What is a Multiplexer? Design a Four-To-One-Line Multiplexer.

A **multiplexer** (often abbreviated as MUX) is a combinational circuit that selects one of several input signals and forwards the selected input to a single output line. The selection of the input is controlled by a set of selection lines (or control lines). Multiplexers are widely used in digital circuits for data routing, signal switching, and resource sharing. ### Four-To-One-Line Multiplexer A **Four-To-One-Line Multiplexer** has four input lines, two selection lines, and one output line. The inputs are typically labeled as \( I_0, I_1, I_2, \) and \( I_3 \), while the selection lines are labeled as \( S_0 \) and \( S_1 \). The output line is labeled as \( Y \). #### Truth Table The truth table for a 4-to-1 multiplexer is as follows: | \( S_1 \) | \( S_0 \) | Output \( Y \) | |-----------|-----------|-----------------| | 0 | 0 | \( I_0 \) | | 0 | 1 | \( I_1 \) | | 1 | 0 | \( I_2 \) | | 1 | 1 | \( I_3 \) | #### Logic Diagram To design a 4-to-1 multiplexer, we can use basic logic gates (AND, OR, NOT). The output \( Y \) can be expressed in terms of the inputs and selection lines as follows: \[ Y = (I_0 \cdot \overline{S_1} \cdot \overline{S_0}) + (I_1 \cdot \overline{S_1} \cdot S_0) + (I_2 \cdot S_1 \cdot \overline{S_0}) + (I_3 \cdot S_1 \cdot S_0) \] Where: - \( \overline{S_1} \) and \( \overline{S_0} \) are the NOT operations on the selection lines. #### Implementation Steps 1. **Inversion of Selection Lines**: Use NOT gates to get \( \overline{S_0} \) and \( \overline{S_1} \). 2. **AND Gates**: Create four AND gates: - The first AND gate takes \( I_0, \overline{S_1}, \overline{S_0} \). - The second AND gate takes \( I_1, \overline{S_1}, S_0 \). - The third AND gate takes \( I_2, S_1, \overline{S_0} \). - The fourth AND gate takes \( I_3, S_1, S_0 \). 3. **OR Gate**: Finally, connect the outputs of the four AND gates to a single OR gate to produce the output \( Y \). #### Schematic Representation Here’s a simplified representation of the schematic: ``` Inputs: I0, I1, I2, I3 Select Lines: S0, S1 +---+ +---+ I0 ------>| | | | | & |-------| | S0 ------>| | | | +---+ | | +---+ | | I1 ------>| | | | | & |-------| | S0 ------>| | | | +---+ | | +---+ | | I2 ------>| | | | | & |-------| | S1 ------>| | | | +---+ | | +---+ | | I3 ------>| | | | | & |-------| | S1 ------>| | | | +---+ | | +---+ | | Y (Output) ``` This design allows the multiplexer to select one of the four inputs based on the values of the selection lines \( S_0 \) and \( S_1 \).