Introduction

MWD technology significantly enhances the capabilities of geological steering and subsurface evaluation in extended-reach wells, challenging horizontal wells, and multilateral wells, holding substantial research significance. As early as 1977, influential continuous wave mud pulse telemetry technology emerged. To date, continuous wave mud pulse telemetry technology has evolved through the optimization of mud pulse generator structures, improvements in drilling tool design, adjustments in mud physical properties, and enhancements in continuous wave signal modulation methods1. However, as the focus of underground exploration increasingly shifts to deep and ultra-deep wells, the complexity of underground conditions has escalated, necessitating timely underground data transmission and ensuring the accuracy of transmitted information2. The current situation imposes heightened demands on the transmission rate of continuous wave mud pulse data3. In terms of enhancing the quality of mud pulse signals, this can be achieved by optimizing the structure of the mud pulse generator. For instance, reference4 transforms the issue of the valve port shape of the mud pulse generator into a topological design methodology, thereby devising a superior structure that ensures the quality of the output mud pulse signal, to some extent reducing demodulation difficulty and improving the accuracy of information transmission. Regarding mitigating the impact of phase drift, reference5, based on two-dimensional axially-symmetric transient flow theory, employs a small signal analysis method to establish a continuous wave phase shift model of drilling fluid. This model analyzes the relationship between the phase shift after channel transmission and drilling parameters as well as signal parameters, providing applicable conditions to reduce the effects of phase drift. To increase the information transmission rate, frequency division multiplexing can be utilized. For example, reference6 implements efficient Orthogonal Frequency Division Multiplexing(OFDM) through a secondary greedy bit allocation algorithm, dividing the channel into multiple sub-channels to transmit different data streams, thereby enhancing the information transmission rate. Therefore, optimizing the structure of the mud pulse generator and employing frequency division multiplexing can effectively facilitate better continuous wave mud pulse data transmission.

In the realm of signal modulation, conventional methodologies include ASK, FSK, and PSK, with FSK and PSK being predominantly utilized in continuous wave transmission technology within drilling fluids7. Furthermore, OFDM, as an emerging modulation technique, has found extensive application in mud pulse data transmission due to its efficacy in mitigating multi-pathway effect and frequency-selective fading. For FSK and PSK, given that mud pulse signals are generated via the rotational motion of mud pulse generator motors8, the pursuit of higher data transmission rates inevitably confronts non-negligible ISI and waveform distortion caused by the motor’s inherent inertia and other indirect factors at high transmission speeds9. Consequently, it is imperative to identify an effective method to eliminate or attenuate such interference and distortion.

In addressing the non-negligible ISI and other waveform distortions indirectly caused by the inherent inertia of motors during high-rate transmissions, existing research predominantly focuses on enhancing the information content per symbol and maintaining waveform stability. To augment the information content per symbol, the approach of increasing the modulation order is employed to boost the information transmission rate. Reference10 utilizes multiple Phase Shift Keying (MPSK) for data modulation, increasing the modulation order based on PSK to ensure symbol transmission rate while enhancing the information content per symbol, thereby achieving an increased information transmission rate. When meeting the same information transmission rate requirements, the motor speed of the mud pulse generator decreases, mitigating the severity of ISI and waveform distortion. Reference7 proposes Offset Quadrature Phase Shift Keying (OQPSK) based on Quadrature Phase Shift Keying (QPSK), which not only increases the information content per symbol but also eliminates the π radians phase shift during symbol information transitions, rendering OQPSK a constant envelope modulation scheme and avoiding potential linear distortions inherent in QPSK.However, the aforementioned modulation methods, along with other modulation techniques, primarily optimize upon the foundations of FSK and PSK, aiming to enhance symbol transmission rates, mitigate ISI or other waveform distortions, and reduce the complexity of modulation and demodulation. In practical applications, they still encounter significant ISI due to the inherent inertia of motors. In terms of demodulation, it is necessary to recognize and match the collected data waveform to extract the symbol information. For waveform recognition and matching, current research predominantly employs deep learning methodologies. Reference11 utilizes Fourier Transform and Wavelet Transform to extract frequency and time-frequency domain features, and employs Principal Component Analysis (PCA) to reduce parameter dimension, leveraging SVM for model training to classify single-cycle effective pulse signals, pseudo-noise, and interference signals with high accuracy. Reference12 performed samples through screening, truncation, and baseline correction, then uses CNN for model training to distinguish between seismic, noise, blasting, and anomalous waveform, achieving high recognition rates. It is evident that deep learning exhibits superior performance in waveform recognition.

In addressing the issue of ISI caused by the inherent inertia of the motor and the practical suitability of various deep learning models, this paper proposes a continuous wave mud pulse data transmission method that combines CGFK modulation with deep learning for demodulation. In this method, CGFK utilizes the rate of frequency change of the continuous wave as the characteristic for modulation and demodulation. Within each symbol period, the frequency of the continuous wave uniformly increases from 0 to a predetermined frequency value and then uniformly decreases back to 0. By maintaining a constant symbol period and altering the rate of frequency change, different symbol information is distinguished.In the context of motor speed, CGFK also exhibits a process where the speed uniformly increases from zero to a set value and then uniformly decreases back to zero, meaning that the motor speed at both the beginning and end of the symbol period is zero. Since ISI is primarily caused by the motor’s inertia preventing the speed from changing as ideally expected, by maintaining consistent conditions such as load and channel, comparing the correlation between the actual and ideal speed waveform of various modulation methods can indirectly reflect the degree of ISI. Through actual testing of the servo motor driving a pulse valve, the normalized mean square error (NMSE) between the actual and ideal speed waveform was calculated for FSK, PSK, and CGFK, with FSK at 0.07638, PSK at 0.07165, and CGFK at 0.05725. It is evident that the actual motor rotational speed of CGFK demonstrates closer alignment with ideal speed variations, thus verifying that its unique motor control methodology effectively mitigates or prevents ISI issues caused by inherent motor inertia through optimized operational paradigms.

In the domain of integrated deep learning demodulation methodologies, conventional mud pulse data transmission systems predominantly employ Matched Filter techniques. Given the superior waveform recognition capabilities inherent in deep learning architectures and the relatively uncomplicated waveform identification requirements within mud pulse communication protocols, rigorous evaluation and analytical assessment have identified SVM, LSTM, and CNN as particularly suitable for waveform recognition-assisted demodulation tasks. To comprehensively investigate the efficacy of deep learning-enhanced CGFK waveform identification, it becomes imperative to conduct comparative studies examining the demodulation performance of SVM, LSTM, CNN, and traditional Matched Filter approaches when applied to CGFK signal processing.

The primary contributions of this paper encompass the following two aspects:

1) Addressing the issue of ISI caused by the inherent inertia of the motor, this paper proposes a CGFK method. This modulation technique utilizes the rate of frequency change of continuous waves as the characteristic for modulation and demodulation. In terms of motor control, within each symbol period, the motor speed uniformly increases from zero to a predetermined value and then uniformly decreases back to zero. This ensures that the motor speed is zero at both the beginning and end of each symbol, fundamentally mitigating or eliminating ISI induced by the motor’s inertia.

2) Regarding the comparative effectiveness of deep learning algorithms versus traditional Matched Filter in assisting CGFK recognition, this study simulates continuous wave mud pulse signals following CGFK protocols. We train models using three deep learning algorithms (SVM, LSTM, CNN) alongside the conventional Matched Filter technique, subsequently evaluating their waveform recognition efficacy. Through comprehensive performance analysis, this research identifies algorithms capable of supplementing CGFK demodulation while demonstrating superior performance to traditional Matched Filter approaches.

