**3. Classification Algorithm Based on the Weighted Features Fusion of Multi-Wave Gates**

By analyzing the characteristics of the helicopter, propeller aircraft and jet aircraft in time and frequency domains, we can classify three kinds of aircraft targets through the micro-Doppler effect caused by rotating parts due to the difference in structure and rotating speed.

#### *3.1. Features Extraction*

According to the difference of echoes in time domain and frequency domain, this paper classifies three types of aircraft targets by extracting amplitude deviation coefficient, time domain waveform entropy and frequency domain waveform entropy. Three feature extraction methods are described as follows:

#### 1. Amplitude deviation coefficient

The amplitude deviation coefficient *gy* of the discrete echo signal **Y** = / *yi* 0 , *i* = 1, 2, ··· , *M* reflects the proportional relationship between the rotating parts of an airplane target and its fuselage, which can be defined as:

$$\mathbf{g}\_y = \sigma\_y / \bar{\mathbf{Y}},\tag{11}$$

where *gy* denotes the amplitude deviation coefficient, <sup>σ</sup>*<sup>y</sup>* <sup>=</sup> *<sup>M</sup>* % *i*=1 (*yi* <sup>−</sup> **¯ Y**) 2 /(*M* − 1) is the variance of

echo amplitude, **¯ <sup>Y</sup>** <sup>=</sup> *<sup>M</sup>* % *i*=1 *yi*/*M* is the mean of echo amplitude, *M* is the length of the echo signal.

Generally speaking, the higher the complexity of the target structure, such as the helicopter and propeller aircraft, the larger the proportion of the micro-Doppler modulation component of the rotating parts to the radar echo, the greater the overall fluctuation of the echo amplitude and the larger the amplitude deviation coefficient of the echo.

#### 2. Waveform entropy

Waveform entropy is usually used to describe the waveform characteristics of radar echo signals. From the analysis of Section 2, it can be seen that there are differences in the blade's number, length and rotating speed of the rotating parts in helicopter, propeller aircraft and jet aircraft, so the micro-Doppler effect of the rotating parts is different in the echo waveform. Therefore, we can distinguish the difference of waveform between different targets by extracting waveform entropy in time domain and frequency domain.

Time domain waveform entropy *Et* and frequency domain waveform entropy *Ef* of echo signal are defined as follows:

$$E\_t = -\sum\_{i=1}^{M'} p\_i \log\_{10}(p\_i),\tag{12}$$

$$E\_f = -\sum\_{i=1}^{M'} q\_i \log\_{10}(q\_i) \tag{13}$$

where *pi* = *yi*/ *M* % *j*=1 *yj* and *qi* = *fi*/ *M* % *j*=1 *fj* are normalized signals in time domain and frequency domain respectively, **F** = / *fi* 0 , *i* = 1, 2, ··· *M* is the result of fast Fourier transform (FFT) of echo signal **Y**.

In this paper, three kinds of aircraft target echo models are established, and the time domain and frequency domain echoes of targets are simulated according to the parameters of the rotor in Table 1. Then, the amplitude deviation coefficient, time domain waveform entropy and frequency domain waveform entropy are extracted. We simulate 200 sparse echo signal samples of three types of aircraft targets respectively from different radar perspectives where one sparse echo signal sample corresponds to one observation angle which denotes the relationship between the aircraft target's flying direction and the radar's line of sight, and it changes uniformly from 0◦ to 360◦ at an interval of 1.8◦ . Therefore, the angle varies from different samples, and the dataset includes 600 samples in all. The results of features extracted from 600 sparse echo signal samples are shown in Figure 7, where the signal to noise ratio (SNR) of target echo before pulse compression is −13dB, which is defined as SNR = **Y** 2 2/(*M* σ2), where σ<sup>2</sup> is the variance of noise.

**Figure 7.** Results of extracted features: (**a**) amplitude deviation coefficient; (**b**) frequency domain waveform entropy; (**c**) time domain waveform entropy.

As can be seen from the above figure, in the case of SNR it is −13 dB, because of the difference among the rotating parts of three types of aircraft targets, the amplitude deviation coefficient, time-domain waveform entropy and frequency-domain waveform entropy are different among targets. Taking the amplitude deviation coefficient as an example, it can be seen from Figure 7a that the amplitude deviation coefficients extracted from the echoes of three kinds of targets have cross-values, which will inevitably lead to erroneous judgment in the process of target classification and reduce the classification probability. The reason may be the low SNR or the change of the angle of view between the aircraft and the radar, which results in small fluctuation of the extracted features. However, we can also see from the graph that the mean values of each feature differ greatly among the three kinds of aircraft targets and are more stable than those extracted from each sparse echo signal sample. Therefore, we need to adopt appropriate methods to make the features extracted from each sparse echo signal sample close to the mean value, so as to eliminate the impact of the target echo fluctuation model and improve the classification probability of the aircraft targets.

