Equivalent Self-Noise Suppression of DAS System Integrated with Multi-Core Fiber Based on Phase Matching Scheme

Abstract

Multi-core fiber (MCF) has drawn increasing attention for its potential application in distributed acoustic sensing (DAS) due to the compact optical structure of integrating several fiber cores in the same cladding, which indicates an intrinsic space-division-multiplexed (SDM) capability in a single piece of fiber. In this paper, a dual-channel DAS integrated with MCF is presented, of which the equivalent self-noise characteristic is analyzed. The equivalent self-noise of the system can be effectively suppressed by signal superposition with the phase matching method. Considering that the noise correlation among the cores is not zero, the signal-to-noise (SNR) gain after signal superposition is less than the theoretical value. The dual-channel DAS system is set up by a piece of 2 km long seven-core MCF, in which the dual-sensing channels are constructed by a four-core series and three-core series, respectively. The total noise correlation coefficient of the seven cores is 11.28, while the equivalent self-noise of the system can be suppressed by 6.32 dB with signal superposition. An equivalent self-noise suppression method based on a linear delay phase matching scheme is proposed for noise decorrelation in the DAS MCF system. After noise decorrelation, the suppression of the equivalent self-noise of the system can reach the theoretical value of 8.45 dB with a time delay of 1 ms, indicating a noise correlation among the seven cores of almost zero. The feasibility of the equivalent self-noise suppression method for the DAS system is verified for both single-frequency and broadband signals, which is of great significance for the detection of weak vibration signals based on a DAS system.

1. Introduction

In the DAS [1] system based on phase-sensitive optical time domain reflectometry (Φ-OTDR) technology, the pulses of light are transmitted along the fiber core. The phase of Rayleigh backscattering (RBS) light will be modulated due to the change in the refractive index of the fiber core and the optical path of the light with an external disturbance. Through coherent detection [2] technology, the information of the disturbance can be reversely calculated. Every single position of the light transmission fiber can independently sense the external information, and then the whole fiber acts as a distributed sensing system. In recent years, DAS has shown great potential in many applications such as oil and gas pipeline monitoring [3,4,5,6], seismic wave detection [7,8,9,10,11], aerospace [12,13], etc.

Due to the limitation of poor consistency and interference fading in the traditional DAS system based on single-mode fiber (SMF) [14], the introduction of special fiber significantly improves the performance of the DAS system. For example, the intensity of RBS light can be enhanced with discrete scattering optical fibers [15,16]. The sensitivity of the system can be enhanced with secondary-coated optical fibers [17] and spiral-structured optical fibers [18,19,20] through different sensitization mechanisms. As a new type of special fiber, MCF greatly improves the information capability of the DAS system with the parallel spatial cores in the same piece of fiber. In addition, combined with SDM technology, lots of key problems have been effectively improved by using MCF, such as interference fading [21,22] and the response frequency band restriction [23] of the DAS system.

However, the performance of the DAS system in the detection of signals is greatly affected by the self-noise level, especially weak signals. The level of the self-noise floor directly determines the detectable accuracy of the signal, the detectable distance, and the reliability of the system. Therefore, it is necessary to reduce the equivalent self-noise of the DAS system and improve the SNR. Among the previous studies, the SNR of the DAS system was improved by promoting the intensity of RBS signals [24,25], denoising the phase signal [26,27], and suppressing both amplitude noise and phase noise by using triple denoising methods [28].

In this paper, a piece of seven-core MCF is utilized to construct the DAS system to improve the performance of the limited equivalent self-noise level of the single-channel DAS system, where each core acts as an independent channel. The influence of noise correlation among different cores on SNR gain is analyzed for the signal superposition of the DAS system, which is verified in the experiment. On this basis, an equivalent self-noise suppression method based on the linear delay phase matching scheme is proposed and theoretically deduced to reduce the noise correlation. The experimental results indicate that the noise decorrelation can be achieved after signal processing. Experiments on the DAS system integrated with 2 km long MCF show that the equivalent self-noise of the system can be suppressed by 6.324 dB with the signal superposition, while the total noise correlation coefficient of the seven cores is 11.28. After noise decorrelation, the suppression of equivalent self-noise of the system can reach the theoretical value of 8.45 dB with a time delay of 1 ms, which means there is almost no noise correlation among different cores. The feasibility of this method is verified for both single-frequency and broadband signals by experiments, and this method has potential application in the detection of weak signals for a DAS system integrated not only with MCF but also multi-fiber cable.

