# Millimeter-Wave Multi-Channel Backscatter Communication and Ranging with an FMCW Radar
## 4 Multi-Tag Scenario  
An overview of the proposed backscatter communication system with FMCW radar structure in a multi-tag scenario is shown in Figure 3.  
When there are multiple tags in the view, each tag modulates individual data to the FMCW signal and sends it back to the radar side. The received signal under this scenario can be represented as  
\[s(t)=\sum_{i=1}^{I}H_{i}\times y_{i}(t-\Delta t_{i})+n(t)+c(t) \tag{9}\]  
where \(c(t)\) represents the reflected signal from other objects and \(n(t)\) is the noise. When it mixes with the local FMCW, it becomes  
\[r(t) =\sum_{i=1}^{I}H_{i}e^{i(\varphi_{i}^{\prime}-2\pi\Delta f_{i}t)} \times\sum A_{i}(n-\Delta n_{i})e^{i\theta_{i}(n-\Delta n_{i})}g(t-\Delta t_{i }-nT_{s})+n^{\prime}(t)+\sum_{k=1}^{K}H_{k}e^{i(\varphi_{k}-2\pi\Delta f_{k} \cdot t)} \tag{10}\]  
Equation (10) shows that reflected signals from tags and objects have different offset frequency \(\Delta f_{i}\) and \(\Delta f_{k}\). Assuming there are two tags and one object in the view, their spectrum can be shown in Figure 4.  
The spectrum of the received signal from the object can be represented as the purple line. There is only a single frequency tone that represents \(\Delta f_{k}\), which can be selected by a bandpass filter (BPF) when its spectral power density is larger  
Figure 4: The spectrum of a down-converted received signal from two tags. Figure 3: The proposed backscatter communication system with FMCW radar in a multi-tag scenario. The spectrum of received signals from two tags are represented as yellow and orange dots. They have offset frequencies of \(\Delta f_{1}\) and \(\Delta f_{2}\), respectively. Their symbol rates are \(1/T_{b1}\) and \(1/T_{b2}\). When the symbol rate is large and the difference between two frequency offsets \(\Delta f_{2}-\Delta f_{1}\) is small, the spectrum of two signals will overlap, as Figure 4 shows. The spectrum of the total signal received from the two tags can be represented by the blue dots. Two signals cannot be separated from their spectrum. Paper [16] studied the requirement of the distance separation of tags without spectrum overlap, which is  
\[\frac{B}{T_{p}}\frac{2\Delta r_{min}}{c}\geq\left[\frac{1}{T_{b1}}+\frac{1}{T _{b2}}\right] \tag{11}\]  
where \(1/T_{b1}\) and \(1/T_{b2}\) are the symbol rate of two transmitted signals from two tags, \(T_{p}\) is the FMCW chirp duration, \(B\) is the chirp bandwidth, \(c\) is the speed of light, and \(\Delta r_{min}\) is the minimum separation range. In this case, the symbol rate of transmitted signals is limited by the distance and FMCW chirp. If the FMCW radar has a chirp bandwidth of 1 GHz, a duration of 1 ms, and the distance between the two tags is 30 m. In this case, the total symbol rate of two tags needs smaller than  
\[\frac{10^{9}\times 2\times 30}{10^{-3}\times 3\times 10^{8}}=2\times 10^{5}=200 \text{ Kbaud} \tag{12}\]  
By using a more advanced FMCW radar or placing two tags further apart from each other, the system will be able to transmit the signal with a higher symbol rate. A commercial FMCW radar normally has a chirp bandwidth smaller than 1 GHz, and a duration time longer than 1 ms. For the smart traffic scenario introduced in Section 2, the distance between two tags is normally a few dozen meters. In this case, it is difficult to further increase the communication symbol rate up to megabaud. In this paper, we propose to use CDMA with backscatter communication and radar sensing system to avoid collisions between multi-tag and jamming between signals from tags and objects. For each tag, an individual signature code is used to modulate the original data at the baseband. By using the signature code to represent the symbol, the spectrum of the baseband signal is spread, and its spectral power density is decreased accordingly. On the radar side, the received signal will correlate to each signature code to recover the data of each tag. Assume the signature code chip \(d(n)\) has a length of \(L\) bits.