## 1. Introduction

In a radar or a sonar system, the received signal, in comparison with the transmitted waveform, often contains both time delay (TD) and Doppler stretch (DS). At present, this topic has attracted many researchers’ attention [

1,

2]. Various algorithms have been proposed to estimate these parameters. For instance, Zhang et al. proposed an expectation-maximization based estimator to estimate delay and Doppler of a moving target in a passive radar [

3]. Qu et al. proposed a method based on the wideband ambiguity function to estimate time delay and Doppler stretch for a wideband signal [

4]. Niu et al. presented a wavelet-based wideband cross ambiguity function (WB-WBCAF) method to estimate the time delay and Doppler stretch between two received signals that are contaminated by noise [

5]. Friedlander proposed an algorithm based on a computationally efficient search-free frequency estimation technique for the sum of complex exponentials for Doppler-Delay Estimation [

6]. These methods may solve such parameter estimation problems in the Gaussian noise environment. However, the performance of these methods degenerates severely in the impulsive noise environment.

To suppress the impulsive noise interference, many parameter estimation algorithms based on the fractional lower-order statistics (FLOS) have been proposed [

7,

8,

9,

10,

11,

12,

13]. Li et al. proposed a new method based on sparse representation of the fractional lower order statistics to estimate Direction of arrival (DOA) in impulsive noise [

7]. Ma et al. proposed the fractional lower order covariance method and least mean

$p$ norm criterion for time delay estimation (TDE), and the fractional lower order ambiguity function for joint time delay of arrival (TDOA) and frequency delay of arrival (FDOA) estimation [

8]. Long et al. presented the applications of fractional lower order time frequency representation to machine bearing fault diagnosis [

9].

Time-frequency distribution is a useful tool to extract helpful information of the received signal. Various time frequency distribution methods based on fractional lower order statistics have been proposed, such as short time Fourier transform based on fractional lower order statistics [

10], Wigner–Ville distributions based on fractional lower order statistics [

11], fractional power spectrum density based on fractional lower order statistics [

12], and fractional correlation based on fractional lower-order statistic [

13]. Li et al. proposed a novel method that combines the fractional lower order statistics and fractional power spectrum density (FLOS-FPSD) for suppressing the impulsive noise and estimating parameters for a bistatic multiple input and multiple output (MIMO) radar system in the impulse noise environment [

12]. The fractional correlation based on fractional lower-order statistic (FLOS-FC) method has been proposed in an impulsive noise environment [

13], where the Doppler stretch and time delay are jointly estimated by peak searching of the FLOS-FC. Therefore, in relevant prior publications, the FLOS theory has been always employed to suppress the impulsive noise interference.

However, the performance of these algorithms based on fractional lower order statistics may degrade seriously for an inappropriate fractional lower order moment

$p$ [

14,

15,

16]. According to the fractional lower order statistics theory, the relationship between the fractional lower order moment

$p$ and characteristic exponent

$\alpha $ must satisfy

$1\le p<\alpha $ or

$0<p<\alpha /2$. Therefore, the methods based on fractional lower order statistics depend on a priori knowledge of the noise.

To overcome these limitations, a novel time-frequency transform, combining fractional Fourier transform and Sigmoid transform and known as Sigmoid fractional Fourier transform, is proposed to estimate the Doppler stretch (DS) and time delay (TD) of wideband echoes for linear frequency modulation (LFM) pulse radar under impulsive noise environment. This technique does not need a priori knowledge of impulsive noise.

This paper is organized as follows.

Section 2 presents a signal model of wideband echoes in impulsive noise environment. In

Section 3, a novel Sigmoid fractional Fourier transform (Sigmoid-FRFT) is defined. In

Section 4, a novel Doppler stretch and time delay estimation method based on Sigmoid-FRFT for impulsive noise is proposed, and performance analysis of the Sigmoid-FRFT method is presented. In

Section 5, the performance of the parameter estimation algorithm is studied through extensive numerical simulations. Finally, conclusions are drawn in

Section 6.

