Threshold Elimination in Compressed Sound

I’ve written quite a few postings in this blog, about sound compression based on the Discrete Cosine Transform. And mixed in with my thoughts about that – where I was still, basically, trying to figure the subject out – were my statements to the effect that frequency-coefficients that are below a certain threshold of perceptibility could be set to zeroes, thus reducing the total number bits taken up, when Huffman-encoded.

My biggest problem in trying to analyze this is, the fact that I’m considering generalities, when in fact, specific compression methods based on the DCT, may or may not apply threshold-elimination at all. As an alternative, the compression technique could just rely on the quantization, to reduce how many bits per second it’s going to allocate to each sub-band of frequencies. ( :1 ) If the quantization step / scale-factor was high enough – suggesting the lowest quality-level – then many coefficients could still end up set to zeroes, just because they were below the quantization step used, as first computed from the DCT.

My impression is that the procedure which gets used to compute the quantization step remains straightforward:

  • Subdivide the frequencies into an arbitrary set of sub-bands – fewer than 32.
  • For each sub-band, first compute the DCTs to scale.
  • Take the (absolute of the) highest coefficient that results.
  • Divide that by the quality-level ( + 0.5 ) , to arrive at the quantization step to be used for that sub-band.
  • Divide all the actual DCT-coefficients by that quantization step, so that the maximum, (signed) integer value that results, will be equal to the quality-level.
  • How many coefficients end up being encoded to having such a high integer value, remains beyond our control.
  • Encode the quantization step / scale-factor with the sub-band, as part of the header information for each granule of sound.

The sub-band which I speak of has nothing to do with the fact that additionally, in MP3-compression, the signal is first passed through a quadrature filter-bank, resulting in 32 sub-bands that are evenly-spaced in frequencies by nature, and that the DCT is computed of each sub-band. This latter feature is a refinement, which as best I recall, was not present in the earliest forms of MP3-compression, and which does not affect how an MP3-file needs to be decoded.

(Updated 03/10/2018 : )

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Which of my articles might paraphrase frequency-domain-based sound compression best.

I have written numerous postings about sound-compression, in which I did acknowledge that certain forms of it are based on time-domain signal-processing, but where several important sound-compression techniques are based in the frequency-domain. Given numerous postings from me, a reader might ask, ‘Which posting summarizes the blogger’s understanding of the concept best?’

And while many people directly pull up a posting, which I explicitly stated, describes something which will not work, but displays that concept as a point-of-view, to compare working concepts to, instead of recommending that posting again, I would recommend this posting.

Dirk

 

I feel that standards need to be reestablished.

When 16-bit / 44.1kHz Audio was first developed, it implied a very capable system for representing high-fidelity sound. But I think that today, we live in a pseudo-16-bit era. Manufacturers have taken 16-bit components, but designed devices which do bot deliver the full power or quality of what this format once promised.

It might be a bit of an exaggeration, but I would say that out of those indicated 16 bits of precision, the last 4 are not accurate. And one main reason this has happened, is due to compressed sound. Admittedly, signal compression – which is often a euphemism for data reduction – is necessary in some areas of signal processing. But one reason fw data-reduction was applied to sound, had more to do with dialup-modems and their lack of signal-speed, and with the need to be able to download songs onto small amounts of HD space, than it served any other purpose, when the first forms of data-reduction were devised.

Even though compressed streams caused this, I would not say that the solution lies in getting rid of compressed streams. But I think that a necessary part of the solution would be consumer awareness.

If I tell people that I own a sound device, that it uses 2x over-sampling, but that I fear the interpolated samples are simply generated as a linear interpolation of the two adjacent, original samples, and if those people answer “So what? Can anybody hear the difference?” Then this is not an example of consumer awareness. I can hear the difference between very-high-pitch sounds that are approximately correct, and ones which are greatly distorted.

Also, if we were to accept for a moment that out of the indicated 16 bits, only the first 12 are accurate, but there exist sound experts who tell us that by dithering the least-significant bit, we can extend the dynamic range of this sound beyond 96db, then I do not really believe that those experts know any less about digital sound. Those experts have just remained so entirely surrounded by their high-end equipment, that they have not yet noticed the standards slip, in other parts of the world.

Also, I do not believe that the answer to this problem lies in consumers downloading 24-bit, 192kHz sound-files, because my assumption would again be, that only a few of those indicated 24 bits will be accurate. I do not believe Humans hear ultrasound. But I think that with great effort, we may be able to hear 15-18kHz sound from our actual playback devices again – in the not-so-distant future.

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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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