Spectral density estimation of stochastic vector processes

Probabilistic Engineering Mechanics 17 (2002) 265±272

www.elsevier.com/locate/probengmech

Spectral density estimation of stochastic vector processes Ka-Veng Yuen a, Lambros S. Katafygiotis b, James L. Beck a,* a

Division of Engineering and Applied Science, California Institute of Technology, Mail Code 104-44, Caltech, Pasadena, CA 91125, USA b Department of Civil Engineering, Hong Kong University of Science and Technology, Hong Kong, People's Republic of China

Abstract A spectral density matrix estimator for stationary stochastic vector processes is studied. As the duration of the analyzed data tends to in®nity, the probability distribution for this estimator at each frequency approaches a complex Wishart distribution with mean equal to an aliased version of the power spectral density at that frequency. It is shown that the spectral density matrix estimators corresponding to different frequencies are asymptotically statistically independent. These properties hold for general stationary vector processes, not only Gaussian processes, and they allow ef®cient calculation of updated probabilities when formulating a Bayesian model updating problem in the frequency domain using response data. A three-degree-of-freedom Duf®ng oscillator is used to verify the results. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Stochastic vector process; Spectral density estimator; Aliasing; FFT; Stationary process

1. Introduction Frequency-domain analysis is widely used in almost every ®eld of research involving analysis of continuous or discrete time-varying signals. In particular, Fourier analysis is a very popular tool for time-history data processing because it provides an ef®cient and systematic procedure to capture important frequency-content information. A common task is to ®t a parameterized theoretical Fourier spectrum to the spectrum obtained from measured time histories. A straightforward least-squares ®t is usually found to give poor results. Therefore, it is desirable to develop a theoretically sound approach for ®tting spectra. In this paper, statistical properties of a spectral density estimator, such as the mean and the covariance matrix of the estimator, are presented for each frequency. Furthermore, it is shown that the spectral density estimator is asymptotically statistically independent at different frequencies for general stationary vector processes. The proof does not assume a Gaussian stochastic model for the signal process. A three-degree-of-freedom Duf®ng oscillator is used to verify the results. These results allow for a mathematically rigorous frequencydomain formulation of the problem of model updating within a Bayesian framework [1,2] using response data. The approach is applicable in the case of a stationary * Corresponding author. Tel.: 11-626-395-4132; fax: 11-626-568-2719. E-mail address: [email protected] (J.L. Beck).

stochastic model for the response of linear or non-linear systems subject to Gaussian or non-Gaussian input.

2. Spectral density estimator Consider a process generating time-history signals that are modeled as realizations of a real continuous-time stationary stochastic vector process x…t† ˆ ‰x1 …t†; x2 …t†; ¼; xd …t†ŠT ; that may be Gaussian or non-Gaussian and that has mean E‰x…t†Š ; m ˆ ‰m1 ; m2 ; ¼; md ŠT ; ;t [ R: Now, consider the corresponding discrete-time stochastic vector process {x…0†; x…Dt†; ¼; ¼; x……N 2 1†Dt†} whose realization corresponds to the sampled data in applications. We de®ne a related frequency-domain stochastic vector process XN with its a th element at frequency vk given by a modi®ed discrete Fourier transform 21 1 NX XaN …vk † ; p e2ivk jDt xa …jDt† N jˆ0

…1†

where vk ˆ kDv; k ˆ 0; 1; ¼; INT…N=2† and Dv ˆ 2p=NDt: Furthermore, denote the real and imaginary parts of XaN …vk † by RNa …vk † and IaN …vk †; respectively, that is: h i RNa …vk † ˆ Re XaN …vk † ;

0266-8920/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0266-892 0(02)00011-5

h i IaN …vk † ˆ Im XaN …vk †

…2†

266

K.-V. Yuen et al. / Probabilistic Engineering Mechanics 17 (2002) 265±272

Note that

rewritten as # "

21 h i ma NX m …1 2 e2ivk NDt †  e2ivk jDt ˆ pa E XaN …vk † ˆ p N …1 2 e2ivk Dt † N jˆ0

