Synchroextracting Transform Based on the Novel Short-Time Fractional Fourier Transform

Abstract

As a generalization of the short-time Fourier transform (STFT), the novel short-time fractional Fourier transform (NSTFRFT) has been introduced recently. In order to improve the concentration of the time–frequency representation (TFR) generated by the NSTFRFT, two post-processing time–frequency analysis methods, two synchroextracting transforms based on the NSTFRFT with two different fractional Fourier transform (FRFT) angles, are proposed in this paper. One is achieved via an equation where the instantaneous frequency satisfies the condition where the FRFT angle takes ${\displaystyle \frac{\pi }{2}}$, and the other one is obtained using the instantaneous frequency estimator in the case that the FRFT angle takes a value related to the chirp rate of the signal. Although the conditions of the two synchroextracting transforms are different, their implementation can be unified into the same algorithm. The proposed synchroextracting transforms supplement existing post-processing time–frequency analysis methods which are based on the NSTFRFT. Experiments are conducted to verify the performance and superiority of the proposed methods.

1. Introduction

The features of non-stationary signals contain significant information, especially the instantaneous frequency (IF) in the time-varying feature, which characterizes the important physical parameter of the input signal. The time–frequency analysis (TFA) method is an effective tool for representing the IF, and it is widely utilized in plenty of fields, such as earthquake prediction [1], financial industry [2], and machinery fault diagnosis [3]. It is generally recognized that classical TFA methods include the short-time Fourier transform (STFT), the continuous wavelet transform (CWT), the S-transform [4] (ST), and the Wigner–Ville distribution (WVD). These TFA methods can well represent some specific signals due to their characteristic. Nevertheless, because of the Heisenberg uncertainty principle, the STFT, the ST, and the CWT cannot achieve high time and frequency resolution, and owing to the interference generated by cross-terms of the multi-component signal, the WVD is unable to obtain a concentrated time–frequency representation (TFR) in the time–frequency plane.

To improve the concentration of the TFRs produced by classical TFA methods, one way is to process further on the basis of the TFR obtained, which is the post-processing method. The reassignment method [5,6], which was introduced in the 1970s, reallocates the location of time–frequency coefficients in each window to the center of gravity, and thus, the TFR provided by the STFT or the CWT is concentrated. In 2011, Daubechies et al. developed another post-processing approach, i.e., the CWT-based synchrosqueezing transform [7] (WSST), which not only enhances the concentration of the TFR but can also reconstruct the signal under analysis. In view of the aforementioned advantages, the idea of the WSST is introduced into the STFT and the ST; hence, the STFT-based synchrosqueezing transform (FSST) and the ST-based synchrosqueezing transform (SSST) were proposed in [8,9], respectively. In order to further improve the concentration of the TFR of a strong frequency-varying signal, Yu et al. presented the time-reassigned multisynchrosqueezing transform [10] (TMSST). Furthermore, the second-order FSST [11] and the second-order WSST [12] have been proposed to enhance the performance of the FSST and the WSST for such a signal. The high-order synchrosqueezing transform [13] that is based on high-order amplitude and phase approximations was presented to deal with very strong amplitude-varying and frequency-varying signals. In 2017, ref. [14] introduced another post-processing approach method, namely the synchroextracting transform (SET), which removes most smeared energy by retaining related information of time-varying features of the signal. For an input signal containing fast-varying IF, the SET has a more concentrated TFR [15]. Subsequently, the SET based on the general linear chirplet transform [16] (GLCT) is given, and it is referred to as the synchroextracting chirplet transform [17] (SECT). In addition, the SET was combined with adaptive ST to produce SET based on adaptive ST [18], which improves the result of the ST. Recently, the SET was applied to handle impulsive-like signals and signals with both harmonic and impulsive components, and the time-synchroextracting general chirplet transform [19], the second-order transient-extracting transform [20], and the synchro-transient-extracting transform [21] were developed.

