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I dont´t understand how works the convolution part of the Chirp z.

I understand how the DFT is transformed

\begin{align*} x(k) = \sum_{n=0}^{N-1} x(n) W_N^{kn} \end{align*}

to this expresion:

\begin{align*} x(k) = W_N^{k^2/2} \sum_{n=0}^{N-1} ( x(n) W_N^{n^2/2} ) W_N^{(k-n)^2/2} \end{align*}

After that it can be expressed as the convulution of two series:

\begin{align*} a_n &= x(n) W_N^{n^2/2}\\ b_n &= W_N^{n^2/2} \end{align*}

  1. Here i get lost. I can see where it is $an$ but the term $bn$ is confusing because it has "lost" its k and it appears outside of the sum.

After that x(k) is expressed by:

\begin{align*} x(k) = b_k * \sum_{n=0}^{N-1} a_n b_{k-n} \end{align*}

I don't understand this last step either.

I read the implementation of this question Chirp z algorithm clarification which truly helped me to undersand how its calculated but

2.I don't understand the answer to it:

When creating the chirp signal used in convolution, the values of (k - n) which result in a negative need to be wrapped around and placed at the end of the signal rather than in the beginning

I think the content of the answer is related to where I get confused because of the first part that I don't understand.

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