William J. Williams

Improved time-frequency representation of multicomponent signals using exponential kernels

The authors introduce a time-frequency distribution of L. Cohens (1966) class and examines its properties. This distribution is called exponential distribution (ED) after its exponential kernel function. First, the authors interpret the ED from the spectral-density-estimation point of view. They then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain, and they analyze the ED for two specific types of multicomponent signals: sinusoidal signals and chirp signals. Next, they define the ED for discrete-time signals and the running windowed exponential distribution (RWED), which is computationally efficient. Finally, the authors present numerical examples of the RWED using the synthetically generated signals. It is found that the ED is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time-frequency distribution. >

Phase space topography and the Lyapunov exponent of electrocorticograms in partial seizures

SummaryElectrocorticograms (ECoGs) from 16 of 68 chronically implanted subdural electrodes, placed over the right temporal cortex in a patient with a right medial temporal focus, were analyzed using methods from nonlinear dynamics. A time series provides information about a large number of pertinent variables, which may be used to explore and characterize the systems dynamics. These variables and their evolution in time produce the phase portrait of the system. The phase spaces for each of 16 electrodes were constructed and from these the largest average Lyapunov exponents (Ls), measures of chaoticity of the system (the larger the L, the more chaotic the system is), were estimated over time for every electrode before, in and after the epileptic seizure for three seizures of the same patient. The start of the seizure corresponds to a simultaneous drop in L values obtained at the electrodes nearest the focus. L values for the rest of the electrodes follow. The mean values of L for all electrodes in the postictal state are larger than the ones in the preictal state, denoting a more chaotic state postictally. The lowest values of L occur during the seizure but they are still positive denoting the presence of a chaotic attractor. Based on the procedure for the estimation of L we were able to develop a methodology for detecting prominent spikes in the ECoG. These measures (L*) calculated over a period of time (10 minutes before to 10 minutes after the seizure outburst) revealed a remarkable coherence of the abrupt transient drops of L* for the electrodes that showed the inital ictal onset. The L* values for the electrodes away from the focus exhibited less abrupt transient drops. These results indicate that the largest average Lyapunov exponent L can be useful in seizure detection as well as a discriminatory factor for focus localization in multielectrode analysis.

Kernel design for reduced interference distributions

The authors present a class of time-frequency signal representations (TFRs) called the reduced interference distribution (RID). An overview of commonly used TFRs is given, and desirable distribution properties are introduced. Particular attention is paid to the interpretation of Cohens class of time-frequency distributions of TFRs in the ambiguity, temporal correlation, spectral correlation, and time-frequency domains. Based on the desirable kernel requirements, the RID is discussed and further defined. A systematic procedure to create RID kernels, (or, equivalently, compute RIDs) is proposed. Some aspects and properties of the RID are discussed. The authors estimate design considerations for RIDs and compare various selections of the primitive window. Some experimental results demonstrating the performance of the RID are presented. >

Alias-free generalized discrete-time time-frequency distributions

A definition of generalized discrete-time time-frequency distribution that utilizes all of the outer product terms from a data sequence, so that one can avoid aliasing, is introduced. The new approach provides (1) proper implementation of the discrete-time spectrogram, (2) correct evaluation of the instantaneous frequency of the underlying continuous-time signal, and (3) correct frequency marginal. The formulation provides a unified framework for implementing members of Cohens class, which was formulated in the continuous-time domain. Some requirements for the discrete-time kernel in the new approach are discussed in association with desirable distribution properties. Some experimental results are provided to illustrate the features of the proposed method. >

Uncertainty, information, and time-frequency distributions

The well-known uncertain principle is often invoked in signal processing. It is also often considered to have the same implications in signal analysis as does the uncertainty principle in quantum mechanics. The uncertainty principle is often incorrectly interpreted to mean that one cannot locate the time-frequency coordinates of a signal with arbitrarily good precision, since, in quantum mechanics, one cannot determine the position and momentum of a particle with arbitrarily good precision. Renyi information of the third order is used to provide an information measure on time-frequency distributions. The results suggest that even though this new measure tracks time-bandwidth results for two Gabor log-ons separated in time and/or frequency, the information measure is more general and provides a quantitative assessment of the number of resolvable components in a time frequency representation. As such, the information measure may be useful as a tool in the design and evaluation of time-frequency distributions.