2. Methods

In the heterodyne coherent detection system, the acquired coherent detection signal is demodulated [29] and the RBS signal in the sensing fiber at the position of $z_1$ and $z_2$ can be expressed as [30].

$$\begin{array}{l}{\widehat{r}}_{z_1}(t,z_1)=A_{z_1}(t,z_1)\exp [j\varphi (t,z_1)]\\ {\widehat{r}}_{z_2}(t,z_2)=A_{z_2}(t,z_2)\exp [j\varphi (t,z_2)]\end{array}$$

(1)

where $t$ is time, $A_{z_1}=2E_L{E}_O{r}_{Ray}{e}^{-\alpha z_1}$, and $A_{z_2}=2E_L{E}_O{r}_{Ray}{e}^{-\alpha z_2}$ are the amplitudes of scattering signals; $E_L$ is the amplitude of the local reference light; $E_O$ is the amplitude of RBS; $r_{Ray}$ is composite Rayleigh scattering coefficient; and $\alpha$ is the attenuation coefficient of the fiber. The phase changes caused by the external disturbance can be obtained by $\Delta \varphi =\varphi (z_2)-\varphi (z_1)$. The SNR of the disturbed signal can be expressed as

$$SNR=10\lg ({\displaystyle \frac{{(\Delta \varphi )}^2}{P_{noise}}})$$

(2)

where $P_{noise}$ is the noise power. The larger the phase change, the higher the SNR.

The SNR can be improved by signal superposition of distributed fiber arrays. For a linear additive array, the gain of the SNR after signal superposition can be achieved by calculating the correlation coefficients of signal and noise between the arrays. Assuming that the signals of $m$ array elements with consistent sensitivities are linearly superimposed, the cross-correlation coefficient between the $i$-th and $j$-th array elements of the signals $s(t)$ and the noises $n(t)$ can be expressed as [31]:

$${({\rho }_s)}_{ij}={\displaystyle \frac{\overline{{\displaystyle \sum s_i(t)s_j(t)}}}{{(\overline{{\displaystyle \sum (s_i(t)}{)}^2{(s_j(t))}^2})}^{1/2}}}$$

(3)

$${({\rho }_n)}_{ij}={\displaystyle \frac{\overline{{\displaystyle \sum n_i(t)n_j(t)}}}{{(\overline{{\displaystyle \sum (n_i(t)}{)}^2{(n_j(t))}^2})}^{1/2}}}$$

(4)

The gain after signal superposition is

$$AG=10\lg {\displaystyle \frac{{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{i=1}^m{({\rho }_s)}_{ij}}}}{{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{i=1}^m{({\rho }_n)}_{ij}}}}}$$

(5)

Therefore, the gain after signal superposition depends on the cross-correlation coefficients of signals and noises among array elements. When the signals are fully correlated and the noises are partially correlated,

$$\left\{\begin{array}{l}{({\rho }_n)}_{ij}={\rho }_{ij},i\ne j\\ {({\rho }_n)}_{ij}=1,i=j\end{array}\right.$$

(6)

The gain can be expressed as

$$AG=10\lg {\displaystyle \frac{m^2}{m+{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{i=1}^m{\rho }_{ij}}}}},i\ne j$$

(7)

