## 2 Rationale  
### Echo Spectral Leakage  
To extract Doppler shift information from echo signals, performing Fourier transform (FT) on them is necessary [16]. The definition of FT is as follows:  
\[F(f)=\int_{-\infty}^{\infty}f(t)exp(-j2\pi ft)dt, \tag{1}\]  
where \(f(t)\) is a continuous time signal, and \(F(f)\) is the spectral function of \(f(t)\). Yet, the traditional FT suits smooth signal spectral analysis, while the echoes from different distance wind speeds are non-smooth [17]. To address the traditional FT's drawbacks in non-smooth signal processing, [PERSON] proposed the short-time Fourier transform (STFT) in 1946 [18], defined as follows:  
\[STFT_{x}(t,f)=\int_{-\infty}^{\infty}[x(\tau)w(\tau-t)]exp(-j2\pi f\tau)d\tau, \tag{2}\]  
where \(x(\tau)\) is the echo signal, and \(w(t)\) is the window function. When \(w(t)=1\), Equation (2) is the conventional Fourier transform. Assuming that \(X[exp(j2\pi f\tau)]\) is the spectral representation of the signal \(x(\tau)\), and \(W[exp(j2\pi f\tau)]\) is the spectral representation of the window function \(w(t)\), Equation (2) can be transformed according to the relationship that the time domain product corresponds to the frequency domain convolution:  
\[STFT_{x}(t,f)=\frac{1}{2\pi}X[exp(j2\pi f\tau)]*W[exp(j2\pi f\tau)], \tag{3}\]In Equation (3), the symbol '\(*\)' denotes the convolution operation. For an ideal single-frequency negative exponential signal, it can be expressed as [19]:  
\[x(f_{0}\tau)=Aexp(j2\pi f_{0}\tau), \tag{4}\]  
where \(A\) denotes the magnitude of the signal amplitude, \(f_{0}\) denotes the signal frequency shift, and the spectrum of \(x(f_{0}\tau)\) is the shock signal with amplitude \(A\) at \(f_{0}\), i.e.,:  
\[X[exp(j2\pi f_{0})]=\mathrm{A}\delta(f-f_{0}), \tag{5}\]  
This can be obtained by inserting Equation (5) into Equation (3):  
\[STFT_{x}(t,f)=\frac{A}{2\pi}W\{\exp[j2\pi(f-f_{0})]\}, \tag{6}\]  
From Equation (6), it can be seen that the essence of the short-time Fourier transform of the echo signal is to translate the windowed spectrum \(W[exp(j2\pi f)]\) by \(f_{0}\) units.  
Combined with Equation (6), the process of performing the FFT on the truncated discrete wind speed echo signal is equivalent to sampling the window function spectrum at each sampling point on the spectrum after translating the window function spectrum by \(f_{0}\) units. To visualize this process, we assume that the discrete wind field return signal \(R(m)\) is [20]:  
\[R(m)=A\cos\{2\pi f_{r}m/f_{s}+\varphi\}+N(m),m=0,1,2,\ldots,M-1, \tag{7}\]  
In Equation (7), \(N(m)\) denotes the system noise, \(A\) denotes the signal amplitude, \(\varphi\) denotes the signal phase, \(f_{r}\) denotes the frequency of the echo signal, its magnitude can be expressed as \(f_{r}=f_{0}+f_{d}\), \(f_{0}\) represents a fixed frequency introduced to distinguish between positive and negative Doppler shifts, \(f_{d}\) is the Doppler shift brought about by the wind speed of the wind field, and the period of the echo signal \(T_{r}\) can be expressed as \(T_{r}=1/f_{r}\). \(f_{s}\) is the signal sampling rate, \(M\) represents the number of signal sampling points within the window function, its relation with \(f_{s}\) can be expressed as \(t=M/f_{s}\), and \(t\) is the signal sampling time. If \(t/T_{r}=L\), where \(L\) is a positive integer, then the signal period and signal sampling time exist integer times. In this case, the echo signal frequency can be extracted by the complete sampling of the frequency point, which allows for the ideal spectrogram to be obtained. However, in the actual wind speed measurement process, it is challenging to align the echo signal frequency with the integer times of the spectral resolution. The relationship between the signal period and the signal sampling time is not an integer multiple, which results in the main spectral line energy being leaked to the window function adjacent to the main flap and the nearby side flap. This phenomenon is referred to as spectral leakage.