Considering that the fractional Fourier transform (FRFT) is regarded as a generalization of the traditional Fourier transform and that the widely used STFT is based on the traditional Fourier transform, the FRFT and the STFT are combined to obtain the fractional Gabor transform [22], the short-time fractional Fourier transform [23], the STFT in the FRFT domain [24], the novel short-time fractional Fourier transform [25] (NSTFRFT), and the sliding short-time fractional Fourier transform [26]. According to [27], the FRFT can be interpreted as a rotation in the time–frequency plane. Therefore, the resolution of the combination between the FRFT and the STFT is limited by the STFT. For the purpose of more concentrated TFR, some post-processing methods based on the combination are developed, such as the synchrosqueezing-based short-time fractional Fourier transform [28], which extends the FSST to the simplified form of the short-time fractional Fourier transform [23], a combination [29] of the SET and the short-time fractional Fourier transform, which is based on the SET and a variant of the short-time fractional Fourier transform [23], a new type of the synchrosqueezing short-time fractional Fourier transform [30], which combines the FSST with modified the NSTFRFT, and the Fractional synchrosqueezing transformation [31], which utilizes the idea of the second-order FSST to the variant of the short-time fractional Fourier transform. Although some effective TFA methods based on fractional-order systems have been provided to deal with practical problems in the existing literature, theoretical research on the nature of these methods needs to be further improved, such as robustness and stability, which can refer to the nature of fractional-order systems [32,33].

The paper is organized as follows. In Section 2, the notations of a signal, the FRFT, and the NSTFRFT are briefly reviewed. Subsequently, the theoretical derivation and the algorithm implementation of the proposed SETs based on the NSTFRFT are detailed in Section 3, and three numerical examples are utilized to validate the performance of the proposed SETs in Section 4. Finally, Section 5 discusses the proposed methods and concludes this paper.

2. Review of the NSTFRFT

In this section, the NSTFRFT, which is treated as a bank of FRFT-domain filters and preserves the properties of the STFT, is recalled following the notions of a signal and the FRFT.

In most cases, a mono-component signal is written as the following form:

$$f(t)=A(t)e^{j\varphi (t)},$$

(1)

where j is the imaginary unit, $A(t)>0$ stands for its instantaneous amplitude, and  $\varphi (t)$ denotes its instantaneous phase. ${\varphi }^{\prime }(t)$ and ${\varphi }^{″}(t)$, respectively, are the IF and the chirp rate of the signal. In practice, some signals do not consist of only one component, and for signals that contain multiple components, they can be modeled as:

$$f(t)=\sum _{n=1}^N{f}_n(t)=\sum _{n=1}^N{A}_n(t)e^{j{\varphi }_n(t)},$$

(2)

where $N$ is a positive integer representing the number of the signal components.

For a signal or function $f(t)\in L^2(\mathbb{R})$, the FRFT, a generalization of the traditional Fourier transform, is defined as [25]:

$$F_{\alpha }(u)={\int }_{-\infty }^{+\infty }f(t){\mathcal{K}}_{\alpha }(t,u)dt$$

(3)

with the transformation kernel:

$$\begin{array}{c}{\mathcal{K}}_{\alpha }(t,u)=\left\{\begin{array}{cc}\sqrt{{\displaystyle \frac{1-jcot \alpha }{2\pi }}}{e}^{j{\displaystyle \frac{t^2+u^2}{2}}cot \alpha -jut csc \alpha }\hfill & \alpha \ne m\pi \hfill \\ \delta (t-u)\hfill & \alpha =2m\pi \hfill \\ \delta (t+u)\hfill & \alpha =(2m-1)\pi ,\hfill \end{array}\right.\end{array}$$

(4)

where $m\in \mathbb{Z}$; $\delta (·)$ and $\alpha$, respectively, stand for the Dirac Delta function and the FRFT angle. Without loss of generality, it is considered that $\alpha \in [0,\pi ]$ in this paper. The argument u is termed as the fractional frequency [25] in the FRFT domain, which corresponds to the frequency in the Fourier transform domain. It is worth noting that the FRFT is simplified as the traditional Fourier transform when $\alpha ={\displaystyle \frac{\pi }{2}}$. In [27], the FRFT is perceived as a rotation by an angle $\alpha$ in the time–frequency plane, as illustrated in Figure 1.

