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Aerospace Systems

, Volume 2, Issue 1, pp 15–20 | Cite as

Atmospheric turbulence detection by PCA approach

  • Qiang ZhangEmail author
  • Gang Xiao
  • Yi-qun Lan
  • Rui-Rui Li
Review
  • 92 Downloads

Abstract

By detecting the severe meteorological situations on flight route, airborne weather radar (WXR) can ensure the safety of the aircraft and on-board personnel. Among these critical weather conditions, atmospheric turbulence is one of the main factors that affect flight safety. Atmospheric turbulence detection method that the current WXR adopts mainly is the pulse pair processing (PPP) method, which estimates Doppler spectrum width of weather target echo and compares it with a threshold to determine whether this weather target is turbulence or not. PPP method is simple and easy to implement, but the performance of this method under the condition of low signal-to-noise ratio (SNR) is poor. In this paper, we propose a new turbulence detection method based on the principal component analysis (PCA) approach. This new method uses PCA approach to preprocess the weather target echo and divides it into two parts: the principal component part as signal and the rest part as noise, so as to realize the de-noising function of PCA approach, and it is then combined with PPP method to estimate the spectrum width. Due to the good de-noising performance of PCA approach, this new method improves the detection performance of traditional PPP method especially under the condition of low SNR.

Keywords

Airborne weather radar Atmospheric turbulence detection Pulse pair processing Principal component analysis De-noising 

1 Introduction

Airborne weather radar (WXR), as a subsystem of the aircraft environment surveillance system (AESS), can help to ensure the safety of aircraft and on-board personnel under severe weather conditions by detecting the weather conditions within a certain fan-shaped area in front of the aircraft. Among these severe weather conditions, atmospheric turbulence is one of the important factors affecting aircraft flight safety. Because of its random fluctuations within a certain period of time and space, severe turbulence will not only make the aircraft jolt but also affect the safety of the on-board personnel.

The detection of meteorological targets by WXR is accomplished through a series of processing of radar echoes reflected from meteorological targets. The three spectrum moments of meteorological target echoes are closely related to the types of meteorological targets [1, 2, 3]: the magnitude of echo power or the zero moment of the Doppler spectrum can be used to determine the intensity of rainfall on the flight path; by calculating the rate of change of the average wind speed with distance, we can determine the degree of danger of wind shear; by comparing the spectrum width of the echo and the spectrum width threshold of turbulence, we can determine whether the front of the aircraft has turbulent area causing bumps. Therefore, the detection of atmospheric turbulence targets is essentially an estimation of the spectrum width of radar echoes.

At present, the methods of estimating spectrum width used in actual airborne weather radars are mainly the pulse pair processing (PPP) method and the fast Fourier transform (FFT) method [4, 5, 6]. The PPP method estimates the Doppler spectrum width by calculating the correlation function of the adjacent pulse echoes; the FFT method calculates the power spectrum of the echoes by the fast Fourier transform, and estimates Doppler spectrum width according to the second order moment of power spectrum. These two methods are simple to compute and provide adequate performance at high signal-to-noise ratio (SNR), but the performance is poor when SNR is low. Therefore, this paper proposes a new spectrum width estimation method using principal component analysis (PCA) approach for noise reduction. Through the principal component analysis of radar echoes, the principal part is regarded as the signal subspace, others as the noise subspace so as to achieve the purpose of removing noise and improving the spectrum width estimation accuracy.

2 Radar echoes simulation

First, we need to simulate the radar echoes of the meteorological target. Compared with the actual radar echoes, the parameters (average frequency and spectrum width) of the simulation signal are adjustable, making it easy to study the feasibility of new methods.

The meteorological targets are distributed targets, which include many scattering points in each radar resolution unit. The echo of each unit is the superposition of the echoes of all these scattering points, so it is generally considered that the echo signal of the meteorological target is Gaussian signal, in which amplitude and power spectrum obey the Gaussian distribution [7, 8].