E S aN;b …vk †

m …1 2 e22kpi † ˆ pa ˆ0 N …1 2 e2ivk Dt †

ˆ

Therefore, E‰RNa …vk †Š ˆ 0 and E‰IaN …vk †Š ˆ 0: The …a; b† element of the spectral density matrix estimator is de®ned for a; b ˆ 1; 2; ¼; d by SNa;b …vk † ; ;

Dt N X …v †X N …v †p 2p a k b k 2 1 NX 21 Dt NX e2ivk …j2l†Dt xa …jDt†xb …lDt† 2pN jˆ0 lˆ0

ˆ

ˆ …3†

where zp denotes the complex conjugate of z: In Section 3, some important statistical properties of the spectral density estimator are introduced. In Section 4, it is shown that SN …v† and SN …v 0 † are independent asymptotically as N ! 1; where v ± v 0 and 0 , v; v 0 , p=Dt; the Nyquist frequency.

2 1 NX 21 Dt Z1 NX ei…V2v k †…j2l†Dt Sa;b …V† dV 2pN 2 1 jˆ0 lˆ0

Z1

NX 2 1 NX 21

2 1 jˆ0

Z1 21

lˆ0

! 1 i…j2v k Dt†…j2l† j e dj Sa;b 2pN Dt

N

F …j 2 v k Dt†Sa;b

! j dj Dt

…7†

where j ; VDt and F N …h† is the Fejer kernel [4]: F N … h† ˆ

sin2 …N h=2† 2pN sin2 …h=2†

…8†

Note that F N …h† is periodic with period 2p: By taking N!1 h i lim E SNa;b …vk †

N!1

3. Statistical properties of the spectral density estimator In this section, the probability density function of the spectral density estimator at a particular frequency vk is presented. First, by taking expectation of Eq. (3), one obtains the following E

h

i

SNa;b …vk †

2 1 NX 21 Dt NX ˆ e2ivk …j2l†Dt fa;b ‰…j 2 l†DtŠ …4† 2pN jˆ0 lˆ0

where fa;b ‰…j 2 l†DtŠ ; E‰xa …jDt†xb …lDt†Š 2 ma mb ; for a; b ˆ 1; 2; ¼; d; is the cross-covariance function between xa and xb with time lag …j 2 l†Dt: By grouping together terms with the same values of …j 2 l†Dt; one can obtain the following expression for the expected value of the spectral density estimator E‰SNa;b …vk †Š ˆ

Dt 4p

NX 21 jˆ0

h

cj fa;b …jDt†e2ivk jDt 1 fa;b …2jDt†eivk jDt

where cj is given by:   j c0 ˆ 1; cj ˆ 2 1 2 ; j$1 N

i …5†

…6†

Note that this relationship is exact and takes care of the aliasing and leakage effects automatically. Furthermore, E‰SNa;b …vk †Š can be calculated ef®ciently using the function FFT in Matlab [3]. R iVt By using fa;b …t† ˆ 1 dV; Eq. (4) can be 2 1 Sa;b …V†e

ˆ

Z1



1 X

2 1 jˆ2 1

ˆ S a ; b …v k † 1 ˆ Sa;b …vk † 1

d…j 2 v k Dt 1 2jp†Sa;b

 j dj Dt

    1  X 2jp 2jp Sa ; b v k 1 1 Sa;b vk 2 Dt Dt jˆ1 1  X jˆ1

 Sa;b

  p  2jp 2jp 1 vk 1 Sa;b 2 vk Dt Dt

…9† where the terms in the summation produce an aliased version of the power spectral density at frequency vk : Now, RNa …vk † and IaN …vk † are Gaussian distributed as N ! 1 by using the Central Limit Theorem [5]. Furthermore, as N ! 1; it was shown [6] that the covariance matrix G X …vk † of the vector ‰RN …vk †T ; IN …vk †T ŠT has the form: 2 6 6 lim G X …vk † ˆ 6 4 N!1

io io 3 1 n h N 1 n h N 2 Im E S …vk † 7 Re E S …vk † 2 2 7 n h i o io 7 5 1 1 n h N N Im E S …vk † Re E S …vk † 2 2 …10†

