🧠 Introduction
Multiplexers (MUX) and Demultiplexers (DEMUX) are powerful combinational circuits that can be used to implement Boolean functions or logical expressions. These devices allow us to use a small number of gates (AND, OR, NOT) to represent complex Boolean equations efficiently.
- Multiplexer (MUX): A MUX selects one of many inputs based on the values of control lines (or select lines).
- Demultiplexer (DEMUX): A DEMUX takes one input and distributes it to one of many output lines based on the values of control lines.
We can implement Boolean functions directly using MUX and DEMUX by treating them as logical selectors for various input combinations.
🎯 1. Implementation of Boolean Equations Using Multiplexer (MUX)
A Multiplexer (MUX) can be used to implement any Boolean function by selecting the appropriate combination of inputs.
General Idea:
To implement a Boolean function using a MUX:
- The inputs to the MUX will correspond to the different combinations of output values of the Boolean function.
- The select lines of the MUX correspond to the variables of the Boolean function.
🔥 Example 1: Implementing a Boolean Function using 4×1 MUX
Let’s take a 4×1 MUX and implement a simple Boolean function.
- Boolean Function: F(A, B) = A’B + AB’
- This function can be implemented using a 4×1 MUX with the following steps:
Truth Table for F(A, B):
| A | B | F(A, B) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
- The inputs to the MUX (I₀, I₁, I₂, I₃) can be taken from the truth table.
- The select lines (S₀, S₁) will be connected to the variables A and B, respectively.
Now, map the Boolean values to the MUX inputs:
- I₀ = 0
- I₁ = 1
- I₂ = 1
- I₃ = 0
MUX Configuration:
- Select lines (S₁, S₀) = A, B
- Inputs (I₃, I₂, I₁, I₀) = 0, 1, 1, 0
Now, the output F(A, B) is given by the MUX logic, where the MUX selects the appropriate input based on the values of A and B.
🔥 Example 2: Implementing a 3-variable Boolean Function using 8×1 MUX
Let’s implement a more complex Boolean function using an 8×1 MUX:
- Boolean Function: F(A, B, C) = AB + A’C
This function can be implemented using an 8×1 MUX with the following steps:
Truth Table for F(A, B, C):
| A | B | C | F(A, B, C) |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
- The inputs to the MUX will correspond to the output values.
- The select lines (S₂, S₁, S₀) will correspond to the variables A, B, and C.
MUX Configuration:
- Select lines (S₂, S₁, S₀) = A, B, C
- Inputs (I₇, I₆, I₅, I₄, I₃, I₂, I₁, I₀) = 1, 1, 1, 0, 1, 1, 0, 0
Now, the MUX selects the appropriate input based on the values of A, B, and C, providing the output F(A, B, C).
🎯 2. Implementation of Boolean Equations Using Demultiplexer (DEMUX)
A Demultiplexer (DEMUX) can be used to distribute an input signal to multiple outputs based on the select lines.
General Idea:
To implement a Boolean function using a DEMUX:
- The input to the DEMUX will correspond to the output signal of the Boolean function.
- The select lines of the DEMUX will be connected to the variables of the Boolean function.
- The outputs of the DEMUX will represent different cases of the Boolean equation.
🔥 Example 1: Implementing a Boolean Function Using 1×4 DEMUX
Let’s consider a 1×4 DEMUX to implement a simple Boolean function:
- Boolean Function: F(A, B) = A’B + AB’
- This function can be implemented using a 1×4 DEMUX with the following steps:
Truth Table for F(A, B):
| A | B | F(A, B) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
- The input to the DEMUX is the signal F(A, B).
- The select lines are A and B.
Now, connect the outputs based on the truth table.
- For A=0, B=0, output will be 0.
- For A=0, B=1, output will be 1.
- For A=1, B=0, output will be 1.
- For A=1, B=1, output will be 0.
🔥 Example 2: Using DEMUX to Route Different Boolean Functions
A DEMUX can also be used to route different Boolean functions to various outputs based on the select lines.
- Input: A signal representing a complex Boolean equation.
- Select Lines: Variables of the Boolean equation.
- Outputs: Individual bits of the output function.
For example, if we have a function like F(A, B, C), the DEMUX can be used to distribute the values based on the select lines and the given Boolean logic.
🎯 Summary of Implementing Boolean Functions
- Multiplexers (MUX) are used to select one of many inputs based on select lines, and this makes them great for implementing Boolean functions where the output depends on the values of multiple variables.
- Demultiplexers (DEMUX) are used to route one input to one of many outputs based on select lines, which can be used for distributing a Boolean output signal to different parts of a system.