Signal representation and recovery under measurement constraints

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.