Audio signal dither and noise shaping

Question

I'm trying to get a handle on the importance of the error feedback term in noise shaping operation in typical audio dither algorithms.

I'm thinking in terms of four signals. The original signal, the output (quantised) signal, the dither signal, and the error signal (the difference between output and original).

A simple approach would be to generate white noise and apply a filter to it to generate a coloured dither signal, add this to the original signal, quantise to the output signal, and disregard the error signal. It appears to be a fairly common approach to use white (unfiltered) or violet (differentiated) noise functions in this configuration.

Another approach I see implemented is to take the error signal and filter that and add white noise to define the dither signal, then add it to the original, quantise, and calculate the next error term for the filter. This implies a tight loop for the data flow -- we cannot compute a new error term without resolving the previous one -- making it hard to use SIMD efficiently.

From those I can infer a third option -- forgo the noise altogether and simply filter and reintegrate the error signal directly. I suppose this comes with some risk of modulation artefacts related to but distinct from the artefacts introduced by quantisation without any dither or shaping at all; however, I suspect this might be how it's done in some high-frequency 1-bit cases.

The point I'm unclear on is that complex filters which attempt to model the threshold of human hearing appear to come under the title of 'noise shaping', and this implies (supported by implementations I have seen) use of the error feedback term.

Is it a flawed or misguided approach to forgo the error feedback but still use the complex filter on white noise for the dither function? Is the error signal genuinely offering something important here?

Answer 1

i dunno what a "Lipshitz filter" is but i do know who Stanley Lipshitz is. now, i don't know for sure what Stanley would say to you, but the best advice i got from him is that it does little good to color the additive dither if you are also using error feedback. if there is no noise shaping feedback, probably the simplest way to get triangular PDF dither is to generate good, independent, rectangular PDF random numbers $v[n]$ and run that through a differentiator.

$$ d[n] = v[n] - v[n-1] $$

the result is high-pass in the spectrum and triangular PDF. if $\Delta$ is the step size of the quantization, $\frac{\Delta^2}{12}$ is the variance (or power) of raw quantizer, but you have two more of them due to your dither and $\frac{\Delta^2}{4}$ is the total power due to the entire dithered quantizer. a 4.77 dB increase in quantization error, but the quantization is fully decoupled from both the mean and variance of the input signal. although this is the same quantization power that you would get from adding two independent random numbers, i think you would prefer that much of the power is steered to higher frequencies (but not all of it, the spectrum of the raw quantizer is still flat).

if you're planning on error feedback, i think the best advice is either use white, triangular PDF dither (requires adding two independently generated rectangular PDF random numbers) and error feedback (designing a good feedback filter takes a bit).

or no dither (if the quantization step size is very small) and error feedback. the simplest no-trivial example of this is "fraction saving" where you simply hack off the low-order bits (the floor() function) and add those bits back in (zero-extended on the left) to the next sample before that sample is quantized (with the floor() function). it has infinite S/N ratio at DC and totally kills any limit cycle in feedback filters (like biquads) that get the filter stuck on a DC value even when the input goes to zero. i've done that a lot.

Answer 2

To get a feel for how to answer my own question I made a -90dB test tone ramp, and applied different processes to it, and plotted the spectrogram with SoX. The z axis is shifted up from the default to make things clearer.

First, the conventional approach (the second one described in my question) of filtering the error and adding this to the signal (shaping) before adding dither and quantising to calculate the new error:

Now to see what happens when we try to avoid reference to the error term, we apply the Lipshitz filter to white noise and use that as the dither before quantisation and discard the error:

What's obvious here is that while the noise floor can be raised in the same place as it's raised in the noise-shaping approach, it seems to hit a limit below which it cannot be lowered.

This is as one might expect, as it's subject to quantisation noise and the only thing special about the signal we're filtering is that it starts out white so that the quantisation noise isn't correlated.

So to complete the picture, we try removing the dither pattern and applying noise shaping in isolation:

Oops! Obviously not a good idea.

So in answer of my own question, if you are going to dither and not noise-shape, there is limited value in using a complex transfer function for the dither pattern, because the quantisation noise floor still exists with dither and the regions that might come out especially clear with shaping are not accessible with plain dither.

That said, there must still be some merit in shaping the dither function, because I can at least reduce the noise in the low frequency region by using violet noise rather than white, and I don't know how far beyond that the theoretical limits of unshaped dither extend. So this answer is incomplete.

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Correction: Originally I messed up a coefficient in the middle image (shaped white noise). I've updated the image, and things work better than before, but the same basic conclusion can still be drawn.