## 2 Methodology  
The subspace methods are based on the spectral decomposition of the covariance matrix of the signals coming from the sensors. They separate the characteristic space of the covariance matrix of the global signal into eigen signal subspace and noise subspace using an eigenvalue decomposition of the covariance matrix. The MUSIC method is an example of this type of method which was first proposed by [PERSON] in 1986 [4] to estimate the directions of arrival of signals. This method can detect several near sources and has better performance than the conventional beam forming method provided that these sources are uncorrelated or weakly correlated. If the sources are strongly correlated, this method requires modification by the uncorrelated algorithms. The LiDAR system receives \(P(P<N)\) signals emitted by radiating sources on which an additive noise is superimposed. It is assumed that the signals emitted by these sources are stationary, centered and not correlated with the additive noise. The \(P\) sources are placed in a far field, therefore assumed to be the point. Let \(X(t)\) be the signal vector observed by the LiDAR system:  
\[X(t)=[x_{0}(t)x_{1}(t)x_{2}(t)\ldots x_{N-1}(t)] \tag{1}\]  
We define the directional vector or the model which corresponds to a perfect transfer signal of the sources \(a(\tau)\) by a Gaussian function. The model signal \(a(\tau)\) corresponds to a perfect signal where the echo is diluted \(\tau\) with samples. The vector \(X(t)\) is in the form:  
\[X(t)=a(\tau)s(t)+b(t) \tag{2}\]  
* \(s(t)=\big{[}s_{1}(t)s_{2}(t)s_{3}(t)\ldots s_{p}(t)\big{]}^{T}\): Vector of the amplitudes of the signals emitted by the \(P\) sources at the instant \(t\);
* \(s(t)=\big{[}a(\tau_{1})a(\tau_{2})a(\tau_{3})\ldots a(\tau_{p})\big{]}^{T}\): Matrix of directional vectors of dimension \((N\times P)\);
* \(b(t)=[b_{0}(t)b_{1}(t)b_{2}(t)\ldots b_{N-1}(t)]^{T}\): Additive noise vector on the \(N\) sensors. The \(P\) signals from the sources are assumed to be independent, the covariance matrix of these sources is then:  
\[R_{s}=E\big{[}s(t)s^{H}(t)\big{]}=diag(p_{1}p_{2}\ldots p_{p}) \tag{3}\]  
With \(p_{i}\) the power of the \(i\)th source (obstacle) that we want to locate. It was also assumed that the signals emitted by these sources are stationary, centered and not correlated with the noise (white noise), we can therefore deduce the covariance matrix:  
\[R=E\big{[}x(t)x^{H}(t)\big{]}=a(\tau)R_{s}a^{H}(\tau)+R_{b} \tag{4}\]  
where  
* \(R_{s}=E\big{[}s(t)S^{H}(t)\big{]}\): covariance matrix of the source signals of dimension \((P\times P)\);
* \(R_{b}=E\big{[}b(t)b^{H}(t)\big{]}=\sigma^{2}I\): ambient noise covariance matrix of dimension \((N\times N)\), \(\sigma^{2}\) where is the noise variance and \(I\) is the identity matrix;  
In practice, we estimate the covariance matrix from a finite number of temporal samples in the form:  
\[\hat{R}_{xx}=\frac{1}{T}\sum_{k=1}^{T}(X_{k}X_{k}^{H}) \tag{5}\]  
where \(X_{k}\) is the signal vector sampled at time \(k\) and \(T\) is the number of samples. The spectral decomposition of the covariance matrix \(R\) into eigen elements to separate the signal subspace from the noise subspace can be expressed in the following form:  
\[R=a(\tau)R_{s}a^{H}(\tau)+\sigma^{2}I=U_{s}\wedge_{s}U_{s}^{H}+\sigma^{2}U_{b} U_{b}^{H} \tag{6}\]where \(\wedge\) is the diagonal matrix of the eigenvalues of the matrix \(R\) and \(U\) is the matrix formed of the eigenvectors corresponding to the eigenvalues of the matrix \(R\) (classified in decreasing order). Therefore, we have:  
\[\wedge=diag\big{[}\lambda_{1}>\lambda_{2}>\ldots\lambda_{p}>\lambda_{p+1}= \lambda_{p+2}=\ldots=\lambda_{N}=\sigma^{2}\big{]} \tag{7}\]  
And \(U=[U_{s}U_{b}]\) with \(U_{s}=\big{[}u_{1}u_{2}\ldots u_{p}\big{]}\), \(U_{b}=\big{[}u_{p+1}u_{p+2}\ldots u_{N}\big{]}\) where \(U_{s}\) and \(U_{b}\) represent, respectively, the matrices of the eigenvalues associated with the signal subspace and with the noise subspace.