Results

Continuous Gradation Frequency Keying modulation model

The generation of underground continuous wave mud pulses is typically achieved through a mud pulse generator. During operation, as the motor drives the rotating assembly, stationary blades remain fixed while rotary blades synchronously revolve. This coordinated motion dynamically modulates the effective flow area through the inter-blade clearance. The consequent variation in fluid velocity through this engineered passage induces corresponding pressure fluctuations, which are captured by pressure sensors positioned at the upper pipeline section to form measurable continuous wave signatures. This operational principle is schematically illustrated in Fig. 1.

Fig. 1
figure 1

Schematic Diagram of Continuous Wave Mud Pulse Generation.

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As observed in Fig. 1, when the rotor blade and stationary blades are fully aligned, the conduction area is maximized, the fluid velocity is the fastest, and the pressure collected by the top pressure sensor is the lowest. As the rotor blades rotate, the conduction area decreases, the fluid velocity slows down, and the received pressure increases. When the rotor blades are completely misaligned with the stationary blades, the conduction area is minimized, and the pressure reaches its maximum. As the rotor blades continue to rotate, the conduction area increases, the pressure begins to decrease, until the rotor blades and stationary blades are fully aligned again, and the cycle repeats. Ultimately, the mud pressure exhibits a continuous waveform change as the motor rotates. Therefore, when the motor speed of the continuous wave generator changes, the frequency of the pressure wave will also change accordingly.

CGFK is a modulation method that controls the rate of frequency change of the carrier signal based on the base-band digital signal. Unlike traditional FSK, CGFK uses the frequency variation rate of the continuous wave as the feature for modulation and demodulation. The frequency of the carrier signal does not remain constant within the symbol period but instead increases uniformly from 0 to a set frequency value and then decreases uniformly back to 0. Therefore, at both the beginning and the end of each symbol period, the carrier signal’s frequency is 0. The frequency variation of the carrier signal can be achieved by controlling the rotor speed of the motor to change uniformly, as shown in the schematic diagram of the motor speed control for carrier frequency variation in Fig. 2.

Fig. 2
figure 2

Schematic Diagram of Frequency Change of Motor Speed Control Waveform.

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In Fig. 2 above, the left side represents the rotor speed of the mud pulse generator motor, and the right side represents the corresponding continuous wave. It can be observed that as the rotor speed, maximum rotational speed, and rotational acceleration and deceleration of the mud pulse generator motor increase, the corresponding frequency, maximum frequency, and frequency variation rate of the continuous wave also increase accordingly. At the start and end of each symbol period, the frequency of the continuous wave is 0, and the motor speed is also 0.Traditional continuous wave mud-pulse modulation usually uses FSK and PSK. Both of these modulation methods inevitably require a sudden change in motor speed during symbol switching.

In FSK, the waveform is required to instantaneously transition from one frequency to another, necessitating an immediate alteration in motor speed. During low-speed data transmission, the symbol period is significantly longer, rendering the buffer interval almost negligible in comparison. However, in high-speed data transmission scenarios, the symbol period is considerably shorter, and the impact of the buffer interval becomes pronounced, leading to substantial ISI, as illustrated in Fig. 3.

Fig. 3
figure 3

Schematic Diagram of FSK ISI.

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The blue waveform represents the standard FSK waveform, and the red waveform represents the FSK ISI distortion waveform. As shown in the figure, during symbol switching, since the motor speed changes continuously, the waveform frequency also undergoes a continuous transition. Therefore, in high-speed data transmission, FSK finds it difficult to generate an ideal modulation waveform, and the higher the transmission rate, the more pronounced the ISI becomes.

PSK distinguishes different symbols by utilizing phase as the demodulation characteristic, thereby necessitating the control of the motor to undergo frequent and rapid alternations between forward and reverse rotations, or to accelerate swiftly to a designated angle. Given that the motor’s speed cannot instantaneously change but rather undergoes a continuous process of acceleration and deceleration, PSK is also susceptible to ISI, as illustrated in Fig. 4.

Fig. 4
figure 4

Schematic Diagram of PSK ISI.

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In Fig. 4, the blue waveform represents the standard waveform, and the red waveform represents the ISI distortion waveform in PSK. It can be observed that, since the motor speed change is a continuous process and cannot change instantaneously, and the phase change is achieved by the motor switching between forward and reverse rotations or accelerating rapidly to a specified angle, the phase does not switch precisely during symbol transitions. Instead, a buffer time is required after entering a new symbol period to complete the transition, resulting in ISI and a phase shift-like effect.

Since CGFK uses the frequency change rate as a demodulation feature, the frequency of the waveform within each symbol period increases uniformly from 0 to a set frequency value and then decreases uniformly back to 0. That is, the motor speed increases uniformly from 0 to the set speed and then decreases uniformly back to 0. At both the beginning and end of each symbol period, the motor speed is 0. This process avoids the sudden changes in motor speed that occur in FSK, PSK, and other similar modulation methods during symbol transitions, thus preventing ISI caused by the motor’s inertia during sudden speed changes.

Certainly, FSK and PSK, as traditional modulation techniques, exhibit certain limitations in specific aspects. As an emerging modulation technology, OFDM may demonstrate superior performance. Indeed, OFDM, as a multi-carrier modulation technique, possesses unique advantages in terms of transmission rate and BER. In underground mud pulse data transmission, OFDM typically achieves a transmission rate ranging from 10 bps to 50 bps, with a BER controlled between \(\:{10}^{-3}\) and \(\:{10}^{-5}\) under signal-to-noise ratio (SNR) conditions of 2 dB to 20 dB. In comparison, CGFK generally attains a transmission rate of 1 bps to 10 bps, with a BER controlled between \(\:{10}^{-3}\) and \(\:{10}^{-3}\) under the same SNR conditions. While the BER performance of both techniques is comparable, OFDM indeed excels in transmission rate. However, the implementation of OFDM often necessitates more complex signal processing, making it more challenging to modulate by controlling motor rotation in mud pulse transmission. In contrast, CGFK offers a simpler approach to motor control, and even in terms of motor control requirements, CGFK imposes fewer demands compared to both emerging and traditional modulation technologies. In scenarios where high transmission rates are not essential, CGFK emerges as a more favorable choice.

Establish a mathematical model for the process of pressure wave generation by the rotation of the CGFK motor. Under ideal conditions, the motor speed uniformly increases from 0 to the maximum speed corresponding to each symbol period and then uniformly decreases to 0 within each symbol period. Assume that the maximum speeds of the motor for the symbols T1, T2, T3, and T4 (00, 01, 10, 11) are \(\:{\text{w}}_{1}\), \(\:{\text{w}}_{2}\), \(\:{\text{w}}_{3}\), and \(\:{\text{w}}_{4}\), respectively, with the corresponding relationships as follows:

$$\:{\text{w}}_{1}=\frac{1}{2}{\text{w}}_{2}=\frac{1}{3}{\text{w}}_{3}=\frac{1}{4}{\text{w}}_{4}$$
(1)

That is, the normal operating speed of the motor must satisfy \(\:{\text{w}}_{4}\). For CGFK, the motor speed gradually changes over time. Let the single symbol period be \(\:{\text{t}}_{0}\), and the relationship of the motor speed \(\:\text{w}\left(\text{t}\right)\) (unit: rps) with respect to time t, the single symbol period \(\:{\text{t}}_{0}\), and the preset maximum motor speeds \(\:{\text{w}}_{\text{i}}\) for each symbol is given by:

$$\:\begin{array}{c}w\left(\text{t}\right)=\left\{\begin{array}{l}\frac{{2\text{w}}_{\text{i}}}{{\text{t}}_{0}}t,\qquad \qquad t\in\:[0,\frac{{\text{t}}_{0}}{2}],\quad i\in\:(\text{1,2},\text{3,4})\\\:2{\text{w}}_{\text{i}}-\frac{{2\text{w}}_{\text{i}}}{{\text{t}}_{0}}t,\quad t\in\:(\frac{{\text{t}}_{0}}{2},{\text{t}}_{0}],\ \ i\in\:(\text{1,2},\text{3,4})\end{array}\right.\end{array}$$
(2)

The aforementioned derivation is conducted under ideal conditions; however, in practical scenarios, the rotational speed of the motor is not simply an ideal linear relationship due to the influence of motor inertia. To account for the impact of motor inertia on the rotational speed, it is decided to introduce the motor’s dynamic equation for correction. The motor’s dynamic equation can be expressed as:

$$\:\text{J}\frac{\text{d}{\upomega\:}\left(\text{t}\right)}{\text{d}\text{t}}={\text{T}}_{\text{m}}\left(\text{t}\right)-{\text{T}}_{\text{l}}\left(\text{t}\right)$$
(3)

where: \(\:\text{J}\) represents the moment of inertia of the motor (unit: kg·m²). ω(t) denotes the angular velocity of the motor (unit: rad/s). \(\:{\text{T}}_{\text{m}}\left(\text{t}\right)\) signifies the output torque of the motor (unit: N·m). \(\:{\text{T}}_{\text{l}}\left(\text{t}\right)\) represents the load torque (unit: N·m).

