Abstract

A novel Brillouin optical time-domain analysis (BOTDA) system without frequency sweep operation is proposed using intensity-modulated direct detection optical orthogonal frequency division multiplexing (IM-DD-OOFDM) probe signal. The influence of peak to average power ratio (PAPR) of OFDM probe signal on the recovery of Brillouin gain spectrum (BGS) is analyzed in theory and experiment. The complex BGS is reconstructed by channel estimation algorithm and Brillouin frequency shift (BFS) is located by curve fitting of intensity spectrum. The IM-DD-OOFDM BOTDA is demonstrated experimentally with 25 m spatial resolution over 10 km standard single mode fibers within much less measurement time.

© 2017 Optical Society of America

1. Introduction

During the past decades, optical fiber sensing based on Brillouin scattering has been turned into one of the most remarkable fields of research in optical fiber sensing, mainly due to its ability of distributed temperature and strain measurements along tens of km of optical fiber [1]. Classical BOTDA system needs to conduct a large number of averaging process to enhance the signal to noise ratio (SNR) with measuring time of ~minute level, which is unsuitable for dynamic distributed sensing. To improve the acquisition speed, several techniques have been proposed. For instance, slope assisted technique is realized by tuning the frequency of probe optical wave to the position with half of the peak intensity in Brillouin gain spectrum (BGS), in such way, BFS is related to gain variations of probe wave [2–6]. However, the dynamic range is limited by the linear range of Brillouin gain. Another sweep-free BOTDA based on multiple probe and pump pulse pairs is proposed [7,8]. These probe tones are arranged in such a way that each probe tone is located in a different region of the BGS of the corresponding pump pulse. Nevertheless, BFS that moves larger than the frequency spacing between adjacent pump tones will introduce ambiguity, thus severely limiting dynamic range. In [9], the scanning-free BOTDA is realized based on an ultra-fine digital optical frequency comb, which is more like the methods based on multiple frequency tones. The comb-based technique generally suffers from optical nonlinearity thus the sensing range and signal SNR are largely constrained. In fact, the traditional frequency-sweep operation can be seen as a kind of discrete spectrum analysis of probe signal, from which point, channel estimation methods show great potential for BGS recovery. On the other hand, digital signal processing (DSP) has significantly improved optical fiber transmission by advanced algorithm design. The optical orthogonal frequency division multiplexing (OOFDM) is a typical example to take advantage of DSP for optical channel estimation, which has been utilized to calculate the combined transfer function of many optical components [10,11]. Lately, coherent optical OFDM (CO-OFDM) is used to demonstrate a single-shot BOTDA system [12]. Despite of the advantage in time saving, it is susceptible to frequency offset and polarization rotation during the coherent detection. Though homodyne coherent detection can be applied to eliminate phase noise, complicated algorithm is still needed for frequency offset estimation/correction. Besides, the IQ modulation and full-version coherent detection front-end are necessary to demonstrate coherent OFDM based BGS reconstruction, which increases the complexity.

In this paper, we propose a cost-effective BOTDA system using optical orthogonal frequency division multiplexing probe signal and direct detection (DD) channel estimation, inspired by [13,14]. Compared to CO-OFDM, DD-OOFDM is characterized of simplicity in modulation/demodulation structure and data processing algorithm. The sweep-free BOTDA system is demonstrated experimentally with discrete multi-tone (DMT) OFDM modulated probe and direct detection, thus only one optical source, one intensity modulator and single photodetector are required. The impact of peak to average power ratio (PAPR) of OFDM signal on BGS reconstruction is theoretically analyzed and experimentally verified. At last, distributed stimulated Brillouin spectrum is measured in a 10 km single-mode fiber within much less measuring time with a spatial resolution of about 25 m.

2. Theory of BGS measurement based on channel estimation

In linear system, the input and output signals of an optical component are related by its transfer function. That is to say, the performance of an optical fiber system is determined by its complete transfer function. In BOTDA system, the probe signal before and after interaction with the pump pulse can be regarded as input and output signals, respectively. Thus, the transfer function (complex BGS) could be calculated by optical channel estimation.

