A Fractional Fourier Transform-Based Channel Estimation and Equalization Algorithm for Mud Pulse Telemetry

Abstract

Mud pulse telemetry (MPT) systems are a promising approach to transmitting downhole data to the ground. During transmission, the amplitudes of pressure waves decay exponentially with distance, and the channel is often frequency-selective due to reflection and multipath effect. To address these issues, this work proposes a fractional Fourier transform (FrFT)-based channel estimation and equalization method. Leveraging the energy aggregation of linear frequency-modulated signals in the fractional Fourier domain, the time delay and attenuation parameters of the multipath channel can be estimated accurately. Furthermore, a fractional Fourier domain equalizer is proposed to pre-filter the frequency-selective fading channel using fractionally spaced decision feedback equalization. The effectiveness of the proposed method is evaluated through a simulation analysis and field experiments. The simulation results demonstrate that this method can significantly reduce multipath effects, effectively control the impact of noise, and facilitate subsequent demodulation. The field experiment results indicate that the demodulation of real data achieves advanced data rate communication (over 12 bit/s) and a low bit error rate (below 0.5%), which meets engineering requirements in a 3000 m drilling system.

1. Introduction

Petroleum is a crucial resource that plays an essential role in industrial production and the functioning of modern society [1]. However, decades of intensive extraction have progressively depleted conventional oil reserves, which were both high-quality and easily accessible. This depletion has brought considerable challenges to the oil industry while simultaneously catalyzing rapid advancements in drilling technology [2]. Among these developments, measurement while drilling (MWD) systems provide real-time data while drilling, cut down the time needed for logging operations after drilling, and ensure the collection of dependable data in high-difficulty wells.

The transmission of data in these systems is broadly classified into wired and wireless telemetry. Wired telemetry primarily includes optical fibers, cables, and so on, which offer high data rates but are costly [3]. Conversely, wireless telemetry mainly includes mud pulse telemetry (MPT), electromagnetic telemetry, and acoustic telemetry, among which mud pulse telemetry is the most widely used [4]. Notably, electromagnetic telemetry suffers from significant signal attenuation and poor noise immunity, rendering it suitable only for short-distance transmission under specific formation resistivity conditions [5]. In addition, acoustic telemetry encounters challenges due to the substantial attenuation of mechanical waves, compounded by the complexities of the drilling environment and the interference from background noise, which hinder the detection and extraction of information from acoustic signals. In contrast, mud pulse telemetry offers longer transmission distances and reliable performance, although its data transmission speed is relatively low compared to other methods [3].

MPT uses the circulating mud within the drill string for communication between the underground environment and the surface system. Activated by the underground control circuit, the generator is modified with the transmitted data, thereby inducing periodic pressure variations and converting electrical signals into pressure signals effectively. After a series of processing steps by the demodulation system, the surface system completes the recovery of underground measured data. With its reliable performance over long communication distances, mud pulse data transmission technology stands as the major wireless transmission manner deployed in real well applications [6]. Additionally, MPT systems do not necessitate specific alterations to the drill string, resulting in relatively low costs. In addition, the technology exhibits robust adaptability to the bottom environment because of the location of the mud channel inside the drill string, which mitigates the influence of geological factors [7]. Consequently, MPT has become one of the most-used wireless drilling transmission techniques [8].

During the transmission process from the bottom of the well to the ground through mud pipelines, significant interference and noise are generated in the transmission of mud pulses. These interferences primarily originate from the periodic motion of piston pumps and various mechanical vibrations underground, causing the useful mud pulse signals to become obscured. Furthermore, over long distances, the mud signal undergoes varying degrees of amplitude attenuation, resulting in significantly weakened signals. In this context, accurately identifying and decoding weak signals from high-energy and irregular noise has become a critical issue in MPT technology. Moreover, in the field environment, pressure signals are subject to frequency-selective attenuation, leading to severe inter-symbol interference (ISI) [9]. Consequently, channel equalization is essential to mitigating the effects of frequency-selective fading and reducing ISI.