Assume that the radar echo power spectrum sequence of the meteorological target is:
$$ G_{\text{n}} = \frac{1}{{\sqrt {2\pi } \sigma_{\text{f}} }}\exp \left[ { - \frac{{(f_{\text{n}} - \bar{f})^{2} }}{{2\sigma_{f}^{2} }}} \right], $$
(1)
where Gn is the discrete power spectrum coefficient corresponding to fn, \( \bar{f} \) is the average Doppler frequency, and σf is the Doppler spectrum width. The echoes generally contain Gaussian white noise, thus meteorological target echo power spectrum sequence superimposed with noise can be expressed as
$$ S_{n} = - \ln (x_{n} )\left[ {KG_{n} + \frac{{P_{N} }}{N}} \right], $$
(2)
where xn is random number following the uniform distribution (0, 1), and K is the parameter to control the SNR:
$$ K = \frac{{P_{N} 10^{{{\text{SNR}}/10}} }}{{\sum\limits_{N} {G_{N} } }}, $$
(3)
where PN is the noise power, generally set to 1. SNR is the signal-to-noise ratio in dB, and N is the length of the spectrum sequence.
To generate the required time signal, i.e., orthogonal two-channel signals I, Q, the phase spectrum of the signal must also be considered, which is irrelevant and uniformly distributed over (− ππ). Decomposing the power spectrum sequence into real parts and imaginary parts to obtain a complex spectrum:
$$ \begin{aligned} A_{n} & = \sqrt {S_{n} } \cos (2\pi y_{n} ), \\ B_{n} & = \sqrt {S_{n} } \sin (2\pi y_{n} ), \\ \end{aligned} $$
(4)
where yn is random number following the uniform distribution (0, 1).
After generating the complex spectrum of the power spectrum, we can generate an orthogonal two-channel signal using the relationship of the Fourier transform pair between the power spectrum and the complex time signal, that is, the inverse Fourier transform of the complex spectrum of the power spectrum:
$$ Z_{n} = I_{n} + jQ_{n} = \sum\limits_{l} {\left( {A_{l} + jB_{l} } \right)\exp \left( {j2\pi \frac{\ln }{N}} \right)} . $$
(5)

At this point, an airborne weather radar echoes with superimposed white noise obeying a Gaussian spectrum distribution can be generated.

3 Traditional PPP method

The traditional PPP method was first proposed by Rummler [9] and has been widely used in actual WXR systems. It estimates the target Doppler mean velocity and spectrum width by calculating the correlation function of the adjacent echo signals [10].

The received radar echo signals of the adjacent pulses are assumed as
$$ \begin{aligned} Z_{n} & = a_{n} e^{{ - j\varphi_{n} }} = I_{n} + jQ_{n} , \\ Z_{n + 1} & = a_{n + 1} e^{{ - j\varphi_{n + 1} }} = I_{n + 1} + jQ_{n + 1} . \\ \end{aligned} $$
(6)
Then
$$ \begin{aligned} Z_{n} Z_{n + 1}^{ * } & = a_{n} a_{n + 1} e^{{ - j(\varphi_{n + 1} - \varphi_{n} )}} \\ & = (I_{n} I_{n + 1} + Q_{n} Q_{n + 1} ) + j(I_{n} Q_{n + 1} - I_{n + 1} Q_{n} ). \\ \end{aligned} $$
(7)
Thus
$$ \varphi_{n + 1} - \varphi_{n} = \arctan \frac{{I_{n} Q_{n + 1} - I_{n + 1} Q_{n} }}{{I_{n} I_{n + 1} + Q_{n} Q_{n + 1} }}. $$
(8)
Set the pulse number of a coherent pulse processing interval to N, then according to the definition of autocorrelation function,
$$ R\left( {T{}_{s}} \right) = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {Z_{n} Z_{n + 1}^{ * } } . $$
(9)
Then
$$ R\left( {T_{s} } \right) = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {\left[ {(I_{n} I_{n + 1} + Q_{n} Q_{n + 1} ) + j(I_{n} Q_{n + 1} - I_{n + 1} Q_{n} )} \right]} , $$
(10)
marked as
$$ R\left( {T_{s} } \right) = R_{e} + jI_{m} . $$
(11)
The average phase difference is
$$ \overline{\Delta \varphi } = \arctan \frac{{I_{m} }}{{R_{e} }}. $$
(12)
The average Doppler frequency and spectrum width are estimated as
$$ \begin{aligned} \hat{f}_{\text{ppp}} & = \frac{1}{{2\pi T_{s} }}\arctan \frac{{I_{m} }}{{R_{e} }}, \\ \hat{\sigma }_{\text{ppp}} & = \frac{1}{{\sqrt 2 \pi T_{s} }}\sqrt {1 - \frac{{\left| {R\left( {T_{s} } \right)} \right|}}{R\left( 0 \right)}} , \\ \end{aligned} $$
(13)
where
$$ R\left( 0 \right) = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {Z_{n} Z_{n}^{ * } } . $$
(14)

According to the Doppler effect, the corresponding values of the average velocity and velocity spectrum width can be obtained.

