Research Article Damped CAPES 2D Spectral Estimation for ...Research Article Damped CAPES 2D Spectral Estimation for Real-Valued Vibration Signals DanhuiDan, 1 JiongxinGong, 1 andYimingZhao

Research ArticleDamped CAPES 2D Spectral Estimation forReal-Valued Vibration Signals

Danhui Dan,1 Jiongxin Gong,1 and Yiming Zhao2

1 Department of Bridge Engineering, Tongji University, Room 711, Bridge Building, 1239 Siping Road, Shanghai 200092, China2 Technology Center of Shanghai Construction Group CO., LTD., Shanghai 200431, China

Correspondence should be addressed to Danhui Dan; [email protected]

Received 5 March 2014; Revised 19 July 2014; Accepted 22 July 2014; Published 26 August 2014

Academic Editor: Mickael Lallart

Copyright © 2014 Danhui Dan et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a 2D representation in the frequency-decay factor plane of an arbitrary real-world vibration signal. The signal isexpressed as the sum of a decayed-attenuation sine term modulated by an amplitude function and a noise residue. We extendthe combined approach of Capon estimation and amplitude and phase estimation (CAPES) to damped real vibration signals (DR-CAPES). In the proposed DR-CAPES method, the high-resolution amplitude and phase are estimated simultaneously for bothangular frequency and decay factor grids. The performance of the proposed approach is tested numerically with noisy vibrationdata. Results show that the DR-CAPES method has an excellent frequency resolution, which helps to overcome difficulties inspectrum estimationwhen vibrationmodes are very close, and a small bias, whichmakes it suitable for obtaining accurate amplitudespectrums. The results also indicate that the proposed method can accurately estimate the amplitude spectrum with the use ofaveraging and denoising processes.

1. Introduction

Thenormal functioning of engineering structures in dynamicenvironments is often affected by large and frequent vibra-tions. These further induce negative effects in terms ofcomfort in use, fatigue durability, dynamic instability, andsafety. Therefore, the structural vibration levels must beassessed under working load conditions and environmentalexcitation. Typically, these assessments should be conductedin the frequency domain, and thus, structural dynamicsexperiments have focused on spectral analysis and identifi-cation of frequency-domain dynamic characteristics, specif-ically the structural vibration signal. The vibration level ofthe structural response is usually described using the Fourierpower spectrum [1, 2]; however, the signal quality couldbe poor because of noise pollution, short signal length, orinappropriate sampling frequency. Because the Fourier powerspectrum frequency often has low resolution and the spectralvalue estimates are inaccurate, a more suitable and accuratespectral estimation algorithm is needed. The power densityspectrum cannot be directly used to assess the vibrationamplitude of a certain vibration response. In some cases, the

amplitude spectrum is even more important than the powerspectrum [3]. Vibrations in the physical world are dampedsignals and the level of vibration is closely related to thedamping.Damping describes the dynamic characteristics andstructural performance of the structure; therefore, it mustbe precisely measured. At present, high-quality dampingestimation is very difficult when using the real responsesignal. In this paper, we propose a nonparametric high-precision spectral estimation algorithm that determines thevibration signal characteristics of engineering structures inorder to analyze data from structural dynamics experiments.

One classic category of spectral estimation algorithms isthe set of nonparametric methods based on the fast Fouriertransform (FFT), in which the frequency resolution is low.However, the FFT has wide applicability, because it has noprespectral shape [3–5]. There is also a category of high-resolution parametric methods that require the observedsignal to be composed of limited number of superposedsinusoidal signals contaminated by white noise. This is diffi-cult to satisfy in structural vibration signal spectral analyses;therefore, these methods often perform poorly [6]. A thirdcategory is the set of nonparametric adaptive filter-bank

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 570347, 11 pageshttp://dx.doi.org/10.1155/2014/570347

2 Shock and Vibration

methods with high resolution capability, including the Caponalgorithm [7] and the recently proposed amplitude andphase estimation spectral (APES) algorithm [8, 9], as wellas variants and hybrids of these techniques. These methodshave been used successfully for almost twenty years. Theymake no assumptions about the spectral structure and do notdepend on the Fourier transform algorithm; as a result, theycan be used for high-resolution spectrum estimation for anystationary signals.

TheCapon algorithmperforms frequency resolutionwithgreater efficiency than other algorithms, and the APES algo-rithm estimates spectral values [10, 11] with greater accuracythan most of the other algorithms. In 2000, Jakobsson andStoica combined the two to give the CAPES algorithm, whichimproves the resolution and estimation accuracy of spectralvalues [12]. The CAPES algorithm has been used for one-and two-dimensional complex-signal spectral estimation inmedical imaging and remote sensing [13–15], and Jakobssonet al. improved the algorithm to handle real one-dimensionalvibration signals without damping [16]. In 2001, Stoica andSundin proposed new versions of the Capon and APES algo-rithms that can deal with complex one-dimensional dampingmodulation signals, whose amplitude decays over time [14].Recently, the researchers focused on the applications and fastcomputation algorithms of APES and CAPON method [17,18]. However, when their algorithms are used to estimate thespectrum of the structural vibration directly the accuracy willbeworse due to the imaginary part of the signals. Actually, thestructural vibration signal is a real one-dimensional dampingmodulation signal.

Based on existing nonparametric estimation algorithms,we propose a two-dimensional spectral estimation algorithmto transform arbitrary real one-dimensional damping vibra-tion signals in the time-domain into a two-dimensionalsignal in a joint domain of circular frequency and dampingfactor. The proposed method is expected to provide a high-resolution frequency, high-precision amplitude spectrum,and high-precision damping factor estimations of the vibra-tion signal. Through numerical examples, we verify theeffectiveness of the proposed algorithm.

