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Abstract
Coding technique has long been investigated as a solution to improve the signal-to-noise ratio (SNR) without sacrificing spatial resolution in optical time domain reflectometry (OTDR) systems. The past researches have been focusing at the construction of new codes, the combination of coding and other techniques, and the application of coding technique to versatile distributed optical fiber sensing systems such as Raman OTDR and Brillouin optical time domain analyzer, where the results are fruitful. Here, we reveal that oversampling after photodetection opens up a new dimension for coded OTDR other than code length and code type. We demonstrate that the coding gain can be further improved by harnessing the oversampling. Furthermore, the photodetector’s bandwidth-limited feature can also be used to select the optimal sampling rate in order to obtain additional SNR enhancement. We believe this principle could be applied to any practical correlation-coded OTDR-based distributed fiber sensing systems with sufficient SNR enhancement. Our findings can serve to update existing instruments based on correlation-coded OTDR in a straightforward manner and at a relatively low cost.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical time domain reflectometry (OTDR) is an important distributed fiber sensing technique, in which backscattered (Rayleigh, Brillouin and Raman) light is acquired to obtain characteristics along the fiber under test. It has been widely deployed in pipeline monitoring [1–3], structural health monitoring [3,4], etc. due to light weight, flexibility for arrangement, immunity to electromagnetic interference and capability for distributed measurement. Another advantage is the ability to utilize common communication fiber directly as sensing fiber, thus granting the possibility of fault detection and link characterization in optical communication systems. However, this technique is born with a major problem: poor signal-to-noise ratio (SNR) due to very low scattering coefficients. The trade-offs between SNR and spatial resolution as well as pulse power and nonlinear effects are the main obstacles that limit the SNR enhancement.
As a fascinating solution to these trade-offs, pulse coding was proposed in 1980s as a salutary lesson learned from radar systems [5]. This solution does not need coherent detection, thus it is cost-effective. Correlation codes were first introduced to OTDR systems beginning with pseudorandom pulse sequence [5], and were improved with complementary correlation codes (Golay pair) by totally eliminating the autocorrelation sidelobes [6]. Nearly ten years after the introduction of correlation codes, linear codes were also brought in represented by Simplex code which was derived from Hadamard matrix [7,8]. Each of these two types of coding schemes has its strengths, and they serve well in improving the SNR in OTDR systems and dramatically extended the dynamic range.
In recent ten years, researchers began to explore the possibility of employing pulse coding in other distributed fiber sensing systems such as Raman OTDR [9], Brillouin OTDR [10,11] and Brillouin optical time domain analyzer (BOTDA) [12–16]. Other SNR enhancement methods such as Raman amplification [13–15] were also employed in these systems combining with pulse coding for further improvement. The sensing fiber can be extended to as long as 240 km with meter scale spatial resolution without coherent detection [14]. On the other hand, cyclic coding is adopted in the linear codes based fiber sensing system to further reduce the measurement time [17,18]. However, in the aspect of increasing the coding gain, the possible approaches have been limited in two dimensions: code length and code type. As will be proved in the following contents, there is still space to improve for coded OTDR infrastructures.
This article aims at further exploiting the SNR enhancement potential in correlation-coded OTDR system by focusing on the sampling rate of ADC after photodetection, whose relationship with SNR has not been revealed yet. A comprehensive analysis will be given and it will demonstrate that oversampling plays a vital role in improving SNR. The bandwidth-limited feature of photodetector (PD) is also investigated and proves to be considerable in selecting the optimal sampling rate. Golay code is taken as an example, but the analysis suits for all correlation code schemes. Experiments were conducted and confirmed our conclusion.
2. Theoretical analysis
2.1 Essence of oversampling
In direct-detection systems, unipolar binary codes can be modulated on light as pulse sequences by on-off modulation. To receive the codes, the sampling rate of ADC after photodetector has to be the same as the modulating rate of modulator, so in every pulse duration, only one sample is obtained to determine the sequence and guarantee that the codes transmitted can be perceived the same at the receiver.
