8-bit Audio Companding (part II)

28/02/2017

A few weeks back, I presented an heuristic for audio companding, making the vague assumption that the distribution of values—sound samples—is somewhat exponentially-distributed. But is it the case?

Let’s then find out the distribution of the samples. As before, I will use the Toréador file and a new one, Jean Michel Jarre’s Electronica 1: Time Machine (the whole CD). The two are very different. One is classical music, the other electronic music. One is adjusted in loudness so that we can here the very quiet notes as well as the very loud one, the other is adjusted for mostly for loudness, to maximum effect.

So I ran both through a sampler. For display as well as histogram smoothing purposes, I down-sampled both channels from 16 to 8 bits (therefore from 0 to 255). In the following plots, green is the left channel and (dashed) red the right. Toréador shows the following distribution:

or, in log-plot,

Turns out, the samples are Laplace distributed. Indeed, fitting a mean $\mu=127$ and a parameter $\beta\approx{7.4}$ agrees with the plot (the ideal Laplacian is drawn in solid blue):

Now, what about the other file? Let’s see the plots:

and in log-plot,

and with the best-fit Laplacian superimposed:

Now, to fit a Laplacian, the best parameters seem to be $\mu=127$ and $\beta\approx{27}$. While the fit is pretty good on most of the values, it kind of sucks at the edge. That’s the effect of dynamic range compression, a technique used to limit a signal’s dynamic range, often in a non-uniform way (the signal values near or beyond the maximum value target get more squished). This explains the “ears” seen in the log-plot, also seen in the (not log-)plot.

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Making the hypothesis that the samples are Laplace-distributed will allow us to devise an efficient quantization scheme for both the limits of the bins and the reconstruction value. In S-law, if we remember, the reconstructed value used is the value in the center of the interval. But, if the distribution is not uniform in this interval, the most representative value isn’t in its center. It’s the value that minimizes the squared expected error. Even if the expression for the moments of a Laplace-distributed random variable isn’t unwieldy, we should arrive at a very good, and parametric, quantization scheme for the signal.

Cleaning Scans

21/02/2017

Scanning documents or books without expensive hardware and commercial software can be tricky. This week, I give you the script I use to clean up a scanned image (and eventually assemble many of them into a single PDF document).

Choosing Random Files

14/02/2017

This week, something short. To run tests, I needed a selection of WAV files. Fortunately for me, I’ve got literally thousands of FLAC files lying around on my computer—yes, I listen to music when I code. So I wrote a simple script that randomly chooses a number of file from a directory tree (and not a single directory) and transcode them from FLAC to WAV. Also very fortunately for me, Bash and the various GNU/Linux utilities make writing a script for this rather easy.

8-bit Audio Companding

07/02/2017

Computationally inexpensive sound compression is always difficult, at least if you want some quality. One could think, for example, that taking the 8 most significant bits of 16 bits will give us 2:1 (lossy) compression but without too much loss. However, cutting the 8 least significant bits leads to noticeable hissing. However, we do not have to compress linearly, we can apply some transformation, say, vaguely exponential to reconstruct the sound.

That’s the idea behind μ-law encoding, or “logarithmic companding”. Instead of quantizing uniformly, we have large (original) values widely spaced but small (original) value, the assumption being that the signal variation is small when the amplitude is small and large when the amplitude is great. ITU standard G.711 proposes the following table: