BLIND SOURCE SEPARATION TOWARDS DECENTRALIZED MODALIDENTIFICATION USING COMPRESSIVE SAMPLING

Ayan Sadhu𝑎, Bo Hu𝑏 and Sriram Narasimhan𝑎

University of Waterloo𝑎 Department of Civil and Environmental Engineering

𝑏 Department of Computer Science200 University Avenue West, Waterloo, ON N2L 3G1

Email-sriram.narasimhan@uwaterloo.ca

ABSTRACT

Wireless sensing technology has gained significant atten-tion in the field of structural health monitoring (SHM).Various decentralized modal identification methods havebeen developed employing wireless sensors. However,one of the major bottlenecks—especially dealing with long-term SHM—is the large volume of transmitted data. Toovercome this problem, we present compressed sensingas a data reduction preprocessing tool within the frame-work of blind source separation. The results of sourceseparation are ultimately used for modal identification oflinear structures under ambient vibrations. When usedtogether with sparsifying time-frequency decompositions,we show that accurate modal identification results are pos-sible with high compression ratios. The main novelty inthe method proposed here is in the application of com-pressive sensing for decentralized modal identification ofcivil structures.

1. INTRODUCTION

Most modal identification methods [13] operate centrallyon measurements obtained from a large array of wiredsensors and process them collectively. Recent advances inMicro-Electro-Mechanical Systems (MEMS) and Wire-less Smart Sensor Networks (WSSNs) have yielded af-fordable hardware that can be rapidly deployed on a largescale [11, 4]. In addition to transmitting data wirelessly,they also contain local processing capabilities. In order toharness their processing capabilities, several de-centralizedmodal identification have recently been developed [21, 17,15]. In long-term monitoring applications, reducing thevolume of transmitted data still remains an issue to be ad-dressed by many of the existing algorithms. This paperpresents a decentralized modal identification algorithm thatutilizes the concept of of compressive sensing [3, 6, 7] toreduce the volume of transmitted vibration data for SHMapplications.

The importance of decentralized processing in WSSNsfor SHM applications has recently been highlighted in thesestudies [21, 17]. Recently, a new de-centralized identifi-cation algorithm based on the principles of Blind SourceSeparation (BSS) [1, 10] was developed by the authors

[15]. In this method, stationary wavelet packet transform(SWPT) is employed to sparsify a signal, and then prin-cipal component analysis (PCA) is employed to identifythe mode shape coefficients. In spite of its simplicity andnumerical efficiency, this method generates a large num-ber of wavelet coefficients corresponding to various scaleswhich needs to be transmitted to the central processor.In the present study, a data compression technique basedon compressive sensing principles is used to reduce theamount of data transmission in the aforementioned algo-rithm.

The objective of most data compression methods isto find the most concise representation of a signal that ispossible with an acceptable level of distortion upon re-construction. Several forms of lossless and lossy datacompression techniques have been adopted in seismic dataand in vibration response data [12, 20, 2]. Recently, apowerful signal processing technique called compressivesensing or compressed sampling [3, 6, 7] has been devel-oped in the field of image processing. It is a powerfultool for signals that have a sparse or compact representa-tion. The number of data points required for reconstruc-tion is usually far lower than the number of measurementsneeded based on Shannon sampling theorem. In the cur-rent study, compressive sensing [8, 6] using 𝑙1 norm min-imization based on primary-dual algorithm is utilized tofacilitate data compression within the framework of BSSmodal identification [15]. By employing a decentralizedframework together with compressive sensing, the result-ing algorithm has a significant potential for long-term andshort-term SHM applications.

2. BASICS OF DATA COMPRESSION USINGCOMPRESSIVE SENSING

Compressive sensing (sampling) is a signal processing tech-nique that is based on two basic criteria: (a) many signalscan be sparsely represented in a linear basis, and (b) max-imum information about the signal can be extracted usingpseudo-random incoherent measurements. Compressivesensing as proposed here seeks to take advantage of thefact that many signals are sparse under some wavelet ba-sis for sampling and reconstruction. Using the 𝑁 × 𝑁

The 11th International Conference on Information Sciences, Signal Processing and their Applications: Special Sessions

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basis matrix 𝜓 = [𝜓1∣𝜓2∣...∣𝜓𝑁 ] with the vectors 𝜓𝑖 ascolumns, a signal 𝑥 can be expressed as:

𝑥 =

𝑁∑𝑖=1

𝜓𝑖𝑠𝑖 (1)

𝑥 = Ψ𝑠

where 𝑠 is the 𝑁 × 1 column vector of transformed coef-ficients 𝑠𝑖 = ⟨𝑥, 𝜓𝑖⟩ = 𝜓𝑇