Research on the characteristics of mud channels primarily focuses on two aspects. Firstly, the Lamb acoustic attenuation model is commonly employed to describe amplitude attenuation as a function of distance, characterizing propagation loss. Secondly, the frequency-selective attenuation caused by factors such as reflection on the transmission path is analyzed to help understand the channel model. Channel estimation is an important method to acquire the channel structure, with current methods performing cross-correlation on linear sweep signals to estimate the delay and attenuation coefficient of the multipath effect [10].

To address the inter-symbol interference caused by multipath effects, researchers have proposed a time-domain equalization method [11]. The commonly used types of equalizers include linear equalizers and decision feedback equalizers (DFEs). At present, time-domain equalization, such as decision feedback equalization based on recursive least squares (RLSs), is extensively used in mud communication systems [12]. In the early 1990s, single-carrier frequency-domain equalization (SC-FDE) technology was proposed, which lowered the computation burden of traditional time-domain equalizers [13]. Although the complexity of SC-FDE is relatively low, it requires the insertion of redundant information such as cyclic prefixes in data frames, leading to a reduction in the communication rate [14]. In addition, due to the frequency-selective attenuation of the mud channel, the attenuation is relatively severe at certain frequency points, and the frequency-domain equalization is greatly affected by noise at deep fading points [15]. With advancements in efficient numerical methods for fractional Fourier transform, fractional Fourier domain equalization has emerged as a solution to address the channel frequency-selective attenuation [16]. It has been shown that fractional Fourier domain equalization is less susceptible to noise at deep fading points and has a computational complexity comparable to Fourier transform.

In recent years, many experts have come up with their own solutions to channel estimation and equalization. Zhang, J. et al. [17] proposed a channel parameter estimation scheme leveraging fractional Fourier transform (FrFT) and compressed sensing (CS) technology to achieve the non-cooperative perception of the FrFT optimal transform order. Zhang, Q. et al. [18] utilized a multi-armed bandit problem to deal with the uncertainty of channel state information (CSI) and proposed a channel estimation algorithm to explore the global changing channel information. Jia, S. et al. [19] adopted a sparse pilot structure and proposed a multi-blocks sparse Bayesian learning channel estimation algorithm to reduce the use of pilots and improve the transmission efficiency. Qi, M. et al. [20] applied the golden section method, parabolic interpolation, and Brent method to search for the optimal fractional-order domain to accurately estimate the parameters of the linear frequency modulation (LFM) signal. Dong, J. et al. [21] studied the FRFT normalized domain of polyphase-coded signals and derived a mathematical model to prove the impulse characteristic of the polyphase-coded signals under optimal order. Fang, Z. et al. [22] studied the energy aggregation characteristics of LFM signal in different FRFT domains, combining the FRFT with fractional power spectrum accumulation to accurately determine the optimal order and capture the signal with the first order Doppler rate of change. Pan, N. et al. [23] created a hybrid carrier communication system based on weighted-type fractional Fourier transform to make the integration of Orthogonal Frequency Division Multiplexing (OFDM) and SC-FDE possible. Li, W. et al. [24] realized feedforward adaptive equalization based on superimposed FrFT training sequences. Pan, C. et al. [25] applied FrFT to channel equalization of the Sweep-Spread carrier (SSC) system and improved its ability to resist the Doppler effect and ISI. Chen, H. et al. [26] proposed a Radio Frequency Fingerprint Identification (RFFI) method integrating Zero-forcing (ZF) equalization to effectively reduce the interference of time-varying channels on RFFI. Yadav, R. K. et al. [27] developed a low-complexity ZF equalizer by exploiting the sparsity of the delay-Doppler cascaded channel and developed an optimization framework to design intelligent reflecting surface coefficients with an objective to minimize the bit error rate (BER). Zischler, L. A. et al. [28] derived statistical models for the minimum mean square error (MMSE) equalizer coefficients and developed analytical solutions for the post-filtering information rate. Dong, P. Y. et al. [29] developed a robust block MMSE-DFE (joint minimum mean square error and decision feedback equalization) framework with joint time–frequency-domain processing for UAV-to-Ground SC-FDE communication systems, which can greatly control the error propagation of DFE and further improve the balancing performance of the DFE equalizer.