In our proposed scheme, channel estimation is conducted in the frequency domain within OFDM signal. In general, the transmitted OFDM signal can be described as follows:

s(t)=ej2πf0t+αej2πf0tsB(t)
where s(t) is the optical OFDM signal, f0is the frequency of optical carrier, α is the scaling efficiency indicating the relative amplitude between the optical carrier and OFDM signal, sB(t) is the baseband OFDM signal given by Eq. (2):
sB(t)=k=0N1ckej2πfkt
where ck represents the mapped symbols on the kth subcarrier, fk and N represent the frequency of the kth subcarrier and total number of subcarriers respectively. Let h(t) be the impulse response function of the system, then the received OFDM signal can be approximated using Eq. (3):
r(t)=s(tτ)h(τ)dτ=ej2πf0th(τ)ej2πf0τdτ+αej2πf0tk=0N1ckej2πfkth(τ)ej2π(f0+fk)τdτ
Note that, channel response function H(f) can be written as the Fourier transform of the impulse response function:

H(f)=h(τ)ej2πftdτ

Thus, Eq. (3) can be simplified as follows:

r(t)=H(f0)ej2πf0t+αej2πf0tk=0N1H(f0+fk)ckej2πfkt

At the receiver, the output current is determined by the square law and can be represented as Eq. (6):

I(t)|r(t)|2|H(f0)|2+2αH(f0)Re{k=0N1[[|H(f0+fk)|ejϕ(fk)ck]rkej2πfkt]}+α2(k=0N1H(f0+fk)ckej2πfkt)(k=0N1H(f0+fk)ckej2πfkt)

In Eq. (6), the first term is DC component which will be removed during DSP process. The second term contains the received OFDM signal with both the phase and intensity response information. Let rk be the received mapped data, then the complex channel estimation can be calculated by H(f0+fk)=rkck and the intensity response is the absolute value of H(f0+fk). For the third term, α can be set to make |α|2 much smaller, so it can be ignored. By calculating H(f0+fk) for each subcarrier, the complex spectrum can be obtained.

In our proposed BOTDA system, the optical OFDM signal serves as the probe signal. If we denoterk as the received mapped data after interaction with pump pulse, the complex response becomes:

H(f0+fk)=rkck=H(f0+fk)HSBS
where HSBSis the complex Brillouin gain spectrum, described by:
HSBS=exp(GSBS+jφSBS)(1+GSBS)exp(jφSBS)=(1+gBΔvB2ΔvB2+4Δv2)exp(j2gBΔvΔvBΔvB2+4Δv2)
withGSBS and φSBSas the Brillouin gain and phase shift, respectively, under a small gain regime for BOTDA sensors, gBis the peak gain, ΔvBis the Brillouin linewidth, Δvis the detuning of the interaction frequency from the center of the Brillouin spectrum. Equation (8) shows that both BGS and Brillouin phase spectrum (BPS) can be easily obtained through channel estimation.

3. Influence of OFDM PAPR on BGS recovery

DD-OOFDM system attracts great attention in short reach optical communication systems because of its high spectral efficiency and robustness against fiber dispersion. However, DD-OOFDM system suffers from the large PAPR that may result in significant distortions to the transmitted signal, leading to power efficiency degradation [15]. Consequently, the optical modulation index of an IM-DD based system is restrained to maintain the linearity, resulting in worse receiver sensitivity and insufficient optical power budget [16].

For the baseband OFDM signal in Eq. (2), the PAPR can be defined as

PAPR(dB)=10log10max(|sB(t)|2)E{|sB(t)|2}

If the amplitude of all subcarriers is normalized, the average power of OFDM signal is E{|sB(t)|2}=N. Therefore,

PAPR(dB)=10log10max(|sB(t)|2)N

The instantaneous power of sB(t) is

P(t)=|sB(t)|2=sB(t)sB(t)=i=0N1k=0N1cickexp[j2πf(ik)t]=N+2Re{i=0N2k=i+1N1cickexp[j2πf(ik)t]}=N+2Re{m=1N1exp(j2πfmt)i=0N1mcic(i+m)}

For any complex z, Re(z)|z|, |zn||zn|. Therefore,

P(t)N+2m=1N1|ρ(m)|
where ρ(m)=i=0N1mcic(i+m),m=0,1,...,N1 is the aperiodic autocorrelation function. Equation (12) shows that if the aperiodic autocorrelation modulus of inverse fast Fourier transform (IFFT) operation input sequence ck is small (i.e. small |ρ(m)| for m1), the peak-power factor of the signal obtained by passing through the multi-carrier combination can also be small [17]. The peak value of the autocorrelation is the average power of input sequence. Then if the number of subcarriers is not changed, this peak value depends on the input sequence. IFFT operation can be viewed as multiplying sinusoidal functions to the input sequence, summing and sampling the results. Thus, the high correlation property of IFFT input causes the sinusoidal functions to be arranged with in-phase form. As a consequence, the sum of these in-phase functions might have a large amplitude. In other words, the PAPR can be reduced by special design of input sequence ck with smaller autocorrelation modulus.

