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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An Observation about Modifying Fourier Transforms

A concept which seems to exist, is that certain standard Fourier Transforms do not produce desired results, and that therefore, They must be modified for use with compressed sound.

What I have noticed is that often, when we modify a Fourier Transform, it only produces a special case of an existing standard Transform.

For example, we may start with a Type 4 Discrete Cosine Transform, that has a sampling interval of 576 elements, but want it to overlap 50%, therefore wanting to double the length of samples taken in, without doubling the number of Frequency-Domain samples output. One way to accomplish that is to adhere to the standard Math, but just to extend the array of input samples, and to allow the reference-waves to continue into the extension of the sampling interval, at unchanged frequencies.

Because the Type 4 applies a half-sample shift to its output elements as well as to its input elements, this is really equivalent to what we would obtain, if we were to compute a Type 2 Discrete Cosine Transform over a sampling interval of 1152 elements, but if we were only to keep the odd-numbered coefficients. All the output elements would count as odd-numbered ones then, after their index is doubled.

The only new information I really have on Frequency-Based sound-compression, is that there is an advantage gained, in storing the sign of each coefficient, notwithstanding.

(Edit 08/07/2017 : )

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About the Amplitudes of a Discrete Differential

One of the concepts which exist in digital signal processing, is that the difference between two consecutive input samples (in the time-domain) can simply be output, thus resulting in a differential of some sort, even though the samples of data do not represent a continuous function. There is a fact which must be observed to occur at (F = N / 2) – i.e. when the frequency is half the Nyquist Frequency, of (h / 2) , if (h) is the sampling frequency.

The input signal could be aligned with the samples, to give a sequence of [s0 … s3] equal to

0, +1, 0, -1

This set of (s) is equivalent to a sine-wave at (F = N / 2) . Its discrete differentiation [h0 … h3] would be

+1, +1, -1, -1

At first glance we might think, that this output stream has the same amplitude as the input stream. But the problem becomes that the output stream is by same token, not aligned with the samples. There is an implicit peak in amplitudes between (h0) and (h1) which is greater than (+1) , and an implicit peak between (h2) and (h3) more negative than (-1) . Any adequate filtering of this stream, belonging to a D/A conversion, will reproduce a sine-wave with a peak amplitude greater than (1).

(Edit 03/23/2017 :

In this case we can see, that samples h0 and h1 of the output stream, would be phase-shifted 45⁰ with respect to the zero crossings and to the peak amplitude, that would exist exactly between h0 and h1. Therefore, the amplitude of h0 and h1 will be the sine-function of 45⁰ with respect to this peak value, and the actual peak would be (the square root of 2) times the values of h0 and h1. )

And so a logical question which anybody might want an answer to would be, ‘Below what frequency does the gain cross unity gain?’ And the answer to that question is revealed by Differential Calculus. If a sine-wave has a peak amplitude of (1), then its instantaneous differential equals (2 π F) , which is also known as (ω) , at zero-crossing. It follows that unit gain will only take place at (F = N / π) . This is a darned low frequency in practice. If the sampling rate was 44.1kHz, this is achieved somewhere around 7 kHz, and music, for which that sampling rate was devised, easily contains sound energy above that frequency.

What follows is also a reason for which by itself, offers poor performance in compressing signals. It usually needs to be combined with other methods of data-reduction, thus possibly resulting in the lossy . And another approach which uses , is , the last of which is a proprietary codec, which minimizes the loss of quality that might otherwise stem from using .

I believe this observation is also relevant to This Earlier Posting of mine, which implied a High-Pass Filter with a cutoff frequency of 1 kHz, that would be part of a Band-Pass Filter. My goal was to obtain a gain of at least 0.5 , over the entire interval, and to simplify the Math.

(Edited 03/21/2017 . )

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