The oversampling which we are talking about should be an integer multiple of modulation rate. Such integer is defined as the oversampling ratio and will be denoted by m in this paper. In this case, each element in the code word is replicated m times in the received sequence. An example where m = 2 is shown in Fig. 1. Consequently, oversampling greatly multiply the length of code without increasing code duration or modulation rate, which allows us to choose components regardless of their matchup, thus lower the total cost. Although the features of code also change along with the increase of code length, we will demonstrate that it is still beneficial in our application.
In communication systems, the modulation rate is usually so high that oversampling has little room to apply. However, in long-range fiber sensing systems where sampling rate is often decided by spatial resolution and required at megahertz level, large oversampling ratio can be realized.
2.2 Oversampled correlation-coded OTDR
In conventional single pulse OTDR systems, the characteristic of fiber under test (FUT) is obtained by probing with a unit optical pulse and detecting the backscattered light. For direct detection systems with non-coherent source, if a pulse sequence where each pulse is identical is injected into FUT, then each pulse will lead to the same response with different time delay and the total response will be the superposition of them. By carefully designing the code sequences, it is possible to recover the single pulse response from the superposed ones by signal processing, and this is the principle of coding OTDR. Since coding technique essentially sends more pulse energy as probe signal, the system SNR can be improved.
For correlation-coded OTDR, the decoding process is based on the code property that the autocorrelation functions of all code words can form delta function by linear combination. Taking Golay code as an example, it has the following property:
in which A(k) and B(k) are the two L-bit code words of Golay code pair, and ⊗ represents discrete correlation operation. The code words consist of bipolar elements, which cannot be implemented in a direct detection scheme, so it is necessary to transmit the unipolar version of these codes as the probe sequences [6]. As a result, four sequences are needed to fulfill one measurement. As shown in Appendix A, by correlating the code words with the corresponding received signals and summing them up, the OTDR response can be recovered. In order to compare the SNR improvement ability of coding technique, it would be convenient to introduce the concept of coding gain, which is defined as the SNR improvement with respect to conventional averaged OTDR when the same total number of traces are averaged. It can be proved that after the implementation of Golay code, the coding gain is [6].It is clear that the coding gain of correlation-coded OTDR is determined by the code length. Considering that oversampling can multiply code length, it is reasonable to infer that oversampling may also help to enhance SNR. In fact, this conjecture is true. The replicated Golay code has a property similar to Eq. (1) as follows
where qm(k) is the autocorrelation function of a single pulse. Usually we choose rectangular pulse as probing pulse, then qm(k) is given byEquation (3) tells that qm(k) is a unit triangular function, with a peak value of 1 and an FWHM of m. Obviously, this function does not depend on the code length. Figure 2 depicts the correlation result when m equals 2 and the code length is 2. T is the pulse width, so the sampling interval is T/m when oversampling ratio is m. Then, the FWHM of qm(k) in the continuous domain is just the same as the width of a single pulse in the pulse sequences, which also means they have the same energy. As a result, oversampling turns the system response from a rectangular pulse response into a triangular pulse response. These two probing pulses have the same FWHM and energy, which implies that the system spatial resolution remains the same and the corresponding OTDR responses should have roughly the same intensity.
Fig. 2 (a) Code word of a Golay pair when the code length is 2 and m = 2. (b) Comparison between a single rectangular pulse (blue) and the equivalent triangular pulse (red).
In this case, the coding gain is derived in to be, where Lm is exactly the code length after replicated (see Appendix B). As a result, oversampling contributes to the SNR improvement of correlation coding, and this principle works for not only Golay code, but also all other coding schemes based on correlation coding [19–22], such as complementary correlated Prometheus orthonormal sequence (CCPONS), hybrid Golay code, etc. However, it does not apply for Simplex code, which belongs to linear combination codes, since their coding gains are in fact not determined by code length, but by the number of code words.