By generalizing the STFT to the FRFT domain, the NSTFRFT of a function $f(t)\in L^2(\mathbb{R})$ is proposed in [25], which is represented as:

$$\begin{array}{cc}\hfill NSTFRF{T}_{f,\alpha }^g(t,u)& =\sqrt{{\displaystyle \frac{2\pi }{1-j cot \alpha }}}{e}^{-j{\displaystyle \frac{t^2+u^2}{2}}cot \alpha }{\int }_{-\infty }^{+\infty }f(\tau )g^{*}(\tau -t){\mathcal{K}}_{\alpha }(\tau ,u)d\tau \hfill \\ \hfill & ={\int }_{-\infty }^{+\infty }f(\tau )e^{j{\displaystyle \frac{{\tau }^2-t^2}{2}}cot \alpha }\times g^{*}(\tau -t)e^{-j\tau u csc \alpha }d\tau ,\hfill \end{array}$$

(5)

where the superscript asterisk ^* is the complex conjugate and $g(t)$ stands for an analysis window.

The window is chosen as $e^{-{\displaystyle \frac{t^2}{2b}}}(b>0)$ in this paper, and the NSTFRFT can be rewritten as:

$$NSTFRF{T}_{f,\alpha }^g(t,u)={\int }_{-\infty }^{+\infty }f(\tau )e^{j{\displaystyle \frac{{\tau }^2-t^2}{2}}cot \alpha }{e}^{-{\displaystyle \frac{{(\tau -t)}^2}{2b}}}{e}^{-j\tau u csc \alpha }d\tau .$$

(6)

3. SET Based on the NSTFRFT

The NSTFRFT that preserves the properties of the FRFT and the STFT can obtain a concentrated TFR of a signal. However, there still exists a bit obscure in the result gained by the NSTFRFT, especially for the crossover point. To achieve a more concentrated energy representation, this section presents post-processing methods, i.e., two new SETs, on the basis of the NSTFRFT.

3.1. Theory Analysis

The subsection starts with a mono-component signal $f(t)=A(t)e^{j\varphi (t)}$. For a small constant $\epsilon >0$ and fixed t, suppose that $A(t)$ and $\varphi (t)$ satisfy the following modulated conditions [18]:

$$|A^{\prime }(t)|<\epsilon ,|{\varphi }^{‴}(t)|<\epsilon ,t\in \mathbb{R},$$

(7)

then $f(\tau )=A(\tau )e^{j\varphi (\tau )}$ can be expressed approximately as the following form:

$$f(\tau )\approx A(t)e^{j(\varphi (t)+{\varphi }^{\prime }(t)(\tau -t)+{\displaystyle \frac{{\varphi }^{″}(t)}{2}}{(\tau -t)}^2)}.$$

(8)

By inserting (8) into (6), the NSTFRFT is written as:

$$\begin{array}{cc}\hfill NSTFRF{T}_{f,\alpha }^g(t,u)& ={\int }_{-\infty }^{+\infty }A(t)e^{j(\varphi +{\varphi }^{\prime }(\tau -t)+{\displaystyle \frac{{\varphi }^{″}}{2}}{(\tau -t)}^2)}{e}^{j{\displaystyle \frac{{\tau }^2-t^2}{2}}cot \alpha }{e}^{-{\displaystyle \frac{{(\tau -t)}^2}{2b}}}{e}^{-j\tau u csc \alpha }d\tau \hfill \\ \hfill & =P_{\alpha }(t){\int }_{-\infty }^{+\infty }{e}^{-{\tau }^2\times {\displaystyle \frac{1-jb(cot \alpha +{\varphi }^{″})}{2b}}}{e}^{\tau [j({\varphi }^{\prime }-t{\varphi }^{″}-u csc \alpha )+{\displaystyle \frac{t}{b}}]}d\tau \hfill \\ \hfill & =P_{\alpha }(t)\sqrt{{\displaystyle \frac{2b\pi }{1-jb(cot \alpha +{\varphi }^{″})}}}{e}^{{\displaystyle \frac{b{[j({\varphi }^{\prime }-t{\varphi }^{″}-u csc \alpha )+\frac{t}{b}]}^2}{2[1-jb(cot \alpha +{\varphi }^{″})]}}},\hfill \end{array}$$

(9)

where $P_{\alpha }(t)=A(t)e^{j(\varphi (t)-t{\varphi }^{\prime }(t)+{\displaystyle \frac{{\varphi }^{″}(t)}{2}}{t}^2)}{e}^{-j{\displaystyle \frac{t^2}{2}}cot \alpha }{e}^{-{\displaystyle \frac{t^2}{2b}}}$ and the argument t has been omitted from the function $\varphi (t)$, ${\varphi }^{\prime }(t)$, ${\varphi }^{″}(t)$ for clarity.