2. Theory

2.1. Representation of a Vibration Signal in a 2D Plane. In an𝑁-degrees-of-freedom (DOF) vibration system, the responseto a steady forced vibration can be written as follows:

𝑥 =

𝑁

∑

𝑗=1

𝑥𝑗=

𝑁

∑

𝑗=1

𝐴(𝜔𝑗, 𝜎𝑗) 𝑒−𝜎𝑗𝑡 sin (𝜔

𝑗𝑡 + 𝜑 (𝜔

𝑗, 𝜎𝑗)) , (1)

where 𝑥𝑗is the contribution of 𝑗th mode of the vibration

system, 𝜔𝑗and 𝜎

𝑗are the 𝑗th mode frequency and damping

factor, and𝐴 and 𝜑 are the amplitude and initial phase of 𝑗thmode. This equation means that the response to any DOFcan be represented by the decayed-attenuation sine function𝑒−𝜎𝑗𝑡 sin(𝜔

𝑗𝑡 + 𝜑(𝜔

𝑗, 𝜎𝑗)). Based on this, we set 𝑥(𝑛) as the

discrete time series of an arbitrary DOF response in thevibrating system, 𝑛 = 0, 1, . . . , 𝑁 − 1, where 𝑁 is the lengthof time series. With interest frequency 𝜔 and damping factor

𝜎

(𝜔, 𝜎)

𝜁 = sin(𝛽(𝜔, 𝜎))

𝛽(𝜔, 𝜎)=

𝜎

√𝜔2 + 𝜎2

𝜔

Figure 1: Schematic 2D representation of the vibration response.

𝜎, 𝑥(𝑛) can be written as the sum of a decayed-attenuationsine termmodulated by the amplitude function 𝑎(𝜔, 𝜎) and aresidual term 𝑤

𝜔,𝜎(𝑛), as follows:

𝑥 (𝑛) = 𝑎 (𝜔, 𝜎) 𝑒−𝜎𝑛 sin (𝜔𝑛 + 𝜑 (𝜔, 𝜎)) + 𝑤𝜔,𝜎 (𝑛) . (2)

In (2), 𝑎(𝜔, 𝜎) and 𝜑(𝜔, 𝜎) are the amplitude functionand the phase function, respectively, of the point (𝜔, 𝜎),which is defined in a 2D plane; 𝑤

𝜔,𝜎(𝑛) includes the decayed

attenuation sine terms of all points except for (𝜔, 𝜎) and therandom noise 𝑤(𝑛).

For the single DOF free response,

𝑎 (𝜔, 𝜎) = √𝑥(0)2+ (

��(0)2− 𝜎𝑥 (0)

𝜔),

𝜑 (𝜔, 𝜎) = arctan( 𝑥 (0) 𝜔

�� (0) − 𝜎𝑥 (0)) ,

(3)

where 𝑥(0) and ��(0) are the initial displacement and theinitial velocity, respectively, when 𝑡 = 0 (or 𝑛 = 0).

We define the relationship between the damping factor 𝜎and the damping ratio 𝜁 as follows:

𝜁 =𝜎

√𝜔2 + 𝜎2=

𝜎

√(2𝜋𝑓)2+ 𝜎2

. (4)

The geometrical meaning can be explained as the sinevector between the point (𝜔, 𝜎) and the original point in a2D plane, as shown in Figure 1.

2.2. Damped CAPES Spectral Estimation for Real-ValuedVibration Response Data. In (2), the time series 𝑥(𝑛) can bewritten as

𝑥 (𝑛) = 𝑎 (𝜔, 𝜎) cos (𝜙 (𝜔, 𝜎)) 𝑒−𝜎𝑛 sin (𝜔𝑛)

+ 𝑎 (𝜔, 𝜎) sin (𝜙 (𝜔, 𝜎)) 𝑒−𝜎𝑛 cos (𝜔𝑛)

+ 𝑤𝜔,𝜎 (𝑛) .

(5)

If we let

𝑎𝑐 (𝜔, 𝜎) = 𝑎 (𝜔, 𝜎) cos (𝜑 (𝜔, 𝜎))

𝑎𝑠 (𝜔, 𝜎) = 𝑎 (𝜔, 𝜎) sin (𝜑 (𝜔, 𝜎))

(6)

Shock and Vibration 3

then

𝑥 (𝑛) = 𝑎𝑐 (𝜔, 𝜎) 𝑒

−𝜎𝑛 cos (𝜔𝑛) + 𝑎𝑠 (𝜔, 𝜎) 𝑒−𝜎𝑛 sin (𝜔𝑛)

+ 𝑤𝜔,𝜎 (𝑛) .

(7)

(i) Hankel MatrixMethod. We can obtain𝑁 practical samplesof the time series 𝑥(𝑛), such as 𝑥(0), 𝑥(1), . . . , 𝑥(𝑁 − 1), 𝑛 =

0, 1, . . . , 𝑁 − 1. According to the nonparametric adaptivefilter-bank spectral estimation method [1], we can constructthe𝑀× 𝐿-order Hankel matrix of observation data 𝑥(𝑛),

Y = [y0 ⋅ ⋅ ⋅ y𝐿−1]

=

[[[[[[

[

𝑥 (0) 𝑥 (1) ⋅ ⋅ ⋅ 𝑥 (𝐿 − 2) 𝑥 (𝐿 − 1)

𝑥 (1) 𝑥 (2) ⋅ ⋅ ⋅ 𝑥 (𝐿 − 1) 𝑥 (𝐿)

.

.

.... d

.

.

....

𝑥 (𝑀 − 2) 𝑥 (𝑀 − 1) ⋅ ⋅ ⋅ 𝑥 (𝑁 − 3) 𝑥 (𝑁 − 2)

𝑥 (𝑀 − 1) 𝑥 (𝑀) ⋅ ⋅ ⋅ 𝑥 (𝑁 − 2) 𝑥 (𝑁 − 1)

]]]]]]

]

.

