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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 the right side of equation (1.1) converges for at least some values of z. We restrict ourselves to the z-transform of sequences with only a finite number N of nonzero points. In this case, we can rewrite eqs.1.1 without loss of generality as

(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 W are arbitrary complex numbers we see that the points lie on an arbitrary straight line segment of arbitrary length and sampling density. Clearly the contour indicated in (6a) is not the most general contour but it is considerably more general than that for which the DFT applies. In Fig. 2, an example of this more general contour is shown in both the z-plane and the s-plane. To compute the Z-Transform along this more general contour would seem to require NM multiplications and additions as the special symmetries of  which are exploited in the derivation of the FFT are absent in the more general case. However, we shall see that by using the sequence  in various roles we can apply the FFT to the computation of Z-transformation along the contour of (1.7). since for W0=1,the sequence   is a complex sinusoid of linearly incresing frequency,and since a similar waveform used in some radar systems has the picturesque name "chirp," we call the algorithm we are about to present the chirp z-transform (CZT). Since the CZT permits computing the z-transform on a more general tour than the FFT permits it is more flexible than the FFT, although it is also considerably slower. The additional freedoms offered by the CZT include the following:

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.