A New Approach To Enhancement Of Background Noise Spectra Characterization Using Accelerometric Data For Tram Movement In Europe

Abstract— Tram vibration signal acquisition and signal component analysis has been carried out and analyzed involving signal separtion for noise measurement in the tram rails when the trams cross the point of observation. In this study, an operation evaluation for vibration induced on the structure of tram rails was conducted at certain measuring points based on Phase II project conducted in Oradea after rehabillation of tram lines. Time domain decomposition of the tram signals and extracted noise spectra data was identified as non-stationary time series data and non-gaussian data sets. In our study, we were able to correlate non-linear independent signal acquired using acceleromets at different spots across the city and extract tram rail vibration noise and model the effect of signal noise to identify the frequency characteristics of the rail by characterizing the spectral content of the noise signal using parametric distribution and then by applying non parametric filters to characterize the signal power spectral density .

Introduction

Oradea is one of the Main cities of Romania where Tram is used as a regular commutation by people and is regular 7 days a week until 10 pm at night. It passes through important crossings and there are interconnected lines across the city forming a network. It is always not possible to analyze the condition of the rail track degradation due to vibration although of great necessity. Acquiring the signal data using accelerometer sensors conducted as in Phase II project in the present scenario and so data has been acquired from Project conducted earlier for 7 locations along Oradea rail tracks along a number of measurement points under Phase II project as shown in Figure 1. Physical deterioration of the tram tracks due to continuous and regular movement of trams is a very critical component of modern day Urban transport in Europe. A number of cities in Romania have regular tram lines as so in many other cities of Europe which serve as essential means of infrastructure in Urban transportation. Virtually, many private and public entities which have ownership or operate in this infrastructure with nearby operations are interested in damage identification through hardware and software tools. Sensors acquire the data and process signals which give many essential parameters for the diagnostic rehabilitation and maintenance of critical urban infrastructure.

Fig. 1. Locations vibration measuring points for Phase II.

In the study of Phase II, location of the measurement points was set taking into account the characteristic features of the areas covered by the trams portion of the stations. In this area certain spots were chosen and the accelerometer K48C was placed in each of the 7 locations as shown in Figure 1 and they have established a number of measurement points. The position of the measurement points is shown in Figure 1 for vibration measurements and measurements of the noise Figure 2. The characteristics of the measurement points are given in Table 1.

Tabel 1. Characteristics of measurement points for data acquisition in Oradea City

2.Data Acquisition

The most commonly used data acquisition instruments for vibration-based damage detection are acceleration and displacement transducers. These instruments measure the acceleration or displacement time history signals of a structure and convert them into voltage signals. The mass and size of the data acquisition system must be small compared with the mass of the structure in order to not interfere with the output response. We have used a K48C piezoelectric accelerometer as part of the work. This is due to their ability to operate over a large range of frequency responses and have a high sensitivity of 1000 mV/g. The output dynamic response is usually measured using transducers that measure position, velocity or acceleration. The spectrum identified in the vibration subspace comprises of three important components which are signal which has a finite nature, sinusoids which are signals that are time varying and harmonics which are multiple order coefficients of the noise data. We further our research with the use of NI cDAQ-9178 and module NI 9234 and connecting an accelerometer to a DAQ device, verifying the signal in NI Measurement & Automation finally taking a measurement in Matlab. We also set the port and from the Data acquisition unit, we set the accelerometer channel and the session rate and duration. We transfer the data to the form “.mat” for transformation and analysis. In the noise subspace, application of simple Direct Frequency Translation[Velchev et al., 2016] or periodogram analysis for spectral identification of the power spectral density[Seydoux et al.,2016] is of disadvantage in many scenarios and thus there is a need for applying better windowing techniques for a clear spectral representation of the noise power for the noise source. Nevertheless, due to the fact that the signal was filtered, the noise contribution due to spectral clustering techniques[Wauthier et al., 2012] is identifying the discontinuities associated with the noise and map the discontinuities on the surface. We also identify the scallop noise [Schmid, 2012] using an exponential decay term due to vibrations to provide an improved method for development of conditions of low SNR for noise signal parameter estimation in the tram rail tracks.

