Analog-to-digital converter

An analog-to-digital converter produces a discrete numerical code from an input voltage. Its specifications include input range, resolution, conversion rate, input impedance, offset, gain error and nonlinearity. A uniform $b$-bit converter covering voltage span $V_{\mathrm{span}}$ has nominal spacing $q\simeq V_{\mathrm{span}}/2^b$. The uniform-quantization error bound applies to rounding inside the range, while clipping is a different error. Sampling and quantization are separate operations: neither sufficient sampling rate nor a large bit count alone guarantees accurate reconstruction.

oversampling takes samples faster than the minimum rate required by the desired signal bandwidth. With suitable uncorrelated conversion noise, digital filtering removes noise outside that bandwidth before decimation. Independent-error averaging improves standard deviation as $N^{-1/2}$; fourfold averaging ideally gains one bit of amplitude resolution. Repetition of an unchanged biased code does not produce the same benefit.

dither is an intentionally introduced fluctuation that makes a quantizer's output vary between codes, reducing signal-correlated quantization artifacts. A suitable distribution can make the averaged code carry information about a constant voltage smaller than one conversion step. The improvement requires filtering or averaging; extra noise does not improve an individual conversion or repair arbitrary calibration errors.

Let a constant input be $x=(k+f)\Delta$ with $0\le f\le 1$, between consecutive quantization levels $k\Delta$ and $(k+1)\Delta$. Add independent centered uniform dither $U\sim \mathrm{Uniform}[-\Delta /2,\Delta /2]$ before nearest-level rounding, and assume that neither code clips.

The upper code is selected exactly when $U\ge (1/2-f)\Delta$. This subinterval occupies fraction $f$ of the uniform distribution's support, so $Q=k\Delta +\Delta B$ with $B$ a Bernoulli random variable of parameter $f$. Thus $\mathbb{E}[Q]=x$ and $\mathrm{Var}(Q)=f(1-f){\Delta }^2$. Averaging $N$ independent conversions givesFor example, an input one quarter of a step above the lower code produces the upper code one quarter of the time. Its averaged code can estimate a sub-step voltage, with a finite remaining variance. Centering, independence and absence of clipping are assumptions of this calculation; a shifted dither distribution can bias the result.

rounding an unclipped voltage to equally spaced levels of separation $\Delta$ leaves error in $[-\Delta /2,\Delta /2]$. If the fractional position within a conversion interval is uniformly distributed, direct integration gives zero mean and expectation value $⟨{ϵ}^2⟩={\Delta }^2/12$. This is a statistical model for the error, not a claim that the converter adds independent white noise for every possible input.

A smooth input probability density varying slowly on the scale of one conversion interval makes the fractional uniform quantization error approximately uniform. Large overall variance is not sufficient: discrete inputs aligned with conversion levels provide a counterexample. Clipping, coarse sampling and correlations between error and signal require separate analysis.

A full-scale range $V_{\mathrm{FS}}$ and allowed quantization step $\Delta$ require $b\ge \lceil {\log }_2(V_{\mathrm{FS}}/\Delta )\rceil$ bits. The ideal uniform-quantizer rms error is approximately $\Delta /\sqrt{12}$. Bandwidth determines the sampling-rate requirement, but does not by itself determine a bit depth.