In this paper, a channel estimation and equalization algorithm based on FrFT is proposed to acquire the mud channel state information, eliminate inter-symbol interference caused by multipath effects, and alleviate frequency-selective attenuation of mud signals. The MPT system model and a detailed introduction of the technological method are shown in Section 2. Both simulations and real well tests have been conducted to test the proposed method’s effectiveness; comparisons and analyses are described in Section 3. The discussion and results are given in Section 4. Conclusions are presented in Section 5.

2. Methods

2.1. The MPT System Model

Figure 1 illustrates a sketch map of a traditional MPT system including sensors and mud pumps [30]. The measurements in such a system are usually carried out by instruments installed in the drill collar, all of which is controlled by electric machines. The generated information signals are subsequently transmitted to the earth’s surface, where they are analyzed by a signal processing unit at the ground [31].

Considering that the drilling speed is usually much slower than the data transmission speed, the MPT system can be considered a linear time-invariant system, and the system model is

$$r\left(t\right)=h\left(t\right)\ast x\left(t\right)+n\left(t\right)$$

(1)

where $\ast$ represents convolution, $r\left(t\right)$ is the output signal, $h\left(t\right)$ represents the impulse response of the mud channel, $x\left(t\right)$ is the original input signal, and $n\left(t\right)$ is the noise during transmission. The goal of channel estimation is to estimate $h\left(t\right)$ as accurately as possible.

Due to the multipath effect in the transmission of mud signals, the received signals may suffer from ISI and frequency-selective fading.

2.2. 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 $x\left(t\right)$ is defined as

$$F_p\left(u\right)=F^p\left[x\left(t\right)\right]\left(u\right)={\int }_{-\infty }^{\infty }x\left(t\right)K_p\left(u,t\right) dt$$

(2)

where p represents the order of FrFT, $F^p$ represents the FrFT operator, and $K_p\left(u,t\right)$ is integral kernel function, which is

$$K_p\left(u,t\right)=A_{\alpha }exp\left[j\pi (u^2\mathit{cot}\alpha -2ut+t^2\mathit{cot}\alpha )\right]$$

(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 $c\left(t\right)$ be expressed as

$$c\left(t\right)=\mathit{exp}\left(j\pi \mu t^2\right)$$

(4)

where $\mu$ is the chirp rate. Under a multipath mud pulse telemetry channel and with $c\left(t\right)$ being the system input, the received multipath signal can be represented as

$$r\left(t\right)={\sum }_{i=1}^N{a}_{i}c\left(t-{\tau }_i\right)+n\left(t\right)$$

(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; $n\left(t\right)$ 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

$$y\left(t\right)=x\left(t\right)p^{*}h\left(t\right)+w\left(t\right)$$

(6)

where $p^{*}$ represents fractional convolution, which is calculated as follows:

$$h\left(t\right)=\left(f{p}^{*}g\right)\left(t\right)$$

(7)

The fractional order form $H_p\left(u\right)$ of $h\left(t\right)$ is

$$H_p\left(u\right)=e^{-j{C}_{\alpha }{u}^2}{F}_p\left(u\right)\ast G_p\left(u\right)$$

(8)

Applying the fractional order form on (6) yields

$$Y_p\left(u\right)=X_p\left(u\right)H_p\left(u\right)e^{-j{\displaystyle \frac{\mathit{cot}\alpha }{2}}{u}^2}+W_p\left(u\right)$$

(9)

where $Y_p\left(u\right)$, $X_p\left(u\right)$, $W_p\left(u\right)$ are the fractional order expressions of the corresponding pressure wave signals, and $H_p\left(u\right)$ 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 $H_p\left(u\right)$. To obtain the minimum MSE of the equalizer, the optimal order should be selected so that the modulus of $H_p\left(u\right)$ is constant 1. Therefore, the objective function can be set as ${\epsilon }_p$, given as

$${\epsilon }_p=\left({\left|\left|{\displaystyle \frac{Y_p\left(u\right)}{X_p\left(u\right)}}\right|-1\right|}^2\right)$$

(10)

By identifying the order p that minimizes ${\epsilon }_p$, the optimized equalizer can be obtained.

