Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Performance of the stochastic MV-PURE estimator in highly noisy settings

Tomasz Piotrowski, Isao Yamada

Abstract

The stochastic minimum-variance pseudo-unbiased reduced-rank estimator (stochastic MV-PURE estimator) has been developed to provide linear estimation with robustness against high noise levels, imperfections in model knowledge, and ill-conditioned systems. In this paper, we investigate the theoretical performance of the stochastic MV-PURE estimator under varying levels of additive noise. We prove that the mean-square-error (MSE) of this estimator in the low signal-to-noise (SNR) region is much smaller than that obtained with its full-rank version, the minimum-variance distortionless estimator, and the gap becomes larger as the noise level increases. These results shed light on the excellent performance of the stochastic MV-PURE estimator in highly noisy settings obtained in simulations so far. Furthermore, we extend previous numerical simulations to show how the insight gained from the results of this paper can be used in practice.

Performance of the stochastic MV-PURE estimator in highly noisy settings

Abstract

The stochastic minimum-variance pseudo-unbiased reduced-rank estimator (stochastic MV-PURE estimator) has been developed to provide linear estimation with robustness against high noise levels, imperfections in model knowledge, and ill-conditioned systems. In this paper, we investigate the theoretical performance of the stochastic MV-PURE estimator under varying levels of additive noise. We prove that the mean-square-error (MSE) of this estimator in the low signal-to-noise (SNR) region is much smaller than that obtained with its full-rank version, the minimum-variance distortionless estimator, and the gap becomes larger as the noise level increases. These results shed light on the excellent performance of the stochastic MV-PURE estimator in highly noisy settings obtained in simulations so far. Furthermore, we extend previous numerical simulations to show how the insight gained from the results of this paper can be used in practice.

Paper Structure

This paper contains 8 sections, 4 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic linear model
  4. MMSE, stochastic BLUE, and stochastic MV-PURE estimators
  5. Variational characterization of eigenvalues of symmetric matrices
  6. Performance analysis under varying levels of signal-to-noise ratio
  7. Numerical example
  8. Concluding remarks

Key Result

Theorem 1

Consider the optimization problem (mvpulreS). The following holds:

Figures (4)

  • Figure 1: MSE [dB] vs. SNR [dB] for a sample channel realization in theoretical case.
  • Figure 2: Monotonic decrease of $\sigma_{15}$ with decreasing $SNR[dB]$ in theoretical case. The 0.5 threshold is crossed between $SNR[dB]=4$ and $SNR[dB]=6.$
  • Figure 3: MSE [dB] vs. SNR [dB] for a sample channel realization in practical case.
  • Figure 4: Monotonic decrease of $\widetilde{\sigma}_{15}$ with decreasing $SNR[dB]$ in practical case. The 0.5 threshold is crossed between $SNR[dB]=4$ and $SNR[dB]=6.$

Theorems & Definitions (5)

  • Theorem 1: $Piotrowski2009$
  • Remark 1
  • Lemma 1
  • Theorem 2
  • Theorem 3