Seda Senay

Reconstruction of nonuniformly sampled time-limited signals using prolate spheroidal wave functions

Shannons sampling theory is based on the reconstruction of bandlimited signals which requires infinite number of uniform time samples. Indeed, one can only have finite number of samples for numerical implementation. In this paper, as a dual of the bandlimited reconstruction, a solution for time-limited signal reconstruction from nonuniform samples is proposed. The system model we present is based on the idea that time-limited signals which are also nearly bandlimited can be well approximated by a low-dimensional subspace. This can be done by using prolate spheroidal wave functions as the basis. The order of the projection on this basis is obtained by means of the time-frequency dimension of the signal, especially in the case of non-stationary signals. The reconstruction requires the estimation of the nonuniform sampling times by means of an annihilating filter. We obtain the reconstruction parameters by solving a linear system of equations and show that our finite-dimensional model is not ill-conditioned. The practical aspects of our method including the dimensionality reduction are demonstrated by processing synthetic as well as real signals.

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.

Regularized signal reconstruction for level-crossing sampling using Slepian functions

In this paper, we propose a method for efficient signal reconstruction from non-uniformly spaced samples collected using level-crossing sampling. Level-crossing (LC) sampling captures samples whenever the signal crosses predetermined quantization levels. Thus the LC sampling is a signal-dependent, non-uniform sampling method. Without restriction on the distribution of the sampling times, the signal reconstruction from non-uniform samples becomes ill-posed. Finite-support and nearly band-limited signals are well approximated in a low-dimensional subspace with prolate spheroidal wave functions (PSWF) also known as Slepian functions. These functions have finite support in time and maximum energy concentration within a given bandwidth and as such are very appropriate to obtain a projection of those signals. However, depending on the LC quantization levels, whenever the LC samples are highly non-uniformly spaced obtaining the projection coefficients requires a Tikhonov regularized Slepian reconstruction. The performance of the proposed method is illustrated using smooth, bursty and chirp signals. Our reconstruction results compare favorably with reconstruction from LC-sampled signals using compressive sampling techniques.

Asynchronous Representation and Processing of Nonstationary Signals : A Time-Frequency Framework

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.

Time encoding and reconstruction of multichannel data by brain implants using asynchronous sigma delta modulators

Recently, information technology and microelectronics have enabled implanting miniature and highly intelligent devices within the brain for in-vitro diagnostic and therapeutic functions. Power and physical size constraints of these devices necessitate novel signal processing methods. In this paper we investigate an effective data acquisition and reconstruction method for brain implants based on Asynchronous Sigma Delta Modulators (ASDMs). The ASDMs are analog non-linear feedback systems capable of time coding signals. The proposed reconstruction algorithm is based on the Prolate Spheroidal Wave Function (PSWF) expansion of the sinc functions and the order of expansion is given by the input signal being coded. Multiplexing and transmission of the different channels of data are accomplished by chirp orthogonal frequency division multiplexing. Computer simulations using multi channel electroen-cephalographic data are performed for wireless transmission by brain implants for monitoring abnormal brain activities of epilepsy patients.

Discrete Prolate Spheroidal Sequences for compressive sensing of EEG signals

Electroencephalography (EEG) is a major tool for clinical diagnosis of neurological diseases and brain research. EEGs are often collected over numerous channels and trials, providing large data sets that require efficient collection and accurate compression. Compressive sensing (CS) emphasizing signal sparseness enables the reconstruction of signals from a small set of measurements, at the expense of computationally complex reconstruction algorithms. In this paper we show that using Discrete Prolate Spheroidal Sequences, rather than sine functions, it is possible to derive a sampling and reconstruction method which is similar to CS. Assuming non-uniform sampling our procedure can be connected with compressive sensing without complex reconstruction methods.

Asynchronous Sigma Delta Modulator based subdural neural implants for epilepsy patients

Although recent advances in neuroscience, information technology and microelectronics have enabled implanting miniature and highly intelligent devices within the brain for in vitro diagnostic and therapeutic functions, novel signal processing methods are required for energy efficient data acquisition due to power constraints accompanying these miniature devices. We present a new method for signal acquisition in brain implants based on Asynchronous Sigma Delta Modulators (ASDM) which requires less power than conventional methods. We also present a computationally efficient signal reconstruction algorithm using the time codes from the output signal of ASDM. Our method and algorithm are validated using subdural EEG signals recorded from an epilepsy patient.

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, equivalent to non-uniform sampling. 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 original signal can be reconstructed from the ASDM time-encoded binary signal. For transmission, 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.

A new spreading code for multi-carrier spread spectrum systems

We present a new spreading code for wireless multi-carrier spread spectrum (MCSS) communication systems that is robust to noise and intentional jammers. In wireless communications , linear time-varying channel spreads the transmitted signal in both time and frequency due to multi-path and Doppler effects. Thus, the modeling and estimation of the communication channel is an important task at the received end. We show that time-frequency analysis can be used to model and estimate the channel of MCSS systems. Using the discrete evolutionary transform (DET) of the noisy channel output, we are able to estimate the spreading function of the channel. Simulations show that the proposed complex and random spreading code helps to increase the channel estimation performance.