3.Structural Rehabilitation of Rails and various associated problems

Experimental modal analysis (EMA)[Allemang,1999] is a process whereby the modal properties of a structure are extracted. Operational modal analysis (OMA) [Mohanty and Rixen,2004] is a technique whereby the modal properties of a structure are extracted. The OMA technique extracts the modal properties of a structure by analyzing its dynamic output response to its operational input excitation. In other words, OMA analyzes a structure's response based solely on ambient loads that it is exposed to [Karbhari et al., 2009]. The lack of a need to generate forced input excitation significantly reduces the cost of Structural Health Monitoring [SHM] and allows for SHM to be performed while the structure remains in operation. For civil applications, OMA has become a very appealing technique for the extraction of modal parameters to be used in long term vibration-based SHM studies. OMA techniques are usually classified as frequency domain OMA or time domain OMA designed to implement as set of empirical tools or rule based functions to extract a specific number of modal frequencies from the structural response. The empirical function designed accepts inputs which are acceleration time history vector, the data sampling frequency and the number of modes desired from a data sample recorded as acceleration. The spectral characteristics obtained using the power spectrum curve are noise dependent and extracted for each node by first computing the Fast Fourier Transform of the acceleration time history signal. The transformed signal is multiplied by its complex conjugate and scaled to get the value by the number of samples points contained in the acceleration time history signal. We find the peaks in the power spectrum plot corresponding to the natural frequencies of the structure, with the largest peak corresponding to the fundamental frequency. In general, as the mode frequency increases, its corresponding peak in the power spectrum curve will decrease in response to Gaussian white noise excitation [Zhu et al., 2013].

4.Damage Detection Techniques Used So far

Modal parameters may be extracted from a dynamic analysis of a structure in order to serve as inputs to a vibration-based damage detection technique. The types of modal parameters used can be modal frequencies and/or modal shapes. Methods that are based on modal frequencies have the inherent assumption that any shifts in the natural frequencies of a given structure indicate that a change in its structural properties has taken place. The advantage of using modal frequencies is that it allows for damage identification to be performed with relatively easy implementations. However, most algorithms that make use of the modal frequencies for damage detection can only provide a qualitative estimation of damage severity and no spatial information on damage.

4.1 Damage Detection based on matrices

Damage detection algorithms can make use of certain structural properties in order to

determine the condition of a structure. Matrices of these structural properties are built from

structural parameters and can be used to identify, locate and quantitatively estimate damage.

These so called matrix methods compare the undamaged to the damaged system matrices in order to detect damage. Typical matrices that are built and used in damage detection are the

stiffness, flexibility, and/or damping matrices.

4.2 Damage Detection based on Machine Learning

Machine learning algorithms are used for damage detection when there are measurement or parameter uncertainties and/or insufficient information. Machine Learning algorithms can be divided into two categories, either supervised learning or unsupervised learning. If data from both the undamaged and damaged state is available, then supervised statistical learning algorithms are implemented. Supervised learning algorithms infer a classifier or regression function from the given training data, which can be the undamaged data features extracted from the structure of interest. Learning algorithms such as artificial neural networks or genetic algorithms are common in the literature. However, if only data from the undamaged state of a structure is available, unsupervised learning algorithms are implemented. Unsupervised learning algorithms must use data mining methods such as clustering or blind source separation as no training data exists to evaluate a potential solution.

4.3 Natural Frequency Detection for rails and corresponding Mode Shape

Our Method is a multi-variate statistical technique to detect the natural frequency of the rail.

