**1. Introduction**

Aircraft target classification is always a difficult problem for traditional narrow-band radar. Even for the three distinct targets, i.e., helicopter, propeller and jet aircraft, the recognition rate of traditional narrow-band radar is not high in practical applications. The main reasons for target recognition probability deteriorating in traditional narrow-band radar include the following three points: (1) it is limited of range resolution due to the narrow bandwidth of radar transmitting signal; (2) a traditional mechanical scanning radar has a short dwell time, which results in limited azimuth resolution; (3) special circumstances such as data missing increase the difficulty of target classification.

In order to overcome the aforementioned problems, many classification methods have been proposed to improve the aircraft recognition rate. An intuitive idea is to increase the signal bandwidth of the radar system. It is well known that the enlargement of signal bandwidth can improve the range resolution of radar, so wide-band radar can provide more information for target classification. The existing methods of aircraft classification in wide-band radar can be generally divided into three types: (1) methods based on image processing [1–5]. In Reference [6], the extracted image, instead of radar

data, was fed into a three-layered feed forward artificial neural network for aircraft classification; (2) methods based on high-resolution range profile (HRRP) [7–11]. Liu et al. [12] proposed a multi-scale target classification method based on the scale-space theory through extracting features from HRRP; and (3) methods based on inverse synthetic aperture radar (ISAR) [13–17]. A shape extraction based aircraft target classification method using ISAR images is proposed in [18]. Although image processing, HRRP- and ISAR-based aircraft classification have achieved good simulation results in wide-band radar, the radar system is more complex and the detectable range is shorter than that in narrow-band radar. Therefore, it is still of great significance to study aircraft classification based on narrow-band radar.

In recent years, many micro-Doppler parameter estimation methods [19–21] were designed to tackle the aircraft target classification problem in narrow-band radar. Two techniques of cubic polynomial fitting and three-point models are developed to estimate the micro-Doppler parameters from a fraction of the period in real-world scenarios [22]. In Reference [23] the minimal mean-square error (MMSE)-based method was proposed for estimating the micro-Doppler parameter from a fraction of the period data. Li et al. [24] proposed the parametric sparse representation and pruned orthogonal matching pursuit to the micro-Doppler parameter estimation for target classification and recognition. There are also many methods used in micro-Doppler parameter estimation for aircraft classification, such as time-frequency transform [25,26], continuous wavelet transform [27], Hough transform [25] and so on. However, these methods have the same common problem of inaccurate parameter estimation while the micro-Doppler signal is weak and only single-wave gate echo data is utilized in the aforementioned methods. Moreover, with echo data missing in narrow-band radar, Wang et al. [28] proposed a complex Gaussian model [29] based and factor analysis model [30]-based signal reconstruction methods. On the basis of the complex Gaussian model, the method of directly reconstructing the time-frequency spectrum of the original is proposed in Reference [31]. Although these methods mentioned above can effectively reconstruct the echo signals of aircraft, they have not studied the problem of classification with missing samples.

In addition, for the limited information of narrow-band radar, deep learning and machine learning [32] based methods have also been introduced to aircraft target classification in recent years. One of the most frequently used methods is the convolutional neural network (CNN) [33,34]. A novel landmark and CNN based aircraft recognition method was proposed [35], in which it alleviates the work of human annotation and can be used for any type of aircraft not contained in the training data set without retraining, thus it is highly accurate and efficient. Zuo et al. [36] proposed a deep convolutional neural network (DCNN) [37,38] based aircraft type recognition framework. Additionally, conditional generative adversarial networks [39], a self-organizing neural network [40] and deep belief net [41] have been used in aircraft targets classification. The networks used in the aforementioned methods, as an intelligent technology developed recently, have achieved good performance on aircraft targets classification, but they should be trained with large-scale datasets, which is quite time-consuming to acquire and mark with label, and difficult to practice in radar equipment. Also there is still a certain gap in the actual application of equipment.

In this paper, an aircraft target classification method is proposed for conventional narrow-band radar with multi-wave gates sparse echo data. By contrast with the previous work where there are only extract features from single wave gate echo for aircraft classification [10], the proposed method in this paper uses multiple wave gates echo data for weight feature fusion, and combines sparse theory to improve the probability of target classification in the case of missing data. Firstly, smoothed *l*<sup>0</sup> norm (SL0) [42] and orthogonal matching pursuit (OMP) [43,44] algorithms are used to reconstruct the sparse echo data in order to solve the low classification probability. Then we analyse the echo data with three kinds of aircraft targets in time domain and frequency domain. According to the difference of micro-Doppler effects [45] of rotating parts due to the difference in structure and rotating speed, features of the amplitude deviation coefficient, time domain waveform entropy and frequency domain waveform entropy are extracted to classify targets. Finally, features extracted from multi-wave gates sparse echo data are weighted and fused to train and test the support vector machine (SVM) [46–48]

model for classification. Experimental results show that the proposed algorithm can improve the classification probability, and four wave gates echo data in weighted features fusion used to extract features is the optimal wave gate number for target classification.

