Showing papers on "Discrete sine transform published in 1973"

Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm†

Abstract: A new method is presented for the measurement of the Wiener kernels of a non-linear system. The method uses the complex exponential functions as a set of orthogonal functions with which to expand the kernels. The fast Fourier transform is then employed in the most efficient measurement of the kernels so far discovered.

On coding and filtering stationary signals by discrete Fourier transforms (Corresp.)

Abstract: This correspondence concerns real-time Fourier processing of stationary data and examines the widespread belief that coefficients of the discrete Fourier transform (DFT) are "almost" uncorrelated. We first show that any uniformly bounded N \times N Toeplitz covariance matrix T_N is asymptotically equivalent to a nonstandard circulant matrix C_N derived from the DFT of T_N . We then derive bounds on a normed distance between T_N and C_N for finite N , and show that \mid T_N - C_N \mid ^ 2 = O(1/N) for finite-order Markov processes. Finally we demonstrate that the performance degradation resulting from the use of DFT (as opposed to Karhunen-Loeve expansion) in coding and filtering is proportional to \mid T_N - C_N \mid and therefore vanishes as the inverse square root of the block size N when N \rightarrow \infty .

The Stabilization of Two-Dimensional Recursive Filters via the Discrete Hilbert Transform

Abstract: Two-dimensional recursive filters are useful only if stable, that is, if their outputs remain bounded for bounded inputs The stability of a recursive filter depends on the phase spectrum of its denominator array A two-dimensional generalization of the discrete Hilbert transform leads to a scheme producing stability with nominal distortions of the filter's desired amplitude spectrum The method is therefore an attractive alternate to a least-squares procedure recently described by Shanks et al

Fully digital spectrum analyzer using time compression and discrete fourier transform techniques

Abstract: A fully digital spectrum analyzer accepting as an input either an analog signal or a series of digital numbers and using time compression and DFT (Discrete Fourier Transform) techniques to provide the spectral component values of the input signal. Novel techniques and means are used in obtaining the power values for selected spectral lines and in averaging these power values. Statistically controlled noise is added to the input of the spectrum analyzer to enhance its resolutin beyond the resolution which would be otherwise available. Advanced and efficient techniques are used for generating and applying trigonometric functions in the course of finding the real and imaginary part of Fourier transforms, and for providing running averages of the power spectra.

Fast Equalization System

Abstract: A discrete frequency domain equalization system is disclosed for utilization in a high-speed synchronous data transmission system where, in a preferred embodiment, samples of an input signal in the time domain are transformed by a discrete fast Fourier transform device into samples in the frequency domain. Reciprocal values of these frequency domain samples are derived from a reciprocal circuit and then transformed by an inverse discrete fast Fourier transform device into time domain samples which are the desired tap gains that are applied to a transversal equalizer in order to minimize the errors in a received signal caused by intersymbol interference and noise.

On deconvolution using the discrete Fourier transform

Abstract: A solution is given to the problem of deconvolving two time sequences using discrete Fourier transform (DFT) techniques when one of the sequences is of infinite duration. Both input- and impulse-response deconvolution problems are considered. Results are summarized of a computer study of the algorithm that performs the deconvolution iteratively, using the fast Fourier transform (FFT) algorithm at each stage.

A modified generalized discrete transform

Abstract: A modified version of the generalized discrete transform described earlier is now developed. The transform matrix of this modified version has a number of zeros as its elements, and consequently its matrix factors are more sparse. This results in fewer arithmetic operations and corresponding savings in computer time, when information is processed.

Time, frequency, sequency, and their uncertainty relations (Corresp.)

Abstract: We study the form assumed by the classical time-frequency uncertainty relations in discrete as well as nontrigonometric spectral analysis. In particular we find that if an N -sample time signal is to contain a fraction \gamma of its energy in T consecutive samples, then the minimum number of frequency components containing that same energy fraction must be greater than N/T(2\gamma - 1)^2 . It is also found that the discrete Walsh transform permits greater energy concentration (less uncertainty) than the discrete Fourier transform.

Class of discrete orthogonal transforms.

Abstract: Abstract For a given N-periodic sequence, a class of log2 N discrete orthogonal transforms ranging from Walsh-Hadamard transform to discrete Fourier transform (DFT) is defined. The power spectra invariant to circular shift of the sampled data for these transforms are developed. Phase spectra, analogous to that of the DFT, for all the discrete transforms are defined and developed. Recursive relations for generating the transform matrices are developed. Generalized expressions for factoring these transform matrices are provided. Based on these matrix factors, efficient algorithms for fast computation of the transform coefficients are developed. By introducing a number of zeros as the elements in the transform matrices, a modified version of the transforms is developed. By using these modified matrices, the power and phase spectra can be computed efficiently. These transforms can be used in the general area of information processing.

A note on exact discrete Fourier transforms

Abstract: It is proved that the z transform of a sequence of data values cannot be exactly computed using a binary number representation for values of z on the unit circle, except z =±1, z = ±j. It is also proved that the discrete Fourier transform (DFT) of a sequence of data values cannot be evaluated with rational numbers.

Spectral modes of the generalized transform

Abstract: Spectral modes for the generalized transform are developed. These spectra are invariant to cyclic shift of the input sequence and possess sequency resolution in their squared terms. They cannot, however, be called energy spectra as there are cross terms, some of which are negative.

DZ transforms and non-linear discrete time systems analysis

Abstract: This study proposes a modified Z transform defined in terms of the regular Z transform and a difference transformation relation for the analysis of classes of nonlinear discrete time systems. The application of this new DZ transform technique is illustrated with an example.

On the Eigenvectors of the Matrix That Performs the Discrete Finite Fourier Transform

Abstract: : The discrete finite Fourier transform can be regarded as a matrix operation, since each element of one member of the pair is a linear combination of all the elements of the other member A remarkably simple relation between a periodic function of a discrete variable and its discrete finite Fourier transform, namely that the absolute values of their expansion coefficients in these eigenvectors are the same, has been demonstrated A canonical form for such functions (with respect to the finite Fourier transform) is suggested in which the transform can be done by inspection