## 3 Simulation and Experiment  
### Simulation Test  
First, the performance of the FrFT channel estimation method is evaluated by comparing it with the time-domain correlation method. This study leverages the autocorrelation properties of LFM signals. The received signal undergoes bottom pressure cancelation, filtering, and denoising processes. The resulting output is a linear sweep frequency signal, as illustrated in Figure 3.  
Figure 2: Flow chart of channel equalization in fractional Fourier domain. To compare the accuracy of the two channel estimation methods, simulations were conducted under a mud pulse channel model, where the noise was assumed to be additive white Gaussian noise (AWGN) with zero mean. The channel duration was set to 1000 ms with a sampling rate of 1000 Hz. A three-path channel was simulated, with each path assigned an attenuation coefficient of 0 dB. Two methods were employed for time delay estimation and evaluated under two conditions: with and without frequency offset. The frequency offset factor, typical for mud signal transmission, was set to 0.025. The input SNR varied from 9 dB to 0 dB. For each SNR level, 1000 Monte Carlo simulations were conducted to assess the relationship between the relative root mean squared error (RMSE) of time delay estimation and the SNR, as shown in Figure 4.  
The results show that, in the presence of an unknown frequency offset, the RMSE of the FrFT-based method is approximately 20% of that of the conventional correlation-based method. This demonstrates that the FrFT-based approach is more robust against frequency offset and achieves significantly higher estimation accuracy. However, when no frequency offset is present, the performance of both methods is comparable in terms of RMSE. Hence, the simulation result proves that the channel estimation method based on FrFT is able to deal with situations of frequency offset effectively. In order to evaluate the usability of the equalization method, a simulation is conducted to testify its effectiveness. The channel is modeled as a deep fading multipath channel model similar to the mud channel, comprising six multipaths with equal delay intervals. The data sequence consists of a pseudo random sequence with a training symbol of 128, modulated by binary frequency shift keying (BFSK), and it undergoes transmission through the channel with a SNR of 10 dB. In order to observe the output noise and ISI, correlation  
Figure 4: RMSE comparisons between two methods. Figure 3: The output signal. analysis is performed between the equalized output data sequence and the transmitted data sequence. Initial estimation of the channels at various orders leads to the determination of the optimal order \(p=1.23\). Subsequently, a corresponding fractional-order domain multiplicative filter is generated, with which the data sequence is processed. Figure 5 shows the signal autocorrelation results without fractional Fourier transform equalization, whereas Figure 6 illustrates the signal autocorrelation results after applying fractional Fourier transform equalization. Figures 5 and 6 reveal that the presence of multipath effect in the channel results in multiple peaks in the autocorrelation results of the signal without fractional Fourier transform equalization, causing inter-symbol crosstalk. In contrast, after fractional Fourier transform equalization, the autocorrelation results exhibit a single peak, indicating a significant reduction in the multipath effect. Meanwhile, the ratio of peak to noise in Figure 6 is much greater than that in Figure 5, showing that fractional Fourier transform equalization can effectively control the influence of noise and is beneficial for the following demodulation. Overall, the proposed method is demonstrated to be effective in noise control and ISI suppression. To access the impact of fractional Fourier domain equalization on demodulation performance, various SNRs ranging from \(-16\) dB to \(0\) dB, with an interval of \(2\) dB, are considered in the channel.