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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper presents a benchmark for peak detection algorithms employed in fiber Bragg grating spectrometric interrogation systems. The accuracy, precision, and computational performance of currently used algorithms and those of a new proposed artificial neural network algorithm are compared. Centroid and gaussian fitting algorithms are shown to have the highest precision but produce systematic errors that depend on the FBG refractive index modulation profile. The proposed neural network displays relatively good precision with reduced systematic errors and improved computational performance when compared to other networks. Additionally, suitable algorithms may be chosen with the general guidelines presented.

Fiber Bragg grating (FBG) interrogation techniques now form a mature field of research where computational techniques must be used to improve the process of monitoring FBG sensors. When used as sensors, FBG are usually subject to uniform fields of certain types of perturbation such as temperature or strain. In this case, the spectrum of the light reflected by the sensor has its peak monitored, indicating the magnitude of the perturbation. Many techniques which are also part of commercial systems use a periodically tunable laser source to illuminate the FBG and produce the signal corresponding to the spectrum of the light that interacts with the device [

A benchmark evaluation of algorithms used in peak detection must be provided to produce guidelines to evaluate the performance of the algorithms, including the least-squares fitting, computational intelligence and simpler techniques. The benchmark for peak detection algorithms in the interrogation of FBG sensors and the adapted algorithms are therefore made publicly available for general applications [

For comparison purposes, a brief description of the most frequently used algorithms to detect peak position in the FBG spectrum will be provided in the following subsections. For testing purposes the spectrum signal is simulated with different levels of additive white Gaussian noise (AWGN). The algorithms were also applied in experimental data obtained with a tunable laser illuminating FBG sensors. The proposal of the neural network algorithm for the approximation of the FBG spectrum is described and the determination of its peak position statistics is then calculated, as equivalently implemented for the other algorithms.

The described algorithms use input spectra with normalized amplitudes from 0 to 1 (0% and 100% reflectivity, respectively). The transfer matrix method [^{−}^{4} for the FBG with a length of 5 mm in a standard single-mode fiber. The simulated spectrum amplitude was normalized. The wavelength peak position of each spectrum is determined by the maximum amplitude value position, and its resolution is then reduced to 1 pm (wavelength distance between two consecutive points) and white-Gaussian noise is added to the simulated signal with different signal-to-noise ratios (SNR). The signals were generated for SNR values of 16 dB up to 60 dB in steps of 2 dB. Sets of 300 input data for each SNR, using the same spectrum as a signal, were generated to determine the statistics of the peak detection error.

The peak detection algorithms listed below were tested and their characteristics were evaluated with the simulated spectra, verifying the accuracy (the mean of the difference between the peak obtained in a noisy spectrum and the noiseless peak in a high resolution spectrum) and the precision (

The computational performance was verified for a sample spectrum, measuring the time needed for the computation of each algorithm implementation using an input signal with an SNR of 60 dB. The implementations were compiled by ^{®} Core^{™}2 Duo T7300

To measure its performance, the proposed neural network training algorithm was also compared to other currently used algorithms in neural networks having the same topology and configuration as the proposed algorithm,, namely the back-propagation [^{6} iterations (epochs), whichever came first. The algorithms were tested using a spectrum from an FBG with gaussian apodization and an SNR of 60 dB.

The tunable laser for the interrogation system is an Erbium-doped fiber (EDF) ring laser with an intra-cavity Fabry-Perot filter (FPF) whose schematic design is depicted in

The experimental data were acquired by a photo-detecting circuit collecting the light reflected by the sensors, which were illuminated by the EDFL sensor interrogation system. Two sensors were periodically monitored at a sweeping frequency of 10 Hz along a wavelength interval of 10 nm. A triangular waveform with a 50% duty-cycle was used as the tuning signal in the laser, while the processing of the acquired signal could be applied during the fall time of the triangular tuning waveform. The data is composed of 20 measurements, where the signal is acquired by a photo-detector circuit during the rise time of the triangular waveform that sweeps the laser wavelength. The acquired data comprise a signal containing the information from the two FBGs, the sensor and the reference FBG, with Bragg wavelength at 1,542.9 nm and 1,547 nm. One of the FBG sensors is connected to a mechanical apparatus able to stretch it in a controlled manner and its length is measured by a micrometer. The other FBG is kept unperturbed and close to the perturbed sensor to compensate temperature variations during the characterization. An example of the tuning signal and the photo-detected signal from the sensors is depicted in

Another test verified the usage of the algorithms for peak detection of experimental data. The experimental data were acquired with an FBG sensor interrogation system based on a tunable Erbium-doped fiber laser. Two sensors were periodically monitored at a sweeping frequency of 10 Hz along a wavelength interval of 10 nm. A triangular waveform with a 50% duty-cycle was used as the tuning signal in the laser, while the processing of the acquired signal could be applied during the fall time of the triangular tuning waveform. The data is composed of 20 measurements, where the signal is acquired by a photo-detector circuit during the rise time of the triangular waveform that sweeps the laser wavelength. The acquired data comprise a signal containing the information from the two FBGs, the sensor and the reference FBG, with Bragg wavelength at 1,542.9 nm and 1,547 nm. One example of the tuning signal and the photo-detected signal from the sensors is depicted in

Similar to the simulated data, the algorithms were evaluated with respect to their accuracy and precision. Seeing that the data is experimental there is no expected peak to calculate the accuracy statistics. Since the relation between strain and peak shift is expected to be linear, the accuracy of the algorithms is defined here as the mean square error between the obtained points in the experimental setup and the linear fitted line.

