We propose the discrete linear chirp transform (DLCT) for decomposing a non‐stationary signal into intrinsic mode chirp functions. The decomposition of a signal into a finite number of intrinsic mode functions (IMFs) was introduced by the empirical mode decomposition (EMD). It exploits the local time‐scale signal characteristics of the signal and provides spectral estimates obtained via the Hilbert transform. Although efficient, the EMD does not provide an analytic representation of the IMFs and is susceptible to noise and to closeness or overlap of the frequency of the IMFs. Using linear chirps as IMFs, the DLCT, a joint frequency instantaneous frequency procedure, provides a parsimonious local orthogonal representation of non‐stationary signals. Moreover, the DLCT allows a parametric estimation of the instantaneous frequency of the signal that is robust to noise and to closeness or overlap in the instantaneous frequency of the modes. More importantly, the DLCT can be used to represent and process signals that are sparse in a joint time–frequency sense. The performance of the DLCT and the EMD are illustrated and compared when used to estimate the instantaneous frequency of individual signal components, to obtain signal decompositions at different frequency bands and to process frequency modulated signals with time‐varying amplitude.

Unlike synchronous processing, asynchronous processing is more efficient in biomedical and sensing networks applications as it is free from aliasing constraints and quantization error in the amplitude, it allows continuous–time processing and more importantly data is only acquired in significant parts of the signal. We consider signal decomposers based on the asynchronous sigma delta modulator (ASDM), a non‐linear feedback system that maps the signal amplitude into the zero‐crossings of a binary output signal. The input, the zero‐crossings and the ASDM parameters are related by an integral equation making the signal reconstruction difficult to implement. Modifying the model for the ASDM, we obtain a recursive equation that permits to obtain the non‐uniform samples from the zero‐time crossing values. Latticing the joint time‐frequency space into defined frequency bands, and time windows depending on the scale parameter different decompositions are possible. We present two cascade low‐ and high‐frequency decomposers, and a bank‐of‐filters parallel decomposer. This last decomposer using the modified ASDM behaves like a asynchronous analog to digital converter, and using an interpolator based on Prolate Spheroidal Wave functions allows reconstruction of the original signal. The asynchronous approaches proposed here are well suited for processing signals sparse in time, and for low‐power applications.

In the new generation wireless communication systems where high data rates are desired, orthogonal frequency division multiplexing (OFDM) has become the standard method because of its advantages over single carrier modulation schemes on multi-path, frequency selective fading channels. However, inter-carrier interference due to Doppler frequency shifts, and multi-path fading severely degrades the performance of OFDM systems. Estimation of channel parameters is required at the receiver. In this paper, we present a time-varying channel modeling and estimation method based on the discrete evolutionary transform that provides a time-frequency procedure to obtain a complete characterization of a multi-path, fading and frequency selective channel. Performance of the proposed method is tested on different levels of channel noise, and Doppler frequency shifts.

In this paper, we propose a low bit rate video coding procedure in the discrete cosine transform (DCT) domain that is based in an embedded zerotree algorithm and uses decimation and interpolation. Theory for decimation/interpolation in the DCT domain is presented, and its implementation with fast algorithms is shown. Motivated by the compositing of several video sources in multipoint video conferencing, we propose a new method for single streams that uses the proposed DCT-decimation in the encoder and the corresponding DCT-interpolation in the decoder. Decimated video frames are encoded using an embedded zerotree and an adaptive arithmetic coder to improve the quality of the decoded videos. Zerotree coding is a progressive coding method that quantizes amplitudes of DCT coefficients of video frames into bit streams with increasing precision. Embedded coding scheme depends on coding a symbol by an adaptive arithmetic encoder as soon as the symbol is obtained by the zerotree coding. Zerotree coder gives flexibility of controlling the bit rate precisely at given target bit rate. Compared to conventional coding our proposed algorithm achieves peak signal-to-noise ratio improvements at low bit rate video coding.