(8)

In (8), 𝑀 is the user parameter, also known as the finiteimpulse response (FIR) filter length [9], 𝑀 ≤ 𝑁/2, and 𝐿 =

𝑁 − 𝑀 + 1 represents the total number of snapshots in thetime series.

If we setA𝑀 (𝜔, 𝜎)

=

[[[[[[

[

1 0

𝑒−𝜎 cos (𝜔) 𝑒

−𝜎 sin (𝜔)...

.

.

.

𝑒−𝜎(𝑀−2)

⋅ cos (𝜔 (𝑀 − 2)) 𝑒−𝜎(𝑀−2)

⋅ sin (𝜔 (𝑀 − 2))

𝑒−𝜎(𝑀−1)

⋅ cos (𝜔 (𝑀 − 1)) 𝑒−𝜎(𝑀−1)

⋅ sin (𝜔 (𝑀 − 1))

]]]]]]

]

,

(9)

Θ (𝜔, 𝜎) = [𝑎𝑐 (𝜔, 𝜎) 𝑎

𝑠 (𝜔, 𝜎)

𝑎𝑠 (𝜔, 𝜎) −𝑎

𝑐 (𝜔, 𝜎)] , (10)

V𝜔,𝜎 (𝑙) = [𝑒

−𝜎(𝑙−1) cos (𝜔 (𝑙 − 1)) 𝑒−𝜎(𝑙−1) sin (𝜔 (𝑙 − 1))] ,

(11)

B𝜔,𝜎 (𝑙) = [

𝑒−𝜎(𝑙−1) cos (𝜔 (𝑙 − 1)) 𝑒

−𝜎(𝑙−1) sin (𝜔 (𝑙 − 1))−𝑒−𝜎(𝑙−1) sin (𝜔 (𝑙 − 1)) 𝑒

−𝜎(𝑙−1) cos (𝜔 (𝑙 − 1))] ,

(12)

𝜃 (𝜔, 𝜎) = [𝑎𝑐 (𝜔, 𝜎) 𝑎𝑠 (𝜔, 𝜎)]

𝑇

= Θ ⋅ [1 0]𝑇

. (13)

Here, in (11) and (12), 𝑙 = 0, 1, . . . , 𝐿 − 1.Then

y𝑙= A𝑀 (𝜔, 𝜎) ⋅ B𝜔,𝜎 (𝑙) ⋅ 𝜃 (𝜔, 𝜎) + n

𝑙

= H𝜔,𝜎 (𝑙) ⋅ 𝜃 (𝜔, 𝜎) + n

𝑙

= A𝑀 (𝜔, 𝜎) ⋅Θ (𝜔, 𝜎) ⋅ V𝜔,𝜎 (𝑙) + n

𝑙

(14)

andH𝜔,𝜎(𝑙) = A

𝑀(𝜔, 𝜎) ⋅ B

𝜔,𝜎(𝑙). Equation (8) becomes

Y = A𝑀 (𝜔, 𝜎) ⋅Θ (𝜔, 𝜎) ⋅ (A𝐿 (𝜔, 𝜎))

𝑇+ n, (15)

where noise matrix is n,

n = [n0 ⋅ ⋅ ⋅ n𝐿−1] . (16)

We now construct the weighted secondary residuals term J asthe least squares estimator, as follows:

J (𝜔, 𝜎) = (Φ−1/2

(𝜔, 𝜎) ⋅W (𝜔, 𝜎))𝑇

× (Φ−1/2

(𝜔, 𝜎) ⋅W (𝜔, 𝜎)) ,

(17)

where Φ(𝜔, 𝜎) is the weight matrix and Φ−1/2 denotes thesquare root of Hermitian matrixΦ.W(𝜔, 𝜎) is defined as

W (𝜔, 𝜎) = Y − A𝑀 (𝜔, 𝜎) ⋅Θ (𝜔, 𝜎) ⋅ (A𝐿 (𝜔, 𝜎))

𝑇. (18)

Then, according to the nonparametric adaptive filter-bank theory on spectral estimation, the weighted leastsquares estimator of parameter Θ, which includes amplitudeinformation, is given by

Θ (𝜔, 𝜎) = arg minΘ(𝜔,𝜎)

‖J (𝜔, 𝜎)‖2. (19)

According to Larsson and Li’s contribution [10], the finalestimation of parameter 𝜃 can be expressed as follows:

�� (𝜔, 𝜎) =G−1(𝜔,𝜎)

⋅ C𝑇(𝜔,𝜎)

⋅Φ−1(𝜔, 𝜎) ⋅ A𝑀 (𝜔, 𝜎)

(A𝑀 (𝜔, 𝜎))

𝑇⋅Φ−1(𝜔, 𝜎) ⋅ A𝑀 (𝜔, 𝜎)

⋅ [1

0] .

(20)

Here, differently with the Larsson and Li’s expressions, A𝑀is

defined as (9) and G(𝜔,𝜎)

, C(𝜔,𝜎)

should be defined by

G(𝜔,𝜎)

=1

𝐿

𝐿−1

∑

𝑙=0

(V𝜔,𝜎 (𝑙))

𝑇V𝜔,𝜎 (𝑙) (21)

C(𝜔,𝜎)

=1

𝐿

𝐿−1

∑

𝑙=0

y𝑙⋅ V𝜔,𝜎 (𝑙) . (22)

The weighting function values are as follows:

ΦCapon = R =1

𝐿YY𝑇 = 1

𝐿

𝐿−1

∑

𝑙=0

y𝑙⋅ y𝑇𝑙, (23)

ΦApes = R − C(𝜔,𝜎)

G−1(𝜔,𝜎)

C𝑇(𝜔,𝜎)

. (24)

(ii) Vectorization of Hankel Matrix. According to (8), wedefine


Yvec = vec (𝑦𝑙) = [𝑦0 𝑦

1⋅ ⋅ ⋅ 𝑦𝐿−1]𝑇

= [𝑥 (0) ⋅ ⋅ ⋅ 𝑥 (𝑀 − 1) 𝑥 (1) ⋅ ⋅ ⋅ 𝑥 (𝑀) ⋅ ⋅ ⋅ 𝑥 (𝐿 − 1) ⋅ ⋅ ⋅ 𝑥 (𝑁 − 1) ]𝑇

.