In our analysis, we apply a number of signal analytical techniques for time domain decomposition for the identification of the components that may help us to identify whether the tram rail would require a retrofitting or replacement. Material testing is not required if the real time accelerometer data can be used to prove where and when the acquired signal is varying and when the acquired signal is non varying. Based on experimental modal analysis, we also extracted the frequency and mode shape of the structure, since this may be useful for determining and monitoring the condition of the said structure. It is found that the sampling rate for the accelerometer used is the same and has a sampling rate of 10KHz. In the proposed work of Phase II project [1], comparative study of the results of measurements taken before and after rehabilitation of tramways to extract vibration values based on time series measurement are more influenced by local characteristics than the type of tram or its inertial mass of the tram rail. Our analysis is made of three components which are identification of the frequency to extract the material aspect of the noise variation due to tram rail and to identify at the same time the prior covariance the signal is likely to have with the traim rails to distinguish each tram signal and finally to extract the dynamic response of the tram rail against the movement of the tram by the analysis of the spectral representation of the tram rail noise. These has been done using two models involving blind signal separation and windowing technique design based on noise amplitude enhancement for the tram rail to identify the change in the moment of inertia of the tram supported beam based on identification of the frequency of the signal detected on the rail. In this study we have applied machine learning based algorithm for feature extraction and source separation problems used to separate useful signal from a mix of signals of interest that remains unvarying with time other than signals and noise. A sudden increase in vibration levels can be an indication of a significant weakening of the structure rail. The travel speed excitation frequency equals the natural frequency of the base-rail-tram system, the system being quasi-resonant mode. We often lack priori knowledge of the structural properties of the tram rail and frequency of vibration induced as time dependent signal. This signal recorded is the accelerometer signal is often a mixture of signal noise in tram rail and frequency induced as a result of the tram movement. In the proposed work, we have applied a windowing technique for the analysis of the noise spectral distribution series to be transformed is multiplied by a window function before the FFT is done. Spectral characteristics of the noise data amplitude is reasonably reduced during frequency translation [Boubela et al., 2013] and windowing as most of the uncorrelated harmonic data is lost and the power spectral density calculated is lesser than the actual spectral estimation. As we apply the signal to calculate the low noise power generation, there is a chance that when we apply frequency translation, amplitude of the noise for the spectral detection would not work rightly. This is a huge disadvantage in the present scenario and there is a need to extrapolate data terms based on the amplitudes and spectral components of the noise signal data for identification of the conditions of low SNR for sinusoidal signal parameter estimation, for precision and stability of noise parameter enhancement. In the case of tram signal identification, this is a basic requirement as the number of sources that produce tram rail vibrations is known in priori but the relation and mutual relationship between the noise signals is hard to identify. Thus on applying a frequency translation, many of the recorded and unrecorded noise data for the tram may be lost from estimation. We propose to use signal estimation based on identification of the mean square error increase as the signal to noise ratio varies and then identification of the noise power spectral density for all frequency bands in the noise signal. Prior to this technique, noise frequency translation for spectrum representation was increased by coherent gain for noise spectrum identification [Vaseghi, 2013] and a correction factor was applied using Blackman Harris [Harris, 1978] algorithm to identify more number of signal sub segments in the noise signal. Leakage effect occur when a Discrete Fourier Transform is performed over a Blackman Harris time window.
Even if the window contains integer multiples of a period erroneous spectral components occur. The further the window size is away from an integer multiple of the contained frequency, the stronger this frequency "leaks" into neighboring frequencies in the spectrum; width and especially amplitude of falsely detected frequencies increase. The second aspect of the research is to identify the covariance’s and classify them for health monitoring and structural health monitoring of tram rails by the identification of the sinusoids in the signal sub space using Eigen value decomposition of the correlation matrix in order to plot the sinusoids in the noise signal data for harmonics. In Blind Signal Separation(BSS) [Belkin et al., 2006, Gribonval and Lesage, 2006] weakness of the prior information is precisely the strength of the BSS model, making it a versatile tool for exploiting the ‘spatial diversity’ [Cardoso,1998]] provided by an array of sensors. Promising applications has already been found in the field of processing of communications signals [Lyons, 2010].

Frequency Analysis and Source signal separation

Analysis of the Signal based on the frequency generation is an important aspect of the simulation algorithm designed for the purpose of the tram track designed identifying when it will cross the natural frequency of the guard rails. It is therefore necessary to identify all the components associated with tram tracks to define the procedure for signal separation of the components.

It is therefore imperative to define certain rule based features including Independent Component Analysis and likelihood identification that is good in the purpose of solving the problem of signal distinction and frequency components generated as a result of signal vibration in the tram tracks. A sudden increase in vibration levels can be an indication of a significant weakening of the structure rail that can sustain the prevailing shock and vibration caused by discontinuities caused by the impact rails.