To simplify the problem, it is assumed that the output torque \(\:{\text{T}}_{\text{m}}\left(\text{t}\right)\) of the motor is constant, and the load torque \(\:{\text{T}}_{\text{l}}\left(\text{t}\right)\) is also constant. Consequently, the angular velocity ω(t) of the motor can be expressed as:

$$\:{\upomega\:}\left(\text{t}\right)={{\upomega\:}}_{0}+\frac{{\text{T}}_{\text{m}}-{\text{T}}_{\text{l}}}{\text{J}}\text{t}$$
(4)

wherein: \(\:{{\upomega\:}}_{0}\) represents the initial angular velocity of the motor (unit: rad/s). \(\:{\text{T}}_{\text{m}}\) denotes the output torque of the motor (unit: N·m). \(\:{\text{T}}_{\text{l}}\) signifies the load torque (unit: N·m). \(\:\text{J}\) is the moment of inertia of the motor (unit: kg·m²). Incorporating the angular velocity formula and accounting for the influence of motor inertia on motor speed, the motor speed \(\:\text{w}\left(\text{t}\right)\) (unit: rps) as a function of time t, single symbol period \(\:{\text{t}}_{0}\), and the preset maximum motor speed \(\:{\text{w}}_{\text{i}}\) for each symbol is expressed as:

$$\:\text{w}\left(\text{t}\right)=\left\{\begin{array}{c}\frac{{2\text{w}}_{\text{i}}}{{\text{t}}_{0}}t+\frac{{\text{T}}_{\text{m}}-{\text{T}}_{\text{l}}}{2{\uppi\:}\text{J}}t,\quad t\in\:[0,\frac{{\text{t}}_{0}}{2}],\quad i\in\:(\text{1,2},\text{3,4})\\ 2{\text{w}}_{\text{i}}-\frac{{2\text{w}}_{\text{i}}}{{\text{t}}_{0}}t+\frac{{\text{T}}_{\text{m}}-{\text{T}}_{\text{l}}}{2{\uppi\:}\text{J}}t,\quad t\in\:(\frac{{\text{t}}_{0}}{2},{\text{t}}_{0}],\quad i\in\:(\text{1,2},\text{3,4})\end{array}\right.$$
(5)

Consequently, the number of revolutions \(\:\text{N}\left(\text{t}\right)\) (unit: r) of the motor rotor blade is given by:

$$\:\begin{array}{c}N\left(\text{t}\right)={\int\:}_{0}^{\text{t}}\text{w}\left(\text{t}\right)\text{d}\text{t}\end{array}$$
(6)

The specific formulas are as follows:

$$\begin{array}{c}N\left(\text{t}\right)=\left\{\begin{array}{c}\frac{{\text{w}}_{\text{i}}}{{\text{t}}_{0}}{\text{t}}^{2}+\frac{{\text{T}}_{\text{m}}-{\text{T}}_{\text{l}}}{4{\uppi\:}\text{J}}{\text{t}}^{2},\quad t\in\:\left(0,{\text{t}}_{0}\right],\quad i\in\:(\text{1,2},\text{3,4})\\ 2{\text{w}}_{\text{i}}t-\frac{{\text{w}}_{\text{i}}}{{\text{t}}_{0}}{\text{t}}^{2}-\frac{{\text{w}}_{\text{i}}{\text{t}}_{0}}{2}+\frac{{\text{T}}_{\text{m}}-{\text{T}}_{\text{l}}}{4{\uppi\:}\text{J}}{\text{t}}^{2},\quad t\in\:\left({\text{t}}_{0},{2\text{t}}_{0}\right],\quad i\in\:(\text{1,2},\text{3,4})\end{array}\right.\end{array}$$
(7)

Thus, the pressure wave \(\:\text{P}\left(\text{t}\right)\) can be represented by the following formula:

$$\:\begin{array}{c}P\left(\text{t}\right)=\frac{{\text{C}}_{2}-{\text{C}}_{1}}{2}\text{sin}\left(2{\uppi\:}\text{N}\left(\text{t}\right)\right)+\frac{{\text{C}}_{1}+{\text{C}}_{2}}{2}\end{array}$$
(8)

In this context, \(\:{\text{C}}_{1}\) denotes the pressure constant when the rotor blades of the motor are fully aligned with the stationary blades, while \(\:{\text{C}}_{2}\) represents the pressure constant when the rotor blades are completely misaligned with the stationary blades. The pressure wave equation is derived as follows:

$$\:\begin{array}{c}P\left(\text{t}\right)=\left\{\begin{array}{c}\frac{{\text{C}}_{2}-{\text{C}}_{1}}{2}\text{sin}\left(\frac{2{\uppi\:}{{\upomega\:}}_{\text{i}}{\text{t}}^{2}}{{\text{t}}_{0}}+\frac{{\uppi\:}({\text{T}}_{\text{m}}-{\text{T}}_{\text{l}})}{2\text{J}}{\text{t}}^{2}\right)+\frac{{\text{C}}_{1}+{\text{C}}_{2}}{2}, \quad t\in\:\left(0,{\text{t}}_{0}\right], \quad i\in\:(\text{1,2},\text{3,4})\\\:\frac{{\text{C}}_{2}-{\text{C}}_{1}}{2}\text{sin}\left(4{\uppi\:}{{\upomega\:}}_{\text{i}}\text{t}-\frac{2{\uppi\:}{{\upomega\:}}_{\text{i}}{\text{t}}^{2}}{{\text{t}}_{0}}-{{\uppi\:}{\upomega\:}}_{\text{i}}{\text{t}}_{0}+\frac{{\uppi\:}({\text{T}}_{\text{m}}-{\text{T}}_{\text{l}})}{2\text{J}}{\text{t}}^{2}\right)+\frac{{\text{C}}_{1}+{\text{C}}_{2}}{2}, \quad \text{t}\in\:\left({\text{t}}_{0},{2\text{t}}_{0}\right], \quad \text{i}\in\:(\text{1,2},\text{3,4})\end{array}\right.\end{array}$$
(9)

\(\:\text{P}\left(\text{t}\right)\) signifies the initial modulated waveform at the transmitting end, wherein \(\:\frac{{\uppi\:}({\text{T}}_{\text{m}}-{\text{T}}_{\text{l}})}{2\text{J}}{\text{t}}^{2}\) accounts for the influence of motor inertia on the phase of the pressure wave. The demodulated waveform actually received at the receiving end is subject to the combined effects of noise and channel attenuation, among other factors.

It is imperative to acknowledge that the derivation of the aforementioned formula is predicated on the assumption of constant load torque and output torque. However, in the actual underground mud pulse data transmission process, the load torque and output torque typically do not remain constant. Instead, they undergo dynamic variations influenced by factors such as mud flow, mechanical friction, pressure wave reaction, temperature, and pressure, thereby exhibiting temporal changes.

