Quantization noise and quantization error

Quantization in an A/D converter maps each sampled value to the nearest of $2^n$ discrete levels. The step size (quantum) is

$$Q = \frac{FSR}{2^n}$$

where $FSR$ is the full-scale range and $n$ the number of bits.

Quantization error

The difference between the actual sample and its quantized value:

$$e(kT) = x(kT) - x_q(kT)$$

With rounding, $-\frac{Q}{2} \le e \le \frac{Q}{2}$ (with truncation, $0 \le e < Q$). It is a fixed, bounded error for each sample.

Quantization noise

When the signal varies enough, the error changes randomly from sample to sample and can be treated as noise added to the signal: uniformly distributed over $\pm Q/2$, zero mean, with variance (noise power)

$$\sigma^2 = \frac{Q^2}{12}$$

The signal-to-quantization-noise ratio improves by about 6 dB for each extra bit.

Example

A 3-bit ADC with $FSR = 8$ V has $Q = 8/2^3 = 1$ V.

Noise power $= 1^2/12 = 0.0833\ \text{V}^2$ (rms $\approx 0.289$ V). Using an 8-bit ADC gives $Q = 31.25$ mV, so the error falls to at most $\pm 15.6$ mV.