(25)

The length of Yvec is𝑀× 𝐿.Let

H𝜔,𝜎 (𝑙)

= A𝑀 (𝜔, 𝜎) ⋅ B𝜔,𝜎 (𝑙)

=

[[[[[[

[

1 0

𝑒−𝜎 cos (𝜔) 𝑒

−𝜎 sin (𝜔)...

.

.

.

𝑒−𝜎(𝑀−2)

⋅ cos (𝜔 (𝑀 − 2)) 𝑒−𝜎(𝑀−2)

⋅ sin (𝜔 (𝑀 − 2))

𝑒−𝜎(𝑀−1)

⋅ cos (𝜔 (𝑀 − 1)) 𝑒−𝜎(𝑀−1)

⋅ sin (𝜔 (𝑀 − 1))

]]]]]]

]

⋅ [𝑒−𝜎(𝑙−1) cos (𝜔 (𝑙 − 1)) 𝑒

−𝜎(𝑙−1) sin (𝜔 (𝑙 − 1))−𝑒−𝜎(𝑙−1) sin (𝜔 (𝑙 − 1)) 𝑒

−𝜎(𝑙−1) cos (𝜔 (𝑙 − 1))] .

(26)

Then, according to (7),

Yvec = H ⋅ 𝜃 (𝜔, 𝜎) + n. (27)

In (27),

H =

[[[[

[

H𝜔,𝜎 (0)

H𝜔,𝜎 (1)

.

.

.

H𝜔,𝜎 (𝐿 − 1)

]]]]

]

, n =

[[[[

[

𝑛1

𝑛2

.

.

.

𝑛𝐿

]]]]

]

,

n𝑙=

[[[[

[

𝑤𝜔,𝜎 (𝑙)

𝑤𝜔,𝜎 (𝑙 + 1)

.

.

.

𝑤𝜔,𝜎 (𝑙 + 𝑀 − 1)

]]]]

]

,

(28)

where Yvec ∈ R𝑀𝐿×1 is the observation vector, H ∈ R𝑀𝐿×2 isthe observation matrix, n ∈ R𝑀𝐿×1, and n

𝑙∈ R𝑀×1, 𝑀𝐿 =

𝑀 × 𝐿.Spectral estimation for a signal at point (𝜔, 𝜎) has become

the problem of parameter estimation of 𝜃, based on Yvec,while minimizing the estimated noise. This is a typical linearestimation problem that can be solved by the followingweighted least squares estimator J∗ as seen in [5]:

J∗ = (Yvec −H ⋅ 𝜃)𝑇Φ−1(Yvec −H ⋅ 𝜃) ,

�� (𝜔, 𝜎) = arg min𝜃(𝜔,𝜎)

J∗ (𝜔, 𝜎) ,(29)

where Φ ∈ R𝑀𝐿×𝑀𝐿 is the weighted values matrix. WhenH𝑇Φ−1H is nonsingular, the former equation is solved by

�� (𝜔, 𝜎) = (H𝑇Φ−1H)−1

H𝑇Φ−1Yvec (30)

and then (21) reaches a minimum.

When the weighted matrixΦ is set to unit matrix, we canget a two-dimensional spectral estimation of the real signal inthe least squares sense as follows:

��sl (𝜔, 𝜎) = (H𝑇H)−1

H𝑇Yvec. (31)

According to the basic idea of Capon spectral estimation,Φ is set to the covariance matrix of data as follows:

ΦCapon = R = Yvec ⋅ Y𝑇

vec. (32)

However, according to the idea of APES spectral estima-tion, Φ is the covariance estimation of the residual 𝑛, givenby

ΦApes = QV = R −H ⋅ ��sl (𝜔, 𝜎) ⋅ ��𝑇

sl (𝜔, 𝜎) ⋅H𝑇. (33)

Thus, we have five 2D damping spectral estimations of areal vibration signal, see in (23), (24), (31), (32), and (33).

3. Algorithm Implementation

Previous methods of estimation required the inverse of largeweighted-matrices to be calculated at each 2D point (𝜔, 𝜎).To reduce the computation time, we can utilize a matrixinversion formula to calculate the inverses in advance, leavingonly a matrix multiplication in every iteration step.

LetA, B,U, andV be arbitrary matrices; according to thematrix inversion formula, we have

(A − UBV)−1 = A−1 − A−1U(I + BVA−1U)−1

BVA−1. (34)

Corresponding to (24), letA be R, let B beG−1(𝜔,𝜎)

, letU beC(𝜔,𝜎)

, and let V be C𝑇(𝜔,𝜎)

; we have

(ΦApes)−1

= (R − C(𝜔,𝜎)

G−1(𝜔,𝜎)

C𝑇(𝜔,𝜎)

)−1

= R−1 − R−1C(𝜔,𝜎)

(I + G−1(𝜔,𝜎)

C𝑇(𝜔,𝜎)

R−1C(𝜔,𝜎)

)

× G−1(𝜔,𝜎)

C𝑇(𝜔,𝜎)

R−1

(35)

and corresponding to (33), let

R𝜃,sl (𝜔, 𝜎) = ��sl (𝜔, 𝜎) ⋅ ��

𝑇

sl (𝜔, 𝜎) . (36)


0 200 400 600 800−30

−20

−10

0

10

20

30Ac

c

Time (s)

Data 1Data 2Data 3

Data 4Data 5

(a)

Effective length, SNR > 0

Mode 1Mode 2 and 4

Mode 3Mode 5

0 200 400 600 800

Time (s)

−30

−20

−10

0

10

20

30

(b)

Figure 2: Free response history of a 5-DOF oscillation system: (a) data history; (b) effective length of each mode’s contribution to the 1stDOF response.

