# Millimeter-Wave Multi-Channel Backscatter Communication and Ranging with an FMCW Radar
## 3 Backscatter Communication with FMCW  
An overview of the proposed backscatter communication system with FMCW radar structure is shown in Figure 2.  
On the radar (vehicle) side, a local oscillator (LO) generates a low-frequency linear frequency modulated (LMF) signal. Then, the LMF signal is multiplied by a frequency  
Figure 1: An illustration of the scenario of interest. Figure 2: The backscatter communication system with FMCW radar structure. multiplier to millimeter-wave frequency. An FMCW signal is generated, for which a single chirp can be represented as:  
\[x(t)=\mathrm{e}^{i(2\pi f_{0}t+\pi\frac{B}{T}t^{2}+\varphi_{0})} \tag{1}\]  
where \(B\) is the chirp bandwidth, \(T\) is the chirp duration, \(f_{0}\) is the initial frequency of the chirp and \(\varphi_{0}\) is the initial phase. The FMCW signal is amplified and then sent out by an antenna. At the tag (infrastructure) side, the received FMCW signal is modulated with data and sent back to the radar. The data is modulated by mixing the FMCW with the baseband signal as  
\[y(t)=x(t)\times\sum A(n)e^{i\theta(n)}g(t-nT_{s}) \tag{2}\]  
where \(g(t)=\left\{ \begin{array}{cc}1,&|t|<T_{s}/2\\ 0,&|t|>T_{s}/2^{\prime}\end{array} \right.\), \(T_{s}\) is the symbol period. When the signal is received by the radar, it becomes  
\[s(t)=H\times y(t-\Delta t)+n(t) \tag{3}\]  
where \(n(t)\) is the noise, \(H\) is the attenuation due to path loss, \(\Delta t\) is the time delay between the transmitted and the received signal at the radar side. After mixing with the local FMCW, we have  
\[ \begin{array}{c}r(t)=H\times y(t-\Delta t)\times\mathrm{e}^{-i(2\pi f_{0}t+ \pi\frac{B}{T}t^{2}+\varphi_{0})}+n^{\prime}(t)\\ =H\mathrm{e}^{i[2\pi f_{0}(t-\Delta t)+\pi\frac{B}{T}(t-\Delta t)^{2}+\varphi_ {0}]}\times\sum A(n-\Delta n)e^{i\theta(n-\Delta n))}g(t-\Delta t-nT_{s})\\ \times\mathrm{e}^{-i(2\pi f_{0}t+\pi\frac{B}{T}t^{2}+\varphi_{0})}+n^{\prime} (t)\\ =H\mathrm{e}^{i(-2\pi f_{0}\Delta t+\pi\frac{B}{T}\Delta t^{2}-2\pi\frac{B}{T} \Delta t\cdot t)}\times\sum A(n-\Delta n)e^{i\theta(n-\Delta n)}g(t-\Delta t -nT_{s})+n^{\prime}(t)\end{array}  \tag{4}\]  
where \(-2\pi f_{0}\Delta t+\pi\frac{B}{T}\Delta t^{2}\) is a fixed phase, \(-2\pi\frac{B}{T}\Delta t\cdot t\) corresponds to a fixed frequency component of \(\frac{B}{T}\Delta t\). \(\sum A(n-\Delta n)e^{i\theta(n-\Delta n)}g(t-\Delta t-nT_{s})\) is the received symbol from the tag. Here, \(\Delta n=\left\lfloor\frac{\Delta t}{Ht}\right\rfloor\), \(\Delta t=2\Delta d/c\), where \(\Delta d\) is the range between the radar and the tag, and \(c\) is the speed of light. Equation (4) can be rewritten as:  
\[r(t)=H\mathrm{e}^{i(\varphi^{\prime}-2\pi\Delta f\cdot t)}\times\sum A(n- \Delta n)e^{i\theta(n-\Delta n)}g(t-\Delta t-nT_{s})+n^{\prime}(t) \tag{5}\]  
where \(\varphi^{\prime}=-2\pi f_{0}\Delta t+\pi\frac{B}{T}\Delta t^{2}\), \(\Delta f=\frac{B}{T}\Delta t\). By analyzing the offset frequency \(\Delta f\) from the received signal \(r(t)\), the range information can be estimated. Then after the removal of the offset frequency, the data can be recovered. The offset frequency is related to chirp bandwidth \(B\), duration \(T\), and the range between the radar and the tag. With wider bandwidth and shorter duration, the offset frequency is larger. Besides the signal reflected by the tag, there is also a radar signal reflected by near objects. The radar signal reflected by other objects can be represented as  
\[c(t)=\sum_{k=1}^{K}H_{k}x(t-\Delta t_{k}) \tag{6}\]  
where \(H_{k}\) and \(\Delta t_{k}\) are the attenuation and the travel time delay of the \(k\)-th object. When \(c(t)\) mixed with the local FMCW signal, it becomes  
\[r_{c}(t)=\sum_{k=1}^{K}H_{k}e^{i(\varphi_{k}-2\pi\Delta f_{k}\cdot t)} \tag{7}\]  
This is a traditionally received signal of an FMCW radar. The range between radar and objects can be estimated from \(\Delta f_{k}\).