The Method of Separation: A Novel Approach for Accurate Estimation of Evolutionary Power Spectra Dominik Schillinger1 and Vissarion Papadopoulos2 1
Chair for Computation in Engineering, Department of Civil Engineering and Surveying, Technische Universität München, Arcisstr. 21, 80333 München., Germany [email protected]
2 Institute of Structural Analysis and Seismic Research, National Technical University of Athens, 9 Iroon Polytechneiou, Zografou Campus, Athens 15780, Greece [email protected]
Abstract One of the most widely used techniques for the simulation of Gaussian evolutionary random fields is the spectral representation method. Its key quantity is the power spectrum, which characterizes the random field in terms of frequency content and spatial evolution in a mean square sense. For the simulation of a random physical phenomenon, the power spectrum can be directly obtained from corresponding measured samples by means of estimation techniques. The present contribution starts with a short review of established power spectrum estimation techniques, which are based on the short-time Fourier, the harmonic wavelet and the Wigner-Ville transforms, and subsequently introduces a method for the estimation of separable random fields, called the method of separation. The characteristic drawbacks of the established methods, i.e. the limitation of simultaneous space-frequency localization or the appearance of negative spectral density, lead to poor estimation results, if the Fourier transform of the input samples consists of a narrow band of frequencies. The proposed method of separation, combining accurate spectrum resolution in space with an optimum localization in frequency, considerably improves the estimation accuracy in the presence of strong narrow-bandedness, which is illustrated by a practical example from stochastic imperfection modeling in structures.
Introduction Within the last three decades, computational stochastic mechanics has evolved into a self-contained and prolific field of research, which has brought forth a wide range of sophisticated and wellestablished methodologies for the stochastic simulation of uncertain engineering systems (see [5,22] and the references therein). One emerging field of application is the stability analysis of thin-walled structures, where the random variability of geometric and material imperfections leads to considerable uncertainty in corresponding buckling loads [13,14,19]. Apart from methodological maturity, the quality of practical stochastic simulations predominantly depends on the accurate reproduction of the random physical key phenomena by corresponding random field models. The most widely used technique for the simulation of imperfections as random fields is the spectral representation method [8,21,23]. The key quantity of spectral representation is the power spectrum [15,17,18], which is related to the average energy of the random field and is obtained in practical applications by estimation from a series of measurements [3,8,14, 22]. Despite its decisive importance for realistic stochastic buckling simulations, only little experience exists so far in transferring experimental imperfection measurements, which typically are strongly narrow-band functions at very low frequencies, into accurate evolutionary power spectra. Against this background, the present contribution intends to shed some light on key issues related to the evolutionary power spectrum estimation of strongly narrow-band random fields, with special emphasis on their application to stochastic imperfection modeling in structures. First, a concise review of the most important existing methods for evolutionary power spectrum estimation is presented, which are based on the short-time Fourier, the harmonic wavelet and the Wigner-Ville transforms. Second, a simple method for evolutionary spectrum estimation of separable random fields is introduced, referred to as the method of separation in the following . Third, the presented methods are applied to the estimation of two benchmark spectra, i.e. the modulated Kanai-Tajimi spectrum  and the power spectrum of geometric imperfections in an I-section
flange . The results demonstrate the considerable advantage of the method of separation in the presence of strong narrow-bandedness due to its accurate simultaneous space-frequency localization.