#### *3.2. Weighted Features Fusion*

On the basis of the above simulation analysis, we propose a target classification algorithm based on the weighted features fusion of multi-wave gates sparse echo data. This algorithm uses multi-wave gates echo data to extract features, which are fused by weighting to improve the correct classification probability. The fused features can be expressed as:

$$\tilde{F} = \sum\_{i=1}^{K} \alpha\_i F\_{i\prime} \tag{14}$$

where *F*˜ is the fused feature, *Fi* is the feature extracted from the echo data of the i-th wave gate, *K* is the number of gates for feature fusion, α*<sup>i</sup>* is the weight of the i-th wave gate feature.

In this paper, we consider that the features extracted from different gates have the same contribution to aircraft target type classification. Therefore, we adopt the same weighting value for feature fusion, that is to say, the weights α*<sup>i</sup>* = 1/*K*. In Section 3.1, we simulate 200 single-wave sparse echo signal samples of one aircraft target where one sparse signal sample corresponds to one radar observation angle. While in the simulation experiment of weighted features fusion, we collect four-wave gates sparse echo data at each radar observation angle which is set the same as that in Section 3.1 during the observation of the aircraft target. That is to say, in each observation angle, we reconstruct four-wave gates sparse echo signal samples, then extract the features from each reconstructed sample and fuse them as a fusion feature. Therefore, each type of feature consists of 200 fusion features for each aircraft target. Figure 8 shows the result of the fusion features extracted and fused from four wave gates echo data. Compared with Figure 7, the cross-value of extracted features between different targets is significantly reduced under the same SNR, we can also say that the fused features are more clustered near the mean of all samples.

**Figure 8.** Result of fusing the features extracted from four wave gates echo data: (**a**) amplitude deviation coefficient; (**b**) frequency domain waveform entropy; (**c**) time domain waveform entropy.

We know that variance is a measure of the degree of dispersion of a set of data. In this paper, we calculate the variance of the fused features extracted from four wave gates sparse echo data with SNR is −13 dB, which is shown in Table 4. For comparison, we also compute the variance of the features that are not fused. We can see from Table 4 that the variance of the fused features is less than that of the features without fusion, no matter which kind of feature. In other words, fusion of extracted features is more conducive to distinguishing the three types of aircraft targets mentioned in this paper.


**Table 4.** Comparison of variance of extracted features whether to fuse or not.

### *3.3. Classification Algorithm*

In this paper, support vector machine (SVM) method is used to classify the extracted fusion features of three types of aircraft targets. SVM was first proposed by Vapink for the classification of two types of liner separable data [51]. By finding the optimal hyperplane which makes the boundary distance between the two classes the maximum, the sample data was divided into two types. Later, it was extended to linear separable data. To solve the problem of three types of aircraft targets classification in this paper, we use the one-vs-one method to construct an SVM classifier between any two types, and construct three classifiers in total, and then obtain the final classification result by voting scheme. Three types of aircraft targets classification method based on SVM that we adopted in this paper are shown in Figure 9.

**Figure 9.** Three-class support vector machine (SVM) model.

In Figure 9, the flow of red dotted box marked is the training process, in which the training dataset labeled in advance are divided into three parts belonging to different aircraft targets, and the two parts of them are combined to train three SVM models, respectively. The flow of purple dotted box marked is the testing process, in which the testing dataset which are completely different from the training dataset are sent to three trained SVM models, and then vote on the results of the SVM model to get the final classification results.

To sum up, with the sparse echo data, the classification algorithm of aircraft targets based on the weighted features fusion of multiple wave gates is summarized in Figure 10. In the proposed algorithm, the multi-wave gates sparse echo data are obtained as described in Figure 3: *K* is the number of wave gates in weighted feature fusion, and the SL0 and OMP methods are used to reconstruct sparse echoes and by which three types of features are extracted: amplitude deviation coefficient, time domain waveform entropy and frequency domain waveform entropy. In addition, in order to improve the classification probability of three types of aircraft targets, a classification algorithm based on the weighted features fusion of multiple wave gates is proposed. Finally, the fused features are used to classify three aircraft targets by three class support vector machine model.

**Figure 10.** Flowchart of the proposed classification algorithm.
