The chirp z-transform algorithm

A computational algorithm for numerically evaluating the -transform of a sequence of samples is discussed. This algorithm has been named the chirp -transform (CZT) algorithm. Using the CZT algorithm one can efficiently evaluate the -transform at points in the -plane which lie on circular or spiral contours beginning at any arbitrary point in the -plane. The angular spacing of the points is an arbitrary constant, and and are arbitrary integers. The algorithm is based on the fact that the values of the -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 and moderately large, the computation time is roughly proportional to as opposed to being proportional to for direct evaluation of the -transform at points.