2. Digital Systems
Section 17
Decoders and Encoders

3. Lecture
Digital Systems
Binary Decoders
A binary decoder can be considered as a black box with n input lines and 2n output lines.
Only one output line is set to 1 for a given input. I n 1 –
inputs E
Enable 2
outputs O
Binary Decoder

4. Lecture
Digital Systems
Binary Decoders
Binary decoders convert an n-bit input to a single output. It uses its n-bit input to determine which of the 2n outputs will be uniquely activated.
Binary decoders can be developed using AND or OR Gates.
Later on, binary decoders can be implemented in logic circuits.
The outputs of a decoder are minterms. That is why decoders are sometimes called as minterm generators.
We can easily use a decoder to implement any sum of minterms expression.
Note: A minterm is a Boolean expression resulting in 1 only for the output of a single row (in a truth table) or a single cell (in a Karnaugh map), and 0s for all other row or cells, respectively.

5. Lecture
Digital Systems
2-to-4 Binary Decoder
A circuit of 2-to-4 binary decoder is shown below.
Binary
Decoder
2-to-4
2 inputs
4 outputs
Enable Y X F0 F1 F2 F3
The truth table shows that for any given input combination, exactly one output will turn to 1.
The enable must be set to 1 to get an output.

6. Lecture
Digital Systems
3-to-8 Binary Decoder
Try to understand the logic circuit of 3-to-8 binary decoder below.
Binary
Decoder
3-to-8
3 inputs
8 outputs
Enable X Y Z F0 F7

7. 3-to-8 Decoder Using Two 2-to-4 Decoders
Lecture
Digital Systems
3-to-8 Decoder Using Two 2-to-4 Decoders Y E
Enable Z X F0 F1 F2 F3 F4 F5 F6 F7 I0 I1 O0 O1 O2 O3
Using Enable signal, which is firstly connected to a combinational circuit
Binary
Decoder
3-to-8
3 inputs
8 outputs
Enable

8. Implementing Functions Using Decoders
Lecture
Digital Systems
Implementing Functions Using Decoders
Any n-variable logic function can be implemented using a single n-to-2n decoder, to generate the minterms (as given by the truth table or the Karnaugh map).
All the minterms are then summed up by an OR Gate.
Any combinational circuit with n inputs and m outputs can be implemented with an n-to-2n decoder and m OR Gates.
A decoder is suitable to use when a circuit has many outputs, where each output function is expressed with its minterms.

9. Implementing Functions Using Decoders
Lecture
Digital Systems
Implementing Functions Using Decoders
Full Adder
Binary Decoder 3-to-8 1 2 X Y Z 3 4 5 6 7 S C
Each OR Gate collects all minterms of respective output
One OR Gate for every output

10. Implementing Functions Using Decoders
Lecture
Digital Systems
Implementing Functions Using Decoders
Making a 4-to-16 Decoder from two 3-to-8 Decoders
In this example, only one decoder can be active at a time.
X, Y, and Z effectively select the output line for a certain value of W.
Binary
Decoder
3-to-8
MSB W is used as Enable signal

11. Implementing Functions Using Decoders
Lecture
Digital Systems
Implementing Functions Using Decoders
Making a Multiplexer from a 2-to-4 Decoder
Output
Enable = 1 E I0 I1 O0 O1 O2 O3 X Y F0 F1 F2 F3
Binary
Decoder
2-to-4

12. Lecture
Digital Systems
Binary Encoders
Binary encoders convert its 2n inputs to an n-bit output.
Only one high value is presented as input. The result is a binary-encoded output of size n.
Binary encoders are useful for compressing data. 2 n
inputs I 1 – O
outputs
Binary Encoder

13. Lecture
Digital Systems
4-to-2 Binary Encoder
In encoder circuit only one input may be set high (1) at a certain time.
The output is a 2-bit number. 1 I0 I1 I2 I3 Y1 Y0 I0 I1 I2 I3 Y0 Y1
Binary
Encoder
4-to-2
2 outputs
4 inputs I0 I1 I2 I3 Y0 Y1

14. Lecture
Digital Systems
8-to-3 Binary Encoder
In this encoder circuit, at any certain time, only one input line has the value of 1.
The output is binary digits of Y2 Y1 Y0.
This circuit is also called octal-to-binary encoder. I0 I1 I2 I3 I4 I5 I6 I7
Y2 = I4 + I5 + I6 + I7
Y1 = I2 + I3 + I6 + I7
Y0 = I1 + I3 + I5 + I7
Binary
Encoder
8-to-3
8 inputs
3 outputs I0 I7 Y0 Y1 Y2