Simulated AWGN were added to a base data set that already incorporates noise, generating spectra with additional SNR values of 16 dB up to 60 dB in steps of 2 dB. Each SNR group is composed by 100 data sets, each formed by 20 spectra, one spectra per strain. The accuracy computed by the previously exposed method is presented for each SNR along with the precision (sample standard deviation instead of the mean value).

If the FBG sensor is subjected to uniform disturbances along its length, the sensor’s modulation index profile will be uniformly altered; this causes the FBG reflectivity spectrum to be uniformly shifted towards lower or higher wavelengths, meaning that any point close to the spectrum peak will determine how the sensor behaves. Due to intrinsic characteristics of the hardware in interrogation systems, the spectrum can incorporate noise and the actual peak wavelength may differ from the peak found in a simple search for the highest value in the photo-detected signal, making it necessary to have suitable peak detection algorithms. In a fiber Bragg grating sensor, the spectrum profile of the light reflected by the sensor also depends on the refractive index modulation profile in the fiber optic core. Due to specific fabrication process characteristics, the modulation index envelope may not be uniform. To explore the behavior of the algorithms, two different spectra will be used: one with symmetrical side-lobes around the spectrum peak (uniform index modulation); and the other with non-symmetrical side-lobes produced by an FBG with a gaussian modulation index profile. An example of a simulated FBG spectrum resulting from the light reflected by a sensor with uniform modulation index profile and signal-to-noise ratio of 30 dB (SNR) is depicted in

The maximum algorithm is based on the search for the wavelength with the highest amplitude in the input data. This method is used as a reference of time execution performance but not of accuracy and precision, due to the naturally high inherent noise sensitivity.

This algorithm uses a linear phase finite impulse response (FIR) low-pass filter to attenuate the high frequency noise and then uses the maximum algorithm to find the peak. This algorithm is designed to have a relatively low order and complexity. The low-pass filter was designed based on a Fourier analysis of the noiseless spectrum signal from uniform and gaussian modulation profiles of FBG sensors. It is an equiripple FIR filter with a normalized passband cutoff frequency at the first zero crossing point of the sensor signal Fourier transform. The FIR filter has an attenuation stopband of 80 dB. This resulted in a filter with order

The centroid algorithm produces a point corresponding to the geometric centroid of a spectrum, calculated by _{i}_{i}

In this algorithm, the spectrum centroid determines how the spectrum is being shifted. Before being fed to the centroid, the input spectrum is centered by removing those points with amplitude lower than 0.4.

Another currently used peak detection algorithm consists of adjusting models to fit the spectrum. With fiber Bragg grating sensors it is natural to use a gaussian or a polynomial function as a model [

In this work, the gaussian fitting is implemented by minimizing the squared errors using the Gauss-Newton algorithm. The adjusted gaussian function is shown in _{i}_{i}

As shown in

The polynomial fitting was also implemented using the Gauss-Newton algorithm. A third order polynomial was used (_{i}_{i}_{j}_{j}

In the polynomial fitting, the amplitudes of the input spectrum are also normalized between 0 and 1, discarding the points with amplitude lower than 0.8. An example of polynomial fitting for a spectrum with gaussian modulation is shown in

A properly constructed artificial neural network is a universal function approximator, as seen in the literature [

The proposed neural network is composed of four neurons, disposed in three layers, as shown in

The network was trained by the ^{T}

Due to its size, which is a function of the number of data points, the full Jacobian matrix needs more memory to be stored than the

The optimization introduced by NBN is based on the fact that matrix multiplication can be implemented by multiplying the columns of the left operand with the rows of the right operand, resulting in a summation of partial matrices, that can be added to a result matrix as soon as they are calculated. As both the Quasi-Hessian and gradient matrices equations involve the transposed Jacobian matrix as left operand, one can calculate them iteratively, after the calculation of each row of the Jacobian matrix.

Only those data points with amplitude equal to or greater than 0.65 were used, with the wavelengths scaled between 0 and 1 to match the activation function operation range. Additionally, the method converged faster when initializing all the weights with random positive values.

Accuracy results for the FBG with uniform modulation profile are shown in

The computational performance of the algorithms is also a critical factor to provide a guideline for the implementation of the algorithm in a real-time system or in embedded hardware.

