Complete signal processing bases and the Jacobi group

The continuous windowed Fourier and wavelet transforms are created from the actions of the Heisenberg and affine groups, respectively. Both wavelet and windowed Fourier bases are known to be complete; that is, the only signal which is orthogonal to every element of each basis is the zero signal. The Jacobi group is a group which contains both the Heisenberg and affine groups, and it can also be used to produce bases for signal processing. This paper investigates completeness for bases of one and two real variables which are produced by the Jacobi group.     

Signal enhancement and the method of successive projections

The signal enhancement algorithm of Cadzow is developed in the context of best approximation theory and Combettes' method of successive projections, which is a generalization of the method of projection on convex sets. The relevant mathematical methods are surveyed. Applications of the signal enhancement algorithm to direction-of-arrival array signal processing for narrow-band and wide-band sources, data interpolation, signal detection, and x-ray fluorescence spectrum processing are presented.     

Linear Kalman–Bucy Filter with Autoregressive Signal and Noise

In the Kalman—Bucy filter problem, the observed process consists of the sum of a signal and a noise. The filtration begins at the same moment as the observation process and it is necessary to estimate the signal. As a rule, this problem is studied for the scalar and vector Markovian processes. In this paper, the scalar linear problem is considered for the system in which the signal and noise are not Markovian processes. The signal and noise are independent stationary autoregressive processes with orders of magnitude higher than 1. The recurrent equations for the filter process, its error, and its conditional cross correlations are derived. These recurrent equations use previously found estimates and some last observed data. The optimal definition of the initial data is proposed. The algebraic equations for the limit values of the filter error (the variance) and cross correlations are found. The roots of these equations make possible the conclusions concerning the criterion of the filter process convergence. Some examples in which the filter process converges and does not converge are given. The Monte Carlo method is used to control the theoretical formulas for the filter and its error.     

Approximation with Impulse Trains

The impulse train obtained by sampling an analog signal is studied directly. It is shown to represent the signal in a weak sense and to converge to the signal as the sampling interval converges to 0. The convergence is in the sense of Sobolev spaces and in the sense of another Hilbert space for which the shifted “delta functions” are orthogonal.     

Quilted Gabor frames – A new concept for adaptive time-frequency representation

Certain signal classes such as audio signals call for signal representations with the ability to adapt to the signalʼs properties. In this article we introduce the new concept of quilted frames, which aim at adaptivity in time-frequency representations. As opposed to Gabor or wavelet frames, this new class of frames allows for the adaptation of the signal analysis to the local requirements of signals under consideration. Quilted frames are constructed directly in the time-frequency domain in a signal-adaptive manner. Validity of the frame property guarantees the possibility to reconstruct the original signal. The frame property is shown for specific situations and the Bessel property is proved for the general setting. Strategies for reconstruction from coefficients obtained with quilted Gabor frames and numerical simulations are provided as well.     

Some methods for determination of the terahertz laser signal and the dynamics of its frequencies

The possibility of restoration of a terahertz signal transmitted through a substance and represented as a set of integral measurements is studied. Being related with substance identification, such problems are topical. In the study, two algorithms are proposed for spline-based signal restoration. Te algorithms enable one to reduce the presence of spurious frequencies. It is important that a spline with a number of nodes approximately equal to the number of measurements does not guarantee a high quality of signal restoration. The proposed algorithm, which involves the choice of the optimal number of measurements, is free from this drawback. Both a model signal and the real response of a medium are considered.     

Sharp adaptation for spherical inverse problems with applications to medical imaging

This paper examines the estimation of an indirect signal embedded in white noise for the spherical case. It is found that the sharp minimax bound is determined by the degree to which the indirect signal is embedded in the linear operator. Thus, when the linear operator has polynomial decay, recovery of the signal is polynomial, whereas if the linear operator has exponential decay, recovery of the signal is logarithmic. The constants are determined for these classes as well. Adaptive sharp estimation is also carried out. In the polynomial case a blockwise shrinkage estimator is needed while in the exponential case, a straight projection estimator will suffice. The framework of this paper include applications to medical imaging, in particular, to cone beam image reconstruction and to diffusion magnetic resonance imaging. Discussion of these applications are included.     

Propagation of electromagnetic waves in a nonlinear dielectric slab

The early phases of propagation of a large amplitude electromagnetic disturbance in a nonlinear dielectric slab embedded between two linear media are investigated by the method of characteristics. This disturbance is triggered by the arrival of an electromagnetic shock wave at one of the interfaces. Reflection and transmission of an arbitrary signal when it arrives at one of the slab boundaries is characterized in terms of nonlinear reflection and transmission coefficients for the interface. No restrictions are placed on the form of the constitutive laws of the material comprising the slab. By introducing, for the nonlinear dielectric, a class of model equations, an exact solution to the characteristic equations which describes the interaction of a centered wave with anarbitrary oncoming signal is obtained. Solutions for the electromagnetic fields are derived for the special case in which the incident disturbance interacts with the reflected signal from the slab interface. A particular case of the disturbance propagating across a nonmagnetic slab is also examined.     