We present a low bit rate video coding procedure in the discrete cosine transform (DCT) domain that is based on fast decimation and interpolation and on embedded zerotree coding algorithms. The theory for decimation/interpolation in the DCT domain is given, and we show how to obtain fast algorithms. Motivated by the compositing of several video sources in multi-point video conferencing, we come up with a new method for single video streams that uses the proposed DCT-decimation in the encoder and the corresponding interpolation in the decoder. The decimated video frames are encoded using an embedded zerotree and an adaptive arithmetic coder to improve the quality of the decoded video and at an exact bit rate. The decimation and interpolation can be done with integer or rational factors, allowing flexibility in choosing the bit rate. Our codec performance is illustrated using different decimation factors, and it is shown that this codec is particularly efficient for low bit rates.

In wireless communications, the channel is typically modeled as a random linear time-varying system that spreads the transmitted signal in both time and frequency due to multipath and Doppler effects. In this paper, we show how time-frequency analysis can be used to model and estimate the channel of a multi-carrier spread spectrum (MC-SS) system with a complex quadratic spreading sequence. We show that in this case the effects of time delays and Doppler frequency shifts can be characterized effectively as time-shifts. Using the discrete evolutionary transform (DET) we are able to estimate these effective time shifts via a spreading function and use them to equalize the channel. To illustrate the performance of the proposed method we perform several simulations with different levels of channel noise, jamming and Doppler frequency shifts.

Transmitting high-quality neurophysiology video via the Internet is a challenging problem in telemedicine. We propose a novel video data processing and compression method based on the discrete wavelet transform (DWT). The DWT is utilized to decompose video frames into subframes, different bandwidth budgets are assigned to important and less important regions in each subframe, and several video encoders are employed in parallel to accelerate computation. When compared with the general-purpose video compression methods, our method allows scalability on real-time network bandwidth allocation, and offers higher video quality within the critical field of neurosurgery

Given the importance of direct sequence spread spectrum (DSSS) communications, the modeling of its transmission channel is of great interest. Due to multipath and Doppler effects in the transmission channel, the transmitted signal is spread in both time and frequency. Transmission channels that spread the message in time and frequency are modeled as random, time-varying systems. It is shown that the estimation of the parameters of such models is possible by means of the spreading function which is related to the time-varying frequency response of the system and the associated evolutionary kernel of the DSSS signal. Applying the time-frequency or frequency-frequency discrete evolutionary transforms, we show how to compute the spreading function from the received DSSS signal. The procedure is efficiently implemented with the discrete evolutionary transform. Once the number of paths, delays, Doppler frequencies and gains characterizing the channel are found, we use this information to obtain an estimate of the pseudo-noise and a decision parameter to determine the bit sent. Our procedure is illustrated with simulations of the process, and the corresponding bit-error rate.

Power spectral density of a signal is calculated from the second order statistics and provides valuable information for the characterization of stationary signals. This information is only sufficient for Gaussian and linear processes. Whereas, most real-life signals, such as biomedical, speech, and seismic signals may have non-Gaussian, non-linear and non-stationary properties. Higher order statistics (HOS) are useful for the analysis of such signals. Time-frequency (TF) analysis methods have been developed to analyze the time-varying properties of nonstationary signals. In this work, we combine the HOS and the TF approaches, and present a method for the calculation of a time-dependent bispectrum based on the positive distributed evolutionary spectrum.

We propose jammer excision techniques for direct sequence spread spectrum communications when the jammers cannot be parametrically characterized. The representation of the non-stationary signals is done using the time-frequency and the frequency-frequency evolutionary transformations. One of the methods, based on the frequency-frequency representation of the received signal, uses a deterministic masking approach while the other, based on nonstationary Wiener filtering, reduces interference in a mean-square fashion. Both of these approaches use the fact that the spreading sequence is known at the transmitter and the receiver, and that, as such, its evolutionary representation can be used to estimate the sent bit. The difference in performance between these two approaches depends on the support rather than on the type of jammer being excised. The frequency-frequency masking approach works well when the jammer is narrowly concentrated in parts of the frequency-frequency plane, while the Wiener masking approach works well in situations when the jammer is spread over all frequencies. Simulations illustrating the performance of the two methods, in different situations, are shown.

