A Clarification on Polynomial Approximations… Refining the Exercise

Some time ago, I posted an idea, on how the concept of a polynomial approximation can be simplified, in terms of the real-time Math that needs to be performed, in order to produce 4x oversampling, in which the positions of the interpolated samples with respect to time, are fixed positions.

In order for the reader to understand the present posting, which is a reiteration, he or she would need to read the posting I linked to above. Without reading that posting, the reader will not understand the Matrix, which I included below.

There was something clearly wrong with the idea, which I wrote above, but what is wrong, is not the fact that I computed, or assume the usefulness, of a product between two matrices. What is wrong with the idea as first posted above, is that the order of the approximation is only 4, thus implying a polynomial of the 3rd degree. This is a source of poor approximations close to the Nyquist Frequency.

But As I wrote before, the idea of using anything based on polynomials, can be extended to 7th-order approximations, which imply polynomials of the 6th degree. Further, there is no reason why a 7×7 matrix cannot be pre-multiplied by a 3×7 matrix. The result will only be a 3×7 matrix.

Hence, if we were to assume that such a matrix is to be used, this is the worksheet, which computed what that matrix would have to be:

Work-Sheet

The way this would be used in a practical application is, that a vector of input-samples be formed, corresponding to

t = [ -3, -2, -1, 0, +1, +2, +3 ]

And that the interpolation should result corresponding to

t = [ 0, 1/4, 1/2, 3/4 ]

Further, the interpolation at t = 0 does not need to be recomputed, as it was already provided by the 4th element of the input vector. So the input-vector would only need to be multiplied by the suggested matrix, to arrive at the other 3 values. After that, a new sample can be used as the new, 7th element of the vector, while the old 1st element is dropped, so that another 3 interpolated samples can be computed.

This would be an example of an idea which does not work out well according to a first approximation, but which will produce high-quality results, when the method is applied more rigorously.

Dirk

 

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