Abstract. Certain solutions to Harper’s equation are discrete analogues of (and approximations to) the Hermite–Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator. 1.

Abstract—The generalized entropies of Rényi inspire new measures for estimating signal information and complexity in the time–frequency plane. When applied to a time–frequency representation (TFR) from Cohen’s class or the affine class, the Rényi entropies conform closely to the notion of complexity that we use when visually inspecting time–frequency images. These measures possess several additional interesting and useful properties, such as accounting and cross-component and transformation invariances, that make them natural for time–frequency analysis. This paper comprises a detailed study of the properties and several potential applications of the Rényi entropies, with emphasis on the mathematical foundations for quadratic TFRs. In particular, for the Wigner distribution, we establish that there exist signals for which the measures are not well defined. Index Terms—Complexity, Rényi entropy, time–frequency analysis, Wigner distribution.

This paper presents a novel approach based on time-frequency distributions (TFDs) for separating signals received by a

Spatial time--frequency distributions (STFDs) have been recently introduced as the natural means to deal with source signals that are localizable in the time--frequency domain. Previous work in the area has not provided the eigenanalysis of STFD matrices, which is key to understanding their role in solving direction finding and blind source separation problems in multisensor array receivers. The aim of this paper is to examine the eigenstructure of the STFDs matrices. We develop the analysis and statistical properties of the subspace estimates based on STFDs for frequency modulated (FM) sources. It is shown that improved estimates are achieved by constructing the subspaces from the time--frequency signatures of the signal arrivals rather than from the data covariance matrices, which are commonly used in conventional subspace estimation methods. This improvement is evident in a low signal-tonoise ratio (SNR) environment and in the cases of closely spaced sources. The paper considers the MUSIC technique to demonstrate the advantages of STFDs and uses it as grounds for comparison between time--frequency and conventional subspace estimates.

In many pattern recognition applications, features are traditionally extracted from standard time--frequency representations (TFRs). This assumes that the implicit smoothing of, say, a spectrogram is appropriate for the classification task. Making such assumptions may degrade classification performance. In general, any time--frequency classification technique that uses a singular quadratic TFR (e.g., the spectrogram) as a source of features will never surpass the performance of the same technique using a regular quadratic TFR (e.g., Rihaczek or Wigner--Ville). Any TFR that is not regular is said to be singular. Use of a singular quadratic TFR implicitly discards information without explicitly determining if it is germane to the classification task. We propose smoothing regular quadratic TFRs to retain only that information that is essential for classification. We call the resulting quadratic TFRs class-dependent TFRs. This approach makes no a priori assumptions about the amount and type of time--frequency smoothing required for classification. The performance of our approach is demonstrated on simulated and real data. The simulated study indicates that the performance can approach the Bayes optimal classifier. The real-world pilot studies involved helicopter fault diagnosis and radar transmitter identification.

Abstract—We deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform (LCT) of a function, from the samples of the original function. Two approaches are presented and compared. The first is based on decomposition of the LCT into chirp multiplication, Fourier transformation, and scaling operations. The second is based on decomposition of the LCT into a fractional Fourier transform followed by scaling and chirp multiplication. Both algorithms take log time, where is the time-bandwidth product of the signals. The only essential deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus, the algorithms compute LCTs with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy. Index Terms—Diffraction integrals, fractional Fourier transform (FRT), linear canonical transform (LCT), time-frequency

The aim of this paper is to propose an adaptive method for suppressing wideband interferences in spread spectrum (SS) communications. The proposed method is based on the time--frequency representation of the received signal from which the parameters of an adaptive time-varying interference excision filter are estimated. The approach is based on the generalized Wigner--Hough transform as an effective way to estimate the instantaneous frequency of parametric signals embedded in noise. The performance of the proposed approach is evaluated in the presence of linear and sinusoidal FM interferences plus white Gaussian noise in terms of SNR improvement factor and bit error rate (BER).

We overview a new non-parametric method for estimating the time-varying spectrum of a non-stationary random process. Our method extends Thomson’s powerful multiple window spectrum estimation scheme to the time-frequency and time-scale planes. Unlike previous extensions of Thomson’s method, we identify and utilize optimally concentrated Hermite window and Morse wavelet functions and develop a statistical test for extracting chirping line components. Examples on synthetic and real-world data illustrate the superior performance of the technique. 2

Recently, Cohen has proposed a construction for joint distributions of arbitrary physical quantities, in direct generalization of joint time-frequency representations. Actually this method encompasses two approaches, one based on operator correspondences and one based on weighting kernels. The literature has emphasized the kernel method due to its ease of analysis; however, its simplicity comes at a price. In this paper, we use a simple example to demonstrate that the kernel method cannot generate all possible bilinear joint distributions. Our results suggest that the relationship between the operator method and the kernel method merits closer scrutiny. I. Introduction By representing signals in terms of several physical quantities simultaneously, joint distribution functions can reveal signal features that remain hidden from other methods of analysis. Distributions measuring joint timefrequency content, such as the Wigner distribution and the spectrogram from Cohen's class [1, 2] ha...

We propose a robust method for estimating the time-varying spectrum of a non-stationary random process. Our approach extends Thomson's powerful multiple window spectrum estimation scheme to the time-frequency and timescale planes. The method refines previous extensions of Thomson's method through optimally concentrated window and wavelet functions and a statistical test for extracting chirping line components. 1. INTRODUCTION Many methods exist for estimating the power spectrum of stationary signals. However, these methods are insufficient for the non-stationary signals that occur in important applications such as radar, sonar, acoustics, biology, and geophysics. These applications demand time-frequency representations that indicate how the power spectrum changes over time. To date research in time-frequency analysis has focused on deterministic signals. Only recently has attention turned to non-stationary random processes [1--4]. Unlike the power spectrum for stationary random proc...