The FBG strain sensor shows a linear response, as the perturbation was within the physical strain limit of the sensor. Examples of experimental calculated points corresponding to the wavelength difference between the reference FBG peak and the strain sensor peak, as a function of the strain level, are depicted in

The values for the different neural network training algorithms are shown in

With respect to the FBG signal corresponding to the sensor with uniform modulation profile, the results presented in

The results presented in

The results of the experimental data test presented in

The computational performance of the algorithms differs by orders of magnitude, preventing the usage of some algorithms in applications that require superior performance. As seen in

When considering the practical implementation of this training algorithm or the trained neural network, a specific processor would be required for the implementation of the previously mentioned matrix calculation algorithms, otherwise the algorithm will not be capable of operating in a real-time manner. Hardware implementation of neural network algorithms or its components may be a direction to the practical implementation of the computationally intelligent signal processing in the interrogation, since there is a current trend to implement neural algorithms in dedicated hardware, for example, using modular neural networks (MNN) [

One strategy to enhance the performance of the neural network algorithm for specific cases is to initialize the weights with previously calculated values, where the network training will be responsible only for the fine-tuning of the weights, performed with a smaller number of training iterations (epochs).

For the FBG sensor with a uniform index modulation profile and the FBGs employed in the experimental data test, it is clear that the centroid and gaussian fitting algorithms have advantages over the other algorithms, due to their higher accuracy, precision and computational performance. However, it is not possible to conclude the same for the gaussian profile. Regarding the FBG with a gaussian index modulation profile, the optimum algorithm choice depends on application circumstances such as the expected SNR in the optoelectronic system, and the performance requirements. However, the centroid and parametric fitting methods can still give good results with applications where the systematic error can be corrected or does not matter.

Due to the longer training phase time to apply the algorithms in experimental data, ordinary neural network training algorithms could be used only in off-line processing of the signal, and not at such tuning frequencies in the laser with this interrogation technique. The largest differences between the fitted line and the measured processed experimental points were obtained for the maximum algorithm. The neural network algorithm shows a smaller amplitude fluctuation than the maximum algorithm, and as previously shown the centroid demonstrates the smallest fluctuations.

In this work a benchmark for algorithms used in FBG interrogation systems was implemented. A new algorithm for training a neural network used as a universal approximator of the FBG reflective spectrum was proposed, and its performance was compared with other algorithms used for the same purpose. The Neuron-by-Neuron training algorithm improves the time performance of the neural networks used to approximate the FBG spectrum. Due to the capacity of neural networks to correct some distortions in the spectrum, they could be used as an alternative to more elementary algorithms. In addition, this training algorithm may establish a lower limit in the time performance enhancement when using neural networks to approximate FBG spectrum, allowing them to be used in real-time processing and embedded systems. As a general rule, the centroid may be considered the fastest and most precise algorithm, even when producing a larger systematic error due to the eventual occurrence of non-symmetrical spectrum signals. This benchmark also provides some guidelines for researchers to choose the proper algorithm for their own application.

The authors gratefully acknowledge the support received from the the National Council of Technological and Scientific Development (CNPq), the Coordination for the Improvement of Higher Education-Personnel (CAPES) and the Photo-refractive Devices Unit (NUFORE) at Federal University of Technology-Paraná (UTFPR), that allowed the implementation of the experimental results.

System schematic diagram for the interrogation of FBG sensors, FBG1 and FBG2, with an Erbium-doped fiber laser (EDFL), a tunable Fabry-Perot filter, EDF in a ring cavity and electronic tuning control.

Representation of the tuning waveform and the photo-detected signal corresponding to the two FBG sensors. The signal is time-based and must be converted to wavelength based on the FBG sensors Bragg wavelength.

Example of optical spectrum for an FBG with uniform modulation index profile and AWGN.

Example of optical spectrum for an FBG with gaussian modulation index profile and AWGN.

Example of gaussian fitting for a spectrum from an FBG with uniform modulation index profile.

Example of polynomial fitting for a spectrum from a gaussian apodized FBG.

Example of neural network fitting for a spectrum from a gaussian modulated FBG.

Topology of the proposed FCC neural network.

Accuracy for the uniform modulation profile.

Precision for the uniform modulation profile.

Accuracy for the gaussian modulation profile.

Precision for the gaussian modulation profile.

Relation between strain and spectrum position by different peak detection algorithms.

Accuracy for the experimental data test calculated using the mean square error between measured points and the fitted straight line.

Precision for the experimental data test based on the square error standard deviation based on the measured points and the fitted straight line.

Computational performance of the evaluated implementations.

Algorithm | Normalized time |
---|---|

Maximum | 1 |

Centroid | 0.67 |

Filter | 27 |

Polynomial Fitting | 340 |

Gaussian Fitting | 943 |

Neural Network | 25,000 |

Performance of different neural network training algorithms.

Training Algorithm | Mean MSE | Mean Epochs |
---|---|---|

NBN | 1.45 | 11 |

iRprop | 1.6 | 10^{6} |

Incremental | 3.5 | 10^{6} |