A Target Tracking Algorithm based on Fractional Ambiguity Function in Impulsive Noise Environment

This paper proposes an airplane tracking algorithm based on study of the problem of interference localization. Firstly, a novel signal model to accurately estimate parameters of the airplane is proposed in impulsive noise environment. A method of instantaneous Doppler frequency estimation based on peak searching of the fractional lower-order ambiguity function based on the fractional Fourier transform (FLOS_FAF) is proposed, and a method of projection approximation subspace tracking using robust m-estimation method based on fractional lowerorder ambiguity function in fractional Fourier transform domain (FF-RLM_PAST) is proposed to estimate the azimuth angle and elevation angle. As a result, the airplane tracking is achieved in bistatic radar, laying the foundations for interference localization. The correctness and effectiveness of the proposed method are verified with the computer simulation.


Introduction
With increase of radio stations quantity, the interfered probability of civil aviation radio frequency increases.It is key point how to make it safe operation of civil aviation radio frequency.Interference localization is realized by utilizing Doppler frequency shifts of the scattered signals from civil airplane in the receiver [1], [2] .If real-time state information of airplane is not obtained, interference localization is not realized.It is shown that accurate estimation of airplane state information is very important.
Studies and experimental measurements have shown that a class of D -stable distributions is more appropriate for modeling impulsive noise than Gaussian distribution in signal processing applications [3] .Since the stable distribution does not have finite second-order moments ( 1 2 d D ), or even first-order moment (  1 ) due to the heavy tails, the performance of the existing parameter estimation and target tracking methods based on secondorder will degrade severely.
In research background of the problem of interference localization, this paper addresses the problem of parameter estimation and target tracking in the presence of impulsive noise.With combination of the fractional ambiguity function with the fractional lower order statistics, a parameter estimation and airplane tracking algorithm is presented in this paper.

The Proposed Signal Model
In this paper, the airplane tracking and localization is realized by employing the principle of parameter estimation in bistatic Multiple-Input Multiple-Output radar system.Fig. 1 illustrates a bistatic MIMO radar system.The considered bistatic MIMO radar is composed of M transmit antennas and N receive antennas with an interelement spacing of 2 O .D is the base line distance between the transmit reference element and the receive reference element.

T M T M T M
A .

Fractional ambiguity function based on fractional lower-order statistics
The ambiguity function is a useful tool for describing the ability of a waveform to simultaneously estimate the range and range-rate (speed) of targets in active (correlation-based) radar and sonar systems [4].In recent years, a new time-frequency analysis tool, the Fractional Fourier transform, attracts increasingly more attention in signal processing and is widely applied to detection, parameter estimation and direction-of-arrival estimation of the linear frequency modulation signal.
With the development of the fractional Fourier transform (FRFT), an important relationship between the FRFT representation of the narrowband ambiguity function and its transform under the rotation operator is studied.In [5], a new ambiguity function theory based on the FRFT is presented to solve the problem of parameter estimation.

Fractional Ambiguity Function
Assume that the cubic phase signal r t is modeled as where 0 b is the signal amplitude, and , 0,1,2,3 i a i are signal phase factors, the amplitude and phase factors are real and unknown .Instantaneous autocorrelation function where W denote delay, U W can be written as , , , , where U is the rotation angle and m is the frequency in FRFT domain, ( , ) where According to ( 6) and ( 7), we can see that

FLOS_FAF
Since the stable distribution does not have finite second-order moments ( 1 2 d D ), or even first-order moment ( 1

D
) due to the heavy tails, the performance of the existing signal selective parameters estimation methods based on second-order statistics, such as Instantaneous autocorrelation function, will degrade severely [7], [8]. .Thus, this paper proposes a novel method fractional ambiguity function based on fractional lowerorder statistics., where p is the order of the fractional lower order, and

