Chirp z-transform

The
z-transform of a sequence of numbers x (n)** **is defined as

(1.1)

a function of the complex variable z. In
general, both ** **and
could be complex. It is assumed
that the sum on

(1.2)

where the sum in (1.2) converges for all z except z= 0.

Equations (1.1) and (1.2) are like the
defining expressions for the Laplace transform of a train of equally spaced
impulses of magnitudes*. *Let
the spacing of the impulses be *T *and let the train of impulses be*. *Then
the Laplace transform is which is the same as if we let

(1.3)

If we are dealing with sampled waveforms
the relation between the original waveform and the train of impulses is well
understood in terms of the phenomenon of aliasing. Thus the z-transform of the
sequence of samples of a time waveform is representative of the Laplace
transform of the original waveform in a way which is well understood. The
Laplace transform of a train of impulses repeats its values taken in a
horizontal strip of the s-plane of width * *in
every other strip parallel to it. The z-transform maps each such strip into the
entire Z-plane, or conversely, the entire z-plane corresponds to any horizontal
strip of the s-plane, e.g., the region where .* *In
the same correspondence, the * *axis
of the s-plane, along which we generally equate the Laplace transform with the
Fourier transform, is the unit circle in the z-plane, and the origin of the
s-plane corresponds to Z=1. The interior of the z-plane unit circle corresponds
to the left half of the x-plane, and the exterior corresponds to the right half
plane. Straight lines in the s-plane correspond to circles or spirals in the
z-plane. Fig.1 shows the correspondence of a contour in the s-plane to a
contour in the z-plane. To evaluate the Laplace transform of the impulse train
along the linear contour is to evaluate the z-transform of the sequence along
the spiral contour.

Values of the z-transform are usually
computed along the path corresponding to the axis, namely the unit circle. This
gives the discrete equivalent of the Fourier transform and has many
applications including the estimation of spectra, filtering, interpolation, and
correlation. The applications of computing z-transforms of the unit circle are
fewer, but one is presented elsewhere namely the enhancement of spectral
resonances in systems for which one has some foreknowledge of the locations of
poles and zeroes. Just as we can only compute (1.2) for a finite set of
samples, so we can only compute (2) at a finite number of points, say*.*

Just as we can only compute (1.2) for a finite set of samples, so we can only compute (1.2) at a finite number of Points,

(1.4)

(1.5)

Where k=0, 1, 2… N-1

For which,

, k=0,1,2…N-1 (1.6)

Equation (1.6) is called the Discrete Fourier transform (DFT).

The DFT has, however have some limitations which can be eliminated using CZT algorithm[1]. Investigations have to be done for the computation of the Z-transform on a more general contour of the following form

,k=0,1,2……………M-1 (1.7)

Where M is an arbitrary integer and both A and W are arbitrary complex number of the form

(1.8)

(1.9)

The case A=1, M=N and corresponds to the DFT.

The general z-plane contour begins with
the point z=A and, depending on the value of *W, *spirals in or out with
respect to the origin. If, the contour is an arc of a
circle. The angular spacing of the samples is. The equivalent s-plane contour
begins with the point

(1.10)

and the general point on the s-plane contour is

(1.11)

Since ** A **and

1) The number of time samples does not have to equal

2) Neither *M *nor ** N **need
be a composite integer.

3) The angular spacing of the ** **is
arbitrary.

4)
The contour need not be a circle but can spiral in or out with respect to the
origin. In addition, the point *zo *is arbitrary, but this is also the
case with the FFT if the* *samples *xn *are multiplied by * *before
transforming.

* *