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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=AWk z k A W k , where A=Aoejθo A Ao θo , W=Woe(jφo) W Wo φo .

We wish to compute MM samples, k=012M1 k 0 1 2 M 1 of Xzk=n=0N1xnzkn=n=0N1xnAnWnk 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 (kn2=n22nk+k2)(nk=12(n2+k2kn2)) k n 2 n 2 2 n k k 2 n k 1 2 n 2 k 2 k n 2 , So Xzk=n=0N1xnAnWn22Wk22Wkn22 X zk n N 1 0 x n A n W n 2 2 W k 2 2 W k n 2 2 Wk22n=0N1xnAnWn22Wkn22 W k 2 2 n N 1 0 x n A n W n 2 2 W k n 2 2

Thus, Xzk X zk can be compared by

  1. Premultiply xn x n by AnWn22 A n W n 2 2 , n= 0 1N1 n 0 1 N 1 to make yn y n
  2. Linearly convolve with Wkn22 W k n 2 2
  3. Post multiply by to get Wk22 W k 2 2 to get Xzk 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)

Wn22 W n 2 2 is required only for (N1)nM1 N 1 n M 1 , so this linear convolution can be implemented with LN+M1 L N M 1 FFTs.

note:

Wrap Wn22 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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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

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Who can create a lens?

Any individual member, a community, or a respected organization.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks