Compressive sensing for vibration signals in high-speed rail

monitoring

Yi-Qing Ni* and Si-Xin Chen

Hong Kong Branch of National Rail Transit Electrification and Automation

Engineering Technology Research Center and Department of Civil and Environmental

Engineering, The Hong Kong Polytechnic University

Hong Kong

+852 27666004

ceyqni@polyu.edu.hk

Abstract

The safety and reliability are critical for the high-speed rail system, and axle box

accelerations are often utilised for inspections of railways. Due to the Nyquist theorem,

there is a compromise between the resolution of defect detection and the amount of

recorded data. As an emerging technique, compressive sensing creates the opportunity

for sub-Nyquist sampling as long as the target signal has a sparse representation in a

known domain. To make use of this advantage, this study proposes a compressive

sensing framework for high-speed rail monitoring. In particular, the process of

compressive sensing is simulated using the axle box acceleration data acquired from a

high-speed train ran on one section of railway in China. The compressed measurements

are received by random projection, and the original signals are reconstructed using

convex optimisation algorithm. Based on the reconstruction results, the influence of

different measuring methods as well as orthogonal bases is evaluated. In addition, a

regression model is formulated to give a recommend equivalent sampling rate according

to the sparsity and the desired accuracy requirement of the target signal. It is found that

the vibration signals are sparser in the discrete cosine transform matrix, leading to better

reconstruction, and the performance of different measuring methods is almost identical.

More importantly, this study proves that the high-speed rail monitoring data can be

satisfactorily obtained through proper sampling rates lower than the Nyquist theorem

requires.

1. Introduction

High-speed rail (HSR) has become an essential component of transportation systems

due to its economic, environmental, and quality-of-life benefits (1)

. It is supported by

European Union, Korea, and China and viewed as the next growth economy wave (2)

.

As the safety and reliability are primary concerns of the high-speed rail system, the

knowledge of track conditions is critical for making a suitable maintenance plan and

performing grinding operations where and when required (3, 4)

. Among various methods

for the track inspection, making use of the axle box accelerations is one of the most

promising, which enables detecting and identifying some singular track defects such as

squats (5, 6)

, bolt tightness of fish-plated joints (7)

and other short track defects (8)

. Other

works also applied axle box accelerations for the detection of rail corrugation (9, 10)

.

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In practice, sampling frequencies are usually lowered down to several kHz (11)

for

speeds up to 300 km/h to avoid a vast amount of data acquired. However, the Nyquist

theorem presents the drawback that those track defects whose excitation frequency is

higher than half of the sampling frequency are not detected. Therefore, there must be a

compromise between the resolution and the amount of recorded data. As an emerging

technique, Compressive Sensing (CS) creates the opportunity for sub-Nyquist sampling.

CS theory confirms that the target signal can be recovered as long as it has a sparse or

compressible representation in a known domain (12, 13)

, and the number of measurements

required is governed by the sparsity degree rather than the bandwidth of the signal.

A number of researchers have studied CS for structural health monitoring (SHM). Bao

et al. explored the use of CS to address data loss common in wireless sensor networks

and achieved more robust signal collection when compressed measurements rather than

time-history samples are lost (14)

. More recently, they demonstrated physical realisation

of CS on the Imote2 wireless sensor using a random demodulator (15)

. O’Connor et al.

explored using CS to reduce power consumption in wireless fatigue life monitoring of

ship hulls (16)

and achieved significant energy reductions in acquiring mode shapes of

the Telegraph Road Bridge (17)

. Studies that combine CS with damage detection have

also existed: Mascarenas et al. showed a CS application for SHM damage detection

using a laboratory testbed structure with a digital prototype of a compressed sensor

embedded into a microcontroller (18)

while Haile et al. studied the use of CS in SHM for

damage detection in composite materials (19)

.

However, there hasn’t been any research that applies this technique to high-speed rail

monitoring. Therefore, this study aims to propose a compressive sensing framework for

HSR monitoring, so that through lower sampling rates the same resolution of defect

detection can be achieved. The process of compressive sensing is simulated using the

axle box acceleration data acquired from a high-speed train ran on one section of

railway in China. The effectiveness of this method is verified by the comparison of the

reconstructed signals and the original ones. Based on the results, recommendations on

the orthogonal basis, the measuring method and the compression ratio in engineering

practice are given.

2. Compressive sensing

The measurement vector My R∈ is acquired by a linear projection of the discrete-time

signal ( )Nf R N M∈ > :

y f=Φ . (1)

When f is represented in terms of an N by N orthogonal basis matrix Ψ with the

basis vectors { }i

ψ as columns, the problem becomes

y = ΦΨx = !Φx (2)

where !Φ = ΦΨ is an M by N matrix called sensing matrix.

