Question

. Assume that we have an arithmetic/logic unit that can carry out 20 distinct operations. Describe exactly what kind of multiplexer circuit would be needed to select exactly one of those 20 operations.

Short answer

A multiplexer with 20 inputs and 5 select lines is needed.

Step-by-step solution

Determine Multiplexer Inputs

To choose from 20 operations, the multiplexer needs to have 20 input lines. Each line will correspond to one operation. So, the multiplexer circuit will require 20 inputs.

Determine the Required Select Lines

To select one of the 20 operations, the multiplexer requires select lines. The number of select lines needed is determined by the formula 2^n = number of inputs. Therefore, we need \(n\) select lines such that \(2^n \geq 20\). The smallest \(n\) is 5 because \(2^5 = 32\).

Design the Multiplexer

The multiplexer circuit should be designed to have 20 input lines and 5 select lines. This design will allow the selection of one out of 20 operations based on the 5-bit binary number provided at the select lines.

Implement Logic Wiring

Wire each of the 20 operations to each of the 20 input lines. Connect the select lines to a device that can output a 5-bit binary value to select the desired operation.

Key concepts

Arithmetic Logic Unit (ALU)

An Arithmetic Logic Unit, often abbreviated as ALU, is a critical component of a computer's central processing unit (CPU). It handles two main types of operations: arithmetic and logic.
This means it performs tasks like addition, subtraction, and logical operations (e.g., AND, OR, NOT) that are fundamental to running programs can produce:

• Arithmetic Operations
• Logical Operations
• Bitwise Operations

The ALU plays a significant role in executing program commands and moving data effectively within a computer. Since it's capable of performing distinct tasks, specific signals need to direct it to the operation being carried out.
In the context of multiplexer circuit design, an ALU can have multiple operations, such as the 20 mentioned, and requires an efficient way to select one specific operation to execute at a time.

Select Lines

Select lines are crucial in the context of a multiplexer, which is a device that can choose one of several input signals and forward the selected input into a single line.
In our multiplexer circuit for the ALU, we need to direct exactly one of 20 operations. This is done through the select lines.
The number of select lines needed is calculated using the formula \(2^n \geq m\), where \(m\) is the number of inputs, in this case, 20. To satisfy this condition, we find that \(n = 5\) select lines are necessary because \(2^5 = 32\), which covers our 20 required inputs.
Select lines thus play a vital role as they allow the multiplexer to control which of the multiple inputs (operations) gets routed to the output. They achieve this by addressing binary numbers, which directly correlate with the inputs.

Binary Number System

The Binary Number System is foundational to the operation of digital computers, including multiplexers and ALUs.
It's based on only two digits: 0 and 1.
This system effectively encodes information that the computer can interpret.
When using a binary number system:

• Each bit represents an increasing power of 2.
• A group of bits can express any number or command in a computer.

For selecting one of the 20 operations in an ALU's multiplexer, the select lines utilize a 5-bit binary number. This binary number acts as an address to choose one of the 20 possible operations.
Understanding binary is critical because it translates high-level operations into electronic signals that guide circuitry in performing the right tasks.