Showing papers on "Discrete sine transform published in 1970"

A linear filtering approach to the computation of discrete Fourier transform

Abstract: It is shown in this paper that the discrete equivalent of a chirp filter is needed to implement the computation of the discrete Fourier transform (DFT) as a linear filtering process. We show further that the chirp filter should not be realized as a transversal filter in a wide range of cases; use instead of the conventional FFT permits the computation of the DFT in a time proportional to N \log_{2} N for any N, N being the number of points in the array that is transformed. Another proposed implementation of the chirp filter requires N to be a perfect square. The number of operations required for this algorithm is proportional to N^{3/2} .

Discrete Hilbert transform

Abstract: The Hilbert transform H\{f(t)\} of a given waveform f(t) is defined with the convolution H{\f(t)} = f(t) \ast (1/\pit) . It is well known that the second type of Hilbert transform K_{0}{\f(x)\}=\phi(x) \ast (1/2\pi)\cot\frac{1}{2}x exists for the transformed function f(tg\frac{1}{2}x)= \phi(x) . If the function f(t) is periodic, it can be proved that one period of the H transform of f(t) is given by the H 1 transform of one period of f(t) without regard to the scale of tbe variable. On the base of the discrete Fourier transform (DFT), the discrete Hilbert transform (DHT) is introduced and the defining expression for it is given. It is proved that this expression of DHT is identical to the relation obtained by the use of the trapezoidal rule to the cotangent form of the Hilbert transform.

The discrete Hilbert transform

Abstract: A Hilbert transformation procedure for discrete data has been developed. This transform could be useful in a variety of applications such as the analysis of sampled data systems and the simulation of filters.

A Generalization of the Fast Fourier Transform

Abstract: A procedure for factoring of the N×N matrix representing the discrete Fourier transform is presented which does not produce shuffled data. Exactly one factor is produced for each factor of N, resulting in a fast Fourier transform valid for any N. The factoring algorithm enables the fast Fourier transform to be implemented in general with four nested loops, and with three loops if N is a power of two. No special logical organization, such as binary indexing, is required to unshuffle data. Included are two sample programs, one which writes the equations of the matrix factors employing the four key loops, and one which implements the algorithm in a fast Fourier transform for N a power of two. The algorithm is shown to be most efficient for Na power of two.

The digital computation of discrete spectra using the fast Fourier transform

Abstract: This paper is devoted to a discussion of discrete spectrum analysis which is important in applicational areas such as sonar and replica correlation. The discrete Fourier transform is shown to arise naturally as a consequence of finite impulsive sampling and the fast Fourier transform is introduced as the most efficient means of computing the discrete Fourier transform. These are described in terms of parameters pertinent to digital sonar signal processing, including resolution, dynamic range, and processing gain. Computational accuracy is investigated as a function of word lengths associated with the data, kernels, and intermediate transforms for both conditional and automatic array scaling. In real-time equipment, it is frequently necessary to employ some sort of automatic gain control and such a device is investigated here. Results are presented which enable specification of word length and automatic gain control requirements as a function of desired dynamic range, input signal-to-noise ratio, and mean-square error at the quantizer output.

Apparatus and methods for deriving in essentially real time continuous electrical representations of the fourier and inverse fourier transform

Abstract: Apparatus and methods for deriving in essentially real time unweighted and weighted continuous electrical representations of the Fourier transform and/or the inverse Fourier transform of a complex waveform. In performing the Fourier transform, the input waveform is sampled at the Nyquist sampling rate and the samples stored in respective sample-and-hold circuits. These samples are applied to signal generating circuitry for deriving harmonically related time-varying cosine and sine signals having peak values corresponding to weighted or unweighted values of respective ones of the sample-and-hold circuit outputs, and having a fundamental frequency which may be chosen independently of the frequency content of the input waveform. These cosine and sine signals are then respectively summed for producing resultant summed sine and cosine signals which respectively correspond to weighted or unweighted representations of the real and imaginary components of the Fourier transform of the input waveform with the frequency variable being simulated by time. In one embodiment, these summed sine and cosine signals are applied to a function generator for generating signals representative of the weighted or unweighted amplitude spectrum and/or phase spectrum of the input waveform for further application to appropriately calibrated and adjusted oscilloscopes for producing visual displays thereof. In another embodiment, these resultant summed sine and cosine signals are in turn sampled at the Nyquist sampling rate to provide samples which may conveniently be modified in accordance with desired criteria. The modified samples are then recombined using the inverse Fourier transform technique of the invention which employs circuitry basically similar to that used for the Fourier transform to produce an output signal representative of the original input signal and containing the modifications produced in accordance with the desired criteria.

Discrete and fast Fourier transformations on N-dimensional lattices

Abstract: The N-dimensional discrete Fourier transform (DFT) may be represented as a matrix of elements of unit magnitude, with the arguments constructed as inner products of lattice vectors in the sampling and wavenumber domains, filling regions inverse to the basic cells on their respective lattices. The fast Fourier transform numerical technique is directly applicable to this configuration.

Factorization of spectra by discrete Fourier transforms (Corresp.)

Abstract: A discrete Fourier transform method for factoring arbitrary spectral density functions is presented. The factorization can be implemented in a straightforward and efficient manner, and it does not require that the spectra be rational. An expression for the absolute error is also presented.

Analog computation of the fast Fourier transform

Abstract: A method is proposed for computing the discrete Fourier transform of complex data whose real and imaginary parts are represented as voltages. Operational amplifiers and resistors are the only computing elements required.

Comment on "Computation of the Fast Walsh-Fourier Transform" 1

Abstract: J. L. Shanks1has given an algorithm for computing the discrete Walsh transform (abstract Fourier transform) of a sampled periodic function whose domain of definition is the set of integers modulo 2n. An algorithm of the same efficiency, using a much simpler notation, was given for the abstract Fourier transform in my correspondence published in 1963 in this TRANSACrIONS.2This transform has an identical matrix representation; the only difference is that the function domain is represented (for computation purposes) by binary coded representations of the integers from 0 to 2n−1. These binary n-tuples form a group under vector addition, modulo two.

The DigitalComputation of Discrete Spectra Using the Fast Fourier

Abstract: This paper is devoted to a discussion of discrete spectrum analysis which is important in applicational areas such as sonar and replica correlation. The discrete Fourier transform is shown to arise naturally as a consequence of finite impulsive sampling and the fast Fourier transform is introduced as the most efficient means of computing the discrete Fourier transform. These are described in terms of parameters pertinent to digital sonar signal processing, including resolution, dynamic range, and processing gain. Computational accuracy is investigated as a function of word lengths associated with the data, kernels, and intermediate transforms for both conditional and automatic array scaling. In real-time equipment, it is frequently necessary to employ some sort of automatic gain control and such a device is investigated here. Results are presented which enable specification of word length and automatic gain control requirements as a function of desired dynamic range, input signal-to-noise ratio, and mean-square error at the quantizer output.