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Expectation Propagation based Line Spectral Estimation

Jiang Zhu, Xupeng Lei, Mihai Alin-Badiu, Fengzhong Qu

TL;DR

This work introduces BiG-LSE, an EP-based bilinear formulation for line spectral estimation that jointly estimates model order, noise variance, and amplitudes from nonlinear measurements by bilinearizing the true model via a Taylor expansion around fixed frequencies. EP is employed to manage nonlinear measurement components while posterior messages are simplified to reduce computational burden, and two initialization schemes (oversampled-grid and greedy) are provided. A von Mises extension enables frequency priors and improved sequential estimation. Through extensive simulations and a real mmWave radar experiment, BiG-LSE (especially the seri variant) achieves competitive NMSE, robust model-order detection, and favorable runtimes compared with state-of-the-art methods.

Abstract

The fundamental problem of line spectral estimation (LSE) using the expectation propagation (EP) method is studied. Previous approaches estimate the model order sequentially, limiting their practical utility in scenarios with large dimensions of measurements and signals. To overcome this limitation, a bilinear generalized LSE (BiG-LSE) method that concurrently estimates the model order is developed. The key concept involves iteratively approximating the original nonlinear model as a bilinear model through Taylor series expansion, with EP employed for inference. To mitigate computational complexity, the posterior log-pdfs are approximated to reduce the number of messages. BiG-LSE automatically determines the model order, noise variance, provides uncertainty levels for the estimates, and adeptly handles nonlinear measurements. Based on the BiG-LSE, a variant employing the von Mises distribution for the frequency is developed, which is suitable for sequential estimation. Numerical experiments and real data are used to demonstrate that BiG-LSE achieves estimation accuracy comparable to current methods.

Expectation Propagation based Line Spectral Estimation

TL;DR

This work introduces BiG-LSE, an EP-based bilinear formulation for line spectral estimation that jointly estimates model order, noise variance, and amplitudes from nonlinear measurements by bilinearizing the true model via a Taylor expansion around fixed frequencies. EP is employed to manage nonlinear measurement components while posterior messages are simplified to reduce computational burden, and two initialization schemes (oversampled-grid and greedy) are provided. A von Mises extension enables frequency priors and improved sequential estimation. Through extensive simulations and a real mmWave radar experiment, BiG-LSE (especially the seri variant) achieves competitive NMSE, robust model-order detection, and favorable runtimes compared with state-of-the-art methods.

Abstract

The fundamental problem of line spectral estimation (LSE) using the expectation propagation (EP) method is studied. Previous approaches estimate the model order sequentially, limiting their practical utility in scenarios with large dimensions of measurements and signals. To overcome this limitation, a bilinear generalized LSE (BiG-LSE) method that concurrently estimates the model order is developed. The key concept involves iteratively approximating the original nonlinear model as a bilinear model through Taylor series expansion, with EP employed for inference. To mitigate computational complexity, the posterior log-pdfs are approximated to reduce the number of messages. BiG-LSE automatically determines the model order, noise variance, provides uncertainty levels for the estimates, and adeptly handles nonlinear measurements. Based on the BiG-LSE, a variant employing the von Mises distribution for the frequency is developed, which is suitable for sequential estimation. Numerical experiments and real data are used to demonstrate that BiG-LSE achieves estimation accuracy comparable to current methods.

Paper Structure

This paper contains 21 sections, 91 equations, 14 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup and Probabilistic Model
  3. BiG-LSE Algorithm
  4. Message Update
  5. Update $m_{z\rightarrow \delta}({\mathbf z})$
  6. Updating of $m_{n\rightarrow l}(\epsilon_l)$
  7. Updating of ${m}_{l\rightarrow n}(\epsilon_l)$
  8. Updating of $m_{n\rightarrow l}(x_l)$
  9. Updating of ${m}_{l\rightarrow n}(x_l)$
  10. Update of $m_{\delta\rightarrow z_n}(z_n)$
  11. Simplifying the Message as BiG-LSE
  12. Approximate $a_{l\rightarrow n}$ and $x_{l\rightarrow n}$
  13. Approximate $\hat{p}_n$ and $\nu^p_n$
  14. Approximate $\hat{r}_l$, $\nu^r_l$, $\hat{q}_l$ and $\nu^q_l$
  15. Initialization
  16. ...and 6 more sections

Figures (14)

  • Figure 1: The whole factor graph. Here we introduce a delta factor node $\delta(\cdot)$, which simplifies the calculation as shown later .
  • Figure 2: The subfactor graph for updating $m_{z_n\rightarrow \delta}(z_n)$.
  • Figure 3: The factor graph for calculating $m_{n\rightarrow l}(\epsilon_l)$.
  • Figure 4: The factor graph for calculating ${m}_{l\rightarrow n}(\epsilon_l)$.
  • Figure 5: The factor graph for calculating $m_{n\rightarrow l}(x_l)$.
  • ...and 9 more figures