## 3 Optimum Mixing Parameters and EMSE
### Stationary Data Model  
Let us consider the following linear regression model:  
\[d\left[n\right]=\mathbf{x}_{n}^{T}\mathbf{w}^{o}+v\left[n\right] \tag{11}\]  
where \(\mathbf{x}_{n}\in\mathbb{R}^{MP\times 1}=\left[ \begin{array}{cc}\mathbf{x}_{1,n}^{T}& \ldots&\mathbf{x}_{P,n}^{T}\end{array} \right]^{T}\) is the first projection (i.e., for \(k=0\)) of the data input matrix \(\mathbf{X}_{n}^{\left(j\right)}\) (for the simplicity of notation, we omit the projection order subscript for \(k=0\)), and it is the same for both MISO filters. Moreover, in (11), \(\mathbf{w}^{o}\in\mathbb{R}^{MP\times 1}=\left[ \begin{array}{cc}\mathbf{w}_{1}^{o,T}& \ldots&\mathbf{w}_{P}^{o,T}\end{array} \right]^{T}\) is the optimal multichannel weight solution. With respect to the model of (11), we consider the following assumptions [6, 41, 42]. **Assumption 1**. : _The signal \(v\left[n\right]\) represents an independent and identically distributed (i.i.d.) noise added to the system output, whose variance is \(\sigma_{v}^{2}\)._  
**Assumption 2**. : _The input signals are assumed to be statistically independent of the noise signal._  
**Assumption 3**. : _We also assume that \(\mathrm{E}\left\{\mathbf{x}_{p,n}\right\}=\mathbf{0}\), \(\mathrm{E}\left\{d\left[n\right]\right\}=0\), \(\mathrm{E}\left\{v\left[n\right]\right\}=0\), and define the covariance matrix \(\mathrm{E}\left\{\mathbf{x}_{p,n}\mathbf{x}_{p,n}^{T}\right\}=\mathbf{R}_{p,n}\)._  
We can define the weight error vector for the \(j^{\text{th}}\) MISO filter and for the multichannel combined scheme, respectively, as:  
\[\widetilde{\mathbf{w}}_{n}^{\left(j\right)} \in\mathbb{R}^{MP\times 1}=\mathbf{w}^{\text{o}}-\mathbf{w}_{n}^{ \left(j\right)} \tag{12}\] \[\widetilde{\mathbf{w}}_{n} \in\mathbb{R}^{MP\times 1}=\mathbf{w}^{\text{o}}-\mathbf{w}_{n}. \tag{13}\]  
We also introduce the a priori and the a posteriori estimation error signals for the \(j^{\text{th}}\) MISO system, respectively, as:  
\[\mathbf{e}_{an}^{\left(j\right)} \in\mathbb{R}^{K_{j}\times 1}=\mathbf{e}_{n}^{\left(j\right)}- \mathbf{v}_{n}^{\left(j\right)}=\mathbf{X}_{n}^{\left(j\right)}\widetilde{ \mathbf{w}}_{n-1}^{\left(j\right)} \tag{14}\] \[\mathbf{e}_{\mathbb{P}_{n}^{\left(j\right)}} \in\mathbb{R}^{K_{j}\times 1}=\mathbf{X}_{n}^{\left(j\right)} \widetilde{\mathbf{w}}_{n}^{\left(j\right)} \tag{15}\]  
and the a priori and the a posteriori estimation error signals for the combination scheme, respectively, as:  
\[\mathbf{e}_{an} \in\mathbb{R}^{K_{\max}\times 1}=\mathbf{e}_{n}-\mathbf{v}_{n}= \mathbf{X}_{n}\widetilde{\mathbf{w}}_{n-1} \tag{16}\] \[\mathbf{e}_{\mathbb{P}_{n}} \in\mathbb{R}^{K_{\max}\times 1}=\mathbf{X}_{n}\widetilde{\mathbf{w}}_{n} \tag{17}\]  
where \(\mathbf{v}_{n}^{\left(j\right)}\in\mathbb{R}^{K_{j}\times 1}=\left[ \begin{array}{cc}v\left[n\right]&v\left[n-1\right]&\ldots&v\left[n-K_{j}+1 \right]\\ \end{array} \right]^{T}\) contains the last \(K_{j}\) samples of the irreducible noise, while \(\mathbf{v}_{n}\in\mathbb{R}^{K_{\max}\times 1}\) contains the last \(K_{\max}\) samples of \(v\left[n\right]\), and \(\mathbf{X}_{n}\in\mathbb{R}^{K_{\max}\times 1}\). It is worth noting that while in the steady-state definition of the model (11), we consider the input at its current time instant, in (14) and (16), we take into account also the projections of the input signal. Considering (10) and (5), the a priori error (16) can be also written as:  
\[\mathbf{e}_{an}\in\mathbb{R}^{K_{\max}\times 1}=\left\{ \begin{array}{ll} \boldsymbol{\lambda}_{n}\odot\mathbf{e}_{an}^{\left(1\right)}+\left(\mathbf{1 }-\boldsymbol{\lambda}_{n}\right)\odot\left[\mathbf{e}_{an}^{\left(2\right)}; \mathbf{0}^{\left(1\right)}\right],&\text{if }K_{\max}=K_{1}\\ \boldsymbol{\lambda}_{n}\odot\left[\mathbf{e}_{an}^{\left(1\right)};\mathbf{0 }^{\left(2\right)}\right]+\left(\mathbf{1}-\boldsymbol{\lambda}_{n}\right) \odot\mathbf{e}_{an}^{\left(2\right)},&\text{if }K_{\max}=K_{2}\\ \end{array} \right.