Parameter Estimation in Low-Rank Models from Small Sample Size and Undersampled Data: DOA and Spectrum Estimation

  • Author / Creator
    Shaghaghi, Mahdi
  • In estimation theory, a set of parameters are estimated from a finite number of measurements (samples). In general, the quality of estimation degrades as the number of samples is reduced. In this thesis, the problem of parameter estimation in low-rank models from a small number of samples is studied. Specifically, we consider two related problems that fit in this system model: direction-of-arrival (DOA) and spectrum estimation. We focus on subspace based DOA estimation methods which present a good compromise between performance and complexity. However, these methods are exposed to performance breakdown for a small number of samples. The reason is identified to be the intersubspace leakage where some portion of the true signal subspace resides in the estimated noise subspace. A two-step algorithm is proposed to reduce the amount of the subspace leakage. Theoretical derivations and simulation results are given to show the improvement achieved by the introduced method. Furthermore, the dynamics of the DOA estimation method in the breakdown region has been investigated, which led to identification of a problem named root-swap where a root associated with noise is mistakenly taken for a root associated with the signal. Then, an improved method is introduced to remedy this issue. Spectrum estimation from undersampled data (samples obtained at a rate lower than the Nyquist rate) is studied next. Specifically, the performance of the averaged correlogram for undersampled data is theoretically analyzed for the finite length sample size as well as asymptotically. This method partitions the spectrum into a number of segments and estimates the average power within each segment from samples obtained at a rate lower than the Nyquist rate. However, the frequency resolution of the estimator is restricted to the number of spectral segments, and the estimation made for each segment has also limited accuracy. Therefore, it is of significant importance to analyze the performance of this method especially in the case that only a finite number of samples is available. We derive the bias and variance of the averaged correlogram for undersampled data for finite-length signals, and we show the associated tradeoffs among the resolution, the accuracy, and the complexity of the method. Finally, spectrum estimation from compressive measurements is studied. The number of such measurements is much less than the number of Nyquist samples, and they are obtained by correlating the signal with a number of sensing waveforms. We specifically consider signals composed of linear combinations of sinusoids. Albeit these type of signals have a sparse model, their representation in the Fourier basis exhibits frequency leakage. This problem results in the poor performance of the conventional compressive sensing recovery algorithms that rely on the Fourier basis. We introduce an improved model-based reconstruction algorithm which has a performance close to the Cramer-Rao bound, which we also derive.

  • Subjects / Keywords
  • Graduation date
    Spring 2015
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.