Well-Posed State/Signal Systems in Continuous Time

We introduce a new class of linear systems, the L ^p -well-posed state/signal systems in continuous time, we establish the foundations of their theory and we develop some tools for their study. The principal feature of a state/signal system is that the external signals of the system are not a priori divided into inputs and outputs. We relate state/signal systems to the better-known class of well-posed input/state/output systems, showing that state/signal systems are more flexible than input/state/output systems but still have enough structure to provide a meaningful theory. We also give some examples which point to possibilities for further study.     

Generalized Signal Decoupling Problem with Stability for Discrete-Time Systems

This paper deals with the decoupling problems of unknown, measurable, and previewed signals. First, well-known solutions of unknown and measurable disturbance decoupling problems are recalled. Then, new necessary and sufficient constructive conditions for the previewed signal decoupling problem are proposed. The discrete-time case is considered. In this domain, previewing a signal by p steps means that the kth sample of the signal to be decoupled is known p steps in advance.The main result is that the stability condition for the mentioned decoupling problems does not change; i.e., the resolving subspace to be stabilized is the same independently of the type of signal to be decoupled, no matter whether it is completely unknown, measured, or previewed.The problem has been studied through self-bounded controlled invariants, thus minimizing the dimension of the resolving subspace which corresponds to the infimum of a lattice. The reduced dimension of the resolving controlled invariant subspace reduces the order of the controller units.     

Vibrational resonance in a time-delayed genetic toggle switch

Biological oscillators can respond in a surprising way when they are perturbed by two external periodic forcing signals of very different frequencies. The response of the system to a low-frequency signal can be enhanced or depressed when a high-frequency signal is acting. This is what is known as vibrational resonance (VR). Here we study this phenomenon in a simple time-delayed genetic toggle switch, which is a synthetic gene-regulatory network. We have found out how the low-frequency signal changes the range of the response, while the high-frequency signal influences the amplitude at which the resonance occurs. The delay of the toggle switch has also a strong effect on the resonance since it can also induce autonomous oscillations.     

Chaos time-domain reflectometry for fault location on live wires

We propose a chaos time-domain reflectometry (CTDR) for locating faults on live wires. This method uses a chaotic output of an improved Colpitts oscillator as probe signal, and detects wire faults by correlating a duplicate with the echo of the probe signal. Benefiting from the anti-jamming of the correlation function of the wideband chaos, fault location on live wires can be achieved. We experimentally demonstrate the detection for live wires in a digital communication system, in which a type of digital signal named high density bipolar of order 3 (HDB3) is transmitted. The effects of the chaotic probe signal on the bit error rate (BER) of the transmitted HDB3 at different rates are analyzed. Meanwhile, the influences of the backward HDB3 reflected by wiring faults on the signal-noise-ratio (SNR) of CTDR measurement are examined experimentally. The results show that fault detection on live wires is achieved when the power of the chaotic probe signal is about from -24.8 dB to -13.5 dB lower than that of the transmitted digital signal. In this case, the BER is kept less than 3E-10, and the SNR of CTDR is higher than 3 dB. Besides, the auto-correlation properties of the improved Colpitts oscillator at different states are investigated experimentally to explore the suitable chaotic states for the CTDR.     

Adjoint translation,adjoint observable and uncertainty principles

The motivation to this paper stems from signal/image processing where it is desired to measure various attributes or physical quantities such as position, scale, direction and frequency of a signal or an image. These physical quantities are measured via a signal transform, for example, the short time Fourier transform measures the content of a signal at different times and frequencies. There are well known obstructions for completely accurate measurements formulated as “uncertainty principles”. It has been shown recently that “conventional” localization notions, based on variances associated with Lie-group generators and their corresponding uncertainty inequality might be misleading, if they are applied to transformation groups which differ from the Heisenberg group, the latter being prevailing in signal analysis and quantum mechanics. In this paper we describe a generic signal transform as a procedure of measuring the content of a signal at different values of a set of given physical quantities. This viewpoint sheds a light on the relationship between signal transforms and uncertainty principles. In particular we introduce the concepts of “adjoint translations” and “adjoint observables”, respectively. We show that the fundamental issue of interest is the measurement of physical quantities via the appropriate localization operators termed “adjoint observables”. It is shown how one can define, for each localization operator, a family of related “adjoint translation” operators that translate the spectrum of that localization operator. The adjoint translations in the examples of this paper correspond to well-known transformations in signal processing such as the short time Fourier transform (STFT), the continuous wavelet transform (CWT) and the shearlet transform. We show how the means and variances of states transform appropriately under the translation action and compute associated minimizers and equalizers for the uncertainty criterion. Finally, the concept of adjoint observables is used to estimate concentration properties of ambiguity functions, the latter being an alternative localization concept frequently used in signal analysis.     

Forecasting continuous-time processes with applications to signal extraction

The paper derives forecasting and signal extraction estimates for continuous time processes. We present explicit formulas for filters and filter kernels that yield minimum mean square error estimates of future values of the process or an unobserved component, based on a continuum of values in the semi-infinite past. The class of processes considered are cumulations of moving average processes, which includes the CARIMA class. Explicit examples are calculated, and some discussion of applications to signal extraction is provided. We also provide an explicit algorithm for spectral factorization of continuous-time moving averages.     

