# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 2 Results
### WSSIS System Function Derivations
\begin{split} dR_{h}(\tau,\tau^{\prime},\Delta t;\tilde{\tau})& =\delta(\tau-\tilde{\tau})\delta(\tau^{\prime}-\tilde{\tau}) \frac{\lambda^{2}\sigma^{0}d\tilde{\tau}}{16\pi^{3}c^{2}\tilde{\tau}^{3}} \cdot\lim_{N\rightarrow\infty}\sum_{n=0}^{N-1}G^{2}(n\Delta\phi,\theta) \mathbb{E}\left[e^{j2\pi\rho(\phi_{n})\Delta t}\right]\Delta\phi\\ &=\delta(\tau-\tilde{\tau})\delta(\tau^{\prime}-\tilde{\tau}) \frac{\lambda^{2}\sigma^{0}d\tilde{\tau}}{16\pi^{3}c^{2}\tilde{\tau}^{3}} \int_{(2\pi)}G^{2}(\phi,\theta)\mathbb{E}\left[e^{j2\pi\rho_{\phi}\Delta t} \right]d\phi.\end{split}  \tag{15}\]  
Note that the expression \(\mathbb{E}\left[e^{j2\pi\rho_{\phi}\Delta t}\right]\) is the characteristic function of the random angular frequency \(2\pi\rho_{\phi,\theta}\)[52]. In this work, we will define the characteristic function \(k(\Delta t|\phi,\theta)\) as:  
\[ \begin{split} k(\Delta t|\phi,\theta)&=\int p(\rho| \phi,\theta)e^{j2\pi\rho\Delta t}d\rho\\ &=\mathcal{F}_{\rho}^{-1}\left[p(\rho|\phi,\theta)\right],\end{split}  \tag{16}\]where \(p(\rho|\phi,\theta)\) is the probability density function (pdf) of the random frequency \(\rho_{\phi,\theta}\). Substituting this expression in Equation (15) yields:  
\[ \begin{split} dR_{h}(\tau,\tau^{\prime},\Delta t;\bar{\tau})& =\delta(\tau-\bar{\tau})\delta(\tau^{\prime}-\bar{\tau})\frac{ \lambda^{2}\sigma^{0}d\bar{\tau}}{16\pi^{3}c^{2}\bar{\tau}^{3}}\cdot\lim_{N \rightarrow\infty}\sum_{n=0}^{N-1}G^{2}(n\Delta\phi,\theta)\mathbb{E}\left[e^{ \rho 2\pi p_{n}\Delta t}\right]\Delta\phi\\ &=\delta(\tau-\bar{\tau})\delta(\tau^{\prime}-\bar{\tau})\frac{ \lambda^{2}\sigma^{0}d\bar{\tau}}{16\pi^{3}c^{2}\bar{\tau}^{3}}\cdot\int_{(2 \pi)}G^{2}(\phi,\theta)k(\Delta t|\phi,\theta)d\phi\end{split}  \tag{17}\]  
One interpretation of the function \(p(\rho|\phi,\theta)\) that is used in the wireless communications literature is that the pdf of the random Doppler shift can be thought of as a normalized power spectral density (PSD) [3; 8]. Thus if one has some model of the local Doppler spectrum, this information is accounted for in Equation (16). To obtain the autocorrelation function for all delays \(\tau\) we will use the US property that the return from each isorange ring \(\bar{\tau}\) is uncorrelated and thus we can integrate over \(\bar{\tau}\) to obtain the total response:  
\[ \begin{split} R_{h}(\tau,\tau^{\prime},\Delta t)&= \int dR_{h}(\tau,\tau^{\prime},\Delta t;\bar{\tau})\\ &=\delta(\tau-\tau^{\prime})\frac{\lambda^{2}\sigma^{0}}{16\pi^{3 }c^{2}\tau^{3}}\int_{(2\pi)}G^{2}(\phi,\theta)k(\Delta t|\phi,\theta)d\phi. \end{split}  \tag{18}\]  
It is clear from Equation (18) that \(A_{h}(\tau,\Delta t)\) is therefore:  
\[A_{h}(\tau,\Delta t)=\frac{\lambda^{2}\sigma^{0}}{16\pi^{3}c^{2}\tau^{3}}\int _{(2\pi)}G^{2}(\phi,\theta)k(\Delta t|\phi,\theta)d\phi. \tag{19}\]  
To find the clutter scattering function \(S_{h}(\tau,\rho)\), we take the Fourier transform of \(A_{h}(\tau,\Delta t)\) with respect to \(\Delta t\):  
\[ \begin{split} S_{h}(\tau,\rho)&=\mathcal{F}_{ \Delta t}\left[A_{h}(\tau,\Delta t)\right]\\ &=\frac{\lambda^{2}\sigma^{0}}{16\pi^{3}c^{2}\tau^{3}}\int_{(2 \pi)}G^{2}(\phi,\theta)\left(\mathcal{F}_{\Delta t}\mathcal{F}_{\rho}^{-1}p( \rho|\phi,\theta)\right)d\phi\\ &=\frac{\lambda^{2}\sigma^{0}}{16\pi^{3}c^{2}\tau^{3}}\int_{(2 \pi)}G^{2}(\phi,\theta)p(\rho|\phi,\theta)d\phi\end{split}  \tag{20}\]  
The expression in Equation (20) is significant because it gives an analytical expression for the full clutter spectrum, not just the mainlobe clutter, for any antenna pattern and provides a mechanism for supplying a priori information about the local Doppler spectra as a function of look angle. In this integral it can be seen that the antenna pattern has the effect of performing a weighted average of the local Doppler spectra over azimuth.