Chirp ZTransform

backlinks: /SRSoftwareOverview/FusionRegistration

Given a signal

latex2png equation

the Z-transform (for a finite number of points) is defined as

latex2png equation

The chirp z-transform computes the z-transform on a contour of the form

latex2png equation

where A and W are arbitrary complex numbers. This path describes a spiral, starting at an arbitrary point A and curving in or outwards depending on the value of W. The transform becomes

latex2png equation

Making the substitution suggested by Bluestein,

latex2png equation

the transform expands to

latex2png equation

The transform can now be broken into three parts:

  1. Construct a signal latex2png equation.
  2. Circularly convolve the signal with latex2png equation.
  3. Weigh the result by latex2png equation.

Bluestein's substitution therefore allows us to write the chirp z-transform in terms of a convolution, which, in turn, can be calculated using the fast Fourier transform.

The fast Fourier transform is especially quick for sequences of square lengths. Given that we'd like to calculate M points, and that our input sentence is of length N, we pad our signals to a length

latex2png equation

Since multiplication in the Fourier domain leads to circular convolution in the discrete-time domain, the signals need to be padded to a length of at least latex2png equation.

  1. Construct latex2png equation of length N and pad to L.
  2. Construct latex2png equation of length L.
  3. Obtain the FFTs of the above signals and multiply (not quite, more on that later).
  4. Select elements N to N+M.
  5. Weigh by latex2png equation.

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