An Observation about the Daubechies Wavelet and PQF

In an earlier posting, I had written about what a wonderful thing Quadrature Mirror Filter was, and that it is better to apply the Daubechies Wavelet than the older Haar Wavelet. But the question remains less obvious, as to how the process can be reversed.

The concept was clear, that an input stream in the Time-Domain could first be passed through a low-pass filter, and then sub-sampled at (1/2) its original sampling rate. Simultaneously, the same stream can be passed through the corresponding band-pass filter, and then sub-sampled again, so that only frequencies above half the Nyquist Frequency are sub-sampled, thereby reversing them to below the new Nyquist Frequency.

A first approximation for how to reverse this might be, to duplicate each sample of the lower sub-band once, before super-sampling them, and to invert each sample of the upper side-band once, after expressing it positively, but we would not want playback-quality to drop to that of a Haar wavelet again ! And so we would apply the same wavelets to recombine the sub-bands. There is a detail to that which I left out.

We might want to multiply each sample of each sub-band by its entire wavelet, but only once for every second output-sample. And then one concern we might have could be, that the output-amplitude might not be constant. I suspect that one of the constraints which each of these wavelets satisfies would be, that their output-amplitude will actually be constant, if they are applied once per second output-sample.

Now, in the case of ‘Polyphase Quadrature Filter’, Engineers reduced the amount of computational effort, by not applying a band-pass filter, but only the low-pass filter. When encoding, the low sub-band is produced as before, but the high sub-band is simply produced as the difference between every second input-sample, and the result that was obtained when applying the low-pass filter. The question about this which is not obvious, is ‘How does one recombine that?’

And the best answer I can think of would be, to apply the low-pass wavelet to the low sub-band, and then to supply the sample from the high sub-band for two operations:

  1. The first sample from the output of the low-pass wavelet, plus the input sample.
  2. The second sample from the output of the low-pass wavelet, minus the same input sample, from the high sub-band.

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An Observation about the Discrete Fourier Transforms

Discrete Fourier Transforms, including the Cosine Transforms, tend to have as many elements in the frequency-domain, as the sampling interval had in the time-domain.

Thus, if a sampling interval had 1024 samples, there will be as many frequency-coefficients, numbered from 0 to 1023 inclusively. One way in which these transforms differ from the FFT, is in the possibility of having a number of elements either way, that are not a power of 2. It is possible to have a discrete transform with 11 time-domain samples, that translate into as many frequency-coefficients, numbered from 0 to 10 inclusively.

If it was truly the project to compute an FFT that has one coefficient per octave, then we would include the Nyquist Frequency, which is usually not done. And in that case, we would also ask ourselves, whether the component at F=0 is best computed as the summation over the longest interval, where it would usually be computed, or whether it makes more sense then, just to fold the shortest interval, which consists of 2 samples, one more time, to arrive at 1 sample, the value of which corresponds to F=0 .

Now, if our discrete transform had the frequency-coefficients

 


G(n) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}


 

Then the fact could be exploited that these transforms tend to act as their own inverse. Therefore I can know, that the same set of samples in the time-domain, would constitute a DC signal, which would therefore have the frequency-coefficients

 


F(n) = {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}


 

If this was taken to be a convolution again, because the discrete transforms are their own inverse, it would correspond to the function

 


F(n) · S(m) == S(m)


 

We would assume that multiplication begins with element (0) and not with element (10). So I have a hint, that maybe I am on the right track. But, because the DCT has an inverse which is not exactly the same, the inverse being the IDCT, the next question I would need to investigate, is whether indeed I should be using the DCT and not the IDCT, to turn an intended set of frequency-coefficients, into a working convolution. And to answer that question, the simple thought does not suffice.

The main advantage with the DCT would be, that we will never need to deal with complex values.

Dirk

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The approximate Difference between a DFT and an FFT

Both the Discreet Fourier Transform and the Fast Fourier Transform produce complex-numbered coefficients, the non-zero amplitudes of which will represent frequency components in the signal. They both produce a more accurate measure of this property of the signal, than the Discreet Cosine Transforms do.

Without getting into rigorous Math,

If we have a 1024-sample interval in the time-domain, then the DFT of that simply computes the coefficients from 0 through to 1023, half-cycles. A frequency component present at one coefficient, let us say an even-numbered coefficient, will also have a non-zero effect on the adjacent, odd-numbered coefficients, which can therefore not be separated fully, by a Fourier Transform that defines both sets. A DFT will generally compute them all.

