## 3 Spatial-Frequency Model of the Point-Wise FRF
### Complex-Valued Gaussian Process  
From a function space view, the impulse response function is seen as an element of the reproducing kernel Hilbert space, and it is searched over a possibly infinite-dimensional functional space. The impulse response function is commonly treated as a zero-mean Gaussian stochastic process in the field of system and control [17]. The zero-mean assumption is typically made most probably due to lack of objective prior information. The prior knowledge of the impulse response function, such as its stability and smoothness, is encoded for inference through the use of a suitable covariance (kernel function). Under these standing conditions, the FRF, which is the Fourier transform of the impulse response function, is a zero-mean complex-valued Gaussian process, but it is not circular complex distributed because the impulse response function is a non-stationary Gaussian process. The probability distribution for a complex-valued Gaussian vector can be obtained as a joint distribution of its real and imaginary parts [21], the probability density function of the FRF is characterized by the covariance matrix \(\underline{\underline{\Sigma}}_{\mathbf{G}}\) of the augmented vector \(\mathbf{G}_{\underset{-0}{\mathbf{G}}}\), and \(\mathbf{G}_{\underset{-0}{\mathbf{G}}}=\left[\mathbf{G}_{\underset{-0}{\mathbf{G }}}^{\mathsf{T}},\mathbf{G}_{\underset{-0}{\mathbf{G}}}^{\mathsf{H}}\right]^{ \mathsf{T}}\) with the superscript \({}^{\mathsf{H}}\) being the conjugate transpose. The covariance matrix reads  
\[\underline{\underline{\Sigma}}_{\mathbf{G}}= \begin{bmatrix}\mathbf{K}& \mathbf{C}\\ \mathbf{C}^{\mathsf{H}}&\mathbf{K}^{*}\\ \end{bmatrix}  \tag{11}\]  
where the superscript \({}^{*}\) denotes the complex conjugate, \(\mathbf{K}=\mathbb{E}\left(\mathbf{G}_{0}\mathbf{G}_{0}^{\mathsf{H}}\right)\), \(\mathbf{C}=\mathbb{E}\left(\mathbf{G}_{0}\mathbf{G}_{0}^{\mathsf{T}}\right)\) with \(\mathbb{E}\) being the expectation operator, and  
\[\mathbf{K}= \begin{bmatrix}K(\omega_{1},\omega_{1})&\cdots&K(\omega_{1},\omega _{F})\\ \vdots&\ddots&\vdots\\ K(\omega_{F},\omega_{1})&\cdots&K(\omega_{F},\omega_{F})\\ \end{bmatrix}  \tag{12}\]  
\[\mathbf{C}= \begin{bmatrix}\mathbf{C}(\omega_{1},\omega_{1})&\cdots&\mathbf{C }(\omega_{1},\omega_{F})\\ \vdots&\ddots&\vdots\\ \mathbf{C}(\omega_{F},\omega_{1})&\cdots&\mathbf{C}(\omega_{F},\omega_{F})\\ \end{bmatrix}  \tag{13}\]The Hermitian covariance matrix \(\mathbf{K}\) is complemented by a second complementary covariance matrix \(\mathbf{C}\) which is called the relation matrix. It can be trivially verified that \(K(\omega_{k},\omega_{l})=C(\omega_{k},\omega_{-l})\). The kernel function of a Gaussian process is crucial to achieve a good bias-variance trade-off for estimation of point-wise FRFs. A few kinds of kernel functions have been invented to embed various types of prior knowledge, such as the smooth and stable properties of the impulse response function. Among these kernels, the so-called stable spline kernel [17] and diagonal correlated (DC) kernel [22] are widely recognized in the system identification community. They are further transformed into the frequency domain using the Fourier transform [23]. Here, the frequency domain DC kernel is adopted for FRF modeling,  
\[K(\omega_{k},\omega_{l})=\gamma_{0}\times\frac{1}{1-e^{-\beta-j(\omega_{k}- \omega_{l})}}\times\frac{1-e^{-2\alpha-\beta-j(\omega_{k}-\omega_{l})}}{(1-e^{ -\alpha-\beta/2-j\omega_{k})}(1-e^{-\alpha-\beta/2+j\omega_{l}})} \tag{14}\]  
where the scaling factor \(\gamma_{0}\in\mathcal{R}^{+}\), and the kernel parameters \(\alpha\), \(\beta\in\mathcal{R}^{+}\), which are called hyperparameters of the kernel. \(\alpha\) controls how markedly the change of \(G_{0}(\omega_{k})\) can be correlated with the variation of \(G_{0}(\omega_{l})\), and \(\beta\) describes the stability (i.e., the impulse response decays to zero exponentially). The basis functions \(\mathbf{g}_{r}\) in Equation (7), \(r=1,\,\cdots,\,n_{\phi}\), are also modeled as complex-valued Gaussian processes.