The Chirp x-Transform Algorithm

A computational algorithm for numerically evaluating the z-transform of a sequence of N samples is discussed. This algorithm has been named the chirp z-transform (CZT) algorithm. Using the CZT algorithm one can efficiently evaluate the z-transform at M points in the z-plane which lie on circular or spiral contours beginning at any arbitrary point in the z-plane. The angular spacing of the points is an arbitrary constant, and M and N are arbitrary integers. The algorithm is based on the fact that the values of the z-transform on a circular or spiral contour can be expressed as a discrete convolution. Thus one can use well-known high-speed convolution techniques to evaluate the transform efficiently. For M and N moderately large, the computation time is roughly proportional to (N+M) log(N+M) as opposed to being proportional to N , M for direct evaluation of the z-transform at M points. Manuscript received December 23, 1968; revised January 16, 1969. This is a condensed version of a paper published in the Bell System Technical Journal, May 1969. Operated with support from the U. S. Air Force. 1, Introduction In dealing with sampled data the z-transform plays the role which is played by the Laplace transform in continuous time systems. One example of its application is spectrum analysis. We shall see that the computation of sampled z-transforms, which has been greatly facilitated by the fast Fourier transform (FFT) [l], [2] algorithm, is still further facilitated by the chirp z-transform (CZT) algorithm to be described in this paper. The z-transform of a sequence of numbers xn is defined as