Azimuth and Elevation Angles Dynamic Recursive Estimation
Bin Yang [9] proposed the method of projection approximation subspace tracking (PAST) based on least squares estimation, which is sensitive to impulse noise and make the performance of PAST algorithm degraded.On the base of the study of the method of PAST, this paper propose a novel RLM_PAST method based on FLOS_FCAF (FF_RLM_PAST) According to (8) Next, the proposed method of FF_RLM_PAST is introduced in detail by taking the subarray

Simulation results
The considered bistatic MIMO radar is composed of 6 M transmit antennas and 8 N receive antennas with an interelement spacing of 0.5m .The base line distance between the transmit reference element and the receive reference element is 5Km D . Since the alphastable process has infinite variance for 2

D
, we use a generalized signal-to-noise ratio (GSNR) measure defined as the ratio of the signal power over the implusive noise dispersion Ȗ , ^

D
, GSNR 12dB .Fig. 2 plots the estimation performance of Doppler.From Fig. 2, we can find the curve of estimated Doppler is more close to the curve of true Doppler.Therefore, the proposed method has good estimation performance about three Doppler parameters.Estimation performance of Doppler Fig. 3 show curves of azimuth angle and elevation angle dynamical recursive estimators with time t .From Fig. 3, we can find the curves of estimated azimuth angle and elevation angle are more close to the curves of true value.From Fig. 2 and Fig. 3, we can find that the proposed method has good estimation performance.In this simulation, the generalized signal to noise ratio is set as GSNR 12dB .Let root mean-squared errors (RMSE) based on average is defined by 2016 APOP 7011 characteristic exponent D .Therefore, the proposed method can suppress the interference of impulse noise, and has good performance.From these figures, we also find that the estimation performance of the Doppler frequency parameters affects the estimation performances of the azimuth angle and elevation angle.Simulation 3: Generalized signal-noise-ratio GSNR In this simulation, the noise is modeled as D -stable noise with 1.4

D
. Fig. 5 show the RMSE of angle and Doppler frequency for various values of the GSNR .From Fig. 5, we can find that the proposed method still has better performance in the condition of the lower SNR.

Conclusion
This paper proposes an airplane tracking algorithm based on study of the problem of interference localization.Firstly, a novel signal model is proposed in impulsive noise environment.A method of instantaneous Doppler frequency estimation based on peak searching of the FLOS_FAF is proposed, and FF-RLM_PAST algorithm is proposed to dynamical recursive estimate the azimuth angle and elevation angle.As a result, the airplane tracking is achieved in bistatic radar, laying the foundations for interference localization.The correctness and effectiveness of the proposed method are verified with the computer simulation.
elevation angle correspond to the transmit array and receive array, respectively.The transmitting antennas emit orthogonal waveforms received signals contain timevariant Doppler due to the three dimension motion of airplane.If the cubic phase is ignored, the performance of parameter estimation and airplane tracking will degrade.This paper presents a new signal model for the bistatic MIMO radar system.The echo of the n th receiving antenna

FRFT
has the characteristics of linear frequency modulation signal, which contains noise.According to the definition of FRFT in[6], fractional ambiguity function (FAF) of the signal r t , namely the 8) where ( , , ) W m U W denotes the FAF of noise.By searching the peak of , ,

Figure 2 .
Figure 2. Estimation performance of DopplerFig.3show curves of azimuth angle and elevation angle dynamical recursive estimators with time t .From Fig.3, we can find the curves of estimated azimuth angle and elevation angle are more close to the curves of true value.From Fig.2and Fig.3, we can find that the proposed method has good estimation performance.

Figure 3 .Simulation 2 :
estimation performance of angle and airplane location Characteristic exponent D .

Fig. 4
Fig.4plots the RMSE of the proposed method versus various values of characteristic exponent D .The smaller the characteristic exponent D is, the stronger the pulse characteristic is.From Fig.4, we can find that the RMSE of the parameters decreases with the increase of the

10 DFigure 4 .Figure 5 .
azimuth angle of receive array elevation angle of receive array azimuth angle of transmit array elevation angle of transmit array Root mean square error as a function of the characteristic exponent D Root mean square error as a function of the GSNR