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Chirp-z Transform

Module by: Douglas L. Jones. E-mail the author

Summary: Efficient scheme for computing samples of the z-transform. (Important special case: DFTs)

Let zk=A⁢W−k z k A W k , where A=Ao⁢ei⁢θo A Ao θo , W=Wo⁢e−(i⁢φo) W Wo φo .

We wish to compute MM samples, k=012…M−1 k 0 1 2 … M 1 of X⁢zk=∑n=0N−1x⁢n⁢zk−n=∑n=0N−1x⁢n⁢A−n⁢Wnk X zk n N 1 0 x n zk n n N 1 0 x n A n W nk

Figure 1
Figure 1 (figure3.png)

Note that (k−n2=n2−2⁢n⁢k+k2)⇒(n⁢k=12⁢(n2+k2−k−n2)) k n 2 n 2 2 n k k 2 n k 1 2 n 2 k 2 k n 2 , So X⁢zk=∑n=0N−1x⁢n⁢A−n⁢Wn22⁢Wk22⁢W−k−n22 X zk n N 1 0 x n A n W n 2 2 W k 2 2 W k n 2 2 Wk22⁢∑n=0N−1x⁢n⁢A−n⁢Wn22⁢W−k−n22 W k 2 2 n N 1 0 x n A n W n 2 2 W k n 2 2

Thus, X⁢zk X zk can be compared by

  1. Premultiply x⁢n x n by An⁢Wn22 A n W n 2 2 , n= 0 1 …N−1 n 0 1 … N 1 to make y⁢n y n
  2. Linearly convolve with W−k−n22 W k n 2 2
  3. Post multiply by to get Wk22 W k 2 2 to get X⁢zk X zk .

1. and 3. require NN and MM operations respectively. 2. can be performed efficiently using fast convolution.

Figure 2
Figure 2 (figure8.png)

W−n22 W n 2 2 is required only for −(N−1)≤n≤M−1 N 1 n M 1 , so this linear convolution can be implemented with L≥N+M−1 L N M 1 FFTs.

note:

Wrap W−n22 W n 2 2 around L when implementing with circular convolution.
So, a weird-length DFT can be implemented relatively efficiently using power-of-two algorithms via the chirp-z transform.

Also useful for "zoom-FFTs".

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