In practice, high PAPR means nonuniform distribution of OFDM probe signal power in time domain. On one hand, the limited optical power leads to a lower SNR of the OFDM probe. On the other hand, nonuniform power may cause lower Brillouin gain along certain fiber section. Both factors affect BGS recovery. If the Brillouin gain configuration is considered for simplicity, the CW probe signal is amplified by stimulated Brillouin scattering (SBS) while the pulsed pump propagates along the fiber, as shown in Fig. 1.

 

Fig. 1 Simplified BOTDA. Pump pulse (with power of Pp) interacts with continuous probe (with power of Ps) through stimulated Brillouin effect and gives rise to a local power transfer ΔPs.

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The Brillouin gain induced probe power increase at fiber location z, is given by [18]:

ΔPs(z)=Ps(z)[exp(gB(z)AeffPp(z)Δz)1]
where gB(z)is the local Brillouin gain coefficient, Aeffrepresents the nonlinear effective area of the fiber and Δz is the interaction length corresponding to the spatial resolution and determined by the pump pulse width.

The power increase expressed by Eq. (13) actually relates to the local sensor response, where the local gain value gB(z) is considered for a given frequency difference and is assumed to be constant over the interaction length Δz. Usually, BOTDA sensors operate in a small gain regime [19], i.e. ΔPs(z)/Ps(z)<<0.1.Therefore, the local sensor response defined by Eq. (13) can be simplified as:

ΔPs(z)=gB(z)AeffPp(z)Ps(z)Δz

In Eq. (14), the local transferred power ΔPs(z)from pump is proportional to the probe power Ps(z), which means the SNR is determined by Ps(z). Although, this relation is derived on the basis of single-frequency probe light, it is still applicable to multi-frequency OFDM probe signal. Substitute Eq. (11) into Eq. (14), then the transferred power becomes:

ΔP(z)=gB(z)¯AeffPp(z)P(z)Δz

Since P(z) consists of multi-frequency signal and gB(z) is related to frequency, gB(z)¯is introduced as constant regardless of frequency difference for simplicity. Different from CW probe, the power of P(z) varies with z, which means the transferred power, ΔP(z) changes as well. Especially with high PAPR, the Brillouin gain may be quite small at most of the fiber locations, resulting in poor SNR. For BGS measurement in BOTDA, BFS is estimated by curve fitting algorithm. The propagated error on the estimated resonance central frequency σν(z) can be expressed as [18]:

σν(z)=σ(z)3δΔνB82(1η)3/2=1SNR(z)3δΔνB82(1η)3/2
whereσ(z) is the noise amplitude, directly represented by the inverse of the local SNR at peak gain frequency, ΔνB is the estimated full-width at half maximum (FWHM) of the resonance, δdenotes the frequency spacing, and η is the fraction of the peak level over which a quadratic least-square fitting is carried out. Equation (16) illuminates that the performance of a certain BOTDA system is determined by SNR, which is proportional to the probe power. Since higher PAPR causes severe nonuniform distribution of probe power, which further affects the SNR distribution along fiber, SNR(z), greater error will be introduced in the central frequency estimation. Thus, it can be inferred that the PAPR can impose great influence to OFDM based system, which will be demonstrated in experiments bellow.

4. Experimental setup

The experimental setup of IM-DD-OOFDM based BOTDA scheme is shown in Fig. 2. The external cavity laser (ECL) working at 1550 nm with less than 100 kHz linewidth is divided by a 3 dB coupler into two parts as probe and pump respectively. At the probe side, the light is first modulated by an electro-optic intensity modulator (EOM1) driven by a microwave source (MS) with 10.65 GHz RF signal to generate dual-sideband optical signal before launched into another intensity modulator (EOM2). Then the OFDM baseband signal is generated from an arbitrary waveform generator (AWG) at the sampling rate of 500 MSa/s and added onto dual-sideband probe light by EOM2. Every OFDM subcarrier carries binary phase shift keying (BPSK) mapped Pseudo Random Binary Sequence (PRBS). The input sequence of IFFT is made Hermitian symmetry to generate real-valued OFDM probe. The OFDM signal has a total number of 128 subcarriers but 63 effective subcarriers, as for real-valued OFDM generation, Hermitian symmetry must be satisfied and the first subcarrier is abandoned to eliminate noise near DC component [20]. Thus, the frequency spacing of adjacent subcarrier is about 4 MHz, corresponding to a time duration of 250 ns. It should be noted that the effective dynamic range is half of the total bandwidth of probe in our IM-DD-OOFDM based BOTDA system.

