New Block Order

March 21, 2017

The idea of reorganizing data before compression isn’t new. Almost twenty five years ago, Burrows and Wheeler proposed a block sorting transform to reorder data to help compression. The idea here is to try to create repetitions that can be exploited by a second compression engine.

But the Burrows-Wheeler transform isn’t the only possible one. There are a few other techniques to generate (reversible) permutations of the input.

Read the rest of this entry »


Much Ado About Nothing

March 7, 2017

A rather long time ago, I wrote a blog entry on branchless equivalents of simple functions such as sex, abs, min, max. The Sing EXtension instruction propagates the sign bit in the upper bits, and is typically used in the promotion of, say, a 16 bits signed value into a 32 bits variable.

But this time, I needed something a bit different: I only wanted the sign-extended part. Could I do much better than last time? Turns out, the compiler has a mind of its own.

Read the rest of this entry »


8-bit Audio Companding (part II)

February 28, 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?

ssound-blocks

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:

toreador

or, in log-plot,

toreador-log

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):

toreador-log-laplace

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

time-machine

and in log-plot,

time-machine-log

and with the best-fit Laplacian superimposed:

time-machine-log-laplace

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.

*
* *

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.


8-bit Audio Companding

February 7, 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.

ssound-blocks

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:

Read the rest of this entry »


Stretching samples

January 31, 2017

So for an experiment I ended up needing conversions between 8 bits and 16 bits samples. To upscale an 8 bit sample to 16 bits, it is not enough to simply shift it by 8 bits (or multiply it by 256, same difference) because the largest value you get isn’t 65535 but merely 65280. Fortunately, stretching correctly from 8 bit to 16 bit isn’t too difficult, even quite straightforward.

stretching-snorlax

Read the rest of this entry »


Whatever sums your floats

January 24, 2017

While flipping the pages of a “Win this interview” book—just being curious, not looking for a new job—I saw this seemingly simple question: how would you compute the sum of a series of floats contained in a array? The book proceeded with the simple, obvious answer. But… is it that obvious?

Read the rest of this entry »


Choosing the Right Pseudoinverse

January 17, 2017

On a number of previous occasions, I have used the pseudoinverse of a matrix solve systems of equations, and do other things such as channel mixing. However, the demonstration I gave before isn’t entirely correct. Let’s see now why it’s important to make the difference between a left and a right pseudoinverse.

otter

Read the rest of this entry »