The rest of this paper is organized as follows. The echo model and reconstruction algorithms are reviewed in Section 2. The proposed algorithm based on multi-wave gates sparse echo data is summarized in Section 3. Section 4 verifies the effectiveness of the proposed algorithm by simulated experiments. Conclusions are presented in Section 5.

#### **2. Reconstruction Algorithm of Sparse Echo Data**

#### *2.1. Echo Model*

In narrow-band radar, the signal wavelength is much smaller than the target size, and the received signal by radar is composed of echoes reflected by multiple scattering points. For targets with rotors, such as a helicopter, propeller aircraft and jet aircraft, the echo can reflect not only the translation of scattering points of the fuselage, but also the fretting characteristics of the scattering points of the rotor blades.

Take a helicopter as an example; the special geometry between radar and a helicopter is shown in Figure 1a, in which the distance between the radar and rotor target center is denoted as *RC*, and the angle of pitch is denoted as β. Considering a 2-D slant-range plane, the simplified geometry is shown in Figure 1b. A radar coordinate system *XOY* and target coordinate system *X O Y* are set up, in which the rotor center is denoted as *O* . The rotation radius of scattering point *P* on the rotor blade is assumed to be *r*—i.e., the distance from *P* to *O* is *r*—and the distance from *P* to the radar is denoted as *RP*. The scattering point *P* rotates around the target coordinate system center *O* with an angular velocity ω, and the rotation angle at the initial time is denoted as θ0. Assume that the radial velocity of the helicopter's translational motion is *v*.

**Figure 1.** Geometry between radar and rotor target: (**a**) space geometry; (**b**) 2-D plane geometry.

In the case of the far field, the instantaneous distance between the scattering point *P* and the radar can be written as:

$$R\_P(t\_m) \approx R\_\mathbb{C} + \upsilon t\_m + r \cos \left(\alpha t\_m + \theta\_0 \right),\tag{1}$$

where *tm* = *mTr* is the slow time, *m* is the *m*-th echo pulse, and *Tr* is the pulse repetition period.

In this paper, we take the linear frequency modulation (LFM) as the transmitted signal, which can be expressed as:

$$\mathbf{s}\_{l}(\hat{t}, t\_{\mathrm{m}}) = \mathrm{rect}(\hat{t}/T\_{\mathrm{P}}) \exp\left(\mathrm{j2\pi}(f\_{\mathrm{f}}t + \mu \hat{t}^{2}/2)\right),\tag{2}$$

where *rect*(·) is the rectangular window, ˆ*t* is the fast time, *Tp* is the pulse width, *fc* is the signal carrier frequency, μ is the chirp rate, *t* is the total time, and *t* = ˆ*t* + *tm*. There are two different time variables ˆ*t* and *t* in the transmitted signal described in Formula (2), the reason is that the signal carrier frequency

*fc* exists on the whole pulse transmission time axis, while the chirp rate μ is used to adjust the change of Doppler frequency within a pulse. The echo signal of scattering point *P* can be expressed as:

$$s\_r(\mathbf{f}, t\_m) = \sigma \text{rect}(t\_m/T\_d) \text{rect}((\mathbf{f} - 2\mathcal{R}\_\mathbb{P}(t\_m)/\mathbf{c})/T\_p) \exp\left(j2\pi (f\_\mathbf{c}(\mathbf{t} - 2\mathcal{R}\_\mathbb{P}(t\_m)/\mathbf{c}) + \mu(\mathbf{f} - 2\mathcal{R}\_\mathbb{P}(t\_m)/\mathbf{c})^2/2)\right), \tag{3}$$

where σ is the scattering coefficient of the scattering point *P*, *Ta* is the observation time, and *c* is the speed of light. The target echo signal after pulse compression can be expressed as:

$$\begin{aligned} s\_P(\hat{t}, t\_m) &= -\sigma T\_p \sin \mathbf{c} [B(\hat{t} - 2R\_P(t\_m)/c)] \text{rect}(t\_m/T\_a) \exp\left(-\mathbf{j}4\pi R\_\odot/\lambda\right) \cdot \\ &\quad \exp\left(-\mathbf{j}4\pi (vt\_m + r\cos\left(\omega t\_m + \theta\_0\right)\right)/\lambda\right) + w(\hat{t}, t\_m) \end{aligned} \tag{4}$$