## 3. Sigmoid Fractional Fourier Transform

#### 3.1. Fractional Fourier Transform

The continuous FRFT [

18,

19] of a signal

$x\left(t\right)$ with a rotation angle

$\beta $ is defined as

where

${F}^{b}$ denotes the FRFT operator,

$b\left(0<b\le 2\right)$ denotes the fractional order,

$\beta \equiv b\pi /2$, and

${K}_{b}\left(t,m\right)$ is the kernel function of the fractional Fourier transform.

${K}_{b}\left(t,m\right)$ can be expressed as

where

${A}_{\beta}=\sqrt{1-j\mathrm{cot}\beta}$ and

$m$ is the frequency in FRFT domain. When

$\beta =\text{}2n\mathsf{\pi}\text{}+\text{}\frac{\mathsf{\pi}}{2}$,

${K}_{b}\left(t,m\right)$ coincides with the kernel of the Fourier transform, and the FRFT reduces to the conventional Fourier transform. The kernel has the following properties:

${K}_{b}\left(t,m\right)={K}_{b}\left(m,t\right)$ and

${K}_{-b}\left(t,m\right)={K}_{b}^{\ast}\left(t,m\right)$.

#### 3.2. FRFT of LFM Signal

From Equations (5) and (6), the FRFT of the LFM signal

$x\left(t\right)$ can be expressed as

When ${\mu}_{0}=-\mathrm{cot}\beta $, $X\left(\beta ,m\right)$ has the best energy-concentrated property and an optimal rotation angle ${\beta}_{0}$ exists to maximize the peak amplitude of $X\left(\beta ,m\right)$. Accordingly, ${b}_{0}=2{\beta}_{0}/\mathsf{\pi}$ is called the optimal fractional order. The $X\left({\beta}_{0},m\right)$ forms a pulse in the FRFT domain and its peak value appears at $\left({\beta}_{0},{m}_{0}\right)$ as

According to Equation (8), we can find that the rotation angle $\beta $ only depends on the frequency modulation rate ${\mu}_{0}$.

Similarly, the FRFT of the echo signal

$y\left(t\right)$ can be written as

where

$N\left(\beta ,m\right)$ denotes the FRFT of the noise

$n\left(t\right)$. If and only if

$Y\left(\beta ,m\right)$ forms

$L$ pulses in the FRFT domain and these peaks appear at

$\left({\beta}_{l},{m}_{l}\right)$. Thus, the estimation of the Doppler stretch and time delay becomes a problem of locating the peak point of

$Y\left(\beta ,m\right)$. Then, it follows directly from Equation (10) that the Doppler stretch and time delay are estimated by

#### 3.3. Sigmoid Transform

Sigmoid is a commonly used nonlinear transformation [

20,

21,

22]. Its definition can be shown as

For a $S\alpha S$ process with $a=0$, the Sigmoid transform has the following properties.

**Property 1**: If $x\left(t\right)$ is a $S\alpha S$ process with $\beta =0$ and $a=0$, then $\mathrm{Sigmoid}\left[x\left(t\right)\right]$ is a symmetric distribution with zero mean in its probability density function, and has the finite second order moment with zero mean (referred as a second order moment process).

**Property 2:**$\mathrm{Sigmoid}\left[x\left(t\right)\right]$ has the same frequency shift as $x\left(t\right)$.

**Property 3:**$\mathrm{Sigmoid}\left[x\left(t\right)\right]$ has the same time delay as $x\left(t\right)$.

Since Properties 1 and Properties 2,3 have been proved in [

4] and [

15], respectively, the relevant proof will be skipped.

