## 2 A Convex Combination Scheme for Adaptive MISO Filters  
An adaptive MISO system is composed of a number \(P\) of adaptive filters in parallel, each one receiving a different input, and yields a single output. A combined system of MISO filters is characterized by the adaptive combination of the outputs of different MISO filters, which can be differentiated in several ways according to the updating rule or some parameter settings. In this paper,we consider the adaptive combination of a number of \(J=2\) MISO filters adapted by using the same updating rule, defined by the regularized multichannel APA [25; 27], but different projection orders. The input signals are the same for each MISO system and are collected into data matrices. However, due to the different projection order, each MISO system uses its own sub-matrix. In particular, the \(p^{\text{th}}\) input sub-matrix of the \(j^{\text{th}}\) MISO system, for \(p=1,\ldots,P\) and \(j=1,2\), is represented as \(\mathbf{X}_{p,n}^{(j)}\in\mathbb{R}^{K_{j}\times M}\):  
\[\mathbf{X}_{p,n}^{(j)} =\left[ \begin{array}{ccc}\mathbf{x}_{p,n}^{(j)}&\mathbf{x}_{p,n -1}^{(j)}&\ldots&\mathbf{x}_{p,n-K_{j}+1}^{(j)}\end{array} \right]^{T}\] \[=\left[ \begin{array}{ccc}x_{p}\left[n\right]&\ldots&x_{p}\left[n -M+1\right]\\ x_{p}\left[n-1\right]&\ldots&x_{p}\left[n-M\right]\\ \vdots&\ddots&\vdots\\ x_{p}\left[n-K_{j}+1\right]&\ldots&x_{p}\left[n-M-K_{j}+2\right]\end{array}  \right] \tag{1}\]  
where \(M\) is the length of the adaptive filters (same for all) and \(K_{j}\) denotes the number of previous entries to keep in memory, i.e., the projection order, for the \(j^{\text{th}}\) MISO system. The concatenation of sub-matrices (1) provides the input data matrix for each MISO system, i.e., \(\mathbf{X}_{n}^{(j)}\in\mathbb{R}^{K_{j}\times MP}=\left[ \begin{array}{ccc} \mathbf{X}_{1,n}^{(j)}&\mathbf{X}_{2,n}^{(j)}&\ldots&\mathbf{X}_{P,n}^{(j)} \end{array} \right]\). Each MISO filter can be represented by \(\mathbf{w}_{n}^{(j)}\in\mathbb{R}^{MP\times 1}\left[ \begin{array}{ccc} \mathbf{w}_{1,n}^{(j),T}&\mathbf{w}_{2,n}^{(j),T}&\ldots&\mathbf{w}_{P,n}^{(j), T}\end{array} \right]^{T}\), with \(j=1,2\). The output vector of each MISO filter \(\mathbf{y}_{n}^{(j)}\in\mathbb{R}^{K_{j}\times 1}=\left[ \begin{array}{ccc}y_{0}^{(j)} \left[n\right]&y_{1}^{(j)}\left[n\right]&\ldots&y_{K_{j}-1}^{(j)}\left[n\right] \end{array} \right]^{T}\) is then obtained as:  
\[\mathbf{y}_{n}^{(j)}=\mathbf{X}_{n}^{(j)}\mathbf{w}_{n-1}^{(j)}. \tag{2}\]  
Taking into account the vector \(\mathbf{d}_{n}^{(j)}\in\mathbb{R}^{K_{j}\times 1}=\left[ \begin{array}{ccc}d \left[n\right]&d\left[n-1\right]&\ldots&d\left[n-K_{j}+1\right]\end{array}  \right]^{T}\) containing the last \(K_{j}\) samples of the desired signal, it is possible to achieve the error signal vector \(\mathbf{e}_{n}^{(j)}\in\mathbb{R}^{K_{j}\times 1}=\left[ \begin{array}{ccc}e_{0}^{(j)} \left[n\right]&e_{1}(j)\left[n\right]&\ldots&e_{K_{j}-1}(j)\left[n\right]\end{array}  \right]^{T}\) for each MISO filter:  
\[\mathbf{e}_{n}^{(j)}=\mathbf{d}_{n}^{(j)}-\mathbf{y}_{n}^{(j)}. \tag{3}\]  
It is worth noting that the difference of notation between the \(k^{\text{th}}\) entry \(d\left[n-k\right]\) of \(\mathbf{d}_{n}^{(j)}\) and the entry \(y_{k}\left[n\right]\) of \(\mathbf{y}_{n}^{(j)}\) (and the same for \(\mathbf{e}_{n}^{(j)}\)) is due to the fact that \(\mathbf{d}_{n}^{(j)}\) is a regression vector collecting the last \(K_{j}\) samples of the desired signal, while all the entries of \(\mathbf{y}_{n}^{(j)}\) and \(\mathbf{e}_{n}^{(j)}\) are computed at each time instant. This means, for example, that the entry of the desired signal for \(k=0\) at the \(n-1\) time instant has the same value of the entry for \(k=1\) at the \(n^{\text{th}}\) time instant, and they both are denoted with \(d\left[n-1\right]\). On the other hand, all the entries of the output and signal vectors may change at each time instant until convergence; hence, for example, \(y_{0}^{(j)}\left[n-1\right]\
eq y_{1}^{(j)}\left[n\right]\).