## 2 The Features of FM-OFDM Communications Scheme
During demodulation by quadrature processing and filtration signals \(x_{I}=y_{I}\) and \(x_{Q}=-y_{Q}\) are formed in channels \(I\) and \(Q\):  
\[\int\limits_{-\infty}^{\infty}\left[\cos(\omega_{0}t)y_{I}-\sin(\omega_{0}t)y_ {Q}\right]\cdot\cos(\omega_{0}t)dt=\frac{y_{I}}{2}=x_{I}\]  
\[\int\limits_{-\infty}^{\infty}\left[\cos(\omega_{0}t)y_{I}-\sin(\omega_{0}t)y_ {Q}\right]\cdot\sin(\omega_{0}t)dt=-\frac{y_{Q}}{2}=x_{Q}\]  
So, for the demodulation of the FM it is necessary to calculate the phase of the received pair of quadrature components and to differentiate the achieved expression with respect to time: \(\overline{s_{RX}(t)}=\frac{1}{\Delta\omega}\left(\frac{dx_{I}}{dt}x_{Q}(t)- \frac{dx_{Q}}{dt}x_{I}(t)\right)\cdot\left(x_{I}{}^{2}(t)+x_{Q}{}^{2}(t) \right)^{-1}\). Taking into attention that \(x_{I}^{2}(t)+x_{Q}^{2}(t)=1\) the expression of demodulation algorithm simplifies [14]:  
\[\overline{s_{RX}(t)}=\frac{1}{\Delta\omega}\left(\frac{dx_{I}}{dt}x_{Q}(t)- \frac{dx_{Q}}{dt}x_{I}(t)\right) \tag{3}\]  
Let us consider the reception of FM-OFDM signal with an arbitrary initial phase:  
\[x_{I\psi}=\int\limits_{-\infty}^{\infty}\left[\cos(\omega_{0}t)y_{I}-\sin( \omega_{0}t)y_{Q}\right]\cdot\cos(\omega_{0}t+\psi)dt\]  
\[=\int\limits_{-\infty}^{\infty}\cos^{2}(\omega_{0}t)\cdot\cos\psi\cdot y_{I} dt-\int\limits_{-\infty}^{\infty}\cos(\omega_  
\[\frac{dx_{l\psi}}{\Delta\omega dt}x_{Q\psi}(t)-\frac{dx_{Q\psi}}{ \Delta\omega dt}x_{l\psi}(t)=\frac{d}{\Delta\omega dt}\Big{(}\frac{y_{I}}{2}\cos \psi+\frac{y_{Q}}{2}\sin\psi\Big{)}\cdot\Big{[}\frac{y_{I}}{2}\sin\psi-\frac{y_ {Q}}{2}\cos\psi\Big{]}\] \[-\frac{d}{\Delta\omega dt}\Big{(}\frac{y_{I}}{2}\sin\psi-\frac{y_ {Q}}{2}\cos\psi\Big{)}\cdot\Big{[}\frac{y_{I}}{2}\cos\psi+\frac{y_{Q}}{2}\sin \psi\Big{]}=\frac{1}{4\Delta\omega}\bigg{(}\frac{dx_{I}}{dt}x_{Q}(t)-\frac{dx_ {Q}}{dt}x_{I}(t)\bigg{)}\]  
Likewise, the influence of the Doppler frequency shift \(\Delta\omega_{D}/2\pi\) can be observed by substituting \(\psi=\Delta\omega_{D}t\) in the above mentioned expressions. During the demodulation, according to the expression (3), the following expression can be obtained:  
\[\overline{s_{RX\Delta\omega_{D}}(t)}=\frac{1}{\Delta\omega}\bigg{(}\frac{dx_{ I\Delta\omega_{D}}}{dt}x_{Q\Delta\omega_{D}}(t)-\frac{dx_{Q\Delta\omega_{D}}}{ dt}x_{I\Delta\omega_{D}}(t)\bigg{)}\]  
\[=\frac{d}{\Delta\omega\cdot dt}\Big{(}\frac{y_{I}}{2}\cos\langle\Delta\omega_ {D}t\rangle+\frac{y_{Q}}{2}\sin\langle\Delta\omega_{D}t\rangle\Big{)}\cdot \Big{[}\frac{y_{I}}{2}\sin\langle\Delta\omega_{D}t\rangle-\frac{y_{Q}}{2}\cos \langle\Delta\omega_{D}t\rangle\Big{]}\]  
\[s_{FM}(t)=U_{0}\cdot\cos\left(\int\limits_{0}^{t}\left[\omega_{0}+2\pi\cdot \Delta fN\cdot m_{FM}\cdot\overline{s(t)}\right]dt\right), \tag{2}\]0  
In that case, the frequency shift equals to the presence of the steady component in OFDM signal. Also, the strong resistance of this demodulation method to the Doppler carrier frequency shift can be noted. Usually, such distortion leads to the presence of a constant component in demodulated signal, which, however, has no effect on the Fast Fourier Transform (FFT) of the OFDM symbol image part. To increase the spectral efficiency of communication system it is rational to use the M-QAM modulation of OFDM subcarriers. However, the implementation of such methods is restricted by such a significant factor as instability of underwater acoustic channel parameters, especially if communicating objects are mobile. This phenomenon leads to substantial variability of signal amplitudes on the symbol transmission time interval, which forces the use of differential methods of data transmission. On the other hand, a bipolar orthogonal system of symbol transmission (QPSK) can be a compromise between the data rate and noise-sustainability. This approach can help in reducing the amplitude ambiguity during the symbol decoding in such conditions.