3. Results

The dual-channel DAS system based on heterodyne detection is shown in Figure 1. The continuous-wave light from the laser is split into one beam of probe light and two beams of local reference light by a coupler (C1) with a splitting ratio of 98:1:1. Then, an acousto-optic modulator (AOM) is used to chop the probe light into pulse light with a width of 100 ns and a frequency of 9985 Hz. The pulse light is split into two beams by a coupler (C2) with a splitting ratio of 50:50. The two beams of probe pulses are amplified by two erbium-doped fiber amplifiers (EDFA1, EDFA2), shaped by filter (Filter1, Filter2), and then injected into the sensing fiber through two circulators (Circulator1, Circulator2). The two beams of RBS from the sensing fiber mix with the two beams of local reference light in couplers (C3, C4), both splitting ratios of which are 50:50. The alternating current (AC) signals output by the BPD1 and BPD2 are received by the data acquisition (DAQ) card. Finally, the phase difference distribution of RBS can be achieved by heterodyne phase demodulation technology. The parameters of the devices used in the system are summarized in

Table 1. Parameters of the devices.

Device	Parameters	Value
Laser	Central wavelength/nm	1550.12
	Line width/kHz	≤1.5
	Relative intensity noise/(dB/Hz)	≤120
AOM	Frequency shift/MHz	80
	Insertion loss/dB	<3
EDFA	Input optical power peak/dBm	1
	Output optical power peak/mW	1000
	Noise coefficient/dB	4.5
Filter	Operating wavelength/nm	900–1700
	Spectral resolution/nm	2–13
	Average light transmittance/%	>90
BPD	Bandwidth/MHz	100
	Responsivity/(A/W) @ 1550 nm	≥0.95
	Transimpedance gain/(KV/A)	≥30

.

The series-connection scheme of the MCF is shown in Figure 2. The pigtails of 2 km long MCF are, respectively, connected to single-mode fibers through fan-in/-out couplers. Core 7 is located at the center, and the remaining six cores are evenly distributed around it. The same pulsed light is injected into the dual channels at the same time, one of which is composed of Cores 1, 2, 3, and 4 connected in series end-to-end, and the other of which is composed of Cores 5, 6, and 7 connected in series end-to-end. A 2 m long fiber at position D is evenly wound around a piezoelectric transducer (PZT) and applied signals by a signal generator. The signals of the seven cores can be superimposed due to the same change in phase difference. The parameters of the MCF are shown in

Table 2. Parameters of MCF.

Parameters	Value
Single core length/m	2000
PZT position/m	1978–1980
Fiber diameter/μm	245
Core diameter/μm	8
Core spacing/μm	41.5
Mold field diameter/μm	9.5
Attenuation/(dB/km) @ 1550 nm	≤0.30

. The maximum detection distance of DAS in a single channel is 10 km. According to Figure 2, the detection distances in the two channels are 8 km and 6 km, respectively, which are both less than the detection upper limit.

3.1. Subsection SNR Gain Analysis with Signal Superposition of Seven Cores

The partial noise of each core comes from the same optical device, such as the laser and AOM. And seven cores in one fiber have the same response to ambient noise, so the noise correlation between each core exists. Based on Equation (4), the coefficient matrix of noise cross-correlation calculated by the noise data of the seven cores for one second is computed as

$${({\rho }_n)}_{ij}=\left[\begin{array}{ccccccc}1& 0.19& 0.09& 0.06& 0.11& 0.08& 0.05\\ 0.19& 1& 0.19& 0.09& 0.08& 0.11& 0.08\\ 0.09& 0.19& 1& 0.19& 0.05& 0.08& 0.11\\ 0.06& 0.09& 0.19& 1& 0.02& 0.01& 0.08\\ 0.11& 0.08& 0.05& 0.02& 1& 0.19& 0.09\\ 0.08& 0.11& 0.08& 0.01& 0.19& 1& 0.19\\ 0.05& 0.08& 0.11& 0.08& 0.09& 0.19& 1\end{array}\right]$$

(8)

