Detection and Parameter Estimation of Multicomponent LFM Signal based on Nonlinear Transformation under the Impulsive Noise

: The linear frequency modulated (LFM) signal has been widely implemented in broadband wireless communications in high-speed vehicles, such as internet of vehicles(IoV), because of its excellent characteristic of long time interval and wide frequency band. In this paper, a novel method, which employs the fractional Fourier transform and the Tuneable-Sigmoid transform, is proposed to estimate parameters of multicomponent LFM signals in Internet of Vehicles(IoV) under the impulsive noise environment. For the optimization in the fractional Fourier domain, an algorithm based on peak searching is proposed. And for multicomponent signals, we further propose a signal separation technique in the fractional Fourier domain which can effectively suppress the interferences on the detection of the weak components brought by the stronger components, and estimate parameters of LFM signals. Moreover, boundedness and the complexity analysis of Tuneable-Sigmoid-FFRT to the S S  noise are presented to evaluate the performance of the proposed method. In additional, the Cramér–Rao bound for parameter estimation is derived and computed in closed form, which shows that better performance has been achieved. Both theoretical analysis and simulations demonstrate the superior performances of the proposed approach over other existing methods.


Introduction
Early the linear frequency modulated (LFM) signal has been widely employed in radar signal processing [1][2][3][4], especially in target detection, tracking, location and radar imaging fields, for its excellent characteristic of long time interval and wide frequency band. In recent years, wireless sensor networks, as well as military ad hoc network, have the requirement of integration with self-localization, sensing, and communication functions [5][6][7][8][9]. At present, broadband wireless communications in high-speed vehicles, such as aircraft and high-speed trains, are much in demand. Thus, the detection and parameter estimation of LFM signal is an attractive topic in internet of vehicles(IoV) [10][11]. Various algorithms have been proposed to estimate the LFM signal in the Gaussian noise environment [12][13][14][15]. For instance, Ahmet Serbes proposed analytical formulations, approximations, upper and lower bounds for the angle sweep of maximum magnitude of fractional Fourier transform of mono-and multicomponent LFM signals. It employs a successive coarse-to-fine grid-search algorithm to estimate the chirp rates of multicomponent nonseparable LFM signals [12]. Aldimashki et al. proposed a fast and powerful method for the estimation of chirp rates in the fractional Fourier domains based on the golden section search [13]. Bai et al. proposed a new chirp rate estimation algorithm by multiple discrete polynomial phase transform (DPT) and weighted combination [14]. Guo et al. proposed two constraints of time-modulation frequency based on time-modulation principle, as well as two conditions of time schemes are drawn according to the law of Nyquist sampling [15]. Studies and experimental measurements showed that the IoV signal may contain impulse noise that are non-Gaussian, primarily owing to impulsive phenomena [16][17][18][19][20]. It is inappropriate to model the noise as Gaussian noise.
To reduce the impulsive noise interference, many parameter estimation algorithms were proposed, especially, based on fractional lower order statistics (FLOS) [18][19][20][21]. According to the fractional lower order statistics theory, the methods based on fractional lower order statistics depend on a priori knowledge of the noise. Therefore, the performance of these algorithms based on FLOS may degrade seriously for an inappropriate fractional lower order moment.
Time-frequency distribution is a useful tool to extract helpful information of the LFM signal [21][22][23][24]. Various time frequency distribution methods are proposed, such as fractional Fourier transform (FRFT), short time Fourier transform and Wigner-Ville distributions. The Sigmoid function is widely used as a common nonlinear transform. The Sigmoid function can suppress impulsive noise interference, and this does not depend on a priori knowledge of the noise [25][26][27][28][29].
To handle this problem, a novel time frequency distribution based on Tuneable Sigmoid transform, referred to as fractional Fourier transform based on tuneable Sigmoid (TS-FRFT), is proposed to estimate parameters of multicomponent LFM signals under the impulsive noise environment, without the requirement of the priori knowledge about impulsive noise. This paper is organized as follows. In section 2, performance analysis of fractional Fourier transform fractional Fourier transform is presented. In section 3, a novel parameters estimation method based on TS-FRFT for impulsive noise is proposed. In section 4, the performance of the TS-FRFT method is analyzed. In section 5, the performance of the parameter estimation algorithm is studied through extensive numerical simulations. Finally, conclusions are drawn in section 6.

Fractional Fourier transform
The fractional Fourier transform (FRFT) is a generalization of the FT and can be interpreted as a rotation of the signal to any angles in the time-frequency plane [23][24].
The continuous FRFT of a signal   xt with a rotation angle  is defined as ,  denotes rotation angle in FRFT domain, m denotes the frequency in FRFT domain, and    Fm  denotes the fractional Fourier transform.

