## 3 Multi-Domain Representation of Human Activity Features
### Time-Frequency Transformation and TDMs  
In reality, we cannot transmit complex-valued signals; thus, the values in the RT matrix \(Z\) obtained in the previous section are real-valued. To obtain a micro-Doppler feature map of human movements, performing time-frequency analysis on real-valued signals alone cannot capture negative frequency information. This limitation prevents us from intuitively understanding the physical meaning of the time-frequency map. Therefore, a Hilbert transform is applied to each pulse of \(Z\), as follows  
\[\hat{z}_{in}=\text{Hilbert}(z_{in})=z_{in}*\frac{1}{\pi t},\ i=1,2,\ldots,m. \tag{5}\]  
Thus, we obtain the matrix \(\hat{Z}\). Let  
\[\hat{Z}=Z+j\hat{Z}. \tag{6}\]  
At this point, \(\hat{Z}\) represents the analytic signal of \(Z\), which is the complex-valued RT matrix we expect.  
The short time Fourier transform (STFT) is simple and efficient for time-frequency analysis. According to the method in [23], the STFT is first performed on each range cell, and then the results are summed. This approach makes the generated time-frequency map very smooth:  
\[s_{tu}=\sum_{i=1}^{n}\text{STFT}\big{[}\hat{z}_{ji}\big{]}=\sum_{i=1}^{n}\sum_{ j=1}^{m}\hat{z}_{ji}w(m-t)e^{-j\omega m}, \tag{7}\]  
Figure 1: RTM of nine different actions. From left to right and top to bottom, the actions represented are walking, running, punching, kicking, stepping, bending, sitting, squatting, and standing up.  
where \(w\) is the window function, typically chosen to have localization properties; in this paper, a Hanning window is used. \(t\) is the time index, used to determine the position of the window function in the signal. \(\omega\) is the frequency variable, which can be expressed as \(2\pi k/N\), where \(k\) is the frequency index and \(N\) is the FFT length. By processing the RT matrix using Equations (5)-(7), a TD matrix can be obtained. As shown in Figure 2, a TDM for nine different actions is illustrated.  
Compared to RTMs, a TDM, which represents micro-Doppler features, often contains richer information about human activities. In addition to their overall movement, a human target also involves various joint flexion and extension, adduction and abduction, and internal and external rotation motions, as well as small vibrations of the body surface. Together, these constitute more complex micro-movements, which introduce additional frequency modulation into the radar echo signal. These additional frequency components are called micro-Doppler frequencies and have an advantage in representing more detailed features. Compared to the actions that are relatively difficult to distinguish in Figure 1, the feature representation in Figure 2 shows more obvious distinctions.  
In Figure 3, we selected the time-frequency map of the punching action to compare the clutter suppression effects of MTI and exponential weighting. It is evident that the exponential weighting method is superior to MTI in terms of clutter suppression.  
Figure 2: TDM of nine different actions. From left to right and top to bottom, the actions represented are walking, running, punching, kicking, stepping, bending, sitting, squatting, and standing up.