This gives

(ΦApes)−1

= (R −H ⋅ R𝜃,sl (𝜔, 𝜎) ⋅H

𝑇)−1

= (R −H ⋅ R𝜃,sl ⋅H

𝑇)−1

= R−1 − R−1 ⋅H

⋅ (I + R𝜃,sl ⋅H

𝑇⋅ R−1 ⋅H) R

𝜃,sl ⋅H𝑇⋅ R−1.

(37)

The large matrix inversions in (30) and (32) do notdepend on the point (𝜔, 𝜎); therefore, these can be calculatedbefore entering the iteration cycle.

Hence, we design the proposed real damped version ofCAPES (DR-CAPES) as follows:

step 1: set the data as 𝑥, and the user data length as𝑀;

step 2: calculate𝐿 = 𝑁−𝑀+1, form the𝑀× 𝐿-orderHankelmatrix R and calculate R−1;

step 3: form the 2Dgrid (𝜔𝑖, 𝜎𝑗), where 𝑖 and 𝑗 are the divisors

of the frequency and damping factor, respectively; 𝑖 =1, 2, . . . , 𝑖𝑖, 𝑗 = 1, 2, . . . , 𝑗𝑗;

step 4: incrementing 𝑖, 𝑗 in a loop, calculate G(𝜔,𝜎)

and C(𝜔,𝜎)

according to (21) and (22); calculate (ΦApes)−1 accord-

ing to (35); calculate ��wsl(𝜔, 𝜎) according to (20);calculate the amplitude and phase of the real dampingvibration signal according to (7).

4. Numerical Case Studies

Consider a 5-DOF linear oscillation system with the follow-ing structural matrix:

M = diag (1, 2, 1, 4, 8) ,

K = diag (0.0987, 0.2843, 0.1934, 1.0107, 2.5582) ,

C = diag (0.004, 0.008, 0.004, 0.024, 0.032) .

(38)

It should be noted that, in order to simplify the problemand focus on the process of vibration signal itself, thisnumerical case of 5-DOF oscillation system is designated asfull decoupling system, with all of its physical matrices K,M, and C being diagonal matrix. The modal parameters ofthe complex modal matrix, angular frequency, and dampingfactor can be calculated analytically as follows:

𝜔 = 2𝜋 ⋅ (0.05, 0.06, 0.07, 0.08, 0.09)𝑇,

𝜎 = (0.003, 0.002, 0.002, 0.003.0.002)𝑇,

|Ψ| =

[[[[[

[

12.3 10.0 5.3 10.0 10.0

5.0 4.0 4.0 2.0 5.0

4 7 3 7 2

1 5 6 8 4

7 4 7 3 1

]]]]]

]

,

∠Ψ =

[[[[[

[

0.10 0.12 0.17 0.16 0.06

0.01 0.08 0.05 0.11 0.19

0.20 0.08 0.04 0.27 0.11

0.21 0.26 0.17 0.38 0.26

0.11 0.23 0.18 0.25 0.27

]]]]]

]

.

(39)

The free vibration responses are generated from theinitial status vector x

0= (2, 3, 4, 5, 1)

𝑇. These responses are


0 1 2 3 4 5×10−3

Damping factor0.1

0.08

0.06

0.04

Frequency (Hz)

0

5

10

Am

plitu

de

(a)

0.05 0.06 0.07 0.08 0.09

Frequency (Hz)

0

1

2

3

4

5×10−3

Dam

ping

fact

or

(b)

Figure 3: 2D amplitude spectrum estimated by the proposed approach (the response of the first DOF 𝑥1is chosen): (a) 2D amplitude of 𝑥

1;

(b) contour of the amplitude spectrum in the 𝑓, 𝜎 plane.

Table 1: Identified spectrum data.

Modes Mode 1 Mode 2 Mode 3 Mode 4 Mode 5Designated frequency 0.05 0.06 0.07 0.08 0.09Identified frequency 0.05 0.06 0.07 0.08 0.09Frequency error 0 0 0 0 0Designated damping factor 0.003 0.002 0.002 0.003 0.002Identified damping factor 0.00301 0.00213 0.00215 0.00302 0.00205Damping factor error (%) 0. 33 6.5 7.5 0. 67 0.25

observed fully under an initial signal-to-noise ratio (SNR) of

RSN,0 =

[[[[[

[

16.6 14.8 9.3 14.8 14.8

13.1 11.1 11.1 5.1 13.1

2.0 6.8 −0.5 6.9 −4.0

−8.6 5.4 7.0 9.5 3.5

1.9 −3.0 1.9 −5.4 −15

]]]]]

]

(40)

and a sampling frequency of 1.0Hz, as shown in Figure 2(a).The total length of the data is 800 samples and there are 40,48, 56, 64, and 72 characteristic periods for each mode. Ifwe define the effective length of the data to be the segmentwith a nonnegative SNR, these numbers are reduced to24, 17, 21, 25, and 38, as illustrated in Figure 2(b). Thesecharacteristic periods are shorter than any other spectralestimation approaches, thereby illustrating the effectivenessof feature extraction in the proposed approach.