Eq. (10) states that RN …vk † and IN …vk † have equal covariance matrices …1=2†Re{E‰SN …vk †Š}; as N ! 1: Also, as N ! 1; the cross-covariance between RN …vk † and IN …vk † has the property 2Im{E‰SN …vk †Š} ˆ Im{E‰SN …vk †Š}T ; i.e. h i h i lim E RNa …vk †IbN …vk † ˆ 2 lim E IaN …vk †RNb …vk †

N!1

N!1

because of the symmetry of G X …vk †: The latter property

K.-V. Yuen et al. / Probabilistic Engineering Mechanics 17 (2002) 265±272

implies also that the diagonal elements of Im{E‰SN …vk †Š} are equal to zero as N ! 1; i.e. lim E‰RNa …vk †IaN …vk †Š ˆ 0;

mators are independent at any two different frequencies v and v 0 as N ! 1:

a ˆ 1; 2; ¼; d:

N!1

Therefore, the complex vector XN …vk † has a complex multivariate Gaussian distribution [7] with zero mean as N ! 1: Assume now that there is a set of M independent and identically distributed time histories whose realizations correspond to sampled data from the same process x…t†: As N ! 1; the corresponding frequency-domain stochastic processes XN;…m† …vk †; m ˆ 1; 2; ¼; M; are independent and follow an identical complex d-variate Gaussian distribution with zero mean [7]. Also, as N ! 1 and if M $ d; the average spectral density estimator M 1 X SN;M …vk † ˆ SN;…m† …vk † …11† M mˆ1 follows a central complex Wishart distribution of dimension d with M degrees-of-freedom [7,8] and mean E‰SN;M …vk †Š ˆ E‰SN …vk †Š given by Eq. (9): M2d 2 2d…d21†=2 M2d1d N;M # " p M S …vk † # " # M p SN;M …vk † ˆ " d Y …M 2 p†! E SN …vk † …12† pˆ1

( "

267

#21

£ exp 2 M tr E SN …vk †

)!

4.1. RNa …v† with RNb …v 0 † First, by using Eqs. (1) and (2), one can easily obtain the following: h h i h ii E RNa …v† 2 E RNa …v† RNb …v 0 † 2 E RNb …v 0 † h i h i h i ˆ E RNa …v†RNb …v 0 † 2 E RNa …v† E RNb …v 0 † ˆ

2 1 NX 21 1 NX …E‰xa …jDt†xb …lDt†Š N jˆ0 lˆ0

2 ma mb † cos…jvDt† cos…lv 0 Dt†

…15†

Let Sa;b …V†; which is assumed to be ®nite ;V [ R; be the cross-spectral density between x a and xb at frequency V: By using the cross-covariance function f a;b …t† ˆ R1 the fact ithat Vt dV and cos…z† ˆ …e iz 1 e2iz †=2, 2 1 Sa;b …V†e h h i h ii E RNa …v† 2 E RNa …v† RNb …v 0 † 2 E RNb …v 0 † ˆ

SN;M …vk †

Z1

2 1 NX 21 1 NX eiV…j2l†Dt Sa;b …V†‰e ijvDt 1 e2ijvDt Š 4N jˆ0 lˆ0

21

0

0

 ‰eilv Dt 1 e2ilv Dt Š dV

where the …a; b† element of the matrix E‰SN …vk †Š is given by Eq. (5). Here, uAu and tr‰AŠ denote the determinant and the trace, respectively, of a matrix A: Note that in the special case of d ˆ 1; this distribution reduces to a Chi-square distribution with 2M degrees-offreedom and mean E‰SN …vk †Š; which is given by: h iM21 ! h i M M SN;M …vk † MSN;M …vk † N;M  p S …vk † ˆ    exp 2  N E S …vk † …M 2 1†! E SN …vk † M …13† Another special case is when M ˆ 1 (i.e. no averaging is performed), then each of the diagonal elements SNa;a …vk †; a ˆ 1; 2; ¼; d follows an exponential distribution as N ! 1 : ! h i SNa;a …vk † 1 N  exp 2  N  …14† p Sa;a …vk † ˆ  N E Sa;a …vk † E Sa;a …vk † In this section, the PDF of the spectral density matrix estimator was given at a particular frequency. In Section 4, it is shown that the spectral density matrix estimators are independent at different frequencies. 4. Asymptotic independence properties of the spectral density estimator In this section, it is shown that the spectral density esti-

