Decomposition of Multicomponent Micro-Doppler Signals Based on HHT-AMD

Micro-Doppler signals analysis has been emerging as an important topic in target identification, and relative research has been focusing on features extraction and separation of the radar signals. As a time-frequency representation, the Hilbert-Huang transform (HHT) could extract the accurate instantaneous micro-Doppler signature from the radar signals by empirical mode decomposition and Hilbert transform. However, HHT has the shortcoming that it cannot decompose the signals with close-frequency components. To solve this problem, an innovative decomposition method for multicomponent micro-Doppler signals based on Hilbert–Huang transform and analytical mode decomposition (HHT-AMD) is proposed. In this method, the multicomponent micro-Doppler signals are firstly decomposed by empirical mode decomposition, and the decomposed signal components are transformed by Hilbert transform to get the Hilbert-Huang spectrum and marginal spectrum. Through the spectrum processing, we get the frequency distribution of each signal component. The next step is to judge whether there exists frequency aliasing in each signal component. If there is aliasing, the AMD method is used to decompose the signal until all the decomposed signals are mono-component signals. Evaluation considerations are covered with numerical simulations and experiments on measured radar data. The results demonstrate that compared with conventional HHT, the proposed method yields accurate decomposition for multicomponent micro-Doppler signals and improves the robustness of decomposition. The method presented here can also be applied in various settings of non-stationary signal analysis and filtering.


Introduction
There has been lots of research focusing on the subject of micro-motion characteristics and application of micro-Doppler (m-D) features in radar systems over the past two decades [1][2][3][4]. The micro-Doppler effect was originally proposed by Zediker [5] in coherent laser systems, and then introduced to radar applications by Victor Chen [6]. In radar jargon, the micro-Doppler effect is the phenomenon of additional sidebands appearing around the central Doppler spectra of the target, which is highly related to its kinematics and geometric features [7]. Micro-motion and micro-Doppler phenomena can be observed everywhere in our lives and contribute to the fields of radar-based exploration and surveillance of humanity activities [8]. Its underlying civilian and military applications include security monitoring, urban warfare, law enforcement, healthcare, kinematics, search/rescue, anti-terrorism surveillance, and so on [9,10].
When a radar returned signal from a vibrating/rotating target is viewed in the frequency domain, its micro-Doppler shifts can be seen by their deviation from the center frequency of the radar returns. The Fourier transform is the most common method to analyze the properties of a signal waveform in the frequency domain. It shows the distribution of the magnitude and the phase at different frequencies contained in the signal during the time interval of analyzing. Although the frequency spectrum may indicate the presence of micro-Doppler shifts, and possibly the relative amount of displacement toward each side, because of the lack of time information, it is not easy to tell the vibration/rotation rate from the frequency spectrum alone. Therefore, the joint time-frequency analysis that provides more insight into localized time-dependent frequency information from the time-varying behavior of the signal is more useful and is complementary to the existing time-domain or frequency-domain methods.
The significant characteristic of micro-Doppler signals is the uniqueness because different micro-motions provide discriminative signatures [11]. Micro-Doppler parameters such as Doppler repetition period, Doppler amplitude, and initial phase reflect distinctive features of the target, which are useful for target classification and recognition. The time-varying trajectories of the m-D signals are revealing in the time-frequency (T-F) domain [12]. Conventionally, joint time-frequency analysis could display the instantaneous characteristic and exploit the time-varying signature, thus it has been widely used in micro-Doppler signals feature extraction [13,14]. However, there are some technical factors to restrain the performance of estimation algorithm for micro-Doppler analysis, e.g., the T-F resolution limitations, multicomponent signals decomposition, and the low ratio of energy in radar echoes [15]. To solve these problems, an approach based on the Hilbert-Huang transform (HHT) method was introduced considering that its distribution does not rely on convolution, avoiding kernel convoluting trouble in Cohen's class T-F representation [16].
After being proposed by Dr. N. E. Huang [17] in 1998, the HHT method has been used in various fields related with nonlinear and non-stationary signals such as seismic data analysis, speech analysis, and blood pressure signal analysis because of its high-resolution characteristic. The HHT method is capable of extracting the complicated signal components from the raw data based on empirical mode decomposition (EMD) [18]. The instantaneous frequencies can be extracted from the Hilbert-Huang spectrum ignoring interference of the frequency resolution and time resolution. Since the obtained instantaneous frequencies reveal exactly time-dependent characteristics of the micro-Doppler signatures, HHT was employed for micro-Doppler signals analysis and became an appropriate tool for analyzing m-D signatures [19,20]. However, considering the previous relative works, some shortcomings of the HHT were also found. The existence of end point effect limits the practical application of the HHT algorithm [21]. Besides, if there are close-frequency components in the signal, the HHT method cannot perform the characteristic frequency extraction effectively. Therefore, some supplementary processes are under investigation [22].
To solve the problems mentioned above, a new signal decomposition approach named Hilbert-Huang transform and analytical mode decomposition (HHT-AMD) for decomposing multicomponent micro-Doppler signals is put forward here. In this method, the potential components of micro-Doppler signals are obtained by the HHT firstly. If there is a frequency aliasing phenomenon, then AMD is applied to decompose the extracted signal into mono-component signals until all the signal components are obtained. Simulation studies and real dataset experiments show that the method can exactly decompose micro-Doppler signals with close-frequency components.
The remainder of this paper is arranged as follows. In Section 2, the micro-Doppler signal model is introduced. In Section 3, the concepts of HHT and AMD and the signal decomposition method based on HHT-AMD for multicomponent micro-Doppler signals is proposed. Typical simulation and experiment results are analyzed to verify the validity of HHT-AMD in Sections 4 and 5, respectively. Finally, concluding remarks are drawn in Section 6.