The response of the first DOF 𝑥1has the biggest initial

SNR of all the responses. Therefore, 𝑥1was chosen to feed

the proposed approach. The 3D amplitude spectrum andits corresponding contour in the (𝑓, 𝜎)-plane are shown inFigure 3.

In Figure 3(a), the five distinct peaks lie in the planeof the natural frequency and damping factor, that is, 𝑓, 𝜎,respectively. The bases of the peaks are well separated fromeach other and the widths in the frequency dimension areso narrow that each peak looks like a thin blade. In thedamping factor dimension, the slopes of the ridges are gentle

and the bases are wide. Each apex is located exactly at itscorresponding designated value, as shown in Figure 3(b).Theidentified spectrum data are listed in Table 1.

The proposed approach performed well in identifyingboth the frequency and damping factor. Under the noiselevels given above, the frequency errors were almost zero.Thedamping factor errors were also smaller with the proposedmethod than with any other approach.

4.1. Frequency Resolution of 3D Amplitude Spectrum. Thetraditional CAPES method is known for its high frequencyresolution and accurate spectral value estimation. To inves-tigate the frequency resolution of the proposed DR-CAPESapproach, we conducted a case study based on the oscillationsystem described above.

The effect of closed peaks was simulated by changing thephysical parameters K,M, or C. In this case, we simply fixedthe second mode to 0.06Hz and gradually adjusted the firstmode frequency from 0.05Hz to 0.06Hz while maintainingM and C. A new 5-DOF oscillation system with a differentK was obtained for each iteration and the free vibrationresponses of the system were calculated under the initialSNRs mentioned above.

Figure 4 shows the amplitude spectra of four typical stepswhen the first mode approaches the second.The other modeswere not identified and are not illustrated in the figures. Asa comparison, the global amplitude spectrum identified by


0

5

10

Am

plitu

de

0.056 0.058 0.06 0.062 0.064Frequency (Hz)

0

0.002

0.004

Dampin

g fac

tor

FFTD.R.CAPES

(a)

0

5

10

Am

plitu

de

0.056 0.058 0.06 0.062 0.064Frequency (Hz)

0

0.002

0.004

Dampin

g fac

tor

FFTD.R.CAPES

(b)

0

5

10

Am

plitu

de

0.056 0.058 0.06 0.062

Frequency (Hz)

0

0.002

0.004

Dampin

g fac

tor

FFTD.R.CAPES

(c)

0

5

10

Am

plitu

de

0.056 0.058 0.06 0.062 0.064

Frequency (Hz)

0

0.002

0.004

Dampin

g fac

tor

FFT

D.R.CAPES

(d)

Figure 4: Comparison of frequency resolution of the amplitude spectrum using DR-CAPES and FFT. ((a) 𝑓1= 0.052, 𝑓

2= 0.06; (b) 𝑓

1=

0.054, 𝑓2= 0.06; (c) 𝑓

1= 0.056, 𝑓

2= 0.06; (d) 𝑓

1= 0.058, 𝑓

2= 0.06; FFT spectrum in each figure was formed from data with a length of 1024

and a zero appending technique was used).

an FFT method was also found and the nine spectral linescorresponding to the same frequency window are plottedin the figures. It can be seen that the frequency resolutionof DR-CAPES is much better than that of the FFT method;from Figures 4(a)–4(c), the two peaks of the FFT spectrumapproach one another quickly and in Figure 4(d) the peakshave overlapped completely, whereas the two 3D peaks givenby the DR-CAPESmethod remain separate and distinct, withthe shape and height of the peaks basically invariant.

To further illustrate the high resolution of the proposedapproach, we define amode closenessmeasure and frequencyresolution in a physical sense as follows.

Let𝜅 = log

10𝛿 (41)

be a measure of the closeness of two adjacent modes (oradjacent peaks) in a spectrum, in which

𝛿 =

𝜔𝑗+1

+ 𝜔𝑗

𝜔𝑗+1

− 𝜔𝑗

(42)

and 𝜔𝑗is the 𝑗th mode or peak in spectrum. Larger 𝜅 values

imply closer distances between two spectrum lines,modes, orpeaks.

When adjacent peaks are close to each other, the physical-sense frequency resolution can be defined as the logarithm ofa ratio of the minimum height of two adjacent peaks and thesaddle value between them, as follows:

𝑅𝜔= log10𝛼,

𝛼 =min (𝐴

1, 𝐴2)

min𝜔1≤𝜔≤𝜔2

(𝐴𝜔).

(43)

3.5 4 4.5 5 5.5 6 6.50

1

2

3

4

5

6

7

D.R.CAPESFFT, 1024

R𝜔=log

𝛼 10

𝜅 = log 𝛿10

Figure 5: Comparison of frequency resolution between DR-CAPESmethod and FFT method.

Here, 𝐴𝜔is the saddle value of two adjacent hills in the

spectrum and𝐴1,𝐴2are the values of the two adjacent peaks.

Figure 5 illustrates how the results of the two processesconverge in terms of the mode closeness and frequencyresolution. As the mode closeness increases, the frequencyresolutions of both the DR-CAPES method and the FFTmethod decrease.The resolution of the FFTmethod dropped


Runs

Am

plitu

de

200 400 600 800 10005

10

15

(a)

Runs

Am

plitu

de

200 400 600 800 10005

10

15

(b)

True amplitude

bFFTiasbDR−APES

ias

𝜇DR−APES𝜇FFT

Runs

Am

plitu

de

200 400 600 800 10005

10

15

(c)

Runs

Am

plitu

de

200 400 600 800 10005

10

15

True amplitude

bFFTiasbDR−APES

ias

𝜇DR−APES𝜇FFT

(d)

Figure 6: Scatterplot of the peak amplitude spectra given by DR-CAPES and FFT ((a) 𝜎 = 0.001; (b) 𝜎 = 0.0015; (c) 𝜎 = 0.0025; (d)𝜎 = 0.003).

close to zero at a relatively small value of mode closeness.Compared to FFT, the proposed DR-CAPES has a higherfrequency resolution, even when the two peaks are extremelyclose to each other.