To investigate the specific impact of varying symbol periods on the BER, a servo motor with a maximum acceleration of 3000 rpm/s was employed to drive the pulse valve for mud pulse modulation, while maintaining the maximum motor speeds for each symbol of CGFK unchanged. Specifically, the maximum motor speeds for symbols T1, T2, T3, and T4 were set at 75 rpm, 150 rpm, 225 rpm, and 300 rpm, respectively, corresponding to maximum pressure wave frequencies of 5 Hz, 10 Hz, 15 Hz, and 20 Hz. To facilitate the statistical analysis of the BER and information transmission rate, the duration of a single symbol period was altered to 0.1 s, 0.2 s, 0.4 s, 0.8 s, and 1.0 s, resulting in corresponding information transmission rates of 2 bps, 2.5 bps, 5 bps, 10 bps, and 20 bps. The variation in the BER is illustrated in Fig. 5.

Fig. 5
figure 5

Schematic Diagram of Bit Error Rate Variation with Respect to Symbol Period.

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As the symbol period decreases, the demand for motor acceleration and deceleration capabilities increases. The maximum acceleration of standard servo motors available on the market is 3000 rpm/s. Under these conditions, the minimum achievable symbol period is 0.1 s. Consequently, as illustrated in the aforementioned figure, when the symbol period exceeds 0.2 s, the CGFK modulation’s requirement for motor rotational acceleration does not surpass the motor’s acceleration capacity, and the BER remains stable at approximately 0.0005. However, when the symbol period is reduced to 0.1 s, the CGFK modulation’s demand for motor rotational acceleration reaches the maximum rotational acceleration of the servo motor, leading to exacerbated waveform distortion and a significant increase in the BER.

Under optimal conditions, the frequency variation rate of continuous wave mud pulse waveform can be altered by modulating the acceleration and deceleration of the motor rotor’s rotation. Consequently, the frequency variation rate can be employed as a demodulation characteristic to distinguish between different symbol information by controlling the motor’s rotational speed. Figure 6 illustrates the schematic representation of four symbols (00, 01, 10, 11) along with their corresponding motor speed variations.

Fig. 6
figure 6

Schematic Diagram of Continuously Gradation Frequency Keying Code Element and Motor Speed.

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From the diagram above, it is clear that at the beginning and end of each symbol period, the frequency of the corresponding waveform is 0, with no frequency jump, and the frequency variation shows a continuous gradation effect. Correspondingly, the motor speed also does not experience any sudden changes, but follows a continuous gradation. Within each symbol period, the motor starts from a stationary state, accelerates uniformly, and then decelerates uniformly back to a stationary state, without switching between forward and reverse rotations, which makes the control simple. At this point, the transmitted information consists of 2-bit binary data, with each symbol carrying 2 bits of information.

Conduct an analysis of CGFK, encompassing both ideal and practical scenarios. Let \(\:{\text{V}}_{\text{m}\text{a}\text{x}}\)denote the maximum rotational speed of the motor in rpm, P represent the number of blades on the stationary and rotor of the motor, and n indicate the number of bits in the transmitted binary data. When transmitting n-bit binary data, the maximum carrier frequency \(\:{\text{f}}_{\text{m}\text{a}\text{x}}\) (in Hz), the average carrier frequency \(\:{\text{f}}_{0}\) (in Hz), and the maximum information transmission rate \(\:{\text{R}}_{\text{b}}\) (in bps) are as follows:

$$\:{\text{f}}_{\text{m}\text{a}\text{x}}={\text{V}}_{\text{m}\text{a}\text{x}}\bullet\:\text{P}$$
(10)
$$\:{\text{f}}_{0}=\frac{\sum\:_{\text{i}=1}^{{2}^{\text{n}}}\text{i}\bullet\:{\text{V}}_{\text{m}\text{a}\text{x}}\bullet\:\text{P}}{{2}^{2\text{n}+1}}$$
(11)
$$\:{\text{R}}_{\text{b}}=\frac{{\text{V}}_{\text{m}\text{a}\text{x}}\bullet\:\text{P}}{{2}^{\text{n}+2}\ast\:15}$$
(12)

Given that the transmitted data consists of n-bit binary data, a single symbol contains n bits of information. The relationship between the maximum information transmission rate \(\:{\text{R}}_{\text{b}}\) (in bps) and the maximum symbol transmission rate \(\:{\text{R}}_{\text{B}}\) (in Baud) is as follows:

$$\:{\text{R}}_{\text{b}}={\text{R}}_{\text{B}}\ast\:{\text{log}}_{2}\text{N}$$
(13)

wherein N denotes the base of the data transmission, specifically \(\:{2}^{\text{n}}\)., the maximum symbol transmission rate \(\:{\text{R}}_{\text{B}}\) is:

$$\:{\text{R}}_{\text{B}}=\frac{{\text{V}}_{\text{m}\text{a}\text{x}}\bullet\:\text{P}}{{2}^{\text{n}+2}\ast\:15\text{n}}$$
(14)

The bandwidth efficiency \(\:{{\upeta\:}}_{\text{b}}\) refers to the achievable bit rate per unit bandwidth, measured in bps/Hz. The effectiveness of a transmission system is generally expressed in terms of bandwidth efficiency:

$$\:{{\upeta\:}}_{\text{b}}=\frac{{\text{R}}_{\text{b}}}{\text{B}}$$
(15)

In the above equation, B denotes the channel bandwidth.

Waveform recognition simulation test

CGFK distinguishes itself from conventional modulation techniques by utilizing the rate of frequency variation as the distinguishing feature for modulation and demodulation of different symbols, rendering traditional demodulation methods unsuitable. To enhance the demodulation process, it is imperative to integrate deep learning methodologies, with waveform recognition at its core. Upon investigation, it is posited that deep learning algorithms such as SVM, LSTM, and CNN may exhibit superior performance compared to the Matched Filter13,14,15,16. In pursuit of an algorithm that exhibits heightened sensitivity to the mud pulse waveform generated by CGFK regulations, it is essential to train models based on the aforementioned four algorithms and perform waveform recognition, followed by a comparative analysis of their suitability. The model training and waveform discrimination for these four algorithms are conducted in accordance with the comprehensive flowchart illustrated in Fig. 7.

Fig. 7
figure 7

Comprehensive Flowchart of Mud Pulse Continuous Wave Decoding and Adjustment.

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The entire process is divided into two parts: model training and waveform classification. Both the model training and waveform classification use datasets that are simulated and generated by a program based on the CGFK rules. Considering that mud pulse transmission in real scenarios will exhibit channel attenuation and mixed noise, in order to make the simulation more realistic and improve the reliability of the test, a system model for the mud channel response is established. This model outputs simulated waveform data containing channel attenuation and noise, making it more representative of actual data.

Since the actual transmission channel is a mud pipeline about 3000 m long, the channel response experimental data is obtained by sending test signals with a bandwidth of 5–30 Hz through a 3000-meter water cycle. The channel response is shown in Fig. 8.

Fig. 8
figure 8

Channel Response Graph.

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From the figure above, it can be observed that the conditions in the mud channel are quite complex, with a very large signal attenuation. The signals in the range of 12 Hz to 18 Hz exhibit a distinct notch phenomenon, where the signals in this frequency range are significantly weakened. This phenomenon is mainly caused by the multi-path propagation of the pulse signal reflected by the bottom hole and the original signal, and the mutual interference of phase and time delay. Since the mud channel responds differently to various frequencies, it exhibits frequency selectivity. For frequencies with good frequency-selective characteristics, the waveform is complete and clear, facilitating demodulation. However, for frequencies with poor frequency-selective characteristics, the waveform is significantly distorted, which increases the difficulty of demodulation.