 

Fig. 2 Experimental setup: ECL: external cavity laser, PC: polarization controller, EOM: electro-optic modulator, MS: microwave source, AWG: arbitrary waveform generator, ISO: optical isolator, SOA: semiconductor optical amplifier, EDFA: Erbium doped fiber amplifier, BPF: optical band-pass filter, PS: polarization controller, FUT: fiber under test, PD: photo detector, TDO: time-domain oscilloscope, DSP: digital signal processing.

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The high extinction ratio pump pulse is generated through a semiconductor optical amplifier (SOA), which is driven by a positive pulse with the width of 200 ns and period of 30 μs from another channel of AWG. An inverse electrical signal from the same channel is used as the trigger of the real-time oscilloscope. After amplified by an Erbium doped fiber amplifier (EDFA), the pump pulse with high peak power is launched into fiber through an optical fiber circulator. An optical band pass filter (BPF) is used to filter out amplifier spontaneous emission noise (ASE) of EDFA. A polarization switch (PS) is placed in front of the circulator to alleviate polarization dependent fluctuations of Brillouin gain. At the receiver, the higher frequency sideband of the probe signal is first filtered out by a narrow band filter for gain-mechanism based BOTDA, then a photodetector (PD) with a bandwidth of 400 MHz is utilized for direct detection of the probe signal. To verify the performance of our proposed scheme, a spool of 2 km standard single mode fiber (SSMF) together with 50 m SSMF heated is used as fiber under test (FUT).

5. Experimental results

Figure 3 depicts the schematic illustration of our designed probe signal in the time-domain. The first four training symbols (TSs) are designed for synchronization and the remaining sensing symbols (SSs) are identical for simplicity. The length of OFDM signal is the same as the pulse period of 30 μs. By adjusting the time delay of pump pulse, the training symbols and the last few sensing symbols are kept unaffected and can be respectively used for synchronization and channel estimation. The received probe OFDM signals are first averaged 512 times to improve the SNR. After that, the received data are divided into many segments, each of which contains one symbol, corresponding to a spatial resolution of 25 m. Then fast Fourier Transform (FFT) is applied to every segment to obtain complex amplitude of each subcarrier. The BGS along the FUT can be obtained after processing all the segments. As aforementioned, both the BGS and BPS can be calculated through channel estimation, though we choose BGS for sensing.

 

Fig. 3 Time-domain OFDM probe signal: TS: Training symbol, SS: Sensing symbol.

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To validate the influence of high PAPR of OFDM signal on BGS reconstruction, two different OFDM signals with PAPR of 15 dB and 9 dB (these two values are not chosen on purpose, other distinct values are suitable as well) are used for probe wave generation, as shown in Figs. 4(a) and 4(b), respectively. As analyzed in Section 3, the power distribution with time is quite different for OFDM signal with different PAPR. Higher PAPR leads to fewer power peaks and gives rise to small values in the majority of sample points, resulting in smaller power gain through SBS.

 

Fig. 4 (a) Normalized baseband OFDM signal with PAPR of 15 dB. (b) Normalized baseband OFDM signal with PAPR of 9 dB.

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The gain spectrum reconstructed in both cases are represented as blue dots in Figs. 5(a) and 5(b), respectively. As can be seen, the spectrum obtained in higher PAPR case fluctuates more seriously, especially for subcarriers with lower Brillouin gain. The red solid lines in Figs. 5(a) and 5(b) indicate the respective Lorentzian curve fitting results, which are obtained using data above the level of 0.2 of peak gain. Thanks to higher SNR, the measured spectrum in lower PAPR case fits better to the Lorentzian curve. Similarly, the phase spectrum of lower PAPR performs better, as denoted in Fig. 5(d) compared to that in Fig. 5(c). Since the SNR of peak gain frequency of probe cannot be precisely estimated by calculating error vector magnitude (EVM) as in communication systems, we adopt the method in reference [21] to evaluate the performance of the system. Accordingly, the SNR of the probe is calculated to be 31.3 dB and 28.5 dB for PAPR of 9 dB and 15 dB, respectively. And the corresponding estimation error of central frequency for BGS curve fitting is about 0.3 MHz and 0.4 MHz, respectively. Thus, the performance of the system degrades at high PAPR of OFDM signal. Since PAPR is kind of statistical property, it is difficult to quantitively analyze the relationship of PAPR and system performance. But it is clear that higher PAPR results in poorer SNR and accuracy. It should be noted that the PAPR is related to the number of subcarriers and the measurement range is in proportion to the bandwidth (i.e. the number of subcarriers for fixed frequency spacing). That is to say, there may be a tradeoff between reducing PAPR and increasing measurement range (there may be exception such as Zadoff-Chu sequence, the PAPR of which will decline with the increase of the number of subcarriers from 128 to 256 for fixed frequency spacing). Therefore, the problem of high PAPR of OFDM signal should be considered carefully for better performance system design.

