Recursive and non-recursive filters for sequential smoothing and prediction with instantaneous phase and frequency estimation applications (extended version)

📄 arXiv: 2311.07089v4 📥 PDF

作者: Hugh Lachlan Kennedy

分类: eess.SP, eess.SY

发布日期: 2023-11-13 (更新: 2025-07-20)

备注: This draft manuscript was reduced in length by: 1) Removing Configurations 2 & 3 from the simulations (only Config 1 remains) 2) Removing most of the basic theory and the design of FIR filters (mostly IIR now) Also emphasised the novel aspects. Changed title. No changes made to this arXiv draft. Final (peer-reviewed and open-access) version in Circuits Syst Signal Process (2025)

DOI: 10.1007/s00034-025-03155-0

💡 一句话要点

提出递归与非递归滤波器以解决瞬时频率估计问题

🎯 匹配领域: 支柱八：物理动画 (Physics-based Animation)

关键词: 递归滤波器 非递归滤波器 瞬时频率估计 信号处理 低信噪比 多普勒测量 相位估计

📋 核心要点

  1. 现有方法在低信噪比条件下，标准FIR估计器在高角变化率信号中表现不佳，容易出现相位或频率解包错误。
  2. 论文提出了一种结合预测滤波器的设计，能够在信号变化迅速时减少解包错误，提高瞬时频率和相位的估计精度。
  3. 实验结果表明，IIR估计器在无解包错误的情况下，其误差方差达到了FIR的下限，且计算成本显著降低。

📝 摘要（中文）

本文描述了一种简单的设计程序，用于递归数字滤波器（IIR）和非递归数字滤波器（FIR）。所设计的固定滞后平滑滤波器旨在无偏跟踪指定度数的多项式信号，同时最小化具有特定功率谱密度的高频噪声增益。IIR变体通过确定最佳滞后（即通带群延迟），实现低复杂度的递归平滑器，具有良好的通带相位线性。该滤波器应用于瞬时频率估计问题，尤其是在多普勒频移测量中，能够在低信噪比下有效估计低频信号的瞬时相位和频率。

🔬 方法详解

问题定义：本文旨在解决在低信噪比条件下，瞬时频率和相位估计中的解包错误问题。现有的FIR估计器在高频变化信号中表现不佳，限制了其应用。

核心思路：通过设计递归和非递归滤波器，尤其是引入预测滤波器，来提高瞬时频率和相位的估计精度，减少解包错误的发生。

技术框架：整体方法包括滤波器设计、参数优化和性能评估三个主要模块。首先设计IIR和FIR滤波器，然后通过模拟实验评估其在瞬时频率估计中的表现。

关键创新：最重要的创新在于结合了预测滤波器的设计，使得在高角变化率信号中，能够有效减少解包错误，提升估计精度。与传统方法相比，提供了更低的计算复杂度和更好的性能。

关键设计：在设计过程中，确定了最佳滞后以实现低复杂度的递归平滑器，并优化了带宽和相位线性度等关键参数。

📊 实验亮点

实验结果显示，结合预测滤波器后，IIR估计器在高角变化率信号中减少了解包错误，误差方差达到了FIR的下限，且计算成本降低了显著。具体性能数据表明，在低信噪比条件下，估计精度有了明显提升。

🎯 应用场景

该研究在瞬时频率和相位估计领域具有广泛的应用潜力，尤其是在信号处理、通信和雷达系统中。通过提高低信噪比条件下的估计精度，能够显著提升相关技术的性能和可靠性，未来可能推动相关领域的进一步发展。

📄 摘要（原文）

A simple procedure for the design of recursive digital filters with an infinite impulse response (IIR) and non-recursive digital filters with a finite impulse response (FIR) is described. The fixed-lag smoothing filters are designed to track an approximately polynomial signal of specified degree without bias at steady state, while minimizing the gain of high-frequency (coloured) noise with a specified power spectral density. For the IIR variant, the procedure determines the optimal lag (i.e. the passband group delay) yielding a recursive low-complexity smoother of low order, with a specified bandwidth, and excellent passband phase linearity. The filters are applied to the problem of instantaneous frequency estimation, e.g. for Doppler-shift measurement, for a complex exponential with polynomial phase progression in additive white noise. For this classical problem, simulations show that the incorporation of a prediction filter (with a one-sample lead) reduces the incidence of (phase or frequency) angle unwrapping errors, particularly for signals with high rates of angle change, which are known to limit the performance of standard FIR estimators at low SNR. This improvement allows the instantaneous phase of low-frequency signals to be estimated, e.g. for time-delay measurement, and/or the instantaneous frequency of frequency-modulated signals, down to a lower SNR. In the absence of unwrapping errors, the error variance of the IIR estimators (with the optimal phase lag) reaches the FIR lower bound, at a significantly lower computational cost. Guidelines for configuring and tuning both FIR and IIR filters are provided.