# Human Activity Signatures Captured Under Different Directions Using SISO and MIMO Radar Systems
## 2 Radar Signal Preprocessing  
In this paper, we have adopted an FMCW \(2\times 2\) MIMO radar to capture the micro-Doppler signatures of a moving person from different perspectives. The fundamental waveform transmitted by the \(i\)th transmitter antenna \(A_{i}^{T_{x}}\) (\(i=1,2\)) of the FMCW radar is the chirp waveform [45]  
\[c_{i}(t^{\prime})=\exp\left[j2\pi\left(f_{0}t^{\prime}+\frac{\gamma}{2}t^{ \prime 2}\right)+j\phi_{i}\right],\qquad 0\leq t^{\prime}<T_{sw} \tag{1}\]  
where \(t^{\prime}\) denotes the fast time, \(f_{0}\) is the initial RF frequency, \(\gamma\) is the slope of the linear chirp in the time-frequency domain, \(\phi_{i}\) is the initial phase, and \(T_{sw}\) is the chirp interval. The two transmitters of the \(2\times 2\) MIMO radar operate in a time division multiple access (TDMA) mode. For \(n=0,1,\ldots,\) the time windows occupied by the \(i\)th transmitter are \((2n+i-1)T_{sw}\leq t^{\prime}<(2n+i)T_{sw}\). The chirp waveform is transmitted periodically by each transmitter in their respective time slot. The overall transmit signal \(s_{i}(t,t^{\prime})\) can be represented as a sum of shifted versions of the fundamental waveform \(c_{i}(t^{\prime})\) according to  
\[s_{i}(t,t^{\prime})=\sum_{n=0}^{\infty}c_{i}(t^{\prime})\delta(t-t_{n,i}) \tag{2}\]  
for \(i=1,2\), where \(\delta(\cdot)\) is a Dirac delta function, \(t\) is the slow time, and \(t_{n,i}\) is the discrete slow time. For TDMA mode, the discrete slow time \(t_{n,i}\) is related to chirp interval \(T_{sw}\) by \(t_{n,i}=(2n+i-1)T_{sw}\). The expression (2) allows us to represent the transmit signal \(s_{i}(t,t^{\prime})\) as function of two separate time variables. The wireless channel link between the \(i\)th transmitter antenna \(A_{i}^{T_{x}}\) and \(j\)th receiver antenna \(A_{j}^{R_{x}}\) is denoted by \(A_{i}^{T_{x}}\)-\(A_{j}^{R_{x}}\), where \(i,j\in\{1,2\}\). When modeling the human body as a cluster of \(\mathcal{L}\) scatterers [46], the beat signal \(s_{b,ij}^{(l)}(t,t^{\prime})\) corresponding to the \(l\)th scatterer and the channel link \(A_{i}^{T_{x}}\)-\(A_{j}^{R_{x}}\) can be represented by  
\[s_{b,ij}^{(l)}(t,t^{\prime})=\sum_{n=0}^{\infty}a_{ij}^{(l)}\exp\Bigl{[}j\Bigl{(} 2\pi f_{b,ij}^{(l)}t^{\prime}+\phi_{ij}^{(l)}\Bigr{)}\Bigr{]}\delta(t-t_{n,i}) \tag{3}\]  
for \(l=1,2,\ldots,\mathcal{L}\). The symbol \(a_{ij}^{(l)}\) stands for the path gain, which is primarily determined by the path loss and the radar cross section. For simplicity, we assume that the path gain \(a_{ij}^{(l)}\) is constant within the observation interval. For the \(l\)th scatterer, the beat frequency \(f_{b,ij}^{(l)}\) and phase \(\phi_{ij}^{(l)}\) in (3) are given as [47]  
\[f_{b,ij}^{(l)}=\frac{2d_{ij}^{(l)}\gamma}{c_{0}} \tag{4}\]  
and  
\[\phi_{ij}^{(l)}=\frac{4\pi d_{ij}^{(l)}}{\lambda} \tag{5}\]  
respectively, where \(d_{ij}^{(l)}\) is the total propagation distance, which is given by  
\[d_{ij}^{(l)}=\frac{1}{2}\Bigl{[}d_{l,j}^{T_{x}}+d_{l,j}^{R_{x}}+L_{i}^{T_{x}} +L_{j}^{R_{x}}\Bigr{]}. \tag{6}\]  
The symbol \(c_{0}\) represents the speed of light in vacuum and \(\lambda\) is the wavelength. The quantity \(d_{l,i}^{T_{x}}\) in (6) is the distance between the transmitter antenna \(A_{i}^{T_{x}}\) and the \(l\)th scatterer. Similarly, the distance between the receiver antenna \(A_{j}^{R_{x}}\) and the \(l\)th scatterer is represented by \(d_{l,j}^{R_{x}}\). The lengths of the RF cables are denoted as \(L_{i}^{T_{x}}\) and \(L_{j}^{R_{x}}\) in (6) for the transmitter antenna \(A_{i}^{T_{x}}\) and receiver antenna \(A_{j}^{R_{x}}\), respectively. In this paper, the transmitter antenna \(A_{i}^{T_{x}}\) and receiver antenna \(A_{j}^{R_{x}}\) are collocated for \(i=j\). As a consequence, the distances from the transmitter antenna \(A_{i}^{T_{x}}\) and receiver antenna \(A_{j}^{R_{x}}\) to the \(l\)th scatterer become identical for \(i=j\); i.e., \(d_{l,j}^{T_{x}}=d_{l,j}^{R_{x}}\) (\(i=1,2\)).