enable

En A B Z0 Z1 Z2 Z3

0 0 0 1 0 0 0

0 0 1 0 1 0 0

0 1 0 0 0 1 0

0 1 1 0 0 0 1

1 x x 0 0 0 0

enabled

disabled

active-low enable

En

2-to-4Decoder

withEnable

A Z0

Z1

Z2

Z3

B

Spring 2011 ECE 301 - Digital Electronics 15

Encoders

Spring 2011 ECE 301 - Digital Electronics 16

Encoders● An encoder has

– 2n inputs

– n outputs

● Outputs the binary value of the selected (or active) input.

● Performs the inverse operation of a decoder.

● Issues

– What if more than one input is active?

– What if no inputs are active?

Spring 2011 ECE 301 - Digital Electronics 17

Encoders

Y0 Y1 Y2 Y3 A B

1 0 0 0 0 0

0 1 0 0 0 1

0 0 1 0 1 0

0 0 0 1 1 1

4-to-2Encoder

A

B

Y0

Y1

Y2

Y3

Spring 2011 ECE 301 - Digital Electronics 18

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 = 1Why is the valid indicator needed?

Spring 2011 ECE 301 - Digital Electronics 19

Priority Encoders

Valid bit

msb

3-to-8Priority

Encoder

Spring 2011 ECE 301 - Digital Electronics 20

Circuit Design using Multiplexers

Spring 2011 ECE 301 - Digital Electronics 21

Using a 2n-input Multiplexer● Use a 2n-input multiplexer to realize a logic circuit for

a function with 2n minterms.

– n = # of control inputs = # of variables in the function

● Each minterm of the function can be mapped to a data input of the multiplexer.

● For each row in the truth table, for the function, where the output is 1, set the corresponding data 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.

Spring 2011 ECE 301 - Digital Electronics 22

Using an 2n-input Mux

Example:

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

F(A,B,C) = Σm(2, 3, 5, 6, 7)

Spring 2011 ECE 301 - Digital Electronics 23

Using an 2n-input Mux

Example:

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

F(A,B,C) = Σm(1, 2, 4)

Spring 2011 24

Using an 2(n-1)-input Multiplexer● Use a 2(n-1)-input multiplexer to realize a logic circuit

for a function with 2n minterms.

– n – 1 = # of control inputs; n = # of variables in function

● Group the rows of the truth table, for the function, into 2(n-1) pairs of rows.

– Each pair of rows represents a product term of (n – 1) variables.

– Each pair of rows is mapped to one data input of the mux.

● Determine the logical function of each pair of rows in terms of the remaining variable.

– If the remaining variable, for example, is x, then the possible values are x, x', 0, and 1.

Spring 2011 ECE 301 - Digital Electronics 25

Using an 2(n-1)-input Mux

Example: F(x,y,z) = Σm(1, 2, 6, 7)

Spring 2011 ECE 301 - Digital Electronics 26

Using an 2(n-1)-input Mux

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

Spring 2011 ECE 301 - Digital Electronics 27

Using a 2(n-2)-input Mux

A similar design approach can be implemented using a 2(n-2)-input multiplexer.

Spring 2011 ECE 301 - Digital Electronics 28

Circuit Design using Decoders

Spring 2011 ECE 301 - Digital Electronics 29

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.

Spring 2011 ECE 301 - Digital Electronics 30

Using an n-output Decoder

Example:

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

F(A,B,C) = Σm(2, 3, 5, 6, 7)

Spring 2011 ECE 301 - Digital Electronics 31

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) = Σm(0, 1, 4, 6, 7)

Spring 2011 ECE 301 - Digital Electronics 32

Hierarchical Design

Spring 2011 ECE 301 - Digital Electronics 33

Hierarchical Design

● Several issues arise when designing large multiplexers and decoders (as 2-level circuits).

– Number of logic gates gets prohibitively large

– Number of inputs to each logic gate (i.e. fan-in) gets prohibitively large

● Instead, design both hierarchically

– Use smaller elements as building blocks

– Interconnect building blocks in a multi-tier structure

Spring 2011 ECE 301 - Digital Electronics 34

Hierarchical Design

Exercise:

Design an 8-to-1 multiplexer using 4-to-1 and 2-to-1 multiplexers only.

Spring 2011 ECE 301 - Digital Electronics 35

Hierarchical Design

Exercise:

Design a 16-to-1 multiplexer using 4-to-1 multiplexers only.

Spring 2011 ECE 301 - Digital Electronics 36

Hierarchical Design

Exercise:

Design a 4-to-16 decoder using2-to-4 decoders only.

Spring 2011 ECE 301 - Digital Electronics 37

Questions?