Memory organization (2ᵏ × n)

Also called: 2ᵏ × n notation, words × bits notation, memory notation, memory configuration, memory geometry

The standard way to describe a memory's shape: number of words × bits per word, such as 1K × 8. It gives the address lines, data lines and capacity.

Memories are described as words × bits per word. A 16K × 4 memory holds 16K words, each 4 bits wide. That pair of numbers is the memory's organization.

Everything follows from it:

First number (words) → address bits: log₂(words). 16K = 2¹⁴ → 14 address lines.
Second number (word size) → data lines: equal to it. 4 data lines.
Product → memory capacity: 16K × 4 = 64K bits = 8 KiB.

In this notation K = 2¹⁰ = 1024, M = 2²⁰ and G = 2³⁰, because memory sizes are powers of 2.

Two memories with the same capacity can be organized differently. 4K × 8 and 8K × 4 both store 32K bits, but the first needs 12 address and 8 data lines, the second 13 address and 4 data lines. When you build a system from chips, the organization tells you how to arrange them: width expansion fixes the second number, depth expansion the first.

The same notation sizes a ROM used as a lookup table: an n-input, m-output function needs a 2ⁿ × m ROM.

Worked examples

Example

Reading a 64K × 16 memory

Find the address lines, data lines and capacity.

1.
Words: 64K = 2⁶ × 2¹⁰ = 2¹⁶ → 16 address lines.
2.
Word size 16 → 16 data lines.
3.
Capacity: 2¹⁶ × 16 = 2²⁰ bits = 2¹⁷ bytes = 128 KiB.

Example

Writing the notation from the lines

A memory has 11 address lines and 8 data lines.

1.
Words: 2¹¹ = 2048 = 2K.
2.
Organization: 2K × 8.

Common mistakes

Reading the numbers the wrong way round. The first is words, the second is bits per word.
Taking log₂ of the word size for address lines. Only the word count sets the address lines.
Reading K as 1000. In memory notation 1K = 1024.

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