## 2 Basic Principles
### Continuous-Wave Radar  
The main characteristic of _continuous-wave (CW) radar systems_ is that they emit a continuous electromagnetic wave using a sine waveform, where the amplitude and frequency remain constant, and process the wave reflected by the target (see Figure 1). Besides information about the reflectability, they contain information about the target's velocity due to the Doppler frequency shift. A common variant of this technique is FMCW radar systems, whose waveforms vary in the time domain. With regard to HAR, FMCW-based radar systems in the mm-wave domain have significant advantages compared to CW radar, and their suitability for human sensing has been proven by numerous works in the last two decades [40; 41; 42; 46; 48; 50; 57]:  
* High sensitivity: For the detection of human motions, especially small-scale motions, e.g., breathing and gestures, a sensitivity close to the wavelength is required. This can be achieved when a high center frequency combined with a high bandwidth (\(B\)) is used. * Minimized risk of multipath propagation and interaction with nearby radar systems due to the high attenuation of the mm-wave RF signal. * Distances and velocities of targets can be measured simultaneously, e.g., when triangular modulation of the chirp signal combined with a related signal processing technique is used. * Thermal noise independence, as the phase is the main carrier containing information about the targets' distances. FMCW-based radar systems generate a sinusoidal power-amplified RF signal (chirp) through a high-frequency oscillating unit, where the frequency varies linearly between two  
Figure 1: Schematic representation of an FMCW-based radar sensor. values, \(f_{\rm min}\) and \(f_{\rm max}\), in a sawtooth-like pattern for a specific duration, \(T_{\rm r}\), according to the following function (see Figure 2):  
\[f_{\rm max}-f_{\rm min}=\frac{\mathrm{d}f}{\mathrm{d}t}\cdot T_{\rm r} \tag{1}\]  
The constant \(K=\mathrm{d}f/\mathrm{d}t=B/T_{\rm r}\) for \(0<t<T_{\rm r}\) determines the slope of the generated signal, whereas the frequency variation is determined by a linear function. This RF signal is emitted via the transmitting antenna, and the echo signal, which results from the scattered reflection of the electromagnetic waves on the objects, is received at the receiving antenna and is low noise-amplified. A mixer processes both the transmitted and received signals and generates a low-frequency beat signal, which, in the following, is preprocessed and used for the analysis. A linear chirp signal that can be defined within the interval \(0<t<T_{\rm r}\) by  
\[s_{\rm T}(t)=A_{\rm t}\mathrm{e}^{(2\pi f_{0}t+\pi Kt^{2})j} \tag{2}\]  
is emitted and mixed with its received echo signal to provide the IF signal  
\[s_{\rm IF}(t)=A_{\rm t}A_{\rm r}\mathrm{e}^{(2\pi f_{0}t_{\rm d}+2\pi Kt_{\rm d }t-\pi Kt_{\rm d}^{2})j} \tag{3}\]  
which, in the following, is preprocessed and used for the calculation of the feature maps. In general, human large-scale kinematics, e.g., the bipedal gait, are characterized by complex interconnected movements, mainly of the body and the limbs. While the limbs have oscillating velocity patterns, the torso can be characterized solely by transitional movements. Figure 2: Time-related characteristics of the chirp signal with sawtooth and triangular shape modulation. According to the Doppler effect, moving rigid-body targets induce a frequency shift in the carrier signal of coherent radar systems that is determined in its simplest form by  
\[f_{\rm D}=-\frac{2 vf_{\rm T}}{c} \tag{4}\]  
where \(v\) is the relative velocity between the source and the target and \(f_{\rm T}\) is the frequency of the transmitted signal. While the torso induces more or less constant Doppler frequency shifts, the limbs produce oscillating sidebands, which are referred to as _micro-Doppler signatures_[33]. In the joint time-frequency plane, these micro-Doppler signatures have distinguishable patterns, which make them suitable for ML-based classification applications.