Micro-Doppler Signal Model
Micro-Doppler signals can be obtained over a variety of sensors, such as linear frequency-modulated (LFM) wave radar, ultra-wide band radar, and pulse Doppler radar [23,24]. To date, researchers have developed micro-Doppler signal models for specific targets. Take the human target as an example. We adopt the Boulic model to represent a man with the detailed geometry, as shown in Figure 1.
For simplicity, some typical point scattering centers of human body parts are selected to build the radar signal model: head, shoulder, torso, arm, hip, leg, and foot [20]. target as an example. We adopt the Boulic model to represent a man with the detailed geometry, as shown in Figure 1. For simplicity, some typical point scattering centers of human body parts are selected to build the radar signal model: head, shoulder, torso, arm, hip, leg, and foot [20]. The returned radar signals are composed of multiple sinusoidal frequency modulated (SFM) signal components expressed by Here, i σ represents the reflectivity of the i micro-motion scatterer and is usually assumed to be constant. 0 f and i β are the carrier frequency and the modulation index of the scattering.

sin(2 )
i m i i f t β π φ + denotes the Doppler frequency of the point scatterer. mi f and i φ are the oscillating frequency and initial phase angle, respectively. According to (1), the instantaneous frequency (m-D) of the i scatterer can be written as Note that the echoes of the target with micro-motions are multicomponent SFM signals and the instantaneous frequencies are sinusoids viewed in the T-F domain [25]. Therefore, m-D signals of target are the combination of sinusoids with distinct frequencies, initial phases, and amplitudes. It should be emphasized that the m-D signals of human body parts may not be purely sinusoids, which will be indicated in the experiment section.