Windowing and period gram analysis

5.1 Application of Windowing Mechanism

PSD is a description of the average power of a signal distribution with frequency. And total power in a specific frequency band can be calculated by integration (summation) over the frequency band. The Periodogram[Kurtz, 1985] graphs a measure of the relative importance of possible frequency values that might explain the oscillation pattern of the observed data. The Welch method provides a variation of the averaged periodogram. Recall that the periodogram is just the average, magnitude squared of the DFT of the signal. The Welch method differs from the averaged periodogram in two ways. First, the data is windowed and secondly, data blocks are overlapped. The data window reduces spectral leakage and by overlapping blocks of data, typically by 50 or 75%, some extra variance reduction is achieved. The Welch PSD estimator uses the DFT. However, it does it in a clever way (overlapping samples) such that the PSD estimates, depending on the data set, are often better than those obtained with the DFT. In our present study, time history analyses the structural response is computed at a number of subsequent time instants. In other words, time histories of the structural response to a given input Response-spectrum analysis (RSA) is a linear-dynamic statistical analysis method which measures the contribution from each natural mode of vibration to indicate the likely maximum response of an essentially elastic structure. Response-spectrum analysis provides insight into dynamic behavior by measuring pseudo-spectral acceleration, velocity, or displacement as a function of structural period for a given time history and level of damping. It is practical to envelope response spectra such that a smooth curve represents the peak response for each realization of structural period. Response-spectrum analysis is useful for design decision-making because it relates structural type-selection to dynamic performance. Structures of shorter period experience greater acceleration, whereas those of longer period experience greater displacement as tram rails. In our present approach, we try to apply response spectral analysis through input time series data for the tram rails.If we consider X(j) as the segments where N=1 be the sample from stationary second order Xi(j)=X(j) where j=0 and L=1. The power spectral density can be realized as a function with the help of the periodogram whereby the spectral estimate is the average of these periodograms . We can apply a reduction of bias and variance depending on the value of 1/K for non overlapping segments when the value of N is large

Reduction of variance by a factor of 1/K is done for non overlapping segments using the summation for the variances. We apply a linear operator using a stochastic differentiation equation where the present values are linearly related with the past values and a additive stochastic shock is formed.

In our proposed mechanism for the study of noise subsequences, we have observed data at n distinct time points, and for convenience we assume that n is even.  Our goal is to identify important frequencies in the data.  To pursue the investigation, we consider the set of possible frequencies ωj = j/n for j = 1, 2,…, n/2 which represent harmonic frequencies.

We will represent the time series as

xt=∑j=1n/2[β1(jn)cos(2πωjt)+β2(jn)sin(2πωjt)].

This is a sum of sin and cosine functions at the harmonic frequencies where β1(j/n) and β2(j/n) as regression parameters.  Then there are a total of n parameters because we let j move from 1 to n/2.  This means that we have n data points and n parameters, so the fit of this regression model will be exact. The first step in the creation of the period gram is the estimation of the β1(j/n) and β2(j/n) parameters.  It’s actually not necessary to carry out this regression to estimate the β1(j/n) and β2(j/n) parameters.  Instead a mathematical device called the Fast Fourier Transform (FFT) is used.  After the parameters have been estimated, we define

P(jn)=β^21(jn)+β^22(jn)

This is the value of the sum of squared “regression” coefficients at the frequency j/n denoting scaled periodogram using correction factors is essential to plot P(j/n) representing the power spectral density of the noise subspace versus j/n along x axis as the observed frequency for j = 1, 2, …, n/2.

5.2 Reading signal RMS values out of a Matlab pwelch periodogram

5.2.1 pwelch with standard parameters

Given a signal x, one can plot a one-sided periodogram using the command pwelch(x) or, if data instead of a plot are required, [Pxx,f]=pwelch(x). This will cut up the signal into eight segments, with 50 % overlap between the segments. Hence, if x is a signal of length 1’000’000, then the length of the single-sided spectrum will not be 62501, as one would expect, but 131073. In practice, we obtain this number by actually executing [Pxx,f]=pwelch(x) and size. Hence, to read a signal magnitude out of a plot generated with pwelch, one would let pwelch calculate the two vectors [Pxx,f] and then determine f as f(2)-f(1). Using this scaling, we can read the RMS-value of a deterministic signal directly off the plot. Depending on the frequency of the signal relative to the centre frequency of the bin showing the highest value, and depending on the window used, there can be read-off errors up to several dB. This effect is called scallop loss and sometimes also picket fence effect. We chose the signal so the 0 dB visible.

5.2.2 pwelch with a different window

The previous example is equivalent to pwelch(x,hamming(1000)). A different window can simply be used by writing pwelch(x,hanning(1000)) or pwelch(x,blackmanharris(1000)) to get a Hanning window or a Blackman-Harris window to reading noise values off the plot, so in order to determine signal levels respectively.

5.2.3 pwelch without overlap

pwelch(x,hanning(1000),0) will set the overlap to zero signal values. For a signal of length 1’000’000, this will reduce n from 1500 above to just 1000.

