## 2 A Convex Combination Scheme for Adaptive MISO Filters
Therefore, the update rule of each MISO filter using the regularized multichannel affine projection algorithm [27; 28]:  
\[\mathbf{w}_{n}^{(j)}=\mathbf{w}_{n-1}^{(j)}+\mu_{j}\mathbf{X}_{n}^{(j),T} \left(\delta_{j}\mathbf{I}+\mathbf{X}_{n}^{(j)}\mathbf{X}_{n}^{(j),T}\right)^ {-1}\mathbf{e}_{n}^{(j)} \tag{4}\]  
where \(\mu_{j}\) is the step size and \(\delta_{j}\) is the regularization factor for the \(j^{\text{th}}\) MISO filter, which is considered the same for both MISO filters. The chosen method for combining MISO filters is the system-by-system combination scheme [8], depicted in Figure 1, since it involves the minimum necessary number of free parameters for single-stage combined architectures. According to this scheme, the overall output of the system is achieved by combining convexly the MISO filter outputs (2). This combination involves the sum of buffers with different lengths, which depends on the projection order \(K_{j}\). This problem is addressed by using zero-padding, thus:  
\[\mathbf{z}_{n}\left\{ \begin{array}{ll}\in\mathbb{R}^{K_{1}\times 1}=&\mathbf{ \lambda}_{n}\odot\mathbf{y}_{n}^{(1)}+(\mathbf{1}-\mathbf{\lambda}_{n})\odot\left[ \mathbf{y}_{n}^{(2)};\mathbf{0}^{(1)}\right],&\text{if }K_{1}\geq K_{2}\\ \in\mathbb{R}^{K_{2}\times 1}=&\mathbf{\lambda}_{n}\odot\left[\mathbf{y}_{n}^{(1)}; \mathbf{0}^{(2)}\right]+(\mathbf{1}-\mathbf{\lambda}_{n})\odot\mathbf{y}_{n}^{(2) },&\text{if }K_{1}<K_{2}\end{array} \right. \tag{5}\]  
where \(\mathbf{0}^{(1)}\in\mathbb{R}^{(K_{1}-K_{2})\times 1}\) and \(\mathbf{0}^{(2)}\in\mathbb{R}^{(K_{2}-K_{1})\times 1}\) are vectors of zeros, \(\mathbf{1}\in\mathbb{R}^{K_{\max}\times 1}\) are vectors of ones, where \(K_{\max}=\max\left\{K_{1},K_{2}\right\}\), and \(\mathbf{\lambda}_{n}\in\mathbb{R}^{K_{\max}\times 1}\) are mixing parameter vectors for \(j=1,2\); finally, \(\odot\) denotes the Hadamard product. The mixing parameters are adaptive weights that balance the combination between the two MISO filters, giving more importance to the best-performing one. Such awareness is obtained according to a mean squared error minimization. The adaptation of \(\mathbf{\lambda}_{n}\) is performed by using an auxiliary vector \(\mathbf{a}_{n}\), whose elements are related to those of \(\mathbf{\lambda}_{n}\) by means of a sigmoidal function defined according to [8; 44]:  
\[\lambda_{k}\left[n\right]=\beta\left(\frac{1}{1+e^{-a_{k}\left[n-1\right]}}- \alpha\right),\qquad\qquad k=0,\ldots,K_{\max}-1 \tag{6}\]  
where:  
\[\alpha=\frac{1}{\left(1+e^{4}\right)},\qquad\qquad\beta=\frac{1}{1-2\alpha}. \tag{7}\]  
The auxiliary vectors are updated by using a gradient descent rule; therefore, for \(k=0,\ldots,K_{\max}-1\):  
\[a_{k}\left[n\right]=a_{k}\left[n-1\right]+\frac{\mu_{c}}{\beta r_{k}\left[n \right]}e_{k}\left[n\right]\left(y_{k}^{(1)}\left[n\right]-y_{k}^{(2)}\left[ n\right]\right)\cdot\left(\lambda_{k}\left[n\right]+\alpha\beta\right)\left(\beta- \alpha\beta-\lambda_{k}\left[n\right]\right) \tag{8}\]  
where \(\mu_{c}/r_{k}\left[n\right]\) represents a normalized step size [45],  
\[r_{k}\left[n\right]=\gamma r_{k}\left[n-1\right]+\left(1-\gamma\right)\left(y_ {k}^{(1)}\left[n\right]-y_{k}^{(2)}\left[n\right]\right)^{2} \tag{9}\]  
Figure 1: System-by-system combination scheme of two adaptive MISO filters. is the estimated power of \(\left(y_{k}^{\left(1\right)}\left[n\right]-y_{k}^{\left(2\right)}\left[n\right]\right)\), and \(\gamma\) is a smoothing factor close to one. The entries \(e_{k}\left[n\right]\) in (8), for \(k=0,\ldots,K_{\max}-1\), of the overall error signal vector \(\mathbf{e}_{n}\) are derived as:  
\[\mathbf{e}_{n}\in\mathbb{R}^{K_{\max}\times 1}=\left\{ \begin{array}{ll} \mathbf{d}_{n}^{\left(1\right)}-\mathbf{z}_{n},&\quad\text{if }K_{\max}=K_{1}\\ \mathbf{d}_{n}^{\left(2\right)}-\mathbf{z}_{n},&\quad\text{if }K_{\max}=K_{2} \end{array}