#### 3.4. Definition of the Sigmoid-FRFT

To overcome the limitations that the performance degradation of the existing methods based on fractional Fourier transform in the impulsive noise and the fractional lower-order statistics-based methods dependencies on the priori knowledge of the noise, this paper proposes a novel Sigmoid fractional Fourier transform (Sigmoid-FRFT), combining the fractional Fourier transform and the Sigmoid transform. The definition of the Sigmoid-FRFT

${X}_{\mathrm{Sigmoid}}\left(\beta ,m\right)$ is expressed as

Since the FRFT spectrum $X\left(\beta ,m\right)$ of the LFM signal $x\left(t\right)$ has the energy-concentrated property, the Sigmoid-FRFT spectrum ${X}_{\mathrm{Sigmoid}}\left(\beta ,m\right)$ of $\mathrm{Sigmoid}\left[x\left(t\right)\right]$ demonstrates the same property according to properties of the Sigmoid transform. Furthermore, $X\left(\beta ,m\right)$ and ${X}_{\mathrm{Sigmoid}}\left(\beta ,m\right)$ form the pulse at the same location in the FRFT domain.

Equation (13) is the definition of the Sigmoid-FRFT that is used throughout this paper. Searching for the peaks of the Sigmoid-FRFT

${X}_{\mathrm{Sigmoid}}\left(\beta ,m\right)$, the

${X}_{\mathrm{Sigmoid}}\left(\beta ,m\right)$ forms a pulse in the FRFT domain and its peak value appears at

$\left({\beta}_{0},{m}_{0}\right)$ as

Figure 1 shows the performance of suppressing the impulsive noise FRFT and the Sigmoid-FRFT under an impulsive noise with

$\mathrm{GSNR}=5\text{}\mathrm{dB}$ with

$\alpha =1.2$.

Figure 1a,c represents the time-frequency distribution of the FRFT and the Sigmoid-FRFT of the impulsive noise.

Figure 1b,d represents the time-frequency distribution of the FRFT and the Sigmoid-FRFT of the LFM signal with impulsive noise. From

Figure 1, it is clearly seen that identifying the correct peak location is not trivial as the FRFT peak cannot be distinguished from the impulsive noise. Accordingly, the estimation performance of the FRFT method degrades severely in the impulsive noise environment. After the application of the Sigmoid transformation, the impulsive noise is suppressed effectively and the Sigmoid-FRFT spectrum forms an obvious pulse in the FRFT domain. Thus, the method based on the Sigmoid-FRFT yields better estimation performance.

## 5. Simulation Results

In this section, we perform three types of simulation experiments to evaluate the relative performance of the FRFT, the FLOS-FC, the FLOS-FPSD, and the Sigmoid-FRFT methods under impulsive noise, respectively.

The parameters of the transmitted LFM signal in the simulation are assumed as follows. The initial frequency

${f}_{0}=0.2{f}_{s}$ and the modulation rate is set to

${\mu}_{0}=0.1{f}_{s}^{2}/N$. The sampling rate is set to

${f}_{s}=1\text{\hspace{0.05em}\hspace{0.05em}}\mathrm{MHz}$ with a sampling length of

$N=1000$. The number of multipath is

$L=2$ and the Doppler stretch and time delay are set to

${\sigma}_{1}=0.8$,

${\sigma}_{2}=1.2$,

${\tau}_{1}=20/{f}_{s}$ and

${\tau}_{2}=40/{f}_{s}$, respectively. The Root Mean Square Error (RMSE) is defined as

where

${\widehat{x}}_{1}$ and

${\widehat{x}}_{2}$ are the estimation of

${x}_{1}$ and

${x}_{2}$, and

$K$ is the number of Monte Carlo. For each simulation, the number of Monte Carlo runs is 500.

#### 5.1. Simulation 1: FRFT, FLOS-FC, FLOS-FPSD, and Sigmoid-FRFT for a Single Estimation for Two Targets

Figure 3 shows the estimation results of the FRFT, FLOS-FC, FLOS-FPSD, and Sigmoid-FRFT for a single trial of data, two targets, under impulsive noise with GSNR = 5 dB and α = 1.2. From

Figure 3a,b, we can find that the FRFT algorithm fails when an impulsive occurs, where the correct peak cannot be obtained and the estimation performance degrades severely in the impulsive noise environment. The reason is that the FRFT method does not have the ability to suppress impulsive noise. The peak of the FRFT is submerged in noise.

