## 4 Overview of Linear Adaptive Filter Techniques  
The MP detector presented in this paper derives from the technique described in [12], based on the Linear Adaptive Filter (LAF) theory. A brief description of the LAF methods is given here, to motivate the new version of the MP detector described in this paper.  
A complete overview of _adaptive algorithms_ is available in [21], where adaptive techniques are presented for channel equalisation. In particular LAF methods are described in [22; 23]. In the LAF theory, a generic measured signal is modeled as \(d[n]=y[n]+n_{0}[n]\), where \(n_{0}[n]\) is a noise sequence and \(y[n]\) is  
\[y[n]=\sum_{k=0}^{M-1}w_{k}^{*}u[n-k]\]  
where \(u[n]\) is a reference signal, and \(w_{k}^{*}\) are the coefficients of a Finite Impulse Response (FIR) filter, which provides an approximated version \(y[n]\) of the input signal \(d[n]\). The length of the filter \(M\) represents the number of delayed components that decompose the incoming signal. The LAF coefficients \(w_{k}^{*}\) are estimated by minimising the cost function  
\[\epsilon=\sum_{n=i_{1}}^{i_{2}}\mid e[n]\mid^{2} \tag{3}\]  
that is the error energy, where \(e[n]=y[n]-d[n]\) is the _residual error_, and \(i_{1}\) and \(i_{2}\) are the limits of the sum. The values assigned to the limits depend on which kind of _data windowing_ is chosen. In [12] we use the so-called covariance method, where \(i_{1}=M\) and \(i_{2}=N\), and no assumption is made on the data outside the interval \([1,N]\). With this approach the minimisation problem can be solved in matrix form, after introducing the quantities  
Figure 7: Zoom around 10 s of \(\rho_{clean}-\rho_{MP}\) in the case \(\Delta T_{\rho}=500\) ms in blue and MP profile in red.  
\[\mathbf{U}^{\mathbf{H}}=\begin{bmatrix}u(M)&u(M+1)&\cdots&u(N)\\ u(M-1)&u(M)&\cdots&u(N-1)\\ \vdots&\vdots&\ddots&\vdots\\ u(1)&u(2)&\cdots&u(N-M+1)\end{bmatrix} \tag{4}\]  
\[\mathbf{\Phi}=\mathbf{U}^{H}\mathbf{U}\]  
and  
\[\mathbf{z}=\mathbf{U}^{H}\mathbf{D}\]  
where \(\mathbf{D}=[d[M],d[M+1],\cdots,d[N]]^{T}\). It is possible to prove that the vector of estimated coefficients \(\hat{\mathbf{w}}=[w_{0},\cdots,w_{M-1}]^{T}\) can be written as  
\[\hat{\mathbf{w}}=\mathbf{\Phi}^{-1}\mathbf{z} \tag{5}\]  
In this way, the problem of estimating \(\hat{\mathbf{w}}\) turns in an inversion matrix problem, by assuming \(\mathbf{\Phi}\) is nonsingular, therefore invertible.  
Another approach to window the data is to define the matrix \(\mathbf{U}\) as  
\[\mathbf{U}^{\mathbf{H}}=\begin{bmatrix}u(1)&u(2)&\cdots&u(N)&u(N)&\cdots&u(N )\\ u(1)&u(1)&\cdots&u(N-1)&u(N)&\cdots&u(N)\\ \vdots&\vdots&\ddots&\vdots&&\\ u(1)&u(1)&\cdots&u(N-M+1)&u(N-M)&\cdots&u(N)\end{bmatrix} \tag{6}\]  
where some a priori assumptions are made, simply replicating the value \(u(1)\) before \(i_{1}=1\) and replicating the value \(u(N)\) after \(i_{2}=N\), as in Equation (6). In this paper we describe a method based on this approach, whose advantages are explained in Section 5. The output provided by the LAF method is then used to define a test statistic for distortions detection. Therefore the 'adaptation' of the coefficients at each new measurement allows us to continuously update the value of the test statistic.