 

Fig. 5 (a) Brillouin intensity gain spectrum for PAPR of 15 dB. (b) Brillouin intensity gain spectrum for PAPR of 9 dB. (c) Brillouin phase spectrum for PAPR of 15 dB. (d) Brillouin phase spectrum for PAPR of 9 dB.

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To verify the performance of our proposed system, the 50 m SSMF spliced at the end of 2 km SSMF is kept in a water bath pot heated from 30 °C to 80 °C with the step of 10 °C. At room temperature, the BFSs of the two fiber sections are 10.775 GHz and 10.738 GHz, respectively. The OFDM signal with PAPR of about 9 dB is used for probe in the experiment.

Figures 6(a) and 6(b) show the reconstructed BGS and measured BFS of the fiber heated in the water pot, respectively. As depicted in Fig. 6(a), the BGS distribution along FUT consists of two sections, corresponding to the long fiber spool and short spliced fiber tail. Because of the 25 m spatial resolution which is dependent on the length of OFDM symbol, it is clear to observe the heated segment as shown by the inset in Fig. 6(a). The spatial resolution can also be estimated by evaluating the fiber length associated with the BFS change from 10% to 90% of the heated area, as shown in Fig. 7. As the symbol length is inversely proportional to the frequency space of adjacent subcarriers, one can simply increase the frequency space for higher spatial resolution, within an acceptable BFS estimation error (σν(z)). The measured BFS of heated fiber section as a function of temperature is plotted as blue dots in Fig. 6(b), which is in good agreement with the linear fitting red solid line. The temperature coefficient for the fiber segment is calculated to be 1.03 °C/MHz.

 

Fig. 6 (a) Reconstructed spectrogram along the fiber. (b) Measured and fit BFS as a function of temperature.

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Fig. 7 Spatial resolution at the end of 2km fiber.

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In our experiment, the time consumption for each acquisition is about 30 μs, and 512 times averaging is needed in a single measurement to improve the SNR together with two measurements for each polarization state of the pump pulse, resulting in a total measurement time of about 30 ms. Actually, the measuring time is mainly dependent on the averaging process to improve SNR. The averaging time can be greatly reduced by choosing proper input sequence with much lower PAPR. Besides, the use of polarization switch doubles the measuring time, which can be replaced by orthogonal polarized probe sidebands or pump technique used in conventional BOTDA. In this way, much progress can be made to reach the limit of single-shot measurement. Note that, the SNR of our system is about 30 dB, thus the sensing length can be easily extended. And by optimizing the probe signal, the detecting range will be further enhanced.

To verify the capability of our system for long range sensing, another experiment is carried out with a sensing fiber of about 10 km. The pulse period (about 130 μs, resulting in longer total measuring time of about 130 ms), OFDM probe length and time delay are modified accordingly with other settings unchanged. As shown in Fig. 8(a), the BGS spectrogram of the fiber is well recovered, as well as the BFS of the heated fiber end with a length of about 25 m. Figure 8(b) depicts the measured BFS of the heated fiber end as a function of temperature, with an estimation error of about 0.6 MHz. Still, the measured data shows an explicit linearity with a calculated temperature coefficient of 1.04 °C/MHz. And the spatial resolution near the fiber end is about 25 m, according to Fig. 9. It has been validated that the IM-DD-OFDM based approach is scalable for long distance distributed sensing without sacrificing the spatial resolution.

 

Fig. 8 (a) 3D plot of Brillouin gain spectrum along 10 km fiber. (b) Measured and fit BFS as a function of temperature at the fiber end.

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Fig. 9 Spatial resolution at the end of 10km fiber.

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Since the performance of the system is largely dependent on the SNR of the obtained probe signal and the high PAPR of OFDM probe is one of the main factors that affects the distributed SNR (SNR(z)), more work should be done for optimal OFDM signal design.

