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For an even greater change of pace, let's try to answer the question

It's only maths, it can't be too hard.

The example always given is the 1 kHz sine wave of 4 LSB peak to peak as used in the paper by Lipshitz and Vanderkooy. When quantised this yields a stairstep waveform best described by a matrix of sample numbers and quantisation values. In turn this waveform is the sum of a series of rectangular pulses of height 1 LSB and width given by the stairstep function. The distortion is simply the difference between this waveform and the original sinewave.

There are two ways around this. The first is to perform the Fourier transform of the sum of the pulses, using the Fourier transform of a rectangular pulse repeated at interval 2Πθ with pulse width 2Πk which is E[k + 2/Π(sinkΠ cosθ + sin2kΠ cos2θ/2 + ... + sin nkΠ cosnθ / n). We can just sum the appropriate rectangular pulses noting that E = 1 LSB for all cases.

The second is to note that the worst error approximates a sawtooth wave of 1 LSB peak to peak at a frequency dependent on the relationship between the signal frequency and the sampling rate (at 1kHz and 44.1 kHz sampling the sawtooth frequency is about 12 kHz). This sawtooth is only present for a fraction of the full cycle. We know that the Fourier transform of a sawtooth wave equals E/Π (sinθ - sin2θ/2 + sin3θ/3 ....) and the largest part of this is the fundamental (which is a harmonic of the original signal) with an amplitude of 2/Π times LSB. Its average value over the full cycle will be about half of this or 1/Π times LSB

This gives us a rule of thumb that the harmonic distortion has relative amplitude of -20 log(bits x Π) in dB. For the example given in Lipshitz and Vanderkooy this is about -22dB. This accords with the values in the graph in the paper, so the rule of thumb looks reasonable and I simply couldn't be bothered ploughing through the exact solution. Note that it is perfectly possible that certain of the harmonics of the different pulses will occur at the same frequency and therefore either add or subtract from one another. My rule of thumb naturally does not take this into account.

As the original paper states the dithered waveform does show statistical correlation to the sampled signal so it does have artefacts that are harmonically related to the signal, but the error signal sounds like white noise and the power spectrum shows no such artefacts above the increased noise floor. I believe this last rider is significant.

Mark Kelly

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