Finite Time Support Research Articles

Asynchronous signal processing for brain-computer interfaces

Brain-computer interfaces (BCIs) provide a way to monitor and treat neurological diseases. An important application of BCIs is the monitoring and treatment of epilepsy, a neurological disorder characterized by recurrent unprovoked seizures, symptomatic of abnormal, excessive or synchronous neuronal activity in the brain. BCIs contain an array of sensors that gather and transmit data under the constrains of low-power and minimal data transmission. Asynchronous sigma delta modulators (ASDMs) are considered an alternative to synchronous analog to digital conversion. ASDMs are non-linear feedback systems that enable time-encoding of analog signals, from which the signal can be reconstructed. An efficient reconstruction of time-encoded signals can be achieved using a prolate spheroidal waveform (PSW) projection. PSWs have finite time support and maximum energy concentration within a given bandwidth. The ASDM time-encoding is related to the duty-cycle modulation and demodulation, which shows that sampling is non-uniform. For transmission of data from BCI, we propose a modified orthogonal frequency division multiplexing (OFDM) technique using chirp modulation. Our method generalizes the chirp modulation of binary streams with non-uniform symbol duration. Our theoretical results relate to recent continuous-time digital signal processing and compressive sampling theories.

Signal Reconstruction from Nonuniformly Spaced Samples Using Evolutionary Slepian Transform-Based POCS

We consider the reconstruction of signals from nonuniformly spaced samples using a projection onto convex sets (POCSs) implemented with the evolutionary time-frequency transform. Signals of practical interest have finite time support and are nearly band-limited, and as such can be better represented by Slepian functions than by sinc functions. The evolutionary spectral theory provides a time-frequency representation of nonstationary signals, and for deterministic signals the kernel of the evolutionary representation can be derived from a Slepian projection of the signal. The representation of low pass and band pass signals is thus efficiently done by means of the Slepian functions. Assuming the given nonuniformly spaced samples are from a signal satisfying the finite time support and the essential band-limitedness conditions with a known center frequency, imposing time and frequency limitations in the evolutionary transformation permit us to reconstruct the signal iteratively. Restricting the signal to a known finite time and frequency support, a closed convex set, the projection generated by the time-frequency transformation converges into a close approximation to the original signal. Simulation results illustrate the evolutionary Slepian-based transform in the representation and reconstruction of signals from irregularly-spaced and contiguous lost samples.

Efficient Wideband Parameter Estimation Using Arbitrary Enveloped LFM Signals via Hermite Decompositions

Signals with finite time support are frequently used in active sonar and radar processing. In these applications, for efficient computations, it is necessary to have a compact description of the signals' shape and time evolution. For this purpose, the Hermite series expansions are used in this paper to approximate an arbitrary enveloped linear frequency modulated signal. The Hermite basis functions are based on the product of Hermite polynomials and a Gaussian function. The analytical expressions for the Fourier transform of linear frequency modulated (LFM) Hermite functions are presented. It is proposed to use the Fourier transform of LFM Hermite functions for efficient computation of the wideband cross-ambiguity function (WCAF) for target parameter estimation in active sonar processing. Efficient 2-D search methods are also proposed to obtain parameters from the WCAF.

Minimum entropy time-frequency distributions

Re/spl acute/nyi entropy has been proposed as an effective measure of signal information content and complexity on the time-frequency plane. The previous work concerning Re/spl acute/nyi entropy in the time-frequency plane has focused on measuring the complexity of a given deterministic signal. In this paper, the properties of Re/spl acute/nyi entropy for random signals are examined. The upper and lower bounds on the expected value of Re/spl acute/nyi entropy are derived and ways of minimizing the entropy of time-frequency distributions by putting constraints on the time-frequency kernel are explored. It is proven that the quasi-Wigner kernel has the minimum entropy among all positive time-frequency kernels with finite time-support and correct marginals. A general class of minimum entropy kernels is presented. The performance of minimum entropy kernels in signal representation and component counting is also demonstrated.

