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.
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:
- 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…
- 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.
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.
Continue reading There has been some confusion about the Sinc-Filter.