# Z-OFDM: A New High-Performance Solution for Underwater Acoustic Communication
## 3 Communication System Model
### Theory
#### 3.1.2 Receiver Model  
The signal is subject to multipath propagation within the limited bandwidth through the underwater channel. Multipath propagation causes ISI, which restricts the data through put and degrades the system's performance. For this reason, the UWA channel impulse response is represented by [40]  
\[h_{c}(t)=\sum_{l=0}^{N}a_{l}\delta(t-\tau_{l}) \tag{10}\]  
in which both \(a_{l}\) and \(\tau_{l}\) are the amplitude gain and Delay on the \(l\) path.  
After passing through the UWA channel, the received signal can be written as [41]  
\[y(t) =h_{c}(t)*x(t)+w(t) \tag{11}\] \[=\sum_{l=0}^{L}a_{l}x(t-\tau_{l})+w(t)\]  
where \(w(t)\) is the additive white Gaussian noise (AWGN) process.  
At the receiver, the first step is frequency down-conversion, which translates the received signal from its carrier frequency to baseband. This is represented by  
\[y_{m}(t)=y(t)e^{i2\pi f_{c}t} \tag{12}\]  
where \(f_{c}\) is the carrier frequency.  
After frequency down-conversion, the signal \(y_{m}(t)\) undergoes serial-to-parallel conversion. This process converts the entire signal into multiple OFDM symbols. Each OFDM symbol contains a CP that must be removed for restoring the original symbol. Once the CP is removed, the signal is frequency-shifted to near zero frequency and then undergoes to low-pass filtering using a Kaiser window function to prevent spectral aliasing during the downsampling. The process is represented as  
\[y_{f}(t)=y_{m}(t)e^{i\omega_{0}t}*h_{r}(t) \tag{13}\]  
where \(h_{r}(t)\) is the impulse response of the low-pass filter.  
After low-pass filtering, the signal is down-sampled to enhance its resolution. Down-sampling involves selecting every \(D\)-th sample, and the process is described as  
\[y_{D}(t)=y_{f}(tD) \tag{14}\]  
where \(D\) is the down-sampling factor.  
This step ensures that the sampling rate of the received signal returns to the original baseband sampling rate.  
The down-sampled signal is then transformed to the frequency domain using FFT. At this step, the refined spectrum of the received signal is obtained. The FFT process is represented as  
\[Y_{z}(\omega)=\sum_{t=0}^{N-1}y_{D}(t)e^{-i\omega t} \tag{15}\]  
where \(Y_{z}(\omega)\) is the frequency-domain representation, and \(N\) is the FFT size.  
Let \(\omega_{c}=2\pi f_{c}\). The complete calculation of the ZoomFFT output at the \(l\) subcarrier is as follows:  
\[\begin{split} Y_{z}(\omega)&=\mathcal{F}(y(tD)e^{ i(\omega_{c}+\omega_{0})tD}*h_{r}(tD))\\ &=H_{C}(\frac{\omega-(\omega_{c}+\omega_{0})}{D})X(\frac{\omega- (\omega_{c}+\omega_{0})}{D})H_{r}(\frac{\omega}{D})+W(\omega)\end{split} \tag{16}\]  
Subsequently, the refined spectrum of the received signal is subjected to channel equalization. This process eliminates the impact of the channel on the data symbols, thereby reducing the BER. The channel's target frequency band spectrum is refined using ZoomFFT. By combining the refined spectrum with the Minimum Mean Square Error (MMSE) equalizer, the influence of the channel on the received signal can be more effectively mitigated. The principle of the equalizer \(G(k)\), obtained through the methods in references [42; 43; 44], is given as shown in Equation (17).  
\[\begin{split}\dot{S}(k)&=G(k)d(k),\\ G(k)&=\Big{(}Y_{z}Y_{z}^{H}+\sigma^{2}I\Big{)}^{-1} Y_{z}e_{i},\\ Y_{z}e_{i}&=E[S_{i}R_{z}^{(i)}]\end{split} \tag{17}\]  
Then, by performing channel equalization on \(Y_{z}(\omega)\), the effects of channel transmission are eliminated. Finally, the modulated information \(X_{p}\) is extracted from the selected refined spectrum \(Y_{z}(\omega)\), and by performing QPSK demapping on \(X_{p}\), the binary data are obtained.