Advances in processing of surface myoelectric signals: Part 2

The paper focuses on the analysis of myoelectric evoked potentials and their progressive scaling, as well as morphological changes using orthogonal basis functions with essentially finite time support; and the compression of the information content using principal component analysis. The application of the Hermite-Rodriguez and the associated Hermite functions is discussed as a means to provide compact information about the shape of the M-wave or of the power spectral density function of either voluntary or electrically elicited myoelectric signals; a means to estimate scaling factors; and a means to describe and classify nonstationarities. The principal component analysis shows the possibility of a compression ratio of at least 10:1 in the storage of M-wave sequences. The paper also describes three methods for the estimation of delay between similar signals, and therefore for estimation of conduction velocity. They are based on normalised integrals. Fourier transform matching and matching in the time domain. In particular conditions they provide different results for the same pair of signals. The concept of delay and the performance of these methods are reviewed and discussed. The paper is not exhaustive. It has the main objective of making the reader aware of the wealth of methods available for nonstationary myoelectric signal analysis and conduction velocity estimation, and of the need to use them with knowledge of their respective advantages, disadvantages, peculiarities and limitations.

Hermite expansions of compact support waveforms: applications to myoelectric signals.

Nonstationary signals with finite time support are frequently encountered in electrophysiology and other fields of biomedical research. It is often desirable to have a compact description of their shape and of their time evolution. For this purpose, Fourier analysis is not necessarily the best tool. The Hermite-Rodriguez and Associated Hermite basis functions are applied in this work. Both are based on the product of Hermite polynomials and Gaussian functions. Their general properties relevant to biomedical signal processing are reviewed. Preliminary applications are described concerning the analysis and description of: a) test signals such as a square pulse and a single cycle of a sinewave, b) electrically evoked myoelectric signals, and c) power spectra of either voluntary or evoked signals. It is shown that expansions with only five to ten terms provide an excellent description of the computer simulated and real signals. It is shown that these two families of Hermite functions are well suited for the analysis of nonstationary biological evoked potentials with compact time support. An application to the estimation of scaling factors of electrically evoked myoelectric signals is described. The Hermite functions show advantages with respect to the more traditional spectral analysis, especially in the case of signal truncation due to stimulation with interpulse intervals smaller than the duration of the evoked response. Finally, the Hermite approach is found to be suitable for classification of spectral shapes and compression of spectral information of either voluntary or evoked signals. The approach is very promising for neuromuscular diagnosis and assessment because of its capability for information compression and waveform classification.

The use of cone-shaped kernels for generalized time-frequency representations of nonstationary signals

Generalized time-frequency representations (GTFRs) which use cone-shaped kernels for nonstationary signal analysis are presented. The cone-shaped kernels are formulated for the GTFRs to produce good resolution simultaneously in time and frequency. Specifically, for a GTFR with a cone-shaped kernel, finite time support is maintained in the time dimension along with an enhanced spectrum in the frequency dimension, and the cross-terms are smoothed out. Experimental results on simulated data and real speech show the advantages of the GTFRs with the cone-shaped kernels through comparisons to the spectrogram and the pseudo-Wigner distribution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An optimal class of generalized time‐frequency representations for nonstationary signal analysis

In this paper, a class of generalized time‐frequency representations (GTFR) that have both good time and good frequency characteristics for nonstationary signal analysis is presented. The basis of the approach used is that a time‐frequency representation can, in some cases, be improved by allowing negative value of the spectrum. The representation (and its associated kernel) can then be optimized to enhance the peaks in frequency while still maintaining finite time support. The finite time support property has the significance of preserving onset times of signals and also providing clear representations of fast‐changing spectral peaks. Experiments were performed on simulated data and real speech for comparison of the GTFR with the spectrogram and the pseudo‐Wigner distribution. The results show distinct advantages of the GTFR. For example, onsets of transitions are clearer and, in the case of speech, close formant peaks are easier to distinguish. [Research supported by NSF and Boeing.]