# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 2 Results
### WSSIS System Function Derivations  
To create a model for the scattering function \(S_{h}(\tau,\rho)\), we need to characterize the signal return as a function of delay and Doppler. To do so, we will consider the Doppler spectrum generated from the backscatter from a single isorange ring and then apply the principle of superposition to obtain the total response from all ranges. The approach taken in this section can be considered a generalization of the Clarke model for flat fading [9; 12], applied to modeling radar surface clutter. We will assume that the impulse response from this ring is the superposition of returns from \(N\) equiangular patches in azimuth with random amplitudes and phases. We will also assume that the  
Figure 3: Return from a single infinitesimal isorange ring on the sea surface. The autocorrelation function of the impulse response of this ring is computed at delay \(\tau\), then the result is integrated over all \(\tau\) to get the total autocorrelation as a function of delay \(\tau\) and time offset \(\Delta t\). Figure 2: Flat earth geometry in the \(X\)-\(Z\) plane. Because the goal is to ultimately produce a range-Doppler map, we need to express all quantities in terms of downrange delay \(\tau\) and absolute time \(t\). scatterers on this ring are located at delay \(\tilde{\tau}\) and write the incremental impulse response of this thin ring as follows:  
\[dh(\tau,t;\tilde{\tau})=\delta(\tau-\tilde{\tau})\sum_{n=0}^{N-1}da(\phi_{n})e^{ j(2\pi\rho(\phi_{n})t+\gamma_{n})} \tag{10}\]  
where \(\phi_{n}=2\pi n/N\) is the azimuth angle to patch \(n\), \(da(\phi_{n})\) is the infinitesimal amplitude gain at angle \(\phi_{n}\), \(\rho(\phi_{n})\) is the Doppler shift at angle \(\phi_{n}\), and \(\gamma_{n}\) is a random phase shift. The incremental autocorrelation function of the impulse response at \(\tilde{\tau}\) is then given by:  
\[ \begin{split} dR_{h}(\tau,\tau^{\prime},t,t+\Delta t;\tilde{ \tau})&=\mathbb{E}\left[dh^{*}(\tau,t;\tilde{\tau})dh(\tau^{ \prime},t+\Delta t;\tilde{\tau})\right]\\ &=\delta(\tau-\tilde{\tau})\delta(\tau^{\prime}-\tilde{\tau}) \sum_{m,n}\mathbb{E}\left[da(\phi_{m})da(\phi_{n})e^{j[2\pi(\rho(\phi_{n})(t+ \Delta t)-\rho(\phi_{m})t)+\gamma_{n}-\gamma_{m}]}\right].\end{split}  \tag{11}\]  
We will also assume that the scatterer amplitudes \(da(\phi_{n})\) are mutually independent, and we will assume that the amplitude \(da(\phi_{n})\) is independent of the Doppler shift \(\rho(\phi_{n})\), therefore:  
\[ \begin{split} dR_{h}(\tau,\tau^{\prime},t,t+\Delta t;\tilde{\tau} )&=\delta(\tau-\tilde{\tau})\delta(\tau^{\prime}-\tilde{\tau}) \sum_{n=0}^{N-1}\mathbb{E}\left[|da(\phi_{n})|^{2}\right]\mathbb{E}\left[e^{j 2\pi\rho(\phi_{n})\Delta t}\right]\\ &=dR_{h}(\tau,\tau^{\prime},\Delta t;\tilde{\tau})\end{split}  \tag{12}\]  
Note that \(\mathbb{E}\left[|da(\phi_{n})|^{2}\right]\) is simply the backscattered power gain from scatterer \(n\), therefore by using Equation (9) and scaling it to account for the fact that the surface area of each patch is smaller by a factor of \(1/N\), we obtain:  
\[ \begin{split}\mathbb{E}\left[|da(\phi_{n})|^{2}\right]& =\frac{1}{N}\frac{dP_{R}}{P_{T}}\\ &=\frac{\lambda^{2}\sigma^{0}d\tilde{\tau}}{8\pi^{2}c^{2}\tilde{ \tau}^{3}}\frac{1}{N}G^{2}(n\Delta\phi,\theta).\end{split}  \tag{13}\]  
Note that \(N=2\pi/\Delta\phi\), where \(\Delta\phi\) is the angular spacing between patches, which upon substituting in Equation (13) yields:  
\[ \begin{split}\mathbb{E}\left[|da(\phi_{n})|^{2}\right]=\frac{ \lambda^{2}\sigma^{0}d\tilde{\tau}}{16\pi^{3}c^{2}\tilde{\tau}^{3}}G^{2}(n \Delta\phi,\theta)\Delta\phi.\end{split}  \tag{14}\]  
Combining this with Equation (12) and taking the limit as \(N\rightarrow\infty\) yields  
\[