Showing papers on "Discrete sine transform published in 1972"

Eigenvalue and eigenvector decomposition of the discrete Fourier transform

Abstract: The principal results of this paper are listed as follows. 1) The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are {1, -1,j, -j} . 2) An eigenvector basis is constructed for the DFT. 3) The multiplicities of the eigenvalues are summarized for an N×N transform as follows.

Discrete Convolutions via Mersenne Transrorms

Abstract: A transform analogous to the discrete Fourier transform is defined in the ring of integers with a multiplication and addition modulo a Mersenne number. The arithmetic necessary to perform the transform requires only additions and circular shifts of the bits in a word. The inverse transform is similar. It is shown that the product of the transforms of two sequences is congruent to the transform of their circular convolution. Therefore, a method of computing circular convolutions without quantization error and with only very few multiplications is revealed.

A fast Fourier transform algorithm for symmetric real-valued series

Abstract: A new algorithm is presented for calculating the real discrete Fourier transform of a real-valued input series with even symmetry. The algorithm is based on the fast Fourier transform algorithm for arbitrary real-valued input series (FTRVI) [1], [2]. By eliminating all unnecessary steps and storage locations, and by rearranging the intermediate results and the operation sequence, it is possible to reduce the computation time and the required core storage by a factor of 2 as compared to the case of arbitrary real input or by a factor of 4 as compared to the general fast Fourier transform for complex inputs.

A note on "Discrete Hilbert transform"

Abstract: An expression for the discrete Hilbert transform (DHT) is given for the case of an odd number of points.

Evaluation of response of nonlinear discrete systems using multidimensional Z transforms

Abstract: Using the concept of a multidimensional Z transform, a method for solving time-invariant nonlinear difference equations is proposed. The solution is obtained in the form of a discrete Volterra series. Using the new technique, the response of a nonlinear sampled-data control system is obtained which compares very favourably with that obtained by other, more tedious and cumbersome, methods.

Automatic equalization and discrete Fourier transform techniques (Ph.D. Thesis abstr.)

Abstract: The primary objective of this thesis is the study and analysis of techniques that use discrete Fourier transforms (DFT) in conjunction with fast Fourier transform (FFT) algorithms for automatic equalization of synchronous data transmission. W e show that the problem of minimizing mean-square intersymbol interference can be analyzed in the discrete f requency domain as an optimization problem with constraints. Various solutions to this problem are studied including Rosen’s gradient projection method, Lagrange multipliers, and direct substitution. W e prove that the rate of convergence toward the opt imum parameter setting is faster for the gradient projection scheme, for channels usually d iscussed in the literature, than for the corresponding t ime-domain technique. W e then develop an alternative gradient projection method that provides savings proport ional to N/log, N in the number of required computat ions, where N is the number of discrete f requency parameters. Finally, we devise a scheme for finding an approximate solution in one iteration using gradient projection. This solution was found to be almost exact for the particular cases we simulated, even in the presence of noise. The speed of convergence for all the schemes is dependent on the overall channel characteristics and each method has advantages in certain special situations. W e show that all the aforement ioned methods converge in the mean in the presence of noise. A var iance bound indicates that the var iance about this setting is finite and can be made as small as desired by reducing the gradient step size and hence the speed of convergence. Our second objective is to use the DFT and FFT algorithms to make time domain equalization computationally more efficient. W e show that the number of computat ions needed to set a time domain equalizer can be made proport ional to M log, M instead of MZ, where M is the number of adjustable parameters, by use of FFT algorithms. Savings always ensue for sufficiently large M and grow rapidly thereafter. The breakeven point is approximately M = 16. W e also derive tight and easily obtainable bounds on the eigenvalues of the t ime-domain iteration matrix in terms of DFT coefficients. This allows us to increase the speed of convergence. Leonard P. W inkler, “Opt imum and adapt ive detector arrays,” Ph.D., Dep. Elec. Eng., Polytech. Inst. Brooklyn, Brooklyn, N.Y., June 1971. Adviser: Mischa Schwartz.

A note on the spectral analysis of periodic functions

Abstract: An analysis of the Fourier transform for a truncated cosine waveform is given that illustrates the dependence of the transform on the phase as the period of the waveform becomes large with respect to the record length. The effects of sampling on the transform are then discussed and illustrated.

Some considerations of the discrete Fourier and Walsh-Hadamard transforms

Abstract: In this Paper, we summarize a set of properties of the discrete Fourier transform (DFT) and compare them with corresponding properties of the Walsh-Hadamard transform (WHT) [see Table I]. It is hoped that such a summary will assist in observing the analogy between Walsh-Hadamard and discrete Fourier analyses for those who are relatively more familiar with the latter.