We propose a method - based on the discrete evolutionary transform (DET) - to estimate the instantaneous frequency of a signal embedded in noise or noise-like signals. The DET provides a representation for non-stationary signals and a time-frequency kernel that permit us to obtain the time-dependent spectrum of the signal. We show the instantaneous phase and the corresponding instantaneous frequency (IF) can also be computed from the evolutionary kernel. Estimation of instantaneous frequency is of general interest in time-frequency analysis, and of special interest in the excision of jammers in the direct sequence spread spectrum. Implementation of the IF estimation is done by masking and a recursive nonlinear correction procedure. The proposed estimation is valid for monocomponent as well as multicomponent signals in the noiseless and noisy situations. Its application to jammer excision in direct sequence spread spectrum communication is considered as an important application. The estimation procedure is illustrated with several examples.

We present a method to capture transient changes in heart-rate variability (HRV) signals. To deal with the nonstationarities in these signals we propose a time-dependent Karhunen-Loeve expansion. Using this expansion together with an entropy criterion we are able to partition the signal into quasi-stationary segments that can be further analyzed. Local eigenvalues and eigenvectors are then computed from a time-varying autocorrelation matrix in each segment. Application of our procedure resides in the dynamic assessment of the activity of the autonomic nervous system. An example is the capture of changes in HRV that might predict the initiation of cardiac arrhythmias. Previously, after performing and analyzing a stationary Karhunen-Loeve expansion of HRV signals, we showed that a significant increase in the dynamic range of the coefficients occurs shortly before the onset of life-threatening disturbances of cardiac rhythm. With the method proposed in this paper, we extend our previous results and we attempt a more precise determination of transients associated with changes in the autonomic nervous system.

In this paper, we present a hybrid rank-order/morphological filtering algorithm for removal of impulse noise. It consists of an impulse noise detector and a weighted rank-order morphological filter. For salt (pepper) noise suppression, a generalized directional opening (closing) filter is selectively used. The detection of impulse noise is easily done using the difference between the signal and its opening and closing. Simulations show that this new filter performs better than other nonlinear filters in preserving desired image features while effectively reducing impulse noise.

Jammer excision enhances the interference immunity in direct sequence spread spectrum (DSSS) communications. We propose an instantaneous frequency (IF) estimation method for a jammer composed of chirps, that uses evolutionary spectrum and Hough transform. The estimation requires no parametric representation of the IF and provides a correction by means of dechirping and low-pass filtering. Synthesis of the multicomponent jammer and its excision from the received signal is made possible with excellent results. Localized processing, linearization of the IF estimate, recursive correction, and no problems due to cross-terms in the time-frequency distribution or in the matching of parametric models make the proposed procedure efficient and practical. The performance of the IF estimation and its application in DSSS communications are illustrated.

In this paper we introduce a neural network implementation of fuzzy mathematical morphology operators and apply it to image denoising. Using a supervised training method and differentiable equivalent representations for the fuzzy morphological operators, we derive efficient adaptation algorithms to optimize the structuring elements. We can then design fuzzy morphological filters for processing multi-level or binary images. The convergence behavior of basic structuring elements for the opening filter and different signals, and its significance for other structuring elements of different shape is discussed. To illustrate the performance of the fuzzy opening filter we consider the removal of impulse noise in multi-level and binary images.