Considering the system response, noise, drift, and attenuation factors, a system model is established based on the physical properties of the mud pulse transmission path and the specific operational parameters of the underground equipment. The model is as follows:

$$\:\text{Y}\left(\text{t}\right)=\text{a}\left(\text{t}\right)\ast\:\text{R}\text{e}\left\{\text{H}\left(\text{f}\right)\ast\:\text{X}\left(\text{f}\right)\ast\:{\text{e}}^{\text{j}2{\uppi\:}\left(\text{f}+\varDelta\:\text{f}\left(\text{t}\right)\ast\:\text{t}+\varDelta\:{\upphi\:}\left(\text{t}\right)\right)}\right\}+\text{b}\left(\text{t}\right)+\text{n}\left(\text{t}\right)$$
(16)

where \(\:\text{H}\left(\text{f}\right)\) is the system response, representing the system’s ability to transmit different frequency components of the signal; \(\:\text{X}\left(\text{f}\right)\) is the frequency-domain representation of the signal’s waveform; \(\:\text{f}\) is the signal frequency; \(\:\varDelta\:\text{f}\left(\text{t}\right)\) is the signal frequency drift; \(\:\varDelta\:{\upphi\:}\left(\text{t}\right)\) is the signal phase drift; \(\:\text{R}\text{e}\left\{\right\}\) represents taking the real part of the overall signal; \(\:\text{b}\left(\text{t}\right)\) is baseline drift, which refers to the non-periodic variation of the measurement signal baseline over time, usually caused by long-term environmental changes or equipment aging; \(\:\text{n}\left(\text{t}\right)\) is random noise, generally Gaussian white noise; \(\:\text{a}\left(\text{t}\right)\) is channel attenuation, representing the signal strength attenuation with time or distance.

When mud propagates through the drill pipe, it continuously attenuates due to various factors such as mud density, internal friction in the pipeline, and drilling pump pressure. Its mathematical model can be represented by the following formula:

$$\:\text{P}\left(\text{x}\right)={\text{P}}_{0}\ast\:{\text{e}}^{-\frac{2\text{x}}{\text{c}\text{D}}\sqrt{\frac{{\uppi\:}\text{f}{\upmu\:}}{{\uprho\:}}}}$$
(17)

where \(\:\text{P}\left(\text{x}\right)\) is the signal strength at a distance of \(\:\text{x}\) meters from the pulse signal source, \(\:{\text{P}}_{0}\) is the signal strength at the signal source, \(\:\text{c}\) is the propagation speed, \(\:\text{D}\) is the inner diameter of the drill pipe wall, \(\:\text{f}\) is the signal frequency, \(\:{\upmu\:}\) is the mud viscosity, and \(\:{\uprho\:}\) is the mud density. To make the simulated mud pulse waveform more realistic, attenuation is applied to the waveform in the frequency range with poor frequency selection characteristics.

Continuous wave mud pulses are susceptible to noise interference during actual transmission, primarily from mud pump noise, motor noise, and phase noise. By capturing signals from a pipeline at a depth of nearly 3000 m at an actual production site, a segment of silent signal was extracted that does not contain modulated signals but only includes motor noise, mud pump noise, and various ground noises during the drilling process. The amplitude distribution is illustrated in Fig. 9.

Fig. 9
figure 9

Statistical Histogram of Noise Amplitude Distribution.

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The primary distribution of noise approximates a normal distribution, with background noise predominantly composed of Gaussian noise, which is uniformly distributed across various frequency domains. A portion of the noise originates from the mud pump. The mud pump serves as the power source for the entire transmission system, exhibiting quasi-periodic characteristics. The noise generated during its operation has a frequency approximately equal to the operational frequency of the mud pump, with its energy uniformly distributed within the frequency band of continuous pressure wave signals. Since the mud pump operates on the surface and is in close proximity to the pressure sensors on the ground, the noise it produces does not undergo channel attenuation, significantly impacting the ground-based acquisition and demodulation-decoding processes. The noise generated by the mud pump primarily consists of fundamental wave components equal to the operational frequency of the mud pump and their higher harmonic signal components, mainly harmonic noise. It is generally considered that the fourth harmonic and higher harmonics are negligible.

Motor noise primarily arises from the use of mud flow power to drive the mud motor for auxiliary drilling. When encountering harder rock formations, the rotation of the mud motor is impeded, causing fluctuations in mud pressure that superimpose onto the mud pulse signal. This noise introduces strong interference signal components between the fundamental frequency and the second harmonic of the mud pump noise, particularly the 4 Hz single-frequency signal component within the motor noise, which has a significant impact when the carrier signal frequency is low.

The operational trajectory of the motor in the mud pulse generator is influenced by the precision of motor control and the complex physical environment underground, resulting in the generation of phase noise during operation, which is particularly noticeable above 100 Hz. This noise exhibits signal energy at both the fundamental frequency of the motor’s operation and its harmonics, directly affecting the demodulation efficacy.

Based on surface experimental results from actual drilling operations, a silent signal devoid of modulation signals but containing only motor noise from the drilling process, mud pump noise, and various surface noises, reveals that the background noise predominantly approximates a normal distribution, primarily composed of Gaussian noise.

To better align with the field environment, noise is added to the simulated waveform. Considering the aforementioned analysis of mud channel transmission, the symbol waveform is filtered and then subjected to noise and baseline drift. Gaussian white noise is employed to simulate environmental noise on-site, and a Butter-worth low-pass filter is utilized to mimic system attenuation.

The program generates a simulated waveform data set consisting of 400,000 processed code element waveform, with an average SNR of 2.54 dB. The specific distribution of the SNR is shown in Fig. 10.

Fig. 10
figure 10

Schematic Diagram of SNR Distribution of Data Set.

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The data set generation adheres to the principle of equilibrium in training samples, with each code element waveform comprising 100,000 instances. The data set, encompassing 400,000 generated waveform samples, is partitioned into training and testing samples at an 8:2 ratio, subsequently loaded into a deep learning program for training and testing, culminating in a model that meets the criteria for accuracy and loss rates.

Following analogous protocols, albeit with slight variations in noise magnitude and attenuation rates, a continuous pressure wave containing 50,000 code elements is generated, with the code elements arranged in a random (shuffled) sequence. This continuous pressure wave is subjected to discriminate and demodulate. The simulation program generates results for four distinct methodologies: SVM, LSTM, CNN, and Matched Filter.

Waveform recognition is conducted for each of the four methods, with 100 recognition sets established to statistically determine accuracy rates. This process yields the accuracy rate line graph for the four methods, as depicted in Fig. 11, and the confusion matrices for the four models, as illustrated in Fig. 12.

Fig. 11
figure 11

Line Graph Depicting The Accuracy Rates of The CGFK Model Across Four Distinct Classification Methodologies.

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Fig. 12
figure 12

CGFK Corresponds to The Confusion matrix of Four Distinct Classification Methodologies.

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As shown in Fig. 11, the accuracy line charts of four methods are presented, where the blue line represents SVM, the green line indicates LSTM, the orange line denotes CNN, and the red line corresponds to Matched Filter. It can be observed that the recognition performance of the Matched Filter is significantly inferior to the other three methods, while SVM also demonstrates poorer recognition effectiveness compared to CNN and LSTM. Figure 12 displays the confusion matrices of the four models, where CNN and LSTM exhibit notably superior performance. Through statistical analysis of 100 datasets, the recognition rates of four discrimination methods for CGFK waveform were calculated, with the results summarized in Table 1.

Table 1 Comparative analysis of four methodologies for assessing the accuracy of CGFK modulation waveform Table.
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From the analysis of the mean accuracy values presented in Table 1, it is evident that both CNN and LSTM significantly outperform the Matched Filter in the recognition of CGFK waveform, and they also exhibit a slight advantage over SVM. To statistically and intuitively demonstrate whether CNN and LSTM possess significant distinctions compared to the other two methods, a pairwise T-test was conducted among the four discriminant methods, with the results illustrated in Fig. 13:

Fig. 13
figure 13

The T-value Tests Corresponding to The Four Discriminant Methods for CGFK.

