# Deep Learning-Enhanced Autoencoder for Multi-Carrier Wireless Systems
## 2 System Model  
In an MC-AE system, the incoming message is passed through the modulation section, which consists of a DL-based encoder, generating a transmitted data vector. This data vector is then processed through a time-domain OFDM (t-OFDM) operation. Specifically, an inverse fast Fourier transform (IFFT) is applied to convert the data into the time domain, and a cyclic prefix is added to combat inter-symbol interference before the signal is transmitted through a fading channel with noise. At the receiver, the cyclic prefix is removed, and an inverse time-domain OFDM operation is performed using a fast Fourier transform (FFT) to convert the signal back into the frequency domain. The M-ary QAM/PSK modulation or several newly developed instant messaging systems [37] are examples of modulation schemes. The resulting received signal is then passed through a demodulation block, which consists of a DL-based decoder that generates the estimated output. The general structure of the proposed MC-AE system is shown in Figure 1. In our proposed system, the incoming message \(s\) is fed as input to the encoder. In the MC-AE encoder, each incoming message \(s\epsilon S=\{s_{1},\ldots,s_{M}\}\), where \(M=2^{m}\) and consists of a bit-stream of \(m\) bits, is mapped into a one-hot vector \(s\) of size \(M\times 1\). This vector has all zero entries except for a single one. Afterward, the DL-based encoder produces the transmitted data vector \(\mathbf{\bar{x}}=[x_{1},x_{2},\ldots,x_{N_{c}}]^{T}\), where \(N_{c}\) denotes the number of sub-carriers. In a manner similar to the IM-based scheme [38], the proposed MC-AE system splits the total \(N_{c}\) subcarriers into \(G\) groups, each containing \(N\) subcarriers, such that \(N_{c}=NG\). The AE structure is then applied independently to each group. Following processing by the t-OFDM block, \(\mathbf{x}\) is transmitted to the receiver where it is impeded by additive noise \(\mathbf{w}\) after first going via the fading channel \(\mathbf{H}\). After performing the inverse t-OFDM operation at the receiver, the received signal \(\mathbf{y}\) is obtained. The received received frequency domain signal \(\mathbf{y}\) is expressed as follows:  
\[\mathbf{y}=\mathbf{H}\odot\mathbf{x}+\mathbf{w}, \tag{1}\]  
In this equation, \(w\) represents the AWGN, where \(\mathbf{w}_{i}\sim\mathcal{CN}(0,\sigma^{2})\). The \(\mathbf{H}=[H_{1},\ldots,H_{N}]\) denotes the Rayleigh fading channel, where \(\mathcal{H}_{i}\sim\mathcal{CN}(0,1)\) and the operator \(\odot\) signifies element-wise multiplication. It is assumed that the average energy of the transmitted M-ary symbol is \(E_{a}\), resulting in an average SNR at the receiver of \(\overline{\gamma}=E_{a}/\sigma^{2}\). In order to decode the signal, we assume that the received signal \(\mathbf{y}\) and the CSI are both fully known at the receiver and are fed into the decoder as inputs. The real vector \(\mathbf{\hat{x}}=[\mathbf{y}_{R},\mathbf{y}_{I},\mathbf{H}_{R},\mathbf{H}_{I }]^{T}\) is specifically created from the complex vectors \(\mathbf{y}\) and \(\mathbf{H}\) where the real and imaginary components of \(\mathbf{y}\) and \(\mathbf{H}\) are denoted, respectively, by \(\mathbf{y}_{R},\mathbf{y}_{I}\) and \(\mathbf{H}_{R},\mathbf{H}_{I}\). The proposed MC-AE significantly differs from traditional RTN-based AE schemes [29]. RTN is designed for block fading channels where sub-carrier channels remain constant over multiple uses. This allows RTN to operate without CSI, relying instead on the domain knowledge of a channel equalizer. These properties make it a model-driven, noncoherent approach. On the other hand, the proposed MC-AE is designed for time-varying channel conditions, in which the channel coefficients fluctuate arbitrarily with each usage. This flexibility enables the proposed scheme to utilize perfect CSI at the receiver in a fully data-driven manner to exploit frequency diversity across different sub-carriers, moving away from the reliance on domain knowledge characteristic of RTN.