Quantization Error & Dynamic Range

Quantization is a many-to-one map: every value inside one decision interval produces the same code. That information loss is controlled by step size only in the ideal model. A real converter adds offset, gain, nonlinearity, noise, reference drift, missing codes, and clipping.

Ideal uniform quantization #

For range $V_{FS}$ and $N$ bits,

$$\Delta=\frac{V_{FS}}{2^N}.$$

With rounding, ideal error $e=x-Q(x)$ lies approximately in $[-\Delta/2,+\Delta/2)$. If error samples are uniformly distributed and uncorrelated with the signal, their variance is

$$\sigma_q^2=\frac{\Delta^2}{12}.$$

For a full-scale sine wave, this gives the familiar ideal ratio

$$SNR_q\approx6.02N+1.76\ \mathrm{dB}.$$

The assumptions matter. A small periodic signal can produce correlated, tonal error rather than white noise; a DC input may toggle among only a few codes; clipping violates the bounded-error model entirely.

Dynamic range terms #

$$ENOB=\frac{SINAD-1.76}{6.02}.$$

A converter can have excellent SNR and poor SFDR, or excellent DC noise-free resolution and poor high-frequency ENOB. State frequency, amplitude, bandwidth, sample rate, input range, and temperature with every performance number.

Dither and oversampling #

Dither deliberately adds noise so quantization error decorrelates from the signal. It can replace deterministic spurs with a smoother noise floor and recover sub-LSB average information, but it does not make individual samples exact. Averaging independent noise reduces RMS uncertainty approximately as $1/\sqrt{M}$; it helps only if noise actually causes code variation and the signal is stable over the average.

Oversampling spreads ideal quantization-noise power across a wider Nyquist band. Filtering and decimation then remove out-of-band noise, yielding roughly 3 dB SNR improvement per doubled sample rate for an unshaped quantizer. Delta-sigma conversion adds noise shaping and can improve faster in-band, at the cost of filtering latency and overload constraints.

Error-budget workflow #

Choose range so expected extremes do not clip but still use enough codes. Convert every error source to common units — volts, LSB, or input-referred RMS — over the relevant bandwidth and temperature. Separate systematic errors that can be calibrated from random noise, and monotonicity from absolute accuracy. Finally verify with a representative source impedance, reference, clock, layout, and processing chain; converter bits alone are not a system specification.