# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 2 Results
### Wssus Processes  
To apply the WSSUS assumption to a radar scattering environment, we will express the impulse response in the following form:  
\[h(\tau,t)=\mathcal{F}_{\rho}^{-1}[\eta(\tau,\rho)]=\int_{-\infty}^{\infty}\eta (\tau,\rho)e^{i2\pi\rho t}d\rho \tag{2}\]  
where \(\mathcal{F}^{-1}\) is the inverse Fourier operator, \(\rho\) is Doppler shift, and \(\eta(\tau,\rho)\) is known as the \"delay-Doppler spread function\" [13; 31]. This function is a stochastic description of the range-Doppler map of the targets. Substituting Equation (2) into Equation (1) yields:  
\[y(t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\eta(\tau,\rho)x(t-\tau)e ^{i2\pi\rho t}d\rho d\tau+w(t). \tag{3}\]  
In Equation (3) it is clear to see that \(y(t)\) can be expressed as the superposition of delayed (by \(\tau\)), frequency-shifted (by \(\rho\)), and scaled (by the complex gain \(\eta(\tau,\rho)d\tau d\rho\)) copies of \(x(t)\). It should be noted that this definition is general enough to include all scatterers in the radar's field of view--clutter as well as useful targets. In this case the overall delay-Doppler spreading function is:  
\[\eta_{\text{total}}=\eta_{\text{targets}}+\eta_{\text{clutter}}, \tag{4}\]  
due to the linearity of the system. To characterize the behavior of \(h\) as a stochastic process, we will assume it is zero mean and compute its second-order statistics:  
\[R_{h}(\tau,\tau^{\prime},t,t+\Delta t)=\mathbb{E}\left[h^{*}(\tau,t)h(\tau^{ \prime},t+\Delta t)\right]. \tag{5}\]  
Here will we apply the two fundamental assumptions of a WSSUS system: (1) The system is wide-sense stationary (WSS), implying that the autocorrelation depends only on \(\Delta t\) along the \(t\) axis, and (2) the scattering at different lags \(\tau\) are uncorrelated (US):  
\[ \begin{split}\mathbb{E}\left[h^{*}(\tau,t)h(\tau^{\prime},t+ \Delta t)\right]&=R_{h}(\tau,\tau^{\prime},\Delta t)\\ &=A_{h}(\tau,\Delta t)\delta(\tau-\tau^{\prime}).\end{split}  \tag{6}\]  
The function \(A_{h}(\tau,\Delta t)\) is known as the system correlation function [48; 3]. By taking the Fourier transform of \(A_{h}(\tau,\Delta t)\) with respect to \(\Delta t\), we can view the Doppler spectrum of the return as a function of delay, referred to in the literature as the \"scattering function\" \(S_{h}(\tau,\rho)\):  
\[S_{h}(\tau,\rho)=\mathcal{F}_{\Delta t}\left[A_{h}(\tau,\Delta t)\right]. \tag{7}\]The scattering function can be thought of as the true range-Doppler map of the scattering environment independent of the waveform used to probe it. It can also be shown that the autocorrelation function of the delay-Doppler spread function \(\eta\) for a WSSUS system is:  
\[\mathbb{E}\left[\eta^{*}(\tau,\rho)\eta(\tau^{\prime},\rho^{\prime})\right]=S_{ h}(\tau,\rho)\delta(\tau-\tau^{\prime})\delta(\rho-\rho^{\prime}). \tag{8}\]  
It is sometimes easier to work with this form when deriving new expressions. This result shows that under a WSSUS model, the scattering not only at different ranges, but also different Doppler shifts are uncorrelated. The full set of WSSUS system functions are shown in Figure 1, all of which are related by Fourier transforms of the different temporal variables. Knowledge of any one of these functions is a sufficient description of the second-order statistics of the system. These functions are standard tools in wireless communication channel modeling, and the diagram shown in Figure 1 can be found in virtually any textbook on the subject [3]. Note that nowhere in this discussion have we specified a distribution for the samples of \(h(\tau,t)\). If we assume that \(h\) is a circularly-symmetric complex Gaussian random process, then its distribution is completely specified by the WSSUS system functions described previously, and the magnitude envelope \(|y|\) of the return will be Rayleigh distributed [3]. However, in many cases a distribution with heavier tails, such as the Weibull or K distribution, is more appropriate, particularly for low grazing angles or when the radar can resolve individual sea spikes [49].