Decomposing ERP time–frequency energy using PCA

OBJECTIVE Time-frequency transforms (TFTs) offer rich representations of event-related potential (ERP) activity, and thus add complexity. Data reduction techniques for TFTs have been slow to develop beyond time analysis of detail functions from wavelet transforms. Cohens class of TFTs based on the reduced interference distribution (RID) offer some benefits over wavelet TFTs, but do not offer the simplicity of detail functions from wavelet decomposition. The objective of the current approach is a data reduction method to extract succinct and meaningful events from both RID and wavelet TFTs. METHODS A general energy-based principal components analysis (PCA) approach to reducing TFTs is detailed. TFT surfaces are first restructured into vectors, recasting the data as a two-dimensional matrix amenable to PCA. PCA decomposition is performed on the two-dimensional matrix, and surfaces are then reconstructed. The PCA decomposition method is conducted with RID and Morlet wavelet TFTs, as well as with PCA for time and frequency domains separately. RESULTS Three simulated datasets were decomposed. These included Gabor logons and chirped signals. All simulated events were appropriately extracted from the TFTs using both wavelet and RID TFTs. Varying levels of noise were then added to the simulated data, as well as a simulated condition difference. The PCA-TFT method, particularly when used with RID TFTs, appropriately extracted the components and detected condition differences for signals where time or frequency domain analysis alone failed. Response-locked ERP data from a reaction time experiment was also decomposed. Meaningful components representing distinct neurophysiological activity were extracted from the ERP TFT data, including the error-related negativity (ERN). CONCLUSIONS Effective TFT data reduction was achieved. Activity that overlapped in time, frequency, and topography were effectively separated and extracted. Methodological issues involved in the application of PCA to TFTs are detailed, and directions for further development are discussed. SIGNIFICANCE The reported decomposition method represents a natural but significant extension of PCA into the TFT domain from the time and frequency domains alone. Evaluation of many aspects of this extension could now be conducted, using the PCA-TFT decomposition as a basis.

Reduced interference distributions: biological applications and interpretations

Time-frequency (TF) signal analysis has recently experienced a slow awakening followed by an accelerating development of interest. This paper discusses a specific concept applied to TF signal analysis, the reduced interference distribution (RID) approach. The RID approach is aimed at achieving high-resolution TF distributions within Cohens class with much reduced cross term or interference activity which often disappoints users of the well-known Wigner distribution (WD). The RID concept is briefly developed and then a number of examples from biomedical and biological settings are discussed. These discussions attempt to accomplish the dual task of illustrating the usefulness of the approach in investigating these problems and to teach one how to interpret the results obtained as well-the art of using the approach. RID results are contrasted with the WD and the spectrogram, another well-known tool. Some of the new and exciting TF approaches that deviate from the RID concept, but provide good results and interesting theoretical frameworks, are also brought in as appropriate.

New time-frequency distributions: theory and applications

A new conceptualization of time-frequency (t-f) energy distributions is discussed. Many new t-f distributions with desirable properties can now be designed with relative ease by approaching the problem in terms of the ambiguity plane representation of the kernel. Careful attention to the design principles yields kernels, which yield high-resolution t-f distributions with a considerable reduction of the sometimes troublesome cross terms observed when using other distributions such as the Wigner distribution. When these new t-f distributions are applied to some common signals, fascinating new details emerge. Examples are provided for speech, cortical evoked potentials, and electromyograms.<<ETX>>

Cross Time-frequency Representation Of Electrocorticograms In Temporal Lobe Epilepsy

Three time-frequency distributions are evaluated in terms of their efficacy in representing nonstationary electrocorticograms (ECOGs) in human temporal lobe epilepsy. The results of a new method, the exponential distribution, are compared with those of the spectrogram and the Wigner distribution. It is shown that the exponential distribution represents a considerable improvement over the spectrogram in terms of resolution and markedly reduces cross-terms present in the Wigner distribution. Exponential distribution representations of ECOGs from different stages of an epileptic record are developed as contour maps. These high-resolution representations offer a lucid display of temporal-spectral features of the rapidly varying signals that constitute ECOGs recorded in temporal lobe epilepsy. >

Renyi information and signal-dependent optimal kernel design

The Renyi uncertainty measure has been proposed to be a measurement of complexity of signals. We further suggest that it can be used to evaluate the performance of different time-frequency distributions (TFDs). We provide two schemes of normalization in calculating the Renyi uncertainty measure. For the first one, TFDs are normalized by their energy, while for the second one, normalized with their volume. The behavior of the Renyi uncertainty measure under several situations is studied. A signal-dependent algorithm is developed to achieve TFDs with a minimal uncertainty measure. For the first normalization scheme, the Wigner distribution is found to be optimal or near-to-optimal under certain constraints. If the second scheme is used, our program can generate minimum uncertainty product kernels which are very effective at suppressing cross terms and maintaining high resolution.