2.3 Extra SNR enhancement with bandwidth-limited PD
The coding gain calculated in the above discussion is based on an assumption that the noise is Gaussian white noise and all noise samples are uncorrelated. In discrete sampling, this is true when the PD has an infinite bandwidth, or when the sampling rate is lower than the PD bandwidth. In OTDR systems, the bandwidth of the PD is often chosen to be higher than the modulating rate of input light pulses. But when the oversampling is applied, the sampling rate may exceed the PD bandwidth. Hence, the noise should be regarded as bandwidth-limited white noise. Under this condition, the coding gain of oversampled correlation-coded OTDR is given by:
where RN(k) is the autocorrelation function of PD noise, and σ represents noise root-mean-square (RMS). Detailed derivation is given in Appendix C. For ideal white noise, the autocorrelation function RN(k) is a δ function, so the summation term in Eq. (4) (which in the following discussion will be called extra term, for convenience) is 0. Similarly, when there is no oversampling, qm(k) is replaced by δ(k) and the extra term also equals 0, which means that the bandwidth-limited feature of the PD will not exert any extra effect in non-oversampled correlation coding schemes. However, when the sampling rate is higher than the bandwidth of the PD—the situation we are mostly interested in—it is not evident whether the extra term will be positive or negative. If it is negative, then Eq. (4) implies that an extra SNR enhancement can be obtained. Next, we will prove that it is possible to achieve this by choosing a proper sampling rate.The spectrum of noise is determined by the frequency response of the PD, which is usually a lowpass filter. For convenience, a rectangular window is adopted as an ideal lowpass filter to represent the noise spectrum, so the autocorrelation of noise is a sinc function, as shown in Fig. 3(a). On the other hand, we have already demonstrated that qm(k) is a triangular function; see Fig. 3(b). The product of them is displayed in Fig. 3(c), where the first three zero-crossing points are indicated. For an ideal lowpass filter, p1, p2, and p3 are 1/(2B), 1/B, and 3/(2B), respectively, where the bandwidth B is the equivalent noise bandwidth (ENB) of the PD, usually larger than the frequency response bandwidth. For a real PD, p1 is also 1/(2B), but the other two points may not follow the same multiple relationship. This curve is defined in the continuous time domain, so the result of Eq. (4) is obtained by sampling it and summing up all the samples.
Fig. 3 (a) Autocorrelation function of noise. (b) Unit triangular function. (c) Their product. The points p1, p2, and p3 are the first three zero-crossings. (d) The sampling of qm(k)RN(k). The sampling rate is equal to 1/τ.
Our primary concern is whether the extra term in Eq. (4) is positive or negative. It can be easily seen from Fig. 3(c) that the zero-crossing points distribute periodically and the curve attenuates rapidly along the horizontal axes. Therefore, the value of the extra term is largely determined by the value of first sampling point next to the vertical axis, the horizontal coordinate of which is also the sampling interval, as shown in Fig. 3(d). When the sampling interval is set at a valley of the curve, the extra term will be negative, resulting in a higher coding gain than the ideal value. This extra SNR enhancement will be maximum if the sampling interval is set between p1 and p2, especially at (p1 + p2)/2. However, for the same reason, if the sampling interval is set between p2 and p3, the bandwidth-limited feature of the PD will reduce the SNR enhancement. Since the curve attenuates rapidly, continuing to increase the sampling interval will make the extra term approach zero, corresponding to the case in which the sampling rate is within the bandwidth of the PD, so the noise can be seen as ideal white noise. On the other hand, when the sampling interval is smaller than p1, the extra SNR enhancement will be smaller. When the sampling interval is sufficiently small, i.e., the oversampling ratio m is sufficiently large, the sum in Eq. (4) will approach an integral. This integral is independent on sampling rate, implying the coding gain will eventually reach an upper limit when increasing the sampling rate, as demonstrated in Appendix C.
As a result, the sampling rate should be set between 1/p1 and 1/p2 in order to get extra SNR enhancement utilizing the bandwidth-limited feature of the PD. For the practical PDs that we tested, p1 was 1/(2B), while p2 and p3 were roughly 3/(2B) and 5/(2B), respectively, so the sampling rate should be chosen between 2B/3 and 2B. The actual value also depends on the pulse width of the input light, since m must be an integer. However, a sampling rate between 2B/5 and 2B/3 will reduce the SNR enhancement. Excessive oversampling will not contribute to coding gain anymore.
Although oversampling has no effect in linear combination coding schemes like the Simplex code, they may also be affected by the bandwidth-limited feature of the PD, depending on the decoding process. However, the conclusions concerning coding gain may be different [8].