Typical orthogonal bases include the wavelet basis (20)

, the discrete Fourier basis (21)

, the

discrete cosine basis (22)

, the curvelet basis (23)

and so on. Besides, redundant dictionaries

also work well in compressive sensing (24–26)

.

Although the problem of recovering the representation x from y with M components

is under-determined, it has been proved that x can be recovered exactly under the

following conditions (27, 28)

:

3

1. The representation x is sufficiently sparse, where a vector is defined as S -sparse

if it has at most S non-zero entries;

2. The matrix !Φ obeys a “restricted isometry property” (RIP).

RIP impose a restricted orthogonality condition on !Φ which enables exact recovery of

sparse x from y . To explain RIP, the S -restricted isometry constant S

δ of the matrix

!Φ given in (29)

is defined as the smallest number such that

(1−δS) c

l2

2

≤ !ΦTcl2

2

≤ (1+δS) c

l2

2

(3)

holds for all subsets !ΦT

with {1,..., }T N⊂ , T S≤ and coefficient sequences ( )j j Tc

∈.

The S-restricted isometry constant shows how close subsets of !ΦT

are to an orthogonal

system when restricted to sparse linear combinations ( T S≤ ).

Focusing on the case where x is sparse, it is desired to find the sparsest solution of !Φx = y and solve

0ˆ argminx x= subject to !Φx = y . (4)

Although this 0l problem has been proved to have a unique S sparse solution if the

minimum requirement that 2

1S

δ < is satisfied (29)

, it is a hard combinatorial problem

that cannot be solved by any algorithm other than brute force search. If a stronger

condition that 2

2 1S

δ < − is satisfied, the convex relaxation is exact and the solution

of the 0l problem can be obtained by solving an

1l problem

(29):

1ˆ argminx x= subject to !Φx = y . (5)

The 1l problem, also named Basis Pursuit

(30) can be recast as a linear programming

problem with computational complexity ( )3O N(31)

.

When the compressed measurements are contaminated with noise Me R∈ bounded

2le ε≤ , the problem can be rewritten as

1ˆ ˆargminx x= subject to !Φx̂-y

2

2

≤ ε (6)

and solved as Second-Order Cone Programs (SOCP) (32)

.

The original signal can be recovered using the optimal basis coefficients x̂ : ˆ ˆf x= Ψ .

However, the convex optimisation is not the only way to reconstruct sparse solutions.

There are at least other two common classes of computational techniques for solving

sparse approximation problems:

1. Greedy pursuit iteratively refines a sparse solution by successively identifying

one or more components that yield the greatest improvement in quality (33, 34)

;

2. Bayesian framework assumes a prior distribution for the unknown coefficients

that favours sparsity first, then develops a maximum a posteriori estimator that

incorporates the observation and finally averages over most probable models (35)

or identifies a region of significant posterior mass (36)

.

Although Ψ is fixed according to the characteristic of the target signal, it is known that

trivial randomised constructions of Φ will enable !Φ to satisfy RIP with overwhelming

probability (12)

. Examples include Gaussian random matrix (where entries of Φ are

4

independently sampled from a normal distribution with mean 0 and variance 1/M ) and

binary random matrix (where entries independently come from a symmetric Bernoulli

distribution ( , / ) 1/ 2P i j Mφ = ± = ).

!Φ can also be constructed by selecting M rows from an N by N orthogonal matrix

uniformly at random, where Φ randomly sub-samples the target signal and Ψ maps the

time domain and the selected domain (12)

.

3. Compressive sensing for high-speed rail monitoring

3.1 Data acquisition

CNERC-Rail (Hong Kong branch) was authorised to monitor the vibration response of

an operating train on Lanzhou−Xinjiang high-speed rail line. The monitoring work

lasted for about one month and mainly inspected the vibration of bogie and car body.

The monitored bogies located in the axis 5, axis 7, axis 8 of car 3 and axle 6 of car 4 and

the accelerometers are installed at the frame, vertical-stop component and axle box

(Figure 1) with the range of ± 1000 g and the sampling frequency of 5000 Hz.

Figure 1 Locations of the accelerometers

As the purpose of this study is to achieve the same resolution of defect detection with

fewer measurements, compressive sensing was applied to the vertical acceleration of

two axle boxes. To ensure the representativeness, 32 signal segments with 5000

components were selected from different time (morning, afternoon and evening) of two

days and under various scenarios (ordinary railway, crossing bridges and crossing

5

tunnels). The signals were obtained when the train achieved the cruising speed

(approximately 200 km/h) so that the influence of train speed was normalised.

