# Human Activity Signatures Captured Under Different Directions Using SISO and MIMO Radar Systems
## 2 Radar Signal Preprocessing
Moreover, the RF cable lengths of the transmitter antenna \(A_{i}^{T_{x}}\) and receiver antenna \(A_{j}^{R_{x}}\) are the same for \(i=j\), i.e., \(L_{i}^{T_{x}}=L_{i}^{R_{x}}\) (\(i=1,2\)). For the \(l\)th scatterer and the aforementioned antenna placement constraints, the total propagation distance \(d_{ij}^{(l)}\) in (6) reduces to the sum of the radar radial range \(r_{ij}^{(l)}\) and the RF cable length \(L_{i}\); i.e.,  
\[d_{ij}^{(l)}=r_{ij}^{(l)}+L_{i} \tag{7}\]where the radar radial range \(r_{ij}^{(l)}\) is the Euclidean distance between the \(l\)th scatterer and the transmitter antenna \(A_{i}^{T_{x}}\) or the receiver antenna \(A_{j}^{R_{x}}\), and \(L_{i}=L_{i}^{T_{x}}=L_{j}^{R_{x}}\) for \(i=j\). Note that we distribute multiple pairs of collocated transmitter-receiver antennas; therefore, the transmitter antenna \(A_{i}^{T_{x}}\) and receiver antenna \(A_{j}^{R_{x}}\) are not collocated for \(i\
eq j\). Consequently, to compute the total propagation distance \(d_{ij}^{(l)}\), the expression in (6) must be used instead of (7) for \(i\
eq j\). With reference to [48], the composite beat signal \(s_{b,ij}(t,t^{\prime})\) is the sum of all beat signals \(s_{b,ij}^{(l)}(t,t^{\prime})\) associated with the cluster of \(\mathcal{L}\) scatterers, which can be expressed as  
\[s_{b,ij}(t,t^{\prime})=\sum\limits_{l=1}^{\mathcal{L}}s_{b,ij}^{(l)}(t,t^{ \prime}). \tag{8}\]  
In an FMCW radar, the composite beat signal \(s_{b,ij}(t,t^{\prime})\) is produced by the quadrature mixer component of the radar. The analog to digital converter (ADC) digitizes the composite beat signal \(s_{b,jj}(t,t^{\prime})\) with a sampling rate of \(F_{s}\) with respect to fast time \(t^{\prime}\). For each chirp interval \(T_{sw}\), the digitized data are stacked in a raw data matrix \(\mathcal{D}_{ij}\). The rows and columns of the raw data matrix \(\mathcal{D}_{ij}\) contain samples of the composite beat signal \(s_{b,ij}(t,t^{\prime})\) in the fast-time and slow-time domain, respectively. The slow-time sampling interval is actually equal to the chirp interval \(T_{sw}\). The fast Fourier transform (FFT) performed on the raw data matrix \(\mathcal{D}_{ij}\) with respect to fast time \(t^{\prime}\) and slow time \(t\) is known as the range FFT and Doppler FFT, respectively. For the channel link \(A_{i}^{T_{x}}\)-\(A_{j}^{R_{x}}\), the expression for the range FFT, also known as the beat frequency profile \(S_{b,ij}(f_{b},t)\), is given as  
\[S_{b,ij}(f_{b},t)=\int\limits_{0}^{T_{sw}}s_{b,ij}(t,t^{\prime})e^{-j2\pi f_{b }t^{\prime}}dt^{\prime} \tag{9}\]  
where \(f_{b}\) represents the beat frequency. The short-time Fourier transform (STFT) is related to the slow-time domain \(t\) and applies on the function resulting from the range FFT in (9). In other words, the data from the beat frequency profile \(S_{b,ij}(f_{b},t)\) are multiplied by a rectangular window function \(W_{r}(\cdot)\) sliding in slow time \(t\) to provide overlapping segments for the FFT operation; i.e.,  
\[X_{ij}(f_{b},f,t)=\int\limits_{-\infty}^{\infty}S_{b,ij}(f_{b},t^{\prime \prime})W_{r}(t^{\prime\prime}-t)e^{-j2\pi ft^{\prime\prime}}dt^{\prime\prime} \tag{10}\]  
where \(f\) is the Doppler frequency and \(t^{\prime\prime}\) denotes the running time. Finally, the TV micro-Doppler signature \(S_{ij}(f,t)\) is obtained by integrating \(X_{ij}(f_{b},f,t)\) over the beat frequencies \(f_{b}\) from zero to the maximum beat frequency \(f_{b,\max}\) and computing the absolute value to the power of 2; i.e., [49]  
\[s_{i}(t,t^{\prime})=\sum_{n=0}^{\infty}c_{i}(t^{\prime})\delta(t-t_{n,i}) \tag{2}\]0  
Note that, according to the Nyquist sampling theorem, the maximum beat frequency \(f_{b,\max}\) is equal to 1/2 of the ADC sampling rate \(F_{s}\); i.e., \(f_{b,\max}=F_{s}/2\).