# Design of a SIMO Deep Learning-Based Chaos Shift Keying (Dlcsk) Communication System
## 3 SIMO DLCSK System Model
### Receiver  
We consider multiple antennas and LSTMs at the receiver end to establish a SIMO design and to obtain a diversity gain. This architecture relies on an ensemble method to fuse several classifiers with the goal of increasing the classification accuracy. In the following, we introduce the structure of a single LSTM-based classifier. The proposed classifier has five base layers: Sequence input layer, LSTM/BiLSTM, fully connected layer (size 2), Softmax, and Classification layer. The sequence input layer is only used to fetch sequential input values of length \(2\beta\). The adopted LSTM cell (unit) is shown in Figure 2.  
The forget gate \(f_{t}\) determines how much of the current cell state should be forgotten, and the output gate \(o_{t}\) controls which part of the information should be sent to the output. \(C_{t-1}\) and \(C_{t}\), respectively, show the state value of the memory unit at the previous step and current step. Then, \(h_{t-1}\) and \(h_{t}\) indicate the output of the previous and the current states, respectively, whereas \(r_{t}\) and \(\bar{\ }\) represents the current input and sigmoid function, respectively. Equation (10) illustrates the LSTM cell calculation process [77]. The forget gate decides which information will be remembered or forgotten based on the last hidden layer output \(h_{t-1}\) and the current input \(r_{t}\). The memory cell value \(C_{t}\) is determined by the current value \(C_{t}\), its own state \(C_{t-1}\), input, and forget gates. The operator (\(*\)) represents element-wise matrix multiplication, while (\(\cdot\)) denotes point multiplication. \(w\) represents the weight and \(b\) is the bias parameter. \[i_{t}=\mu\ (w_{i}\cdot[h_{t-1},r_{t}]+b_{i}), \tag{10a}\] \[f_{t}=\mu\ (w_{f}\cdot[h_{t-1},r_{t}]+b_{f}),\] (10b) \[C_{t}=\tanh\ (w_{c}\cdot[h_{t-1},r_{t}]+b_{c}),\] (10c) \[C_{t}=f_{t}*C_{t-1}+i_{t}*C_{t},\] (10d) \[o_{t}=\mu\ (w_{o}\cdot[h_{t-1},r_{t}]+b_{o}),\] (10e) \[h_{t}=o_{t}*\tanh(C_{t}). \tag{10f}\]  
In this work, motivated by the features of LSTM, a Bidirectional (BiLSTM) arrangement is implemented for classification tasks. The hidden state of BiLSTM at times \(t\) can be calculated by the weighted sum of the forward hidden state \(\overrightarrow{h_{t}}\) and the backward hidden state \(\overleftarrow{h_{t}}\) as follows:  
\[\overrightarrow{h_{t}}=\text{LSTM}(r_{t},\overrightarrow{h_{t-1 }}), \tag{11a}\] \[\overleftarrow{h_{t}}=\text{LSTM}(r_{t},\overleftarrow{h_{t-1 }}),\] (11b) \[U_{t}=w_{t}\overrightarrow{h_{t}}+v_{t}\overleftarrow{h_{t}}+b_ {t}, \tag{11c}\]  
where \(w_{t}\) and \(v_{t}\) denote the weights corresponding to \(\overrightarrow{h_{t}}\) and \(\overleftarrow{h_{t}}\), respectively. The number of hidden units indicates the number of BiLSTM units that need to be placed in the hidden layer of the network. The fully connected layer receives the output of the BiLSTM layer in order to increase the stability of the output by performing more non-linear operations. There are two fully connected layers for two output classes. The Softmax layer is an activation function that  
\[i_{t}=\mu\ (w_{i}\cdot[h_{t-1},r_{t}]+b_{i}), \tag{10a}\]0 The LSTM cell structure. calculates a probability for each sequence and sends results to the next layer. This layer contains two nodes, which is the same as the number of output classes. The utilized Softmax function can be written as [78],  
\[i_{t}=\mu\ (w_{i}\cdot[h_{t-1},r_{t}]+b_{i}), \tag{10a}\]1  
where \(\gamma(U_{t})_{j}=p_{n,j}\) is the probability that vector \(U_{t}\) is a member of \(j\)th class \((j\in\{0,1\})\), and \(I=\{0,1\}\) is a set of all possible classes. In other words, \(p_{n,0}\) and \(p_{n,1}\), \((1\leq n\leq N)\), represents the probability that the transmitted chaotic signal is \"0\" or \"1\", respectively.