Then ${\displaystyle \sum _{j=1}^{m=7}{\displaystyle \sum _{i=1}^{m=7}{({\rho }_n)}_{ij}}}=m+{\displaystyle \sum {\rho }_{ij}}=11.28$. The noise level experiences an increase of 10.52 dB with the signal superposition of the seven cores, while the SNR gain can attain a value of $AG=6.38\mathrm{dB}$ according to Equation (7). Considering a sinusoidal vibration at 500 Hz with an amplitude of 5 V on a sampling frequency of 9985 Hz, Figure 3 shows the time domain signals and power spectral density (PSD) curves of the demodulated phase signals both before and after the signal superposition process. The PSD value at the 500 Hz frequency point is uniformly −33.68 dB, and the mean noise level is −80.92 dB for the seven individual cores. Consequently, the SNR is 47.24 dB. The PSD at 500 Hz rises to −16.77 dB, and the mean noise level increases to −70.29 dB when summing the seven-core signals. This SNR is 53.52 dB, signifying a 6.28 dB enhancement. Meanwhile, this implies that the equivalent self-noise of the system has been reduced by 6.28 dB.

Table 3. SNR of each frequency before and after seven-core signal superposition (dB).

PZT Sinusoidal Frequency/Hz	PZT Amplitude /V	Before Signal Superposition	After Signal Superposition	Gain	Mean Gain
100	5	36.05	42.41	6.36	6.32
200	5	50.86	57.18	6.32
500	5	47.24	53.52	6.28
800	2	30.98	37.33	6.35
1000	2	33.66	40.02	6.36

shows the SNR both before and after the signal superposition of the seven-core signals at five frequency points. Evidently, the average gain of SNR after the signal superposition reaches 6.324 dB. This indicates that the equivalent self-noise of the system has been reduced by 6.32 dB, which is close to the calculated prediction.

Because of the correlation between each core’s noise, the equivalent self-noise of the system with the signal superposition of seven-core signals is only reduced by 6.32 dB, which is less than the theoretical value (the noise of the seven cores is uncorrelated, which means ${\rho }_{ij}=0$ and $10\lg 7=8.45 \mathrm{dB}$ according to Equation (7)). It is easy for this to be overshadowed by noise when detecting weak signals, resulting in missed detections. The weak signals can emerge from the background noise by effectively lowering the system’s equivalent self-noise. This improvement can enhance the system’s capability to accurately discern the presence of signals, minimize the false-alarm rate, and mitigate the risk of overlooking genuine signals. Furthermore, a high SNR offers a multitude of benefits, such as supporting steady signal and improving measurement accuracy. Given these remarkable advantages, removing the noise correlation of each core further to suppress the equivalent self-noise is an urgent problem.

3.2. SNR Gain Analysis After Noise Decorrelation

Firstly, the time correlation of each core’s noise is analyzed to eliminate the noise correlation and further reduce the equivalent self-noise of the system. In an acoustic field, the degree of time correlation of noise can be quantitatively characterized by the time correlation coefficient,

$${({\rho }_n)}_{ij}(\tau )={\displaystyle \frac{\overline{{\displaystyle {\int }_{-\infty }^{\infty }{n}_i(t)n_j(t+\tau )dt}}}{\sqrt{\overline{{\displaystyle {\int }_{-\infty }^{\infty }{n}_i^2(t)dt}{\displaystyle {\int }_{-\infty }^{\infty }{n}_j^2(t+\tau )dt}}}}}$$

(9)

The time correlation coefficient of noise is a function of the delay time $\tau$. In the analysis of the system’s pure noise, the duration $\tau$ for correlation analysis should not exceed one-fifth of the recording duration $T$. We opt to analyze data with $T=1 \mathrm{s}$ and set the duration $\tau =0.1 \mathrm{s}$ for correlation analysis. Figure 4 shows the variation trends of the noise correlation coefficients ${\rho }_n$ with respect to $\tau$ for several core pairs in the MCF at point D: Core 1 and Core 5 (from the same sensing distance in the dual channels), Core 1 and Core 6 (from different sensing distances in the dual channels), Core 1 and Core 2 (two adjacent cores within the same channel), and Core 1 and Core 3 (two cores separated by one intermediate core within the same channel). When $\tau =0$, the correlation coefficient ${\rho }_n$ attains its peak value. As $\tau$ progressively increases, ${\rho }_n$ undergoes a steep decline, approaching zero rapidly. Subsequently, it fluctuates in the vicinity of zero and gradually reaches a stable state. As clearly shown in Figure 4, when $\tau =0.5 \mathrm{ms}$, the noise correlation coefficients for all the aforementioned core pairs drop to near zero, signifying a substantial reduction in correlation. Figure 4 illustrates that by introducing a certain time delay between the signals of each core before signal superposition, the influence of noise correlation can be effectively reduced. Theoretically, when the time delay approaches infinity, the noise of the seven cores becomes completely uncorrelated, and the SNR gain for signal superposition can reach 8.45 dB.

