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Nonstationary spectral analysis based on timefrequency operator symbols and underspread approximations
 IEEE TRANS. INF. THEORY
, 2006
"... We present a unified framework for timevarying or time–frequency (TF) spectra of nonstationary random processes in terms of TF operator symbols. We provide axiomatic definitions and TF operator symbol formulations for two broad classes of TF spectra, one of which is new. These classes contain all m ..."
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Cited by 14 (6 self)
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We present a unified framework for timevarying or time–frequency (TF) spectra of nonstationary random processes in terms of TF operator symbols. We provide axiomatic definitions and TF operator symbol formulations for two broad classes of TF spectra, one of which is new. These classes contain all major existing TF spectra such as the Wigner–Ville, evolutionary, instantaneous power, and physical spectrum. Our subsequent analysis focuses on the practically important case of nonstationary processes with negligible highlag TF correlations (socalled underspread processes). We demonstrate that for underspread processes all TF spectra yield effectively identical results and satisfy several desirable properties at least approximately. We also show that Gabor frames provide approximate Karhunen–Loève (KL) functions of underspread processes and TF spectra provide a corresponding approximate KL spectrum. Finally, we formulate simple approximate input–output relations for the TF spectra of underspread processes that are passed through underspread linear timevarying systems. All approximations are substantiated mathematically by upper bounds on the associated approximation errors. Our results establish a TF calculus for the secondorder analysis and timevarying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes.
Improved Optimization of TimeFrequency Based Signal Classifiers
, 2001
"... TimeFrequency Pepresentations (TFPs) are efficient tools for nonstationary signal classification. However, the choice of the TFP and of the distance measure employed is critical when no prior information other than a learning set of limited size is available. In this letter, we propose to jointly o ..."
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Cited by 8 (4 self)
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TimeFrequency Pepresentations (TFPs) are efficient tools for nonstationary signal classification. However, the choice of the TFP and of the distance measure employed is critical when no prior information other than a learning set of limited size is available. In this letter, we propose to jointly optimize the TFP and distance mea sure by minimizing the (estimated) probability of classifi cation error. The resulting optimized classification method is applied to multicomponent chirp signals and real speech records (speaker recognition). Extensive simulations show the substantial improvement of classification performance obtained with our optimization method.
A discrete model for the efficient analysis of timevarying narrowband communication channels
, 2006
"... We derive an efficient numerical algorithm for the analysis of certain classes of Hilbert–Schmidt operators that naturally occur in models of wireless radio and sonar communications channels. A common shorttime model of these channels writes the channel output as a weighted superposition of time a ..."
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Cited by 7 (3 self)
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We derive an efficient numerical algorithm for the analysis of certain classes of Hilbert–Schmidt operators that naturally occur in models of wireless radio and sonar communications channels. A common shorttime model of these channels writes the channel output as a weighted superposition of time and frequency shifted copies of the transmitted signal, where the weight function is usually called the spreading function of the channel operator. It is often believed that a good channel model must allow for spreading functions containing Dirac delta distributions. However, we show that many narrowband finite lifelength systems such as wireless radio communications can be well modelled by smooth and compactly supported spreading functions. Further, we exploit this fact to derive a fast algorithm for computing the matrix representation of such operators with respect to well timefrequency localized Gabor bases (such as pulseshaped OFDM bases). Hereby we use a
Classification of Chirp Signals using Hierarchical Bayesian Learning and MCMC Methods
, 2001
"... This paper addresses the problem of classifying chirp signals using hierarchical Bayesian learning together with Markov Chain Monte Carlo (MCMC) methods. Bayesian learning consists of estimating the distribution of the observed data conditional upon each class from a set of training samples. Unfortu ..."
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Cited by 5 (5 self)
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This paper addresses the problem of classifying chirp signals using hierarchical Bayesian learning together with Markov Chain Monte Carlo (MCMC) methods. Bayesian learning consists of estimating the distribution of the observed data conditional upon each class from a set of training samples. Unfortunately, this estimation requires to evaluate intractable multidimensional integrals. This paper studies an original implementation of hierarchical Bayesian learning which estimates the class conditional probability densities using MCMC methods. The performance of this implementation is first studied via an academic example for which the class conditional densities are known. The problem of classifying chirp signals is then addressed by using a similar hierarchical Bayesian learning implementation based on a Metropoliswithin Gibbs algorithm.
Twelfth International Congress on Sound and Vibration A CRITERION FOR DETECTING NONSTATIONARY EVENTS
"... This paper tackles the problem of detecting nonstationary events in a signal. An ergodic process is stationary if its law is invariant whatever the time translation is. So the control of the invariance of the law parameters can lead to a stationarity measure. We propose such a test stated from a tim ..."
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This paper tackles the problem of detecting nonstationary events in a signal. An ergodic process is stationary if its law is invariant whatever the time translation is. So the control of the invariance of the law parameters can lead to a stationarity measure. We propose such a test stated from a timefrequency representation so as to localize both in time and frequency nonstationary events. At each frequency we define two observation subsets. The first one is the subset of time frequency points, whose process is random and stationary eventually added with a deterministic process. The second one gathers all timefrequency points, whose process is non stationary. The problem is thus simplified into two competing sets or hypotheses between which we have to choice. A binary hypothesis test is stated from the NeymanPearson criterion. We construct a decision rule to have maximum probability of detection while not allowing the probability of false alarm to exceed a given value. Given that no a priori information is known, a recursive algorithm is performed in order to estimate the unknown parameters of the decision rule. The test converges when the subsets become steady. To be independent of the choice of the false alarm probability, the decision rule is applied for different values of this probability. To initialize the algorithm, we only assume that a time stationary part exists at each frequency. Under all these assumptions, all types of nonstationarities can be detected. The proposed detector is a postprocessing to a timefrequency estimator. In this paper, we use a spectrogram or a gliding correlogram, which sets the size of the nonstationary events to detect. The detector could be adapted to any other timefrequency estimator if its statistical law is known. Applications on real signals are carried out and show that the proposed method performs well. 1 ha l0