𝑖 𝑥 and 𝑇 denotes the transpose.Clearly, 𝑥 and 𝑠 are equivalent representations of the sig-nal, with 𝑥 in the time domain (for vibration data) and 𝑠 inthe 𝜓 domain. Consider the problem of reconstructing avector 𝑥𝜖𝑅𝑁 from linear measurements 𝑦𝜖𝑅𝑝 of 𝑥 in thefollowing under-sampled form, where 𝑝≪ 𝑁 : [6, 3, 7]

𝑦𝑘 = ⟨𝑥, 𝜙𝑘⟩, 𝑘 = 1, ..., 𝑝 (2)

𝑦 = Φ𝑥 = ΦΨ𝑠 = Θ𝑠

where Φ is the 𝑝×𝑁 measurement matrix. If 𝑥 is sparsein 𝜓 basis, one can recover the signal by solving the fol-lowing convex optimization (∣∣𝑠∣∣𝑙1 =

∑𝑁𝑖=1 ∣𝑠𝑖∣) [8, 5],

where the estimated 𝑠 is to find the 𝑠 which yields theminimum 𝑙1 norm:

𝑠 = argmin ∣∣𝑠∣∣𝑙1with an equality constraint of Θ𝑠 = 𝑦.(3)

Then the reconstructed signal is:

𝑥𝑟 = Ψ𝑠 (4)

The above convex optimization problem can be solvedeither by using basis pursuit [5, 9], or by using least angleregression (LAR) [18] and one of its derivatives, LASSO(Least Absolute Shrinkage and Selection Operator) [19].In the current study, we used primary-dual linear program-ming algorithm [5] and its associated 𝑙1 magic tool. In or-der to illustrate the data compression technique, considera damped sine wave (𝜔 = 1 Hz and 𝜉 = 1%) with aduration of 2 seconds. The sampling frequency is 50 Hzand the total number of data points is 100. Suppose thatthe data is randomly undersampled at 40 points as shownin Figure 1. The measurement matrix Φ is formed corre-sponding to the random indices of the data, and the basismatrix Ψ is formed using Fourier basis. Once Φ and Ψ areformed, the matrix Θ can be constructed using Equation2. Then, using Equation 3, the transformed coefficientsare found using the measured undersampled signal 𝑦. Thesignal 𝑥𝑟(𝑡) is then reconstructed using Equation 4 and isshown in Figure 1. It can be seen that even for a compres-sion ratio of 60%, the technique adequately captures theoriginal sine wave. Of course, for this example, signifi-cantly higher compression ratio is possible in the Fourierdomain, but this example is just for illustration of the nu-merical technique.

2.1. Data Compression for Vibration Signals

In this section, a set of numerical and experimental stud-ies are presented to illustrate the advantages of using datacompression for vibration signals.

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2.1.1. Experimental Cantilever Model

A wooden cantilever beam with a dimension of 110 ×20 × 2.5 cm is used for illustration first. The beam isclamped at one end and is impacted with a hammer at thefree end. A wireless sensor board and a base receiver sta-tion (developed at Waterloo using commercially availablecomponents) is used as a sensor platform and to embedthe compression algorithm. An analog high-sensitivityaccelerometer (manufactured by PCB electronics with asensitivity of 10V/g) was connected to the sensor board.The wireless components are shown in Figure 2 and Fig-ure 3. The sensor board communicates with the base sta-tion wirelessly, while the sensor itself is a wired sensor.Though not pursued here, integrating MEMS sensors di-rectly to the sensor board is also possible. The commu-nication protocol is Zigbee, which is designed to targetradio-frequency applications that require low data rates,long battery life, and secure networking.

Fig. 2. Sensor board

Fig. 3. Gateway

The compressed sensing algorithm is embedded on thesensor board and the under-sampled data is transmittedwirelessly to the gateway (base station) connected to a lap-top. The signal is reconstructed from this incomplete data

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using 𝑙1 norm minimization. To verify the performance ofdata reconstruction, two sensors are placed at the tip of thecantilever; one of them is a conventional wired data acqui-sition system, while the other is the aforementioned wire-less compressive sampling method (a compression ratio of75% was embedded on the board). The beam is subjectedto three consecutive hammer impacts. The original dataand the wireless reconstructed data are compared in Fig-ures 4 and Figure 5. The results show the accuracy of theembedded random under-sampling method in the wirelesssensor, both in terms of time as well as frequency domainfeatures (amplitude spectrum).

