## 3 Numerical Experiments with FM-OFDM-QPSK in Multipath Channels  
In this part of the paper, the analysis of effectiveness of the FM-OFDM-QPSK mode (with the spread spectrum coefficient \(m_{FM}\) of 1, 2, 4, and 10) with precise frame synchronization of OFDM symbols based on maximum amplitude of the matched filter response on pseudorandom sequence (PRS), which is located in front of data packet, is presented. The results of numerical modeling of BER performance for different multipath channel responses are presented. To estimate the effectiveness of the work of FM-OFDM-QPSK system numerical models with different spread spectrum the coefficients were designed. The parameters of numerical models were adapted to experimental underwater acoustic equipment (central frequency of 400 Hz, total bandwidth \(\Delta F=200\) Hz) to simplify the following sea trials. Models for each spread spectrum coefficient \(m_{FM}\) of 1, 2, 4, and 10 were designed. The parameters of designed models are presented in Table 2, where \(T_{guard}\) is the time guard interval and \(f_{b}\) is the data transmission rate. The time guard interval between OFDM symbols was chosen based on previously conducted experiments at the distances more than 100 km aimed at the estimation of the channel impulse response and data transmission at 400 Hz frequency in current aquatic area, and it is equal to 200 ms [15; 16]. The ratio of the duration of multi-frequency symbol to the time guard interval \(\frac{T_{OFDM}}{T_{guard}}\) was chosen equal to 5 for each mode, where \(T_{OFDM}=\frac{1}{\Delta f}\). When considering the modes of FM-OFDM-QPSK with spread spectrum, it is expected to observe the decrease of the spectral efficiency calculated by formula \(n_{f}=\frac{2N}{(T_{OFDM}+T_{guard})\Delta F}\). However, one of the advantage of the systems exploiting parallel frequency multiplexing of information symbols (OFDM) is the slower decrease of the spectral efficiency depending on the value of the time guard interval \(T_{guard}\) compared to methods based on sequential frequency multiplexing, where the effectiveness of spectral utilization is estimated by the expression \(n_{f}\approx\frac{1}{\sqrt{2\Delta F\cdot T_{guard}}}\)[8; 9]. Also, this advantage of OFDM systems appears when comparing it to other known modulation methods. For example, mode K10 (bandwidth of 17 bit/s) reached BER \(<10^{-2}\) in channel with AWGN (SNR = 0 dB). In that case, the protection from reverberation distortions with the length of 200 ms was provided. When comparing FM-OFDM to DSSS on similar SNR in the 200 Hz bandwidth, it can be noted that the DSSS system with chip speed of 100 chip/s and zero guard intervals reaches only 5 bit/s of effective information speed for the code of 20 elements not providing proper noise-sustainability due to short code. For the modes K1, K2, K4, and K10 (depending on the spectrum spread coefficient \(m_{FM}\)) numerical simulations were made to estimate the dependence of total BER from SNR under AWGN on the input of FM demodulator. The BER values were measured without any noise-sustainable coding. The results of the modelling are presented in Figure 3.  
The character and values of BER (SNR) for chosen modes correspond with analytical expressions, according to the expression (5) [13]. To evaluate the noise-sustainability of the method under study, a number of multipath channel impulse responses corresponding to the underwater acoustic channel were chosen [17]. \begin{table}
\begin{tabular}{c c c c c} \hline  & **K1** & **K2** & **K4** & **K10** \\ \hline \(N\) & 100 & 50 & 25 & 10 \\ \(\Delta f\) & 1 Hz & 1 Hz & 1 Hz & 1 Hz \\ \(T_{guard}\) & 200 ms & 200 ms & 200 ms & 200 ms \\ \(f_{b}\) & 167 bps & 83 bps & 42 bps & 17 bps \\ \(n_{f}\) & 0.835 bps/Hz & 0.418 bps/Hz & 0.21 bps/Hz & 0.083 bps/Hz \\ \hline \end{tabular}
\end{table} 
Table 2: The parameters of FM-OFDM-QPSK numerical models for the central frequency of 400 Hz and bandwidth of 200 Hz.