3,376 research outputs found
Control Theoretic Approach To Sampling And Approximation Problems
Thesis (Ph.D.) University of Alaska Fairbanks, 2009We present applications of some methods of control theory to problems of signal processing and optimal quadrature problems. The following problems are considered: construction of sampling and interpolating sequences for multi-band signals; spectral estimation of signals modeled by a finite sum of exponentials modulated by polynomials; construction of optimal quadrature formulae for integrands determined by solutions of initial boundary value problems. A multi-band signal is a function whose Fourier transform is supported on a finite union of intervals. The approach used in Chapter I is based on connections between the sampling and interpolation problem and the problem of the controllability of a dynamical system. We prove that there exist infinitely many sampling and interpolating sequences for signals whose spectra are supported on a union of two disjoint intervals, and provide an algorithm for construction of such sequences. There exist numerous methods for solving the spectral estimation problem. In Chapter II we introduce a new approach to this problem based on the Boundary Control method, which uses the connection between inverse problems of mathematical physics and control theory for partial differential equations. Using samples of the signal at integer moments of time we construct a convolution operator regarded as an input-output map of a linear discrete dynamical system. This system can be identified, and the exponents and amplitudes of the signal can be found from the parameters of the system. We show that the coefficients of the signal can be recovered by solving a generalized eigenvalue problem as in the Matrix Pencil method. Our method allows to consider signals with polynomial amplitudes, and we obtain an exact formula for these amplitudes. In the third chapter we consider an optimal quadrature problem for solutions of initial boundary value problems. The problem of optimization of an error functional over the set of solutions and quadrature weights is a problem of optimal control of partial differential equations. We obtain estimates for the error in quadrature formulae and an optimality condition for quadrature weights
Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples
This paper presents a novel power spectral density estimation technique for
band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The
technique employs multi-coset sampling and incorporates the advantages of
compressed sensing (CS) when the power spectrum is sparse, but applies to
sparse and nonsparse power spectra alike. The estimates are consistent
piecewise constant approximations whose resolutions (width of the piecewise
constant segments) are controlled by the periodicity of the multi-coset
sampling. We show that compressive estimates exhibit better tradeoffs among the
estimator's resolution, system complexity, and average sampling rate compared
to their noncompressive counterparts. For suitable sampling patterns,
noncompressive estimates are obtained as least squares solutions. Because of
the non-negativity of power spectra, compressive estimates can be computed by
seeking non-negative least squares solutions (provided appropriate sampling
patterns exist) instead of using standard CS recovery algorithms. This
flexibility suggests a reduction in computational overhead for systems
estimating both sparse and nonsparse power spectra because one algorithm can be
used to compute both compressive and noncompressive estimates.Comment: 26 pages, single spaced, 9 figure
Connections between signal processing and complex analylis
Donem una descripció d'una de les línies de recerca del Grup de Teoria de Funcions de la UAB i la UB, que tracta els problemes de mostreig i interpolació en anàlisi del senyal i les seves connexions amb la teoria de funcions de variable complexa.We describe one of the research lines of the Grup de Teoria de Funcions de la UAB UB, which deals with sampling and interpolation problems in signal analysis and their connections with complex function theory
Sub-Nyquist Sampling: Bridging Theory and Practice
Sampling theory encompasses all aspects related to the conversion of
continuous-time signals to discrete streams of numbers. The famous
Shannon-Nyquist theorem has become a landmark in the development of digital
signal processing. In modern applications, an increasingly number of functions
is being pushed forward to sophisticated software algorithms, leaving only
those delicate finely-tuned tasks for the circuit level.
In this paper, we review sampling strategies which target reduction of the
ADC rate below Nyquist. Our survey covers classic works from the early 50's of
the previous century through recent publications from the past several years.
The prime focus is bridging theory and practice, that is to pinpoint the
potential of sub-Nyquist strategies to emerge from the math to the hardware. In
that spirit, we integrate contemporary theoretical viewpoints, which study
signal modeling in a union of subspaces, together with a taste of practical
aspects, namely how the avant-garde modalities boil down to concrete signal
processing systems. Our hope is that this presentation style will attract the
interest of both researchers and engineers in the hope of promoting the
sub-Nyquist premise into practical applications, and encouraging further
research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin
Non-uniform sampling and reconstruction of multi-band signals and its application in wideband spectrum sensing of cognitive radio
Sampling theories lie at the heart of signal processing devices and
communication systems. To accommodate high operating rates while retaining low
computational cost, efficient analog-to digital (ADC) converters must be
developed. Many of limitations encountered in current converters are due to a
traditional assumption that the sampling state needs to acquire the data at the
Nyquist rate, corresponding to twice the signal bandwidth. In this thesis a
method of sampling far below the Nyquist rate for sparse spectrum multiband
signals is investigated. The method is called periodic non-uniform sampling,
and it is useful in a variety of applications such as data converters, sensor
array imaging and image compression. Firstly, a model for the sampling system
in the frequency domain is prepared. It relates the Fourier transform of
observed compressed samples with the unknown spectrum of the signal. Next, the
reconstruction process based on the topic of compressed sensing is provided. We
show that the sampling parameters play an important role on the average sample
ratio and the quality of the reconstructed signal. The concept of condition
number and its effect on the reconstructed signal in the presence of noise is
introduced, and a feasible approach for choosing a sample pattern with a low
condition number is given. We distinguish between the cases of known spectrum
and unknown spectrum signals respectively. One of the model parameters is
determined by the signal band locations that in case of unknown spectrum
signals should be estimated from sampled data. Therefore, we applied both
subspace methods and non-linear least square methods for estimation of this
parameter. We also used the information theoretic criteria (Akaike and MDL) and
the exponential fitting test techniques for model order selection in this case
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