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Sampling and Distortion Tradeoffs for Band-limited Periodic Signals

Mohammadi, Elaheh | 2017

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 50546 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Marvasti, Farokh
  7. Abstract:
  8. One of the funadmental problems in signal processing is finding the best sampling strategy of a continuous time signal, and then finding the best strategy for compressing the obtained signal samples. The sampling and compression steps are designed with the aim of minimizing the distortion of the reconstructed signal. The problem of finding the best sampling strategy has been widely studied in the signal processing literature. In particular, the Nyquist–Shannon sampling theorem gives a sufficient condition for perfect reconstruction of a continuous-time signal of finite bandwidth. Most of the signal processing literature deals with deterministic signals, with relatively less attention paid to random signal models. On the other hand, if we are interested in compressing the sample values, Shannon’s rate distortion theory gives an answer to this problem when the signal samples are random. The goal of this thesis is to study the optimal sampling strategies(uniform or nonuniform) of random signals.In the first chapter we motivate the problem and review the existing literature. In the second chapter, optimal sampling strategies (uniform or nonuniform) and distortion tradeoffs for Gaussian bandlimited periodic signals with additive white Gaussian noise are studied.Our emphasis is on characterizing the optimal sampling locations as well as the optimal pre-sampling filter to minimize the reconstruction distortion. We first show that to achieve the optimal distortion, no pre-sampling filter is necessary for any arbitrary sampling rate. Then, we provide a complete characterization of optimal distortion for low and high sampling rates (with respect to the signal bandwidth). We also provide bounds on the reconstruction distortion for rates in the intermediate region. It is shown that nonuniform sampling outperforms uniform sampling for low sampling rates. In addition, the optimal nonuniform sampling set is robust with respect to missing sampling values. On the other hand, for the sampling rates above the Nyquist rate, the uniform sampling strategy is optimal.An extension of the results for random discrete periodic signals is discussed with simulation results indicating that the intuitions from the continuous domain carry over to the discrete domain. Sparse signals are also considered, where it is shown that uniform sampling is optimal above the Nyquist rate.In chapter four, we consider a continuous signal that cannot be observed directly. Instead,one has access to multiple corrupted versions of the signal. The available corrupted signals are correlated because they carry information about the common remote signal. The goal is to reconstruct the original signal from the data collected from its corrupted versions.Known as the indirect or remote reconstruction problem, it has been mainly studied in the literature from an information theoretic perspective. A variant of this problem for a class of Gaussian signals, known as the “Gaussian CEO problem”, has received particular attention; for example, it has been shown that the problem of recovering the remote signal is equivalent with the problem of recovering the set of corrupted signals (separation principle). The information theoretic formulation of the remote reconstruction problem assumes that the corrupted signals are uniformly sampled and the focus is on optimal compression of the samples. On the other hand, in this paper we revisit this problem from a sampling perspective. More specifically, assuming restrictions on the sampling rate from each corrupted signal, we look at the problem of finding the best sampling locations for each signal to minimize the total reconstruction distortion of the remote signal. In finding the sampling locations, one can take advantage of the correlation among the corrupted signals. The statistical model of the original signal and its corrupted versions adopted in this paper is similar to the one considered for the Gaussian CEO problem; i.e., we restrict to a class of Gaussian signals. Our main contribution is a fundamental lower bound on the reconstruction distortion for any arbitrary nonuniform sampling strategy. This lower bound is valid for any sampling rate. Further- more, it is tight and matches the optimal reconstruction distortion in low and high sampling rates. Moreover, it is shown that in the low sampling rate region, it is optimal to use a certain nonuni- form sampling scheme on all the signals. On the other hand, in the high sampling rate region, it is optimal to uniformly sample all the signals. We also consider the problem of finding the optimal sampling locations to recover the set of corrupted signals, rather than the remote signal.Unlike the information theoretic formulation of the problem in which these two problems were equivalent, we show that they are not equivalent in our setting
  9. Keywords:
  10. Sampling ; Nonuniform Sampling ; Compression ; Distortion Index ; Landa Criterion ; Nyquist Diagram ; Continuous Spectrum Signal

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