3. Experiment
Experiments were performed to verify the theoretical analysis, using the setup shown in Fig. 4. We employed a broadband amplified spontaneous emission (ASE) light source, operating at 1559 nm and with a 3-dB spectral width of 12 nm. To modulate the CW light, an acousto-optic modulator (AOM) was used and it was controlled by an arbitrary function generator (AFG) that generated the code sequences or single pulse. The backscattered light was converted into electrical signal by a 150 MHz highly sensitive photodetector (PD). Finally, the electrical signal was received in the oscilloscope and processed by a computer. The bandwidth of the AOM was 10 MHz, so the single pulse width could be no shorter than 100 ns. The maximum sampling rate of the oscilloscope was 20 gigasamples per second (GS/s). To ensure that the oversampling ratios were integer, in the following two experiments we chose 200 ns and 100 ns as pulse width, respectively. All the fibers under test were standard single-mode fibers (SMFs).
3.1 Proving the triangular pulse phenomenon
The goal of the first experiment was to prove that the equivalent probing pulse in correlation-coded OTDR with oversampling is a triangular pulse. Since the reflection peak in an OTDR curve should have the same shape as the probing pulse, we chose for our tests two 2 km SMFs connected by a flange plate carefully screwed to generate a small reflection in the junction. To reach a high enough SNR, the Simplex code was used to obtain the single pulse OTDR response. The length of the Simplex and Golay codes was 255 and 2048, respectively. The pulse width was 200 ns, and the sampling rate was 50 Megasamples per second (MS/s) in both cases, so that the oversampling ratio was 10. Both measurements were also averaged 256 times for best SNR. The results are shown in Fig. 5. The two curves are almost identical, proving that the decoding process exactly recovered the OTDR response. Moreover, the shapes of the two reflection peaks are also in agreement with our analytical results.
Fig. 5 Comparison between the intensities of the Golay-coded OTDR (blue) and Simplex-coded OTDR (red) response signals.
3.2 Coding gain verification
The second experiment was conducted to verify the coding gain analysis. Since the signal intensity is constantly changing along the fiber while the RMS of noise remains constant, the coding gain is obtained as follows:
where Sc and Sp are the signal intensities and Nc and Np the noise intensities in the Golay-coded and single-pulse cases. As has been explained, Sc and Sp can be regarded as equivalent, so we only have to calculate the RMS noise in both cases. Since implementing the Golay code takes 4 measurements, the noise of the single-pulse measurement Np should be recorded 4 times and averaged for a fair comparison.
The fiber length in this case was 23.8 km, consuming 238 μs in a period of 500 μs. We chose 100 ns as the pulse width, so the maximum code length could not be over 2048 bits, as the Golay code length must be a power of 2. The non-signal section of the OTDR curve was used for calculating noise and further obtaining the coding gain through Eq. (5). The sampling rate ranged from 10 MS/s to 2.5 GS/s, corresponding to an oversampling ratio from 1 to 250. Golay sequences with code length ranging from 32 to 2048 were injected into the fiber, and each of the backscattered signals was acquired at the different sampling rates mentioned above.
Figure 6(a) shows a comparison between the intensities of the single-pulse and Golay-coded OTDR response signals. The blue curve gives the averaged single-pulse OTDR results calculated from 10000 measurements. We see that the signal can hardly be distinguished from the massive background noise. The red and yellow curves represent the results of 2048 bit Golay-coded OTDR with oversampling ratios of 10 and 100, respectively, without averaging. The SNR for the red curve is already better than that of the blue curve, and that for the yellow curve is even better. Note that both coded results take 4 periods, or 2 ms, to complete one measurement, while the single pulse result takes 10000 periods, or 5 s, to acquire all data, and its SNR is still worse than the coded cases. Implementing an averaging method in the coding scheme could serve to enhance the SNR even further.
Fig. 6 (a) Comparison between the intensities of the single-pulse OTDR (blue) and 2048 bit Golay-coded OTDR (red & yellow, for two different oversampling ratios) response signals. The single pulse results were averaged, while the coded results were not. (b) Measured coding gain versus code length for different sampling rates. The curves represent theoretical values, while the crosses correspond to the experimental results.
