## 3 The Proposed Hybrid Demodulation Technique  
### Modulation Intensity Distribution  
MID is a kind of signal demodulation method that has cyclostationary properties. It is very suitable for the analysis of first-order modulation (discrete carrier frequency) and is also applicable to the analysis of a two-order modulation (random carrier frequency) signal [20]. MID can reveal the relationship between the cyclic frequency and carrier frequency in the bi-frequency domain. In order to better interpret the principle of MID, a simple modulated signal can be assumed as below:  
\[x(t)=\sin(2\pi ft)[1+A_{1}\sin(2\pi t)+A_{2}\sin(2\pi 2\beta t)+\cdots+A_{m} \sin(2\pi n\beta t)] \tag{4}\]  
where \(A_{m}\) is the amplitude, \(m=\{1,2,\cdots,M\}\), and \(n\beta\) is frequency doubling. The spectrum of the above signal \(x(t)\) consists of a spectrum line located at the center frequency \(f\) (carrier signal) together with some spectrum lines symmetrically located around the frequency \(f\) and spaced by cyclic frequency \(\beta\) (modulation components) in Figure 4.  
Figure 3: Multi-harmonic signal analysis: (**a**) results of VMD with the initialization of the center frequency \(\mathcal{P}_{u}\) and their spectra (**b**); (**c**) results of VMD with the initialization of the center frequency \(\mathcal{P}_{z}\) and their spectra (**d**); (**e**) results of empirical mode decomposition (EMD) and their spectra (**f**). IMF = intrinsic mode function; BLIMF = band-limited intrinsic mode function. Actually, the core ideal of MID is the sideband filter that allows us to extract the potential carrier signal together with the corresponding modulating signals. Thus, the filtered signal by this technique only has specific signal components and reduced noise. The filtered signal can be given by [27]  
\[x_{i}=x_{\Delta f}(t,f-i\beta),i=\{-1,0,1\} \tag{5}\]  
where \(x_{\Delta f}(t,f)\) represents the filtered version of \(x(t)\) in a narrow frequency band \([f-\Delta f/2,f+\Delta f/2]\). The result of the proposed sideband filtering might be then utilized for computing the relationship between three spectral components spaced by the frequency \(\beta\) that is indicated in the presence of modulation [11]. The spectral correlation (SCor) density can be used as an indicator to detect modulation components in the signal. SCor density is defined by  
\[\text{SCor}_{x}^{\beta}(f)=\underset{\Delta f\to 0\Delta t\to\infty}{\lim}\frac{1}{\Delta t}\int_{- \Delta t/2}^{\Delta t/2}X_{1/\Delta f}\bigg{(}t,f+\frac{\beta}{2}\bigg{)}*X_{ 1/\Delta f}^{*}\bigg{(}t,f-\frac{\beta}{2}\bigg{)}dt \tag{6}\]  
where \(X_{1/\Delta f}\) is the completed envelope of the filtered version of the signal \(x(t)\) in a narrow frequency band \([f-\Delta f/2,f+\Delta f/2].X_{1/\Delta f}(t,f)\) can be calculated as an STFT with a rectangular time window [28]. \[X_{1/\Delta f}(t,f)\triangleq\int_{t-1/2\Delta f}^{t+1/2\Delta f}x(t)e^{-j2 \pi ft}dt \tag{7}\]  
As is given in Equation (8), the relationship between \(X_{1/\Delta f}\) and \(x_{\Delta f}\) is  
\[x_{\Delta f}\ (t,f)=X_{1/\Delta f}(t,f)e^{-j2\pi ft}. \tag{8}\]  
Therefore, Equation (6) may be written as a time average of a cyclic periodogram. \[\text{SC}_{x}^{\beta}(t,f)=\frac{1}{\Delta f}x_{\Delta f}\bigg{(}t,f+\frac{ \beta}{2}\bigg{)}x_{\Delta f}^{*}\bigg{(}t,f-\frac{\beta}{2}\bigg{)}e^{-j2\pi \beta t} \tag{9}\]  
Then, SCor density can be represented as follows:  
\[\text{SCor}_{x}^{\beta}(f)=\underset{\Delta f\to 0T\to\infty}{\lim}\frac{1}{T \Delta t}\int_{T}x_{\Delta f}\bigg{(}t,f+\frac{\beta}{2}\bigg{)}x_{\Delta f} ^{*}\bigg{(}t,f-\frac{\beta}{2}\bigg{)}e^{-j2\pi\beta t}dt \tag{10}\]  
where \(x_{\Delta f}(t,f)\) is defined by Equation (5). It should be noted that multiplication by \(e^{-j2\pi\beta t}\) in Equation (9) performs a frequency shifting operation by cyclic frequency \(\beta\) so as to offer the same central frequency for components \(x_{\Delta f}(t,f+\beta/2)\) and \(x_{\Delta f}(t,f-\beta/2)\).