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Fig. 4. Uniform completely-sampled data and randomunder-sampled data from the wired and wireless sensors,respectively

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Fig. 5. Performance of the embedded reconstruction algo-rithm and the Fourier spectrum of the reconstruction

2.1.2. 5-storey Building model

The next example is a typical building frame with a shear-beam approximation. Synthetic data is generated usingthis model using numerical simulations on this 5-storeymodel [15]. A zero mean unit variance Gaussian whitenoise was used to excite the structure at all the floor lev-els. The floor responses containing 4000 samples origi-nally are randomly under-sampled to 1600 data points, re-sulting in a compression ratio of 60%. The floor responsesare reconstructed using 𝑙1 norm minimization. The floorresponse corresponding to the first floor is compared withthe reconstructed response in Figure 6. The normalizedFourier spectrum for each of these cases confirms excel-lent reconstruction in terms of the frequency content.

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Fig. 6. Reconstruction of first floor response

3. COMPRESSED SENSING INDECENTRALIZED MODAL IDENTIFICATION

Recently, the authors presented a de-centralized algorithmbased on time-frequency transforms to infer global struc-ture mode information using measurements obtained us-ing a small group of sensors at-a-time [15]. In this method,the problem of identification is cast within the frameworkof under-determined blind source separation invoking trans-formations of measurements to the time-frequency domainto yield a sparse representation. The partial mode shapecoefficients so identified are then combined to yield com-plete modal information. The transformations are under-taken using stationary wavelet packet transform (SWPT).Principal component analysis (PCA) is then performed onthe resulting wavelet coefficients, yielding the partial mix-ing matrix coefficients from a few measurement channelsat-a-time. This process is repeated using measurementsobtained from multiple sensor groups, and the results soobtained from each group are concatenated to obtain theglobal modal characteristics of the structure. A short de-scription of the decentralized approach is given in nextsection for the sake of completeness.

We consider a linear, classically damped, and lumped-mass 𝑛𝑠 degrees-of-freedom (DOF) structural system, sub-jected to an excitation force, F(𝑡):

Mx(𝑡) +Cx(𝑡) +Kx(𝑡) = F(𝑡) (5)

where, x(𝑡) is a vector of displacement coordinates at theDOF. The matrices, M, C and K are the mass, damp-ing and stiffness of the structural system. Assuming thedamping to be of proportional type and the excitation tobe broadband, the solution to Equation 5 can be written interms of an expansion of vibration modes. In matrix form,

x = Φq =

𝑛𝑠∑𝑙=1

𝜙𝑟𝑞𝑟(𝑡) (6)

where, x ∈ ℜ𝑛𝑚×𝑁 is the trajectory matrix composed ofthe sampled components of x, q ∈ ℜ𝑛𝑠×𝑁 is a matrix ofthe corresponding modal coordinates, and Φ𝑛𝑚×𝑛𝑠

is themodal transformation matrix or the mode shape matrix.

Assuming the measurements can be represented in theform expressed in Equation 6, the measured signal at the

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𝑖𝑡ℎ floor level can be expressed as:

𝑥𝑖(𝑡) =

𝑛𝑠∑𝑙=1

𝐴𝑖𝑙𝑠𝑙(𝑡) 𝑖 = 1, 2, ...., 𝑛𝑚 (7)

In the above equation, 𝐴𝑖𝑙 is a vector of mixing ma-trix coefficients corresponding to the 𝑖𝑡ℎ floor level, and𝑠 are the sources. This is consistent with the definitionsemployed in BSS based methods [15]. Considering thewavelet packet coefficients of the sensor responses andsources at any specific node (𝑗, 𝑣), and finally applyingthe orthogonality condition for wavelets, we get [15]:

𝑓 𝑗,𝑣𝑘,𝑖 (𝑡) =

𝑛𝑠∑𝑙=1

𝐴𝑖𝑙𝑒𝑗,𝑣𝑘,𝑙(𝑡) 𝑖 = 1, 2, ...., 𝑛𝑚 (8)

where 𝑗, 𝑘, 𝑣, 𝑖 and 𝑙 represent scale, shift, modulation,sensor and source index respectively. 𝑒𝑗,𝑣𝑘,𝑙 and 𝑓 𝑗,𝑣𝑘,𝑖 denotethe wavelet packet coefficient of 𝑖𝑡ℎ and 𝑙𝑡ℎ source at aspecific node (𝑗, 𝑣). 𝜓(𝑡) is the chosen wavelet. Consider-ing only the coefficients corresponding to the highest scalelevel (𝑗 = 𝑠), various coefficients of the 𝑠𝑡ℎ scale levelcontain the sparse modal responses and high frequencynoise components. Thus, assuming the existence of only𝑙𝑡ℎ source component in the 𝑓𝑠,𝑣𝑘,𝑖 , Equation 8 becomes:

