Synchrosqueezing Fractional S-transform: Theory, Implementation and Applications

Abstract

The synchrosqueezing transform (SST) is an advanced post-processing method to sharpen the time–frequency representation (TFR). However, it still processes the signal in frequency domain. Therefore, it cannot effectively analyze signals whose energy is not well concentrated in frequency domain. The fractional S-transform (FrST) inherits the merits of the short-time fractional Fourier transform and the continuous wavelet transform, processing signals in fractional frequency domain. In this paper, a novel non-stationary signal processing tool, synchrosqueezing fractional S-transform (SSFrST) has been proposed, which combines the advantages of SST and FrST. First, we introduce a novel FrST and derive its fundamental properties. Second, based on the novel FrST, we propose SSFrST and discuss its reconstruction formula and implementation. It can not only improve time–frequency resolution, but also process signals in the time-fractional–frequency plane. Finally, we provide several applications to validate the effectiveness of our methods, including chirp signal parameters estimation, signal separation, filtering and noise separation.

Appendices

Appendix A

Proof

Firstly, let us calculate FrFT of the kernel function ${\varphi _{\alpha ,a,b}}(t)$.

$$\begin{aligned}{} & {} \int \limits _{ - \infty }^{ + \infty } {{\varphi _{\alpha ,a,b}}(t)} K_\alpha (u,t)dt\\{} & {} \quad = \int \limits _{ - \infty }^{ + \infty } {a\varphi (a(t - b)){e^{jat - \frac{j}{2}{t^2}\cot \alpha }}} {A_\alpha }{e^{\frac{j}{2}({u^2} + {t^2})\cot \alpha - jut\csc \alpha }}dt\\{} & {} \quad = a{A_\alpha }\int \limits _{ - \infty }^{ + \infty } {\varphi (a(t - b)){e^{jat - jut\csc \alpha }}} {e^{\frac{j}{2}{u^2}\cot \alpha }}dt\\{} & {} \quad = {A_\alpha }\int \limits _{ - \infty }^{ + \infty } {\varphi (z){e^{ja(\frac{z}{a} + b) - ju(\frac{z}{a} + b)\csc \alpha }}} {e^{\frac{j}{2}{u^2}\cot \alpha }}dz\\{} & {} \quad = {e^{-\frac{j}{2}{b^2}\cot \alpha }}{A_\alpha }{e^{\frac{j}{2}({u^2} + {b^2})\cot \alpha - jbu\csc \alpha }}{e^{jab}}\int \limits _{ - \infty }^{ + \infty } {\varphi (z)}{e^{jz - j\frac{u}{a}\csc \alpha z}}dz\\{} & {} \quad = {e^{-\frac{j}{2}{b^2}\cot \alpha }}{e^{jab}}F_{\varphi }(\frac{u-a}{a}\csc \alpha ){K_\alpha }(u,b).\\ \end{aligned}$$

Secondly, according to Parseval’s theorem of FrFT [27], we have

$$\begin{aligned} \begin{aligned} ST^\alpha _s(a,b)&= \left\langle {s(t),{\varphi _{\alpha ,a,b}}(t)} \right\rangle =\left\langle {{F^\alpha _s}(u),{F^\alpha _{\varphi _{\alpha ,a,b}}}(u)} \right\rangle \\&= {e^{\frac{j}{2}{b^2}\cot \alpha }}{e^{ - jab}}\int \limits _0^{ + \infty } {{F^\alpha _s}(u)} F^*_{\varphi }(\frac{u-a}{a}\csc \alpha ) K_\alpha ^*(u,b)du. \end{aligned} \end{aligned}$$

$\square$

Appendix B

Proof

Setting and , we have

Setting and ${C_\psi } = - \int \limits _0^{ + \infty } {\frac{{\Psi ^* (\zeta )}}{\zeta }} d\zeta$, we have

$$\begin{aligned} \begin{aligned} \int \limits _0^\infty {ST^\alpha _s(a,b){e^{jab - \frac{j}{2}{b^2}\cot \alpha }}\frac{{da}}{a}}&=-s(b)\int \limits _0^\infty { {\Psi ^*(\zeta )} \frac{{d\zeta }}{\zeta }}\\&={C_\psi }s(b). \end{aligned} \end{aligned}$$

Then, we can obtain

$$\begin{aligned} s(b) = \frac{1}{{{C_\psi }}}\int \limits _0^\infty {ST^\alpha _s(a,b)({e^{jab}}{e^{ - \frac{j}{2}{b^2}\cot \alpha }})\frac{{da}}{a}}. \end{aligned}$$

$\square$

Appendix C

According to Fubini’s Theorem and Eq. (14), we have

$$\begin{aligned} \begin{aligned}&\int \limits _{ - \infty }^\infty {\frac{a}{{\sqrt{2\pi } }}\int \limits _{ - \infty }^\infty {s(t){e^{\frac{j}{2}{t^2}\cot \alpha }}\varphi (a(t - b)){e^{ - jat}}} dtdb}\\&\quad = \int \limits _{ - \infty }^\infty {\frac{a}{{\sqrt{2\pi } }}\int \limits _{ - \infty }^\infty {s(t){e^{\frac{j}{2}{t^2}\cot \alpha }}\varphi (a(t - b)){e^{ - jat}}} dbdt} \\&\quad = \frac{a}{{\sqrt{2\pi } }}\int \limits _{ - \infty }^\infty {{e^{ - \frac{{{a^2}{{(t - b)}^2}}}{2}}}db} \int \limits _{ - \infty }^\infty {s(t){e^{\frac{j}{2}{t^2}\cot \alpha }}{e^{ - jat}}} dt\\&\quad = \int \limits _{ - \infty }^\infty {s(t){e^{\frac{j}{2}{t^2}\cot \alpha }}{e^{ - jat}}} dt\\&\quad = {F_{{\tilde{s}}}}(a)\\ \end{aligned} \end{aligned}$$