Hilbert-Huang Transform (HHT)
The main concept of the Hilbert-Huang transform is to decompose the original signal into several intrinsic mode functions (IMFs) via EMD, and then the Hilbert spectrum is obtained by applying Hilbert transform on IMFs [26]. The HHT process includes two parts: EMD and Hilbert spectral analysis (HSA). The combination of EMD and HT has led to definitions of instantaneous frequency and amplitude, as well as a more physically meaningful description of the dataset.
Generally, EMD is a sifting process to obtain the IMFs from a given dataset [27]. For a given dataset ( ) s t , the first step is to identify the local extrema (maxima and minima) and connect them to produce the upper and lower envelopes, which cover the whole data. Their mean is  The returned radar signals are composed of multiple sinusoidal frequency modulated (SFM) signal components expressed by Here, σ i represents the reflectivity of the i micro-motion scatterer and is usually assumed to be constant. f 0 and β i are the carrier frequency and the modulation index of the scattering. β i sin(2π f mi t + φ i ) denotes the Doppler frequency of the point scatterer. f mi and φ i are the oscillating frequency and initial phase angle, respectively.
According to (1), the instantaneous frequency (m-D) of the i scatterer can be written as Note that the echoes of the target with micro-motions are multicomponent SFM signals and the instantaneous frequencies are sinusoids viewed in the T-F domain [25]. Therefore, m-D signals of target are the combination of sinusoids with distinct frequencies, initial phases, and amplitudes. It should be emphasized that the m-D signals of human body parts may not be purely sinusoids, which will be indicated in the experiment section.

Hilbert-Huang Transform (HHT)
The main concept of the Hilbert-Huang transform is to decompose the original signal into several intrinsic mode functions (IMFs) via EMD, and then the Hilbert spectrum is obtained by applying Hilbert transform on IMFs [26]. The HHT process includes two parts: EMD and Hilbert spectral analysis (HSA). The combination of EMD and HT has led to definitions of instantaneous frequency and amplitude, as well as a more physically meaningful description of the dataset.
Generally, EMD is a sifting process to obtain the IMFs from a given dataset [27]. For a given dataset s(t), the first step is to identify the local extrema (maxima and minima) and connect them to produce the upper and lower envelopes, which cover the whole data. Their mean is M 1 , and h 1 is given by the difference between M 1 and original data Since h 1 is not an IMF all the time, the sifting process should be performed iteratively. h 1 is regarded as the original data in the subsequent steps, The h 1j turns into an IMF until the fixed constrains are satisfied. Thus, h 1j is the first IMF component extracted from the data We separate im f 1 from the data, and the residual part is The residue r 1 may contain longer-period components, and it should be subjected to the above sifting process as the new data. The procedure should be repeated for all subsequent r m The process stops when no more IMF can be extracted and the residue r m is a monotonic function or a function with single extremum. Now, the original data is decomposed into M IMFs and a residue. In summary, s(t) can be reconstructed by the summation of all extracted IMFs and a final residue For IMFs, the Hilbert spectrum can be obtained by Hilbert transform. The instantaneous frequency of IMF is physical significant [17].
By integrating with respect to time, we get the marginal spectrum [28] h(ω) = T o H(ω, t)dt (10)

Analytical Mode Decomposition (AMD)
Chen [28] proposed a new signal decomposition method called analytical mode decomposition (AMD) in 2012, which can decompose the frequency aliasing signals with closely spaced frequency components into several single-frequency signal components.
Suppose s(t) is the temporal signal with n components. Its Fourier spectrum is S(w). s(t) can be written as In the spectral domain, each signal component has narrow-band character x i (t) = sin(ω bi t)H[s(t) cos(x i (ω bi t))] − cos(ω bi t)H[s(t) sin(x i (ω bi t))] (i = 1, . . . , n − 1) where x 0 (t) = 0, and H[ ] donates the Hilbert transform of the function [22]. Then the temporal sequence can be regarded as the sum of two signals Suppose ω b is a positive dimidiate frequency, then By denoting x c (t) = cos(ω b t), x s (t) = sin(ω b t), the Hilbert transform of x k (t)s(t) is According to Bedrosian theorem, the relation turns to be When k is evaluated to be c and s, respectively, we get The Hilbert transform of x c (t) and x s (t) can be expressed as Then we have Now, (19) and (20) turn to be According to (14), we can obtain Finally, we get two signals x 1 (t) and x 1 (t), which are decomposed from s(t) by AMD.

