## 2 Transmission and Multipath Effect of OFDM Signals  
Orthogonal frequency division multiplexing (OFDM) is a parallel transmission technique that divides the broadband data information flow into multiple independent narrowband data information flows [14]. In a practical OFDM-based system, the frequency band available is first partitioned into N orthogonal frequency sub-bands F\({}_{0}\), F\({}_{1}\)\(\ldots\) F\({}_{\text{N-1}}\). Each sub-band can independently and flexibly transmit the data information through different modulations such as amplitude shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK). Through different selection of OFDM discrete frequency domain values, different modulation methods can be used to transmit information, such as introducing different frequency shifts for discrete frequency domain values \(X\)(s), and the combination of FSK and OFDM can be realized. The mutually superposed sub-bands are orthogonal, and do not interfere with each other, so that a guard band is not needed between them [15]. Therefore, OFDM achieves a higher frequency band utilization rate than the existing single carrier frequency shift keying communication system [16]. The OFDM signals can be modulated and demodulated by inverse fast Fourier-transform (IFFT) and fast Fourier-transform (FFT), respectively [17]. We need to generate the digital modulation signals \(x(n)\) by IFFT, which are sampled at the rate\(f_{sam}\). For its convenience, the time duration \(T\) of each symbol is set to be \(V/f_{sam}\), where \(V\) is an integer power of 2. Based on the correspondence of time frequency domain to \(V\)-point IFFT, it is known that up to \(V/2\) mutually orthogonal subcarriers can be generated for modulation under \(V\)-point IFFT [18]. Moreover, the frequency interval between two adjacent subcarriers is \(\Delta f=f_{sam}/V\). In practical applications, \(N\) out of \(V/2\) subcarriers are selected in terms of the available bandwidth. The \(N\) subcarriers have the frequency calculated by:  
\[f_{i}=f_{0}+i\Delta f=m_{0}\Delta f+i\Delta f=(m_{0}+i)\cdot f_{sam}/V \tag{1}\]  
where \(f_{i}\) stands for the frequency of the \((i+1)\)th subcarrier, and \(f_{0}\) represents the initial subcarrier frequency. Based on the frequency domain of IFFT, the frequency \(f_{i}\) of each subcarrier under OFDM is an integral multiple of \(\Delta f\). When IFFT is used for the modulation of OFDM signals, the complex baseband sub-symbol \(X_{i}\) of each subcarrier can be determined by the bit \(b_{i}\) allocated to the subcarrier and the mode of modulation adopted for the subcarrier [19]. Moreover, a discrete frequency domain value \(X(s)\) (where \(s\) ranges from \(m_{0}\) to \(m_{0}+N-1\), and \(m_{0}\) is the discrete frequency domain serial number of \(f_{0}\)) is assigned in the IFFT operation. Subsequently, \(X(s)\) is expanded by setting the frequency outside the signal frequency band to zero. In order to obtain the sampled signals \(x(n)\) of the real number under OFDM, frequency domain conjugate symmetry is imposed on \(X(s)\) to determine the required discrete frequency domain value of \(V\) points. In the end, \(V\)-point IFFT is performed for \(X(k)\) to obtain the sampled OFDM signals after discrete modulation at the sampling frequency \(f_{sam}\) as shown in Equation (2):  
\[x(n)=\frac{1}{\sqrt{V}}\sum_{k=0}^{V-1}X(k)e^{j\frac{2\pi}{V}kn} \tag{2}\]  
After \(x(n)\) signals are transmitted in the channel, the receiver is only required to perform the \(N\)-point FFT operation for the received sampled signals of a symbol. In this way, the information \(X^{\prime}(k)\) transmitted from the source can be conveniently restored. Thus, signals are demodulated. Nevertheless, severe multipath effect has a practical impact on an underwater acoustic channel [20]. It may cause the distortion of the OFDM signals \(x(n)\) and then a large number of symbol errors in the communication system, which even results in failed communication.