## 2 Methods
### Channel Estimation Based on Fractional Fourier Transform  
Fractional Fourier transform is an extensively used transformation technique in signal analysis which is founded on the principles of the Fourier transform. This algorithm is attractive due to its simplicity, low computational complexity, and typically high output signal-to-noise ratio. The FrFT of signal \(x(t)\) is defined as  
\[F_{p}(u)=F^{p}[x(t)](u)=\int_{-\infty}^{\infty}x(t)K_{p}(u,t)\ dt \tag{2}\]  
where \(p\) represents the order of FrFT, \(F^{p}\) represents the FrFT operator, and \(K_{p}(u,t)\) is integral kernel function, which is  
\[K_{p}(u,t)=A_{a}exp\left[j\pi\Big{(}u^{2}cot\alpha-2 ut+t^{2}cot\alpha\Big{)}\right] \tag{3}\]  
where \(\alpha=p\pi/2\) is equivalent to the angle of rotation in the time-frequency domain. Let a single-component linear frequency modulation (LFM) signal \(c(t)\) be expressed as  
\[c(t)=exp\Big{(}j\pi\mu t^{2}\Big{)} \tag{4}\]  
where \(\mu\) is the chirp rate. Under a multipath mud pulse telemetry channel and with \(c(t)\) being the system input, the received multipath signal can be represented as  
\[r(t)=\sum_{i=1}^{N}a_{i}c(t-\tau_{i})+n(t) \tag{5}\]  
where \(i\) is the multipath index; \(N\) is the total number of multipaths; \(a_{i}\) is the amplitude of the \(i\)-th multipath; \(\tau_{i}\) is the delay of the \(i\)-th multipath relative to the receiver time window; \(n(t)\) is the additive white Gaussian noise. In the actual MPT system, the frequency modulation rate varies based on sampling frequency offset, optimal order, modulation rate, and so on. Therefore, determining the optimal order is crucial when employing the fractional Fourier transform for channel estimation [32]. Within the fractional Fourier domain, multipath signals manifest multiple peaks resulting from the optimal order transformation [33]. By calculating the difference between the fractional Fourier domain sampling points corresponding to these peaks and normalizing the dimensions, the delay differences between multipath channels can be determined, which is equivalent to calculating the channel's impulse response in the time domain. A channel model based on fractional Fourier convolution is defined. Assume that the mud channel is a time-invariant channel, similar to an ordinary time-domain convolution model; the received pressure wave signal is  
\[y(t)=x(t)p^{*}h(t)+w(t) \tag{6}\]  
where \(p^{*}\) represents fractional convolution, which is calculated as follows:  
\[h(t)=(fp^{*}g)(t) \tag{7}\]  
The fractional order form \(H_{p}(u)\) of \(h(t)\) is  
\[H_{p}(u)=e^{-iC_{a}u^{2}}F_{p}(u)*G_{p}(u) \tag{8}\]  
Applying the fractional order form on (6) yields  
\[Y_{p}(u)=X_{p}(u)H_{p}(u)e^{-i\frac{\alpha t}{2}u^{2}}+W_{p}(u) \tag{9}\]where \(Y_{p}(u),\;\;X_{p}(u),\;\;W_{p}(u)\) are the fractional order expressions of the corresponding pressure wave signals, and \(H_{p}(u)\) represents the channel expression in the fractional Fourier domain. According to the system model, the least squares channel estimation is similar to traditional frequency-domain equalization, yielding the estimated value of \(H_{p}(u)\). To obtain the minimum MSE of the equalizer, the optimal order should be selected so that the modulus of \(H_{p}(u)\) is constant 1. Therefore, the objective function can be set as \(\varepsilon_{p}\), given as  
\[\varepsilon_{p}=\left(\left|\left|\frac{Y_{p}(u)}{X_{p}(u)}\right|-1\right|^{ 2}\right) \tag{10}\]  
By identifying the order \(p\) that minimizes \(\varepsilon_{p}\), the optimized equalizer can be obtained. Applying the FrFT of order \(p_{0}\), which matches the LFM chirp rate, to \(r(t)\), yields  
\[R_{p_{a}}(u)=\mathcal{F}_{p_{a}}\{r(t)\}(u)=\sum_{l=1}^{L}a_{l}\mathcal{F}_{p_ {a}}\{c(t-\tau_{l})\}(u)+N_{p_{a}}(u) \tag{11}\]  
where \(N_{p0}(\mu)\) is the transformed noise component. Owing to the shift property of FrFT and the energy compaction of LFM signals, each delayed replica \(c(t-\tau_{l})\) produces a sharp peak at a corresponding FrFT coordinate [34].