AMD Signal Extraction Algorithm
When the signals are not completely decomposed by Hilbert-Huang transform, the signal components are expected to be extracted by the filter method before processing AMD [22]. The AMD method can decompose the signal into the sum of any two signals. Suppose there is a time-series s(t) = s 1 (t) + s 2 (t) + · · · + s n (t), and its relevant frequencies are f 1 , f 2 · · · f n , which are incremental. If we want to extract the signal component with frequency f 1 , we should select a value ω k between f 1 and f 2 as the dimidiate frequency. If we want to extract the signal component with frequency f i , a value between the f i−1 and f i is used as the dimidiate frequency to decompose the first half of the signal; then a value between the f i and f i+1 is used as the dimidiate frequency to decompose the first half part of the signal. The signal component with frequency f i could be obtained by their subtraction. The schematic diagram is shown in Figure 2.

Signal Decomposition Algorithm
The signal decomposition algorithm combined HHT with AMD method is proposed to decompose multicomponent micro-Doppler signals. Firstly, EMD is applied to decompose the signal into several intrinsic mode functions. The frequency spectrum of the signal and the frequency values are obtained by Hilbert transform. Then, AMD method separates the signal component with frequency aliasing to single-frequency components. In detail, the algorithm process is described as the following steps.
Step 1: To decompose the original signals by EMD and get the time spectrum and marginal spectrum by Hilbert transform.
Step 2: The frequency components are obtained by searching the peak values of the spectrum. Thus, the signal components are extracted.
Step 3: To distinguish whether there is frequency aliasing phenomenon in each extracted signal.
Step 4: If there is frequency aliasing in a signal component, we can firstly use the frequency value as the dimidiate frequency to decompose it by AMD. By the analysis of the correlation coefficient, the frequency range turns narrow and the search step is reduced, until the best dimidiate frequency value is found, and the signal is successfully decomposed into two signals.
Step 5: To repeat the steps 2, 3, and 4 for the two signals, until there is no frequency aliasing for all the extracted signals.
The process of this algorithm is shown in Figure 3.

Signal Decomposition Algorithm
The signal decomposition algorithm combined HHT with AMD method is proposed to decompose multicomponent micro-Doppler signals. Firstly, EMD is applied to decompose the signal into several intrinsic mode functions. The frequency spectrum of the signal and the frequency values are obtained by Hilbert transform. Then, AMD method separates the signal component with frequency aliasing to single-frequency components. In detail, the algorithm process is described as the following steps.
Step 1: To decompose the original signals by EMD and get the time spectrum and marginal spectrum by Hilbert transform.
Step 2: The frequency components are obtained by searching the peak values of the spectrum. Thus, the signal components are extracted.
Step 3: To distinguish whether there is frequency aliasing phenomenon in each extracted signal.
Step 4: If there is frequency aliasing in a signal component, we can firstly use the frequency value as the dimidiate frequency to decompose it by AMD. By the analysis of the correlation coefficient, the frequency range turns narrow and the search step is reduced, until the best dimidiate frequency value is found, and the signal is successfully decomposed into two signals.
Step 5: To repeat the steps 2, 3, and 4 for the two signals, until there is no frequency aliasing for all the extracted signals.
The process of this algorithm is shown in Figure 3.

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 ≤ t < 1 s and sampled with ∆T = 1/512 s.
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 2 s and 3 s ) in the IMF1 signal, and there is monocomponent signal (representing 1 s ) 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.

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 ≤ t < 1 s and sampled with ∆T = 1/512 s. s(t) = exp(−j20 sin(5πt)) + exp(−j10 cos(10πt)) + exp(−j10 cos(12πt)) = s1 + s2 + s3 (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 signal, and there is monocomponent signal (representing s1) 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 π in s2 and 12 π in s3) are aliasing components which result in false decomposition by HHT.  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 π in s2 and 12 π in s3) are aliasing components which result in false decomposition by HHT. 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 (10π in s2 and 12π in s3) are aliasing components which result in false decomposition by HHT. 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 π and s3 with the characteristic frequency 12 π , respectively. Thus, all three components of the original signals were extracted after processing HHT-AMD, which validates the effectiveness of the proposed method.  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 10π and s3 with the characteristic frequency 12π, respectively. Thus, all three components of the original signals were extracted after processing HHT-AMD, which validates the effectiveness of the proposed method. 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 π and s3 with the characteristic frequency 12 π , respectively. Thus, all three components of the original signals were extracted after processing HHT-AMD, which validates the effectiveness of the proposed method.