Applying the FrFT of order $p_0$, which matches the LFM chirp rate, to $r\left(t\right)$, yields

$$R_{p_{\mathfrak{o}}}\left(u\right)={\mathcal{F}}_{p_{\mathfrak{o}}}\left\{r\left(t\right)\right\}\left(u\right)=\sum _{l=1}^L{a}_l{\mathcal{F}}_{p_{\mathfrak{o}}}\left\{c\left(t-{\tau }_l\right)\right\}\left(u\right)+N_{p_{\mathfrak{o}}}\left(u\right)$$

(11)

where $N_{p0}\left(\mu \right)$ 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]. Therefore,

$${\mathcal{F}}_{p_0}\left\{c\left(t-{\tau }_l\right)\right\}\left(u\right)\approx \delta \left(u-u_l\right)$$

(12)

resulting in

$$R_{p_0}\left(u\right)\approx \sum _{l=1}^L{a}_l\delta \left(u-u_l\right)+N_{p_0}\left(u\right)$$

(13)

The estimated channel impulse response is then reconstructed as

$$\widehat{h}\left(t\right)=\sum _{l=1}^{\widehat{L}}{\widehat{a}}_l\delta \left(t-{\widehat{\tau }}_l\right)$$

(14)

The channel estimation method based on fractional-order Fourier transform requires only a single FrFT calculation. Since the computational complexity of the FrFT decomposition is comparable to that of the fast Fourier transform (FFT), its complexity is lower than that of the traditional method.

2.3. Channel Equalization Based on Fractional Fourier Transform

Channel equalization is essential for addressing ISI during demodulation. Equalizers can be broadly classified into linear and nonlinear categories. Linear equalizers, such as ZF and MMSE equalizers, aim to remove ISI under ideal conditions. In contrast, nonlinear equalizers, including the decision feedback equalizer (DFE), adaptively adjust filter parameters to accommodate time-varying or nonlinear system conditions. Furthermore, equalization techniques can be categorized based on their implementation domain: time-domain, frequency-domain, and time–frequency-domain equalization. Due to the deep fading of mud channels in the frequency domain, the signal-to-noise ratio (SNR) is very low at certain frequencies, and the signal is significantly affected by noise interference. In some cases, the exclusive use of decision feedback equalizers fails to adequately mitigate the impact of frequency-selective attenuation, which results in a high BER during data decoding. Therefore, an effective channel compensation scheme is needed.

Various channel compensation strategies are available, with frequency-domain equalization constituting a common method, categorized into linear equalization and nonlinear equalization. When using a linear equalizer, the equalizer based on the ZF criterion can completely eliminate inter-symbol interference under ideal conditions. However, deep fading points in the channel frequency domain can result in increased noise and reduced SNR. The equalizer based on MMSE can find a compromise between channel noise and inter-symbol crosstalk, but it cannot completely eliminate inter-symbol crosstalk. The frequency-domain DFE based on decision feedback trades complexity for improved performance [35].

The recursive least squares decision feedback equalizer (RLS-DFE) is a type of adaptive equalizer that combines the benefits of decision feedback equalization with the RLS algorithm for efficient adaptive filtering. The RLS-DFE equalizer consists of two main components: the feedforward filter and the decision feedback filter. The RLS algorithm is applied to adaptively update the filter coefficients based on the received signal. The feedforward filter is responsible for mitigating the linear distortion caused by the channel. It processes the incoming signal by applying adaptive filter coefficients that are updated recursively using the RLS algorithm. After the feedforward filter processes the received signal, the feedback filter uses the decisions made about the transmitted symbols to remove the interference caused by past symbols.

The least squares algorithm sums the squares of errors as the optimization objective, and the cost function is expressed as

$$J_n=\sum _{i=0}^n{\mu }^{n-i}{\left|e_i\right|}^2=\sum _{i=0}^n{\mu }^{n-i}{|I_i-{\omega }_n^T{r}_i|}^2$$

(15)

where $\mu$ is the forgetting factor, with its value ranging (0,1), $e_i$ is the error vector, ${\omega }_n$ is the equalizer tap at time n, $I_i$ is the output vector, and $r_i$ is the input vector (the received signal).

