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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Guessing at the Number of Coefficients Filters Might Need

There probably exist Mathematically-more-rigorous ways to derive the following information. But just in order to be able to understand concepts clearly, I often find that I need to do some estimating, that will give some idea, of how many zero-crossings, for example, a Sinc Filter should realistically have, on each side of its center sample. Or, of what kind of cutoff-performance the low-pass part of a Daubechies Wavelet will have, If it only has 8 coefficients…

If the idea is accepted that a low-pass filter is supposed to be of some type, based on the ‘Sinc Function’, including filters that only have 2x / 1-octave over-sampling, then a question which Electronics Experts will face, is what number of zero-crossings is appropriate. This question is especially difficult to find a precise answer to, because the series does not converge. It is a modified series of the form Infinite Sum (1/n) .

Just to orient ourselves within the Sinc Function when applied this way, the center sample is technically one of the zero-crossings, but is equal to 1, because it has the only coefficient of the form (0/0). After that, each coefficient twice removed is a zero-crossing, and the coefficients displaced from those are the standard non-zero examples.

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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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