Two inherently different types of interpolations, that exist.

One observation which I made about certain people has been, that they are able to conceive that given a certain audio sample-rate, the signal-processing operation on it, to perform some sort of interpolation, may be needed, especially when up-sampling, or, otherwise resampling the stream. But what I seemed to notice was, that those people failed to distinguish between two different categories of interpolation, which I would split as follows:

  1. There exist interpolations, in which the samples to be interpolated, have fixed positions in time, between the input samples.
  2. There exist interpolations, where for every interpolated sample, the time between the two adjacent, input samples, is not known, up to the very instant when the interpolation is finally computed, and where this time-position needs to be defined by an additional parameter, which may be called (t), and which would typically span the interval [0.0 .. 1.0).

For the type (1) above, if polynomials are going to be used, then all the values of (t) are known in advance, and therefore, all the values of (x) that define the polynomial, are also known in advance. This also means, that all the powers of (x) are known in advance. In that case:

To compute the interpolation, can be applied.

However, for the type (2) of interpolation above, IF polynomials are going to be used, then to derive the actual polynomial will become necessary, as well as, the need to ‘Plug parameter (t) in to the resulting polynomial.’ Because the polynomial could be of the 6th degree, this can become an expensive computation to perform in real-time, and implementors are likely to look for alternatives to using polynomials, that are also cheaper to compute.

Also, if the polynomial is to be plotted, then the positions along the X-axis are assumed to form a continuous interval, for which reason, the actual polynomial needs to be derived.

Dirk