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The T-test is employed to assess the significance of differences between the means of two datasets. The T-value quantifies the standardized difference between the sample data and the hypothesized mean; a larger T-value indicates a more significant difference, with a T-value exceeding 5 generally denoting a significant disparity. The P-value represents the probability of observing the current data under the assumption that the null hypothesis is true; a higher P-value suggests a lack of significant evidence, and a P-value greater than 0.05 typically indicates that the difference is not significant. According to Fig. 13, the T-values for CNN and LSTM are less than 5, and the P-values exceed 0.05, which, in conjunction with the accuracy statistics presented in Table 1, indicates that there is no significant difference between the two, and their recognition performance is comparable. In contrast, the T-values for other methods exceed 5, and the P-values approach 0, signifying significant differences. The analysis reveals that CNN and LSTM exhibit superior and similar recognition performance compared to SVM and Matched Filter, with CNN demonstrating slightly better recognition stability. Consequently, CNN is ultimately selected to assist in the demodulation of CGFK waveform.

The supplementary illustration of the CNN model’s network architecture is provided as follows:

Fig. 14
figure 14

Architectural Diagram of CNN.

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Figure 14 illustrates the training process of a single sample, where each sample consists of a waveform data segment comprising 128 sampling points. During actual training, the input batch size is set to 4096, meaning that 4096 samples are processed simultaneously in each training iteration. The single sample data initially undergoes two convolution layers. The first layer, Conv1D_1, employs a convolution kernel size of 5 and 128 filters. Following the convolution operation, the data undergoes normalization and is subjected to a non-linear transformation via the ReLU activation function. The second layer, Conv1D_2, also utilizes a convolution kernel size of 5 but with 64 filters, and similarly undergoes normalization before being processed by the ReLU activation function. The purpose of these two convolution layers is to progressively extract relevant features from the waveform data (time series), capturing local patterns and structures within the signal. After each convolution layer, a pooling operation is performed with a pooling window size of 2, aimed at reducing data dimension, minimizing computational load, and retaining the most critical feature information. To prevent model over-fit, a dropout layer, Dropout_1, with a dropout rate of 0.2 is introduced after the second pooling layer, Max_pooling1D_2, thereby enhancing the model’s generalization capability. Another dropout layer, Dropout_2, with a dropout rate of 0.5, is placed after the fully connected layer, Dense_1, to further mitigate over-fit. The fully connected layer, Dense_1, maps 1856 dimensions to 64 dimensions with the ReLU activation function, aiming to combine and transform the features extracted by the convolution layers for improved classification. The final output layer, Dense_2, employs the Softmax activation function to classify the waveform data into four distinct symbols: T1, T2, T3, and T4. The optimizer used is Adam, with a fixed learning rate of 0.001. Through this architecture, the model effectively extracts features from the waveform data and outputs the classification results.

Physical test of motor control and waveform modulation

To validate whether CGFK exhibits theoretical performance gaps compared to FSK and PSK under practical conditions, servo motor operation under FSK, PSK, and CGFK modulation protocols was implemented with mechanical loading. The loaded condition was established by attaching a metallic cylindrical guide rail (measuring 1.5 m in length and weighing 2 kg) to a YASKAWA Σ−7 series servo motor, which drives the rail to rotate about its central axis, thereby applying mechanical load. Rotational speed variations under loaded conditions were monitored, with the FSK motor speed profile illustrated in Fig. 15.

Fig. 15
figure 15

Actual Motor Speed Curve Diagram of FSK.

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Due to the external analog speed mode control of the servo motor, the motor speed control exhibits slight dithering influenced by the stability of the external analog voltage. By controlling the servo motor according to the FSK rules and observing the speed curve as shown in Fig. 15, it can be observed that the motor speed curve corresponding to symbol switching exhibits distinct peaks that differ from the dithering caused by voltage stability within the symbol period. These peaks are attributed to the inherent inertia of the servo motor under load, which further leads to waveform distortion in the mud pulse generator, resulting in ISI.

In the process of implementing FSK and PSK through the motor, both essentially involve the switching of the motor speed between two different velocities. Consequently, PSK, similar to FSK, is affected by the motor’s inherent inertia, producing distinct peaks during symbol switching that differ from the dithering caused by external analog instability, thereby generating ISI.

By controlling the servo motor under load according to the CGFK rules and reading the motor speed, the corresponding motor speed curve for CGFK, as illustrated in Fig. 16, is obtained.

Fig. 16
figure 16

Actual Motor Speed Curve Diagram of CGSK.

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As observed in the figure above, although affected by external analog voltage instability, the motor speed control still exhibited oscillations. However, at both the start and end boundaries of each symbol period, the motor speed remained at zero. No significant waveform distortion or inter-symbol interference (ISI) was detected during symbol transitions.

Theoretically, CGFK prevents abrupt speed variations during symbol switching, thereby reducing control precision requirements for the motor.In order to observe whether the actual situation is consistent with the theoretical analysis, the motor is controlled to rotate for 4000 s according to the rules of FSK, PSK and CGFK, that is, each contains 10,000 symbol information.Waveform conformity was quantified through comparative analysis with ideal rotational speed profiles using normalized mean square error (NMSE), where lower NMSE values indicate closer approximation to theoretical models. The measured NMSE values were: FSK 0.07638, PSK 0.07165, and CGFK 0.05725. These results demonstrate that CGFK achieves superior waveform conformity, confirming its unique motor control methodology effectively mitigates ISI caused by inherent motor inertia during symbol transitions.Experimental validation conclusively demonstrates that CGFK outperforms conventional FSK and PSK modulation in suppressing inertia-induced ISI, thereby providing empirical validation of the theoretical postulates regarding the interference suppression capabilities of CGFK.

To verify the practical effectiveness of deep learning-assisted demodulation methods, it is necessary to construct a physical testing environment. Since both FSK and PSK essentially manifest as switching between different motor speed states, this study specifically examines the practical demodulation performance of FSK and CGFK integrated with CNN. The research team built a ground testing environment for a mud pulse telemetry system, comprising water pipes, pumps, pressure sensors, pulse valves, servo motors, and water tanks. The system configuration is as follows: the water pipe inlet is connected to a pump that draws water from the tank, the middle section is equipped with a pressure sensor, and the outlet is connected to a pulse valve driven by a servo motor, with discharged water returning to the tank to form a closed-loop circulation. When the pulse valve aperture changes according to predefined rules, the hydraulic pressure within the pipe fluctuates, and the pressure sensor captures these variations.

The testing environment uses standard 10-meter-long PVC rigid pipes to simulate mud transmission channels. The pump at the inlet is a SHIMGE Pump Industry QDX3-20-0.55 submersible sewage pump, which continuously draws water from the tank to maintain fluid circulation. The PT210B-M20 pressure transducer, driven by an STM32F407VET6 main control board, collects real-time fluid pressure data. The pulse valve is a standard adjustable-aperture valve whose opening degree varies with servo motor rotation to simulate mud pulse modulation. The servo motor driver and servo motor belong to the YASKAWA Σ−7 series.

For FSK modulation, the servo motor is driven according to the FSK protocol to actuate the pulse valve’s degree of opening and closing, thereby inducing fluid pressure variations for pressure data acquisition. Since CGFK employs a 0.4-second symbol period with 2-bit binary encoding, the FSK configuration also adopts a 0.4-second symbol period to facilitate direct comparison with CGFK. To avoid increased modulation order requiring the motor to switch between multiple speed states, two distinct motor speeds were selected to represent binary symbols 0 and 1. By switching motor speeds within a single symbol period, four symbol combinations (00, 01, 10, 11) are encoded. Representative data from a stabilized operational phase are shown in Fig. 17, illustrating the correlation between motor speed transitions and hydraulic pressure fluctuations.

Fig. 17
figure 17

Schematic Diagram of FSK Water Flow Pressure and Motor Speed.