Comparison with the Conventional HHT
Aiming at investigating the superiority of the proposed algorithm, comparison between conventional HHT and HHT-AMD in decomposing multicomponent micro-Doppler signals is provided [28]. The considered signals are s(t) = exp(−j cos( f 1 t)) + exp(−j cos( f 2 t)) = s1 + s2. The decompose success rate (DSR) is considered here, as a function of close-frequency ratio. The DSR represents the percentage of the successful cases decomposing close-frequency components in the all simulation tries. The close-frequency ratio is the ratio of the two close frequencies f 1 and f 2 (for example, the ratio of two frequencies 10 and 15 is 1.5). One-hundred Monte-Carlo-type simulations were carried out for a set of ratios 1.1~2. The DSRs of HHT-AMD were compared with those of conventional HHT, which were plotted in percentage, see Figure 8.
From Figure 8, the DSRs of HHT-AMD were continually higher than those of HHT. There are still two main points to notice. On one hand, when the close-frequency ratios are lower than 1.5, the DSRs of HHT methods were in the low level, which means the decompose process was problematic. Meanwhile, the DSRs of HHT-AMD were obviously higher than those of HHT. On the other hand, when close-frequency ratios were higher than 1.5, the DSRs of HHT and HHT-AMD were considerable and acceptable. Consequently, there was a threshold effect at 1.5 of close-frequency ratio. Thus, HHT-AMD method showed its superiority over HHT method on decomposing close-frequency components especially when the close-frequency ratio was under 1.5.

Comparison with the Conventional HHT
Aiming at investigating the superiority of the proposed algorithm, comparison between conventional HHT and HHT-AMD in decomposing multicomponent micro-Doppler signals is provided [28]. The considered signals are 1 2 ( ) exp( cos( )) exp( cos( )) 1 2 s t j ft j ft s s = − + − = + . The decompose success rate (DSR) is considered here, as a function of close-frequency ratio. The DSR represents the percentage of the successful cases decomposing close-frequency components in the all simulation tries. The close-frequency ratio is the ratio of the two close frequencies 1 f and 2 f (for example, the ratio of two frequencies 10 and 15 is 1.5). One-hundred Monte-Carlo-type simulations were carried out for a set of ratios 1.1~2. The DSRs of HHT-AMD were compared with those of conventional HHT, which were plotted in percentage, see Figure 8. From Figure 8, the DSRs of HHT-AMD were continually higher than those of HHT. There are still two main points to notice. On one hand, when the close-frequency ratios are lower than 1.5, the DSRs of HHT methods were in the low level, which means the decompose process was problematic. Meanwhile, the DSRs of HHT-AMD were obviously higher than those of HHT. On the other hand, when close-frequency ratios were higher than 1.5, the DSRs of HHT and HHT-AMD were considerable and acceptable. Consequently, there was a threshold effect at 1.5 of close-frequency ratio. Thus, HHT-AMD method showed its superiority over HHT method on decomposing close-frequency components especially when the close-frequency ratio was under 1.5.

Influence of the SNR
Signal-to-noise ratio (SNR) is a significant environmental factor in the signal processing [30]. In some cases, the micro-Doppler feature might not be presented in a recognizable and correct form because of the severe noise or weak modulation problem. In order to simulate a practical situation and understand the effect of noise on the proposed method, Gaussian white noise is added to the simulated signals here. Simulations were repeated for SNR of 0~30 dB. Suppose the signals are ( ) exp( 20sin(10 )) exp( 10 cos(15 )) ( ) s t j t j t nt π π = − + − + over time interval 0 ≤ t < 1 s and sampled with ∆T = 1/256 s, where ( ) n t is Gaussian white noise with variance 2 n σ . The DSR is considered, as a function of SNR plotted in decibel, see Figure 9.