Deterministic constructions of compressed sensing matrices

Compressed sensing is a new area of signal processing. Its goal is to minimize the number of samples that need to be taken from a signal for faithful reconstruction. The performance of compressed sensing on signal classes is directly related to Gelfand widths. Similar to the deeper constructions of optimal subspaces in Gelfand widths, most sampling algorithms are based on randomization. However, for possible circuit implementation, it is important to understand what can be done with purely deterministic sampling. In this note, we show how to construct sampling matrices using finite fields. One such construction gives cyclic matrices which are interesting for circuit implementation. While the guaranteed performance of these deterministic constructions is not comparable to the random constructions, these matrices have the best known performance for purely deterministic constructions.     

Highly sparse representations from dictionaries are unique and independent of the sparseness measure

The purpose of this paper is to study sparse representations of signals from a general dictionary in a Banach space. For so-called localized frames in Hilbert spaces, the canonical frame coefficients are shown to provide a near sparsest expansion for several sparseness measures. However, for frames which are not localized, this no longer holds true and sparse representations may depend strongly on the choice of the sparseness measure. A large class of admissible sparseness measures is introduced, and we give sufficient conditions for having a unique sparse representation of a signal from the dictionary w.r.t. such a sparseness measure. Moreover, we give sufficient conditions on a signal such that the simple solution of a linear programming problem simultaneously solves all the nonconvex (and generally hard combinatorial) problems of sparsest representation of the signal w.r.t. arbitrary admissible sparseness measures.     

Signal estimation using stochastic velocity models and irregular arrays

Consider the situation where a plane wave signal is received by a spatial arrangement of recorders. Information derived from observations on such a process can be used to determine the speed and direction of the signal together with properties of the medium through which the signal is being propagated. Certain models for the case where the signal velocity can be regarded as stochastic and where the array is irregular are investigated and estimation procedures proposed. A major practical property of these models is that, unlike their deterministic counterparts, coherence decays to zero as distance between recorders increases.     

Discrete-Time Linear-Quadratic Dynamic Games

Finite-dimensional, time invariant, linear quadratic dynamic games are perhaps the best understood and researched class of dynamic games. This is particularly true for continuous-time linear quadratic differential games. In this paper, the application of the theory of dynamic games to signal processing is considered. We are interested in digital signal processing and therefore we confine our attention to discrete-time linear-quadratic dynamic games (LQDG). In discrete-time the cost function contains product terms between the decision variables which complicates the analysis compared to its continuous-time analogue. With a view to facilitate the application of the theory of dynamic games to digital signal processing, and in particular, disturbance rejection, the complete solution of the discrete-time LQDG is worked out and explicit results are obtained. Thus,discrete-time LQDGs have the distinct advantage of being amenable to analysis, closed-form solutions are possible, and one is in tune with modern digital signal processing techniques. In this paper, minimal necessary and sufficient conditions for the existence of a solution to the discrete-time LQDG are provided and its explicit, closed-form, solution is worked out. This opens the way to designing novel digital signal processing algorithms for disturbance rejection. Information plays a critical role in game theory and in particular in dynamic games. Using our explicit solution of the deterministic LQDG, a hierarchy of three zero-sum stochastic LQDGs characterized by a sequence of information patterns which increase in complexity is analyzed.     

“Chaos” in superregenerative receivers

The superregenerative principle has been known since the early 1920s. The circuit is extremely simple and extremely sensitive. Today, superheterodyne receivers generally supplant superregenerative receivers in most applications because there are several undesirable characteristics: poor selectivity, reradiation, etc. Superregenerative receivers undergo a revival in recent papers for wireless systems, where low cost and very low power consumption are relevant: house/building meters (such as water, energy, gas counter), personal computer environment (keyboard, mouse), etc. Another drawback is the noise level which is higher than that of a well-designed superheterodyne receiver; without an antenna input signal, the output of the receiver hears in an earphone as a waterfall noise; this sound principally is the inherent input noise amplified and detected by the circuit; however, when the input noise is negligible with respect of an antenna input signal, we are faced to an other source of “noise” self-generated by the superregenerative working. The main objective of this paper concerns this self-generated noise coming from an exponential growing followed by a re-injection process for which the final state is a function of the phase of the input signal.     

Some issues in improving quality of noisy periodic signals and estimating their parameters and characteristics numerically by using a cluster approach: Problem statement

In this paper, a technique is proposed to process and analyze noisy periodic signals recorded at discrete moments of time. The technique includes two stages: (a) signal quality improvement (processing) with the use of weighted order statistics filters, and (b) cluster analysis of processing results. Basic definitions and specific features of this type of filtration are given, as well as formulations and definitions of cluster analysis, which enable problems to be stated for periodic signal analysis. The efficiency of cluster analysis with weighted order statistics filters is proved based on results of numerical modeling of a noisy frequency-modulated signal.