15. ? 4-to-2 Priority Encoder ⇔
Lecture
Digital Systems
4-to-2 Priority Encoder
For some application, there can be more than one input line that have a value of 1.
One way to handle these inputs is to ignore the lower inputs and only process the highest input (‘priority input’). 1 X I0 I1 I2 I3 Y1 Y0
This encoder has 4 input lines: I3, I2, I1, and I0.
I3 has the highest priority, I0 has the lowest priority. I0 I1 I2 I3 Y0 Y1
Draw the Karnaugh maps for both outputs Y1 and Y0. ?
Y1 = I3’·I2 + I3
Y0 = I3’·I2’·I1 + I3
Y1 = I2 + I3
Y0 = I2’·I1 + I3 ⇔

16. Lecture
Digital Systems
8-to-3 Priority Encoder
The following is the truth table of an 8-to-3 priority encoder.
X indicates don’t cares.

17. Lecture
Digital Systems
8-to-3 Priority Encoder
As one alternative solution, we can use a priority circuit that implements the following logic functions:
H0 = I7’·I6’·I5’·I4’·I3’·I2’·I1’·I0
H1 = I7’·I6’·I5’·I4’·I3’·I2’·I1
H2 = I7’.I6’·I5’·I4’·I3’·I2
H3 = I7’.I6’·I5’·I4’·I3
H4 = I7’·I6’·I5’·I4
H5 = I7’·I6’·I5
H6 = I7’·I6
H7 = I (Highest Priority)

18. Lecture
Digital Systems
8-to-3 Priority Encoder
The overall circuit of the priority encoder will be as shown below.
The equations that express the binary encoder are:
Y0 = I1 + I3 + I5 + I7 Y1 = I2 + I3 + I6 + I7 Y2 = I4 + I5 + I6 + I7 I1 I2 I3 Y1 Y2 I4 I5 I6 I0 Y0 I7
Binary encoder
Priority Circuit H1 H2 H3 H4 H5 H6 H0 H7
IDLE
Priority Encoder

19. Design Exercise: 8-to-3 Priority Encoder
Lecture
Digital Systems
Design Exercise: 8-to-3 Priority Encoder
Design an 8-to-3 priority encoder that will deliver only the lowest input and process it to becomes the output.

20. Priority Encoder Navigation
Lecture
Digital Systems
Priority Encoder Navigation
One possible application of encoder is in positional control as used on ships or robotics.
The angular position of a compass or the rotary position of a joint is converted into a digital code by a 8-to-3 priority encoder.
An example of a simple 8-position to 3-bit output encoder is shown below.

21. Trivia: Disreputable Numbers
Lecture
Digital Systems
Trivia: Disreputable Numbers
Find your birthday on the cards.
Tell me the color of the cards.
I can guess your birthday.

22. Lecture
Digital Systems
Homework 12 (1/2)
Sketch the output waveforms of a 8-to-3 priority encoder based on the given waveforms below. I6 I7 1
I0 = I1 = I2 = I3 = I4 = I5 = 1
Binary
Decoder
3-to-8
Enable I0 I1 I2 O0
... O7
i. Given one of such decoder, create the following equation F(A,B,C) = Σm(1,3,5,6) G(A,B,C) = Σm(2,3,4,7)
ii. Given two of such decoder, create the following equation: F(A,B,C,D) = Σm(2,4,6,14)

23. Homework 12 (2/2) Enable I0 I1 I2 O0 ... O7 O2 Enable I0 ... I7 O1 O0
Use 3-to-8 binary decoders and/or 8-to-3 binary encoders and logic gates (where necessary) to create a circuit that will be able to memorize the birth dates of 8 people, as shown in the next table. Realize the connections for Input 1 and Input 5 only.
Input
Output 1 16 2 23 3 05 4 14 5 17 6 03 7 09 8 27
Binary
Decoder
3-to-8
Enable I0 I1 I2 O0
... O7 O2
Binary
Encoder
8-to-3
Enable I0
... I7 O1 O0
Please write your Class number after your Student ID.
Deadline: 1 day before class.
Monday, 11 December 2017 (Class 2).
Tuesday, 12 December 2017 (Class 1).