We present a new technique for estimating the generalized transfer function (GTF) of a time-varying filter from time-frequency representations (TFRs) of its output. We use the fact that many of these representations can be written as blurred versions of the GTF. The approach consists in minimizing the error between the TFR found from the data and that found by blurring the GTF. The problem as such has many solutions. We, therefore, additionally constrain it to minimize the distance between the GTF-based spectrum and the autoterms of the Wigner distribution, suppressing the cross terms using an appropriate signal dependent mask function. To illustrate the performance of the proposed procedure we apply it to the spectral representation of speech and to signal masking and demonstrate its superior performance over the existing methods.

We consider solutions to the non-stationary Wiener filtering problem using the evolutionary spectral theory. Two cases of interest result from the fact that the desired signal and the noise is uncorrelated. One constrains the support of the generating kernels of the signals and the other imposes orthogonality on their innovation processes. The latter condition is more general and our solution coincides with the one presented previously by Abdrabbo and Priestley (1969). For the first case, we develop a new solution that depends directly on the Wold-Cramer models of the desired and noisy processes. Implementation is achieved in both cases by estimating the kernels for the Wold-Cramer representations from the spectra using the evolutionary maximum entropy spectral estimation. An example illustrating the filtering is given.

A method for generating an informative prior when constructing a positive time-frequency distribution (TFD) by the method of the minimum cross-entropy (MCE) is developed. The prior is obtained from a combination of the Wigner distribution (WD) and the evolutionary periodogram, and results in a more informative MCE-TFD, as quantified via the mutual information of the distribution. The procedure allows any of the bilinear distributions to be used in the prior. Examples illustrate the performance of the new technique.

In this paper, we present a Gabor representation based on a nonrectangular tiling of the time-frequency plane and use it to improve the time and frequency resolutions of evolutionary spectra. In the traditional Gabor expansion, a signal is decomposed into a weighted combination of sinusoidally modulated windows resulting in a rectangular time-frequency plane tiling. Poor time and frequency localizations occur in the evolutionary spectrum when the corresponding signal is not modeled well by this fixed-window analysis. We are thus proposing the warped Gabor representation based on a linear chirp model for the signal. By means of a frequency transformation we are able to use the previous sinusoidal representation and choose the Gabor coefficients according to either a frequency masking or an energy concentration measure. Examples are given to illustrate our procedures.

Signals originating from a class of time-varying systems are modeled as dynamic fuzzy sets, i.e. fuzzy sets with membership functions that change in time. A signal trajectory in feature space is mapped into a dynamic fuzzy set which quantifies and characterizes the most significant aspects of the system's dynamics. A dynamic fuzzy set is visualized as a trajectory within a corresponding fuzzy information space. An example involving modeling of electroencephalographic signals during sleep is presented to illustrate the applicability of the method.

We apply time-frequency (TF) spectral analysis techniques, namely evolutionary spectral estimators, to postural sway data gathered during quiet standing and in response to external visual stimuli. These techniques provide insight into the time-varying properties of the human balance control systems during standing. We demonstrate by means of individual and group examples that the results of the TF methods can be used to characterize the behavior of the balance system for groups of patients and controls. Specifically we show that, for healthy control subjects, sway at a visual stimulus frequency toward and away from the subject shows an amplitude which decays in time. On the other hand, patients display a response whose amplitude at the stimulus frequency increases with time. Thus TF analysis yields insights into the time-varying nature of the postural control system.

We present a connection between the discrete Gabor expansion and the evolutionary spectral theory. Including a scale parameter in the Gabor expansion, we obtain a new representation for deterministic signals that is analogous to the Wold-Cramer decomposition for non-stationary processes. The energy distribution resulting from the expansion is easily calculated from the Gabor coefficients. By choosing Gaussian windows and appropriate scales, the expansion can represent narrow-band and wide-band signals, as well as their combination. As an application, we consider the masking of signals in the time-frequency space and provide an approximate implementation using the new Gabor expansion. Examples illustrating the time-frequency analysis and the masking are given.