4.2. Bias and Accuracy of the Amplitude Spectrum Givenby the DR-CAPES Method. The bias and accuracy of theamplitude spectrum identified by the DR-CAPES methodcan be revealed by numerical simulations. Four simulationswere performed under a fixed observation noise level; that is,the initial SNR was 16.6 dB for the first mode component ofthe response corresponding to the 1st DOF. Each simulationwas run with the same parameters, except for the dampingfactor, which was set as 𝜎 = 0.001, 0.0015, 0.0025, and 0.003.In each simulation, the following processes were repeated1000 times: generation of random response, simultaneousspectrum estimation with the proposedmethod, and the FFTmethod.

Figure 6 shows the scatterplots of the simulation results.Each subplot corresponds to a designated damping fac-tor, that is, 𝜎 = 0.001, 0.0015, 0.0025, and 0.003. The

mean and bias of these estimates by these two methods,indicated as 𝜇DR-APES, 𝜇FFT, 𝑏

FFTias , and 𝑏

DR-APESias , and the

true values of the amplitude (designated value) are alsoplotted.

The results show that the spectrum bias of theDR-CAPESmethod is smaller than that of the FFT method. Regardlessof the designated damping factor, the bias of the DR-CAPESmethod stabilized to a certain level below the true amplitude,whereas the FFT estimationwas biased on either side depend-ing on the designated damping factor. Additionally, as shownin Figure 6, the extent of scatter of the FFT result is smallerthan that of the proposed method; this may mean that theaccuracy of the proposed DR-CAPES method can be furtherimproved.

4.3. Influence of Noise on the Amplitude Spectrum. In theabove discussion, the noise is fixed at a very low level.Further examination investigated the influence of noise levelon the quality of the estimated amplitude spectrum. Similarto the numerical simulation case studies above, we simulated


Mea

n of

ampl

itude

SNR (dB)−20 −10 0 10 200

5

10

15

20

25

(a) 𝑓 = 0.05HZ, 𝜎 = 0.003

Std.

of a

mpl

itude

0

2

4

6

8

10

SNR (dB)−20 −10 0 10 20

(b) 𝑓 = 0.05HZ, 𝜎 = 0.003

Mea

n of

ampl

itude

SNR (dB)−20 −10 0 10 200

5

10

15

20

25

(c) 𝑓 = 0.05HZ, 𝜎 = 0.002

Std.

of a

mpl

itude

0

2

4

6

8

SNR (dB)−20 −10 0 10 20

(d) 𝑓 = 0.05HZ, 𝜎 = 0.002

Mea

n of

ampl

itude

SNR (dB)−20 −10 0 10 200

5

10

15

20

25

D.R.CAPESFFTDesignated

(e) 𝑓 = 0.05HZ, 𝜎 = 0.001

Std.

of a

mpl

itude

0

2

4

6

8

SNR (dB)−20 −10 0 10 20

D.R.CAPESFFT

(f) 𝑓 = 0.05HZ, 𝜎 = 0.001

Figure 7: Influence of noise on the identified amplitude spectrum, that is, mean and standard deviation. ((a), (b) mode 1, 𝜎 = 0.003; (c), (d)mode 2, 𝜎 = 0.002; (e), (f) mode 3, 𝜎 = 0.001).

four different damping factors. In each simulation, the noiselevel was divided into 100 steps with an initial SNR intervalof [−20, 20] dB. At each noise level, the simulation processincluded the generation of a random response and estimationof the amplitude spectrum using the proposed method andthe FFT method. The mean and standard deviation of the

results given by bothmethods at different SNRs are illustratedin Figure 7. For brevity, we only present three of the typicaldamping factors in this paper; that is, 0.003, 0.002, and 0.001.

Figures 7(a), 7(c), and 7(e) show that the proposed DR-CAPES method produces less bias than the FFT methodand that damping factor effects were separated from the


amplitude effects in the amplitude spectrum estimation. Theproposed method is not affected by the damping factor,whereas the FFT method performed poorly with a swing-intolerant bias when the damping factor was changed.

The curves for both methods gently plateau when theinitial SNR is set to a specific value, that is, −6.0 dB forFigure 7(a) and −10 dB for both Figures 7(c) and 7(e). Beforethis plateau, the mean of the estimated amplitude droppedsharply from a value that is far higher than the true amplitudeor designated value. The plateau in the curves proves that anoptimum workable SNR zone exists for each individual free-vibration response, which implies that a denoising process isneeded if the initial SNR of the raw response data is belowthis workable zone.

Figures 7(b), 7(d), and 7(f) show that the observationnoise level has a strong influence on the statistical qualitiesof the amplitude spectrum estimation, that is, the standarddeviation 𝜎. As the initial SNR was increased, the standarddeviation of both methods decreased. Under most noiselevels, the standard deviation of the proposed method wassignificantly greater than that of the FFT method. Thereason for this may be that the spectrum estimation of theproposed method is more complicated than that of the FFTmethod, where variance is propagated. A higher SNR meansa smaller standard deviation and statistically better spectrumestimation; this conclusion indicates that we should focus onimproving the quality of the amplitude spectrum.