6. Conclusions

We have proposed and experimentally demonstrated a novel sweep-free BOTDA system based on IM-DD-OOFDM channel estimation, which is cost effective with improved measurement efficiency. Through channel estimation algorithm, the complex BGS is obtained and the BFS is located by BGS curve fitting with the uncertainty of less than 1 MHz. Moreover, the influence of PAPR on system performance is theoretically analyzed and experimentally verified. It implies that OFDM probe should be deliberately designed for better performance. Distributed temperature measurement over 10 km SSMF is conducted and the total measurement time is within 130 milliseconds with the spatial resolution of 25 m, which means the proposed scheme has the obvious advantage in time saving over conventional BOTDA system. It is worth stressing that the spatial resolution and measurement time can still be improved, which will be explored in future works.

Funding

National Natural Science Foundation of China (NSFC) (61331010); the 863 High Technology Plan of China (2013AA013402); the Program for New Century Excellent Talents in University (NCET-13-0235); Fundamental Research Funds for the Central Universities (2016YXZD038).

References and links

1. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995). [CrossRef]  

2. Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011). [CrossRef]   [PubMed]  

3. Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012). [CrossRef]   [PubMed]  

4. Y. Peled, A. Motil, I. Kressel, and M. Tur, “Monitoring the propagation of mechanical waves using an optical fiber distributed and dynamic strain sensor based on BOTDA,” Opt. Express 21(9), 10697–10705 (2013). [CrossRef]   [PubMed]  

5. D. Ba, B. Wang, D. Zhou, M. Yin, Y. Dong, H. Li, Z. Lu, and Z. Fan, “Distributed measurement of dynamic strain based on multi-slope assisted fast BOTDA,” Opt. Express 24(9), 9781–9793 (2016). [CrossRef]   [PubMed]  

6. D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017). [CrossRef]  

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12. J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), paper AT5C.3.

13. X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012). [CrossRef]  

14. B. Guo, T. Gui, Z. Li, Y. Bao, X. Yi, J. Li, X. Feng, and S. Liu, “Characterization of passive optical components with ultra-fast speed and high-resolution based on DD-OFDM,” Opt. Express 20(20), 22079–22086 (2012). [CrossRef]   [PubMed]  

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16. C. Wei, H. Y. Chen, H. Chu, Y. Chen, C. Song, I. Lu, and J. Chen, “32-dB Loss Budget High-Capacity OFDM Long-Reach PON over 60-km Transmission without Optical Amplifier,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th3G.1. [CrossRef]  

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References

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  1. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
    [Crossref]
  2. Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
    [Crossref] [PubMed]
  3. Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012).
    [Crossref] [PubMed]
  4. Y. Peled, A. Motil, I. Kressel, and M. Tur, “Monitoring the propagation of mechanical waves using an optical fiber distributed and dynamic strain sensor based on BOTDA,” Opt. Express 21(9), 10697–10705 (2013).
    [Crossref] [PubMed]
  5. D. Ba, B. Wang, D. Zhou, M. Yin, Y. Dong, H. Li, Z. Lu, and Z. Fan, “Distributed measurement of dynamic strain based on multi-slope assisted fast BOTDA,” Opt. Express 24(9), 9781–9793 (2016).
    [Crossref] [PubMed]
  6. D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017).
    [Crossref]
  7. A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS Based Fiber Optical Sensing Using Frequency-Domain Simultaneous Tone Interrogation,” J. Lightwave Technol. 29(11), 1729–1735 (2011).
    [Crossref]
  8. A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011).
    [Crossref] [PubMed]
  9. C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
    [Crossref] [PubMed]
  10. X. Yi, W. Shieh, and Y. Ma, “Phase Noise Effects on High Spectral Efficiency Coherent Optical OFDM Transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
    [Crossref]
  11. W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express 15(2), 350–356 (2007).
    [Crossref] [PubMed]
  12. J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), paper AT5C.3.
  13. X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
    [Crossref]
  14. B. Guo, T. Gui, Z. Li, Y. Bao, X. Yi, J. Li, X. Feng, and S. Liu, “Characterization of passive optical components with ultra-fast speed and high-resolution based on DD-OFDM,” Opt. Express 20(20), 22079–22086 (2012).
    [Crossref] [PubMed]
  15. L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
    [Crossref]
  16. C. Wei, H. Y. Chen, H. Chu, Y. Chen, C. Song, I. Lu, and J. Chen, “32-dB Loss Budget High-Capacity OFDM Long-Reach PON over 60-km Transmission without Optical Amplifier,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th3G.1.
    [Crossref]
  17. C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
    [Crossref]
  18. M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
    [Crossref] [PubMed]
  19. L. Thévenaz, S. F. Mafang, and J. Lin, “Effect of pulse depletion in a Brillouin optical time-domain analysis system,” Opt. Express 21(12), 14017–14035 (2013).
    [Crossref] [PubMed]
  20. Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
    [Crossref]
  21. A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
    [Crossref]

2017 (1)

2016 (1)

2015 (2)

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

2013 (4)

2012 (3)

2011 (3)

2008 (1)

2007 (1)

1999 (1)

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

1997 (1)

C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
[Crossref]

1995 (1)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Ba, D.

