# ATLAS: Latest Advancements and First Observations
## 2 Materials and Methods
### Signal Model  
Let us consider a radar system emitting a pulse with a carrier at frequency \(f_{0}\), modulated with a complex baseband linear frequency modulation (LFM)  
\[u(t)=\text{rect}\bigg{[}\frac{t}{T_{p}}\bigg{]}e^{i\pi\frac{B}{T_{p}}t^{2}}e^{ i(2\pi f_{s})}, \tag{3}\]  
where \(T_{p}\) is the pulse duration, \(t\) is the fast time, \(B\) is the signal bandwidth, and \(j=\sqrt{-1}\). Now let us assume that the radar is illuminating a target with a constant radial velocity, causing the range to change according to  
\[r(t_{s})=r_{0}+v_{r}t_{s},t_{s}\in[0,\text{CPI}], \tag{4}\]  
where \(t_{s}\) is the slow time, \(r_{0}\) is the initial range, \(v_{r}\) is the radial velocity, and CPI is the coherent processing interval. The slow time can be written as dependent on the pulse number  
\[t_{s}=nT_{r},n\in[0,1,2,\ldots,M-1], \tag{5}\]  
where \(T_{r}\) is the pulse repetition interval, and \(M\) is the number of pulses in a CPI.  
If the radar emits a burst of identical pulses, given by (3), the received echoes after carrier demodulation are given by  
\[u_{r}(t,t_{s})=A_{r}\text{rect}\bigg{[}\frac{t-2r(t_{s})/c}{T_{p}}\bigg{]}e^{ i\pi\frac{B}{T_{p}}(t-2r(t_{s})/c)^{2}}\times e^{-j\frac{4\pi r(t_{s})}{ \lambda}}+n(t), \tag{6}\]  
where \(A_{r}\) is the complex amplitude of the echo, \(c\) is the speed of light, \(\lambda=c/f_{0}\), and \(n(t)\) is the additive noise term. The sources of noise arise from inside and outside a circuit: solar noise, cosmic and galactic noise, interference from other antennas, and instrumental noise. Different noise sources are modeled by different distributions, with the most predominant one being thermal noise [(23)] (Chapter 1), which is white and follows a normal distribution in amplitude. Given that the final noise signal is a byproduct of interference from potentially many different sources, the Central Limit Theorem can be applied, and it is reasonable to assume additive white Gaussian noise (AWGN) with a zero mean and a variance \(\sigma^{2}\): \(\mathcal{N}(0,\,\sigma^{2})\).  
After passing the signal through a matched filter, the target's echoes are given by  
\[u_{pc}(r_{i},t_{s})=A_{pc}\text{s}inc\bigg{[}\pi B\frac{2(r_{i}-r_{0}-v_{r}t_{ s})}{c}\bigg{]}\times e^{-j\frac{4\pi(t_{0}+v_{r}t_{s})}{\lambda}}+n(r_{i}), \tag{7}\]  
where \(A_{pc}\) is the complex amplitude of the compressed echoes, and \(r_{i}=ct/2\) is the range. Equation (7) contains a _sinc_ term [(23)] (Chapter 20), which is centered at \(r_{0}\) and migrates during the CPI.  
Real examples of the signal model in Equation (7) are presented in Figures 4 and 5 (\(r_{i}\) is the range axis, and \(t_{s}\) is the Illumination time axis). These figures correspond to the squared magnitude of the pulse-compressed echoes received during the observation of two satellites. In both cases, it is possible to see a distinct line that corresponds to range migration during the illumination time.  
The complex exponential term at the end of Equation (7) can be developed into  
\[e^{-j\frac{4\pi(r_{0}+r_{s}t_{s})}{\lambda}}=e^{-j\frac{4\pi r_{0}}{\lambda}}e^{- 2\pi f_{D}t_{s}}, \tag{8}\]  
where \(f_{D}=\frac{2r_{0}}{\lambda}\) is the Doppler shift. The sequence of phase-change terms along the slow time forms a complex sinusoid at the Doppler frequency and is sometimes called the spatial Doppler signal.