There has been some confusion about the Sinc-Filter.

I have read descriptions about the Sinc-Filter somewhere, which predicted that it would become unstable, if the frequency of the input stream, happened to correspond to the spacing, between its non-zero coefficients. As far as I can tell, this prediction was based on a casual inspection of the Sinc Function, but overlooks something which is easy to overlook about it. This case also happens to correspond, to the input stream having a frequency equal to the Nyquist Frequency, of certain practical applications, such as over-sampling.

The Sinc Function has zero-crossings at regular intervals, including the center-point, where its coefficient is stated as being equal to (1.0) . This happens because the value at the center-point, is the solution to a limit equation, that corresponds to (0/0) .

This center coefficient is symmetrically flanked by two positive ones, one of which is only positive, because it forms as a division of the sine of x by the corresponding negative value of x. At frequencies below the Nyquist Frequency, the sum of their products starts to reinforce the center element. Above Nyquist, they start to cancel the product with the center coefficient.

sincplot_2

This can be complicated to plot using Computer Algebra Systems, because plotting functions are always numerical, and at (x=0), there is no numerical solution (only the Algebraic solution given lHôpitals Rule). So, a CAS typically needs to have the Sinc Function defined as a special case, to be able to plot it, otherwise requiring a complex workaround.

So it is possible that the frequency of the incoming stream aligns to the spacing between the maxima and minima of the Sinc Function. If that happens, there are two behaviors to bear in mind:

  1. The peak of the input stream could be aligned with the center-point. In that case, all the other waves will have zero-crossings, where the Sinc Function has maxima. The fact that the single input-sample seems to produce (1.0) as the output amplitude, is due to how the function is frequently normalized for practical use. According to that, maximum output should reach (2.0) at a frequency of zero…
  2. The input stream could have a zero-crossing, at the center-point of the Sinc Function, so that its product from there should equal (0.0) . In that case, the input stream will have positive peaks on one side of the center-point, that all correspond to negative peaks on the other side of the center-point. According to that, the instantaneous output should equal (0.0) .

All of this would suggest to me, that the Sinc-Filter will work properly.

sincplot_3

One way in which people can misinterpret the plot of the curve, would be to notice it has a positive peak in the center, to notice that after a zero-crossing, it forms two negative peaks, and then to conclude that those negative peaks are also the two closest non-zero coefficients to the center.

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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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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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About The History of Sinc Filters

A habit of mine which betrays my age, is to use the term ‘Sinc Filter’. I think that according to terminology today, there is no such thing. But there does exist a continuous function called ‘the Sinc Function’.

When I use the term ‘Sinc Filter’, I am referring to a convolution – a linear filter – the discreet coefficients of which are derived from the Sinc Function. But I think that a need exists to explain why such filters were ever used.

The Audio CDs that are by now outdated, were also the beginning of popular digital sound. And as such, CD players needed to have a Digital-to-Analog converter, a D/A converter. But even back when Audio CDs were first invented, listeners would not have been satisfied to listen to the rectangular wave-patterns that would come out of the D/A converter itself, directly at the 44.1 kHz sample-rate of the CD. Instead, those wave-patterns needed to be put through a low-pass filter, which also acted to smooth the rectangular wave-pattern.

But there was a problem endemic to these early Audio CDs. In order to minimize the number of bits that they would need to store, Electronic Engineers decided that Human Hearing stopped after 20 kHz, so that they chose their sampling rate to be just greater than twice that frequency. And indeed, when the sample-rate is 44.1 kHz, the Nyquist Frequency, the highest that can be recorded, is exactly equal to 22.05 kHz.

What this meant in practice, was that the low-pass filters used needed to have an extremely sharp cutoff-curve, effectively passing 20 kHz, but blocking anything higher than 22.05 kHz. With analog circuits, this was next to impossible to achieve, without also destroying the sound quality. And so here Electronics Experts first invented the concept of ‘Oversampling’.

Simply put, Oversampling in the early days meant that each analog sample from an D/A converter would be repeated several times – such as 4 times – and then passed through a more complex filter, which was implemented at first on an Analog IC.

This analog IC had a CCD delay-line, and at each point in the delay-line it had the IC equivalent to ‘a potentiometer setting’, that ‘stored’ the corresponding coefficient of the linear filter to be implemented. The products of the delayed signal with these settings on the IC, were summed with an analog amplifier – on the same IC.

Because the Sinc Function defines a brick-wall, low-pass filter, if  a 4x oversampling factor was used, then this linear filter would also have a cutoff-frequency at 1/4 the new, oversampled Nyquist Frequency.

What this accomplished, was to allow an analog filter to follow, which had 2 octaves of frequency-separation, within which to pass the lower frequency, but to block this oversampled, Nyquist Frequency.

Now, there is a key point to this which Electronics Experts were aware of, but which the googly-eyed buyers of CD players were often not. This type of filtering was needed more, before the Analog-to-Digital conversion took place, when CDs were mastered, than it needed to take place in the actual players that consumers bought.

The reason was a known phenomenon, by which If a signal is fed to a sample-and-hold circuit running at 44.1 kHz, and if the analog, input frequency exceeded the Nyquist Frequency, these excessive input frequencies get mirrored by the sample-and-hold circuit, so that where the input frequencies continued to increase, the frequencies in the digitized stream would be reflected back down – to somewhere below the Nyquist Frquency.

And what this meant was, that if there was any analog input at an supposedly-inaudible 28.05 kHz for example, it would wind up in the digital stream at a very audible 16.05 kHz. And then, having an oversampling CD player would no longer be able to separate that from any intended signal content actually at 16.05 kHz.

Therefore, in studios where CDs were mastered, it was necessary to have the sample-and-hold circuit also run at 4x or 8x the final sample-rate, so that this could be put through a homologous low-pass filter, only 1/4 or 1/8 the samples of which would actually be converted to digital, through the A/D converter, and then stored…

Now today, that sort of filter design has been replaced completely, through the availability of better chips, that do all the processing numerically and therefore digitally. Hence, if 4x oversampling is being used, the digital version of the signal and not its analog version, are being ‘filtered’, through specialized digital chips.

Back in the 1980s, the types of chips and the scale of integration required, were not yet available.

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