An FFT has as a premise, a specific number of coefficients per octave. That number could be (1), but seldom actually is. In general, an FFT will at first compute (2 * n) coefficients over the full sampling interval, will then fold the interval, and will then compute another (n) coefficients, and will fold the interval again, until the highest-frequency coefficient approaches 1/2 the number of time-domain samples in the last computed interval.

This will cause the higher-octave coefficients to be more spread out and less numerous, but because they are also being computed for successively shorter sampling intervals, they also become less selective, so that all the signal energy is eventually accounted for.

Also, with an FFT, it is usually the coefficients which correspond to the even-numbered ones in the DFT which are computed, again because one frequency component from the signal does not need to be accounted for twice. Thus, whole-numbers of cycles per sampling interval are usually computed.

For example, if we start with a 1024-sample interval in the time-domain, we may decide that we want to achieve (n = 4) coefficients per octave. We therefore compute 8 over the full interval, including (F = 0) but excluding (F = 8). Then we fold the interval down to 512 samples, and compute the coefficients from (F = 4) through (F = 7).

A frequency component that completes the 1024-sample interval 8 times, will complete the 512-sample interval 4 times, so that the second set of coefficients continues where the first left off. And then again, for a twice-folded interval of 256 samples, we compute from (F = 4) through (F = 7)…


 

After we have folded our original sampling interval 6 times, we are left with a 16-sample interval, which forms the end of our series, because (F = 8) would fit in exactly, into 16 samples. And, true to form, we omit the last coefficient, as we did with the DFT.

210  =  1024

10 – 6 = 4

24  =  16


So we would end up with

(1 * 8) + (6 * 4) =  32  Coefficients .

And this type of symmetry seemed relevant in this earlier posting.

Dirk

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Emphasizing a Presumed Difference between OGG and MP3 Sound Compression

In this posting from some time ago, I wrote down certain details I had learned about MP3 sound compression. I suppose that while I did write, that the Discreet Cosine Transform coefficients get scaled, I may have missed to mention in that same posting, that they also get quantized. But I did imply it, and I also made up for the omission in this posting.

But one subject which I did mention over several postings, was my own disagreement with the practice, of culling frequency-coefficients which are deemed inaudible, thus setting those to zero, just to reduce the bit-rate in one step, hoping to get better results, ‘because a lower initial bit-rate also means that the user can select a higher final bit-rate…’

In fact, I think that some technical observers have confused two separate processes that take place in MP3:

  1. An audibility threshold is determined, so that coefficients which are lower than that are set to zero.
  2. The non-zero coefficients are quantized, in such a way that the highest of them fits inside a fixed maximum, quantized value. Since a scale-factor is computed for one frequency sub-band, this also implies that close to strong frequency coefficients, weaker ones are just quantized more.

In principle, concept (1) above disagrees with me, while concept (2) seems perfectly fine.

And so based on that I also need to emphasize, that with MP3, first a Fast-Fourier Transform is computed, the exact implementation of which is not critical for the correct playback of the stream, but the only purpose of which is to determine audibility thresholds for the DCT transform coefficients, the frequency-sub-bands of which must fit the standard exactly, since the DCT is actually used to compress the sound, and then to play it back.

This FFT can serve a second purpose in Stereo. Since this transform is assumed to produce complex numbers – unlike the DCT – it is possible to determine whether the Left-Minus-Right channel correlates positively or negatively with the Left-Plus-Right channel, regarding their phase. The way to do this effectively, is to compute the dot-product between two complex numbers, and to see whether this dot-product is positive or negative. The imaginary component of one of the sources needs to be inverted for that to work.

But then negative or positive correlation can be recorded once for each sub-band of the DCT as one bit. This will tell, whether a positive difference-signal, is positive when the left channel is more so, or positive if the right channel is more so.

You see, in addition to the need to store this information, potentially with each coefficient, there is the need to measure this information somehow first.

But an alternative approach is possible, in which no initial FFT is computed, but in which only the DCT is computed, once for each Stereo channel. This might even have been done, to reduce the required coding effort. And in that case, the DCT would need to be computed for each channel separately, before a later encoding stage decides to store the sum and the difference for each coefficient. In that case, it is not possible first to determine, whether the time-domain streams correlate positively or negatively.

This would also imply, that close to strong frequency-components, the weaker ones are only quantized more, not culled.

So, partially because of what I read, and partially because of my own idea of how I might do things, I am hoping that OGG sound compression takes this latter approach.

Dirk

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