Fractional Fourier equalization provides a low-complexity solution to the aforementioned issues, effectively addressing the challenge of deep fading points in the channel. It avoids the need for recursive operations, making it faster in many cases. This technique resolves the trade-off between noise power control and ISI, offering a computationally simple and easily implementable solution [25]. The channel compensation method designed in this article is to preprocess the input signal using linear equalization in the fractional Fourier domain for pre-filtering and then feed the signal into the decision feedback equalizer.

After transformation into the FrFT domain with optimal order, we obtain

$$R_{p_0}\left(u\right)=\sum _{n}x\left[n\right]\cdot H_{p_0}\left(u\right)\cdot e^{-j2\pi un{T}_s}+N_{p_0}\left(u\right)$$

(16)

To equalize the frequency-selective channel, we apply a frequency-domain equalizer

$$\tilde{X}\left(u\right)={\displaystyle \frac{R_{p_0}\left(u\right)}{H_{p_0}\left(u\right)+\epsilon }}$$

(17)

where $\epsilon$ is a regularization constant to mitigate division by deep-fade components.

Next, we perform an inverse FrFT to recover the time-domain signal

$$\tilde{x}\left(t\right)={\mathcal{F}}_{-p_0}\left\{\tilde{S}\left(u\right)\right\}$$

(18)

The whole flow chart of channel equalization in fractional Fourier domain is shown in Figure 2. The specific steps for implementing fractional Fourier domain channel equalization are outlined as follows:

(1)

Separate the training sequence and data sequence of the baseband data.

(2)

Transform the training sequence into fractional Fourier domains of varying orders for least squares channel estimation and select the optimal order.

(3)

Set the tap coefficients of the equalizer based on the least squares channel estimation of the optimal order.

(4)

Partition the received signal sequence into blocks for the optimal-order fractional Fourier transform.

(5)

Apply a multiplicative filter and revert the optimal order back to the time domain.

(6)

Convert the segmented processed data into serial data, which can then be fed into a decision feedback equalizer in the time domain for decision making.

3. Simulation and Experiment

3.1. 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.

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 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.

Figure 5 and Figure 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. The BER of the signal subjected to FrFT plus DFE is compared with that of the signal directly demodulated by DFE. Figure 7 shows the comparison of demodulation effects between the two methods. It reveals that the BER of the signal that subjected to FrFT plus DFE is much lower than that of the signal directly demodulated by DFE, highlighting the significantly improved demodulation performance with fractional Fourier transform equalization compared to direct decision feedback equalization. Hence, the simulation result shows that the channel equalization method based on FrFT can suppress multipath effects and obtain a lower BER, shows better performance and applicability in demodulation.

3.2. Real Well Test

The experimental data collected in Xinjiang, China, in October 2017 was used to further validate the present technique. Figure 8 shows the abridged general view of the MPT experimental site and Figure 9 is the actual experimental site [30]. In the real well experiment, the drilling rig was equipped with two mud pumps and worked at depths exceeding 3000 m.

The input signal is modulated by the BFSK with 1000 Hz sample rate and 10 bit/s symbol rate. The main steps of demodulation include filtering, denoising, synchronization, channel estimation, and equalization. Following the initial processing steps, Figure 10 displays the received signal after denoising, corresponding to a well depth of 2799 m, and then LFM signals are utilized to estimate the mud pulse channel. The channel estimates obtained using the fractional Fourier transform method are shown in Figure 11, demonstrating that there are three or four obvious multipath paths within a frequency of 10 Hz and the ISI is quite severe. Therefore, the following channel equalization is essential for addressing ISI during demodulation.

In order to quantitatively evaluate the suggested equalization method, further processing was performed on the equalization output in the demodulation process. The same noise signal still exists, with variations in symbol rate and utilization of BFSK modulation.

Table 1. BER comparisons between two techniques.

