## 2 The Features of FM-OFDM Communications Scheme
In the general case, the FM-OFDM signal propagating through a non-stationary multi-path communication channel and arriving at the receiver input can be expressed as:  
\[s_{FM}(t)=U_{0}\cdot\cos\left(\int\limits_{0}^{t}\left[\omega_{0}+2\pi\cdot \Delta fN\cdot m_{FM}\cdot\overline{s(t)}\right]dt\right), \tag{2}\]4  
where \(A_{k}(t),\ \tau_{k}(t),f_{k}(t)\), \(\varphi_{k}(t)\) are time-variant magnitude, signal delay, frequency, and phase shifts of ray k from ray ensemble L, correspondingly. Substituting expression (4) in (6) and solving it according to (3) we get a complex relationship with a large number of variables, which is easier to solve using the numerical form. Let us take a closer look at the stationary two-ray channel model passing a signal (4). For the propagation model with one reflection component with amplitude of \(\rho\) (\(\rho\leq 1\)) and delay of \(\tau\), the channel impulse response will be described by expression \(h(t)=1+\rho\cdot e^{-i\omega\tau}\). After substituting it to the expression (3) and some transformations, we obtain the following expression:  
\[s_{FM}(t)=U_{0}\cdot\cos\left(\int\limits_{0}^{t}\left[\omega_{0}+2\pi\cdot \Delta fN\cdot m_{FM}\cdot\overline{s(t)}\right]dt\right), \tag{2}\]5 The structural scheme of the orthogonal frequency division multiplexing in combination with additional frequency modulation (FM-OFDM-QPSK) transmitting-receiving processes. \[ \begin{split} s_{RX}(t)=\overline{s(t)}\Bigg{[}1+\rho\cdot\cos\left( \Delta\omega\int\limits_{0}^{t}\left(\overline{s(t)}-\overline{s(t-\tau)} \right)dt\right)\Bigg{]}+\overline{s(t-\tau)}\\ \cdot\rho\left[\rho+\cos\left(\Delta\omega\int\limits_{0}^{t} \left(\overline{s(t)}-\overline{s(t-\tau)}\right)dt\right)\right]\end{split}  \tag{7}\]  
Here, it can be seen that the demodulated OFDM signal is exposed to a parasitic amplitude modulation when FM detector has a direct and strong delayed signal, which leads to the corruption of OFDM subcarriers' orthogonality. The fragment of FM-OFDM symbol under the influence of strong reflected component of the signal (\(\rho=1,\ \tau=100\) ms) is shown in Figure 2. In that case, the sum of the direct and reflected signal is modulated on amplitude by the function \(\alpha(t)\):  
\[ \begin{split} s_{RX}(t)=\left(\overline{s(t)}+\overline{s(t- \tau)}\right)\left[1+\cos\left(\Delta\omega\int\limits_{0}^{t}\left(\overline {s(t)}-\overline{s(t-\tau)}\right)dt\right)\right]=\alpha(t)\left[\left( \overline{s(t)}+\overline{s(t-\tau)}\right)\right]\\ \alpha(t)=\left[1+\cos\left(\Delta\omega\int\limits_{0}^{t} \left(\overline{s(t)}-\overline{s(t-\tau)}\right)dt\right)\right],\rho=1,\ \tau=100\;\text{ms}\end{split} \]  
For the substantial weakening of this effect the adaptive amplitude clipper must be implemented at the FM demodulator input. Some numerical BER estimations were made under conditions of different multipath propagation profiles, depending on the saturation parameter \(\gamma=\frac{U_{limit}}{U_{0}}\) taking into account the fundamental analytical difficulties in determining the optimal level of signal clipping. In the Table 1 below BER values for FM-OFDM-QPSK under convolution with multipath channel profiles with frequency characteristics \(H\dot{1}(\omega)=1+0.9\cdot e^{-j\omega\times 10.625\times 10^{-3}}\) and \(H\dot{2}(\omega)=1+0.9\cdot e^{-j\omega\times 11.25\times 10^{-3}}\) and depending on saturation parameter \(\gamma\) are presented. The parameters of the numerical model were chosen similar to marine experiments conditions; the center frequency was equal to 400 Hz, OFDM-QPSK bandwidth--100 Hz, and sampling frequency--8 kHz. Using this method, the input signal amplitude clipping at the levels corresponding to \(\gamma=[0.3...0.5]\) BER values are minimal. Also, an effective clipping of strong impulse noises, which are usual for shallow and noisy aquatic areas, is implemented in this method. Figure 2: A fragment of the FM-OFDM symbol in the presence of a strong reflected signal component.