A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >

Linear and quadratic time-frequency signal representations

http://www.norbertwiener.umd.edu/Research/Duhamel_Vetterli_FFT_90.pdf

Fast fourier transforms: a tutorial review and a state of the art

Note: V. K. Madisetti, D. B. Williams, Eds. Reference LCAV-CHAPTER-2005-009 Record created on 2005-06-27, modified on 2017-05-12

Fast Fourier transforms: a tutorial review and a state of the art

Efficient signal processing algorithms are important for embedded and power-limited applications since, by reducing the number of computations, power consumption can be reduced significantly. Similarly, efficient algorithms are also critical to very large scale applications such as video processing and four-dimensional medical imaging. This self-contained guide, the only one of its kind, enables engineers to find the optimum fast algorithm for a specific application. It presents a broad range of computationally-efficient algorithms, describes their structure and implementation, and compares their relative strengths for given problems. All the necessary background mathematics is included and theorems are rigorously proved, so all the information needed to learn and apply the techniques is provided in one convenient guide. With this practical reference, researchers and practitioners in electrical engineering, applied mathematics, and computer science can reduce power dissipation for low-end applications of signal processing, and extend the reach of high-end applications.

/pdf/fast-algorithms-for-signal-processing-53jk24t0yb.pdf

Fast Algorithms for Signal Processing

Algorithms to compute coefficients of the finite double sum expansion of time-varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented. The algorithms are based on the computation of the discrete Zak transform (DZT). Fast algorithms to compute DZT are presented. Modifications to the algorithms which increase robustness are given. >

The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals

We exhibit new algorithms for DFT(p; k), the discrete Fourier transform on a k-dimensional data set with p points along each array, where p is a prime. At a cost of additions only, these algorithms compute DFT(p; k) with (pk- 1)/(p - 1) distinct DFT(p; 1) computations.

New algorithms for the multidimensional discrete Fourier transform

A fast algorithm computing decimated finite cross-ambiguity functions is designed, which is tailored for implementation on parallel computer architectures. Several standard methods carry out cross-ambiguity function computations in two stages, a coarse mode which estimates the location of peaks, and a fine mode relying on FFT algorithms which produces greater detail. The coarse mode computation is not required, but only decimated values ranging over the entire domain of the cross-ambiguity function are computed. >

L. Auslander

Papers

New algorithms for the multidimensional discrete Fourier transform

Computing decimated finite cross-ambiguity functions