Browsing by Subject "signal recovery"

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Open Access
Signal representation and recovery under measurement constraints
(2012) Özçelikkale Hünerli, Ayça
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.
Open Access
Signal representation and recovery under partial information, redundancy, and generalized finite extent constraints
(2009) Öktem, Sevinç Figen
We study a number of fundamental issues and problems associated with linear canonical transforms (LCTs) and fractional Fourier transforms (FRTs). First, we study signal representation under generalized finite extent constraints. Then we turn our attention to signal recovery problems under partial and redundant information in multiple transform domains. In the signal representation part, we focus on sampling issues, the number of degrees of freedom, and the timefrequency support of the set of signals which are confined to finite intervals in two arbitrary linear canonical domains. We develop the notion of bicanonical width product, which is the generalization of the ordinary time-bandwidth product, to refer to the number of degrees of freedom of this set of signals. The bicanonical width product is shown to be the area of the time-frequency support of this set of signals, which is simply given by a parallelogram. Furthermore, these signals can be represented in these two LCT domains with the minimum number of samples given by the bicanonical width product. We prove that with these samples the discrete LCT provides a good approximation to the continuous LCT due to the underlying exact relation between them. In addition, the problem of finding the minimum number of samples to represent arbitrary signals is addressed based on the LCT sampling theorem. We show that this problem reduces to a simple geometrical problem, which aims to find the smallest parallelogram enclosing a given time-frequency support. By using this equivalence, we see that the bicanonical width product provides a better fit to the actual number of degrees of freedom of a signal as compared to the time-bandwidth product. We give theoretical bounds on the representational efficiency of this approach. In the process, we accomplish to relate LCT domains to the time-frequency plane. We show that each LCT domain is essentially a scaled FRT domain, and thus any LCT domain can be labeled by the associated fractional order, instead of its three parameters. We apply these concepts knowledge to the analysis of optical systems with arbitrary numbers of apertures. We propose a method to find the largest number of degrees of freedom that can pass through the system. Besides, we investigate the minimum number of samples to represent the wave at any plane in the system. In the signal recovery part of this thesis, we study a class of signal recovery problems where partial information in two or more fractional Fourier domains are available. We propose a novel linear algebraic approach to these problems and use the condition number as a measure of redundant information in given samples. By analyzing the effect of the number of known samples and their distributions on the condition number, we explore the redundancy and information relations between the given data under different partial information conditions.