For a structural dynamics engineer assessing a system’s vibration characteristics, the frequency response function (FRF) is indispensable, irrespective of whether the setup being tested is experimental, numerical or analytical. This paper outlines the different FRF estimation techniques that have been developed over the years. Algorithms that compute an ordinary least squares (OLS) estimate of the FRF, assuming uncorrelated measurement noise to be present either on the output (${\mathbf{H}}_1$) or the input (${\mathbf{H}}_2$) signals, have been compared with those that employ total least squares (TLS) equations by making use of an augmented input–output auto and cross power (${\mathbf{G}}_{FFX}$/${\mathbf{G}}_{XFF}$) matrix at every frequency such as the ${\mathbf{H}}_v$ (using eigenvalue decomposition) and ${\mathbf{H}}_{SVD}$ (using singular value decomposition) algorithms. Another FRF estimation method based upon Cholesky decomposition (${\mathbf{H}}_{CD}$) is also discussed. Further discussion has been included of TLS algorithms computing the FRFs for one output at a time, as historically presented in the development of the ${\mathbf{H}}_v$ algorithm, with a case that evaluates the FRF matrix computed for all the outputs simultaneously. The development of the corresponding coherence functions has been presented, highlighting the method dependent paradigms that led to concepts such as virtual coherence and partial coherence while underscoring their equivalence with ordinary and multiple coherence calculations. It has been shown that some of the existing $conditioned$ coherence metrics are inconsistent from an input–output standpoint, for which the corrected interpretations have been subsequently described. This article is unique in that no previous work summarizes all of the previously developed FRF estimation and coherence algorithms and no previous paper explains the inconsistencies in the conditioned coherence functions with respect to multiple coherence.

对于评估系统振动特性的结构动力学工程师来说，频率响应函数 (FRF) 是必不可少的，无论被测试的设置是实验性的、数值的还是分析性的。本文概述了多年来开发的不同 FRF 估计技术。计算 FRF 的普通最小二乘 (OLS) 估计的算法，假设不相关的测量噪声出现在输出 (${\mathbf{H}}_1$) 或输入 (${\mathbf{H}}_2$) 信号，已经与那些通过使用增强的输入输出自动和交叉幂使用总最小二乘 (TLS) 方程的信号进行了比较（${\mathbf{G}}_{FFX}$/${\mathbf{G}}_{XFF}$) 矩阵在每个频率，例如 ${\mathbf{H}}_v$ （使用特征值分解）和 ${\mathbf{H}}_{秒伏D}$（使用奇异值分解）算法。另一种基于 Cholesky 分解的 FRF 估计方法（${\mathbf{H}}_{CD}$) 也进行了讨论。进一步讨论了 TLS 算法，一次计算一个输出的 FRF，正如历史上在开发${\mathbf{H}}_v$算法，在一个案例中同时评估为所有输出计算的 FRF 矩阵。已经介绍了相应相干函数的开发，强调了导致虚拟相干和部分相干等概念的方法依赖范式，同时强调了它们与普通和多重相干计算的等效性。已经表明，一些现有的$C○nd\mathrm{一世}吨\mathrm{一世}○n\mathrm{电子}d$从输入-输出的角度来看，一致性度量是不一致的，随后对其进行了更正的解释。这篇文章的独特之处在于，以前的工作没有总结所有以前开发的 FRF 估计和相干算法，也没有以前的论文解释条件相干函数在多重相干性方面的不一致性。