## 3 Receiver Structure  
### FMT Demodulation  
The receiver structure of the proposed method is shown in Figure 2. After being transmitted through the UWA channel, the received signal, the input of the receiver, can be expressed as  
\[\textit{\large{/}}\left(k\frac{T}{K}\right)=\sum_{p=0}^{N_{c}-1}\textit{\large{ \large{s}}}\left[(k-p)\frac{T}{K}\right]\textit{\large{/}}\left(p\frac{T}{K} \right)+\eta\textit{\large{/}}\left(k\frac{T}{K}\right) \tag{2}\]  
where \(\textit{\large{/}}\left(k\frac{T}{K}\right)\) denotes the discrete time domain response of the UWA channel with length \(N_{c}\), and \(\eta\textit{\large{/}}\left(k\frac{T}{K}\right)\) denotes the discrete sample of channel noise. Combing Equation (1) with Equation (2), the received signal can be further expressed as  
\[\textit{\large{/}}\left(k\frac{T}{K}\right)=\sum_{p=0}^{N_{c}-1}\sum_{m=0}^{M -1}\sum_{n=-\infty}^{+\infty}g_{\textit{\large{/}}}\left[(k-p)\frac{T}{K}-nT \right]\textit{\large{/}}\left(p\frac{T}{K}\right)\textit{\large{/}}\left(nT \right)+\eta\textit{\large{/}}\left(k\frac{T}{K}\right) \tag{3}\]  
Figure 1: Block diagram for transmitter structure. Figure 2: Block diagram for receiver structure. Referring to the receiver structure shown in Figure 2, the received signal is first processed by the FMT demodulator that consists of carrier demodulators, receiver filters, and \(K\)-times down-samplers. The \(m\)-th output signal of the FMT demodulator can be expressed as  
\[ \begin{array}{ll}y^{m}(nT)&=\sum\limits_{i=0}^{M-1}\sum\limits_{i=-\infty}^{+ \infty}\sum\limits_{k=-\infty}^{+\infty}\sum\limits_{p=0}^{N-1}g_{t}\Big{[}(k-p )\frac{T}{K}\Big{]}c\Big{(}p\frac{T}{K}\Big{)}e^{-j\frac{2\pi}{M}ip}g_{T}\Big{(} nT-IT-k\frac{T}{K}\Big{)}e^{j\frac{2\pi}{M}(i-m)(k+lK)}x^{i}(IT)\\ &+\sum\limits_{k=-\infty}^{+\infty}\eta\Big{(}k\frac{T}{K}\Big{)}g_{T}\Big{(} nT-k\frac{T}{K}\Big{)}e^{-j\frac{2\pi}{M}mk}\end{array}  \tag{4}\]  
where \(g_{r}(t)\) denotes the discrete time domain response of the receiver filter, that is matched to the transmit filter. Equation (4) can be further expressed as  
\[\begin{array}{ll}y^{m}(nT)&=\sum\limits_{i=0}^{M-1}\sum\limits_{l=-\infty}^ {+\infty}h^{(m,i)}(nT-IT)x^{i}(IT)+w^{m}(nT)\\ &=h^{(m,m)}(0)x^{m}(nT)+\sum\limits_{l=-\infty}^{+\infty}h^{(m,m)}(nT-IT)x^{m} (IT)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad l \
eq n\\ &+\sum\limits_{ \begin{array}{ll}\sum\limits_{i=0}^{M-1}\sum\limits_{l=-\infty }^{+\infty}h^{(m,i)}(nT-IT)e^{j\frac{2\pi}{M}(i-m)lK}x^{i}(IT)+w^{m}(nT)\\ i\
eq m\end{array}  \tag{5}\]  
where \(h^{(m,i)}(nT)\) denotes the discrete time domain response of the subchannel between the \(i\)-th transmitted sequence \(x^{i}(nT)\) and the \(m\)-th demodulated sequence \(y^{m}(nT)\), and \(w^{m}(nT)\) denotes the \(i\)-th output noise of the FMT demodulator. \(h^{(m,i)}(nT)\) and \(w^{m}(nT)\) can be expressed as  
\[ \begin{array}{ll}h^{(m,i)}(nT)=\sum\limits_{i=0}^{M-1}\sum\limits_{k=-\infty }^{+\infty}\sum\limits_{p=0}^{N-1}g_{t}\Big{[}(k-p)\frac{T}{K}\Big{]}c\Big{(}p \frac{T}{K}\Big{)}e^{-j\frac{2\pi}{M}ip}g_{r}\Big{(}nT-k\frac{T}{K}\Big{)}e^{j \frac{2\pi}{M}(i-m)k}\\ w^{m}(nT)=\sum\limits_{k=-\infty}^{+\infty}\eta\Big{(}k\frac{T}{K}\Big{)}g_{r} \Big{(}nT-k\frac{T}{K}\Big{)}e^{-j\frac{2\pi}{M}mk}\end{array}  \tag{6}\]  
It can be observed from Equation (5) that the interference in the demodulated signal \(y^{m}(nT)\) includes three types, namely, the ISI from other symbols on the \(m\)-th subband, the ICI from symbol sequences on the other subbands, and the noise. In FMT modulation, the subbands are separated and have high spectral containment, and therefore the ICI is a minor concern even when the Doppler spread, caused by the water motion, exists in UWA communications [13; 14; 15; 16; 17]. This paper focus on exploiting turbo equalization to suppress the ISI in FMT modulation UWA communications. The ICI is slight, and therefore the effect of ICI is not taken into consideration. When the ICI is neglected, the demodulated signal \(y^{m}(nT)\) can be expressed as:  
\[