# A Survey on Physical Layer Techniques and Challenges in Underwater Communication Systems
## 3 Modeling of UWA Channel Problems and Challenges
### Doppler Shift  
From transmitter to receiver, each path encounters variable time inconsistency effects because of reflections from moving objects, platform motion, currents, surface waves, tides, and internal waves. Doppler spread/shift is prompted by each variability source that moves the entire ranges of frequencies either up/down or where neighboring frequencies are spread mutually. Doppler spread in the range of 0 to 30 Hz is highly influenced by the communication system features, such as the motion of the sea surface, climate, the motion of the platform, and transmission frequency. As the sound speed is considerably lower compared to the speed of light, the Doppler effect detected in the underwater channels is comparatively severe compared to the RF channels. To demonstrate the Doppler effect, consider a pulse \(p(t)\), modulated through carrier frequency \(f_{c}\) and transmitted across an acoustic communication channel with a fixed sound speed \(v_{s}\), moving at a constant velocity \(v\) transmitter to receiving end. The delay in the propagation of the received signal is \(\tau(t)=\tau_{0}-v_{t}/v_{s}\), where \(\tau_{0}\), defined as a reference delay. The equation, i.e., \(a=v/v_{s}\) entails the proportionality of the Doppler effect to this mathematical function. As given in [171], the signal transmitted, i.e., \(s(t)\) is:  
\[s(t)=\left\{Rep(t)e^{(2\pi f_{c}t)}\right\} \tag{34}\]  
The signal received at the end of the receiver as stated in [171] is:  
\[r(t)=s(t+vt/v_{s}-\tau_{0})=s(t+at-\tau_{0})=Re\Big{\{}p(t+at-\tau_{0})e^{2\pi f _{c}(t+at-\tau_{0})}\Big{\}} \tag{35}\]  
Considering the central frequency, the baseband receives the signal (the signal received after demodulation and low pass filtering) [171]. Two signal distortions are observed while ignoring the phase shift \(2\pi f_{c}\tau\):  
1. There is time dilation of the signal by a factor of \(1+a\). Hence, the function \(p^{\prime}(t)=p(t(1+a))\) can be used to model the signal. The frequency response is contracted due to dilated response in the time domain. 2. The signal shows a frequency offset \(af_{c}\) that is identified as a Doppler shift. When the signal bandwidth is appropriately small, the dilation effect can be overlooked. If the bandwidth \(p(t)\) is signified as \(B_{p}\) the signal is almost time-invariant for a time \(B_{p}^{-1}\). If the total duration of the data packet is \(T\), the total packet dilation \(vT/v_{s}\) needs to be much less than \(B_{p}^{-1}\) to ignore the dilation. Therefore, dilation can be ignored if the product of time and bandwidth \(TB_{p}\) satisfies the relation \(TB_{p}=v_{s}/v\); otherwise, the signal received needs sampling [172]. Another method to describe the Doppler is the scattering function. Suppose, the coefficient of the channel at a specific delay \(\tau\) and time \(t\) is \(g(t,\tau)\). The relation is considered a time difference function if the identified channel is wide-sense stationery. Therefore, the function for channel temporal correlation is shown as:  
\[R_{g}(Vt;\tau,\tau_{2})=E\{g(t,\tau_{1})g^{*}(t+Vt,\tau_{2})\}, \tag{36}\]  
that is considered a function of three different variables; the time difference \(Vt\) and the two delays \(\tau_{1}\)\(\tau_{2}\). The function for the scattering of the channel is named the Fourier transform of the function for temporal correlation:  
\[s_{g}(\lambda_{d};\tau_{1},\tau_{2})=\int_{-\infty}^{\infty}R_{g}(Vt;\tau_{1},\tau_{2})e^{-2\pi\lambda_{d}Vt}dVt \tag{37}\]  
where, \(\lambda_{d}\) is the variable for Doppler spreading. At a specific delay, \(\tau_{1}=\tau_{2}\) is a demonstration of the channel Doppler spread. This results in a relationship between the Doppler spread \(B_{d}\) and the channel coherence time \(\left(Vt\right)_{c}\)[32]:  
\[\left(Vt\right)_{c}\approx\frac{1}{B_{d}} \tag{38}\]  
The time at which the channel at a specific delay is correlated, i.e., coherence time \(\left(Vt\right)_{c}\), is the inverse of the channel Doppler spread \(B_{d}\).