where *B* is the signal bandwidth, λ is the wavelength, and *w*(ˆ*t*, *tm*) denotes the Gaussian white noise signal. By taking the derivative of the phase, the micro-Doppler frequency can be obtained as:

$$f\_{d-P} = \frac{1}{2\pi} \frac{d\phi}{dt\_m} = \frac{1}{2\pi} \frac{\mathrm{d}[-4\pi(vt\_m + r\cos\left(\omega t\_m + \theta\_0\right)\big)/\lambda]}{\mathrm{d}t\_m} = -2(v - \alpha r \sin\left(\omega t\_m + \theta\_0\right))/\lambda,\tag{5}$$

It can be seen from the above formula that the Doppler frequency of the scattering point echo on the target rotor blade is related not only to the radial velocity *v* of the scattering point echo on the target rotor blade is related not only to the radial velocity of the translational motion, but also to the angular velocity ω of the rotating component and the blade length *r*. Because the rotational motion of scattering points on the fuselage is negligible, it is equivalent to only translational motion. Therefore, the instantaneous distance between scattering point *F* on fuselage and the radar can be written as:

$$R\_F(t\_m) \approx R\_C + \upsilon t\_{\text{m}} \tag{6}$$

The echo signal of scattering point *F* on fuselage after pulse compression can be expressed as:

$$s\_{\mathbb{F}}(\hat{t}, t\_{\mathfrak{m}}) = \sigma T\_{\mathbb{F}} \sin \mathfrak{c} [B(\hat{t} - 2\mathcal{R}\_{\mathbb{F}}(t\_{\mathfrak{m}})/\mathfrak{c})] \text{rect}(t\_{\mathfrak{m}}/T\_{\mathfrak{a}}) \exp\left(-\text{j}4\pi \mathcal{R}\_{\mathbb{C}}/\lambda\right) \exp\left(-\text{j}4\pi \text{wt}\_{\mathfrak{m}}/\lambda\right) + w(\hat{t}, t\_{\mathfrak{m}}), \quad (7)$$

Compared with the echo of blade scatterer in Formula (4), the fuselage echo lacks only the fretting term. In addition, the Doppler frequency of echo is only related to translational radial velocity, that is *fd*−*<sup>F</sup>* = −2*v*/λ.

The frequency domain echo of helicopter, propeller aircraft and jet aircraft are simulated as shown in Figure 2. The simulation parameters of radar and transmitting signal are set as follows, the pulse repetition frequency *Fr* is 5000 Hz, the pulse-repetition period *Tr* is 0.2 ms, the pulse width *Tp* is 50 μs, the signal carrier frequency *fc* of LFM signal is 1 GHz, the signal bandwidth *B* is 2 MHz, and the observation time *Ta* is 0.05 s. The parameters of three types of aircraft targets are shown in Table 1.

**Figure 2.** Target frequency domain echo: (**a**) helicopter; (**b**) propeller; (**c**) jet.


**Table 1.** Three kinds of aircraft targets simulation parameters.

The rotation plane of the helicopter main rotor is parallel to the ground, so the micro-Doppler effect produced can be observed easily by conventional ground radar. It can be seen from Figure 2a that in addition to the fuselage echo, there are strong echo components caused by micro-Doppler motion of rotor blades in the frequency domain echo of the helicopter, and the micro-Doppler spectrum width of the helicopter is higher than that of the propeller in Figure 2b, which is due to the fact that the length of the helicopter rotating parts is significantly longer than that of propeller. In addition, because the rotating plane of the propeller's engine blade is perpendicular to the flight direction of the aircraft, the blade is easily obscured by the fuselage, and its micro-Doppler effect is relatively weak. Because of the small size of jet engine blades and the particularity of its position, the micro-Doppler effect caused by blade rotation can hardly be observed by ground radar. It can be seen from Figure 2c that the echo of the jet aircraft only contains the fuselage component, but not the micro-Doppler component caused by blade rotation.

#### *2.2. Reconstruction Algorithm*

The multi-wave gates echo data can be obtained from the echo reflected by the transmitted signal after encountering the target during each resolution of the radar antenna in the continuous observation of the aircraft target by the radar. The definition of multi-wave gates echo data is shown in Figure 3.

