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Widely-Linear MMSE Estimation of Complex-Valued Graph Signals

Alon Amar, Tirza Routtenberg

TL;DR

The graph signal processing (GSP)-WLMMSE estimator is developed, which minimizes the MSE among estimators that are represented as a two-channel output of a graph filter, i.e. widely-linear GSP estimators.

Abstract

In this paper, we consider the problem of recovering random graph signals with complex values. For general Bayesian estimation of complex-valued vectors, it is known that the widely-linear minimum mean-squared-error (WLMMSE) estimator can achieve a lower mean-squared-error (MSE) than that of the linear minimum MSE (LMMSE) estimator. Inspired by the WLMMSE estimator, in this paper we develop the graph signal processing (GSP)-WLMMSE estimator, which minimizes the MSE among estimators that are represented as a two-channel output of a graph filter, i.e. widely-linear GSP estimators. We discuss the properties of the proposed GSP-WLMMSE estimator. In particular, we show that the MSE of the GSP-WLMMSE estimator is always equal to or lower than the MSE of the GSP-LMMSE estimator. The GSP-WLMMSE estimator is based on diagonal covariance matrices in the graph frequency domain, and thus has reduced complexity compared with the WLMMSE estimator. This property is especially important when using the sample-mean versions of these estimators that are based on a training dataset. We then state conditions under which the low-complexity GSP-WLMMSE estimator coincides with the WLMMSE estimator. In the simulations, we investigate two synthetic estimation problems (with linear and nonlinear models) and the problem of state estimation in power systems. For these problems, it is shown that the GSP-WLMMSE estimator outperforms the GSP-LMMSE estimator and achieves similar performance to that of the WLMMSE estimator.

Widely-Linear MMSE Estimation of Complex-Valued Graph Signals

TL;DR

The graph signal processing (GSP)-WLMMSE estimator is developed, which minimizes the MSE among estimators that are represented as a two-channel output of a graph filter, i.e. widely-linear GSP estimators.

Abstract

In this paper, we consider the problem of recovering random graph signals with complex values. For general Bayesian estimation of complex-valued vectors, it is known that the widely-linear minimum mean-squared-error (WLMMSE) estimator can achieve a lower mean-squared-error (MSE) than that of the linear minimum MSE (LMMSE) estimator. Inspired by the WLMMSE estimator, in this paper we develop the graph signal processing (GSP)-WLMMSE estimator, which minimizes the MSE among estimators that are represented as a two-channel output of a graph filter, i.e. widely-linear GSP estimators. We discuss the properties of the proposed GSP-WLMMSE estimator. In particular, we show that the MSE of the GSP-WLMMSE estimator is always equal to or lower than the MSE of the GSP-LMMSE estimator. The GSP-WLMMSE estimator is based on diagonal covariance matrices in the graph frequency domain, and thus has reduced complexity compared with the WLMMSE estimator. This property is especially important when using the sample-mean versions of these estimators that are based on a training dataset. We then state conditions under which the low-complexity GSP-WLMMSE estimator coincides with the WLMMSE estimator. In the simulations, we investigate two synthetic estimation problems (with linear and nonlinear models) and the problem of state estimation in power systems. For these problems, it is shown that the GSP-WLMMSE estimator outperforms the GSP-LMMSE estimator and achieves similar performance to that of the WLMMSE estimator.
Paper Structure (29 sections, 4 theorems, 100 equations, 8 figures)

This paper contains 29 sections, 4 theorems, 100 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. WLMMSE Estimation
  4. Graph Signal Processing (GSP)
  5. GSP-LMMSE Estimator
  6. WLMMSE for Graph Signals
  7. GSP-WLMMSE Estimator
  8. Discussion: Performance and Complexity
  9. Order relation between the estimators
  10. Performance
  11. Orthogonality principle
  12. Computational complexity
  13. Chebyshev Polynomial Approximation
  14. General design of the graph frequency response
  15. Alternative derivation by real and imaginary parts
  16. ...and 14 more sections

Key Result

Theorem 1

Let us assume that $\mathbf{C}_{\mathbf{y}\mathbf{y}}\neq {\bf{0}}$ and $\rho_n \neq 1$, $n=1,\ldots,N$. Then, the GSP-WLMMSE estimator, which is the estimator that achieves the minimum mse within the class of widely-linear GSP estimators with the form WLMMSE_general_form, for zero-mean complex grap where the diagonal coefficient matrix is

Figures (8)

  • Figure 1: The relationships between the classes of estimators: linear, widely-linear, GSP-linear, and GSP-widely-linear estimators. The numbers in the brackets ($4 N^2$, $4N$, $2 N^2$, and $2N$) are the degrees of freedom, i.e. the number of real parameters that we need to set in order to find the estimators for the recovery of an $N$-dimensional graph signal.
  • Figure 2: GSP linear estimator \ref{['GSPlinear']}
  • Figure 3: Widely-linear GSP estimator \ref{['WLMMSE_general_form']}
  • Figure 4: The considered network of Example 1 with $N=100$ sensors and an average of 350 edges.
  • Figure 5: The theoretical MSEs, $\varepsilon_{L}^2$, $\varepsilon_{WL}^2$, $\varepsilon_{GSP-L}^2$, and $\varepsilon_{GSP-WL}^2$, and the empirical MSEs of the sLMMSE, sWLMMSE, sGSP-LMMSE, and sGSP-WLMMSE estimators, for the linear system of Example 1 versus the size of the training data, $K$, for $\eta=0.1,0.3,0.7$ and $0.9$.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Claim 1
  • Theorem 1
  • Remark
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4