Figure 6(b) shows the measured coding gain versus code length for different sampling rates. It is apparent that the measured results (represented by the crosses) agree perfectly with the theoretical curves. Note that for a sampling rate of 10 MS/s there is no oversampling, so Fig. 6(b) also proves that oversampling contributes to the SNR enhancement.
However, if we continue increasing the sampling rate, the experimental coding gain will diverge from the theoretical curve, as shown in Fig. 7. Here, apart from the 150 MHz PD which has been used in the previous experiments, another 300 MHz PD is also utilized for comparison. By recoding the PD outputs on no-load status at 20 GS/s, the autocorrelation functions of their noise properties were calculated as Figs. 7 (a) and 7(c). Using these two PDs, the measured coding gain versus sampling rate are shown in Figs. 7 (b) and 7(d), respectively.
Fig. 7 (a) Autocorrelation function of the 150 MHz PD noise and (b) the corresponding measured coding gain versus sampling rate; (c) Autocorrelation function of another 300 MHz PD noise and (d) the corresponding measured coding gain versus sampling rate. The curves represent theoretical values, while the crosses correspond to the experimental results.
For the 150 MHz PD, the values of the zero-crossings p1 and p2 were calculated to be 1.85 and 5.9 ns, respectively, as can be seen in Fig. 7(a). According to p1 = 1/(2B), the equivalent noise bandwidth was calculated to be about 270 MHz. That means the best sampling rate section in order to obtain extra SNR enhancement was from 180 to 540 MS/s. The extra SNR enhancement can be up to 3 dB when sampling rate is 400MS/s or 500MS/s as shown in Fig. 7(b), along with the over 20 dB coding gain already gained by correlation coding. When the sampling rate was over 1 GS/s, the coding gain did not increase any more. For the 300 MHz PD, the p1, p2 and p3 values were calculated to be 0.9, 2.6, and 4.7 ns, respectively, as can be seen in Fig. 7(c). Thus, the equivalent noise bandwidth was about 550 MHz, and the preferred sampling rate section ranged from 370 to 1100 MS/s, while the “negative” sampling rate section (where the SNR is not enhanced, but reduced) went from 220 to 370 MS/s. Indeed, for this PD, when sampling rate is 250 MS/s, the coding gain was lower than theoretical value, as shown in Fig. 7(d). The experimental results from these two PDs provide sufficient and convincing verification of our theoretical analysis.
4. Summary
We have made a thorough investigation of the sampling criteria for correlation-coded OTDR. Oversampling has been proven to further improve the coding gain without extending the measurement time, while it also turns the received OTDR curve into a triangular pulse response instead of the conventional, rectangular one. However, the coding gain is not always improved with the increase of the sampling rate due to the bandwidth-limited feature of the PD. Interestingly, one can take advantage of this feature to obtain extra SNR enhancement by choosing the sampling rate between 2B/3 and 2B. To continue increasing the sampling rate would make no sense since the coding gain would reach a limit value. When the pulse width of Golay codes is fixed, larger PD bandwidth can support larger oversampling ratio, resulting in better coding gain improvement. Therefore, a large bandwidth PD is preferred in correlation-coded OTDR to fully utilize the oversampling scheme. Since the oversampling does not require any change in the optical transmitter end, it can be conveniently applied to the existing systems. As an extra dimension in the coding scheme, oversampling can compatibly work with other SNR-enhancement approaches, such as extending the code length and averaging measurement.
Although the Golay code was taken as an example for our purposes, the sampling criteria proposed in this paper is also valid for other correlation coding schemes. Furthermore, this principle can also serve to significantly improve the performance of other fiber sensing systems based on direct detection OTDR, such as BOTDA and ROTDR.
Appendix A Decoding process for correlation-coded OTDR
All the discussions in the appendixes are described in the discrete time domain.
Defining f(k) as the backscatter response to an optical pulse signal and c(k) as code sequence, then the code response is given by [6]
where * denoted discrete convolution operation.Golay code is a pair of L-elements codes A(k) and B(k), which have the following relation
where δ(k) is impulse function. By probing the system under test with Golay codes A(k) and B(k), the output signals are expressed asCorrelating these outputs with their respective probe sequences and summing the results, one obtains
where ⊗ represents discrete correlation operation. Thus, we can precisely recover f(k) from yA(k) and yB(k), meanwhile the signal intensity is enhanced by 2L.