ˆ

Z1

2 1 NX 21 0 1 NX ‰e ij…V1v†Dt1il…v 2V†Dt 4N jˆ0 lˆ0

21

0

0

1 eij…V1v†Dt1il…2v 2V†Dt 1 eij…V2v†Dt1il…v 2V†Dt 0

1 eij…V2v†Dt1il…2v 2V†Dt ŠSa;b …V† dV ˆ

Z1 21

1 1

H aN;b …V; v; v 0 † dV 1

Z1 21

Z1 21

Z1 21

H aN;b …V; v; 2v 0 † dV

H aN;b …V; 2v; v 0 † dV H aN;b …V; 2v; 2v 0 † dV

(16)

where H N …V; v; v 0 † is de®ned as: H aN;b …V; v; v 0 † ;

21 NX 21 0 S a;b …V† NX e ij…V1v†Dt eil…v 2V†Dt 4N jˆ0 lˆ0

…17† If uVu ± v and uVu ± v

0 0

H aN;b …V; v; v 0 † ˆ

‰1 2 e iNDt…V1v† Š‰1 2 eiNDt…v 2V† Š Sa;b …V† 0 4N‰1 2 eiDt…V1v† Š‰1 2 eiDt…v 2V† Š …18†

268

K.-V. Yuen et al. / Probabilistic Engineering Mechanics 17 (2002) 265±272

By using sin…z† ˆ …e iz 2 e2iz †=2i,

tend to zero as N ! 1 and so from Eq. (16): h h i h ii lim E R aN …v† 2 E RNa …v† RNb …v 0 † 2 E RNb …v 0 †

HaN;b …V; v; v 0 † #  " 0 …V 1 v†NDt …v 2 V†NDt i…N21†Dt…v1v 0 †=2 e sin sin 2 2 # ˆ S a;b …V†   " 0 …V 1 v†Dt …v 2 V†Dt sin 4N sin 2 2 

…19† Then, the following inequality can be obtained since ue ir u ˆ 1 and usin…r†u # 1 …r [ R† : S … V † a ; b N 0 " # Ha;b …V; v; v † #   0 …V 1 v†Dt …v 2 V†Dt 4N sin sin 2 2 …20† By taking the limit as N ! 1 lim H aN;b …V; v; v 0 † ˆ 0

if uVu ± v and uVu ± v 0

N!1

…21†

Similarly, H aN;b …V; 2v; v 0 †; H aN;b …V; v; 2v 0 † and H aN;b …V; 2v; 2v 0 † also tend to zero as N ! 1; if uVu ± v and uVu ± v 0 : Now, we consider H aN;b …V; v; v 0 † at uVu ˆ v or uVu ˆ v 0 : First, at V ˆ 2v H aN;b …V; v; v 0 † ˆ

ˆ

21 0 S a;b …V† NX e il…v 2V†Dt 4 lˆ0

Sa;b …V†‰1 2 e

iNDt…v 0 2V†

Š

N!1

ˆ 0; if v ± v 0 4.2. IaN …v† with IbN …v 0 †

Similarly, it can be proved that limN!1 E‰…IaN …v† 2 E‰IaN …v†Š†…IbN …v 0 † 2 E‰IbN …v 0 †Š†Š ˆ 0 as follows " #! " #!# " IaN …v† 2 E IaN …v†

E ˆ

H aN;b …V; v; v 0 † ˆ

2 1 NX 21 1 NX …E‰xa …jDt†xb …lDt†Š N jˆ0 lˆ0

0

0 0

1 eij…V2v†Dt1il…2v 2V†Dt ŠSa;b …V† dV " 21 NX 21 Z1 1 NX 0 ˆ2 e ij…V1v†Dt eil…v 2V†Dt 4N 21 jˆ0 lˆ0 2 2 1 ˆ2 1