Influence of the SNR
Signal-to-noise ratio (SNR) is a significant environmental factor in the signal processing [30]. In some cases, the micro-Doppler feature might not be presented in a recognizable and correct form because of the severe noise or weak modulation problem. In order to simulate a practical situation and understand the effect of noise on the proposed method, Gaussian white noise is added to the simulated signals here. Simulations were repeated for SNR of 0~30 dB. Suppose the signals are s(t) = exp(−j20 sin(10πt)) + exp(−j10 cos(15πt)) + n(t) over time interval 0 ≤ t < 1 s and sampled with ∆T = 1/256 s, where n(t) is Gaussian white noise with variance σ 2 n . The DSR is considered, as a function of SNR plotted in decibel, see Figure 9.  Figure 9 describes the comparison of DSRs between HHT-AMD and HHT under varying SNRs. As we can see, when the SNR increases, the DSR increases. The DSRs of HHT are comparatively at a low level, which means that IMFs of the signals become corrupted, resulting in a loss of information. The DSRs of HHT-AMD are relatively higher than those of HHT all the time. A threshold effect appears at 12 dB SNR for the HHT and 8 dB SNR for the HHT-AMD. The increasing speed of DSRs of HHT-AMD is slower than that of HHT, which verifies the stability and the superior performance of proposed algorithm against noise. The results prove the robustness of HHT-AMD in decomposing multicomponent micro-Doppler signals corrupted by noisy circumstance as in a real environment.

Selection of the Dimidiate Frequency and Searching Step Length
As aforementioned analysis in Section 3.4, the search range, dimidiate frequency and search step play important roles in the HHT-AMD process. In order to investigate the influence of these parameters on the algorithm, the simulated signals ( ) exp( 10sin (20 )) exp( 20 cos(25 )) ( ) s t j t j t nt π π = − + − + are used here. After HHT, we get the cursory components of the signals. Based on the decomposition results by HHT, the frequency range of the search is set to be 5 π~50 π initially for the signals. We choose a set of step lengths [1][2][3][4][5] and the dimidiate frequency is set to be 25 initially. The search step length should be determined according to the frequency range. The larger step length may be used firstly, and then the small step length is used during searching process gradually. Dynamic simulations are tested for different combination of the three parameters, respectively. The relationship of these parameters is utilized to analyze the performance of HHT-AMD. The results of the measurements are shown in Table 1. It is clear that the different parameter selections lead to different results of decomposition. Generally, large search range is applied at first, and it tends smaller during the process. The dimidiate frequency should be modified according to the change of the search range. When the search range becomes smaller and the dimidiate frequency is fixed, the search step should also decrease. Therefore, suitably chosen parameters yield good performance of HHT-AMD to decompose multicomponent micro-Doppler signals.  As we can see, when the SNR increases, the DSR increases. The DSRs of HHT are comparatively at a low level, which means that IMFs of the signals become corrupted, resulting in a loss of information. The DSRs of HHT-AMD are relatively higher than those of HHT all the time. A threshold effect appears at 12 dB SNR for the HHT and 8 dB SNR for the HHT-AMD. The increasing speed of DSRs of HHT-AMD is slower than that of HHT, which verifies the stability and the superior performance of proposed algorithm against noise. The results prove the robustness of HHT-AMD in decomposing multicomponent micro-Doppler signals corrupted by noisy circumstance as in a real environment.

