The Chirp z-Transform Algorithm and Its Application

We discuss a computational algorithm for numerically evaluating the z-transform of a sequence of N samples. This algorithm has been named the chirp z-transform algorithm. Using this 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; 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) log2(N + M) as opposed to being proportional to N · M for direct evaluation of the z-transform at M points. Applications discussed include: enhancement of poles in spectral analysis, high resolution narrow-band frequency analysis, interpolation of band-limited waveforms, and the conversion of a base 2 fast Fourier transform program into an arbitrary radix fast Fourier transform program.