The partial derivative of (9) with respect to time is calculated as:

$${\displaystyle \frac{\partial }{\partial t}}NSTFRF{T}_{f,\alpha }^g(t,u)=NSTFRF{T}_{f,\alpha }^g(t,u)[-{\displaystyle \frac{t}{b}}(1+jb cot \alpha )+{\displaystyle \frac{j({\varphi }^{\prime }(t)-t{\varphi }^{″}(t)-u csc \alpha )+\frac{t}{b}}{1-jb(cot \alpha +{\varphi }^{″}(t))}}].$$

(10)

For a fixed $(t,u)$, when $NSTFRF{T}_{f,\alpha }^g(t,u)\ne 0$, Equation (10) can be rewritten as the following form:

$${\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,u)}{NSTFRF{T}_{f,\alpha }^g(t,u)}}=-{\displaystyle \frac{t}{b}}(1+jb cot \alpha )+{\displaystyle \frac{j({\varphi }^{\prime }(t)-t{\varphi }^{″}(t)-u csc \alpha )+\frac{t}{b}}{1-jb(cot \alpha +{\varphi }^{″}(t))}}.$$

(11)

Evidently, the above equation is relevant to parameter $\alpha$, which is not easy to gain. To circumvent the problem, two special cases of $\alpha$ are considered here, and they are $\alpha ={\displaystyle \frac{\pi }{2}}$ and $\alpha =arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$. The reason to choose ${\displaystyle \frac{\pi }{2}}$ as the value of $\alpha$ is that the time-fractional-frequency representation [25] of the signal obtained by the NSTFRFT becomes a familiar TFR. The reason that $\alpha$ takes $arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$ is that the value can eliminate ${\varphi }^{″}(t)$, a characteristic of the signal under analysis, in Equation (11), which is hard to get. In addition, $arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$ can be achieved by the method in [28].

In the case of $\alpha ={\displaystyle \frac{\pi }{2}}$, Equation (11) can be simplified as:

$$\begin{array}{c}\hfill {\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,u)}{NSTFRF{T}_{f,\alpha }^g(t,u)}}=-{\displaystyle \frac{t}{b}}+{\displaystyle \frac{j({\varphi }^{\prime }(t)-t{\varphi }^{″}(t)-u)+\frac{t}{b}}{1-jb{\varphi }^{″}(t)}},\end{array}$$

(12)

and it can be turned into:

$$\begin{array}{c}\hfill {\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,u)}{NSTFRF{T}_{f,\alpha }^g(t,u)}}={\displaystyle \frac{j({\varphi }^{\prime }(t)-u)}{1-jb{\varphi }^{″}(t)}}.\end{array}$$

(13)

where u is the frequency $\omega$ in the time–frequency plane, and therefore, in accordance with Equation (13), the equation that the IF $\omega ={\varphi }^{\prime }(t)$ satisfies can be written as:

$${\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,\omega )}{jNSTFRF{T}_{f,\alpha }^g(t,\omega )}}=0.$$

(14)

Let $\tilde{\omega }(t,\omega )$ be the solution of Equation (14), and then under the condition of $\alpha ={\displaystyle \frac{\pi }{2}}$, the synchroextracting transform based on the novel short-time fractional Fourier transform (SET based on the NSTFRFT) is defined by:

$$S(t,\omega )=NSTFRF{T}_{f,\alpha }^g(t,\omega )\zeta (\omega -\tilde{\omega }(t,\omega )),$$

(15)

where $\zeta (\omega -\tilde{\omega }(t,\omega ))$ stands for the synchroextracting operator, defined as:

$$\begin{array}{c}\zeta (\omega -\tilde{\omega }(t,\omega ))=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\omega =\tilde{\omega }(t,\omega ),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(16)

In addition, in the case of $\alpha =arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$, Equation (11) can be written as:

$$\begin{array}{c}\hfill {\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,u)}{NSTFRF{T}_{f,\alpha }^g(t,u)}}=j({\varphi }^{\prime }(t)-u csc \alpha ).\end{array}$$

(17)

According to the relationship [28] between the frequency and the fractional frequency, the instantaneous frequency estimator is defined as:

$$\widehat{\omega }(t,\omega )={\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,u)}{jNSTFRF{T}_{f,\alpha }^g(t,u)}}+u csc \alpha ={\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,\omega · sin \alpha )}{jNSTFRF{T}_{f,\alpha }^g(t,\omega · sin \alpha )}}+\omega .$$