One approach to eliminate noise correlation is to introduce sequential time delays to the signals of each fiber core before performing signal superposition. However, phase differences inevitably arise among the signals from different cores with the time delay operation, which can cause the amplitude of the aggregated signals to fail to reach its maximum potential, and consequently, the SNR may not experience the desired improvement.

To address this issue, while effectively removing noise correlation and ensuring that the delayed signals can be combined in-phase, a novel method of SDM for phase linear delay matching signals using MCF (SDM-PLDMS-MCF) is presented in this paper. Assume that at time $t_0$, the exponential representations of the demodulated seven-core time domain signals are

$$y_c(t)={\displaystyle \frac{A}{2j}}(e^{j({\omega }_{0}t+{\phi }_0)}-e^{-j({\omega }_{0}t+{\phi }_0)})+{\displaystyle \sum _k{B}_{ck}{e}^{j{\omega }_{k}t}}$$

(10)

where the values of $c$ are 1, 2, 3, 4, 5, 6, and 7, representing fiber cores 1–7; $A$ is the signal amplitude; ${\omega }_0$ is the signal frequency; ${\phi }_0$ is the initial phase of the signal; $B_{ik}$ is the amplitude of the frequency component $k$ in the noise of the $i$-th core; and ${\omega }_k$ is the noise frequency. The signals of seven cores sequentially delayed by $\Delta t$ are obtained as follows

$$y_c(t)={\displaystyle \frac{A}{2j}}(e^{j({\omega }_0(t+(c-1)\Delta t)+{\phi }_0)}-e^{-j({\omega }_0(t+(c-1)\Delta t)+{\phi }_0)})+{\displaystyle \sum _k{B}_{ck}{e}^{j{\omega }_k(t+(c-1)\Delta t)}}$$

(11)

Then, by performing Fourier transforms on these signals in sequence, the single sideband spectra are, respectively, obtained as follows

$$Y_c(\omega )=\pi A\delta (\omega -{\omega }_0)e^{j({\omega }_0(c-1)\Delta t+{\phi }_0)}+{\displaystyle \sum _k\pi B_{ck}\delta (\omega -{\omega }_k)e^{j(c-1){\omega }_k\Delta t}}$$

(12)

Taking the phase of $Y_1$ as the reference, the signals of the remaining cores need to be multiplied by $e^{-j\omega \Delta t}$, $e^{-j2\omega \Delta t}$, $e^{-j3\omega \Delta t}$, $e^{-j4\omega \Delta t}$, $e^{-j5\omega \Delta t}$, and $e^{-j6\omega \Delta t}$ in sequence to perform phase-shifting. When $\omega ={\omega }_0$, phase-shifting is performed on the frequency of the signal. When $\omega ={\omega }_k,k\ne 0$, phase-shifting is carried out on the noise. After phase-shifting, the signals of each core are, respectively, as follows

$${Y_c}^{\prime }(\omega )=\pi A\delta (\omega -{\omega }_0)e^{j{\phi }_0}+{\displaystyle \sum _k\pi B_{ck}\delta (\omega -{\omega }_k)e^{j{\phi }_{ck}}},(c=2,3,4,5,6,7)$$

(13)

In the case of a single-frequency signal with periodic characteristics, the phases of the signals in Cores 2 to 7 are adjusted to align precisely with the phase of the signal in Core 1. Due to noise with inherent randomness, there is no quantifiable correlation between the post-phase-shifting noise phases of each core and the pre-phase-shifting phases described in Equation (10). The emerged phase differences are, respectively, expressed as ${\phi }_{2k}$, ${\phi }_{3k}$, ${\phi }_{4k}$, ${\phi }_{6k}$, and ${\phi }_{7k}$ as a result. Through this process, the objective of summing the signals in-phase while decorrelating the noise is successfully attained. Ultimately, the signals are reverted from the frequency domain back to the time domain, where the effectiveness of the signal superposition can be verified.