As N ! 1 and V ˆ v or 2v 0 , …24†

1

That is, H aN;b …V; v; v 0 † is ®nite at V ˆ ^v and V ˆ ^v 0 : Similarly, it can be shown that H aN;b …V; v; 2v 0 †; H aN;b …V; 2v; v 0 † and H aN;b …V; 2v; 2v 0 † are ®nite at V ˆ ^v and V ˆ ^v 0 : Therefore, H aN;b …V; v; v 0 †; H aN;b …V; v; 2v 0 †; H aN;b …V; 2v; v 0 † and H aN;b …V; 2v; 2v 0 † tend to zero as N ! 1 if uVu ± v and uVu ± v 0 and they are ®nite as N ! 1 if uVRu ˆ v or uVu ˆ v 0 : ItR can be concluded 1 1 N 0 N 0 that 2 1 H a;b …V; v; v † dV; R 2 1 H a;b …V; v; 2v † dV; R 1 1 N 0 N 0 2 1 H a;b …V; 2v; v † dV and 2 1 H a;b …V; 2v; 2v † dV

2

lim H aN;b …V; v; v 0 † ˆ 0

N!1

0

2 eij…V1v†Dt1il…2v 2V†Dt 2 eij…V2v†Dt1il…v 2V†Dt

iNDt…v 0 2V†

‰1 2 e Š‰1 2 e Š Sa;b …V† iDt…V1v† iDt…v 0 2V† Š‰1 2 e Š 4N‰1 2 e …23†

0

 ‰e ijvDt 2 e2ijvDt Š‰eilv Dt 2 e2ilv Dt Š dV 2 1 NX 21 Z1 1 NX 0 ˆ2 ‰e ij…V1v†Dt1il…v 2V†Dt 2 1 4N jˆ0 lˆ0

which is ®nite. Similarly, it can be shown that HaN;b …V; v; v 0 † is ®nite at V ˆ v 0 : Next, consider V ˆ v or V ˆ 2v 0 iNDt…V1v†

IbN …v 0 † 2 E IbN …v 0 †

2 ma mb † sin…jvDt† sin…lv 0 Dt† 2 1 NX 21 1 Z1 NX eiV…j2l†Dt Sa;b …V† ˆ2 4N 2 1 jˆ0 lˆ0

…22†

0

4‰1 2 eiDt…v 2V† Š

…25†

NX 21

eij…V1v†Dt

NX 21

jˆ0

lˆ0

NX 21

NX 21

eij…V2v†Dt

jˆ0

lˆ0

NX 21

NX 21

e

jˆ0 Z1 21 Z1 21 Z1 21 Z1 21

ij…V2v†Dt

0

eil…2v 2V†Dt 0

eil…v 2V†Dt # e

il…2v 0 2V†Dt

lˆ0

Sa ; b … V † dV

H aN;b …V; v; v 0 † dV H aN;b …V; v; 2v 0 † dV H aN;b …V; 2v; v 0 † dV H aN;b …V; 2v; 2v 0 † dV

…26†

where H aN;b is given by Eq. (17). R1 R1 N 0 As shown in Section 4.1, 2 1 Ha;b …V; v; v † dV; 2 1 R 1 N 0 N 0 H and 2 1 H a ; b … V ; 2 v ; v † dV R1a;b …V;Nv; 2v † dV; 0 2 1 H a;b …V; 2v; 2v † dV tend to zero as N ! 1; so h h i h ii lim E I aN …v† 2 E IaN …v† IbN …v 0 † 2 E IbN …v 0 † ˆ 0; N!1

if v ± v 0

(27)

K.-V. Yuen et al. / Probabilistic Engineering Mechanics 17 (2002) 265±272

269

4.3. RNa …v† with IbN …v 0 † Finally, we prove: limN!1 E‰…RNa …v† 2 E‰RNa …v†Š† £ 2 E‰IbN …v 0 †Š†Š ˆ 0

…IbN …v 0 † "

" RNa …v†

E

ˆ

2E

#!