Experimental results

We formulate the problem of BSS as one of solving a generalized eigenvalue problem [9], where one of the matrices is the covariance matrix of the observations and the other is chosen based on the underlying statistical assumptions on the sources. This view unifies various approaches in simultaneous decorrelation using the using prior probability approach for supervised learning methods. Independent component analysis is highly directional and depends on the eigen values. The source signals which are extracted as noise and signal of the train are independent and non-gaussian. The typical use of Independent Component Analysis [Parra and Sajda, 2003] is centering, whitening and dimensionality reduction as pre-processing. The ICA however cannot identify the actual number of source signals but we can ensure that whitening can be done for static signal processing as linear transformation between the signals into independent components measured by some factor of independence. Hence, we need to identify vibrations generated by the running rail is an important component of the noise excitation. We first need to demonstrate that whether for a statistical mixture, the signal satisfies the various statistical assumptions, the different choices for the noise causing the un-mixing of the values. We found that if the damping values are increasing beyond a certain point mixed signal satisfies all the three components of the statistical estimators including non-stationary, non-white and non-Gaussian. The recovered matrix was found to be independent for all values of the noise subspace. Figure 2 shows when the correlation subspace was identified as a matrix that can define the signal detection and isolation for the frequency range and design a scenario for feature extraction of the required points for change in stiffness of the tram rail. These noises are due and existing sources on the vehicle (engines, transmissions and other mechanical components). Obviously, vibrations generated by the running rail is an important component of the noise excitation.

NATURAL FREQUENCY AND PEAK DETECTION ANALYSIS

7.1 TIME DOMAIN IMPLEMENTATION RESULTS

7.1.1 MODEL DESCRIPTION

The data was taken by a simulated signal where the frequency has been varied from 30Hz-60Hz whereby the signal was simulated and an additive noise was added to the signal only to separate the signal from the noise. We first calculate the error variance of the signal and then compute the covariance matrix to calculate the eigen value and eigen vectors using kurtosis method[Cardoso,1990]. Finally, we apply the whitening matrix and apply an orthogonal transform [Comon,1992] to extract the main signal from the noise vibration. We then subtract the noise from the main data as the noise is additive for non-Gaussian and no-stationarity to identify the noise. The following observations have been made in our study to yield the spectral characteristics of the signal which propounded new research avenues.

1. The sensors placed on the rails observed tram data movement of the rails and found that maximum amplitudes occur in the range 40-50 Hz for all three types of trams across all points in the city.

2. Many of the spectral components in the rail show that the frequencies do not cross 300 Hz.
3. In all type of trams, the recorded value for ultra-low frequencies in the spectrum do not extend beyond the observed frequencies above 200Hz.
4. In one instance it has been found by sensors which recorded for point of observation located on Dacia bridge that the spectral peaks are observed at a frequency of about 4 Hz are sometimes accompanied by harmonics. This data set is the observation of the noise frequency by the sensor as a result of vibration in the tram lines which proves the need for a function to identify the natural frequency of the tram lines.