No.	Modulation Mode	Depths (m)	Data Rate (bit/s)	BER
				DFE	DFE + FrFT
1	BFSK	887	8	0.004	0
2	BFSK	887	10	0.12	0.04
3	BFSK	2515	10	0	0
4	BFSK	2602	9	0.50	0.05
5	BFSK	2632	6	0.12	0.01
6	BFSK	2890	12	0.14	0.03
7	BFSK	2980	10	0.23	0.02
8	BFSK	3016	12	0.08	0

shows the BER results after demodulation with two techniques (only DFE and DFE + FrFT) at different depths. From the table, it can be seen that both schemes can successfully demodulate in cases 1 and 3. However, for cases 2 and 4–8, the method employing only uses time-domain DFEs fails to achieve correct demodulation, resulting in a BER significantly higher than the standard requirements. When using channel estimation based on FrFt plus DFE, BER is decreased by 90%, which is shown in Table 1. In general, integration of FrFT equalization reduces the BER to below 0.5%, which meets the engineering requirements. The results demonstrate the effectiveness of the presented technique, indicating enhanced performance and usability in real well tests.

4. Discussion and Results

Section 3 presents the numerical simulation results that validate the performance of the channel estimation and equalization methods based on the FrFT. Additionally, communication conducted at a real well site further confirms that the proposed method significantly reduces the BER in the 3000 m experiment and achieves the recovery of the original signal. A comparison with traditional DFE methods also highlights the superior performance of the fractional Fourier transform-based approach in terms of accuracy and robustness.

Numerous iterative channel estimation techniques have been proposed in the wireless communication domain. However, these methods typically incur high computational costs due to their iterative nature and often exhibit degraded performance under strong interference conditions. In contrast, the proposed FrFT-based method achieves comparable estimation accuracy using a single transform operation, thereby significantly reducing computational complexity. By jointly performing channel estimation and equalization in the fractional Fourier domain, our approach delivers superior bit error rate (BER) performance, particularly in the presence of severe multipath and structured interference as observed in real-world downhole environments.

Beyond comparisons with advanced iterative algorithms, we also benchmark our method against conventional equalization schemes such as time-domain RLS-DFE, MMSE, and ZF equalizers. While RLS-DFE is capable of adaptively tracking channel variations and performs adequately under moderate fading, it is susceptible to instability and requires intensive computation for recursive coefficient updates and matrix operations. In contrast, our FrFT-based approach reformulates the equalization problem in the fractional Fourier domain, effectively pre-whitening the channel response and mitigating multipath distortion without the need for iterative adaptation.

MMSE equalization offers a trade-off between noise suppression and ISI mitigation but suffers from residual inter-symbol interference in channels with deep fades. ZF equalizers can eliminate ISI under ideal channel conditions; however, their sensitivity to spectral nulls leads to significant noise amplification in practice. The proposed method circumvents these limitations by leveraging the energy compaction properties of linear frequency-modulated (LFM) signals in the FrFT domain. This enables precise channel impulse response estimation and efficient spectral flattening prior to time-domain DFE processing.

In addition to its effectiveness in handling multipath and frequency-selective fading, the proposed FrFT-based method demonstrates intrinsic robustness against line-spectral interference—a prevalent challenge in mud pulse telemetry due to the periodic activity of mud pumps and rotary machinery. Such interference introduces narrowband components that overlap with the signal spectrum, severely impacts traditional time- and frequency-domain estimation methods.

By design, the FrFT provides optimal energy localization for LFM signals when the transform order aligns with the chirp rate. Consequently, the desired signal components are sharply focused in the FrFT domain, whereas narrowband interferers remain dispersed or shifted in the time–frequency plane. This intrinsic separation facilitates enhanced discrimination and extraction of useful signals without requiring interference modeling or adaptive filtering. Field data further verify the method’s robustness under periodic pump induced interference, underscoring its practical applicability in telemetry systems operating under harsh and noise-prone downhole conditions.

To illustrate the superiority of our work, we make quantitative comparisons between the proposed equalization schemes and some of the mentioned equalization works in Section 1 to create a scientific discussion. We use the same data in Table 1, for which the traditional DFE method fails to achieve correct demodulation.

