2,967 research outputs found
Signal recovery from partial fractional fourier domain information and pulse shape design using iterative projections
Cataloged from PDF version of article.Signal design and recovery problems come up in a wide variety of applications in signal
processing. In this thesis, we first investigate the problem of pulse shape design
for use in communication settings with matched filtering where the rate of communication,
intersymbol interference, and bandwidth of the signal constitute conflicting
themes. In order to design pulse shapes that satisfy certain criteria such as bit rate,
spectral characteristics, and worst case degradation due to intersymbol interference,
we benefit from the wellknown Projections Onto Convex Sets. Secondly, we investigate
the problem of signal recovery from partial information in fractional Fourier
domains. Fractional Fourier transform is a mathematical generalization of the ordinary
Fourier transform, the latter being a special case of the first. Here, we assume
that low resolution or partial information in different fractional Fourier transform
domains is available in different intervals. These information intervals define convex
sets and can be combined within the Projections Onto Convex Sets framework. We
present generic scenarios and simulation examples in order to illustrate the use of
the method.Güven, H EmreM.S
Optimal Filtering with Linear Canonical Transformations
Cataloged from PDF version of article.Optimal filtering with linear canonical transformations allows smaller mean-square errors in restoring signals degraded
by linear time- or space-variant distortions and non-stationary noise. This reduction in error comes at no additional
computational cost. This is made possible by the additional flexibility that comes with the three free parameters of linear
canonical transformations, as opposed to the fractional Fourier transform which has only one free parameter, and the
ordinary Fourier transform which has none. Application of the method to severely degraded images is shown to be
significantly superior to filtering in fractional Fourier domains in certain cases
Recovering edges in ill-posed inverse problems: optimality of curvelet frames
We consider a model problem of recovering a function from noisy Radon data. The function to be recovered is assumed smooth apart from a discontinuity along a curve, that is, an edge. We use the continuum white-noise model, with noise level .
Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level only as as . A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to . However, as we show here, this rate is not optimal.
In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.
We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE as noise level . This rate of convergence holds uniformly over a class of functions which are except for discontinuities along curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example
Repeated filtering in consecutive fractional Fourier domains
Ankara : Department of Electrical and Electronics Engineering and the Institute of Engineering and Science of Bilkent University, 1997.Thesis (Ph. D.) -- Bilkent University, 1997.Includes bibliographical references leaves 96-105.In the first part of this thesis, relationships between the fractional Fourier
transformation and Fourier optical systems are analyzed to further elucidate
the importance of this transformation in optics. Then in the second part, the
concept of repeated filtering is considered. In this part, the repeated filtering
method is interpreted in two different ways. In the first interpretation the
linear transformation between input and output is constrained to be of the
form of repeated filtering in consecutive domains. The applications of this
constrained linear transformation to signal synthesis (beam shaping) and signal
restoration are discussed. In the second interpretation, general linear systems are
synthesized with repeated filtering in consecutive domains, and the synthesis of
some important linear systems in signal processing and the .synthesis of optical
interconnection architectures are considered for illustrative purposes. In all of the
examples, when our repeated filtering method is compared with single domain
filtering methods, significant improvements in performance are obtained with
only modest increases in optical or digital implementation costs. Similarly, when
the proposed method is compared with general linear systems, it is seen that
acceptable performance may be possible with significant computational savings
in implementation costs.Erden, M FatihPh.D
Generalized filtering configurations with applications in digital and optical signal and image processing
Ankara : Department of Electrical and Electonics Engineering and Institute of Engineering and Sciences, Bilkent Univ., 1999.Thesis (Ph.D.) -- Bilkent University, 1999.Includes bibliographical refences.In this thesis, we first give a brief summary of the fractional Fourier transform which
is the generalization of the ordinary Fourier transform, discuss its importance in
optical and digital signal processing and its relation to time-frequency representations.
We then introduce the concept of filtering circuits in fractional Fourier domains.
This concept unifies the multi-stage (repeated) and multi-channel (parallel) filtering
configurations which are in turn generalizations of single domain filtering in fractional
Fourier domains. We show that these filtering configurations allow a cost-accuracy tradeoff
by adjusting the number of stages or channels. We then consider the application
of these configurations to three important problems, namely system synthesis, signal
synthesis, and signal recovery, in optical and digital signal processing. In the system
and signal synthesis problems, we try to synthesize a desired system characterized by its
kernel, or a desired signal characterized by its second order statistics by using fractional
Fourier domain filtering circuits. In the signal recovery problem, we try to recover or
estimate a desired signal from its degraded version. In all of the examples we give,
significant improvements in performance are obtained with respect to single domain
filtering methods with only modest increases in optical or digital implementation costs.
Similarly, when the proposed method is compared with the direct implementation of
general linear systems, we see that significant computational savings are obtained with
acceptable decreases in performance.Kutay, Mehmet AlperPh.D
Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations
It is possible to obtain either exact realizations or useful approximations of linear systems or matrix-vector products arising in many different applications, by synthesizing them in the form of repeated or multi-channel filtering operations in fractional Fourier domains, resulting in much more efficient implementations with acceptable decreases in accuracy. By varying the number and configuration of filter blocks, which may take the form of arbitrary flow graphs, it is possible to trade off between accuracy and efficiency in the desired manner. The proposed scheme constitutes a systematic way of exploiting the information inherent in the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent
Solution and cost analysis of general multi-channel and multi-stage filtering circuits
The fractional Fourier domain multi-channel and multi-stage filtering configurations that have been recently proposed enable us to obtain either exact realizations or useful approximations of linear systems or matrix-vector products in many different applications. We discuss the solution and cost analysis for these configurations. It is shown that the problem can be reduced to a least squares problem which can be solved with fast iterative techniques
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