## 3 Methodology
### CNN-Based Joint Channel Estimation, Equalization, and Demodulation  
The model we designed for OFDM symbol handling was inspired by a CNN with depthwise convolutions [4]. Our model takes the place of a traditional receiver's channel estimation, equalization, and OFDM demodulation blocks by taking the output of the Fast Fourier Transform as input and then generating each coded bit's Log-Likelihood Ratio (LLR) as an output. The model input is similar to that outlined in [3], with the real and imaginary matrices for the received signal \(Y\), received pilot signals \(H\), and expected pilots \(X\) stacked along the third dimension. For this paper, we examine different model sizes to show the merits of not only the achievable performance gains but also the potential for reduced complexity.  
The model architecture is comprised of only four layer types: 2D convolution, 1 \(\times\) 1 2D convolution, Batch Normalization, and Rectified Linear Unit (ReLU). These layers combined together perform a Depthwise Separable Convolution in the convolutional block. For this paper, three different models of varying sizes are evaluated for our comparative analysis, with their fundamental architecture shown in Figure 2.  
In this study, we use an OFDM transceiver system with 64 subcarriers, out of which 11 are for the guardband regions and 1 is the DC subcarrier, plus 6 subcarriers used for pilots in a comb pattern, leaving 46 data subcarriers, with a frame comprised of a variable number of OFDM symbols to closely match the bits of the encoded LDPC code. We used QPSK, 16-QAM, and 64-QAM for data modulation, in a SISO system, tested with the Rician channel model as well as the four different LOS models provided by 3 GPP TR 38.900 (CDL-D/E, TDL-D/E) [15]. Matlab's 5G toolbox was utilized to generate the dataset, with each test constituting a dataset of 500,000 frames for each modulation scheme and channel model for training. The train, test, and validation split was configured to be 40%, 10%, and 50%, respectively, to ensure a sufficiently sized validation set for the purposes of this evaluation. Three models were trained to compare the parameter size against the achieved BER performance, and show the feasibility of this implementation on resource-constrained devices.  
Hyper-Parameter Tuning  
For hyper-parameter tuning, layer channels were tested for values [4; 16; 64; 128; 256], and dilation values were tested with values {(1, 1), (2, 3), and (3, 6)}, and the number of convolutional blocks was selected between 1, 3, and 5. Each model was tested for convergence against a 4-QAM AWGN dataset. Convergence occurred on most models with 3 and 5 convolutional blocks. Thus, the smallest, largest, and a representative mid-sized model were selected for comparative evaluation. The model overview of the highest-performing models is shown in Table 2.  
Figure 2: CNN-based joint OFDM receiver signal operations.  
The model output represents the bit LLRs, obtained as a 3-dimensional array of dimensions \(I\times J\times K\), where \(I\) is the number of data+pilot subcarriers, \(J\) is the number of OFDM symbols, and \(K\) is the number of bits per subcarrier. Pilot signals are discarded from the output of the demodulator deep learning model, as they are not considered in the final data bit array and do not have the same modulation as the data subcarriers. Each output data bit \(b_{ijk}\;\;\forall\;i\in I,j\in J,k\in K\) is then translated from the LLR to a probability utilizing the Sigmoid function \(x_{ijk}=\frac{1}{1+e^{-i\beta_{jk}}}\). We use a Binary Cross-Entropy loss function for model training (Equation (1)), where \(B\) is the number of bits per subcarrier, \(R\) is the set of data subcarriers in a transmitted frame, and \(y_{ijk}\) are the target coded bits.  
\[L=-\frac{1}{RB}\sum_{k}^{B}\sum_{ij}^{R}(y_{ijk}(log(x_{ijk}))+(1-y_{ijk})(log(1 -x_{ijk}))) \tag{1}\]