An equivalent self-noise suppression method based on a linear delay phase matching scheme using MCF is shown in Figure 5. As shown in Figure 5a, assuming that the pulse interval is $\Delta t$, the pulses numbered from 1 to 8 are applied to Core 1. The pulses sequentially delayed by one pulse are applied to Cores 2 through 7. For Core 2, pulses 2 to 8 are all advanced by a phase difference of $\Delta {\phi }_2=2\pi f\Delta t$ corresponding to one pulse interval upon applying the Fourier transform to transition the signals into the frequency domain. Following this, an inverse Fourier transform is employed to restore the signals to the time domain, yielding pulses 2′, 3′, 4′, 5′, 6′, 7′, and 8′. The phases of these transformed pulses precisely align with those of pulses 1 to 7 in Core 1. In the frequency domain, Cores 3 to 7 are, respectively, advanced by phase shifts of $\Delta {\phi }_3=2\pi f*2\Delta t$, $\Delta {\phi }_4=2\pi f*3\Delta t$, $\Delta {\phi }_5=2\pi f*4\Delta t$, $\Delta {\phi }_6=2\pi f*5\Delta t$, and $\Delta {\phi }_7=2\pi f*6\Delta t$ to align their phases with those of Core 1. As shown in Figure 5b, the phases of the signals across all cores are initially coherent before the introduction of the delay at 500 Hz. However, after a delay of one pulse interval, as shown in Figure 5c, the phase consistency among the signals of each core is disrupted. If these out-of-phase signals are directly summed, the increasing multiple of amplitude is less than 7. Therefore, it is necessary to shift phase in the frequency domain, ensuring the signals of all cores are once again in-phase, as shown in Figure 5d.

To verify this proposed method, the SNR gains with a time delay $\Delta t=0.5 \mathrm{ms}$ for various single-frequency signals are shown in

Table 4. Comparison of SNR gain before and after noise decorrelating (dB).

PZT Sinusoidal Frequency /Hz	PZT Amplitude /V	Before Noise Decorrelating		After Noise Decorrelating
		Gain	Average Gain	Gain	Average Gain
100	5	6.36	6.32	8.26	8.26
200	5	6.32		8.33
500	5	6.28		8.21
800	2	6.35		8.21
1000	2	6.36		8.30

. Before the noise decorrelation, the average SNR gain following the signal superposition of the seven-core signals stands at 6.32 dB. After the noise decorrelation, the SNR gains at each individual frequency point all surpass 8 dB, and the overall average gain reaches 8.26 dB. This improvement implies that the equivalent self-noise pressure of the system is further diminished by 1.94 dB.

We selected a series of values, 0 ms, 0.3 ms, 0.5 ms, 0.7 ms, and 1 ms, to explore the relationship between the SNR gain and the time delay parameter. The resulting curves, which illustrate the variation in the SNR gain at each frequency point as $\Delta t$, are shown in Figure 6. As the value of $\Delta t$ increases, the SNR gain shows a gradual upward trend and reaches the peak value at $\Delta t=1\mathrm{ms}$, approaching the theoretical maximum of 8.45 dB. This outcome indicates that the noise correlation is basically eliminated under this condition.

The research presented above is predicated on the assumption that the disturbance takes the form of a single-frequency signal. Our findings indicate that, in the case of a single-frequency signal, it is possible to stagger the signals from each core by one signal period before signal superposition. This approach not only effectively reduces noise correlation but also ensures that the signals from all fiber cores are in-phase. Nonetheless, the nature of the signals is often unknown in real-world applications. Consequently, the simplistic strategy of delaying by the signal period is not a viable solution. As we can see from Equation (13), the phases of all frequency points in the frequency band can be shifted in the method presented in this paper, so it is applicable to unknown signals at any frequency point in the frequency band.

