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Line Spectral Estimation Using a G-Filter: Atomic Norm Minimization with Multiple Output Vectors

Jiale Tang, Bin Zhu

TL;DR

This work tackles gridless line spectral estimation when prior band-selectivity is available through Georgiou's G-filter. It introduces MOV-ANM, defining a continuous atom set tied to the G-filter and formulating the estimation as an SDP via a generalized Carathéodory–Fejér decomposition. The authors show that allowing multiple output vectors enhances robust recovery and resolution, particularly at low SNR, and they demonstrate superior performance over standard ANM and single-output G-filter ANM. The approach broadens applicability to band-limited frequency estimation and offers a principled, convex optimization framework for joint estimation of the number of components and their frequencies.

Abstract

We propose an atomic norm minimization (ANM) estimator of frequencies in a noisy complex sinusoidal signal that integrates Georgiou's filter bank (G-filter) with multiple output vectors (MOV). Unlike our previous work on the G-filter version of ANM which is restricted to a single filtered output vector, the proposed method in this paper uses MOV to improve data utilization and robustness of the estimate. The ANM problem with MOV can be reformulated as a semidefinite program thanks to a Carathéodory--Fejér-type decomposition for output covariance matrices of the G-filter. Numerical simulations demonstrate that the proposed approach significantly outperforms the standard ANM and the G-filter version of ANM with a single output vector in recovering the correct number of frequency components when the frequencies fall within the band(s) selected by the G-filter, particularly in the low SNR regime.

Line Spectral Estimation Using a G-Filter: Atomic Norm Minimization with Multiple Output Vectors

TL;DR

This work tackles gridless line spectral estimation when prior band-selectivity is available through Georgiou's G-filter. It introduces MOV-ANM, defining a continuous atom set tied to the G-filter and formulating the estimation as an SDP via a generalized Carathéodory–Fejér decomposition. The authors show that allowing multiple output vectors enhances robust recovery and resolution, particularly at low SNR, and they demonstrate superior performance over standard ANM and single-output G-filter ANM. The approach broadens applicability to band-limited frequency estimation and offers a principled, convex optimization framework for joint estimation of the number of components and their frequencies.

Abstract

We propose an atomic norm minimization (ANM) estimator of frequencies in a noisy complex sinusoidal signal that integrates Georgiou's filter bank (G-filter) with multiple output vectors (MOV). Unlike our previous work on the G-filter version of ANM which is restricted to a single filtered output vector, the proposed method in this paper uses MOV to improve data utilization and robustness of the estimate. The ANM problem with MOV can be reformulated as a semidefinite program thanks to a Carathéodory--Fejér-type decomposition for output covariance matrices of the G-filter. Numerical simulations demonstrate that the proposed approach significantly outperforms the standard ANM and the G-filter version of ANM with a single output vector in recovering the correct number of frequency components when the frequencies fall within the band(s) selected by the G-filter, particularly in the low SNR regime.
Paper Structure (10 sections, 2 theorems, 23 equations, 6 figures)

This paper contains 10 sections, 2 theorems, 23 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Signal model and problem statement
  3. Atomic norm minimization approach
  4. Atomic norm for MOV of the G-filter
  5. C--F-type decomposition of output covariance matrices
  6. Reformulation via semidefinite programming
  7. Simulations
  8. Choice of the regularization parameter $\lambda$
  9. Simulation results
  10. Conclusion

Key Result

Theorem 1

Let $\Sigma$ satisfy state_cov_mat for some nonnegative measure $\mathrm{d}\mu_y$ and have rank $r<n$. Then it admits a unique decomposition where, $\boldsymbol{\rho} = \left[ \right]^\top \in\mathbb R^r$ has each component $\rho_k > 0$, $\mathop{\mathrm{diag}}\nolimits({\boldsymbol{\rho}})$ is a diagonal matrix with diagonal entries in $\boldsymbol{\rho}$, and $\boldsymbol{\theta} = \left[ \ri

Figures (6)

  • Figure 1: The squared gain $\|G^{(1)}(e^{i\theta})\|^2$ of a G-filter of size $n=20$ versus the frequency $\theta\in [0, 2\pi)$. The filter parameters $(A^{(1)}, \mathbf b^{(1)})$ are constructed as per Example \ref{['trans_func_filter_bank']} with a repeated pole $p_1=0.58e^{i2}$.
  • Figure 2: The squared gain $\|G^{(2)}(e^{i\theta})\|^2$ of a G-filter in Example \ref{['filter-gaintwo-example']} of size $n=20$ versus the frequency $\theta\in [0, 2\pi)$. The filter parameters $(A^{(2)}, \mathbf b^{(2)})$ are constructed with two poles $p_{k}=0.58e^{i\varphi_k}$, $k = 1,2$ where $\varphi_1 = 1.7$ and $\varphi_2 = 3.3$, respectively, each having multiplicity $n/2=10$.
  • Figure 3: Panels (a) and (b): Probabilities of successfully recovering the number $m=3$ of cisoids versus $\theta_0\in \{1.1, 1.3, \dots, 3.9\}$ with two G-filters $G^{(1)}$ and $G^{(2)}$, respectively. Notice that the markers represent computed values and lines have no meaning.
  • Figure 4: Panels (a)--(d): Absolute errors $\|\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}\|$ of frequency estimation using the G-filter $G^{(1)}$ for $\theta_0^{(1)}\in \{1.5, 1.6, \dots, 2.5\}$ in Subfig. \ref{['subfig:recov_G-filter1']} under different SNRs.
  • Figure 5: Panels (a)--(d): Absolute errors $\|\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}\|$ of frequency estimation using the G-filter $G^{(2)}$ for $\theta_0^{(2)}\in \{1.5, 1.7, 1.9, 3.1, 3.3, 3.5\}$ in Subfig. \ref{['subfig:recov_G-filter2']} under different SNRs.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Example 1: A bandpass filter bank
  • Example 2: G-filter with multiple passbands
  • Remark 1
  • Remark 2: Filtering finite-length signals
  • Theorem 1: C--F-type decomposition georgiou2000signal
  • Theorem 2