𝑓𝑠,𝑣𝑘,𝑖 (𝑡) = 𝐴𝑖𝑙𝑒𝑠,𝑣𝑘,𝑙 (𝑡) 𝑖 = 1, 2, ...., 𝑛𝑚 (9)

Above expression relates the SWPT coefficient of 𝑖𝑡ℎ floorresponse with the SWPT coefficient of sparse 𝑙𝑡ℎ source.Therefore, following the same logic for 𝑖 = 𝑞 and 𝑖 = 𝑟in Equation 6, the partial mixing matrix coefficients of 𝑞𝑡ℎ

floor normalized with the 𝑟𝑡ℎ floor can be obtained as:

𝑓𝑠,𝑣𝑘,𝑞

𝑓𝑠,𝑣𝑘,𝑟

=𝐴𝑞𝑙𝑒

𝑠,𝑣𝑘,𝑙 (𝑡)

𝐴𝑟𝑙𝑒𝑠,𝑣𝑘,𝑙 (𝑡)

=𝐴𝑞𝑙

𝐴𝑟𝑙= ∣𝑎𝑞𝑙∣; 𝑙 = 1, 2, .....𝑛𝑠

(10)𝑎𝑞𝑙 =

𝐴𝑞𝑙

𝐴𝑟𝑙represents the estimated normalized mixing

matrix coefficient of 𝑙𝑡ℎ mode at 𝑞𝑡ℎ DOF. Therefore, thenormalized partial mixing matrix coefficients can be esti-mated using the SWPT coefficients of the partial floor re-sponses at the highest scale level. A thresholding based onaverage root-mean-square (RMS) value of the coefficientsis then employed to retain coefficients with significant en-ergy.

Filter banks constituted out of the wavelet packets haveoverlapped frequency contents due to imperfect filtering[14]. This creates the possibility of having multi-componentsources in the WPT coefficients at the last scale level.However, Equation 9 and Equation 10 are valid only whenthe coefficients are mono-component signals. In order toidentify the mono-component sources in the wavelet coef-ficients, PCA is performed using these coefficients.

However, note that that the number of transmitted co-efficients is 𝑚 and the signal length of each coefficient is𝑁 , which is same as the original signal length due to theredundant SPWT implementation. Thus the transmissionrequirement in each channel is significantly higher in thismethod, even when compared to the raw data. To reduce

this increased transmission overhead, in the present study,the compressive sensing scheme is applied to the SWPTcoefficients as follows:

𝑔𝑠,𝑣𝑘,𝑞(𝑝×1) = Φ(𝑝×𝑁)𝑓𝑠,𝑣𝑘,𝑞(𝑁×1) (11)

where 𝑔𝑠,𝑣𝑘,𝑞 is the undersampled signal of 𝑓𝑠,𝑣𝑘,𝑞 and Φ is therandom measurement matrix. If we consider the similaritybetween Equation 11 and 2, then 𝑓𝑠,𝑣𝑘,𝑞 can be reconstructedusing:

𝑓𝑠,𝑣𝑘,𝑞(𝑁×1) = Ψ(𝑁×𝑁)𝑠𝑓𝑠,𝑣𝑘,𝑞(𝑁×1) (12)

where 𝑠𝑓 are the coefficients of 𝑓 in Ψ domain which areobtained using the convex optimization [5] and Ψ is theFourier basis matrix for the coefficients. It may be ob-served that 𝑔 has the dimension of 𝑝 where 𝑝 << 𝑁 .Therefore, for 𝑚 coefficients, the reduction in the amountof transmission is of the order of ( 𝑝×𝑚

𝑁×𝑚 ) = 𝑝𝑁 = 𝐶,

where 𝐶 is the compression ratio.

4. RESULTS

In order to illustrate the proposed method, 5𝑡ℎ and 4𝑡ℎ

floor responses are used. Figure 7 shows the plots of theFourier spectrum (FS) of the floor responses with addedmeasurement noise having 𝑆𝑁𝑅 = 100. Both the floor

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Fig. 7. Fourier Spectra of the floor responses

responses are decomposed using SWPT up to a scale level𝑠 = 6. 𝑑𝑏5 wavelet basis is chosen for the analysis. Figure8 shows the wavelet transform of the measurement signalsat the 5𝑡ℎ and 4𝑡ℎ floor levels, 𝑥5(𝑡) and 𝑥4(𝑡), and showsthe first low-pass (LP) waveform at each scale level, 𝑓 𝑗,𝑣𝑖 ,where 𝑣 = 0, 𝑗 = 1 − 6 and 𝑖 = 4, 5. It can be seen thatthe lowest frequency sources become dominant at higherscale levels, whereas the high-pass (HP) waveforms suc-cessively contain the higher frequency waveforms and noise.