"

RNa …v†

IbN …v 0 †

#!#

IbN …v 0 †

2E

2 1 NX 21 1 NX …E‰xa …jDt†xb …lDt†Š N jˆ0 lˆ0

2 ma mb † cos…jvDt† sin…lv 0 Dt† ˆ

Z1 21

2 1 NX 21 1 NX eiV…j2l†Dt Sa;b …V†‰e ijvDt 1 e2ijvDt Š 4Ni jˆ0 lˆ0 0

0

 ‰eilv Dt 2 e2ilv Dt Š dV ˆ

Z1 21

2e

R1

21

1e

H aN;b …V; 2v; 2v 0 † dV tend to zero as N ! 1; so

h h i h ii lim E R aN …v† 2 E RNa …v† IbN …v 0 † 2 E IbN …v 0 †

2 1 NX 21 0 1 NX ‰e ij…V1v†Dt1il…v 2V†Dt 4Ni jˆ0 lˆ0

ij…V1v†Dt1il…2v 0 2V†Dt

Fig. 1. Three-DOF Duf®ng oscillator.

N!1

if v ± v 0

ˆ 0;

ij…V2v†Dt1il…v 0 2V†Dt

(29)

0

2 eij…V2v†Dt1il…2v 2V†Dt ŠSa;b …V† dV ˆ

Z1 21

2

1 4Ni

NX 21

"

NX 21 jˆ0

eij…V1v†Dt

jˆ0

1

NX 21

2

eij…V2v†Dt

e

Z1

1i 2i 1i

0

eil…v 2V†Dt

We conclude that any element in the set {RNa …v†; IaN …v†; RNb …v†; IbN …v†} with any element in the set {RNg …v 0 †; IgN …v 0 †; RNd …v 0 †; IdN …v 0 †} gives an uncorrelated pair, where a; b; g; d ˆ 1; 2; ¼; d and v ± v 0 : Furthermore, RNa …V† and I aN …V† are Gaussian distributed ;V [ R and a ˆ 1; 2; ¼; d as N ! 1 even if the stochastic process x is not Gaussian (Section 3). Since uncorrelated Gaussian random variables are independent, as N ! 1 each element in the set {R aN …v†; IaN …v†; RNb …v†; IbN …v†} is statistically independent of each element in the set {RNg …v 0 †; IgN …v 0 †; RNd …v 0 †; IdN …v 0 †}; where a; b; g; d ˆ 1; 2; ¼; d and v ± v 0 : By using Eqs. (1)±(3), the following can be obtained:

lˆ0 NX 21

0

eil…2v 2V†Dt

NX 21

0

eil…v 2V†Dt

lˆ0 ij…V2v†Dt

jˆ0

ˆ 2i

4.4. SNa;b …v† with SNg;d …v 0 †

lˆ0

jˆ0 NX 21

e ij…V1v†Dt

NX 21

21

NX 21

# e

il…2v 0 2V†Dt

lˆ0

Sa;b …V† dV

H aN;b …V; v; v 0 † dV

Z1 21

Z1 21

Z1 21

Dt N X …v†XbNp …v† 2p a # (" Dt N N N N Ra …v†Rb …v† 1 Ia …v†Ib …v† ˆ 2p

SNa;b …v† ˆ

H aN;b …V; v; 2v 0 † dV H aN;b …V; 2v; v 0 † dV H aN;b …V; 2v; 2v 0 † dV

" (28)

where H aNR;b is given by Eq. (17). R 1 1 N 0 N 0 2 1 Ha;b …V; v; v † dV; 2 1 H a;b …V; v;2v † dV; R1Again, N 0 2 1 H a;b …V; 2v; v † dV and

1i

…30†

#) IaN …v†RNb …v†

2

RNa …v†IbN …v†

Therefore, SNa;b …v† and SNg;d …v 0 † are statistically independent if a; b; g; d ˆ 1; 2; ¼; d; v ± v 0 and 0 , v; v 0 , p=Dt; the Nyquist frequency.

