The computation of a minimum and sufficient set of addresses, along with corresponding bit-reversed addresses, is necessary to do unscrambling in conventional FFT (fast Fourier transform) procedures. A matrix-based interpretation of this problem is presented, and typical software and hardware solutions are described. An overall relationship between the two sets of primary and secondary indices is shown, obviating any additional relationship at individual pairwise level. Although the algorithms have been discussed on radix-2 basis, they can be modified for a radix if necessary. >

Bit reversal in FFT from matrix viewpoint

This paper presents a model of the Southeastern part of the UCTE Interconnection that is used to assess interarea oscillation modes and the transient loading, after a severe disturbance, of the tie-lines connecting the Hellenic System to its Northern neighbours. Eigenvalue analysis and FFT are used to identify the interarea modes involving the UCTE Interconnection, as well as the Hellenic System. Damping is found adequate but depending upon external stabilizers and governor modelling. It was shown by simulation that a severe disturbance may cause load shedding to protect the most critical tie-line.

Interarea Oscillations and Tie-line Transients in the Hellenic Interconnected System

A recent class of continuous transforms is generalized to give an infinite number of new transforms. A frequency interpretation of the transform base functions gives fast generalized transform (FGT) matrices using standard digit reversal techniques. Examples of FGT factorizations are given. Advantages of the generalized transforms include continuous transform properties for system design and analysis and FGT algorithms for practical applications using digital hardware.

http://ieeexplore.ieee.org/iel6/29/26114/01162746.pdf

Generalized transforms

The generalized discrete transform (GT) r and its modified version (MGT) r have been defined and their properties have been developed. The phase or position spectra and the shift-invariant power spectra for these transforms have been developed. For the (GT) r family, these transforms range from the Walsh-Hadamard transform (WHT) to the discrete Fourier transform (DFT) whereas for the (MGT) r group, the range is from the modified WHT (MWHT) to the DFT. The power and phase spectra for these transforms represent groups of frequencies unlike the individual frequency composition characteristic of the DFT. Recently, it was shown that the input data can be recovered from the power and phase spectra of the WHT. This concept is now extended to the generalized transforms and it is shown that the original signal can be reconstructed from the phase and power spectra of any of the members of the (GT) r family. Also an efficient algorithm for evaluating the generalized phase spectrum for various cyclic shifts of the data sequence is developed.

Generalized phase spectrum

Slant transform image coding (Ph.D. Thesis abstr.)

A system is developed to detect tracks crossing two-dimensional noise fields. This is accomplished by filtering and integration of signal power in the frequency domain of a new class of generalized transforms which include the FFT and FBT as limiting forms. If the track is essentially parallel to a transform axis and if filters are derived from hard-limited linear approximations to tracks, the FBT offers considerable hardware economy over the FFT in the mechanization algorithm.

Low-Level Signal Detection Using a New Transform Class

Publisher Summary The fast Fourier transform (FFT) computes the discrete Fourier transform (DFT) using a reduced number of arithmetic operations as compared to brute-force evaluation of the DFT. The method is efficient, because it eliminates redundancies that result from adding certain data sequence values after they have been multiplied by the same factors of fixed complex constants during the evaluation of different DFT transform coefficients. The efficiency is achieved at the expense of reordering the data sequence and/or transform sequence, but the additional expense is small compared to the reduction in multiplications and additions. The chapter reviews one-dimensional (1-D) and two-dimensional (2-D) DFTs, presents 2-D flow diagrams and the equivalent operations in matrix format, and explains DFT and FFT matrix representation. The first FFTs result from a mixed-radix integer representation (MIR) that includes binary, decimal, and octal integers. The chapter presents a comparison of number of arithmetic operations required to compute various FFT algorithms. Many real-time FFTs are mechanized with dedicated, fixed-point hardware. The chapter also discusses the determination of the number of bits to fully utilize these FFT processors and presents a simple, easy to visualize graphical techniques i to specify digital word lengths in a typical spectral analysis system.

Fast Fourier Transforms

Publisher Summary This chapter provides an overview of transforms and transform properties. It reviews the nature of the sampled data and transforms and transform properties for the analysis of data sequences. The integrals defining the series coefficients correspond to the inverse discrete-time Fourier (IDTFT) and considers one- and two-dimensional series. The aliasing phenomenon leads to a periodic spectrum for data sequences so that the spectrum has a Fourier representation in terms of the data. This representation can be found from the data by using the discrete-time Fourier transform (DTFT). The DTFT is generalized to the z-transform, which is a powerful tool for data sequence analysis. The chapter discusses the reason for the periodicity of the discrete-time spectrum and presents DTFT properties. It also reviews the discrete Fourier transform (DFT) and highlights the Laplace transform. The chapter reviews discrete-time random sequences and discusses correlation and covariance sequences and their power spectral densities.

Douglas F. Elliott

Papers

Low-Level Signal Detection Using a New Transform Class

Fast Fourier Transforms

Transforms and Transform Properties