## 2 Concept  
### Fundamentals  
The following part explains the basic idea of the frequency coded radar sub-channel concept as it is schematically shown in Figure 1. The initial point is a bi-static radar system with divided transmit (TX) and receive (RX) paths. Although we will utilize a frequency-modulated continuous-wave (FMCW) radar in the later measurements, the fundamental radar system design is not of relevance as long as complex-valued raw data of the transfer function is recorded and available for signal processing. Following the TX path, the transmit signal \(s_{\text{t}}(t)\) is fed to a power amplifier (PA) and a sub-subsequent radar transmit antenna. While the TX antenna is chosen in a way that the whole AOI is illuminated, the PA ensures sufficient signal power. In contrast to the TX antenna, which provides a broad beam pattern, the RX antenna array consists of narrow beam antennas that point to different directions within the AOI [20]. Each RX antenna is connected to its own RX filter with a unique filter function within the illustrated filter bank. This approach can also be interpreted as the coding of the radar sub-channel and alignment of the corresponding cross-range direction. Finally, all filter bank outputs are combined into a single RX signal, which is connected to the RX path of the radar system. As combining losses increase with an increasing number of sub-channels, low noise amplifiers as initial stages of the filter bank are useful for multiple sub-channel designs. The range resolution, i.e., the minimum distance between two equally strong targets that allows for target separation in the radar diagram, \(\Delta R_{\mathrm{r}}\) of a radar is defined by:  
\[\Delta R_{\mathrm{r}}=\frac{c\cdot k}{2\cdot B}. \tag{1}\]  
In (1), \(c\) is the speed of light, \(B\) is the utilized bandwidth, and \(k\) is a forming factor of the applied window function in signal processing. Obviously, the utilized signal bandwidth has a direct influence on the range resolution. Consequently, the RX-filter design has a significant influence on the later radar performance and resolution. Therefore, in order to preserve maximum range resolution, it is advisable that RX-coding filters should not narrow the absolute signal bandwidth. Hence, the chosen filters could be of the notch or allpass type. Using the described setup allows for an instantaneous scan of the whole AOI with a single measurement. The receiving signal \(s_{\mathrm{r}}(t)\) is the summation of all signals of the \(N\) sub-channels:  
\[s_{\mathrm{r}}(t)=\sum_{i=1}^{N}s_{\mathrm{r}i}(t), \tag{2}\]  
while every sub-channel signal \(s_{\mathrm{r}i}(t)\) can be modeled as follows:  
\[s_{\mathrm{r}i}(t)=s_{\mathrm{t}}(t)*h_{\mathrm{pa}}(t)*h_{\mathrm{rx}}(t)*h_{ \mathrm{fs}}(t-\tau)*\sigma*h_{\mathrm{rx}}(t)*h_{\mathrm{i}}(t). \tag{3}\]  
In (3), \(s_{\mathrm{t}}(t)\) represents the transmit signal while \(h_{\mathrm{pa}}(t)\), \(h_{\mathrm{tx}}(t)\), \(h_{\mathrm{fs}}(t-\tau)\), \(h_{\mathrm{rx}}(t)\), and \(h_{\mathrm{i}}(t)\) describe the time domain transfer functions of the PA, the TX antenna, free space propagation, RX antenna, and \(i\)-_th_ coding filter, respectively. The connecting mathematical operation \"\(*\)\" is the convolution operator. Furthermore, the parameters \(\sigma\) and \(\tau\) stand for the target's radar cross section (RCS) and the time of flight which is defined by  
\[\tau=\frac{2R}{c}. \tag{4}\]  
Here, \(R\) is the line-of-sight (LOS) distance to the target. In good approximation, we may consider that the system-related transfer functions do not alter while in operation. Therefore, they will be summarized in the following and described by the system's time domain transfer function \(h_{\mathrm{sys}}(t)\):  
\[h_{\mathrm{sys}}(t)=h_{\mathrm{pa}}(t)*h_{\mathrm{rx}}(t)*h_{\mathrm{rx}}(t) \tag{5}\]  
Figure 1: Block diagram of the proposed bi-static radar setup with RX-channel coding.