## 3 Time-Varying Carrier Frequency Offset Estimation  
The previous section discussed a model OFDM system with ICI caused by the CFO, which was analogous to the orthogonal MC-CDMA was discussed. However, because of the complexity of underwater acoustic environments, it is inappropriate to assume that both transducers will maintain uniform motion over a long period. Their motion can be impacted by multiple factors, including sea surface undulation, turbulence, and internal wave motion. More often, this relative motion is a variable speed motion with random accelerations. Therefore, the ICI in the received OFDM signal has a TVCFOrather than a constant CFO. In general, the duration of one OFDM block of underwater acoustic communication lasts from tens of milliseconds to approximately one hundred milliseconds. Considering the actual environment for underwater acoustic communication, we can reasonably assume that the acceleration of the relative motion is constant over the duration of an OFDM block; that is, the CFO changes within the duration of an OFDM frame but remains constant in the very short OFDM time block. Therefore, the model mentioned above should be modified slightly to change the constant CFO in the frame time to a variable value, while allowing it to remain steady during the block duration. The estimation of a single CFO value within a data frame is changed to finding the CFO values for each block within a data frame. Water works as a waveguide in communication; however, it is also a viscous fluid, making it impossible for channel conditions to change quickly. Therefore, there are dependencies between the CFO values in neighboring blocks of the received signal because the velocity of relative motion corresponding to the latter time block changes based on that of the previous time block, which is adjacent to it. As previously mentioned, relevant research has shown that the DE algorithm is particularly suitable for dealing with problems in which there are dependencies between the adjacent parameters to be optimized. Therefore, we used the DE algorithm to jointly optimize the estimated TVCFO values in each block of the received signal. Based on the above analysis, the specific steps for estimating the TVCFO using the DE algorithm are given below. _(1)_ _Determine optimization variables and their limits_  
Based on the previous analysis, this study investigated TVCFO estimation, where the CFO values on multiple OFDM symbol blocks varied over time and were unequal. The CFO on each OFDM symbol in a frame can be regarded as a variable to be optimized. If a data frame has \(N\) symbol blocks, then the \(i-\)th individual of the DE can be expressed as follows:  
\[\epsilon_{i}=(\epsilon_{i,1},\ \epsilon_{i,2},\ \ldots,\ \epsilon_{i,N}) \tag{11}\]  
for \(i=1,\ 2,\ \ldots,\ Np\), where \(Np\) is the population size of the DE. According to the introduction of Formula (3), the normalized CFO is in the range of \([0,1]\). In other words, the minimum value, \(\epsilon_{min}\), is \(0\), and the maximum value, \(\epsilon_{max}\), is \(1\). If it exceeds the limit, it should be readjusted back to this range; otherwise, it is meaningless. Real number DE coding was adopted in this study. _(2)_ _Determine the fitness function_  
The fitness function can generally be regarded as a cost function in an optimization problem or its equivalent transformation and is mainly used for measuring the quality of solutions. For minimization problems, a smaller fitness function value indicates a higher quality for the individual solution. When the fitness function value is zero, the solution has the highest quality, and the individual is therefore the optimal solution. When the optimal solution was introduced and the Euclidean distance between the reconstructed and received signals was zero, the estimated value of the CFO was the actual value of the received OFDM signal. Although the fitness value could not be zero because of the influence of noise, there was still a relationship between the quality of the solution and its fitness.