Bao, X.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Bao, Y.

X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
[Crossref]

B. Guo, T. Gui, Z. Li, Y. Bao, X. Yi, J. Li, X. Feng, and S. Liu, “Characterization of passive optical components with ultra-fast speed and high-resolution based on DD-OFDM,” Opt. Express 20(20), 22079–22086 (2012).
[Crossref] [PubMed]

Bremner, T. W.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Brown, A. W.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Chen, W.

Chi, N.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Chitgarha, M.

DeMerchant, M. D.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Deng, L.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Dong, Y.

Fan, Z.

Feng, X.

Feng, Y.

Feng, Z.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Fu, S.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Gui, T.

Guo, B.

Guo, N.

Horiguchi, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Ji, Y.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Jiang, T.

Jin, C.

Koyamada, Y.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Kressel, I.

Kurashima, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Lau, A. P. T.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Li, H.

Li, J.

Li, Z.

Liang, H.

Lin, J.

Lin, R.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Liu, D.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Liu, J.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Liu, S.

Lu, C.

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Lu, Z.

Ma, Y.

Mafang, S. F.

Motil, A.

Nuccio, S. R.

Peled, Y.

Pendock, G.

Qiu, K.

X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
[Crossref]

Shamee, B.

Shieh, W.

Shimizu, K.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Shum, P.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Soto, M. A.

Tang, M.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Tao, L.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Tateda, M.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Tellambura, C.

C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
[Crossref]

Thévenaz, L.

Tucker, R. S.

Tur, M.

Voskoboinik, A.

Wang, B.

Wang, J.

Wang, L.

Wang, R.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Willner, A. E.

Willner, A. W.

Wu, Q.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Xu, P.

Yaron, L.

Yi, X.

Yilmaz, O. F.

Yin, M.

Yu, C.

Zhang, H.

Zhang, L.

Zhou, D.

Electron. Lett. (1)

C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
[Crossref]

IEEE Netw. (1)

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (2)

X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
[Crossref]

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (11)

W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express 15(2), 350–356 (2007).
[Crossref] [PubMed]

A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011).
[Crossref] [PubMed]

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
[Crossref] [PubMed]

Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012).
[Crossref] [PubMed]

Y. Peled, A. Motil, I. Kressel, and M. Tur, “Monitoring the propagation of mechanical waves using an optical fiber distributed and dynamic strain sensor based on BOTDA,” Opt. Express 21(9), 10697–10705 (2013).
[Crossref] [PubMed]

D. Ba, B. Wang, D. Zhou, M. Yin, Y. Dong, H. Li, Z. Lu, and Z. Fan, “Distributed measurement of dynamic strain based on multi-slope assisted fast BOTDA,” Opt. Express 24(9), 9781–9793 (2016).
[Crossref] [PubMed]

D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017).
[Crossref]

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

L. Thévenaz, S. F. Mafang, and J. Lin, “Effect of pulse depletion in a Brillouin optical time-domain analysis system,” Opt. Express 21(12), 14017–14035 (2013).
[Crossref] [PubMed]

B. Guo, T. Gui, Z. Li, Y. Bao, X. Yi, J. Li, X. Feng, and S. Liu, “Characterization of passive optical components with ultra-fast speed and high-resolution based on DD-OFDM,” Opt. Express 20(20), 22079–22086 (2012).
[Crossref] [PubMed]

Proc. SPIE (1)

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Other (2)

C. Wei, H. Y. Chen, H. Chu, Y. Chen, C. Song, I. Lu, and J. Chen, “32-dB Loss Budget High-Capacity OFDM Long-Reach PON over 60-km Transmission without Optical Amplifier,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th3G.1.
[Crossref]

J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), paper AT5C.3.

