## 3 Reducing Transmission Complexity in MIMO-WCDMA Networks Using PCA  
As it was shown in [6], the selection and processing of all multipath components in frequency-selective MIMO-WCDMA networks can improve mean BER. In this context, an iterative transmission approach was proposed that calculates transmission matrices in Equation (8). This algorithm, which is described in Algorithm 1 for the \(k\)th user, successively calculates the RAKE multiplying vectors for each multipath component, the total signal power from Equation (8) as well as the transmit weight vector of the desired user, until convergence is achieved. Note that \(\mathbf{I}_{dim}\) is the \(dim\times dim\) identity matrix, tr(X) is the trace of matrix X and \(\mathbf{D}\) is a diagonal matrix indicating the power ratio per transmission mode. However, it is evident from Figure 1, where the pdf of the number of repeats of the Maximization of Signal-to-Noise Ratio (MSNR) algorithm of Algorithm 1 is presented (i.e., mean value of parameter \(n\) that indicates total repetitions for convergence), that the mean value can be up to 285 repeats. Note that the curve of Figure 1 is depicted for quanta of \(L^{2}\), since the multi\(-\)cation steps to calculate matrix \(\mathbf{A}\) in Algorithm 1 are also included in the total number of repeats of the MSNR algorithm. In our proposed approach, the correlation matrix of the received data is first formulated, with dimensions \(L\times L\) and sorted afterwards. The (\(i\),\(j\)) entry of this matrix indicates the degree of correlation among the signals of two different multipaths. Therefore, instead of considering all \(L^{2}\) terms in matrix \(\mathbf{A}\) formulation, \(\mathbf{C}\) terms are selected that correspond to the first \(\mathbf{C}\) dominant terms of the correlation matrix. In this approach, it is assumed that all received data for an arbitrary user after RAKE processing can be stacked as an \(L\times S\) matrix (denoted as \(\mathbf{Z}\) throughout the rest of the manuscript), where \(S\) indicates the samples per observation [13]. The received data matrix per transmission mode will be given by:  
\[\mathbf{Z}_{k,d}(l,s)=z_{k,d,l,s},1\leq d\leq dim,1\leq l\leq L,1\leq s\leq S \tag{21}\]  
In Equation (21), \(z_{k,d,l,s}\) is the \(s\)th received symbol from the \(k\)th user in the \(l\)th RAKE finger and \(d\)th transmission mode (\(S\) denotes the total number of transmitted symbols). Hence, the correlation matrix will be given by:  
\[\mathbf{C}_{\mathbf{Z}_{k,d}}\big{(}l,l^{\prime}\big{)}=\frac{1}{S}\sum_{s=1} ^{S}z_{k,d,l,s}\overline{z_{k,d,l,s}},\ 1\leq d\leq dim,1\leq l,l^{\prime}\leq L,1\leq s,s^{\prime}\leq S \tag{22}\]  
Figure 1: Probability density function (PDF) curve of the number of repeats of the Maximization of Signal to Noise Ratio (MSNR) algorithm (Algorithm 1). With the help of PCA, a different representation of \(\mathbf{Z}_{k,d}\) can be formulated (i.e., \(\mathbf{Y}_{k,d}=\mathbf{P}\mathbf{Z}_{k,d}\)), so that the new correlation matrix, namely \(\mathbf{C}_{\mathbf{Y}_{k,d}}=(1/S)\mathbf{Y}_{k,d}\mathbf{Y}_{k,d}^{\mathrm{H}}\) will be a diagonal one. In this case, the final set of data can be decomposed to individual orthogonal representations, also known as principal components. The importance of each principal direction is now measurable. Namely, the variances associated to each direction (i.e., rows of \(\mathbf{P}\)) quantify how \"principal\" each direction is by rank-ordering each basis vector according to the corresponding variances. The reconstruction of matrix \(\mathbf{C}_{\mathbf{Z}_{k,d}}\) is performed according to the following equation:  
\[\mathbf{C}_{\mathbf{Z}_{k,d}}=\mathbf{V}(1:L,1:w)^{\mathrm{H}}\Sigma(1:w,1:w) \mathbf{V}(1:L,1:w) \tag{23}\]  
where \(w\) (\(1\leq w\leq L\)) is the number of dimensions that are considered at matrix reconstruction, while \(\mathbf{V}\) and \(\mathbf{\Sigma}\) correspond to the eigenvector and diagonal eigenvalue matrix, respectively, of \(\mathbf{C}_{\mathbf{Z}_{k,d}}\).