5. Conclusion

The vibration response of structures in the real physicalworld is evaluated and modulated by amplitude damping. Byrepresenting the free vibration response in two dimensions(frequency and damping factor), we extended the hybridCAPES approach, which is a high-resolution amplitudespectrum estimation method, to estimate the amplitudespectrum of the structural vibration response. Comparedto the FFT method, the proposed DR-CAPES has a muchbetter resolution in the frequency domain and is, therefore,more suitable for overcoming the difficulties in spectrumestimation when vibration modes are close together. Thedownward bias of the amplitude spectrum estimated byDR-CAPES is smaller than that given by the FFT methodand is not influenced by the damping factor; therefore, theproposed method is more accurate than FFT-like approacheswhen determining amplitude. However, with higher SNRs,secondary operations, such as averaging and denoising, areneeded to improve the amplitude spectrum estimate.

Notations

𝜎 : Damping factor, decay𝜁: Damping ratio𝛿: Logarithmic decrement𝑐: Damping coefficient𝜔𝑑: Damped natural angular frequency

𝜔𝑛: Natural angular frequency

𝑚𝑗: Lumped mass at the 𝑗th degree of freedom(modal)

𝑘𝑗: Modal stiffness at the 𝑗th degree of free-

dom𝑥(0): Initial displacement𝐿: Total number of snapshots in the time

seriesΦCapon: Weight matrix for Capon spectral esti-

mation methodYvec: Vectorization of Hankel matrix𝜔: Angular frequency vector∠Ψ: Modal phase matrix𝛼: Frequency resolutionA𝑀: Defined as (9)

𝑅𝜔: Logarithm of frequency resolution

𝑎𝑠: Sinusoidal component of amplitude

𝑐𝑗: Modal stiffness at the 𝑗th degree of free-

domm: Diagnosis mass matrix (lumped mass

matrix)k: Modal stiffness matrix (diagnosis)c: Modal damping matrix (diagnosis)M: Mass matrixC: Damping matrixK: Stiffness matrix𝜔: Angular frequency (measured)��(0): Initial velocityY: Hankel matrix of observation dataJ: Least squares estimator matrix of the

weighted secondary residuals term 𝐽

ΦApes: Weight matrix for APES spectral estima-tion method

H: Observation matrix𝜎: Damping factor vectorRSN,0: Initial signal-to-noise ratio of each con-

tribution in each responseG(𝜔,𝜎)

: Defined by (21)B𝜔,𝜎

: Defined by (12)𝑎𝑐: Cosine component of amplitude

𝑒: Exponential function𝑛: Number of degrees of freedom𝑥: Response of steady forced vibration𝐴(⋅), 𝑎(⋅): Amplitude function𝜑(⋅): Phase function𝑡: Time series𝑓: Natural frequency𝑤: Residual term𝑁: Length of the time series𝑀: Finite impulse response (FIR) filter

length𝜃, ��wsl: Solution to the weighted least squares

parameter and its estimatorΘ, Θ: Weighted least squares parameter and its

estimator��sl: 2D spectral estimation of real signal in

the sense of least squares|Ψ|: Modal amplitude matrix𝜅: Closeness of two adjacent modes (or

adjacent peaks) in spectrum


C(𝜔,𝜎)

: Defined by (22)V𝜔,𝜎

: Defined by (11)n: Noise matrix.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This research is supported by the State High-Tech ResearchandDevelopment Plans (863), Grant no. 2014AA110402; StateMeteorological Administration Special Funds ofMeteorolog-ical Industry Research, Grant no. 201306102; the Project ofNational Key Technology R&D Program in the 12th Five YearPlan of China (Grant no. 2012BAJ11B01); the FundamentalResearch Funds for the Central Universities.

References

[1] J. Penzien, “Dynamics of structures,” Computers and Structures,Inc., 2003.

[2] D. J. Ewins, “Modal Testing,Theory, Practice, and Application,”2000.

[3] B. P. Lathi, Linear Systems and Signals, Oxford University Press,New York, NY, USA, 2nd edition, 2004.

[4] P. Stoica and R. Moses, Introduction to Spectral Analysis,Prentice Hall, Upper Saddle River, NJ, USA, 1997.

[5] D. P. Kalogeras,AnOverview of the Spectral EstimationMethods,vol. 12 ofMathematics andComputers in Science andEngineeringSeries, 2013.

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[8] J. Li and P. Stoica, “An adaptive filtering approach to spectralestimation and SAR imaging,” IEEE Transactions on SignalProcessing, vol. 44, no. 6, pp. 1469–1484, 1996.

[9] P. Stoica, H. Li, and J. Li, “New derivation of the APES filter,”IEEE Signal Processing Letters, vol. 6, no. 8, pp. 205–206, 1999.

[10] E. G. Larsson and J. Li,High-Resolution Nonparametric SpectralAnalysis: Theory and Applications, 2003.

[11] H. Li, J. Li, and P. Stoica, “Performance analysis of forward-backward matched-filterbank spectral estimators,” IEEE Trans-actions on Signal Processing, vol. 46, no. 7, pp. 1954–1966, 1998.

[12] A. Jakobsson and P. Stoica, “Combining Capon and APESfor estimation of spectral lines,” Circuits, Systems, and SignalProcessing, vol. 19, no. 2, pp. 159–169, 2000.

[13] E. G. Larsson, G. Liu, P. Stoica, and J. Li, “High-resolution SARimagingwith angular diversity,” IEEETransactions onAerospaceand Electronic Systems, vol. 37, no. 4, pp. 1359–1372, 2001.

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[15] Z.-S. Liu, R. Wu, and J. Li, “Complex ISAR imaging of maneu-vering targets via the Capon estimator,” IEEE Transactions onSignal Processing, vol. 47, no. 5, pp. 1262–1271, 1999.

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Research Article Damped CAPES 2D Spectral Estimation for ...Research Article Damped CAPES 2D Spectral Estimation for Real-Valued Vibration Signals DanhuiDan, 1 JiongxinGong, 1 andYimingZhao