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Figures (9)

Fig. 1
Fig. 1 Simplified BOTDA. Pump pulse (with power of Pp) interacts with continuous probe (with power of Ps) through stimulated Brillouin effect and gives rise to a local power transfer ΔPs.
Fig. 2
Fig. 2 Experimental setup: ECL: external cavity laser, PC: polarization controller, EOM: electro-optic modulator, MS: microwave source, AWG: arbitrary waveform generator, ISO: optical isolator, SOA: semiconductor optical amplifier, EDFA: Erbium doped fiber amplifier, BPF: optical band-pass filter, PS: polarization controller, FUT: fiber under test, PD: photo detector, TDO: time-domain oscilloscope, DSP: digital signal processing.
Fig. 3
Fig. 3 Time-domain OFDM probe signal: TS: Training symbol, SS: Sensing symbol.
Fig. 4
Fig. 4 (a) Normalized baseband OFDM signal with PAPR of 15 dB. (b) Normalized baseband OFDM signal with PAPR of 9 dB.
Fig. 5
Fig. 5 (a) Brillouin intensity gain spectrum for PAPR of 15 dB. (b) Brillouin intensity gain spectrum for PAPR of 9 dB. (c) Brillouin phase spectrum for PAPR of 15 dB. (d) Brillouin phase spectrum for PAPR of 9 dB.
Fig. 6
Fig. 6 (a) Reconstructed spectrogram along the fiber. (b) Measured and fit BFS as a function of temperature.
Fig. 7
Fig. 7 Spatial resolution at the end of 2km fiber.
Fig. 8
Fig. 8 (a) 3D plot of Brillouin gain spectrum along 10 km fiber. (b) Measured and fit BFS as a function of temperature at the fiber end.
Fig. 9
Fig. 9 Spatial resolution at the end of 10km fiber.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

s ( t ) = e j 2 π f 0 t + α e j 2 π f 0 t s B ( t )
s B ( t ) = k = 0 N 1 c k e j 2 π f k t
r ( t ) = s ( t τ ) h ( τ ) d τ = e j 2 π f 0 t h ( τ ) e j 2 π f 0 τ d τ + α e j 2 π f 0 t k = 0 N 1 c k e j 2 π f k t h ( τ ) e j 2 π ( f 0 + f k ) τ d τ
H ( f ) = h ( τ ) e j 2 π f t d τ
r ( t ) = H ( f 0 ) e j 2 π f 0 t + α e j 2 π f 0 t k = 0 N 1 H ( f 0 + f k ) c k e j 2 π f k t
I ( t ) | r ( t ) | 2 | H ( f 0 ) | 2 + 2 α H ( f 0 ) Re { k = 0 N 1 [ [ | H ( f 0 + f k ) | e j ϕ ( f k ) c k ] r k e j 2 π f k t ] } + α 2 ( k = 0 N 1 H ( f 0 + f k ) c k e j 2 π f k t ) ( k = 0 N 1 H ( f 0 + f k ) c k e j 2 π f k t )
H ( f 0 + f k ) = r k c k = H ( f 0 + f k ) H S B S
H S B S = exp ( G S B S + j φ S B S ) ( 1 + G S B S ) exp ( j φ S B S ) = ( 1 + g B Δ v B 2 Δ v B 2 + 4 Δ v 2 ) exp ( j 2 g B Δ v Δ v B Δ v B 2 + 4 Δ v 2 )
P A P R ( d B ) = 10 log 10 max ( | s B ( t ) | 2 ) E { | s B ( t ) | 2 }
P A P R ( d B ) = 10 log 10 max ( | s B ( t ) | 2 ) N
P ( t ) = | s B ( t ) | 2 = s B ( t ) s B ( t ) = i = 0 N 1 k = 0 N 1 c i c k exp [ j 2 π f ( i k ) t ] = N + 2 Re { i = 0 N 2 k = i + 1 N 1 c i c k exp [ j 2 π f ( i k ) t ] } = N + 2 Re { m = 1 N 1 exp ( j 2 π f m t ) i = 0 N 1 m c i c ( i + m ) }
P ( t ) N + 2 m = 1 N 1 | ρ ( m ) |
Δ P s ( z ) = P s ( z ) [ exp ( g B ( z ) A e f f P p ( z ) Δ z ) 1 ]
Δ P s ( z ) = g B ( z ) A e f f P p ( z ) P s ( z ) Δ z
Δ P ( z ) = g B ( z ) ¯ A e f f P p ( z ) P ( z ) Δ z
σ ν ( z ) = σ ( z ) 3 δ Δ ν B 8 2 ( 1 η ) 3 / 2 = 1 S N R ( z ) 3 δ Δ ν B 8 2 ( 1 η ) 3 / 2

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