3.2 Procedures and implementations

Although the process of acquiring the compressed measurements was simulated on the

computer rather than realised at the sensor, the data received in conventional ways can

be utilised for comparison. Firstly, the compressed measurements were obtained by

projecting the original vibration signal onto the selected M by N matrix Φ . With the

prior knowledge that the vibration signal is sparse or compressible in a specific

orthogonal basis, convex optimisation techniques were then applied to reconstruct the

target signal in the sparse domain and the time domain. Finally, the reconstructed signal

was compared with the original one for verification purposes.

One way to obtain the compressed measurements is to project the target signal onto the

Gaussian or Bernoulli random matrix Φ . These measuring methods have been achieved

physically using special sensors, so further experiments are feasible. Gaussian random

projection has been performed by a method called random demodulator (RD) (37, 38)

and

Bernoulli projection has been achieved in single-pixel camera (39)

by spatial light

modulators. Another way is to randomly sub-sample the target signal, which can be

realised by randomly triggering the ADC (17)

.

As the sampling rate of the data acquisition system is 5000 Hz, it has been determined

that the vibration responses are band-limited to 2500 Hz. However, by inducing the

technique of compressive sensing, signals with the same bandwidth can be acquired by

a sampling rate equivalently lower than the Nyquist rate.

The number of compressed measurements M was set as 40%, 50%, 60%, 70%, 80% of

N for different trials and /M N was defined as the compression ratio. The discrete

cosine basis (DCT) and the discrete Fourier basis (DFT) were selected as orthogonal

bases Ψ so that the sensing matrix !Φ can be obtained. Based on the compressed

measurements, the l1-magic package (40)

was used to reconstruct the target signal.

4. Results

4.1 Sparsity level of signals in different bases

The sparsity level of the signal in different domains can be known beforehand as the

complete time history is recorded, although in real life the orthogonal basis is chosen

according to the characteristic of the target signal without knowing the exact sparsity

level. In this study, the sparsity level is calculated as the ratio between the number of

zeros and the segment length. Indeed, as the vibration signal is contaminated with noise,

none of the coefficients of each data segment is originally zero. To generate spectrally

sparse signals, those coefficients that have a value smaller than 1% of the maximum are

eliminated from the spectrum. The statistics of sparsity level is defined as 0.01/

sL N N= .

Figure 2 gives a 1-second segment of an acceleration signal in the time domain as well

in other domains. The sparsity level is 63.6% and 82.2% when the signal is represented

by the DCT and DFT bases, respectively.

6

Figure 2 One segment of the acceleration signal in three domains

The same investigations are also done for other signals. Figure 3 shows the comparison

of average sparsity level in two orthogonal bases. It is shown that the axle box vertical

acceleration signals are sparser in the discrete cosine transform domain, where at least

60% of coefficients are close to zero.

The reason why the DCT general performs better for sparse coding of signals than the

DFT is explained as follows. The DCT implies different boundary conditions from the

DFT or other related transforms: the DCT implies an even extension at both left and

right boundaries while the DFT extend the original signal periodically to positive and

negative infinity. As any random segment of a signal is unlikely to have the same value

at both the left and right boundaries, discontinuities usually occur using the DFT. It is

well known that the smoother a discrete signal is, the fewer coefficients in its DFT or

DCT domain are required to represent it accurately.

Figure 3 Sparsity levels of vertical accelerations in different bases

4.2 Influence of sparsity level on reconstruction accuracy

The metric of residual sum-of-squares (RSS), normalised by the traditional Nyquist

sampled signal f , is used to assess the reconstruction error:

7

2

2

2

2

ˆ

l

l

f fRSS

f

−

= . (7)

To investigate the influence of sparsity level on the reconstruction quality, 32 vibration

segments are reconstructed using convex optimisation based on 2500 random sub-

samples of the target signals with 5000 entries. Figure 4 shows the relationship between

sparsity level and reconstruction error and an obvious trend: the sparser the signal is in

the orthogonal basis, the better it can be reconstructed.

Figure 4 Influence of sparsity levels on reconstruction errors

4.3 Reconstruction accuracy via different measuring method

It has been mentioned that the compressed measurements can be obtained by projecting

the target signal onto random matrices or randomly sub-sampling. To investigate the

influence of adopting different measuring approaches, 32 segments of acceleration are

assumed to be sparse in the DCT domain and recovered using convex optimisation and

50% of samples. As shown in Figure 7, the average reconstruction errors using different

methods are almost identical, which means that the selection of Φ does not make a

difference to the reconstruction as long as it is sufficiently incoherent with Ψ . Among

these methods, randomly sub-sampling is preferred as it is more intuitive and easier to

implement.