270

K.-V. Yuen et al. / Probabilistic Engineering Mechanics 17 (2002) 265±272

5. Illustrative example Consider a three-degree-of-freedom Duf®ng oscillator as shown in Fig. 1: 2

x1

3

2

c1 1 c2

6 7 6 7 6 M6 4 x2 5 1 4 2c2 x3 0

k1 1 k2 0

Fig. 2. PDF of the system response at the 1st DOF.

0

2c3 2k2

k2 1 k3 2k3

0

32

x_1

3

76 7 6 7 2c3 7 54 x_2 5

c2 1 c3

0

B 1B @ 2k2 2

2c2

x_3

c3 10

x1

1

CB C B C 2k3 C A@ x2 A k3

m1 x31 1 m2 …x1 2 x2 †3

x3 3

6 7 6 7 1 6 m2 …x2 2 x1 †3 1 m3 …x2 2 x3 †3 7 4 5 m3 …x3 2 x2 †3 2

f1 …t†

3

6 7 7 ˆ6 4 f2 …t† 5

…31†

f3 …t† where M ˆ I kg is the mass matrix, cj ˆ 2:0 £ 1023 N s/cm, kj ˆ 0:5 N/cm and mj ˆ 0:1 N/cm 3 are the damping coef®cient, linear stiffness and third order stiffness for the jth ¯oor

Fig. 3. PDF of the real part of the FFT for the 1st DOF response.

K.-V. Yuen et al. / Probabilistic Engineering Mechanics 17 (2002) 265±272

Fig. 4. CDF of the real part of the FFT for the 1st DOF response.

Fig. 5. Correlation coef®cients of SN1;1 …v†:

271

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K.-V. Yuen et al. / Probabilistic Engineering Mechanics 17 (2002) 265±272

… j ˆ 1; 2; 3†; respectively. The small amplitude fundamental frequency of the structure is 0.5 Hz. The external force ‰f1 ; f2 ; f3 ŠT is assumed to be independent Gaussian white noise with spectral intensity 5:0 £ 1024 N 2/s. The response time history is simulated using the function `ODE45' in Matlab [3]. The duration and sampling time interval are 5000 s and 0.02 s, respectively. Fig. 2 shows the PDF of the 1st DOF response p…x1 †; estimated using 250 000 samples (solid) and its best ®tting curve in the class of Gaussian distributions. It can be seen that the structural response is non-Gaussian [9]. The standard deviation of these samples is 1.2 cm. Two thousand realizations, each with duration 100 s and sampling time interval 0.02 s, are generated and their FFTs are calculated. Fig. 3 shows the PDF for RN1 …v† at v ˆ 0:25; 0.5, 1.0, 1.5 Hz. It can be seen that the Gaussian approximation for the real part of the FFT is accurate although the structural response is not Gaussian. Furthermore, the corresponding cumulative distribution function (CDF) is shown in Fig. 4. The solid line corresponds to the best ®tting Gaussian CDF and the dashed line corresponds to the CDF estimated using simulation. It can be seen that the two curves lie virtually on top of each other, implying that the real part of the FFT of the spectral density is well approximated by the Gaussian distribution. The same conclusion can be drawn for the imaginary part and for the response of other ¯oors. Fig. 5 shows the sample correlation coef®cients for the spectral density of the response at the 1st DOF SN1;1 …v† at 0.25, 0.5, 1.0, 1.5 Hz with other frequencies. It can be seen that the coef®cients of correlation are around zero at each frequency, except when the frequencies match where it is unity. The same conclusion can be drawn for the other components (auto or cross-terms) of the spectral density matrix.

6. Concluding remarks A spectral density matrix estimator is de®ned based on a ®nite number, N, of data points which takes care of aliasing and leakage effects automatically. Furthermore, the probability density function of this spectral density matrix estimator for a general stationary stochastic vector process as N ! 1 is presented. It is also proved that for such a process, the spectral density estimators corresponding to different frequencies are asymptotically independent as N ! 1: This implies that the spectral density estimators of the response of a linear or non-linear dynamical system subject to input modeled as a stationary stochastic process are always asymptotically independent at different frequencies as N ! 1:

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