Quantization

Rounding a continuous-amplitude signal to a finite set of discrete levels — the source of quantization error, and the tradeoff behind every bit-depth choice.

Quantization rounds a smooth signal onto the nearest of a finite set of levels — the gap is quantization error

Definition

Sampling captures a signal's value at discrete times, but each value is still a real number with infinite precision. Quantization is the second step: rounding each sample's amplitude to the nearest of a finite set of allowed levels, so it can be stored as a finite number of bits.

The difference between the true value and its rounded level is the quantization error. With more levels (more bits), the error shrinks; with fewer levels, it grows — a direct tradeoff between storage/bandwidth and fidelity.

Bit depth and levels

An 8-bit quantizer has $2^8 = 256$ levels. A 16-bit quantizer (standard for CD audio) has $2^{16} = 65{,}536$ levels — vastly finer resolution, at the cost of twice the storage per sample.

Try it

If a signal ranges from 0 to 10 volts and is quantized with 4 bits, how many distinct levels are available, and what's the gap between adjacent levels?

Solution

4 bits gives $2^4 = 16$ levels. Spread evenly across the 10-volt range, adjacent levels are $10/15 \approx 0.67$ volts apart (15 gaps between 16 levels).

Related concepts

Calculus· Discrete-Time Signal Processing

Sampling TheoremThe Nyquist-Shannon theorem: a signal can be perfectly reconstructed from samples taken at twice its highest frequency — and what goes wrong (aliasing) if you sample slower.

Statistics· Descriptive

VarianceThe average squared distance of each value from the mean — a measure of spread.

Statistics· Descriptive

Standard DeviationThe square root of variance, expressing spread in the same units as the data.