The code words consist of bipolar elements, which cannot be implemented in a direct detection scheme, so it is necessary to transmit the unipolar version of these codes as the probe sequences. The construction of unipolar codes is achieved through
Transmitting these four unipolar sequences to the system, and then subtracting the received signals between each pair of unipolar codes, we get
With (9) and (11), the fiber response is recovered by using unipolar codes.
Appendix B Coding gain analysis considering oversampling
When oversampling is considered, A(k) and B(k) are replaced by their replicated versions A’(k) and B’(k), then Eqs. (7) and (9) are replaced by
where qm(k) is the is the autocorrelation function of a single pulse. Usually we choose rectangular pulse as probing pulse, then qm(k) is given byNote that f’(k) in (13) is the OTDR response to a single rectangular pulse of width T/m, (rather than T, as for f(k)). Then ftri(k) can be seen as the OTDR response to the unit triangular pulse qm(k). From (13), we see that oversampling is equivalent to probing the fiber with a triangular pulse rather than the conventional rectangular one. These two probing pulses have the same FWHM and energy, which implies that the system spatial resolution remains the same and the corresponding OTDR responses, i.e., f(k) and ftri(k), should have roughly the same intensity.
Now let us turn to the coding gain analysis. The signal voltage at the PD is proportional to the light intensity, so it is reasonable to use signal voltage and noise voltage to calculate the SNR. Only white noise is taken into consideration, so the noise voltage has a normal distribution with mean value 0 and standard deviation σ. Hence, a noise signal Ni(k) is added to each of the four received raw signals, so Eq. (11) is revised as
whereas, according to Eq. (9), Eq. (13) becomesThe last two terms in Eq. (16) represent noise, and they will be grouped together and denoted by Ntotal in the following discussion. Note that A(k) and B(k)here have Lm elements due to oversampling. Since noise is evaluated by its root-mean-square (RMS) value, our objective is to find out the RMS of Ntotal:
An and Bn are made up by −1’s and 1’s, so the RMS of Ntotal is given by
where the operator D denotes variance. Thus, the SNR of the Golay code correlation process is given byIn order to obtain the coding gain, it is necessary to compare the SNRs of the Golay-coded and single-pulse probe signals under similar measurement conditions, i.e., for the same peak power and measurement time. Thus, we should perform four single-pulse measurements and calculate the final result as the average. This process is expected to enhance the SNR by a factor 2, so we have
As discussed before, f(k) and ftri(k) should have roughly the same intensity, so the coding gain is given by
Appendix C Noise analysis considering bandwidth-limited feature of PD
Bandwidth-limited white noise has the following features:
where RN(k) is the autocorrelation function of noise. From Eq. (22), noise samples are uncorrelated for different a and b, so the four components in N(total) can be separately calculated and summed to obtain the final result. Taking e.g. the first component, since the mean value of noise is always 0, we haveThe range of (j − i) in Eq. (23) is from −(Lm − 1) to (Lm − 1); then, we can simplify this equation by replacing j with (i + k):
where RA(k) represents the autocorrelation function of An. Similarly, the amended version of Eq. (18) is given bySo the ratio of coding gain between system with bandwidth-limited PD and one with ideal PD can be obtain according to Eqs. (18) and (25):
When the sampling interval is sufficiently small, i.e., the oversampling ratio m is sufficiently large, the sum in Eq. (25) will approach an integral. Note that an oversampling ratio m means that the time interval between adjacent sampling points is T/m, so we have
Again, the sum in Eq. (27) approaches an integral when m is sufficiently large. The value of such integral depends only on the autocorrelation function of the PD noise and thus can be regarded as a constant, denoted by C. Then the coding gain is given by
Equation (28) implies the coding gain will eventually reach an upper limit when increasing the sampling rate in the case with bandwidth-limited PD.
Funding
National Natural Science Foundation of China (NSFC) (61331010, 61722108); Fundamental Research Funds for the Central Universities (2016YXZD038); Major Program of the Technical Innovation of Hubei Province of China (2016AAA014); the Open Fund of State Key Laboratory of Optical Fiber and Cable Manufacture Technology, YOFC (SKLD1706).
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