# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 2 Results
### Important Special Cases
#### 2.5.2 Side-Looking Antenna, Level Flight Path  
In many airborne radars, the antenna broadside is pointed at \(\phi=-\pi/2\). In this case, we will create a new angular variable \(\psi=\phi+\pi/2\) to represent angular deviation from broadside. The  
Figure 4: Constant-\(\tau\) cut of the no-internal clutter motion (ICM) scattering function Equation (21) with the antenna pattern removed (i.e., \(G=1\)) and no vertical motion (\(\rho_{Z}^{\prime}=0\)). The true scattering function will be windowed by the antenna pattern to focus on a specific region of \(\rho\)—a side-looking antenna will be focused near \(\rho=0\), whereas a nose-aspect antenna will be focused near \(\rho=\rho_{X}^{\prime}\). antenna pattern and local Doppler spectrum with respect to this coordinate will be denoted \(\tilde{G}\) and \(\tilde{p}\), respectively. If we assume a level flight path (i.e., \(\rho_{Z}=0\)), then:  
\[ \begin{split}\beta(\phi,\theta)&=\rho_{X}\cos\theta \cos\phi\\ &=\rho_{X}\cos\theta\sin\psi\\ \approx&(\rho_{X}\cos\theta)\psi=\rho_{X}^{\prime}\psi, \end{split}  \tag{23}\]  
which, for an antenna with beamwidth of less than \(20^{\circ}\), is a very good approximation. Substituting into Equation (20) to compute the scattering function yields:  
\[ \begin{split} S_{h}^{\text{(side)}}(\tau,\rho)&=\frac {\lambda^{2}\sigma^{0}}{16\pi^{3}c^{2}\tau^{3}}\int_{-\pi}^{\pi}\tilde{G}^{2}( \psi,\theta)\tilde{p}(\rho|\psi,\theta)d\psi\\ \approx&\frac{\lambda^{2}\sigma^{0}}{16\pi^{3}c^{2} \tau^{3}}\int_{-\infty}^{\infty}\tilde{G}^{2}(\psi,\theta)\tilde{p}(\rho|\psi, \theta)d\psi\\ &=\frac{\lambda^{2}\sigma^{0}}{16\pi^{3}c^{2}\tau^{3}}\frac{1}{ \rho_{X}^{\prime}}\int_{-\infty}^{\infty}\tilde{G}^{2}(\beta/\rho_{X}^{\prime},\theta)\tilde{p}(\rho|\beta/\rho_{X}^{\prime},\theta)d\beta,\end{split}  \tag{24}\]  
where the second equality is due to the fact that the antenna gain is nearly zero near the edges of the angular limit, so extending the limits of the integral to infinity does not change the result, and the third equality converts the integral from the angular domain to the Doppler domain. If we simplify the integrals by defining \(\underline{G}(\beta|\tau)=\tilde{G}(\beta/\rho_{X}^{\prime}(\tau),\theta(\tau))\) and \(\underline{p}(\rho|\beta,\tau)=\tilde{p}(\rho|\beta/\rho_{X}^{\prime}(\tau), \theta(\tau))\), which are just the antenna pattern and local Doppler spectrum projected into delay-Doppler space, we can see that the scattering function is just a linear transformation of the antenna pattern with the local Doppler spectrum acting as the kernel function:  
\[S_{h}^{\text{(side)}}(\tau,\rho)=\frac{\lambda^{2}\sigma^{0}}{16\pi^{3}c^{2} \tau^{3}}\frac{1}{\rho_{X}^{\prime}}\int_{-\infty}^{\infty}\underline{G}^{2} (\beta|\tau)\underline{p}(\rho|\beta,\tau)d\beta. \tag{25}\]  
Note that \(\underline{p}(\rho|\beta,\tau)\) is the Doppler spectra observed by a moving platform, meaning that its center frequency is being modulated by the platform Doppler shift \(\beta(\phi,\theta)\). If we assume that \(\underline{p}\) does not change much relative to \(\beta\), which is a reasonable assumption due to the fact that radar antennas usually have a very small azimuthal beamwidth, we can express \(\underline{p}\) in terms of a \"baseband\" Doppler spectrum, i.e., the Doppler spectrum caused only by ICM that would be seen by a stationary radar with an infinitesimal beamwidth. The ICM spectrum is usually what is discussed in papers on Doppler spectrum modeling (e.g., [28; 29; 53]). We will denote this quantity as \(\underline{b}(\rho|\tau)\) and note that under the narrow beamwidth assumption, \(\underline{p}(\rho|\beta,\tau)\approx\underline{b}(\rho-\beta|\tau)\), therefore:  
\[