**Figure 3.** The definition of multi-wave gates echo data.

As we can see from Figure 3, it is a radar display that denotes the position relationship between the radar and the aircraft target from the perspective of top view. Let us assume that the aircraft target flies in a positive direction along the x-axis with a velocity of *v*, and the blue two-way arrow line in the Figure is the signal transmission route. In the rotation of radar antenna, when the aircraft target is seen for the first time, the aircraft target is located at position A, when the antenna comes to the aircraft target after a rotation cycle, the aircraft target is located at position B, the next position is C, and so on. While the radar irradiates the aircraft target, it will receive the echo data in the area, marked by red, green and purple square boxes in this Figure, where the aircraft target is located, and we count it as a wave gate echo. After multiple irradiations, we can obtain multi-wave gates echo data.

It is difficult for conventional narrow-band radar to obtain continuous observation of the same target for a long time, and it may lead to the loss of target echo pulse in one observation time which can be called sparse echo data. The description of complete and sparse echo data are shown in Figure 4.

**Figure 4.** Description of complete and sparse echo data: (**a**) complete echo data; (**b**) sparse echo data.

As we can see from Figure 4a, there are *M* received echo pulses in one observation time which are called the single-wave gate echo data, among which the pulses marked in red randomly indicate the echo data that may be lost when the radar receives the echo. From Figure 4b, it can be seen that the number of pulses of sparse echo data is less than that of complete echo data in one observation time, we can also say that less echo information is available in sparse echo data, which is not conductive to aircraft target classification. Therefore, it is feasible and necessary to reconstruct sparse echo data by appropriate sparse signal recovery methods.

Since the emergence and development of compressed sensing (CS) [49,50] technology, sparse signal recovery algorithms have mainly been divided into greedy algorithms, non-convex function minimization algorithms, and Bayesian algorithms. The most typical and widely used greedy algorithm is orthogonal matching pursuit (OMP), which finds the best matching dictionary unit by solving the maximum inner product of the residual and the dictionary matrix, then obtains the approximate value of the sparse vector by using the least square method, and finally obtains the reconstruction signal by alternately updating the support set and solving the sparse vector. The smoothed *l*<sup>0</sup> norm (SL0) algorithm is one of the most famous non-convex function minimization algorithms, which transforms the *l*<sup>0</sup> norm minimization problem into an optimization problem by introducing a smooth Gaussian function to approach the *l*<sup>0</sup> norm. The reason for doing this is that we can avoid the non-deterministic polynomial (NP) time hard problem caused by the direct solution of *l*<sup>0</sup> norm minimization. Therefore, we use the SL0 and OMP algorithms to reconstruct the sparse echo signal, respectively in this paper.

Assuming that the number of pulses of sparse echo data is *M* , and the typical model of CS can be expressed as:

$$
\Upsilon = \Phi \mathfrak{X},
\tag{8}
$$

where **Y** is an *M* × 1 measurement vector. Actually, **Y** is the superposition of sparse echo data of scattering point and fuselage, which can be expressed as:

$$\mathbf{Y} = \mathbf{s}\_P(t\_m) + \mathbf{s}\_F(t\_m) \quad m = 1, 2, \dots, M',\tag{9}$$

**Φ** is an *M* × *N* dictionary matrix, and **X** is an *N* × 1 sparse vector to be determined. According to CS theory, the sparse solution **X** can be obtained by:

$$\hat{\mathbf{X}} = \operatorname\*{argmin}\_{\mathbf{X}} \|\mathbf{X}\|\_{0} \quad \text{s.t.} \quad \|\mathbf{Y} - \boldsymbol{\Phi} \cdot \mathbf{X}\|\_{2}^{2} \le \varepsilon,\tag{10}$$

where ·<sup>0</sup> and ·<sup>2</sup> donate L0 and L2 norms respectively, ε is the error threshold in the sparse recovery processing. The solution for (10) can be obtained by the SL0 and OMP algorithms through iteration. The main steps of the two algorithms are summarized in Tables 2 and 3.

