## Signal representation and recovery under measurement constraints

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##### Author

Özçelikkale Hünerli, Ayça

##### Advisor

Özaktaş, Haldun M.

##### Date

2012##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis

##### Metadata

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http://hdl.handle.net/11693/15806##### Abstract

We are concerned with a family of signal representation and recovery problems
under various measurement restrictions. We focus on finding performance
bounds for these problems where the aim is to reconstruct a signal from its direct
or indirect measurements. One of our main goals is to understand the effect
of different forms of finiteness in the sampling process, such as finite number of
samples or finite amplitude accuracy, on the recovery performance. In the first
part of the thesis, we use a measurement device model in which each device has a
cost that depends on the amplitude accuracy of the device: the cost of a measurement
device is primarily determined by the number of amplitude levels that the
device can reliably distinguish; devices with higher numbers of distinguishable
levels have higher costs. We also assume that there is a limited cost budget so
that it is not possible to make a high amplitude resolution measurement at every
point. We investigate the optimal allocation of cost budget to the measurement
devices so as to minimize estimation error. In contrast to common practice which
often treats sampling and quantization separately, we have explicitly focused on
the interplay between limited spatial resolution and limited amplitude accuracy.
We show that in certain cases, sampling at rates different than the Nyquist rate
is more efficient. We find the optimal sampling rates, and the resulting optimal
error-cost trade-off curves. In the second part of the thesis, we formulate a set of
measurement problems with the aim of reaching a better understanding of the relationship
between geometry of statistical dependence in measurement space and
total uncertainty of the signal. These problems are investigated in a mean-square
error setting under the assumption of Gaussian signals. An important aspect of
our formulation is our focus on the linear unitary transformation that relates the
canonical signal domain and the measurement domain. We consider measurement
set-ups in which a random or a fixed subset of the signal components in
the measurement space are erased. We investigate the error performance, both We are concerned with a family of signal representation and recovery problems
under various measurement restrictions. We focus on finding performance
bounds for these problems where the aim is to reconstruct a signal from its direct
or indirect measurements. One of our main goals is to understand the effect
of different forms of finiteness in the sampling process, such as finite number of
samples or finite amplitude accuracy, on the recovery performance. In the first
part of the thesis, we use a measurement device model in which each device has a
cost that depends on the amplitude accuracy of the device: the cost of a measurement
device is primarily determined by the number of amplitude levels that the
device can reliably distinguish; devices with higher numbers of distinguishable
levels have higher costs. We also assume that there is a limited cost budget so
that it is not possible to make a high amplitude resolution measurement at every
point. We investigate the optimal allocation of cost budget to the measurement
devices so as to minimize estimation error. In contrast to common practice which
often treats sampling and quantization separately, we have explicitly focused on
the interplay between limited spatial resolution and limited amplitude accuracy.
We show that in certain cases, sampling at rates different than the Nyquist rate
is more efficient. We find the optimal sampling rates, and the resulting optimal
error-cost trade-off curves. In the second part of the thesis, we formulate a set of
measurement problems with the aim of reaching a better understanding of the relationship
between geometry of statistical dependence in measurement space and
total uncertainty of the signal. These problems are investigated in a mean-square
error setting under the assumption of Gaussian signals. An important aspect of
our formulation is our focus on the linear unitary transformation that relates the
canonical signal domain and the measurement domain. We consider measurement
set-ups in which a random or a fixed subset of the signal components in
the measurement space are erased. We investigate the error performance, both We are concerned with a family of signal representation and recovery problems
under various measurement restrictions. We focus on finding performance
bounds for these problems where the aim is to reconstruct a signal from its direct
or indirect measurements. One of our main goals is to understand the effect
of different forms of finiteness in the sampling process, such as finite number of
samples or finite amplitude accuracy, on the recovery performance. In the first
part of the thesis, we use a measurement device model in which each device has a
cost that depends on the amplitude accuracy of the device: the cost of a measurement
device is primarily determined by the number of amplitude levels that the
device can reliably distinguish; devices with higher numbers of distinguishable
levels have higher costs. We also assume that there is a limited cost budget so
that it is not possible to make a high amplitude resolution measurement at every
point. We investigate the optimal allocation of cost budget to the measurement
devices so as to minimize estimation error. In contrast to common practice which
often treats sampling and quantization separately, we have explicitly focused on
the interplay between limited spatial resolution and limited amplitude accuracy.
We show that in certain cases, sampling at rates different than the Nyquist rate
is more efficient. We find the optimal sampling rates, and the resulting optimal
error-cost trade-off curves. In the second part of the thesis, we formulate a set of
measurement problems with the aim of reaching a better understanding of the relationship
between geometry of statistical dependence in measurement space and
total uncertainty of the signal. These problems are investigated in a mean-square
error setting under the assumption of Gaussian signals. An important aspect of
our formulation is our focus on the linear unitary transformation that relates the
canonical signal domain and the measurement domain. We consider measurement
set-ups in which a random or a fixed subset of the signal components in
the measurement space are erased. We investigate the error performance, both in the average, and also in terms of guarantees that hold with high probability,
as a function of system parameters. Our investigation also reveals a possible relationship
between the concept of coherence of random fields as defined in optics,
and the concept of coherence of bases as defined in compressive sensing, through
the fractional Fourier transform. We also consider an extension of our discussions
to stationary Gaussian sources. We find explicit expressions for the mean-square
error for equidistant sampling, and comment on the decay of error introduced by
using finite-length representations instead of infinite-length representations.