Presents a method that is a generalization of hidden Markov modeling for the situations where elementary events cannot be clearly defined. A family of fuzzy sets, induced on a temporal universe, is used to model the dynamic trajectory of a physical system as a collection of hidden processes that coexist at the same time, but to different degrees. An algorithm based on unsupervised pattern recognition that estimates the prototypes and activities of the hidden processes is presented. The performance of the method is illustrated using experimental data obtained from electroencephalographic (EEG) signals recorded during sleep.

We propose an adaptive procedure to model non-stationary signals using autoregressive systems with time-varying parameters. A non-stationary signal that is representable by a time-varying autoregressive system has parameters which are expandable in terms of a set of basis functions. The parameters can be found by posing a minimum least-squares modeling problem and solving a large set of normal equations. The costly calculations involved in this problem make an adaptive solution quite desirable. Using the parameter expansions, we convert the modeling into a linear prediction problem and solve it adaptively for a given set of basis functions. We apply our procedure in the modeling of a segment of speech and in the estimation of the evolutionary spectrum of a non-stationary signal.>

We extend maximum entropy (ME) spectral analysis to non-stationary signals using the theory of the Wold-Cramer evolutionary spectrum. The evolutionary maximum entropy (EME) problem reduces to the fitting of a time-varying autoregressive model to the Fourier coefficients of the evolutionary spectrum. The model parameters are efficiently found by means of the Levinson algorithm. In the non-stationary case it is not the autocorrelation function that provides the appropriate data for the EME analysis, but rather the Fourier coefficients of the evolutionary spectrum. An estimator of these coefficients is proposed. By means of examples we show the EME estimator provides higher frequency resolution and better sidelobe behavior than existing estimators of the evolutionary spectrum.>

The use of a sum-of-cumulants as data instead of cumulants is proposed for reducing the computational complexity of two-dimensional cumulant-based identification procedures. The sum-of-cumulants are the Fourier coefficients of a frequency slice of the bispectrum. The dimensionality is thus reduced from four to two without loss of information about the system, and the identification is converted into a rational approximation problem. Non-separable asymmetric half-plane support two-dimensional systems represented by autoregressive moving average (ARMA) models are considered. Determining the region of support of a function of the frequency slice of the bispectrum, the authors obtain, a set of equations for the AR parameters. Cepstral procedures are then used to calculate the MA parameters. The procedure is illustrated by means of examples.>

An algorithm solution to the noisy image restoration problem under assumptions that the image is nonstationary and that the noise process is a superposition of white and impulsive noises is proposed. A composite model is used for the image in order to consider its nonstationarities, in the mean and the autocorrelation. Separating the gross information about the image from its textural information, the authors exploit the advantages of median, range, and Levinson filters in restoring the image. Median statistics are used to estimate the image's gross information and to filter the impulsive noise. Range statistics are used to segment the textural image into approximately locally stationary images to be filtered by Levenson filters. The proposed restoration algorithm adapts to the nonstationarity of the image, and, thus, it performs well. The algorithm is compared with others based on either median or linear filtering alone.>

A novel unbiased estimator for the time-dependent spectrum of a nonstationary signal is presented. The signal is modeled at a given frequency, as having a time-varying amplitude accurately represented by an orthonormal basis expansion. After a minimum-variance estimate of this time-varying amplitude is computed, and depending on assumptions made on the signal model, different time-varying estimators analogous to the ones used in the stationary case are developed. Presented is the time-varying periodogram estimator. By repeating the process over all frequencies, a power distribution is obtained as a function of time and frequency that is consistent with the evolutionary spectrum. The approach is guaranteed to produce nonnegative spectra without the cross-term behavior of the Wigner distribution, and it does not require windowing of data in the time domain, as the short time Fourier transform does. Examples illustrating the estimator are given.>

