Design a 4-to-1 multiplexer using 2-to-1 multiplexers (tree structure)

Design a 4-to-1 multiplexer using 2-to-1 multiplexers (tree structure). Provide truth table, k-map, logic equations, block diagram and circuit diagram.

The correct answer and explanation is :

A 4-to-1 multiplexer (MUX) can be designed using 2-to-1 multiplexers in a tree structure. This design allows us to use basic building blocks (2-to-1 MUXes) to achieve the functionality of a larger MUX. Let’s break this down step by step.

Truth Table for 4-to-1 Multiplexer:

A 4-to-1 multiplexer has 4 input lines (I0, I1, I2, I3), 2 select lines (S1, S0), and 1 output line (Y). The select lines determine which of the 4 inputs is passed to the output.

S1	S0	I0	I1	I2	I3	Y
0	0	0	0	0	0	I0
0	1	0	0	0	0	I1
1	0	0	0	0	0	I2
1	1	0	0	0	0	I3

K-map for 4-to-1 Multiplexer:

The 4-to-1 MUX can be represented by a 2-variable K-map (since there are 2 select lines). The function for the output is a selection between the four inputs.

S1 \ S0	0	1
0	I0	I1
1	I2	I3

From this, we can write the Boolean expression for the output (Y) in terms of the inputs and select lines as:

[
Y = (S1′ \cdot S0′ \cdot I0) + (S1′ \cdot S0 \cdot I1) + (S1 \cdot S0′ \cdot I2) + (S1 \cdot S0 \cdot I3)
]

Block Diagram of the 4-to-1 MUX:

The 4-to-1 MUX can be constructed using 2-to-1 MUXes in a tree structure.

• First, use two 2-to-1 MUXes to select between I0 and I1, and between I2 and I3.
• Then, use another 2-to-1 MUX to select between the outputs of the first two MUXes, based on the select line S1.

Circuit Diagram:

• The first 2-to-1 MUX (MUX1) takes inputs I0 and I1 and is controlled by S0.
• The second 2-to-1 MUX (MUX2) takes inputs I2 and I3 and is controlled by S0.
• A third 2-to-1 MUX (MUX3) takes the outputs of MUX1 and MUX2, controlled by S1, to produce the final output Y.

Explanation:

This design takes advantage of the hierarchical structure of multiplexers. By combining smaller MUXes (2-to-1) in stages, we can build up a larger 4-to-1 MUX. The advantage of using this tree structure is that it reduces the number of logic gates and makes the design more modular and manageable.

Block Diagram:

Below is the block diagram of the 4-to-1 multiplexer designed using 2-to-1 multiplexers:

         +-------+
I0 ---->|       |
         | MUX1  |---->+
I1 ---->|       |     |
         +-------+     |
                      |
         +-------+     |
I2 ---->|       |     | 
         | MUX2  |----+
I3 ---->|       |
         +-------+     
                      |
         +-------+     |
         |       |     |
S1 ---->| MUX3  |----> Y
         |       |
S0 ---->|       |
         +-------+

Here is the block diagram of the 4-to-1 multiplexer designed using 2-to-1 multiplexers. It shows the hierarchical structure where the select lines (S1, S0) control which input (I0, I1, I2, I3) is passed through the output (Y). The tree structure of the multiplexers (MUX1, MUX2, and MUX3) is clearly illustrated.