Why can adding a small random signal before rounding samples to a coarse grid improve the result, despite raising total error power?

## The noise model has a precondition Every convenient statement about quantization error - that it is uniform across the bin, zero-mean, independent of the signal, with power equal to the square of the step over twelve - is a **model**, and the model has a condition attached: the signal must move across several levels between consecutive samples so that successive errors are effectively unrelated. Where that holds, the error behaves like a low, featureless floor. Where it does not hold, the error is what it always literally was: a **deterministic function of the input**. Rounding is not random. Feed it a signal that spans one or two steps, or one that changes very slowly, and the error becomes a repeating, signal-shaped pattern. ## What correlated error looks like The symptoms differ by medium but the mechanism is identical: - a **fading tail** quantized to few levels breaks into a stepped staircase, heard as tonal artifacts rather than hiss, and the artifacts change pitch with the signal; - a **smooth gradient** rounded to a coarse value grid shows **contour bands**, because a whole region of nearly equal values collapses onto one level and the boundary between levels becomes a visible edge; - a **slowly drifting measurement** rounded to a coarse grid produces long runs of identical values and then a jump, which downstream statistics read as a step change that never happened. In all three, the total error power may be small. The problem is that it is **organised**, and organised error is far more noticeable than unorganised error of the same size. ## What dither actually does Dither adds a small random value to each sample **before** the rounding, and does not subtract it afterwards. The randomness moves the rounding decision off the signal's own value, so which level a given input lands on stops being a fixed function of that input. | | Undithered rounding | With triangular dither | |---|---|---| | Error vs signal | deterministic, correlated | independent in mean and variance | | Character | harmonics, bands, staircases | featureless floor | | Total error power | the square of the step over twelve | about three times that, roughly 5 dB more | | Low-level detail | collapses onto one level | survives as a modulation of the floor | The last row is the part that surprises people. A signal smaller than one step is not simply lost under dither: it shifts the *proportion* of samples rounding up versus down, so information below the step survives in the statistics of the output even though no single sample resolves it. The choice of distribution matters: 1. **One uniform draw spanning one step** makes the error's average independent of the signal, but its variance still moves with the signal, which is perceived as the floor pumping in time with the content. 2. **Two independent uniform draws, summed** - a triangular distribution spanning two steps - makes the second moment independent too, which is why it is the usual default. 3. **Anything much larger** simply adds noise without buying further decorrelation, so more is not safer. ## Where it belongs in a chain Dither goes **immediately before the final, irreversible rounding onto the delivery grid, and only there**. Applied earlier, it is re-rounded by later stages, which both re-correlates the error and stacks another noise contribution on top of the first. Each irreversible rounding earns at most one dither of its own, which makes dither placement another argument for a chain that rounds once, late. It is also not universal. It is a technique for data that is **perceived** - waveforms, images, anything a human judges - where structured error is worse than louder error. For a measurement pipeline, a counter, or a currency amount, structure in the error is not the enemy and a documented rounding rule is worth more than injected randomness; auditability beats decorrelation there. ## The answer an interviewer wants State the trade in one line: dither raises total error power in exchange for removing correlation between the error and the signal, and correlation is what is actually noticeable. Then name the precondition that makes it necessary - a signal small or smooth relative to the step - and the placement rule that keeps it from being applied several times over.