Question

How to design 4 1 multiplexer?

Answer 1

A 4-to-1 multiplexer (4:1 MUX) is a combinational circuit that selects one of the four input data lines and forwards it to a single output line. This selection is controlled by two selection lines. Here's a detailed step-by-step guide to design a 4:1 multiplexer:

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### **Step 1: Understand the Functionality**
- **Inputs**: Four data inputs: \( I_0, I_1, I_2, I_3 \).
- **Select Lines**: Two select lines: \( S_0 \) and \( S_1 \), which determine which input is selected.
- **Output**: One output: \( Y \), which reflects the selected input.

The relation between the select lines and the inputs is as follows:
- When \( S_1S_0 = 00 \), \( Y = I_0 \).
- When \( S_1S_0 = 01 \), \( Y = I_1 \).
- When \( S_1S_0 = 10 \), \( Y = I_2 \).
- When \( S_1S_0 = 11 \), \( Y = I_3 \).

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### **Step 2: Write the Truth Table**
Create a truth table to define the behavior of the multiplexer.

| \( S_1 \) | \( S_0 \) | Output \( Y \) |
|-----------|-----------|----------------|
| 0         | 0         | \( I_0 \)      |
| 0         | 1         | \( I_1 \)      |
| 1         | 0         | \( I_2 \)      |
| 1         | 1         | \( I_3 \)      |

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### **Step 3: Derive the Logic Equation**
The output \( Y \) can be expressed as:
\[
Y = S_1'S_0'I_0 + S_1'S_0I_1 + S_1S_0'I_2 + S_1S_0I_3
\]
Where:
- \( S_1' \) and \( S_0' \) represent the complements (NOT) of \( S_1 \) and \( S_0 \).
- The terms \( S_1'S_0'I_0 \), \( S_1'S_0I_1 \), \( S_1S_0'I_2 \), and \( S_1S_0I_3 \) represent the contribution of each input based on the select line conditions.

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### **Step 4: Design the Circuit Using Logic Gates**
#### **Components Required**:
1. **AND gates**: To combine the select line conditions with the corresponding input.
2. **OR gate**: To sum up the contributions of all inputs.
3. **NOT gates**: To generate the complements of \( S_1 \) and \( S_0 \).

#### **Implementation**:
1. **NOT Gates**:
   - \( S_1' = NOT(S_1) \)
   - \( S_0' = NOT(S_0) \)

2. **AND Gates**:
   - \( A_0 = S_1' \cdot S_0' \cdot I_0 \)
   - \( A_1 = S_1' \cdot S_0 \cdot I_1 \)
   - \( A_2 = S_1 \cdot S_0' \cdot I_2 \)
   - \( A_3 = S_1 \cdot S_0 \cdot I_3 \)

3. **OR Gate**:
   Combine all the AND gate outputs to produce the final output:
   \[
   Y = A_0 + A_1 + A_2 + A_3
   \]

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### **Step 5: Draw the Logic Diagram**
1. **Inputs**: Four data lines (\( I_0, I_1, I_2, I_3 \)) and two select lines (\( S_1, S_0 \)).
2. **NOT Gates**: Generate \( S_1' \) and \( S_0' \).
3. **AND Gates**: Four AND gates, each combining the select lines and one data input.
4. **OR Gate**: A single OR gate combining the outputs of the four AND gates to produce the final output \( Y \).

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### **Step 6: Verify the Design**
To confirm the functionality, check the truth table against the circuit diagram:
- For \( S_1S_0 = 00 \), ensure \( Y = I_0 \).
- For \( S_1S_0 = 01 \), ensure \( Y = I_1 \).
- For \( S_1S_0 = 10 \), ensure \( Y = I_2 \).
- For \( S_1S_0 = 11 \), ensure \( Y = I_3 \).

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### **Practical Considerations**
1. Use Integrated Circuits (ICs) like the 74LS157 if a ready-made multiplexer chip is acceptable.
2. If designing from scratch, ensure proper power supply and logic level compatibility.
3. For complex systems, cascaded multiplexers can be used for higher input counts (e.g., an 8:1 MUX using two 4:1 MUXes).

By following these steps, you can design and implement a functional 4-to-1 multiplexer.