Implementing Logic Functions with Decoders and MUXs
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Implementing Boolean Functions with Truth Tables
Problem 1: Implement the following function F(A, B, C) = Σ(2, 5, 6, 7) using a truth table with:
- a. Decoder
- b. Multiplexer
Truth Table for Function F(A, B, C)
| A | B | C | F |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
a. Decoder Implementation
A decoder is used to select one of the outputs based on the input combination. In this case, we have 3 inputs (A, B, C) and 8 possible combinations. The truth table shows that F = 1 for input combinations (010), (101), (110), and (111), and F = 0 for all other combinations. To implement this, we connect the corresponding decoder outputs to an OR gate.
b. Multiplexer Implementation
A multiplexer (MUX) selects one of the input lines and passes it to the output. The selection is controlled by select lines. Since there are 3 inputs (A, B, C), we can use an 8-to-1 multiplexer. The truth table shows that for inputs (010), (101), (110), and (111), the output should be 1, and for all other inputs, the output should be 0. We connect the data lines corresponding to these minterms to a logic high signal.
Advanced Logic Function Implementation
Problem 2: Implement the following function with a truth table: F = Σ(0, 1, 3, 4, 7, 8, 9, 10, 12, 15) using:
- Decoder
- Multiplexer
- PLA (Programmable Logic Array)
Truth Table for Function F
| A | B | C | F |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
1) Decoder Method
To implement this, we use a 3-to-8 decoder (or a 4-to-16 decoder for 4-variable functions) and then apply an OR gate to the outputs based on the function F. The decoder outputs corresponding to the inputs that yield F = 1 are: 0, 1, 3, 4, 7, 8, 9, 10, 12, and 15. We take these specific outputs and apply OR logic to achieve the final result.
2) Multiplexer Method
We can use a multiplexer where the select lines are A, B, and C. The data inputs will be set to 1 or 0 depending on whether the corresponding row in the truth table has F = 1 or F = 0.
3) PLA (Programmable Logic Array)
We can program the PLA to directly implement the truth table. The input lines will be A, B, and C, and the output line will be F. We program the PLA to produce logic 1s for the rows where F = 1 and logic 0s for the rows where F = 0.