Signal Reconstruction From Nonuniform Samples Using Prolate Spheroidal Wave Functions: Theory and Application

Nonuniform sampling occurs in many applications due to imperfect sensors, mismatchedclocks or event-triggered phenomena. Indeed, natural images, biomedical responses andsensor network transmission have bursty structure so in order to obtain samples that correspondto the information content of the signal, one needs to collect more samples when thesignal changes fast and fewer samples otherwise which creates nonuniformly distibuted samples.On the other hand, with the advancements in the integrated circuit technology, smallscale and ultra low-power devices are available for several applications ranging from invasivebiomedical implants to environmental monitoring. However the advancements in the devicetechnologies also require data acquisition methods to be changed from the uniform (clockbased, synchronous) to nonuniform (clockless, asynchronous) processing. An important advancementis in the data reconstruction theorems from sub-Nyquist rate samples which wasrecently introduced as compressive sensing and that redenes the uncertainty principle. Inthis dissertation, we considered the problem of signal reconstruction from nonuniform samples.Our method is based on the Prolate Spheroidal Wave Functions (PSWF) which can beused in the reconstruction of time-limited and essentially band-limited signals from missingsamples, in event-driven sampling and in the case of asynchronous sigma delta modulation.We provide an implementable, general reconstruction framework for the issues relatedto reduction in the number of samples and estimation of nonuniform sample times. We alsoprovide a reconstruction method for level crossing sampling with regularization. Another way is to use projection onto convex sets (POCS) method. In this method we combinea time-frequency approach with the POCS iterative method and use PSWF for the reconstructionwhen there are missing samples. Additionally, we realize time decoding modulationfor an asynchronous sigma delta modulator which has potential applications in low-powerbiomedical implants

Time-frequency BSS of biosignals

Time–frequency (TF) representations are very important tools to understand and explain circumstances, where the frequency content of non-stationary signals varies in time. A variety of biosignals such as speech, electrocardiogram (ECG), electroencephalogram (EEG), and electromyogram (EMG) show some form of non-stationarity. Considering Priestley's evolutionary (time-dependent) spectral theory for analysis of non-stationary signals, the authors defined a TF representation called evolutionary Slepian transform (EST). The evolutionary spectral theory generalises the definition of spectra while avoiding some of the shortcomings of bilinear TF methods. The performance of the EST in the representation of biosignals for the blind source separation (BSS) problem to extract information from a mixture of sources is studied. For example, in the case of EEG recordings, as electrodes are placed along the scalp, what is actually observed from EEG data at each electrode is a mixture of all the active sources. Separation of these sources from a mixture of observations is crucial for the analysis of recordings. In this study, they show that the EST can be used efficiently in the TF-based BSS problem of biosignals

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

<p/> <p>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.</p

An evolutionary time-frequency approach to multi-carrier wireless communications

This paper provides a time-frequency approach to modeling and estimating the communication channel, and to symbol transmission in multi-carrier wireless systems. The method is based on the evolutionary spectral theory of signals and systems. Multi-carrier systems, such as orthogonal frequency division multiplexing (OFMD) and multi-carrier spread spectrum (MCSS), are very efficient in fast fading channels. However, the basic pulse used in the modulation causes dispersion in time-frequency, complicating inter-symbol and inter-channel interferences. Our time-frequency approach deals separately with the channel modeling and estimation, and with the symbol transmission. Using the properties of the response of the channel to a linear chirp signal it is shown that the typical linear time-variant channel model, needed to characterize multi-path and Doppler, is simplified into a linear time-invariant model of minimal order. Time and Doppler shifts are represented by equivalent time-shifts, and are estimated in a blind fashion from the evolutionary kernel of the received signal. Thus, the linear chirp signal is used as a pilot sequence to characterize the channel. A coherent receiver uses such information to detect the sent symbols. A multi-user OFDM system is obtained using a linear chirp as the modulating signal for basic pulses shifted in time, and by choosing the instantaneous frequency of the linear chirp to be of unity slope for an optimal time-frequency lattice. This optimal lattice increases the transmission rate, diminishes the inter-symbol and inter-channel interference, and provides a new way of looking at OFDM. This approach is extended to MCSS. To illustrate the concepts, simulations, with different signal to noise ratios, are performed. The results are encouraging and worth of further investigation

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 (PS W) 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

Asynchronous Representation and Processing of Nonstationary Signals

Nonstationarity relates to the variation over time of the statistics of a signal. Therefore, signals from practical applications that are realizations of nonstationary processes are difficult to represent and to process. In this article, we provide a comprehensive discussion of the asynchronous representation and processing of nonstationary signals using a time-frequency framework. Power consumption and type of processing imposed by the size of the devices in many applications motivate the use of asynchronous, rather than conventional synchronous, approaches. This leads to the consideration of nonuniform, signal-dependent level-crossing (LC) and asynchronous sigma delta modulator (ASDM)-based sampling. Reconstruction from a nonuniform sampled signal is made possible by connecting the sinc and the prolate spheroidal wave (PSW) functions-a more appropriate basis. Two decomposition procedures are considered. One is based on the ASDM that generalizes the Haar wavelet representation and is used for representing analog nonstationary signals. The second decomposer is for representing discrete nonstationary signals. It is based on a linear-chirp-based transform that provides local time-frequency parametric representations based on linear chirps as intrinsic mode functions (IMFs). Important applications of these procedures are the compression and processing of biomedical signals, as it will be illustrated in this article