## 2 Rationale
### HSCW Overview  
In the process of spectrum correction using the energy center method, the number of spectral lines within the main lobe of the window function affects the overall algorithm complexity of the system. The narrower the main lobe, the simpler the energy centroid algorithm.  
The comparison diagram of classic window functions is shown in Figure 1. Compared to other typical window functions such as RW, Blackman window, and five-term MSD window, the Hanning window, as a representative of traditional cosine combination windows, has a main lobe width of \(8\pi/N\), second only to the RW.  
However, the side-lobe attenuation level of the Hanning window is \(-32\) dB, and the attenuation rate is \(18\) dB/oct. Compared with other window functions, its suppression effect on spectrum leakage is poor, which is not conducive to achieving high-precision wind speed measurement. How to solve the contradiction between the side-lobe attenuation level, side-lobe attenuation rate, and window function main-lobe width is a key research direction to improve the accuracy of the CDWL.  
To address this issue, this article utilizes window function self-convolution to enhance the spectral leakage suppression capability of the Hanning window. The window function self-convolution can be defined as [21]:  
\[W^{p}(n)=\frac{\mathrm{w}(M)*\mathrm{w}(M)*\mathrm{w}(M)\ldots*\mathrm{w}(M)}{ \mathrm{P}}, \tag{8}\]  
In Equation (8), \(\mathrm{p}\) denotes the self-convolution order, and \(\mathrm{w}(M)\) is a window function of length \(M\). According to the convolution theorem, it can be seen that the convolution of \(\mathrm{p}\) identical window functions can be obtained as a sequence \(W^{p}(n)\) of the length \(\mathrm{pM}-1\), and the \(\mathrm{p}\) order self-convolutional window function of the length \(\mathrm{pM}\) can be obtained by padding the zero. The Hanning window frequency domain expression is:  
\[W(n)=0.5W_{R}\left(\frac{2n\pi}{H}\right)-0.25\left[W_{R}\left(\frac{2n\pi}{H }-\frac{2\pi}{M}\right)+W_{R}\left(\frac{2n\pi}{H}+\frac{2\pi}{M}\right) \right], \tag{9}\]  
In Equation (9), \(W_{R}(n)\) denotes the RW spectral function, and \(H\) is the number of cosine terms of the window function. The corresponding \(\mathrm{p}\) order self-convolutional window function frequency domain expression is:  
\[W^{p}(n)=\left\{0.5W_{R}\left(\frac{2n\pi}{H}\right)-0.25\left[W_{R}\left( \frac{2n\pi}{H}-\frac{2\pi}{M}\right)+W_{R}\left(\frac{2n\pi}{H}+\frac{2\pi}{ M}\right)\right]\right\}^{p}, \tag{10}\]  
Figure 1: Comparison of classic window function spectrograms: (**a**) Rectangular window; (**b**) Hanning window; (**c**) Blackman window; (**d**) Five-term MSD window.  
The p order Hanning window satisfies \(H=1\), \(\left|W_{.p}(n)\right|=0\) at the zeros of the spectral lines in the spectrogram of the self-convolutional window, at this point [22]:  
\[\left\{\begin{array}{l}\frac{u\pi}{p}=d\pi\\ \pi\left(\frac{n}{pM}-\frac{1}{M}\right)\
eq d\pi\end{array}\right.\qquad d= \pm 1,\pm 2,\pm 3,\ldots,\pm\infty\,, \tag{11}\]  
Equation (11) holds when \(n=2 dp,d>0\), and when \(d=1\), \(n=2p\). This leads to a p-order Hanning window self-convolutional window function with a principal flap width \(B_{W}\) of:  
\[B_{W}=\frac{8p\pi}{pM}=\frac{8\pi}{M} \tag{12}\]  
When \(M=200\), the spectrogram of the HSCW first- to third-order convolutional window function is shown in Figure 2.