(18)

Therefore, when $\alpha =arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$, the SET based on the NSTFRFT is defined as:

$$S(t,\omega )=NSTFRF{T}_{f,\alpha }^g(t,\omega · sin \alpha )\zeta (\omega -\widehat{\omega }(t,\omega )).$$

(19)

3.2. Algorithm Implementation

From (14) and (18), the realization of ${\displaystyle \frac{\partial }{\partial t}}NSTFRF{T}_{f,\alpha }^g(t,u)$ is crucial in the proposed SETs, and consequently, this process is derived as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}NSTFRF{T}_{f,\alpha }^g(t,u)& ={\displaystyle \frac{\partial }{\partial t}}{\int }_{-\infty }^{+\infty }f(\tau )e^{j{\displaystyle \frac{{\tau }^2-t^2}{2}}cot \alpha }\times g^{*}(\tau -t)e^{-j\tau u csc \alpha }d\tau \hfill \\ \hfill & =-jt cot \alpha {\int }_{-\infty }^{+\infty }f(\tau )e^{j{\displaystyle \frac{{\tau }^2-t^2}{2}}cot \alpha }{g}^{*}(\tau -t)e^{-j\tau u csc \alpha }d\tau \hfill \\ \hfill & -{\int }_{-\infty }^{+\infty }f(\tau )e^{j{\displaystyle \frac{{\tau }^2-t^2}{2}}cot \alpha }{(g^{*}(\tau -t))}^{\prime }{e}^{-j\tau u csc \alpha }d\tau \hfill \\ \hfill & =-jt cot \alpha NSTFRF{T}_{f,\alpha }^g(t,u)-NSTFRF{T}_{f,\alpha }^{g^{\prime }}(t,u).\hfill \end{array}$$

(20)

It is evident that ${\displaystyle \frac{\partial }{\partial t}}NSTFRF{T}_{f,\alpha }^g(t,u)$ can be achieved by calculating the NSTFRFT with various windows.

It is worth mentioning that in the case of $\alpha =arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$, $u csc \alpha ={\varphi }^{\prime }(t)$ is a solution of the following equation:

$${\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,u)}{jNSTFRF{T}_{f,\alpha }^g(t,u)}}=0.$$

(21)

In essence, the above equation and (14) maintain consistency, and that means that with either $\alpha ={\displaystyle \frac{\pi }{2}}$ or $\alpha =arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$, $u csc \alpha ={\varphi }^{\prime }(t)$ is always the solution of Equation (21). Accordingly, the implementation of the SETs based on the NSTFRFT can be summed up in Algorithm 1. In particular, for the initialization parameter $\alpha$, the value is taken as ${\displaystyle \frac{\pi }{2}}$ when a signal varies slowly and $arctan(-{\displaystyle \frac{1}{{\varphi }^{″}(t)}})$ if the chirp rate of the signal has a large value.

The implementation of the SETs based on the NSTFRFT mainly depends on the NSTFRFT and the synchroextracting operator that is founded on Equation (21). Under the assumption of the signal with N samples, the computational complexity of the NSTFRFT is $O(N^2{log}_{2}N)$ according to [25], and the synchroextracting operator requires $O(N^2)$ operations. Accordingly, the computational complexity of the proposed SETs based on the NSTFRFT is $O(N^2{log}_{2}N)$.

Algorithm 1 SETs based on the NSTFRFT algorithm

1: Input signal $f(t)$, $\alpha$, window $g(t)$, $g^{\prime }(t)$, positive constants $\eta$ and $\xi$;   2: Calculate $NSTFRF{T}_{f,\alpha }^g(t,u)$ and $NSTFRF{T}_{f,\alpha }^{g^{\prime }}(t,u)$ according to [25];   3: Compute ${\displaystyle \frac{\partial }{\partial t}}NSTFRF{T}_{f,\alpha }^g(t,u)$ from (20);   4: Let $W(t,u)=0$ for all t and u;   5: for each point $(t,u)$   6:       if  $|NSTFRF{T}_{f,\alpha }^g(t,u)|>\eta$   7:             if  $|{\displaystyle \frac{\frac{\partial }{\partial t}NSTFRF{T}_{f,\alpha }^g(t,u)}{jNSTFRF{T}_{f,\alpha }^g(t,u)}}|<\xi$   8:             Define $W(t,u)=1$;   9:             end if 10:       end if 11: end for 12: for each point $(t,u)$ 13: Calculate $S(t,u)=NSTFRF{T}_{f,\alpha }^g(t,u)W(t,u)$; 14: Output $S(t,u)$.