4 PPP based on PCA de-noising

The method proposed in this paper is first using the PCA approach to de-noise and then combining with PPP method to estimate the spectrum width. The initial application of the PCA is dimension reduction, that is, through orthogonal transformation, a set of high-dimensional data with correlation is converted into a set of low-dimensional data composed of principal components that have no correlation. In this paper, we use principal component analysis to decompose the autocorrelation matrix of the echo data into principal components and minor components. The principal components are considered as signal subspaces and the rest are noise subspaces. The purpose of noise reduction is achieved by retaining the signal subspace only [11]; Benchebha et al. [12, 13].

The weather radar echo sequence is assumed as
$$ Z = {\rm I} + jQ = (Z_{1} , \ldots ,Z_{N} ) \in C^{N} , $$
(15)
where
$$ Z_{i} = s_{i} + n_{i} \quad i = 1, \ldots ,N, $$
(16)
where s represents the signal, n represents the additive noise, and N represents the number of pulses, i.e., the sequence length. Then the autocorrelation matrix of the echo data can be expressed as a sum of the autocorrelation matrices related to the signal part and the autocorrelation matrices related to the noise part. In practice, the autocorrelation matrix is usually unknown but can be estimated from the sample data. If the matrix size is assumed as N × N, then
$$ {\mathbf{R}}_{Z} = \frac{1}{r}\sum {{\mathbf{Z}}(r){\mathbf{Z}}^{H} (r)} = {\mathbf{R}}_{s} + {\mathbf{R}}_{n} , $$
(17)
where r is the number of realization of the process, which means Z(r) = defZ.
The eigenvalue decomposition of this autocorrelation matrix can be expressed as below
$$ {\mathbf{R}}_{Z} = \sum\limits_{i = 1}^{N} {\lambda_{i} {\mathbf{V}}_{i} {\mathbf{V}}_{i}^{H} } , $$
(18)
where λi denotes the eigenvalues and is assumed to be in descending order (λ1 ≥ λ2 ≥ … ≥ λN), Vi is an eigenvector corresponding to the eigenvalue λi.
Assuming that the autocorrelation matrix has p major components, then
$$ {\mathbf{R}}_{Z} = \sum\limits_{i = 1}^{P} {\lambda_{i} {\mathbf{V}}_{i} {\mathbf{V}}_{i}^{H} } + \sum\limits_{i = P + 1}^{N} {\lambda_{i} {\mathbf{V}}_{i} {\mathbf{V}}_{i}^{H} } , $$
(19)
the first term of (19) represents the autocorrelation function matrix related to the signal alone, and the second term of (19) is the autocorrelation function matrix related to the noise alone. Only the principal component, i.e., the signal part is retained and the noise part is eliminated, then the autocorrelation matrix of the signal is formed:
$$ {\hat{\mathbf{R}}}_{s} = \sum\limits_{i = 1}^{P} {\lambda_{i} {\mathbf{V}}_{i} {\mathbf{V}}_{i}^{H} } . $$
(20)

Since we only keep the signal part and remove the noise, the performance of the spectrum width estimation will be enhanced. Now, any spectrum moment estimator can use this autocorrelation matrix to estimate the spectrum width. In this paper, the PPP method is combined with PCA to estimate the spectrum width.

5 Simulation and analysis

In this section, we will focus on comparing the estimated performance of PPP method before and after PCA de-noising. We introduce the relative error as a measure criterion, the smaller the error, the better the accuracy of the estimation.

In order not to be affected by the influence of radar pulse repetition frequency, the parameters in this simulation adopt normalized values. First, we set the power as 1, average Doppler frequency as 0.1, Doppler spectrum width within 7 range cells as [0.07, 0.075, 0.08, 0.085, 0.09, 0.095, and 0.1]. According to Eqs. (1)–(5), radar echo sequences with different spectrum widths are generated.

