What is decoder? Design a 3x8 decoder.

What is a Decoder?

A decoder is a combinational logic circuit that converts binary information from n input lines to a maximum of 2^n unique output lines. Its main function is to detect the presence of a particular binary code on its input and to indicate that presence by activating the corresponding output line. Decoders are widely used in digital electronics, such as in memory address decoding, data multiplexing, seven-segment displays, and more.

Designing a 3x8 Decoder

A 3x8 decoder has 3 input lines and 8 output lines. The 3 input lines can represent any binary number from 0 to 7 (2^3 = 8), and the decoder will activate one of the 8 outputs corresponding to the binary number on the input.

The truth table for a 3x8 decoder is as follows:

Inputs	Outputs
A2 A1 A0	O7 O6 O5 O4 O3 O2 O1 O0
0 0 0	0 0 0 0 0 0 0 1
0 0 1	0 0 0 0 0 0 1 0
0 1 0	0 0 0 0 0 1 0 0
0 1 1	0 0 0 0 1 0 0 0
1 0 0	0 0 0 1 0 0 0 0
1 0 1	0 0 1 0 0 0 0 0
1 1 0	0 1 0 0 0 0 0 0
1 1 1	1 0 0 0 0 0 0 0

Where A2, A1, and A0 are the input lines, and O7 to O0 are the output lines.

To design the 3x8 decoder, we need to create the logic expressions for each output. Each output is a min-term of the inputs, which means it is a product (AND) of the inputs or their complements. The expressions for each output are as follows:

• O0 = A2' A1' A0'
• O1 = A2' A1' A0
• O2 = A2' A1 A0'
• O3 = A2' A1 A0
• O4 = A2 A1' A0'
• O5 = A2 A1' A0
• O6 = A2 A1 A0'
• O7 = A2 A1 A0

Here, the prime (') denotes the NOT operation.

Now, let's draw the logic diagram for the 3x8 decoder using AND, NOT, and OR gates:

  A2 ───┐
        │
        ├─[NOT]──┐
        │        │
  A1 ───┼──┐     │
        │  │     │
        ├─[NOT]──┼──┐
        │        │  │
  A0 ───┼──┼──┐  │  │
        │  │  │  │  │
        ├─[NOT]─┼──┼──┼──[AND]── O0
        │       │  │  │
        ├───────┼──┼──┼──[AND]── O1
        │       │  │  │
        ├───────┼──┼──┼──[AND]── O2
        │       │  │  │
        ├───────┼──┼──┼──[AND]── O3
        │       │  │  │
        ├───────┼──┼──┼──[AND]── O4
        │       │  │  │
        ├───────┼──┼──┼──[AND]── O5
        │       │  │  │
        ├───────┼──┼──┼──[AND]── O6
        │       │  │  │
        ├───────┼──┼──┼──[AND]── O7

In the above diagram, the NOT gates generate the complements of the inputs. Each AND gate corresponds to one of the outputs and combines the inputs or their complements according to the min-term expressions.

Example

Let's say the inputs are A2=1, A1=0, and A0=1. This corresponds to the binary number '101', which is 5 in decimal. According to the truth table, only output O5 should be activated (high), and all other outputs should be low.

Input: A2=1, A1=0, A0=1 Expected Output: O7 O6 O5 O4 O3 O2 O1 O0 = 0 0 1 0 0 0 0 0

The logic expressions confirm this:

• O0 = A2' A1' A0' = 0
• O1 = A2' A1' A0 = 0
• O2 = A2' A1 A0' = 0
• O3 = A2' A1 A0 = 0
• O4 = A2 A1' A0' = 0
• O5 = A2 A1' A0 = 1 (activated)
• O6 = A2 A1 A0' = 0
• O7 = A2 A1 A0 = 0

Thus, the decoder correctly decodes the input '101' and activates output O5.