AI 中文总结
本文提出基于拉盖尔-傅里叶展开的L-ETFE方法，解决了标准ETFE在激励带限或谱零下的数值病态问题，通过仿真实验实现了脉冲响应的准确恢复。
AI 中文摘要
经验传递函数估计（ETFE）是声学、音频工程、地震学、断层成像等工程领域中，用于线性时不变（LTI）系统辨识的广泛使用方法。然而，当激励信号为带限或在某些频率处消失时，ETFE会出现数值限制，这是实际应用中激励的常见物理约束。在这种情况下，频域中的除法会变得严重病态，微小的测量扰动或数值误差都会劣化解。本文提出了L-ETFE，这是一种基于拉盖尔-傅里叶展开的ETFE泛化方法，用于解决这些限制。经过适当变换后，即使原始ETFE系统病态，该方法也能产生条件良好的循环问题。通过经典ETFE求解该问题可得到系统传递函数的离散拉盖尔-傅里叶系数，随后通过后续变换流程可恢复待辨识系统的期望脉冲响应（IR）。我们推导了执行这些变换的新颖高效算法，并分析了变换后问题的条件，明确表征了其对输入及拉盖尔-傅里叶展开所用参数的依赖性。我们在两个不同复杂度的仿真离散时间LTI系统上评估了该方法，实验表明，在标准ETFE失效的谱零和带限激励情况下，该方法能实现准确的IR恢复。
英文摘要
The empirical transfer function estimate (ETFE) is a widely used method for system identification of linear time-invariant (LTI) systems in engineering disciplines such as acoustics, audio engineering, seismography, and tomography. However, ETFE suffers from numerical limitations when the excitation signal is band-limited or vanishes at certain frequencies, which is a common physical constraint of the excitation in practice. In such cases, division in the frequency domain becomes heavily ill-conditioned, and small measurement disturbances or numerical inaccuracies can degrade the solution. This paper presents L-ETFE, a generalization of ETFE based on Laguerre-Fourier expansions that addresses these limitations. After a suitable transformation, the method can yield a well-conditioned circulant problem even when the original ETFE system is ill-conditioned. Solving this problem via classical ETFE yields the discrete Laguerre-Fourier coefficients of the system's transfer function. The desired impulse response (IR) of the system-to-be-identified can then be recovered by a subsequent transformation pipeline. We derive novel and efficient algorithms for performing these transformations and analyse the conditioning of the transformed problem, explicitly characterizing its dependence on the input and a parameter used in the Laguerre-Fourier expansion. We evaluate the method on two simulated discrete-time LTI systems of varying complexity. The experiments demonstrate accurate IR recovery for spectral-zero and band-limited excitation, where standard ETFE fails.
