Wiener-Khinchin Theorem

Given a wide-sense stationary (WSS) stochastic process, its autocorrelation function and its power spectral density function are a pair of Fourier transform.

The frequency domain correlation can be derived as

\begin{align} \mathcal{E}\left[ H(f_1) H^{*}(f_2) \right] &= \mathcal{E} \left[ \int_{-\infty}^{+\infty}h(\tau_1)e^{-j2\pi f_1\tau_1} d\tau_1 \cdot \int_{-\infty}^{+\infty}h^{*}(\tau_2)e^{j2\pi f_2\tau_2} d\tau_2 \right] \nonumber \\ &= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \mathcal{E}\left[ h(\tau_1)h^{*}(\tau_2)\right]e^{-j2\pi(f_1 \tau_1 - f_2 \tau_2)} d\tau_1 d\tau_2, \label{eq:corr-f} \end{align}

where \(h(\tau)\) and \(H(f)\) are the channel impulse response (CIR) and channel frequency response (CFR) of the channel, respectively.

If the individual paths of the channel are uncorrelated, i.e., uncorrelated scatering (US), \(\mathcal{E}\left[h(\tau_1) h^{*}(\tau_2)\right] = h(\tau_1) h^{*}(\tau_2) \delta(\tau_1 - \tau_2)\), then equation \eqref{eq:corr-f} can be further written as

\begin{align} \mathcal{E}\left[ H(f_1) H^{*}(f_2) \right] &= \int_{-\infty}^{+\infty}|h(\tau)|^2 e^{-j2\pi(f_1-f_2)\tau}d\tau \nonumber \\ &=\int_{-\infty}^{+\infty}P_{\mathrm{delay}}(\tau) e^{-j2\pi(f_1-f_2)\tau}d\tau, \label{eq:corr-f-us} \end{align}

where \(P_{\mathrm{delay}}(\tau) \triangleq |h(\tau)|^2\) is the power delay profile (PDP) of the channel.

Particularly, if PDP is uniformly distributed within cyclic prefix (CP), denoted by \(L_{\mathrm{cp}}\), i.e.,

\begin{align*} P_{\mathrm{delay}}(\tau) &= \frac{1}{L_{\mathrm{cp}}}, \quad \tau \in [0, L_{\mathrm{cp}}]; \\ \mathcal{E}\left[ H(f_1) H^{*}(f_2) \right] &=\mathrm{sinc} \left[ \left( f_1 - f_2 \right)L_{\mathrm{cp}} \right] e^{-j\pi\left( f_1 - f_2 \right)L_{\mathrm{cp}}}. \end{align*}

Without loss of generality, we consider a frequency-flat fading channel for instance, and the channel can be represented as

\begin{align} h(t) = \int_0^{2\pi} a(\theta)e^{j2\pi f_m t \cos\theta} d\theta, \label{eq:h-t} \end{align}

where \(a(\theta)\) is the complex-valued gain of the ray arrived in the direction of \(\theta\), and \(f_m \triangleq \dfrac{v}{\lambda}\) is termed maximum Doppler frequency.

Then, the temporal domain correlation can be expressed as

\begin{align} \mathcal{E}\left[h(t_1)h^{*}(t_2)\right] &= \mathcal{E}\left[ \int_0^{2\pi} a(\theta_1)e^{j2\pi f_m t_1 \cos\theta_1} d\theta_1 \cdot \int_0^{2\pi} a^{*}(\theta_2)e^{-j2\pi f_m t_2 \cos\theta_2} d\theta_2\right] \nonumber \\ &=\int_0^{2\pi}\int_0^{2\pi}\mathcal{E}\left[a(\theta_1)a^{*}(\theta_2)\right]e^{j2\pi f_m(t_1\cos\theta_1 - t_2\cos\theta_2)}d\theta_1 d\theta_2\label{eq:corr-t}. \end{align}