Figure 5 Reconstruction errors via different measuring methods

8

4.4 Recommended equivalent sampling ratio via regression

The Nyquist sampling theorem states that a signal must be sampled at least two times

faster than its bandwidth in order not to lose information. The theory of compressive

sensing provides a new data acquisition paradigm that the equivalent sampling rate is

determined by the sparsity level in the selected domain instead of the bandwidth. Thus,

a trade-off between reconstruction accuracy and sampling rate reduction can be

achieved. In this study, to estimate the recovery error according to the compression ratio

/M N (i.e. /se sf f when measuring method is randomly sub-sampling) and the sparsity

level of the target signal in selected domain, a linear regression model is formulated:

1 2( / )se se f f sα β β= + × + × (8)

where e : reconstruction error;

s : sparsity level of the target signal;

/se sf f : equivalent sampling rate/ sampling rate of recovery signal.

To learn the parameter α and β in this regression formula, 32 signal segments are

investigated, and their sparsity levels in the DCT basis are calculated. The compression

ratio /se sf f is set as 40%, 50%, 60%, 70% and 80%, which means that sef equals to

2000Hz, 2500Hz, 3000 Hz, 3500Hz and 4000Hz respectively. Signals are recovered

based on the compressed measurements. In this way, 160 experiments with different

configurations are conducted, and 160 results of reconstruction are obtained for the

regression. Finally, the formula becomes:

1.548 0.78 (4 1.00( ) ( / ) 7)se se f f s= + − × + ×− . (9)

This formula gives a recommend equivalent sampling rate based on the sparsity and the

desired accuracy requirement of the target signal. Once the accepted error is determined,

the equivalent sampling rate depends on the sparsity level instead of the bandwidth.

Typical reconstruction results with two error levels (0.20 and 0.30) are illustrated in the

following two figures. The results prove that when the proper compression ratio is set,

signals can be satisfactorily obtained through sampling rates lower than the Nyquist

theorem requires. The adoption of compressive sensing can reduce the sampling rate

and the communication payload of a data acquisition system for high-speed rail

monitoring.

Figure 6 Reconstructed signal versus target signal (M/N = 0.5; RSS = 0.20)

9

Figure 7 Reconstructed signal versus target signal (M/N = 0.5; RSS = 0.30)

Figure 8 illustrates the actual and expected reconstruction errors corresponding to

different compression ratios.

Figure 8 Actual and expected reconstruction errors of different compression ratios

5. Conclusions

This study is the first application of compressive sensing to high-speed rail monitoring

area. Specifically, the process of compressive sensing is simulated on the vibration

signals collected from two axle boxes of a high-speed train, and reconstruction results

are used to evaluate different configurations. It is found that the axle box acceleration

signals are sparser in the discrete cosine transform basis and can be better reconstructed

if they are sparser in the selected orthogonal basis. Additionally, all measuring methods

including projecting signals onto random matrices and randomly sub-sampling enable

similar reconstruction results. The findings are useful for engineering practice.

10

A regression model is also formulated to give a recommend equivalent sampling rate

based on the sparsity and the desired accuracy requirement of the target signal. Thus, a

trade-off between reconstruction accuracy and sampling rate reduction can be achieved.

When the proper compression ratio is set, signals from high-speed rail monitoring with

the same bandwidth can be satisfactorily obtained through sampling rates lower than the

Nyquist theorem requires. This new data acquisition paradigm has the potential to

improve the resolution of defect detection, save the energy consumption and deal with

the issue of data loss in high-speed rail monitoring.

As sensors that can directly acquire compressed measurements from analogue signals

have already existed, it is expected that compressive sensing can be implemented in the

real practice of monitoring rather than simulations. Apart from that, this study only

utilises the prior knowledge that signals are sparse and reconstruct them one by one. To

make use of the common characteristics that they could learn from each other,

dictionary learning technique will be employed in the future work. Most signals are

expected to be sparser in the dictionary learned from a training set, and thus the

reconstruction performance can be improved.

Acknowledgements

The work described in this paper was (in part) supported by a grant from the Research

Grants Council of the Hong Kong Special Administrative Region, China (Grant No.

PolyU 152767/16E). The authors would also like to appreciate the funding support by

the Innovation and Technology Commission of Hong Kong SAR Government to the

Hong Kong Branch of National Transit Electrification and Automation Engineering

Technology Research Center (Project No.: K-BBY1).

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