Parameter estimation based on fractional Fourier transform in the Gaussian noise
Assume that the LFM signal () xt with Gaussian noise is modeled as , 0 f denotes the initial frequency, 0  denotes chirp rate, and   wt denotes the Gaussian noise with zero-mean.
According to (1) and (3), the FRFT of the LFM signal () xt can be expressed as, When 0 = cot   ,

 
, Xm  has the best energy-concentrated property and an optimal rotation angle 0  exists to maximize the peak amplitude of   Then, parameters 0 f and 0  could be estimated directly from (6) by

The SS  Distribution Noise
In this section, we introduce the statistical model that will be used to describe the additive noise. The model is based on the class of bivariate symmetric  -stable ( SS  ) distribution and is well-suited for describing noise processes that are impulsive in nature. Its characteristic function of the SS  distribution is defined as [17][18][19]: where the parameter  is usually called the characteristic exponent. It can be proven that, in order for (7) to define a characteristic function, the values of  must be restricted to the interval 02  . When 02  , the distribution is algebraic tailed with tail constant  , implying infinite variance. When =2  , the distribution is Gaussian, implying lighterthan-algebraic tails.
The parameter  , usually called the dispersion, is a positive constant related to the scale of the distribution. For a fixed  , larger values of  correspond to larger strengths of the process. It is easy to see that 1   is, in fact, a scale parameter of the distribution.

Performance analysis of FRFT to the SS  noise
We consider   xt as an observed signal, defined as where   st denotes the LFM signal and the noise   nt is a sequence of i.i.d isotropic complex SS  random variable.
The fractional Fourier power spectrum function is defined as [23]  According to the properties of   stable distribution, if the signal   xtcontains the SS  distribution noise, then   xt does not have finite second-order moments. Therefore,

 
xx Pm  is unbounded for SS  stable distribution noise. Thus, the algorithm based on the FRFT becomes unbounded when the signal contains alpha-stable distribution noise, and this peak location algorithm may fail.
Therefore, we present a nonlinear transform, tuneable-Sigmoid transform, to suppress the alpha-stable distribution noise interference.

Tuneable Sigmoid transform
In fact, many nonlinear functions can be used to suppress the lower order  -stable noise. The ideal nonlinear transform function should have following features: it can eliminate the impact of the impulsive noises, but it does not cause a severe distortion to the normal LFM signal. The tuneable-Sigmoid function widely used in the artificial neural network is a very good nonlinear function for both purposes. By using the tuneable Sigmoid function [25], this paper proposes a nonlinear transform-based parameter estimation algorithm.
The nonlinear transform, tuneable Sigmoid function, is defined as where  is used as a scale factor to fit various signals and noises.

Definition of Tuneable-Sigmoid-FRFT
In section, a novel tuneable Sigmoid fractional Fourier transform (TS-FRFT), combining the fractional Fourier transform and the tuneable Sigmoid transform is proposed to suppress the interference of the impulsive noise. The definition of the xt has the energy-concentrated property, the tuneable-  Sigmoid-FRFT spectrum of the LFM signal with impulsive noise forms two obvious pulses that is because the nonlinear transform restrains impulsive noise interference, as illustrated is Fig.1 (c)-(d). Therefore, the proposed method based on the tuneable-Sigmoid-FRFT can effectively suppress impulsive noise interference, yields an accurate peak estimation, and has a better estimation performance.

Parameters estimation of LFM Signal based on TS-FRFT
Suppose the signal   yt with alpha-stable distribution noise as where the noise () nt denotes a sequence of i.i.d isptropic complex SS  random variable.
According to (12), the tuneable-Sigmoid-FRFT of the signal   yt can be expressed as

Boundedness of TS-FRFT to the SS  noise
Assumed the signal   xt contains the impulsive noise, we can defined as where   st is the signal and   nt is the SS  random variable noise.
The fractional power spectrum function based on TS-FRFT is defined as [23]    

Complexity Analysis
In this section, we evaluate the computational complexity of the proposed method, FRFT method, and FLOS-FPSD method. Suppose that the data length is N, the computational complexity of the FRFT method is  

The Cramer-Rao Bound
In this section, we derive a novel explicit expressions for the exact Cramer-Rao Bound (CRB) on the accuracy of estimating the LFM signal with impulsive noise.
The CR bound expresses a lower bound for the variance of an unbiased estimate and is, in general, not too difficult to compute [30][31][32][33][34]. By comparing the performance of an estimator to the CR bound, we can often have an indication on how close the estimator is to the optimum.
The LFM signal with impulsive noise can be expressed as x n (19) where   xt is the LFM signal and   nt is the SS  random variable noise..
The two parameters to be estimated are the chirp rate 0  and initial frequency 0 f , which form the parameter vector ξ such that

