## 3 Architecture of the Proposed HDL Detector  
Figure 2 illustrates the structure of the proposed HDL model. Similar to existing OFDM-IM detection techniques, the receiver is assumed to have knowledge of the channel information. As a result, both the channel \(\mathbf{h}\) and received signal \(\mathbf{y}\) are considered as inputs to the HDL model. To enhance detection performance, \(\mathbf{y}\) and \(\mathbf{h}\) undergo preprocessing using domain knowledge from OFDM-IM before being fed into the HDL detector. To obtain the equalized received signal vector, the widely used zero-forcing (ZF) equalizer is applied [38]. This is represented as follows:  
\[\mathbf{\hat{y}}=\mathbf{y}\odot\mathbf{h}^{-1} \tag{4}\]  
where the received signal \(\mathbf{y}\) is multiplied element-wise by the inverse of the channel \(\mathbf{h}\). Along with this equalizer, we also utilized the minimum mean square error (MMSE) and decision feedback equalizer (DFE) to compare their performance. By minimizing the mean square error (MSE) between the transmitted and received signals, the MMSE equalizer optimizes the tradeoff between inter-symbol interference (ISI) suppression and noise enhancement [39]. The equalization process of the MMSE equalizer is expressed as follows:  
\[\mathbf{\hat{y}}=\frac{y\cdot h}{|h|^{2}+\sigma^{2}} \tag{5}\]  
where \(\mathbf{\hat{y}}\) represents the received signal equalized by MMSE, \(|h|^{2}\) represents the squared magnitude of the channel coefficient, and \(\sigma^{2}\) is the noise variance. The DFE equalizer achieves equalization by using the received signal along with feedback from previously detected symbols to reduce the ISI [40]. The equalization process of the DFE equalizer can be represented as follows:  
\[\mathbf{\hat{y}}=\mathbf{y}-\mathbf{d}\cdot\mathbf{\hat{y}} \tag{6}\]  
where \(\mathbf{\hat{y}}\) is the equalized output signal, \(\mathbf{d}\) represents the feedback filter applied to the previously detected symbols, and \(\mathbf{\hat{y}}\) denotes the decision (or detected) output from the previous symbol decisions. To generate the HDL's input, the energy of the received signal \(\mathbf{y}_{c}=\left[|y_{1}|^{2},|y_{2}|^{2},\dots,|y_{N}|^{2}\right]\) is calculated and then added to \(\mathbf{\hat{y}}\). The \(3N\)-dimensional input vector \(\mathbf{1}=\left[\mathbf{\hat{y}}_{R},\mathbf{\hat{y}}_{I},\mathbf{y}_{c}\right]\) is created by concatenating the real and imaginary parts of \(\mathbf{\hat{y}}\), respectively denoted as \(\mathbf{y}_{R}\) and \(\mathbf{y}_{I}\), with the received energy vector \(\mathbf{y}_{c}\), as illustrated in Figure 2.  
Our proposed HDL detector employs an organized methodology consisting of several essential elements: a 1D-CNN layer, a batch normalization layer, a Bi-LSTM layer, a one-dimensional max-pooling layer, and a dense layer with a sigmoid activation function in the output layer. The processed data \(I\) are fed into the 1D-CNN layer. A total of \(F=64\) filters of  
Figure 2: Structure of the proposed HDL detector for OFDM-IM systems. various \(3\times 3\) sizes are used in the convolutional layer to extract features [41]. The function of the 1D-CNN model with the data \(I\) can be explained as follows:  
\[\mathbf{U}=f(\mathbf{I}_{-1}\otimes\mathbf{w}_{i}+\mathbf{b}_{i}) \tag{7}\]  
where the weight and bias vectors of the \(i\)-th layer are indicated by \(\mathbf{w}_{i}\) and \(\mathbf{b}_{i}\), respectively. The batch normalization (BN) layer follows next. The BN technique is used to normalize the input of each layer in order to minimize internal covariate shifts and stabilize the learning process. This enables neural networks to achieve faster training, greater stability, and improved performance [42]. The Bi-LSTM layer takes its input from the output of the BN layer, denoted as \(B\). The proposed Bi-LSTM layer is configured with \(Q=128\) hidden units. Two LSTM layers (forward and backward layers) make up a Bi-LSTM hidden layer. Figure 3 shows that each LSTM cell contains three gates, namely, the input, forget, and output gates [43].