3.3. Suitability Analysis for Broadband Signals

Notably, for the method of SDM-PLDMS-MCF, the phase-shifting size is related to the delay time, and the phase-shifting object is all of the frequency points of the concerned frequency band. Moreover, there is no frequency aliasing effect between each frequency point, and it is independent for every frequency point in the phase-shifting process. The SDM of the seven-core signals is realized by the form of signal superposition after they are restored to the time domain, and the signals of seven cores do not affect each other. Thus, this method is also applicable to broadband signals.

Considering that the signal measured in practical applications is usually broadband, it is necessary to research the SNR gain of broadband signals. For example, acoustic wave measurements in several kHz frequency bands need to be achieved in the field of underwater inter-well imaging [32]. To verify the suitability of this method for broadband signals, a broadband sweep sinusoidal signal spanning from 800 Hz to 1000 Hz with an amplitude of 2 V is applied to the PZT. The sweep cycle is 100 ms, consisting of three steps with a consistent frequency interval of 100 Hz. The demodulated time domain signal and its corresponding frequency spectrum are shown in Figure 7. Positions I, II, and III, respectively, represent frequencies of 800 Hz, 900 Hz, and 1000 Hz, whose periods are 0.0012 s, 0.0011 s, and 0.0010 s in Figure 7a, and the corresponding frequency spectrum is shown in Figure 7b. Subsequently, the method of SDM for phase linear delay matching signals using MCF is applied in this broadband signal. Figure 8 shows how the SNR gain varies with the time delay $\Delta t$. As the time delay $\Delta t$ increases, the SNR gain experiences a continuous upward trend. The SNR gain approaches the theoretical value of 8.45 dB at $\Delta t=1 \mathrm{ms}$. This outcome is in line with the results obtained from single-frequency signals, effectively verifying that the proposed method is equally applicable to broadband signals.

4. Discussion

Similarly, this method can be promoted for use in multi-strand optical fiber cables in DAS systems, which are widely used today. This method does not require complex computational processes, such as modal decomposition, which means it is applicable for real-time monitoring systems. The improvement of SNR promotes the detection accuracy of weak signals for DAS systems, such as seismic monitoring and oil–gas exploration. It has to be admitted that the SNR gain is limited by the number of accumulations. The upper limit of the SNR gain is $10\log m$.

The method of SDM-PLDMS-MCF mainly removes noise correlation by delaying the signals of each core. The ${\rho }_n$ in Equation (8) is calculated in a controlled laboratory environment, which is not universal and may differ significantly in different conditions. The ${\rho }_n$ needs to be updated based on the noise in the context of the application. Then, the appropriate $\Delta t$ can be introduced to remove the noise correlation. The delay can be adjusted according to the specific systems and the temporal correlation variations in noise among the cores, which are very flexible.

In addition, under real-world field conditions, environmental noise is uncontrolled and may lead to a significant reduction in the SNR, even leading to negative SNR. For example, the field effect amplifies low-frequency noise in seismic monitoring; the noise will increase under wind and rain conditions in perimeter security, and so on. Whether this method is effective in the condition of low SNR or even negative SNR needs further verification.

5. Conclusions

This work demonstrates a method for the suppression of equivalent self-noise for the DAS system integrated with MCF and an experimental test is conducted. The influence of noise correlation among cores on the SNR after signal superposition is studied theoretically and experimentally. An equivalent self-noise suppression method based on a linear delay phase matching scheme using MCF is proposed. The feasibility of this method is verified for both single-frequency and broadband signals through experiments. Before noise decorrelating, the noise correlation coefficient after signal superposition is 11.28, and the SNR experiences an enhancement of 6.32 dB. After noise decorrelating, the SNR gain approaches the theoretical maximum of 8.45 dB with a 1 ms delay, corresponding to an 8.45 dB reduction in the equivalent self-noise of the system. The research findings hold significant implications for achieving precise detection of weak signals.