When the sequence length N is 256 and the SNR is 20 dB, the simulation results are shown in the following figures:

Figure 1 shows the actual values of the spectral width and the estimated values of the two methods with a sequence length of 256, a high SNR (20 dB). σreal represents the true normalized spectral width of 7 range cells, σppp is the spectral width value obtained by the PPP method, and σpca is the spectral width value estimated by the proposed PCA method. It can be generally seen from the result of Fig. 1 that the estimated value obtained by the proposed method is closer to the true value. Figure 2 shows the relative error between the real value and the estimated value of the spectrum width obtained by the two estimation methods. Eppp is the relative error of the PPP method and Epca is the relative error of the PCA method. It can be clearly seen that the proposed method has a smaller error, that is the higher estimation accuracy.
Fig. 1

Comparison of the spectral width estimation

Fig. 2

Comparison of the spectral width estimation relative error

When the sequence length N is 256 and the SNR is 10 dB, the simulation results are shown in the following figures:

Figure 3 shows the actual values of the spectral width and the estimated values of the two methods with a sequence length of 256, a low SNR (10 dB). σreal represents the true normalized spectral width of 7 range cells, σppp is the spectral width value obtained by the PPP method, and σpca is the spectral width value estimated by the proposed PCA method. From the results in Fig. 3, it can be seen that although the two estimation methods are very different from the real values, the proposed method estimates are closer to the real spectral width. Figure 4 shows the relative error between the real value and the estimated value of the spectrum width obtained by the two estimation methods. Eppp is the relative error of the PPP method and Epca is the relative error of the PCA method. Although the estimation errors of the two methods are both high in the case of low SNR, the proposed method can still improve the estimation error of the PPP method and the estimation result is more accurate.
Fig. 3

Comparison of the spectral width estimation

Fig. 4

Comparison of the spectral width estimation relative error

When the sequence length N is 128 and the SNR is 20 dB, the simulation results are shown in the following figures:

Figure 5 shows the actual spectrum width and the estimated values of the two methods with a high SNR (20 dB) when the sequence length is 128. Figure 6 shows the relative error between the real value and the estimated value of the spectrum width obtained by the two estimation methods, the results in the figure show that the error of the proposed method is still smaller than that of the traditional PPP method even when the echo data sequence is short.
Fig. 5

Comparison of the spectral width estimation

Fig. 6

Comparison of the spectral width estimation relative error

Table 1 shows the relative error of the spectral width estimation by the PPP method and the proposed method under different sequence lengths and different SNR values. The average relative error of the PCA method is reduced by 40.6% compared to the PPP method when sequence length is 256, the SNR value is 20 dB; reduced by 8.1% when sequence length is 256, the SNR value is 10 dB, and reduced by 47.0% when sequence length is 128, the SNR value is 20 dB.
Table 1

The relative error of spectral width estimation (%)

Range cell

N = 256

SNR = 20 dB

N = 256

SNR = 10 dB

N = 128

SNR = 20 dB

PPP

PCA

PPP

PCA

PPP

PCA

1

4.55

1.45

33.77

31.73

6.24

2.48

2

3.95

1.45

29.07

27.48

6.39

2.54

3

3.51

1.69

26.29

23.07

5.55

2.42

4

3.66

1.86

21.11

19.90

5.49

2.68

5

3.10

2.53

18.07

16.65

5.48

3.12

6

4.13

3.39

16.15

14.16

5.53

3.83

7

4.35

3.83

13.11

11.68

6.01

4.47

Average

3.89

2.31

22.51

20.67

5.81

3.08

6 Conclusion

To improve the performance of spectral width estimation under low SNR in traditional PPP method, this paper proposes a spectral width estimation method using PCA approach to de-noise. Through the principal component analysis of autocorrelation function matrix of radar echo data, the autocorrelation function matrix related to the signal alone is obtained, and then the spectrum width estimation is performed with the PPP method.

The simulation results show that the estimated error of the proposed method is smaller than that of the PPP method regardless of whether the SNR is high or low. And when the data length is short, the proposed method is still valid. The analysis and simulations show that the proposed method can improve the accuracy of traditional PPP method’s spectral width estimation, and thus can be expected to accurately detect atmospheric turbulence in practical applications.

Although the proposed method can improve the estimation performance under low SNR, the estimation error is still high. Next we will use this new method to process the actual radar data to find out whether some parameters need to be adjusted.

Notes

Acknowledgments

This paper is sponsored by National Program on Key Basic Research Project (2014CB744903), National Natural Science Foundation of China (61673270), Shanghai Pujiang Program (16PJD028), Shanghai Industrial Strengthening Project (GYQJ-2017-5-08), Shanghai Science and Technology Committee Research Project (17DZ1204304), and Shanghai Engineering Research Center of Civil Aircraft Flight Testing.

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Copyright information

© Shanghai Jiao Tong University 2018

Authors and Affiliations

  • Qiang Zhang
    • 1
    Email author
  • Gang Xiao
    • 1
  • Yi-qun Lan
    • 2
  • Rui-Rui Li
    • 2
  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Department of Aviation MaintenanceShanghai Civil Aviation CollegeShanghaiChina

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