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In the aforementioned diagram, green represents motor speed variations while blue indicates hydraulic pressure fluctuations. After the rules are preset, the motor speed combines and switches between two predetermined rotational velocities within a single symbol period per FSK requirements. Corresponding hydraulic pressure waveform exhibit anticipated patterns as theoretically predicted.

A total of 2,500 actual waveform were collected for each of the four symbol types, forming a data-set containing 10,000 symbol instances. To address the limited data-set size, synthetic data generation was implemented, expanding the data-set to 100,000 symbol instances with 25,000 instances per symbol type. The synthetic data were generated using the channel response characteristics and noise analysis methodology described in the waveform recognition simulation section. Base symbol waveform were filtered through a Butter-worth low-pass filter to simulate system attenuation, followed by the addition of Gaussian white noise (emulating environmental interference) and baseline drift to enhance fidelity with ground-testing empirical data. The augmented data-set was partitioned into training and test sets at an 8:2 ratio, and a CNN model was subsequently trained. Post-training validation was conducted using pressure sensor-acquired pulse waveform data, with the decoded symbol information schematically presented in Fig. 18.

Fig. 18
figure 18

Schematic Diagram of FSK Water Flow Pressure Wave and Code Element Information.

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It is observable that the combination of two rotational speeds within a single symbol period facilitates the expression of four distinct symbol information types. Under this operational mode, the CNN maintains robust recognition capabilities for continuous wave mud pulse signals generated during actual testing, successfully identifying waveform and outputting corresponding symbol information. For the purpose of BER analysis, a predefined set of 20,000 symbols for transmission was utilized. The mud pulse generator’s motor rotor blades were controlled according to FSK principles to produce mud pulses. These mud pulse waveform were then input into a trained model for discrimination, demodulating the symbol information. A comparison between the demodulated symbol information and the intended transmission data revealed an information recognition accuracy of 0.9865, with a BER of 0.0135.

Under the CGFK scheme, the CNN model was trained using the same methodology. Following CGFK protocols, the servo motor was driven to control the opening and closing of the pulse valve, thereby enabling the pressure sensor to detect continuous pressure variations and capture corresponding waveform. Data from a sustained operational period was extracted, with the relationship between motor speed and hydraulic pressure illustrated in Fig. 19.

Fig. 19
figure 19

Schematic Diagram of CGFK Water Flow Pressure and Motor Speed.

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In the above figure, green represents the motor speed variation, and blue represents the water flow pressure variation. After the modulation rules are preset, the motor speed increases uniformly from 0 to the maximum value and then decreases uniformly back to 0 within one symbol period, as required by CGFK. It can be observed that the water flow pressure also exhibits the expected waveform.

The water flow pressure waveform is then imported into the trained CNN for judgment. The symbol information determined by the CNN is shown in Fig. 20.

Fig. 20
figure 20

Schematic Diagram of CGFK Water Flow Pressure Wave and Code Element Information.

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It is observable that CNN demonstrates commendable recognition capabilities for continuous wave mud pulses generated during actual testing, successfully identifying waveform and outputting corresponding symbol information, with an information transmission rate of 5bit/s. An analysis of the BER was conducted by initialize 20,000 symbols of information to be transmitted. The mud pulse generator motor rotor blades were controlled according to the CGFK rules to produce mud pulses, which were then imported into a trained model for discrimination, demodulating the symbol information. Comparing the demodulated symbol information with the information to be transmitted, the information recognition accuracy reached 0.9996, with a BER of 0.0004. It can be discerned that CGFK, compared to FSK, also exhibits advantages in terms of demodulation BER.

Similarly, to verify whether the CGFK combined with CNN method demonstrates a lower BER in actual testing environments compared to the CGFK combined with Matched Filter method, the aforementioned method was employed, increasing the initialize information to be transmitted to 20,000 symbols, and using the Matched Filter method for discrimination. Experimental testing revealed an information recognition accuracy of 0.9954, with a BER of 0.0046. Comparing with the CGFK combined with CNN method, it is evident that the CNN model indeed surpasses the traditional Matched Filter method in assisting CGFK with waveform discrimination.

It is noteworthy that the environment setup for the aforementioned physical testing differs from the actual underground operating environment, where the length of the mud pipeline is approximately 3000 m, and even longer in ultra-deep wells. Mud pulses are subject to channel attenuation and background noise, mud pump noise, and motor noise from drill bit operation obstructions, with the degree of attenuation related to pipeline length, mud properties, and signal frequency. The pipeline length in the physical testing section was only 10 m, and clear water was used instead of mud for the experiment, resulting in channel attenuation and noise impact levels that are not comparable to the actual underground operating environment. In the actual underground operating environment, the BER would further increase.

Certainly, actual underground operations would employ signal enhancement, noise suppression, and error correction methods to reduce the BER. The purpose of this physical testing is merely to illustrate that CGFK, compared to traditional modulation methods such as FSK and PSK, can effectively reduce ISI and lower the BER.

Discussion

.In this research, we propose a continuous wave mud pulse data transmission method that integrates CGFK with CNN demodulation.

Compared to traditional modulation methods such as FSK and PSK, CGFK modulates the waveform by controlling the frequency variation within each symbol period: the frequency uniformly increases from 0 to a set frequency and then uniformly decreases back to 0, with the rate of frequency change serving as the modulation and demodulation characteristic. Consequently, the waveform frequency at the beginning and end of each symbol period is 0, corresponding to a motor speed of 0. This design effectively mitigates the ISI problem caused by the motor’s inherent inertia during symbol transitions.

In comparison with the emerging technology OFDM, the BER of CGFK and OFDM are relatively similar. In terms of symbol transmission rate, OFDM indeed exhibits superior performance; however, the implementation of OFDM typically necessitates more complex signal processing, making it challenging to control motor rotation for modulation in mud pulse transmission. In contrast, the motor control method of CGFK is simpler to implement, and even in motor control, whether for emerging or traditional modulation technologies, CGFK imposes lower requirements on the motor. In scenarios where the transmission rate requirements are not excessively high, CGFK would be a more preferable choice.

To evaluate the efficacy of various deep learning algorithms in comparison to the traditional Matched Filter for the recognition of CGFK waveform, we employed four distinct methodologies: SVM, LSTM, CNN, and Matched Filter. Each method was utilized to determine CGFK waveform across 100 test sets, with accuracy rates meticulously recorded. In each test set, the four methods were required to discern 50,000 symbol units. The outcomes of these tests are illustrated in Figs. 11 and 12, and detailed in Table 1. A T-test was conducted on the statistical results from Table 1 to ascertain statistical significance, with the T-values and P-values depicted in Fig. 13. Upon analysis, it was observed that among the four methods, CNN and LSTM exhibited comparable and superior recognition capabilities, whereas the traditional Matched Filter method demonstrated the least effectiveness. Consequently, both CNN and LSTM are deemed suitable for assisting in the demodulation of CGFK. Further analysis of the accuracy rates in Table 1 suggests that the CNN model’s recognition accuracy is more concentrated, potentially indicating greater stability in recognition. Therefore, CNN was ultimately selected to aid in the demodulation process.

To further substantiate the superiority of CGFK in mitigating ISI caused by motor inertia during symbol transitions, a rudimentary mud pulse transmission system was constructed. This system primarily comprises a pipeline, a water pump, a pressure sensor, and a pulse valve. The water pump at the pipeline’s forepart furnishes the fluid environment, with a pressure sensor connected midstream and a pulse valve at the terminus. By regulating the pulse valve’s operation in accordance with modulation protocols, the water flow velocity is altered, enabling the pressure sensor to capture continuous pressure wave data, thereby facilitating the transmission and acquisition of mud pulse data, and subsequently, its recognition based on a CNN model.