Selection of the Dimidiate Frequency and Searching Step Length
As aforementioned analysis in Section 3.4, the search range, dimidiate frequency and search step play important roles in the HHT-AMD process. In order to investigate the influence of these parameters on the algorithm, the simulated signals s(t) = exp(−j10 sin(20πt)) + exp(−j20 cos(25πt)) + n(t) are used here. After HHT, we get the cursory components of the signals. Based on the decomposition results by HHT, the frequency range of the search is set to be 5π~50π initially for the signals. We choose a set of step lengths [1][2][3][4][5] and the dimidiate frequency is set to be 25 initially. The search step length should be determined according to the frequency range. The larger step length may be used firstly, and then the small step length is used during searching process gradually. Dynamic simulations are tested for different combination of the three parameters, respectively.
The relationship of these parameters is utilized to analyze the performance of HHT-AMD. The results of the measurements are shown in Table 1. It is clear that the different parameter selections lead to different results of decomposition. Generally, large search range is applied at first, and it tends smaller during the process. The dimidiate frequency should be modified according to the change of the search range. When the search range becomes smaller and the dimidiate frequency is fixed, the search step should also decrease. Therefore, suitably chosen parameters yield good performance of HHT-AMD to decompose multicomponent micro-Doppler signals.

Experimental Data
In the operative system, the micro-Doppler signals are usually embedded in low signal-to-clutter ratio (SCR) and we should consider how to extract the features from the radar echo [31,32]. In this section, an experiment was conducted to investigate the effectiveness and physical significance of decomposing micro-Doppler signals based on HHT-AMD.
Experimental and simulated studies on micro-Doppler signatures of a human walking have been previously reported in literature. For humans observed by radar, micro-Doppler signatures are related to the motion of various parts of the body while a person is walking, running, or performing other movements. We take walking as a typical example of human motion that is comprised of many movements of individual parts.
The measured radar echoes from a walking man with swinging arms are provided for characteristic analysis of micro-Doppler signals. The setup of the experiment can be seen in Figure 10a. Experimental testing was conducted with a human walking towards the radar to collect the return signals using an X-band CW (continuous wave) radar operating at 10.48 GHz. A sampling rate of 100 kHz was used to collect 100,000 samples in the time interval of 1 s. The SNRs are ranging from 10 dB to 40 dB related to the range between man and radar. The time-frequency signature of the radar echo is given using STFT in Figure 10b. The T-F distribution consists of the trajectories associated with different micro-motion components: the head, torso, arms, hip, legs, and foot from up to down.

Experimental Data
In the operative system, the micro-Doppler signals are usually embedded in low signal-to-clutter ratio (SCR) and we should consider how to extract the features from the radar echo [31,32]. In this section, an experiment was conducted to investigate the effectiveness and physical significance of decomposing micro-Doppler signals based on HHT-AMD.
Experimental and simulated studies on micro-Doppler signatures of a human walking have been previously reported in literature. For humans observed by radar, micro-Doppler signatures are related to the motion of various parts of the body while a person is walking, running, or performing other movements. We take walking as a typical example of human motion that is comprised of many movements of individual parts.
The measured radar echoes from a walking man with swinging arms are provided for characteristic analysis of micro-Doppler signals. The setup of the experiment can be seen in Figure  10a. Experimental testing was conducted with a human walking towards the radar to collect the return signals using an X-band CW (continuous wave) radar operating at 10.48 GHz. A sampling rate of 100 kHz was used to collect 100,000 samples in the time interval of 1 s. The SNRs are ranging from 10 dB to 40 dB related to the range between man and radar. The time-frequency signature of the radar echo is given using STFT in Figure 10b. The T-F distribution consists of the trajectories associated with different micro-motion components: the head, torso, arms, hip, legs, and foot from up to down. The human data was decomposed into seven IMFs and the residue by EMD. The TFD of the first three IMFs are shown in Figure 11a-c. We can see that, IMF1 signal shows the micro-Doppler signature of foot motion. The IMF2 and IMF3 signals have undistinguishable multiple components. The human data was decomposed into seven IMFs and the residue by EMD. The TFD of the first three IMFs are shown in Figure 11a-c. We can see that, IMF1 signal shows the micro-Doppler signature of foot motion. The IMF2 and IMF3 signals have undistinguishable multiple components. Work was done to further decompose the signals of IMF2 and IMF3 by HHT-AMD, and the results are presented in Figures 12 and 13. As expected, the IMF2 signals are decomposed into two signal components which represent the micro-Doppler signatures of the arms and legs motion. Furthermore, the motion directions of the arms and legs are opposite as apparently exhibited in the TFD. The IMF3 signals are decomposed into three signal components which are corresponding to the micro-Doppler signatures of the rest body parts motion of head, torso, and hip, respectively Work was done to further decompose the signals of IMF2 and IMF3 by HHT-AMD, and the results are presented in Figures 12 and 13. As expected, the IMF2 signals are decomposed into two signal components which represent the micro-Doppler signatures of the arms and legs motion. Furthermore, the motion directions of the arms and legs are opposite as apparently exhibited in the TFD. The IMF3 signals are decomposed into three signal components which are corresponding to the micro-Doppler signatures of the rest body parts motion of head, torso, and hip, respectively. (a) (b) (c) Figure 11. Decomposition results of the HHT: (a-c) TFD of IMF1-IMF3.
Work was done to further decompose the signals of IMF2 and IMF3 by HHT-AMD, and the results are presented in Figures 12 and 13. As expected, the IMF2 signals are decomposed into two signal components which represent the micro-Doppler signatures of the arms and legs motion. Furthermore, the motion directions of the arms and legs are opposite as apparently exhibited in the TFD. The IMF3 signals are decomposed into three signal components which are corresponding to the micro-Doppler signatures of the rest body parts motion of head, torso, and hip, respectively