The coefficients at the last scale level (2𝑠 = 64) ofdecomposition are used to estimate the partial mixing ma-trix. Of these, the coefficients containing low-energy noisycomponents are discarded using a thresholding criterionbased on RMS values. The RMS values of the coefficientscorresponding to 𝑥5(𝑡) and 𝑥4(𝑡) are shown in Figure 9.The average (𝜇𝑅𝑀𝑆) of all the RMS values is shown as adotted horizontal line in Figure 9. The values of 𝜇𝑅𝑀𝑆 areobtained as 0.022 and 0.021 for 𝑥5(𝑡) and 𝑥4(𝑡), respec-tively. The total number of coefficients with RMS values

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Fig. 8. LP-wavelet packet coefficients of measurements atvarious scale levels

greater than 𝜇𝑅𝑀𝑆 , are 10 and 11 in 𝑥5(𝑡) and 𝑥4(𝑡), re-spectively. To ensure that all the significant energy coeffi-cient pairs from both the floors are present, 11 coefficientpairs (i.e., higher of the two values 10 and 11) are retained.

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Fig. 9. RMS values of the last scale level coefficients of𝑥5(𝑡) and 𝑥4(𝑡)

Higher compression ratios can be achieved in this methodcompared to raw time series due to the sparse nature ofthe SWPT coefficients. This is illustrated in Figure 10.In this figure, the correlation coefficient (𝜌) between thetrue and the reconstructed signal is used as a performancemeasure. It can be seen that typically high compressionratios (about 75%) can be achieved, which corresponds to𝜌 values > 0.95. On the other hand, the performance ofthe compression algorithms operating on the original rawtime series signal reduces significantly using the same (oreven less) compression ratios. This shows that the decen-tralized method using SWPT is particularly suitable foruse with compressive sensing.

The compressed data of the SWPT coefficients canbe reconstructed and PCA can be performed on the re-constructed data. The PCA results of the reconstructedSWPT coefficients for the modal responses are shown inFig. 11. The left column corresponds to the true SWPTcoefficients, whereas reconstructed SWPT coefficients are

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ρ

Original SignalSWPT coefficient

Fig. 10. Increasing amount of compression ratio in SWPTcoefficients

shown in the right column. The results indicate that thescatter diagrams from the reconstructed and the originalsignals are nearly identical. The modal parameters are es-

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Fig. 11. Performance of data compression on the PCAcoefficients

timated using the PCs and the corresponding principal di-rections [15]. The MAC (Modal Assurance Criteria) num-bers of the estimated mode shapes are greater than 0.98 forall the five modes. The calculated mode shapes are rela-tively insensitive to noise (reflected by the relatively highMAC values) even for 𝑆𝑁𝑅 as high as 10 (as shown inTable 1). This can be attributed to the inherent de-noisingcapability of the wavelet filter-bank implementation.

Table 1. Effect of noise level in MAC numbers𝑗 SNR = 100 SNR = 20 SNR = 10

1 1.0 1.0 0.992 0.996 0.981 0.983 1.0 0.99 0.9854 1.0 0.99 0.995 0.992 0.984 0.981

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5. CONCLUSIONS

A decentralized modal identification technique is proposedwith the aid of compressed sensing. The quality of thereconstructed data is verified using various numerical andexperimental models utilizing embedded wireless sensors.The proposed decentralized modal identification methodwhen used in conjunction with compressive sensing al-lows us to achieve significant reductions in the transmis-sion overhead. This has significant potential in long termSHM applications.

6. ACKNOWLEDGEMENTS

The authors thank Prof. Keshav Srinivasan and Alan Ka-plan for their support in the development of the wirelesssensors. Financial support by the Natural Sciences En-gineering Research Council of Canada is gratefully ac-knowledged.

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[3] R. G. Baraniuk, ”Compressed sensing,” IEEE Signalprocessing magazine, vol. 24, no. 4, 118–120, 2007

[4] M. Bocca, L. M. Eriksson, A. Mahmood, R. Jantti,and J. Kullaa, ”A synchronized wireless sensor net-work for experimental modal analysis in structuralhealth monitoring,” Computer-aieded Civil and In-frastucture Engineering, vol. 26, 483–499, 2011

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