## 3 Proposed ESM Block-DNN Detector
### B-DNN Parameters and Training  
An array of L fully connected neural layers, including L-1 hidden layers, is designed to decode the signals emanating from each active transmit antenna at the transmitter. Table 2 delineates the parameters of the proposed DNN, including the count of layers, where the \(l\)th layer is endowed with \(\delta_{l}\) nodes. We use \(\lambda\) to represent the comprehensive collection of parameters within the DNN, where \(\lambda\) is defined as \(\lambda=\{\lambda_{1},\lambda_{2},...,\lambda_{L}\}\). The parameters specific to the \(l\)th layer are signified by \(\lambda_{l}=\{\mathbf{W}^{l},\mathbf{b}_{l}\}\). Consequently, the \(l\)th layer is delineated as  
\[\mathbf{Z}_{l}=\sigma(\mathbf{W}^{l}\mathbf{Z}_{l-1}+\mathbf{b}_{l}), \tag{11}\]  
where the weight matrix can be expressed as \(\mathbf{W}^{l}\in\mathbb{R}^{\delta-1}\times\delta_{l}\) and the deviation vector \(\mathbf{b}_{l}\in\mathbb{R}^{\delta\times 1}\), \(\sigma(\cdot)\) is the activation function. Apart from the final layer, the end of each layer uses a famous method, called Rectified linear unit (ReLU), and the expression is \(\sigma(x)=max(0,x)\). The derivative of this method consistently yields a singular value,oscillating between the binary states of zero and one, thereby preventing the exponential diminution of gradient magnitude as it traverses the depth of numerous layers. The ReLU function facilitates rapid learning within the DNN. This in turn renders the efficient training of a profound supervised network feasible, obviating the need for unsupervised pre-training, as elucidated in [20]. In the penultimate layer, we employ the renowned softmax function to map the output onto a continuum that spans the interval \([0,1]\), and the mapping of the input and output of the DNN can be represented by the following equation:  
\[\mathbf{Z}_{L} =\sigma(\mathbf{W}^{(L)}(\sigma(\mathbf{W}^{(L-1)}(...\sigma( \mathbf{W}^{(1)}\mathbf{Z}_{0}+\mathbf{b}_{1})...)\] \[+\mathbf{b}_{L-1}))+\mathbf{b}_{L})). \tag{12}\]  
We utilize categorical cross-entropy to ascertain the discrepancy in the cost function between the actual data and the predicted outcomes, thereby facilitating the identification of parameters that enhance our network's optimization. We let \(\mathbf{Z}_{T}\) represent the one-hot encoded vector signifying the labels for the supervised training phase, which contrasts with the output of the predictions, denoted as \(\mathbf{Z}_{L}\). The expression for this method is given as follows:  
\[L(\mathbf{Z}_{T},\mathbf{Z}_{L})=-\sum_{n=1}^{M}Z_{T_{n}}log(Z_{L_{n}}), \tag{13}\]  
where the difference between \(Z_{T_{n}}\) and \(Z_{L_{n}}\) should be minimized. The objective is to effectuate its reduction to the greatest extent possible. To achieve this, we employ the stochastic gradient descent (SGD) optimization technique. This process involves the strategic adjustment of the existing weight values by either incrementing or decrementing them in accordance with the gradient's magnitude, modulated by the learning rate, thus facilitating an efficient traversal towards the minimum of the optimization target. The parameter \(\eta\), known as the learning rate, is a crucial hyper-parameter that is constrained to the interval from zero to one. It plays a significant role in the SGD algorithm, which incrementally refines the parameters over iterations through the application of gradient information. The following equation illustrates this process of updating the parameters:  
\[\mathbf{W}_{t+1}^{(l)}=\mathbf{W}_{t}^{(l)}-\eta\frac{\partial L(\mathbf{Z}_{ T},\mathbf{Z}_{L})}{\partial\mathbf{W}_{t}^{(l)}}. \tag{14}\]  
The iterative refinement of weights persists in a cyclical fashion until the cost function's value indicates a state of equilibrium over successive iterations. Upon attaining the optimal set of weights, the capability to make predictions is enabled through the utilization of the trained feed-forward DNN. Throughout the training phase, it is essential to have the input vectors for training and their corresponding tags.