# A Novel Transfer Learning-Based OFDM Receiver Design for Enhanced Underwater Acoustic Communication
## 2 Overview of the UWA OFDM Communication System  
### Conventional UWA OFDM Communication System  
A schematic representation of the UWA OFDM system, as discussed in [36; 37; 38; 39; 40; 41], is shown in Figure 1. The process begins with the generation of a binary data stream, to which pilot tones are added for the purpose of estimating the channel impulse response (CIR). This data is then modulated as shown in Figure 1. Following this, the inverse fast fourier transform (IFFT) is applied to shift the signal from the frequency domain to the time domain, utilizing \(N\) orthogonal narrowband subcarriers. The resulting time domain signal \(x(n)\) is computed from the frequency domain symbols \(X(k)\) as follows [23]:  
\[x(n)=\text{IFFT}\{X(k)\}=\frac{1}{N}\sum_{k=0}^{N-1}X(k)e^{j2\pi\frac{uk}{N}}, \quad n=0,1,\ldots,N-1. \tag{1}\]  
where:  
* \(X(k)\) is the frequency-domain representation of the OFDM symbol
* \(x(n)\) is the time-domain signal obtained after IFFT,
* \(N\) is the total number of subcarriers,
* \(k\) is the subcarrier index, ranging from \(0\) to \(N-1\),
* \(n\) is the time-domain sample index, ranging from \(0\) to \(N-1\).  
After the IFFT operation, a cyclic prefix (CP) is appended to the time domain signal to help counteract inter symbol interference (ISI) as depicted in Figure 1. The CP extended signal, \(x_{g}(n)\), is defined as [23]:  
\[x_{g}(n)=\begin{cases}x(N+n),&\text{for }n=-N_{g},-N_{g}+1,\ldots,-1\\ x(n),&\text{for }n=0,1,\ldots,N-1.\end{cases} \tag{2}\]  
Figure 1: The conventional UWA OFDM communication system.  
Here, \(N_{g}\) indicates the number of samples in the CP. This step involves copying the last \(N_{g}\) samples of \(x(n)\) and placing them at the beginning of the symbol to form a guard interval. The resulting signal \(x_{g}(n)\) has a total duration of \(N+N_{g}\).  
Once the signal propagates through the UWA channel, the received signal \(y_{g}(n)\) is expressed as:  
\[y_{g}(n)=x_{g}(n)\otimes h(n)+w(n),\quad-N_{g}\leq n\leq N-1. \tag{3}\]  
In this expression, \(\otimes\) represents circular convolution, \(h(n)\) denotes the CIR, and \(w(n)\) is zero-mean Additive White Gaussian Noise (AWGN). The CIR can be modeled as:  
\[h(n)=\sum_{i=0}^{r-1}h_{i}\delta(n-\tau_{i}), \tag{4}\]  
where \(\delta(\cdot)\) is the Dirac delta function, \(r\) is the number of channel paths, and \(h_{i}\), \(\tau_{i}\) represents the complex gain and corresponding delay of the \(i\)-th path, respectively.  
At the receiver side is in Figure 1, the cyclic prefix is removed, and the time-domain signal \(y(n)\) is transformed back into the frequency domain using the Fast Fourier Transform (FFT) as follows:  
\[Y(k)=\text{FFT}\big{\{}y(n)\big{\}}=\frac{1}{N}\sum_{n=0}^{N-1}y(n)e^{-\beta \pi\frac{nk}{N}},\quad k=0,1,\ldots,N-1. \tag{5}\]  
Assuming ISI is effectively eliminated, the frequency-domain received signal can be modeled as [23]:  
\[Y(k)=X(k)H(k)+W(k),\quad k=0,1,\ldots,N-1, \tag{6}\]  
where \(H(k)\) and \(W(k)\) are the Fourier transforms of the CIR \(h(n)\) and noise \(w(n)\), respectively. This equation establishes the relationship between the transmitted and received signals in the frequency domain over an UWA channel.