For US, similarly, we have \(\mathcal{E}\left[a(\theta_1)a^{*}(\theta_2)\right] = a(\theta_1)a^{*}(\theta_2)\delta(\theta_1-\theta_2)\), and equation \eqref{eq:corr-t} becomes

\begin{align} \mathcal{E}\left[h(t_1)h^{*}(t_2)\right] &= \int_0^{2\pi} |a(\theta)|^2 e^{j2\pi f_m (t_1-t_2)\cos\theta} d\theta \nonumber \\ &= \int_0^{2\pi}P_{\mathrm{angle}}(\theta) e^{j2\pi f_m (t_1-t_2)\cos\theta} d\theta \label{eq:corr-t-us-angle} \\ &= \int_{-f_m}^{f_m} P_{\mathrm{doppler}}(f)e^{j2\pi f(t_1-t_2)} df, \label{eq:corr-t-us-doppler} \end{align}

where \(P_{\mathrm{angle}}(\theta) \triangleq |a(\theta)|^2\) is the power angle spectrum of the channel, and \(P_{\mathrm{doppler}}(f) \triangleq 2P_{\mathrm{angle}}(\theta) \left|\dfrac{d\theta}{df}\right| = \dfrac{2P_{\mathrm{angle}}(\theta)}{\sqrt{f_m^2 - f^2}}\) is the well-known Doppler power spectrum of the channel.

Particularly, if all the arriving paths are uniformly around the receiver, a.k.a. uniform ring scattering in Clarke model, we have

\begin{align*} P_{\mathrm{angle}}(\theta) &= \frac{1}{2\pi}, \quad \theta \in [0,2\pi); \\ P_{\mathrm{doppler}}(f) &= \frac{1}{\pi \sqrt{f_m^2 - f^2}}, \quad f \in [-f_m, f_m]; \\ \mathcal{E}\left[h(t_1)h^{*}(t_2)\right] &= J_0 \left[ 2\pi f_m(t_1-t_2) \right], \end{align*}

where \(J_0(\cdot)\) is the zero-order Bessel function of the first kind. Accordingly, the Doppler power spectrum becomes U-shaped.

Without loss of generality, we consider a uniform antenna array housing \(N\) antenna elements spaced by \(d\). The spatial-domain channel can be denoted by

\begin{align} \mathbf{h} = \int_0^{2\pi} a(\theta) \mathbf{v}(\theta) d\theta, \end{align}

where \(a(\theta)\) is a complex-valued amplitude of a path in the direction of \(\theta\), and \(\mathbf{v}(\theta)\) is the corresponding steering vector, i.e.,

\begin{align} \mathbf{v}(\theta) = \begin{bmatrix} 1 \\ e^{{jkd\cos\theta}} \\ e^{j2kd\cos\theta} \\ \vdots \\ e^{j(N-1)kd\cos\theta} \end{bmatrix}, \end{align}

where \(k\triangleq \dfrac{2\pi}{\lambda}\) is the wave number and \(\lambda\) the wave length. Then, the spatial domain correlation can be derived as

\begin{align} \mathcal{E}\left[ \mathbf{h}(n_1) \mathbf{h}^{*}(n_2) \right] &= \mathcal{E} \left[ \int_{0}^{2\pi} a(\theta_1) e^{jn_1kd\cos\theta_{1}} d\theta_{1}\cdot \int_{0}^{2\pi} a^{*}(\theta_{2}) e^{-jn_2kd\cos\theta_{2}} d\theta_{2}\right]; \quad 0 \leq n_1, n_2 \leq N-1; \nonumber \\ &= \int_{0}^{2\pi}\int_{0}^{2\pi} \mathcal{E} \left[a(\theta_1)a^{*}(\theta_2) \right]e^{jkd(n_1\cos\theta_{1} - n_2\cos\theta_{2})} d\theta_{1}d\theta_{2}. \label{eq:corr-s} \end{align}

Similarly, in US case, we have \(\mathcal{E}\left[a(\theta_{1}) a^{*}(\theta_{2}) \right]=a(\theta_{1}) a^{*}(\theta_{2}) \delta(\theta_{1}-\theta_{2})\), and equation \eqref{eq:corr-s} becomes

\begin{align} \mathcal{E}\left[ \mathbf{h}(n_1) \mathbf{h}^{*}(n_2) \right] &= \int_0^{2\pi}|a(\theta)|^2 e^{jk(n_1-n_2)d\cos\theta} d\theta \nonumber \\ &= \int_0^{2\pi} P_{\mathrm{angle}}(\theta)e^{jk(n_1-n_2)d\cos\theta} d\theta. \end{align}

Particularly, for uniform ring scattering, we have

\begin{align*} P_{\mathrm{angle}}(\theta) &= \frac{1}{2\pi}, \quad \theta \in [0, 2\pi); \\ \mathcal{E}\left[ \mathbf{h}(n_1) \mathbf{h}^{*}(n_2) \right] &= J_0 \left[ k(n_1-n_2)d \right]. \end{align*}