# Intelligent Transmit Antenna Selection Schemes for High-Rate Fully Generalized Spatial Modulation
## 3 Antenna Selection Schemes for FGSM
### Generation of a Dataset  
Elements of channel matrices are used to generate the training dataset. A multiclass classification technique is then used to divide the channel matrix into numerous corresponding classes, each of which represents the optimal antenna subset. ML methods are used to build the model. The developed models can also be used to predict the class labels of test data. Four major steps are involved in the collection of the dataset: (1) Generation of training data. (2) Extraction of feature vectors. (3) Evaluation of KPI. (4) Use of the KPI information to obtain the class labels. 1. Generation of training data. To form the training data, the \(N\) number of channel matrices of size \(N_{rx}\times N_{T}\) are randomly generated. \[\mathbb{H}=\Big{[}\mathbf{H}_{T}^{1},\mathbf{H}_{T}^{2},\ldots,\mathbf{H}_{T}^{N}\Big{]}.\] (16)
2. Extraction of feature vectors The training performance is generally influenced by the feature vectors. In this work, four different features, i.e., \(\Big{|}h_{i,j}^{n}\Big{|}\), \(\measuredangle\Big{(}h_{i,j}^{n}\Big{)}\), channel gain \(\Big{|}h_{i,j}^{n}\Big{|}^{2}\), and channel gain of every element of the correlation matrix \(\Big{|}(\mathbf{H}_{T}^{n})^{H}(\mathbf{H}_{T}^{n})\Big{|}^{2}\), are considered to train the models. From each sample \(\mathbf{H}_{T}\), a feature of size \(N_{Q}\)\((N_{rx}\times N_{T})\) is extracted. The process of extracting these four features is as follows: (a) \(\Big{|}h_{i,j}^{n}\Big{|}\): The absolute value of each element of the channel matrix is considered as a dominating feature, which is used throughout this work. The extraction of the feature vector, in this case, is given as follows: \[\mathbf{f}^{n}=\Big{[}\big{|}h_{1,1}^{n}\big{|},\ldots,\big{|}h_{1,N_{T}}^{n}\big{|},\ldots,\big{|}h_{i,j}^{n}\big{|},\ldots,\big{|}h_{N_{rx},N_{T}}^{n}\big{|} \Big{]},n\in\{1,2,...N\},\] (17) where \(h_{i,j}^{n}\) is a \((i,j)\)th complex element of \(\mathbf{H}_{T}^{n}\), whose absolute value is calculated as follows: \[\Big{|}h_{i,j}^{n}\Big{|}=\sqrt{\Re\Big{(}h_{i,j}^{n}\Big{)}^{2}+\Im\Big{(}h_ {i,j}^{n}\Big{)}^{2}}.\] (18) (b) \(\measuredangle\Big{(}h_{i,j}^{n}\Big{)}\): This is the second feature that is extracted to build the model from the elements of \(\mathbf{H}_{T}\). The feature vector is constructed as follows: \[\mathbf{f}^{n}=\Big{[}\measuredangle\Big{(}h_{1,1}^{n}\big{)},\ldots,\measuredangle \Big{(}h_{1,N_{T}}^{n}\Big{)},\ldots,\measuredangle\Big{(}h_{i,j}^{n}\Big{)}, \ldots,\measuredangle\Big{(}h_{N_{rx},N_{T}}^{n}\Big{)}\Big{]},n\in\{1,2,...N\},\] (19) where \[\measuredangle\Big{(}h_{i,j}^{n}\Big{)}=\tan^{-1}\Bigg{(}\frac{\Im\Big{(}h_{i,j}^ {n}\Big{)}}{\Re\Big{(}h_{i,j}^{n}\Big{)}}\Bigg{)}.\] (20)Channel gain \(\left|h_{i,j}^{n}\right|^{2}\): The channel gain of an individual element of the matrix is considered to be another feature to train the models whose feature vector is given by: \[\mathbf{f}^{n}=\left[\left|h_{1,1}^{n}\right|^{2},\ldots,\left|h_{1,N_{T}}^{n}\right| ^{2},\ldots,\left|h_{i,j}^{n}\right|^{2},\ldots\ \left|h_{N_{rx},N_{T}}^{n}\right|^{2}\right],n\in\{1,2,...N\}.\] (21) (d) Squared element norm of \(\left(\mathbf{H}_{T}^{n}\right)^{H}(\mathbf{H}_{T}^{n})\): In addition, the similarity between two distinct column vectors is a major feature. Similar column vectors introduce detection ambiguity while estimating the transmit antenna index. The feature vector extraction is conducted as follows: \[\left|\mathbf{H}_{T}^{nH}\mathbf{H}_{T}^{n}\right|^{2}=\left[ \begin{array}{c}(\mathbf{h}_ {1}^{n})^{H}\\ (\mathbf{h}_{2}^{n})^{H}\\ \vdots\\ (\mathbf{h}_{N_{T}}^{n})^{H}\end{array} \right]\left[ \begin{array}{c}\mathbf{h}_{1}^ {n}\ \mathbf{h}_{2}^{n}\ \ldots\ \mathbf{h}_{N_{T}}^{n}\end{array} \right].\] (22) To avoid bias, features are normalized using \[\bar{f}^{n}(q)=\frac{f^{n}(q)-E[(\mathbf{f}(q)]}{max(\mathbf{f}(q))-min((\mathbf{f}(q)))},n =1,\ldots,N,q=1,\ldots,N_{Q}.\] (23)
3. Evaluation of KPI. Input data samples are labeled using KPI. FD is considered as the KPI in this work, which is calculated using (14). 4.