# Pilot-Based Adaptive Channel Estimation for Underwater Spatial Modulation Technologies
## 5 Recursive Least-Squares Adaptive Channel Estimation  
RLS channel estimation serves as a distinct adaptive filter method employed for finding the coefficients recursively to reduce the cost function of a weighted linear least-squares corresponding to the input signal. The mentioned algorithm is highly contrasting to the other algorithms, just as LMS minimizes the MSE. This paper uses the RLS method in SMTs for channel estimation using pilot symbols, which is represented by \(\mathbf{x}_{p,i}\). The pilot symbols initially undergo sequential transmission via transmit antennas along with the channel coefficients for the optimum transmission as well as antenna receiver, and then pilots are estimated by the proposed algorithm on an individual level for the whole duration of every pilot symbol. In this work, the received signal pilot symbols are represented by \(\mathbf{y}_{r_{x},p,i}\), where \(r_{x}=1,2,...,N_{r}\) and \(P\) is pilot. The received signals go through sequential processing at the receiver end, and the up-gradation of channel estimations is carried out upon the arrival of new samples. This section briefly illustrates the recursive computation of least-squares channel estimations. The transmitted pilot symbols can be written as:  
\[\mathbf{x}_{p,i}=[x_{P,1},x_{P,2},...,x_{P,N_{P}}], \tag{22}\]  
where \(N_{P}\) is the total number of pilots for active antennas. The received pilots from sub-channels of active antennas can be represented as:  
\[\mathbf{y}_{r_{x},p,i}=[y_{r_{x},P,1},y_{r_{x},P,2},...,y_{r_{x},P,N_{P}}]. \tag{23}\]  
The set of linear filters output are defined by the following equation:  
\[\Im_{r_{x},g,i}=\widehat{\mathbf{h}}_{r_{x},g,i}\mathbf{x}_{P,i},\qquad i=1,2,...,N_{P} \tag{24}\]  
where \(\widehat{h}_{r_{x},g,i}\) are the channel coefficients between the \(g^{th}\) transmitted antennas and received antennas \(r_{x}\) which are recursively estimated to minimize the sum of square errors denoted as:  
\[e_{i}=\sum_{n=1}^{i}\lambda(y_{r_{x},P,n}-\widehat{\mathbf{h}}_{r_{x},g,n}x_{ P,n})^{2}, \tag{25}\]  
where \(\lambda\) is the weighting factor, which reduces the interference of the old data and its selected value \(0<\lambda\leq 1\). Figure 2 shows the basic adaptive channel estimation model which is used to estimate the UWA-MIMO channel of the SMTs. Here, \(u_{i}\) is the input of the unknown sampled system as well as an adaptive filter. The response of the unknown sample system \(d_{i}\) is measured in the presence of noise \(v\). The adaptive filter response \(y_{i}\) is subtracted from the unknown sample system output \(d_{i}\). The error signal \(e_{i}\) is the difference of adaptive filter and unknown sample system outputs, which is used for updating the adaptive filter coefficients. If the system is correctly devised, the transfer function of the adaptive filter will converge to the transfer function of the unknown sample system after a known number of Monte Carlo iterations. Here, the adaptive filter has been employed to estimate the coefficients of the UWA-MIMO channel using the RLS algorithm. The complexity of adaptive filter RLS is \(O(N^{2})\) per iteration, where \(N\) is the filter size, but it is well known because of fast convergence [42; 43]. The RLS steps are defined in Algorithm 1. **Input:** The individual entries of \(\mathbf{y}\) are represented by \(d_{i}\) as \(\mathbf{y}=[d_{1},d_{2},...,d_{i}]^{T}\), and the individual rows of \(\mathbf{H}\) can be denoted by \(u_{i}\) as \(\mathbf{H}=[\mathbf{u}_{1},\mathbf{u}_{2},...,\mathbf{u}_{i}]^{T}\). **Initialization:**\(\mathbf{w}_{0}=[\quad]\), \(\mathbf{P}_{0}=\epsilon\mathbf{I}\) the initial covariance matrix \(\mathbf{P}\), here \(\epsilon\) is a large positive number and \(\lambda=0.995\).