The Chirp z-Transform Algorithm and Its Application

Abstract

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.

Fast computation of the ambiguity function and the Wigner distribution on arbitrary line segments

“…In order to compute the samples of the AF on an arbitrarily positioned segment of a radial slice, the chirp -transform (CZT) algorithm [37] can be used. Here, we will use a special version of this algorithm (which is also called the chirp transform algorithm) to compute uniformly spaced samples of a radial slice on the interval for arbitrary values of the parameters and .…”

Hedging of Asian options under exponential Lévy models: computation and performance

“…We compute numerically the Greeks convolutions (16) (2011), respectively, about the construction of the Fourier transforms and their discretization for a fast numerical implementation using the so-called chirp z-transform (see also Bluestein, 1968;Rabiner et al, 1969Rabiner et al, andČerný, 2004, which is readily available in common numerical computing plat-forms like Matlab. This powerful approach allows us to gauge explicitly the precision of the scheme and, by further exploiting its smooth convergence in the number of discretization points of the Fourier transforms, achieve highly accurate results as we show in the following section.…”

Glottal Source Estimation Using an Automatic Chirp Decomposition

“…The Chirp Z-Transform (CZT), as introduced by Rabiner et al [13] in 1969, allows the evaluation of the Z-transform on a spiral contour in the Z-plane. Its first application aimed at separating too close formants by reducing their bandwidth.…”