Memory Addressing Fundamentals

Memory addressing is a fundamental concept in computer architecture that involves assigning unique addresses to each storage location within a memory module. This allows the processor to access specific data stored at a particular address. The question asks about addressing a \(32\text{K} \times 16\) memory.

Let's break down the memory specification:

• The term "\(32\text{K}\)" refers to the total number of unique addressable locations or words in the memory. Here, 'K' stands for Kilo, which in digital systems typically means \(2^{10}\) (1024).
• The term "\(\times 16\)" indicates that each memory location (or word) can store \(16\) bits of data. This part of the specification is relevant for the data bus width but not directly for determining the number of address lines or decoder gates.

To find the number of address lines required, we first need to convert the total number of memory locations into a power of \(2\).

We have \(32\text{K}\) memory locations.

Since \(32\) can be expressed as \(2^5\) and \(1024\) as \(2^{10}\), we can rewrite the expression:

Using the rules of exponents (\(a^m \times a^n = a^{m+n}\)):

If a memory has \(2^N\) locations, then \(N\) address lines are required to uniquely identify each location. In this case, since we have \(2^{15}\) locations, the number of address lines required is \(15\).

Decoder Implementation for Memory

A decoder is a combinational logic circuit that is essential for memory addressing. It converts a binary input code (from the address lines) into a single active output line that corresponds to that code. An \(n\)-to-\(2^n\) decoder has \(n\) input lines and \(2^n\) output lines. Only one of the \(2^n\) output lines is active (typically high) at any given time, corresponding to the unique binary input applied.

In our scenario, the \(15\) address lines serve as the input to the decoder. Therefore, it is a \(15\)-to-\(2^{15}\) decoder.

This decoder will have \(2^{15}\) distinct output lines. Each output line is responsible for selecting a specific memory location. For instance, if the input address is all zeros, the first output line will be activated, selecting the first memory location. If the input address is all ones, the last output line will be activated, selecting the last memory location.

AND Gates Count for Decoder

Every output of a standard decoder is generated by an AND gate. Each AND gate takes a unique combination of the input address lines (or their complements) to produce a high output for its corresponding address.

For example, consider a small \(2\)-to-\(4\) decoder with input address lines \(A_1\) and \(A_0\). The outputs would be:

• Output 0 (for address \(00_2\)): \(A_1' \cdot A_0'\)
• Output 1 (for address \(01_2\)): \(A_1' \cdot A_0\)
• Output 2 (for address \(10_2\)): \(A_1 \cdot A_0'\)
• Output 3 (for address \(11_2\)): \(A_1 \cdot A_0\)

Each of these \(4\) outputs is generated by a separate AND gate.

Since our decoder needs to produce \(2^{15}\) distinct output signals, one for each unique memory address, it will require precisely \(2^{15}\) AND gates. Each AND gate will be responsible for activating one specific memory location based on the \(15\) input address lines.

Thus, the minimum number of AND gates required for the decoder is \(2^{15}\).