# Decomposition of Multicomponent Micro-Doppler Signals Based on HHT-AMD
## 4 Simulation Studies and Performance Evaluation  
### Example  
In this section, a set of simulations were carried out in order to better understand the expected performance of the proposed HHT-AMD algorithm for micro-Doppler signals decomposition. The micro-Doppler signals considered here are synthetic data of a walking man model, which is represented by three typical individual parts with different frequencies [29]. The simulated signals are SFM signals over time interval \(0\leq t<1\) s and sampled with \(\Delta\)T = 1/512 s.  
\[s(t)=\exp(-j20\sin(5\pi t))+\exp(-j10\cos(10\pi t))+\exp(-j10\cos(12\pi t))=s1 +s2+s3 \tag{27}\]  
To reveal the time-frequency characteristic of the extracted micro-Doppler signal components, the short time Fourier transform (STFT) is adopted here. The waveform and the Fourier spectrum of the signal and its time-frequency distribution (TFD) obtained by STFT are shown in Figure 4a-c respectively. We can see from Figure 4b,c that, although the frequency spectrum may indicate the presence of micro-Doppler shifts and possibly the relative amount of displacement toward each side, but we cannot know how the frequencies are varying with time. The time-varying trajectories of the m-D signals are revealing in the time-frequency (T-F) domain, but three components of the signals are not able to recognize. By EMD, the original signals are decomposed into seven IMFs from the high-frequency signal to the low-frequency signal and the residue. We discard the residue as it contains lowest frequencies of the original signals. The Hilbert-Huang spectrums (HH spectrum) of IMF1 and IMF2 obtained by HHT method are demonstrated in Figure 5a,b, and the TFDs of IMF1 and IMF2 are demonstrated in Figure 5c,d, respectively. Viewed in the T-F domain, the IMF2 signal captures clearly the sinusoid curve of the micro-Doppler signature, while the IMF1 signal does not. The main underlying significance in the IMFs is visibly obvious: there are multicomponent signals (representing \(s2\) and \(s3\)) in the IMF1  
Figure 3: Flow chart of signal decomposition. signal, and there is monocomponent signal (representing \(s\)1) in the IMF2 signal. For IMF1 signal, the TFD is confused, which means that the two micro-Doppler signals are hard to recognize and separate. The Hilbert spectral analysis is given in the following. After applying Hilbert transform, the Hilbert-Huang spectrum of the signals is shown in Figure 6a, as well as the marginal spectrum in Figure 6b. By finding the peak values in the marginal spectrum, the frequency of each component can be obtained. From Figure 6b, we observe that there are two frequency components in the decomposition result, but the original signals contain three components. This is because the two frequencies (10\(\pi\) in s2 and 12\(\pi\) in s3) are aliasing components which result in false decomposition by HHT. Figure 4: The simulated signal: (**a**) waveform; (**b**) Fourier spectrum; (**c**) STFT (short time Fourier transform). Figure 5: Decomposition results of HHT (Hilbert–Huang transform): (**a,b**) HH (Hilbert–Huang) spectrum of IMF1 (intrinsic mode function 1) and IMF2; (**c,d**) TFD (time-frequency distribution) of IMF1 and IMF2. Now, the proposed procedure is used to producing satisfactory results. The AMD was performed to extract the signals of frequency-aliasing components. For the signal extracted from IMF1, the estimated frequency can be used as the dimidiate frequency to observe the decomposition results. Figure 7 shows the AMD decomposition results (named AMD1 and AMD2) of the IMF1 signal. It can be seen from the graphs that, the upper subplots and the lower subplots reveal the components of s2 with the characteristic frequency \(10\pi\) and s3 with the characteristic frequency \(12\pi\), respectively. Thus, all three components of the original signals were extracted after processing HHT-AMD, which validates the effectiveness of the proposed method. Figure 6: Hilbert spectral analysis: (**a**) Hilbert-Huang spectrum; (**b**) the marginal spectrum.