Talk:Chirp Z-transform

This page was also cited as a source in the following academic paper:

• Igor Aizenberg and Jaakko T. Astola, "Discrete Generalized Fresnel Functions and Transforms in an Arbitrary Discrete Basis," IEEE Transactions on Signal Processing 54 (11), pp. 4261-4270 (2006).

Dubious

If one considers what the formulae actually say, it becomes clear that $b_{n}$ really has length at least $2N-1$, because all indices between $-(N-1)$ and $N-1$ inclusive are used at least once. It is true than an implementation could make do with making b_n a vector of length N since $b_{-n}=b_{n}$, but that is an optimisation that is inappropriate to make that early in the text, since it makes the text confusing.

A better approach would probably be to consider $a_{n}$ and $b_{n}$ both as defined for all $n\in \mathbb {Z}$, but with $a_{n}=0$ for $n<0$ and $n\geq N$, and possibly $b_{n}=0$ for $|n|\geq N$. Then the convolution becomes a straightforward (linear) convolution, and it is clear that it can be computed as a periodic convolution of sufficient length M. The shortest length that suffices is $M=2N-1$, but the idea of the algorithm is that one can instead pick a slightly larger M for which there is an efficient FFT using some more basic algorithm. 81.170.129.141 (talk) 21:03, 18 September 2012 (UTC)