Figure 2: Signal separation into tram movement and rail vibration

In order to read signal values directly off the plot; second, to read the noise power spectral density directly off the plot; and thirdly to quantitatively determine the power in any frequency band by adding the values of all bins in that band. We measure the power spectrum curve for each measured value by first computing the Fast Fourier Transform[FFT] of the noise data time history signal. If we want to analyze peaks in the noise floor, then we need not optimize the FFT window as every single frequency bin of the transformed signal is a linear combination of N time samples. If the signal to be transformed is a sine function, then, ideally, all these N samples add up in one bin and cancel out in all other bins. We do not need to measure or simulate a longer data sequence or rather a distribution. For the purpose of deconvolution of the frequency sets, we multiply the transformed signal by its complex conjugate and scale the value by the number of samples points contained in the noise time history signal. Once the power spectrum curve is obtained, the proposed algorithm finds peaks correspond to the natural frequencies of the system. Lastly, the resultant frequency values (in radians per second) that correspond to the power spectrum curve, fs returns a frequency vector, f, in cycles per unit time gives the signal component in the noise rails. An algorithm design for the entire work process is shown below as in Figure 3 for the spectral extraction of noise signal. A noise spectrum suffers from several losses of the harmonic components due to frequency translation done using Direct frequency translation during the process of windowing. In order to apply for corrections in windowing before spectrum sensing, we need to achieve either resampling of the noise data characteristics and so windowing before FFT decreases the amplitude compared to the non-windowed data. Estimation of the Power spectral density of signals series by classical methods cannot be done directly from the time series decomposition of the acceleration signal. Instead, it requires resampling to achieve uniform time intervals. This resampling, required in order to use the well-known methods of PSD estimation of evenly sampled signals, introduces low-pass filtering and possible artifacts in the estimated spectrum. For noise and noise stationary data analysis and frequency domain decomposition, we need to apply non-parametric method and so we have applied periodogram and Welch’s method for better extraction of the noise components into shorter overlapping time windows for extraction of the correlation in the noise subspace. Windowing any time series is affected by leakage in the noise correlation as the frequencies are not known. As the natural frequencies of the signal and the resolution Fs/N (sampling frequency to the number of sample) such that frequencies will be all multiple of this resolution and there will be leakage. There is a 1/N term in the DFT/FFT formula in order to estimate the amplitudes of spectral components of the signal, where N is the transform length. In case of signal sensing by using FFT, windowing decreases the amplitude of the spectral components. For windowed DFT/FFT, we replaced the average sampling rate N into a more generalized term for identification of the aggregate of the summation of the windows that correspond to the noise data. We need to calculate the damping due to the source and Delta defines the amount of damping and is usually defined as the logarithmic decrement, which is the natural log of the ratio of any two successive peak amplitudes in free vibration. With the base displacement and base acceleration analysis types static loads only have an inertial effect on the structure. With the harmonic and amplitude analysis types, the static loads applied within the time history case are varied dynamically according to what is specified. It is useful to know that a time history, convolved with the transfer function of s single degree of freedom (sdof) system is what produces the response spectrum. The frequency axis of the response spectrum is not ordinary frequency but rather the natural frequency of the sdof system. The procedure simplifies downstream estimation of maximum response and allows combination of multiple time histories. In essence, what is needed is simple modal analysis, the multiplication of the modal response by the response spectrum amplitude at the natural frequency of the computed modes. However there is an inherent spectral leakage associated with the system which will be comprised of noise and other components.

Figure 3: Program Execution of spectrum analysis and non-parametric analysis and signal noise detection

In order to avoid this leakage, we have to develop a methodology as the proper spectral representation based on amplitude of the signal is very important. We add a signal obtained by periodogram to estimate the power spectral density signal s1 and then find the power spectrum corresponding to the maximum variance.

–(2)

The mean square error is given as

(3)

We extract further information on signal and noise levels using through the use of windowing using Pwelch function.by the use of periodograms in the noise data matrix. It is found that the RMS Value of the acceleration was already recorded and values recorded were having less significant fluctuations. We apply a Blackman Harris window function as the noise levels were found to be very small recorded at 0.05 and 0.13 of the RMS values for Phase 1.

RESULTS

Our effort in this paper was directed to identify the characteristics of the acquired data from the accelerometer for the tram. The signal acquired has been separated using machine learning algorithm of Independent component analysis of Kurtosis method to separate the signal of the tram rail and the noise. A set of measurements was made for each of the tram data sets with sample points from 1.1C-N and gol, 2.1 C,N and gol and further data recorded from 3.1-7C,N and 4,5,6,7 as shown in Table 1.

Table 2: Elaborate results for outputs resultant of the Tram rail vibration and associated frequency from Power Spectrum

The mechanism of detection of spectrum estimation and noise detection process has been updated in this piece of study using a parametric filter development for the detection of the frequency of the rail and finally to identify the characteristics of the noise signal spectrum and presence of sinusoids or harmonics in the noise data for the application of welch algorithm with correction factor. In all this cases the sampling frequency is 10 KHz.There are three ways to normalise the resulting spectrum, depending on how one wants to use the PSD: first, to read signal values directly off the plot; second, to read the noise power spectral density directly off the plot; and third, to quantitatively determine the power in any frequency band by adding the values of all bins in that band. We apply non parametric methods to acquire signal directly from Power spectral density. The most common spectral estimation are periodogram and blackman-tukey[Comon, 1994] method. The Welch’s [Welch, 1967] method however yields higher resolution than the other non-parametric methods. We model the noise data to estimate the frequency of the vibration of the rail as a linear system to identify the frequency of the rail and also identify the mean square error for the rail. Auto-regressive methods is used to give accurate spectra for data where we need to identify peaks for identification of data with large peaky data as shown in the Figure 4. A pre-processing Kalman filter using an auto-regressive parameter estimation was also designed [Dutta and Mishra, 2016] for the purpose of identification of missing measurements in the seismic data set in our previous study. However, as the associated non linearity’s in the noise data characterization cannot be known apriori, a similar data set is hard to define. The frequency of the rail determined using the same process is found to be more effective as it will identify sets of linear equations. If we want to be able to read the root mean square value of deterministic signals from an FFT plot, we have to divide the FFT by N times the coherent gain and then calculate the power spectral density. So for an input signal of the noise we were able to acquire the power spectral density map, find out the value of the power corresponding to the maximum horizontal frequency for the rail and also identify mean square error for the signal x. We were able to identify the peaks in the noise spectrum plot correspond to the irregularities in the structure, with the largest peak corresponding to the fundamental frequency as shown in Figure 5.

