## 2 Methods
### FMCW Radar Signal Processing  
To extract the distance and velocity information contained in the received signal described in (3) using an FMCW radar, a signal processing procedure is required. First, a Range FFT is performed along the \(n\)-dimension of the data in (3), which corresponds to the fast time (sampling within a single chirp). By removing symmetric components, the result becomes (6), where \(n\) represents information about distance \(r\). Next, a Velocity FFT is performed along the \(m\)-dimension, corresponding to the slow time (across chirps), resulting in (7). Here, \(m\) represents information about velocity \(v\). By accumulating the results from these two FFT processes, a three-dimensional matrix is obtained, representing distance, velocity, and time. Figure 1: TX and RX signal of FMCW radar. Figure 2: Functional block diagram of FMCW radar. \[X(r,m,t)=\sum_{n=0}^{N-1}x(n,m,t)\cdot e^{-j2\pi f_{n}} \tag{6}\]  
\[Z(r,v,t)=\sum_{m=0}^{N-1}X(r,m,t)\cdot e^{-jvm} \tag{7}\]  
To observe the distance and velocity information of a moving object over time, the range map \(R(r,t)\) and the Doppler map \(D(v,t)\) are defined as follows. \[R(r,t)=\sum_{v}|Z(r,v,t)| \tag{8}\]  
\[D(v,t)=\sum_{r}|Z(r,v,t)| \tag{9}\]  
The range map represents distance information over time, where the corresponding values indicate the signal strength. If there is a moving object at a certain distance, the signal strength appears strong; otherwise, it appears weak. The Doppler map represents velocity information over time, and like the range map, its values also indicate signal strength. Figure 3 shows the range map and Doppler map for a walking motion from approximately 1 m to 5.5 m.  
The Doppler map contains velocity information for all moving objects. During walking, different parts of the body--Spine, Leg, and Foot--move at different speeds, which can be verified using motion sensors (Model: Perception Neuron Studio, Noitom). As shown in Figure 4, which presents the results obtained from the motion sensor, Foot moves the fastest (approximately 3 m/s), followed by the Leg (approximately 2 m/s), while Spine moves the slowest (approximately 1 m/s). Figure 4: Velocity by body part measured using a motion sensor: (**a**) Spine, (**b**) Leg, (**c**) Foot. Figure 3: During walking: (**a**) range map and (**b**) Doppler map. In the Doppler map shown in Figure 3, various patterns reflecting differences in body part velocities are present, and the speeds corresponding to Foot, Leg, and Spine can be visually distinguished. Therefore, by analyzing these distinct patterns, it is possible to extract the velocities of individual body parts. For gait analysis, it is essential to extract multiple Doppler envelopes that represent the velocity variations of each body part. In previous studies, percentile-based methods have been used to extract Doppler envelopes [16; 17; 18; 19]. To remove noise from the Doppler map in (7), a denoising process is applied, as described in (8) and (9), resulting in an improved Doppler map \(P(v,t)\). Figure 5 shows the denoised Doppler map \(P(v,t)\), based on Figure 3b. \[\overline{D}(v,t)=D(v,t)-\mu(t),\quad\mu(t)=\mathbb{E}[D(t,\cdot)] \tag{10}\]  
\[P(v,t)= \begin{cases}0,&\overline{D}(v,t)<\max(\overline{D}(v,t))\cdot\alpha\\ \overline{D}(v,t),&\text{otherwise}\end{cases}  \tag{11}\]  
In conventional gait analysis, a probability density function (PDF) of the velocity distribution in the Doppler map is first computed. Then, specific percentiles are manually selected to extract Doppler envelopes that represent the velocity information of individual body parts. For example, the velocities corresponding to the toe, ankle, and knee are experimentally chosen at the 95 th, 75 th, and 50 th percentiles, respectively [12].