# mmHSE: A Two-Stage Framework for Human Skeleton Estimation Using mmWave FMCW Radar Signals
## 3 Radar Data Acquisition and Processing  
### Operating Principle of FMCW Radar  
The mmWave FMCW radar system transmits a continuous wave signal with linearly increasing frequency, commonly referred to as a chirp signal. As shown in Figure 2a,b, the transmitted signal can be expressed as  
\[s_{T}(t)=A\cos\left(2\pi f_{c}t+\pi\mu t^{2}+\phi_{0}\right), \tag{1}\]  
where \(A\) denotes the amplitude, \(f_{c}\) is the initial frequency, \(\mu=\frac{BW}{I_{m}}\) is the chirp rate with bandwidth \(BW\) and chirp duration \(T_{m}\), and \(\phi_{0}\) is the initial phase.  
When the signal reflects from a target at range \(R\) moving with relative velocity \(v\), it experiences a time delay  
\[\tau(t)=\frac{2(R+vt)}{c}, \tag{2}\]  
where \(c\) is the speed of light. The received signal is then given by  
\[s_{R}(t)=A\cos\left(2\pi f_{c}(t-\tau(t))+\pi\mu(t-\tau(t))^{2}+\phi_{0}\right). \tag{3}\]  
Figure 1: The architecture of the mmHSE system, consisting of three core modules: data acquisition, data processing, and the two-stage network.  
By mixing the transmitted and received signals, an IF signal is generated with instantaneous frequency  
\[f_{\rm IF}(t)=\mu\tau(t)=\frac{2\mu(R+vt)}{c}. \tag{4}\]  
The resulting IF signal is  
\[s_{\rm IF}(t)=A\cos(2\pi f_{\rm IF}(t)t+\phi_{\rm IF}), \tag{5}\]  
where \(\phi_{\rm IF}\) represents the phase accumulated during the mixing process.  
In practical systems, a sequence of chirps is transmitted, and the received signals are sampled by multiple antenna elements. For the \(i\)-th chirp, the \(m\)-th antenna element, and the \(n\)-th ADC sample, the sampled IF signal can be modeled as  
\[s_{\rm IF}(i,m,n)=A\cos(2\pi(f_{\rm D}iT_{m}+f_{\rm R}nT_{s}+f_{\rm\theta}m)+ \phi_{\rm total}), \tag{6}\]  
where \(f_{\rm D}=\frac{2\pi f_{\rm c}}{c}\) is the Doppler frequency, \(f_{\rm R}=\frac{2\mu R}{c}\) is the range frequency, The angular frequency formula assumes the use of a linear antenna array, as is the case with the IWR1642 BOOST and IWR6843 ISK radar modules, which employ linear antenna arrays. Here, \(T_{\rm s}=\frac{T_{m}}{N_{\rm s}}\) is the ADC sampling interval with \(N_{\rm s}\) samples per chirp, \(l\) is the antenna spacing, as shown in Figure 2c. The angle \(\theta\) represents the angle between the target's line of sight and the normal to the antenna array. The positive and negative values of this angle are defined relative to the orientation of the antenna array. In the case of a linear array, this angle is measured from the axis perpendicular to the array. This measurement method is illustrated in Figure 2d, where the angle is calculated geometrically through \(f_{\rm\theta}=\frac{f_{\rm c}l\sin\theta}{c}\) based on the relative positions of the radar and the target. The term \(\phi_{\rm total}\) aggregates constant phase shifts, which include any phase contributions from the system hardware, antenna alignment, or signal propagation characteristics.  
Following the signal acquisition, the radar data are organized into a 3D tensor--commonly referred to as a Radar Data Cube (RDC)--with dimensions corresponding to chirps (slow time), ADC samples (fast time), and antenna channels. Specifically, each entry \(s_{\rm IF}(i,m,n)\) denotes the IF signal sampled at the \(n\)-th fast-time index (within chirp), from the \(m\)-th antenna, during the \(i\)-th chirp.  
To extract meaningful features such as range, velocity, and angle, a sequence of 1D or 2D Fourier Transforms is applied along appropriate dimensions of the RDC: first, along the fast-time axis (for range), then slow-time (for Doppler/velocity), and finally, across antenna elements (for angle). This staged transformation produces a 3D frequency-domain representation suitable for downstream tasks such as detection, tracking, and pose estimation.  
Figure 2: The working principle of the mmWave FMCW radar.