## 3 Multi-Domain Representation of Human Activity Features  
### Clutter Suppression and RTM  
The data collected by IR-UWB radar are directly represented in range-time (RT) form, where the range dimension is the fast-time axis and the time dimension is the slow-time axis. At this stage, the data obtained are a raw RT numerical matrix without any processing. In through-wall scenarios, the raw RT matrix contains not only the target echo components but also the strong direct current clutter and multipath components caused by the wall ([PERSON] et al., 2018). To obtain an RTM that represents human activity features, clutter suppression is necessary. In non-through-wall radar activity recognition research, moving target indication (MTI) ([PERSON] et al., 2018) technology is widely used for static clutter suppression. However, under through-wall conditions, this method tends to excessively attenuate the target amplitude and is less effective at suppressing multipath components, resulting in a low signal-to-noise ratio after filtering. To better suppress wall-induced clutter, this paper employs an exponential weighting method, an improved version of the average background subtraction method ([PERSON] et al., 2018). The differences between MTI and the exponential weighting method will become more evident in the TDM representation in Section 2.2. Therefore, in Section 2.2, this paper will demonstrate how the exponential weighting method outperforms MTI under through-wall conditions. The average background subtraction method estimates the background of a detected area based on the echo signals. By subtracting the background estimate from the current echo, a filtered signal can be obtained. Let \(X\) denote the radar echo RT matrix, and denote the element in the \(m\)-th row (slow-time) and \(n\)-th column (fast-time) of \(X\). The background estimate \(Y\) can be expressed as [22]  
\[ \begin{split} y_{mn}&=\frac{1}{m}\sum_{i=1}^{m}x_{in} \\ &=\frac{1}{m}\left(\sum_{i=1}^{m-1}x_{in}+x_{mn}\right)\\ &=\frac{1}{m}((m-1)y_{m-1,n}+x_{mn})\\ &=\left(1-\frac{1}{m}\right)y_{m-1,n}+\frac{1}{m}x_{mn},\ i=1,2, \ldots,m.\end{split}  \tag{1}\]  
As can be seen from Equation (1), the current background estimate consists of the previous background estimate and the current echo. However, in through-wall environments, the radar echo background has significant fluctuations. To prevent the background from earlier times from excessively contributing to the current background, Equation (1) can be adjusted by adding a weighting factor \(\alpha\) to smooth the background estimate:  
\[ \begin{split} y_{mn}&=(1-\alpha)x_{mn}+\alpha y_{m- 1,n}\\ &=(1-\alpha)\left(x_{mn}+\alpha x_{m-1,n}+\cdots+\alpha^{m-2}x_{2, n}\right)+\alpha^{m-1}y_{1,n}.\end{split}  \tag{2}\]  
The method described in Equation (2) is known as the exponential weighting method. It is easy to observe that if \(\alpha\) is assigned a suitable fixed value, the contribution of the background from earlier times decreases as \(t\) increases. Using Equation (2), a radar echo matrix with suppressed clutter can be easily obtained:  
\[Z^{\prime}=X-Y. \tag{3}\]  
By applying range compensation to the \(Z^{\prime}\) matrix and displaying it as a heatmap, a range-time map (RTM) of human activity can be obtained. As shown in Figure 1, a RTM for nine different actions is illustrated. The purpose of range compensation is to enhance the distant features of the RTM, while relatively further suppressing near-range clutter. However, this suppression effect is only effective for RTMs, and the use of range compensation does not have a significant impact on TDMs, as discussed in Section 3.2 of this paper. Here, the range compensation can be expressed as  
\[z_{mn}=z^{\prime}_{mn}\cdot n^{k}, \tag{4}\]  
where \(z_{mn}\) represents the element in the \(m\)-th row and \(n\)-th column of the range-compensated human activity RT matrix. \(k\) is a constant and, in this paper, \(k\) is set to 2. The RTM represents the features of human activity in the range-time domain.