 
T 00 , ,where   T denotes the transpose of a vector. Suppose that the number of snapshots is N .
We may approximate the conditional probability density function (PDF) according to the model in (20) as (20) where K is a constant, and 0 T is large enough such that The CR bound for an unbiased estimate  of real deterministic parameters ξ is given by [30]         The power of a second-order process,   2 EX has been widely accepted in signal processing as a standard measure of signal strength. In the especial case of heavy algebraic tails 0 2  , the second-order power is always infinite and does not give useful information about process strength. In order to develop signal processing tools for the class of logarithmicorder processes, it is necessary to present the definition of geometric power.
The geometric power of the symmetric  -stable random variables [18] as defined in (25), is given by , is the exponential of the Euler constant.
It can be easily shown that the geometric power is a scale parameter, and as such, it can be effectively used as an indicator of process strength or 'power' in situations where second-order methods are inadequate. The geometric power 0 S is used to represent the power of symmetric  -stable random noise, i.e., Monte Carlo runs is 100 in simulation 2 and 3. Through the analysis, the inclined coefficient for the TS-FRFT is set as 1   in all the later simulations of this paper. We use the generalized signal-noise-ratio (GSNR) [20], which is defined as: where 2 x  and  are the variance of the underlying signal and dispersion of the SS  noise, respectively. Compared with the FRFT spectrum of the LFM signal, the FRFT peak cannot be easily separated from the impulsive noise in the FRFT spectrum of the LFM signal with impulsive noise. Thus, the correct peak cannot be obtained and the estimation performance degrades severely in the impulsive noise environment. Compared with the FRFT spectrum of the LFM signal, the TS-FRFT spectrum of the LFM signal with impulsive noise also forms three obvious pulses, that is because the tuneable-Sigmoid transform restrains impulsive noise interference, as illustrated is Fig.2 (c), the TS-FRFT of the LFM signal with impulsive noise both form three obvious pulses. Therefore, the proposed method based on the TS-FRFT can effectively suppress impulsive noise interference, yields three accurate peaks, and has a better estimation performance.

Simulation 2：Estimation accuracy with respect to GSNR
To evaluate the performance of initial frequency and chirp rate, in this simulation, the characteristic exponent is set to , combining the fractional lower order statistics theory with the fractional power spectrum density, can effectively suppress the SS  noise interference. On the contrary, the TS-FRFT method can suppress the SS  noise interference employing tuneable-Sigmoid transform, and the estimation performance of the TS-FRFT cannot be affected by the fractional lower-order moment p value. Therefore, the performance of the TS-FRFT method outweighs those methods.

Simulation 3：Estimation accuracy with respect to Characteristic Exponent 
In this simulation, the GSNR is set to 5dB GSNR  and the fractional lower order moment is set to for the FLOS-FPSD method. Fig.4 shows the performance versus characteristic exponent  . From Fig.4, we can find that the FPSD algorithm has a better estimation performance when the characteristic exponent  is close to 2. The FLOS-FPSD method may suppress  -stable distribution noise interference employing the fractional lower order statistic theory. The performance of the FLOS-FPSD method is shown to be better than that of the FRFT method. Since the FLOS and tuneable-Sigmoid transform methods can both suppress impulsive noise, the suppression capacity of the FLOS method is insufficient, and the tuneable Sigmoid function suppresses the outliers much harder than the FLOS [21].
Therefore, the estimation performance of the TS-FRFT algorithm is superior to that of FLOS-FPSD algorithm.

Simulation 4：Estimation accuracy with respect to
Step size b V In this simulation, the GSNR is set to GSNR 10dB  and the characteristic exponent  is set to Fig.5 shows the performance versus the step size b V of fractional order b . From Fig.5, we can find that the step size directly affects the RMSE of the two parameters. The smaller the fractional step-size, the smaller the RMSE of parameter estimation, but the time complexity will increase.

Conclusion
The linear frequency modulated (LFM) signal has been widely implemented in broadband wireless communications in high-speed vehicles, such as internet of vehicles(IoV). In this paper, a novel method, which employs the fractional Fourier transform and the tuneable-Sigmoid transform, is proposed to estimate parameters of multicomponent LFM signals in internet of vehicles(IoV) under the impulsive noise environment. For multicomponent signals, we propose a signal separation technique in the fractional Fourier domain which can effectively suppress the interferences on the detection of the weak components brought by the stronger components through searching peaks, and estimate parameters of LFM signal.
Moreover, boundedness and complexity analysis of tuneable-Sigmoid-FFRT to the SS  noise are presented, and the Cramér-Rao bound for parameter estimation is derived and computed in closed form. Both theoretical analysis and simulations demonstrate the superior performances of the proposed approach over other existing methods. Therefore, the proposed method does not need a priori knowledge of noise with higher estimation accuracy under alpha stable distribution noise environment.