## 2 System Model of the TR Acoustic Communication Using FMT
### Receive Structure
#### 2.2.2 TR Process  
It is observed from Equations (3) and (4) that the channel response for the \(i\)-th subcarrier is:  
\[H_{{}_{i}}\left(\omega\right) =H\left(\omega+\frac{2\pi}{M}i\right) \tag{6}\]  
Equation (6) shows that the channel responses for all the subcarriers are different from each other. Therefore, in order to process the output signals of FMT demodulator using TR technique, the channel response of each subcarrier should be estimated individually.  
From Figure 3, it is observed that the output signal of the \(m\)-th matched filter \(\hat{H}_{{}_{m}}^{*}\left(\omega\right)\) can be expressed as:  
\[z_{{}_{m}}\left(nK\right) =\sum_{i=1}^{M}\sum_{l=\infty}^{\infty}\sum_{\lambda=\infty}^{ \infty}\sum_{k=\infty}^{\infty}\sum_{p=\infty}^{\infty}g\left(k-p-lK\right)h_ {{}_{i}}\left(p\right)g^{*}\left(\lambda K-k\right)\hat{h}_{{}_{m}}^{*}\left( nK-\lambda K\right)e^{\frac{2\pi}{M}\left(i-m\right)k}a_{{}_{i}}\left(lK\right)\] \[+\sum_{\lambda=\infty}^{+\infty}\eta_{{}_{m}}\left(\lambda K \right)\hat{h}_{{}_{m}}^{*}\left(nK-\lambda K\right) \tag{7}\] \[=\sum_{i=1}^{M}\sum_{l=\infty}^{\infty}q^{\left(m,i\right)}\left( l,n\right)a_{{}_{i}}\left(lk\right)+\zeta_{{}_{m}}\left(nK\right)\]  
where \(q^{\left(m,i\right)}\left(l,n\right)\) denotes the time domain response of the composite channel between \(a_{{}_{i}}\left(nK\right),\) the \(i\)-th input signal of FMT modulator and \(z_{{}_{m}}\left(nK\right),\) the \(m\)-th output signal of the TR process, \(\hat{h}_{{}_{m}}^{*}\left(nK\right)\) denotes the time domain response of the matched filter \(\hat{H}_{{}_{m}}^{*}\left(\omega\right),\) and \(\zeta_{{}_{m}}\left(nK\right)\) denotes the output noise of the TR process. The frequency domain expressions of \(q^{\left(m,i\right)}\left(l,n\right)\) and \(\zeta_{{}_{m}}\left(nK\right)\) are:  
\[Q^{\left(m,i\right)}\left(\omega\right) =G\left(\omega\right)H_{{}_{i}}\left(\omega\right)G^{*}\left( \omega+\frac{2\pi i}{M}-\frac{2\pi m}{M}\right)\hat{H}_{{}_{m}}^{*}\left( \omega+\frac{2\pi i}{M}-\frac{2\pi m}{M}\right) \tag{8}\] \[\zeta_{{}_{m}}\left(\omega\right) =W\left(\omega\right)G^{*}\left(\omega-\frac{2\pi}{M}m\right)\hat{ H}_{{}_{m}}^{*}\left(\omega-\frac{2\pi}{M}m\right) \tag{9}\]  
For the convenient analysis, the \(z_{{}_{m}}\left(nK\right)\) in Equation (7) can be further expressed as:\[\begin{split} z_{m}\left(nK\right)=q^{\left(m,m\right)}\left(n,n \right)a_{m}\left(nK\right)+\sum_{\begin{subarray}{c}l=-\infty\\ l\
eq n\end{subarray}}^{l\leftrightarrow\infty}q^{\left(m,m\right)}\left(l,n \right)a_{m}\left(lK\right)\\ +\sum_{\begin{subarray}{c}l=1\\ i\
eq m\end{subarray}}^{M}\sum_{l=-\infty\end{subarray}}^{l\leftrightarrow \infty}q^{\left(m,i\right)}\left(l,n\right)a_{i}\left(lK\right)+\zeta_{m} \left(nK\right)\end{split} \tag{10}\]  
Equation (10) indicates that the interference in output signals of the TR process is composed by ISI, ICI, and noise. Combining Equation (10) with Equation (8), it is concluded that since ICI can be considered as zero in the stable acoustic channel, ISI is the main factor degrading the performance of the TR process when the transmitted symbol energy is high enough.  
It is indicated from Equations (4) and (8) that since the response of each filter for the TR process is matched to the estimated response of channel for each subcarrier, ISI can be compressed using the TR process. However, it should be mentioned that since the frequency domain response of the \(m\)-th filter for the TR process is \(\hat{H}_{m}^{\ast}\left(\omega\right)\) but not \(\hat{H}_{m}^{-1}\left(\omega\right),\) residual ISI is still existing even when the channel is estimated without any error. Residual ISI reduces the performance of the TR process.