## 2 FMCW LiDAR
### FMCW LiDAR Key Parameters
#### 2.3.4 Random Noise and Signal Processing  
In practice, FMCW LiDAR is subjected to various types of interference, and random noise is also an important influencing factor that leads to a decrease in range accuracy. Since the laser transmission loses a large amount of optical energy in the atmosphere, the received echo signal has a very low signal-to-noise ratio and is susceptible to noise interference. The main noise in the ranging process is generated by the internal components of the system, as well as noise entering the system through the receiver, including time-domain waveform continuity, amplitude, and phase of random noise, belonging to the Gaussian noise. The power spectral density distribution of these noises is again a constant value; thus, the noise in the differential frequency signal can be treated as Gaussian white noise. Figure 6 illustrates the actual signal waveform detected by the FMCW LiDAR. The detected echo signal in the figure contains a sinusoidal signal; however, the superposition of random noise leads to varying degrees of distortion in the amplitude and phase of the waveform, thereby reducing the signal-to-noise ratio of the signal. Excessive noise power can lead to incorrect estimation of spectral peaks in spectral analysis, resulting in a reduction in range accuracy. Therefore, denoising the signals in the processing of the acquired signals is necessary to ensure that the frequency values of the differential frequency signals can be correctly extracted. Due to the advantages of digital signal processing, the fast Fourier transform (FFT) spectral analysis method is often employed to process the beat frequency signals. Following discrete processing, the expression for the sequence of the beat frequency signal is obtained as follows [18]:  
\[\left\{ \begin{matrix}x(n)=A(cos\omega_{0}n+\varphi_{0}),n\in[0,N-1]\\ \omega_{0}=\frac{2\pi\cdot f_{0}}{f_{s}},\omega_{0}\epsilon[0,2\pi].\end{matrix} \right. \tag{12}\]  
In the equation, \(x(n)\) represents the discretely processed beat frequency signal, and \(N\) is the sampling length. \(f_{0}\) is the frequency of the beat frequency signal, and \(f_{s}\) is the sampling frequency. \(\omega_{0}\) denotes the signal's discrete angular frequency, and \(\varphi_{0}\) represents the initial phase of the signal. A represents the amplitude of the signal, which indicates the strength or magnitude of the difference frequency signal. The variable \(n\) denotes the discrete time index, which is the sequence number of the signal at discrete time points, typically ranging from \(0\) to \(N-1\). Based on the conjugate symmetry property of the FFT spectrum of the real signal, it is generally sufficient to analyze only the first \(N/2\) points of the discrete spectrum. The expression for the power spectrum \(G(K)\) of the signal \(x(n)\) for the first \(N/2\) points is given by  
\[G(K)=\frac{A^{2}}{2}\cdot\left|\frac{sin(\pi(K-K_{0}))}{\pi(K-K_{0})}\right|,K\in\left[0,\frac{N}{2}-1\right],K_{0}=\frac{Nf_{0}}{f_{s}}. \tag{13}\]  
In the formula, \(K_{0}\) represents the position of the signal frequency within the discrete spectrum; \(N\) is the sampling length; \(f_{0}\) is the frequency of the beat frequency signal; \(f_{s}\) is the sampling frequency; \(A\) represents the amplitude of the signal, which indicates the strength or magnitude of the difference frequency signal; and the variable \(K\) represents an index in the discrete frequency spectrum, indicating the \(K_{th}\) frequency bin in the spectrum, that is, the frequency coordinate corresponding to the spectral peak. The power spectrum \(G(K)\) of the signal \(x(n)\) is illustrated in Figure 7, where \(m\) denotes the index of the spectral line with the peak amplitude. When the frequency under test coincides exactly with the peak spectral line of the discrete spectrum, i.e., \(K=K_{0}\), the amplitude of the power spectrum reaches its maximum value. In this case, the frequency estimation equals the exact true value, resulting in zero estimation error.