**Table 2.** Main steps of orthogonal matching pursuit (OMP) reconstruction algorithm.

**Input:** estimated signal **<sup>Y</sup>** <sup>∈</sup> <sup>C</sup>*M* , dictionary matrix **<sup>Φ</sup>** <sup>∈</sup> <sup>C</sup>*M* <sup>×</sup>*N*, error threshold ε0. **Initialization:** let the iterative counter *<sup>k</sup>* = 1, residual matrix <sup>γ</sup><sup>0</sup> = **<sup>Y</sup>**, the index set <sup>Λ</sup><sup>0</sup> = ∅. **Iteration:** at the *k*-th iteration (1) Update the index set Λ*<sup>k</sup>* = Λ*k*−<sup>1</sup> ∪ λ*k*, where λ*<sup>k</sup>* = argmax *i*=1,2,··· ,*N* γ*k*−1,ϕ*<sup>i</sup>* . , <sup>ϕ</sup>*<sup>i</sup>* is the *<sup>i</sup>*-th column of **<sup>Φ</sup>**. (2) Update the support set **<sup>Φ</sup>**Λ*<sup>k</sup>* = **Φ**Λ*k*−<sup>1</sup> ,ϕλ*<sup>k</sup>* , and calculate the signal **<sup>X</sup>**<sup>ˆ</sup> *<sup>k</sup>* <sup>=</sup> argmax**<sup>Y</sup>** <sup>−</sup> **<sup>Φ</sup>**Λ*k***X***k*<sup>2</sup> = (**Φ**Λ*<sup>k</sup>* <sup>H</sup>**Φ**Λ*<sup>k</sup>* ) −1 **Φ**Λ*<sup>k</sup>* <sup>H</sup>**Y**. (3) Update the residual matrix <sup>γ</sup>*<sup>k</sup>* = **<sup>Y</sup>** <sup>−</sup> **<sup>Φ</sup>**Λ*k***X**<sup>ˆ</sup> *<sup>k</sup>*. (4) Increment *k*, and return to Step (1) until the stopping criterion γ*k*<sup>2</sup> ≤ ε<sup>0</sup> is met. The selection of the error threshold ε<sup>0</sup> is related to the precision requirement. **Output:** Reconstructed signal **X**ˆ = **X**ˆ *<sup>k</sup>*.

**Table 3.** Main steps of smoothed *l*<sup>0</sup> norm (SL0) reconstruction algorithm.

**Input:** estimated signal **<sup>Y</sup>** <sup>∈</sup> <sup>C</sup>*M* , dictionary matrix **<sup>Φ</sup>** <sup>∈</sup> <sup>C</sup>*M* <sup>×</sup>*N*, the search step length α. **Initialization:** Choose an appropriate standard deviation parameter decrement sequence [σ1, σ2, ··· , σ*I*], the outer loop number is *I*, the inner loop number is *J*. The initial solution is the minimum L2 norm of **Y** = **ΦX**, that is **X**<sup>0</sup> = (**Φ**H**Φ**) −1 **Φ**H**Y**. **Iteration:** (1) The *i*-th outer iteration, *i* = 1, 2, ··· , *I*, at this time σ = σ*i*, **X** = **X***i*−1. (2) The *j*-th inner iteration, *j* = 1, 2, ··· , *J* 1. Update the signal with **X** = **X** + α**d**, where **d** = −*x*<sup>1</sup> exp (−*x*<sup>1</sup> 2/2σ2), ··· ,−*xn* exp (−*xn* 2/2σ2) . 2. Project **<sup>X</sup>** onto the feasible domain, that is **<sup>X</sup>** <sup>←</sup> **<sup>X</sup>** <sup>−</sup> **<sup>Φ</sup>**H(**ΦΦ**H) −1 (**ΦX** − **Y**). (3) Update the reconstructed signal **X**ˆ *<sup>i</sup>* = **X**. **Output:** Reconstructed signal **X**ˆ = **X**ˆ *<sup>I</sup>*.

SL0 and OMP algorithms are used to reconstruct the sparse frequency domain echoes of three types of aircraft targets. In order to simulate the sparse echo data in real radar equipment, we randomly cut half of the pulses in the complete echo data as sparse echo data in this paper, that is the pulse number of sparse echo data equals *M* = *M*/2. The reconstructed results of SL0 and OMP algorithms are shown in Figures 5 and 6, respectively, where the dictionary matrix is the Fourier transform matrix because the time domain echoes are reconstructed to obtain the frequency domain echoes in this paper and the error threshold is set as ε<sup>0</sup> = 0.05**Y** 2 <sup>2</sup>, that is the iteration process is stopped when the residual energy is equal to or smaller than 5% of the received signal energy. Comparing the complete frequency domain echoes in Figure 2, we can see that SL0 and OMP algorithms can realize the reconstruction of sparse echo data. Compared with the reconstructed results of SL0 and OMP algorithms in Figures 5 and 6, it can be seen that the reconstructed result of SL0 algorithm is better than the OMP algorithm in the similarity to the complete frequency domain echo of Figure 2. The echo data reconstructed by these

two algorithms are used to extract features, and the simulation experiment of classification probability will be given in Section 4.

**Figure 5.** Reconstructed frequency domain echoes of SL0 algorithm: (**a**) Helicopter; (**b**) Propeller; (**c**) Jet.

**Figure 6.** Reconstructed frequency domain echoes of OMP algorithm: (**a**) helicopter; (**b**) propeller; (**c**) jet.
