## 4 Ultrasonic-Assisted RIS Phase Shift Matrix Design Based on Subspace Self-Organizing Iterations
### RIS Initial Phase Shift Matrix Solving Based on Ultrasonic-Assisted PF Algorithm  
We estimate the ultrasonic DOA arriving at the ultrasonic array and design a rough phase shift matrix of RIS accordingly. The particle filter (PF) algorithm is an algorithm for real-time localization and tracking, and its goal is to recursively update the results by continuously updating the particle weights based on the measurement model and the state transition model. In this algorithm, the observation likelihood function can be expressed as  
\[P(\hat{\textbf{Y}}^{\prime}(k)|\varsigma_{i}(k))=\left\{\frac{\widetilde{P}( \hat{\textbf{Y}}^{\prime}(k)|\varsigma_{i}(k))}{\max_{\varsigma_{i}(k)} \widetilde{P}(\hat{\textbf{Y}}^{\prime}(k)|\varsigma_{i}(k))}\right\}^{\omega} \tag{38}\]  
where \(\omega\) is the exponential factor determined by the simulation experiment [25]. \(\widetilde{P}(\hat{\textbf{Y}}^{\prime}(k)|\varsigma_{i}(k))\) denotes the likelihood function determined by the maximum likelihood estimation method. \(\varsigma_{i}(k)\) is the particle, and \(i=1,2,\ldots,\rho\); the number of particles is \(\rho\). The initial weight of each particle is \(1/\rho\), and the initial value is generated randomly. Since the conventional likelihood function is greatly affected by environmental conditions such as a low signal-to-noise ratio, the update of particles will become difficult. We apply the propagation operator method in MUSIC spectrum estimation to improve the likelihood function. The covariance matrix of the ultrasonic transducer array output data in (39) is expressed as  
\[\mathfrak{S}=E[\hat{\textbf{Y}}^{\prime}(k)\hat{\textbf{Y}}^{\prime}H(k)] \tag{39}\]  
The covariance matrix is chunked into the following form:  
\[\mathfrak{S}= \begin{bmatrix}\mathfrak{S}_{1}\\ \mathfrak{S}_{2}\end{bmatrix}  \tag{40}\]  
where \(\mathfrak{S}_{1}\) and \(\mathfrak{S}_{2}\) denote \(N_{r}\times N_{r}\) and \((N_{t}-N_{r})\times N_{r}\) dimensional matrices, respectively. \(\mathfrak{S}_{1}\) is a non-singular matrix with linear independent column vectors. According to the propagation operator method, when noise is not considered, there exists a matrix that acts as a propagation operator, such that  
\[\mathfrak{S}_{2}=\boldsymbol{\Xi}^{H}\mathfrak{S}_{1} \tag{41}\]  
Considering the presence of Gaussian noise in the real situation and the finite length of the array output, the problem of solving the propagation operator matrix \(\boldsymbol{\Xi}\) can be formulated as the following minimization problem:  
\[ \begin{array}{l}\underset{\Omega}{\text{min}}||\mathfrak{S}_{2}-\boldsymbol{ \Xi}^{H}\mathfrak{S}_{1}||_{F}\\ \text{s.t.}\ \mathfrak{S}_{2}\rightarrow\boldsymbol{\Xi}^{H}\mathfrak{S}_{1} \end{array}  \tag{42}\]  
The propagation operator matrix \(\boldsymbol{\Xi}\) can be found as follows:  
\[\boldsymbol{\Xi}=(\mathfrak{S}_{1}\mathfrak{S}_{1}^{H})^{-1}\mathfrak{S}_{1} ^{H}\mathfrak{S}_{2} \tag{43}\]  
Construct the matrix \(\boldsymbol{\Gamma}\)\(=[\boldsymbol{\Xi}^{H},\textbf{I}_{\textbf{N}_{t}-N_{r}}]\), where \(\textbf{I}_{\textbf{N}_{t}-N_{r}}\) is a \(N_{t}-N_{r}\) dimensional unit matrix, such that it satisfies the following condition  
\[\boldsymbol{\Gamma}^{H}\mathfrak{S}=0 \tag{44}\]  
We use the idea of the MUSIC algorithm, i.e., the signal is orthogonal to the noise subspace so that the spectral function can be estimated.