A parametric approach, based on sums of cumulants, is proposed for the identification of linear shift-invariant systems. The identification of the autoregressive parameters of the system is related to the rational approximation of a frequency slice of the output bispectrum. For a causal or a noncausal system it is shown that it is sufficient to solve Hankel-type matrix equations. The procedure can be efficiently implemented by means of recursive Pade approximants or modified least-squares rational approximations algorithms. The moving-average parameters of the system are calculated by solving some nonlinear equations. Examples illustrating the procedure and its advantages are given.>

Images are modeled as second-order nonstationary random fields, using a composite model that consists of a space-varying mean value function and a zero-mean nonstationary random field. The first component represents the underlying structure of an image, whereas the residual component provides the image's fine information. The authors develop space-varying estimators for the mean value and the variance functions and obtain conditions for such estimators to be statistically unbiased and consistent. The residual image component is modeled as an autoregressive system with space-varying coefficients. Identification of these coefficients is accomplished using a complete set of orthonormal basis functions to represent them. An efficient recursive procedure is proposed to solve the normal equations resulting from the minimum-mean square identification. Applications for the proposed modeling procedure can be found in the restoration and coding of images. Examples illustrating some aspects of the procedure and its application to image restoration are given.>

The two-dimensional (2-D) linear prediction "covariance" method is considered for the general case of a predictor with an asymmetric half-plane support lattice. The extension of a previous one-dimensional algorithm, based on conjugate directions, for solving recursively the symmetric set of normal equations for the 2-D covariance method is discussed. The effects of modeling caused by the choice of support lattices and of the ordering for the predictor coefficients are also studied. Examples are provided illustrating the various issues involved in modeling and the application of the "covariance" method in image coding.

This letter presents an efficient least squares procedure to obtain autoregressive integrated moving average (ARIMA) models for non-stationary, quasi-periodic signals. This new procedure is applied to the modeling of voiced speech and textural images.

The nonstationarity of images and the lack of a factorization theorem in two dimensions make the implementation of recursive imagerestoration procedures very difficult. A restoration procedure that partially avoids these issues by modeling the image differently is presented. The image is decomposed into additive components, and each one is modeled as a second-order random process with a separable exponential autocovariance and a constant mean. These components are obtained by means of a masking function relating the regional luminance in the image and the image sensitivity to noise. The present model facilitates the application of a two-dimensional Wiener filter in the form of a bank of filters. A restoration example illustrating the issues involved is given.

There are many industrial, biomedical, and military applications for which there is a need to compress images using techniques that are as closely information-preserving as possible. The fact that images cannot be accurately modeled as stationary processes and that the image statistics are usually not Gaussian reduces the efficiency of commonly used predictive and transform-based coding procedures. An efficient multiple-source coding procedure capable of preserving sharp features in the image with small computational load is presented. The procedure uses a composite model to deal with the nonstationarity in the space variation of the mean function. The image is then modeled as the addition of two components: an "approximate" image, which represents the underlying structure of the image, and a "difference" image, which corresponds to random variations superimposed on that structure. The approximate image is constructed using a binary image that describes the boundaries of its homogeneous regions, the local means of these regions, and an interpolation between the regions. The difference image is the result of subtracting the approximate image from the original. The coding of the binary and the difference images can be done very efficiently: the binary image can be coded by a facsimile technique, and the difference image can be coded using a predictive technique since it can be modeled accurately as a stationary random field. A lower bound for the compressibility of an image using the proposed procedure is given. To illustrate the procedure, as well as some issues involved, an example is shown.

In this paper we present a procedure to obtain two-dimensional (2-D) Planar Least-Squares Inverses (PLSIs) in the general form of asymmetric half-plane (AHP) filters, and which are bounded input output (BIBO) stable in most cases. These PLSIs are obtained by transforming the problem into a one-dimensional PLSI problem using the slice projection theorem, solving it by means of the Levinson's algorithm and transforming the solution back into an AHP filter. The procedure guarantees a good matching of the autocorrelation data and the BIBO stability of the filter in most cases. We discuss the application of this procedure to the stabilization of 2-D recursive filters and provide examples illustrating this application.