4. Numerical Experiment

In this section, the superiority of the SETs based on the NSTFRFT is verified in three examples by comparing them with other TFA methods.

As the first example, a cross signal consisting of two linear frequency modulation components is applied to illustrate the performance of the proposed SETs, and it is expressed as:

$$\begin{array}{cc}\hfill & f_{11}(t)=e^{j80t}\hfill \\ \hfill & f_{12}(t)=e^{j60t+j25t^2}\hfill \\ \hfill & f_1(t)=f_{11}(t)+f_{12}(t),0\le t<1.\hfill \end{array}$$

(22)

The ideal TFR of this cross signal is displayed in Figure 2 where the crossover point is located at 0.4 s. Figure 2 shows the TFRs produced by the ST, the GLCT [16], the STFT [4], the NSTFRFT [25], and some advanced TFR methods containing the WSST [7], the SSST [9], the second-order FSST [11], the SET [14], and the proposed SETs based on the NSTFRFT. As illustrated in Figure 3a, the ST generates a blurry TFR where components in the cross signal cannot be distinguished. By comparison, the result provided by the GLCT can show that the signal under analysis includes two components, and yet it suffers from a bad frequency resolution, as displayed in Figure 3b. The STFT (Figure 3c) with a wide window gives more concentrated energy measures than the GLCT, and meanwhile, it can be found that the crossover point is near 4 s. As a rotation version of the TFR supplied by the STFT, the TFR obtained by the NSTFRFT has a similar resolution. Because of the property of the CWT and the ST, which is a narrow window at high frequencies and a wide window at low frequencies, the post-processing methods based on the CWT and the ST, i.e., the WSST and the SSST, are not able to deal with the linear frequency modulation signal, which is neither an impulse signal nor a harmonic signal well. Therefore, the result generated by the WSST and the SSST result have worse time–frequency resolution than the result achieved by the second-order FSST, as illustrated in Figure 3e–g. Figure 3h shows that the SET provides a blurry TFR where two components are unable to be recognized. Comparing with the TFRs produced by the above TFA methods, the TFR obtained by the proposed SETs based on the NSTFRFT (Figure 3i) is the most concentrated, and it is acquired by the superposition of the TFR in Figure 4a and the TFR in Figure 4b.

For the purpose of quantitatively measuring the performance of the SETs based on the NSTFRFT, IF estimation results of different TFA methods which contain the WSST, the SSST, the second-order FSST, the SET, and the proposed methods are tested by the cross signal under diverse noise levels. The test results are evaluated by the mean relative error (MRE), which is acquired by:

$$MRE={\displaystyle \frac{1}{N_t}}{\Vert {\displaystyle \frac{\overline{IF}-IF}{IF}}\Vert }_1,$$

(23)

where $N_t$ denotes the discrete length of the IF, ${\Vert ·\Vert }_1$ stands for the $l_1$-norm, $\overline{IF}$ represents the estimated IF, and $IF$ is the true IF of the signal. Figure 5 displays the errors of estimated IF of two components in the cross signal, which is under different noise levels (2 dB, 5 dB, 8 dB, 11 dB, and 14 dB), by five methods, and every error is calculated by the mean of errors conducted by running 10 times in the same Signal-to-Noise Ratio (SNR) environment. For $f_{11}(t)$, the proposed SET based on the NSTFRFT with $\alpha ={\displaystyle \frac{\pi }{2}}$ achieves a more accurate IF estimation than other methods, as illustrated in Figure 5a. From Figure 5b, the proposed SET based on the NSTFRFT with $\alpha =arctan(-{\displaystyle \frac{1}{50}})$ and the second-order FSST provide a similar result, which obtains better accuracy than the WSST, the SSST, and the SET, for $f_{12}(t)$ in a high SNR environment (e.g., ≥8). Furthermore, the proposed method produces the best accuracy among the five methods in a low SNR environment (e.g., ≤5) for $f_{12}(t)$. Consequently, the SETs based on the NSTFRFT have a stronger noise robustness.