Under the assistance of the CNN model for demodulation, the disparities in data before and after transmission using CGFK versus FSK were compared, with assessments made on accuracy and BER. Empirical testing revealed that CGFK effectively circumvents or diminishes ISI attributable to motor inertia during symbol transitions. In terms of demodulation error rates, CGFK demonstrated a marked advantage over FSK.

To validate the simulation test conclusion that CNN outperforms the traditional Matched Filter in aiding CGFK demodulation, additional physical tests employing the Matched Filter for CGFK demodulation were conducted. The findings corroborated that CNN indeed surpasses the Matched Filter in assisting CGFK demodulation.

Conclusion

In response to the issue of ISI caused by the motor’s own inertia and the practical suitability of various deep learning models, this paper proposes a continuous wave mud pulse data transmission method that combines CGFK modulation with deep learning for demodulation. As a modulation method based on the frequency change rate as the demodulation characteristic, CGFK significantly reduces the motor control difficulty and accuracy requirements due to the fact that at the beginning and end of each symbol period, the waveform frequency is zero, meaning the motor speed is zero. Regarding waveform distortion, Symbol period of CGFK starts and ends with a frequency of zero, which effectively avoids ISI. Since CGFK uses frequency change rate as a demodulation characteristic, it is insensitive to phase changes, thus greatly reducing the impact of phase drift. In terms of modulation order, compared with FSK and PSK, CGFK is simpler to implement with increased modulation order, as it only requires distinguishing between different frequency change rates. This can be achieved by simply adjusting the motor’s acceleration and deceleration, making control straightforward and preventing confusion between different symbols. In terms of information transmission rate, CGFK distinguishes different symbols by frequency change rate, which allows for both a high symbol transmission rate and a relatively large amount of information per symbol, thereby ensuring a good transmission rate.In summary, CGFK modulation, while maintaining a high information transmission rate, has distinct advantages in motor control, modulation-demodulation difficulty, and ISI prevention. By using CGFK modulation, the abrupt changes in motor speed during symbol transitions are fundamentally avoided, which in turn eliminates the ISI caused by the motor’s inertia.

For the deep learning-assisted demodulation aspect, CNN were chosen. CNN were initially designed for image processing, but the characteristics of convolution operations make them highly suitable for handling any data with local patterns or structures, such as sound, telecommunications signals, or other time-series waveform. Thus, CNN have advantages in waveform classification and recognition, such as local feature extraction, translation invariability, and parameter sharing. In waveform recognition, a 1D CNN is typically used. This network structure is similar to the 2D CNN used in image processing but performs convolution operations in a single dimension (time dimension).

In the simulation tests, the CNN demonstrated significantly superior performance in recognizing CGFK waveform compared to SVM and Matched Filter. Additionally, LSTM also exhibited excellent performance in the simulation tests. When the recognition effects of both were comparable, the accuracy rate of the CNN model was more concentrated, suggesting a slight advantage in recognition stability, leading to the adoption of the CNN network.

In practical tests, with the assistance of the CNN model for demodulation, CGFK exhibited a lower BER compared to FSK. The experimental results indicate that the continuous wave mud pulse data transmission method, which combines CGFK with CNN for demodulation, achieves a higher information transmission rate and a lower BER. This method effectively avoids ISI caused by the motor’s inherent inertia during symbol switching, presenting unique advantages in terms of modulation and demodulation difficulty. Furthermore, due to its distinctive method of distinguishing symbols, it offers simpler motor control and less confusion among symbols when increasing the modulation order, compared to other modulation methods.

The proposed integration of CGFK with CNN demonstrates considerable promise for industrial implementation. From a deployment cost perspective, the hardware infrastructure primarily comprises underground mud pulse generators, surface receivers, and data processing units. Given the extensive adoption of existing mud pulse telemetry systems within drilling operations, hardware expenditures would predominantly involve system upgrades and optimization rather than complete replacement. On the software front, the CNN architecture - while being a well-established technology - proves functionally adequate for auxiliary CGFK demodulation requirements, featuring a streamlined architecture yet sufficiently robust to support demodulation tasks. Although initial investments in model development and training are necessary, subsequent deployment and iterative refinement incur comparatively minimal costs. Notably, this hybrid approach exhibits superior data transmission accuracy and operational efficiency compared to conventional CGFK implementations utilizing matched filter techniques. Such enhancements translate to measurable reductions in temporal and material resource wastage during drilling operations, ultimately yielding long-term operational cost efficiencies.

In terms of compatibility within commercial drilling scenarios, the integration of CGFK with CNN methodologies can seamlessly amalgamate with existing mud pulse data transmission systems. Through software enhancements, current surface receivers and data processing units can facilitate the deployment of CNN models without necessitating extensive hardware overhauls. This compatibility enables the swift assimilation of this technology into existing drilling operations, thereby mitigating the complexity and cost associated with technological upgrades.

In the application of deep well exploration, the challenges of signal attenuation and noise interference are particularly pronounced, while the high-temperature and high-pressure environments impose heightened demands on the durability of hardware equipment. Regarding signal attenuation and noise mitigation, CNN models demonstrate superior capability compared to conventional matched filter techniques in effectively extracting discriminate signal features and enhancing anti-interference robustness. Concerning hardware implementation, the distinctive motor control methodology employed in CGFK systems simplifies motor operation protocols and reduces performance requirements for electrical machine, thereby substantiating the technical viability of integrating CNN approaches with CGFK configurations for deep borehole exploration scenarios.

However, the widespread adoption of CGFK integrated with CNN technology still encounters several challenges. Primarily, the training of CNN models demands substantial quantities of high-quality data. This issue could be addressed through collaborative partnerships with drilling enterprises to utilize their historical operational data for model optimization. Secondly, significant discrepancies in data distribution across diverse drilling environments and geological conditions may impair model generalization capabilities. To mitigate this, transfer learning techniques could be implemented to fine-tune trained models for specific scenarios, complemented by data augmentation strategies to enhance data-set diversity and reinforce model robustness. Furthermore, computational constraints at field sites might be overcome through deployment of lightweight CNN architectures or offloading intensive processing tasks to cloud-based systems. Lastly, the industry’s inherent conservatism toward technological innovation may create adoption barriers. This could be countered through demonstration pilot programs showcasing the methodology’s efficacy, coupled with comprehensive training and technical support to facilitate operational understanding and implementation.

However, the widespread implementation of integrated CGFK and CNN technology still faces several challenges. First, training CNN models demands large volumes of high-quality data. This issue can be addressed through collaboration with drilling companies to utilize their historical operational data for model training. Second, significant data distribution discrepancies across different drilling environments and operational conditions may compromise model generalization capabilities. For CNN models, variations in physical environments, equipment differences, and operational parameter modifications can induce characteristic shifts and domain drift in mud pulse waveform, leading to increased misjudgment rates in new scenarios.Transfer learning methodologies enable fine-tuning of preliminary train source domain models through strategies such as freezing lower-layer feature extraction modules while adjusting top-layer classifiers to adapt to novel environments, potentially enhancing recognition accuracy. Data augmentation techniques can artificially expand training datasets by simulating signal variations across diverse drilling environments, thereby improving CNN model robustness and adaptability to complex operational scenarios. When onsite computational resources are limited, lightweight CNN architectures or cloud-based distributed computing frameworks could be implemented for partial task offloading.Finally, the drilling industry’s conservative adoption tendencies toward emerging technologies may create implementation barriers. Targeted solutions include demonstrating the advantages of the integrated CGFK and CNN methodology through pilot projects and success case studies, coupled with comprehensive training programs and technical support systems to facilitate industrial stakeholders’ comprehension and practical application of this innovation.

The synergistic combination of CGFK and CNN demonstrates remarkable potential for industrial applications, significantly enhancing data transmission accuracy and operational efficiency while reducing drilling costs and associated risks. Although technical implementation presents certain obstacles, strategic deployment of targeted solutions positions this methodology for broad industry adoption, ultimately driving technological advancement across the drilling sector through improved decision-making capabilities and optimized resource allocation.