As shown in

Figure 3c, the spectrum of the FLOS-FC does not have the energy-concentrated property at a specific rotation angle. The accurate rotation angle cannot be obtained. Therefore, the FLOS-FC method fails in this noise environment. The spectrum of the FLOS-FPSD is shown in

Figure 3d, and we can find that the FLOS-FPSD algorithm also fails in the impulsive noise environment because the peak of the FLOS-FPSD cannot be easily separated from the impulsive noise in the FLOS-FPSD spectrum of the echo signal with impulsive noise. The FLOS-FPSD algorithm, combining the fractional lower-order statistics theory with the fractional power spectrum density function, can effectively suppress the Alpha-stable noise interference under certain impulsive noise environments. However, when impulsiveness is stronger or GSNR is lower, the FLOS-FPSD method fails. As shown in

Figure 3c,d, the FLOS-FPSD algorithm fails in the impulsive noise environment with GSNR = 5 dB and α = 1.2 because the correct peak of the FLOS-FPSD cannot be obtained. On the other hand, the Sigmoid-FRFT spectrum of the echo signal with impulsive noise forms the obvious pulse, that is because the Sigmoid transform can restrain impulsive noise interference, as illustrated in

Figure 3e. Compared with the FRFT of the echo signal with no noise, it is clearly seen that the results obtained from the Sigmoid-FRFT of the echo signal with impulsive noise matches quite well and with peaks at the same location. As analyzed in

Section 4.3, the Sigmoid transform can suppress impulsive noise better than the FLOS. Therefore, the proposed method based on the Sigmoid-FRFT can effectively suppress impulsive noise interference, yields an accurate peak estimation and has a better estimation performance.

#### 5.2. Simulation 2: Estimation Accuracy with Respect to GSNR

To evaluate the performance of TD and DS in this simulation, the characteristic exponent

$\alpha $ is set to α = 1.2 and the fractional lower order moment

$p$ is set to

$p=1.1$ and

$p=1.5$ for the FLOS-FPSD method, respectively. The resulting RMSE performance versus GSNR is illustrated in

Figure 4.

From

Figure 4, we can find that the FRFT method has a poor estimation performance with impulsive noise interference. The FLOS-FPSD method on the other hand, combining the fractional lower order statistics theory with the fractional power spectrum density, can effectively suppress the Alpha-stable noise interference. Accordingly, the FLOS-FPSD method yields a clear peak under an impulsive noise. However, the performance is affected by the fractional lower-order moment

$p$ value. According to the fractional lower order statistics theory, the characteristic exponent of the noise must be estimated to ensure

$1\le p<\alpha $ or

$0<p<\alpha /2$. The methods employing the FLOS theory cannot accurately estimate the parameters if there is no a priori knowledge of the characteristic exponent. On the contrary, the Sigmoid-FRFT has clear peaks because the Sigmoid-FRFT cannot be affected by the fractional lower-order moment

$p$. Therefore, we can accurately obtain the peaks of the Sigmoid-FRFT in impulsive noise environment.

#### 5.3. Simulation 3: Estimation Accuracy with Respect to Characteristic Exponent $\alpha $

To measure the estimation performance of TD and DS, the fractional lower order moment

$p$ is set to

$p=1.1$ and

$p=1.5$ for the FLOS-FPSD method, respectively. The GSNR is set to

$\mathrm{GSNR}=5\text{}\mathrm{dB}$.

Figure 5 shows the performance versus characteristic exponent

$\alpha $. From

Figure 5, we can find that the FRFT algorithm has a better estimation performance when the characteristic exponent

$\alpha $ is close to 2. The FLOS-FPSD method may suppress impulsive 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 suppression impulsive noise performance of the Sigmoid transform outweighs that of the FLOS theory, the estimation performance of the Sigmoid-FRFT algorithm is superior to that of the FLOS-FPSD algorithm.