The second example is a signal with SNR = 10 dB on the basis of the following form described as:

$$\begin{array}{cc}\hfill & f_{21}(t)=\delta (t-0.4)\hfill \\ \hfill & f_{22}(t)=\delta (t-0.45)\hfill \\ \hfill & f_2(t)=f_{21}(t)+f_{22}(t),0\le t<1,\hfill \end{array}$$

(24)

and it consists of two impulse signals and a Gaussian white noise. The TFRs of the noisy impulse signal provided by various TFA methods are displayed in Figure 6. Figure 6a shows the result obtained by the GLCT, which has a poor resolution, and the result in Figure 6b is given by the ST, which provides a good time–frequency resolution at high frequencies and a bad resolution at low frequencies. The NSTFRFT overcomes the above shortcomings, and gives a concentrated TFR in the time–frequency plane, as displayed in Figure 6c. The SSST (Figure 6e) produces a better resolution than the WSST (Figure 6d) and the second-order FSST (Figure 6f), whereas the SET (Figure 6g) provides a more concentrated result than the SSST. By contrast, the proposed SET based on the NSTFRFT with $\alpha ={\displaystyle \frac{\pi }{2}}$ gives a higher resolution and a more concentrated TFR than other TFA methods, as displayed in Figure 6h.

An echolocation signal is the significant basis of identifying the object for a bat, and thus, it is necessary to analyze the echolocation signal. To illustrate the effectiveness of the proposed SET, the third experiment is a bat echolocation signal with SNR = 5 dB, and its TFRs obtained by six TFA methods (i.e., the STFT, the WSST, the SSST, the second-order FSST, the SET, and the SET based on the NSTFRFT with $\alpha ={\displaystyle \frac{\pi }{2}}$) are displayed in Figure 7.

The TFR provided by the second-order FSST (Figure 7d) is more concentrated than the results obtained by the STFT (Figure 7a), the WSST (Figure 7b), and the SSST (Figure 7c). In spite of this, the information at high frequencies is polluted by the noise, as illustrated in Figure 7d. In comparison with the second-order FSST, the SET (Figure 7e) and the SET based on the NSTFRFT with $\alpha ={\displaystyle \frac{\pi }{2}}$ produce clear TFRs, and by using a wide window, the proposed SET has a more satisfied result at high frequencies than the SET, as shown in Figure 7f.

In order to evaluate the energy concentration of six TFA methods, the Rényi entropies of these methods [13] are calculated and listed in

Table 1. Rényi entropy comparison of the six TFA methods.

TFA Methods	STFT	WSST	SSST	Second-Order FSST	SET	SET Based on the NSTFRFT with $\mathit{\alpha }=\frac{\mathit{\pi }}{2}$
Rényi entropy	18.5202	16.4763	18.3826	14.5749	12.7497	12.5070

. As a lower value of the Rényi entropy means a more concentrated TFR, the SET based on the NSTFRFT with $\alpha ={\displaystyle \frac{\pi }{2}}$ has the most concentrated result among the six TFA methods.

5. Discussion and Conclusions

In this paper, on the basis of the NSTFRFT with two different FRFT angles, two synchroextracting transforms are developed. One is obtained under the condition that the FRFT angle takes ${\displaystyle \frac{\pi }{2}}$, and the other one is given in the case that the FRFT angle takes a value related to the chirp rate of the signal. Two synchroextracting transforms are realized by the same algorithm because the equations that the IF of the signal satisfies in two cases are the same. Experimental results verify the effectiveness of the proposed methods. For a cross signal, it is hard to obtain the high-resolution TFR, but the proposed SETs acquire a concentrated TFR. Considering that a signal is locally regarded as a linear frequency modulation signal and that two components of the cross signal are linear frequency modulation signals, the proposed methods have vital practical significance for a general cross signal.

According to Figure 7, the SET generates a more concentrated result at low frequencies than the SET based on the NSTFRFT, and it means the proposed method is not always the best choice. In practical application, the approach used depends on various factors, for example, the characteristics of the signal under analysis. It is worth mentioning that the proposed SETs possess some advantages when the signal consisting of linear frequency modulation components is dealt with. In addition, the complete theory of the proposed methods is worthy of continued investigation, such as the local sensitivity of the NSTFRFT to time variations and the accuracy of the proposed SETs.