Figure 4: Power spectral Density estimation for data 4.2_N_56_cg.mat using Pwelch

We consider the decay of the signal to be an exponential decay and then we average the value. The sample points of the noise signal shown that the maximum nodal frequencies located at the peak times of the tram noise data. The amplitude of each point is computed as the inverse of the time difference between consecutive peaks and is placed at the instant of the second peak.

Figure 5: Identification of peaks in the unevenly sampled noise data from rail for sinusoids.

Conclusion

Our work is a useful tool for the purpose of defining the process of signal noise extraction and noise power spectral density measurement from the vibrating rails. We are able to define the natural frequency of the tram rails and also identify the spectral components of the power spectral density from the frequencies of the rail tracks. In this work we have presented a detailed analysis of the frequency analysis of the tram rail vibration by analysis of the power spectrum estimation of non-stationary signals. We have developed analytical expressions to find the performance of the estimate and we have corroborated this study applying Independent component analysis based on signal separation and information detection for the estimation of the power spectral density for the spectrum of the tram data.We have done a computational approach to observe the spectrum signal components for frequency ranges higher (300 Hz) for vibration caused by the rolling of wheels on rails and resonance wheels caused by frictional forces are minor and occur only sporadically compared to the vibrations of low frequency generated type of impact forces. In particular, we have taken measurements along 7 spots along the Oradea city for each different type of tram and identified the nature of the power spectral density and frequency content of the noise which corresponds to the tram rail vibration. Estimation of the signal characteristics for noise is a difficult proposition as most of the signal spectra including the noise amplitude due to the sinusoids and harmonics are lost due to signal averaging when we apply frequency translation. In order to better estimate the power spectral density for the associated tram rail measurement for the accelerometer, we apply a number of machine learning approaches which gives us a better spectral estimation. Our theoretical predictions for non- linear spectral estimation for noise were confirmed experimentally using simulated and real data acquired signals. We were also able to apply a parametric signal processing for the extraction of the frequency of the rail vibration when the tram passes the rail and compare with the actual data. In place of resampling the noise data, we applied a welch correction factor to reduce distortions to correct the amplitude of spectral components. We have found a periodic behavior of the noise spectrum, which corresponds to the frequency of the mean sampling period. For higher frequency, auto-regressive estimators behave better that linear resampling, but the estimation with parameter estimation approaches are better than the FFT as the mean noise frequency increases. We have also designed the methodology for identification of the power spectrum energy loss during windowing. As the plausible approach, we find that we were able to enhance the output noise signal by applying a summation of all components of the noise signal than frequency averaging using a Direct frequency translation and the mean square error is lower. Further using a Blackman Harris power correction, we increased the resolution of the sub segments of the signal before normalizing them or applying the Welch Function. This approach is suitable in the future in identifying signals where signal separation is difficult and identification of the upper noise level is a difficult approach to identify cross correlations to identify the eigen vectors in the signal sub-space and identify the coefficients from the correlation matrix and design a band pass filter in future works around the given input frequency that are more than 3% above or below the band-pass frequency will be attenuated.

Acknowledgment

The paper published has been sponsored under the Erasmus Mundus partnership program agreement vide number 2014-0855/001-001 coordinated by and between University of Oradea and City University of London Under Action Plan 2 for the year 2016 for Post-Doctoral Fellowship Programme under the Supervision of Prof Dr Radu Tarca from University of Oradea. This work was conducted with the guidance from Prof. Tiberiu Vesselenyi with report Phase II and data acquired as part of the service contract between RA OTL RESEARCH no. Reg. 155 of 10.06.2010 and the University of Oradea no. Reg. 8072 of 25.06.2010 with new set of data acquired also for OTL, Oradea, Romania.

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