Table 2. BER comparisons between three equalizers.

No.	Depths (m)	Data Rate (bit/s)	BER
			ZF	MMSE	FrFT
1	887	10	0.02	0.04	0.04
2	2602	9	0.13	0.07	0.05
3	2632	6	0.08	0.01	0.01
4	2890	12	0.15	0.14	0.03
5	2980	10	0.12	0.17	0.02

displays the BER results after demodulation with three different equalizers (FrFT, ZF, and MMSE) at various depths. From the table, it can be seen that all equalizers can successfully demodulate in case 1. However, for cases 2–6, the ZF equalizer cannot demodulate properly due to the increase in well depth, while the MMSE equalizer suffers from residual ISI in channels with deep fades and fails to achieve correct demodulation at data rates of 10 and 12 bit/s. The results indicate that our proposed equalizer has better performance in improving the BER in a real well test demodulation.

In general, the main innovation points of this article are as follows:

(1)

A channel estimation method tailored to the characteristics of mud channels is proposed using FrFT, along with an optimal order search strategy that significantly reduces computational complexity.

(2)

A compensation scheme for channel frequency-selective fading is developed using FrFT domain equalization, which enhances the overall communication performance. The BER of experimental results after demodulation has decreased by 90% compared to the traditional DFE method. Moreover, compared with ZF and MMSE equalization, our method obtains a lower BER in real well tests. In general, the BER after demodulation is reduced to below 0.5%, which meets the engineering requirements.

Although the current study has achieved some good results, the overall work is still not perfect and exists limitations which needs to be further researched. Specifically, when using FrFT for LFM signal detection, currently there is no adaptive threshold method due to the impact of noise. In addition, the high computational complexity for selecting the optimal fractional order in FrFT should be reduced. In addition, we need to test and compare the performance of this study with more advanced published works. Finally, further experiments are needed to verify the proposed method in different drilling environments.

5. Conclusions

This paper proposed the channel estimation and equalization method based on fractional Fourier transform. The architecture of mud pulse telemetry is introduced, along with a brief overview of the processes involved in signal generation, transmission, and acquisition. The continuous pressure wave signal used in this project is generated by a swinging mud pulse generator and is transmitted to the ground through mud fluid. During the transmission process, the pressure wave decays exponentially with distance and exhibits significant frequency-selective attenuation.

A systematic analysis of the noise and channels within the mud of the actual well was conducted using field data. We developed a channel estimation method based on FrFT to estimate path delays and attenuation coefficients. Compared to conventional LFM-based time-domain estimation techniques, the proposed method offers lower computational complexity and improved robustness, especially in the presence of unknown frequency offsets, and achieves significantly higher estimation accuracy.

Moreover, an FrFT domain equalizer is designed to mitigate the effects of frequency-selective fading. The equalizer effectively compensates for channel distortion without amplifying background noise. The overall effectiveness of the proposed technique is validated through data collected from a real well site, demonstrating enhanced decoding performance. The BER of experimental results after demodulation is decreased by 90% compared to the traditional DFE method and is reduced to below 0.5%, which meets the engineering requirements, ensuring that our approach is not only theoretically but also practically feasible. In addition, our proposed equalizer achieves better performance in improving the BER compared to the ZF and MMSE equalizers. Finally, we achieve advanced data rate communication (over 12 bit/s) in a 3000 m drilling system.

Future research will focus on expanding and refining our current methodology. Specifically, we should explore the potential of FrFT-based equalization algorithms to reduce training sequence length and eliminate redundant information, thereby lowering computational complexity. In addition, practical deployment remains an important direction for continued investigation. Key aspects include analyzing the sensitivity of fractional order selection under dynamic channel conditions, developing lightweight and adaptive order estimation strategies, and implementing FrFT-based signal processing on embedded or low-power DSP platforms. Finally, more experiments will be conducted to verify the proposed method in the changing behavior of the mud and drilling environment and test the channel characteristics with varying temperature, pressure, and depth. These efforts are expected to bridge the gap between theoretical algorithm design and real-time field deployment, ultimately enabling robust and efficient telemetry systems for harsh downhole environments.