Conclusions
The method for decomposing multicomponent micro-Doppler signals based on AMD-HHT is proposed in this paper. This method solves the problem that Hilbert-Huang transform cannot identify and separate the signal with close-frequency components. The process is as follows. Firstly, the signal is decomposed by EMD, and Hilbert-Huang spectrum and marginal spectrum are obtained by Hilbert transform. If there is frequency aliasing in the decomposed signal, the close-frequency components are separated by AMD. A set of simulations and the experimental testing on radar echo are used to assist in the interpretation of the method analytically. By Work was done to further decompose the signals of IMF2 and IMF3 by HHT-AMD, and the results are presented in Figures 12 and 13. As expected, the IMF2 signals are decomposed into two signal components which represent the micro-Doppler signatures of the arms and legs motion. Furthermore, the motion directions of the arms and legs are opposite as apparently exhibited in the TFD. The IMF3 signals are decomposed into three signal components which are corresponding to the micro-Doppler signatures of the rest body parts motion of head, torso, and hip, respectively

Conclusions
The method for decomposing multicomponent micro-Doppler signals based on AMD-HHT is proposed in this paper. This method solves the problem that Hilbert-Huang transform cannot identify and separate the signal with close-frequency components. The process is as follows. Firstly, the signal is decomposed by EMD, and Hilbert-Huang spectrum and marginal spectrum are obtained by Hilbert transform. If there is frequency aliasing in the decomposed signal, the close-frequency components are separated by AMD. A set of simulations and the experimental testing on radar echo are used to assist in the interpretation of the method analytically. By

Conclusions
The method for decomposing multicomponent micro-Doppler signals based on AMD-HHT is proposed in this paper. This method solves the problem that Hilbert-Huang transform cannot identify and separate the signal with close-frequency components. The process is as follows. Firstly, the signal is decomposed by EMD, and Hilbert-Huang spectrum and marginal spectrum are obtained by Hilbert transform. If there is frequency aliasing in the decomposed signal, the close-frequency components are separated by AMD. A set of simulations and the experimental testing on radar echo are used to assist in the interpretation of the method analytically. By comparing with the conventional HHT, the proposed method is capable of yielding correct decomposition of the effective data. The work presented here extends the prior work in this area. The main limitation of the method is the optimal selection problem of the parameters, which will be further studied and reported in a separate paper in the near future.
Author Contributions: All authors discussed the contents of the manuscript. W.L. conceived the experiments, conducted the theoretical analyses, and wrote the paper; W.L. and B.X. provided the experimental equipment and analysis tools; B.X. and G.K. revised the manuscript.