## 2 The Proposed Method
### Time-Frequency Compression Fusion
#### 2.1.3 Comparison of the Two Methods  
TMSST and MSST adopt different forms of short-time Fourier transform in principle from the perspective of signal reconstruction. Let the direct current component of the window function not equal to zero, that is, \(G^{*}(0)\
eq 0\). \(*\) is the conjugate symbol. The inverse transformation formula of TMSST can be expressed as:  
\[x(t)=\frac{1}{G^{*}(0)}\mathcal{F}^{-1}\bigg{\{}\int_{-\infty}^{+\infty}TMSST ^{[N]}(\tau,v)d\tau\bigg{\}} \tag{8}\]  
where \(\mathcal{F}^{-1}\{\cdot\}\)represents the inverse Fourier transform operator. Assuming \(g^{*}(0)\
eq 0\), the inverse transformation formula of MSST can be expressed as:  
\[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]  
The proof process of combined Equations (1) and (6) is as follows:  
\[\frac{1}{G^{*}(0)}\mathcal{F}^{-1}\bigg{\{}\int_{-\infty}^{+\infty }TMSST^{[N]}(\tau,v)d\tau\bigg{\}}\] \[=\frac{1}{2\pi G^{*}(0)}\int_{-\infty}^{+\infty}\bigg{[}\int_{- \infty}^{+\infty}TMSST^{[N]}(\tau,v)d\tau\bigg{]}e^{jvt}dv\] \[=\mathcal{F}^{-1}\bigg{\{}\frac{1}{G^{*}(0)}\int_{-\infty}^{+ \infty}\int_{-\infty}^{+\infty}TSTFT(t,v)\delta(\tau-\hat{t}^{[N]}(t,v))dtd \tau\bigg{\}}\] \[=\mathcal{F}^{-1}\bigg{\{}\frac{1}{G^{*}(0)}\int_{-\infty}^{+ \infty}TSTFT(t,v)dt\bigg{\}}\] \[=\mathcal{F}^{-1}\bigg{\{}\frac{1}{G^{*}(0)}\int_{-\infty}^{+ \infty}TSFT(\tau,v)dt\bigg{\}}\] \[=\mathcal{F}^{-1}\bigg{\{}\frac{1}{G^{*}(0)}\int_{-\infty}^{+ \infty}\int_{-\infty}^{+\infty}x(t)g^{*}(t-\tau)e^{-jvt}dtd\tau\bigg{\}}\] \[=\mathcal{F}^{-1}\bigg{\{}\frac{1}{G^{*}(0)}\int_{-\infty}^{+ \infty}x(t)\bigg{[}\int_{-\infty}^{+\infty}g(\tau)e^{-jvt}d\tau\bigg{]}^{*}e^{ -jvt}dt\bigg{\}}\] \[=\mathcal{F}^{-1}\bigg{\{}\frac{1}{G^{*}(0)}\int_{-\infty}^{+ \infty}x(t)G^{*}(0)e^{-jvt}dt\bigg{\}}\] \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]0 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]1 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]2The proof process of combined Equations (3) and (6) is as follows:  
\[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]3 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]4 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]5 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]6 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]7 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]8 \[x(t)=\frac{1}{2\pi g^{*}(0)}\int_{-\infty}^{+\infty}MSST^{[N]}(\tau,v)dv \tag{9}\]9 \[\frac{1}{G^{*}(0)}\mathcal{F}^{-1}\bigg{\{}\int_{-\infty}^{+\infty }TMSST^{[N]}(\tau,v)d\tau\bigg{\}}\]0 \[\frac{1}{G^{*}(0)}\mathcal{F}^{-1}\bigg{\{}\int_{-\infty}^{+\infty }TMSST^{[N]}(\tau,v)d\tau\bigg{\}}\]1  
From the derivation of Equations (7) and (8), it can be known that whether the reassignment transform can be reconstructed depends on the short-time Fourier transform it selects. The reconstruction of the traditional STFT is completed by integrating along the time axis. Before reconstruction, the redistribution of the time-frequency coefficients along the time axis does not affect the final reconstruction result, and the traditional STFT is suitable for TMSST. The reconstruction of the improved STFT is completed by integrating it along the frequency axis. Before reconstruction, the redistribution of time-frequency coefficients along the frequency axis does not affect the final reconstruction result, and the improved STFT applies to MSST. The reconstruction properties of TMSST and MSST determine their redistribution mode to redistribute the time-frequency point coefficients of the time-frequency image to the group delay estimation and instantaneous frequency estimation of the signal, respectively.