Relevant elements of stochastic process theory A Gaussian random field h(x), equivalently known as a stochastic process in a time context, represents an ensemble of spatial functions, whose exact values are a-priori indeterminate, but follow a Gaussian probability distribution. It can be split into a deterministic mean μ(x) and a zero-mean Gaussian random field f(x) in the form h(x)=μ(x)+f(x). In view of the spectral decomposition of its realizations into trigonometric functions, the Gaussian zero-mean random field f(x) can be characterized by a two-sided power spectrum S, which specifies the stochastic frequency content and its correlation along the spatial axis [12,15,17,18,25]. It is called homogeneous, if S(ω) depends only on frequency ω, and evolutionary, if S(ω,x) depends on both frequency ω and space x. An intuitive approach to the power spectrum is its interpretation as the distribution of the mean square of the random field over the space-frequency domain, so that it follows E ⎡ f ( x) ⎢⎣
⎤ = 2 ∞ S (ω , x) dω ∫0 ⎥⎦
Analogous to Eq. (1), the incremental energy in frequency reads ⎡ E ⎢ F (ω ) ⎣
⎤ ⎥⎦ =
S (ω , x) dx
Eqs. (1) and (2) are also known as the marginal spectral densities of a random field . The power spectrum is called narrow-band, if the bulk of its energy is located only within a very small frequency band [12,15]. It satisfies spectral separability, if it can be multiplicatively decomposed into a homogeneous spectrum S(ω) and a modulating envelope g(x) in the form
S (ω , x) = S (ω ) ⋅ g ( x)
If the power spectrum S(ω,x) of a random field is known, an arbitrary number m of corresponding random samples can be generated by the spectral representation method [21,23], which reads for a onedimensional univariate zero-mean Gaussian random field f ( i ) ( x) =
cos ω n x + φ n(i )
⎧ An = 2 ⋅ S (ω n , x) ⋅ Δω ⎪ ⎪ ω = n ⋅ Δω with ⎨ n ⎪ Δω = ωup N ⎪ A = 0 or S (ω = 0, x) = 0 0 ⎩ 0
where i=1,2,…,m and n=0,1,2,…(N-1). The parameter ωup is the cutoff frequency, beyond which the power spectrum is assumed to be zero, the integer N determines the discretization of the active frequency range, and φn(i) denotes the (i)th realization of N independent phase angles uniformly distributed in the range [0,2π]. For nonGaussian random fields, the translation field theory can be used to generate random samples from a simple transformation of an underlying Gaussian field . The spectrum estimation methods to be presented in the following are tested by a standard separable benchmark problem, the uniformly modulated Kanai-Tajimi spectrum [9,22], which is defined in view of Eq. (3) by its separable components ω ⎞ ⎟⎟ 1 + 4ς ⎜⎜ ω ⎝ 0⎠ 2⎛
S (ω ) =
⎡⎛ ⎢ ⎜1 − ⎛⎜ ω ⎢ ⎜ ⎜ ω0 ⎢⎣ ⎝ ⎝
⎞ ⎟⎟ ⎠
2 2⎤ ⎞ ⎟ + ⎛⎜ 2ς ω ⎞⎟ ⎥ ⎜ ω ⎟ ⎥ ⎟ 0 ⎠ ⎝ ⎠ ⎥⎦
Power Spectrum Magnitude [mm 3/rad]
15 20 Frequency ω [rad/mm]
(a) Analytical homogeneous spectrum S(ω)
20 Length x [mm]
(b) Analytical modulating envelope g(x)
Fig. 1. Components of the separable Kanai-Tajimi benchmark spectrum
g ( x) =
e −0.25 x − e −0.5 x 0.25
Parameters ω 0 = 10 rad mm and ς = 0.24 represent the natural frequency and the damping ratio, respectively. Their specific values in conjunction with Fig. 2. Analytical reference solution of the exponential modulating the separable Kanai-Tajimi spectrum function of Eq. (7) are adopted from  and yield a power spectrum with equally pronounced evolution in space and frequency directions, therefore representing a suitable benchmark for evolutionary estimation techniques (see Figs. 1 and 2). The Kanai-Tajimi spectrum of Fig. 2 is used to generate 10,000 corresponding random field samples f (i)(x), i=1,2 ,…,10,000 via the spectral representation formula Eq. (4). Estimates of the KanaiTajimi spectrum are obtained by substituting samples f (i)(x) into the estimation methods introduced in the following, whose results can be assessed by comparing them to the analytical reference of Fig. 2.
Fig. 3. Periodogram based estimate of the Kanai-Tajimi spectrum
Fig. 4. STFT based estimate of the Kanai-Tajimi Spectrum
Existing Methods for Evolutionary Spectrum Estimation The estimation of a homogeneous Fourier power spectrum is a standard method [12,15], which can be obtained from a series of samples f (i)(x) by the so-called periodogram ⎡ 1 ~ ⋅ S h (ω ) = E ⎢ ⎢⎣ 2π L
2 ⎤ f (i ) ( x) ⋅ e − I ω x dx ⎥ ⎥⎦
with L being the total sample length. The homogeneous spectrum es~ timate S h (ω ) of the evolutionary Kanai-Tajimi spectrum is shown in Fig. 3. Whereas the energy distribution in frequency direction is predicted correctly, the spatial location of the energy peak is lost, since the Fourier transform in Eq. (8) averages the energy variation in space over the whole length L. To preserve this information, samples f (i)(x) have to be transferred into evolutionary power spectra.
The short-time Fourier transform The most common approach for evolutionary power spectrum estimation is based on the short-time Fourier transform (STFT), also re-
ferred to as moving window or Gabor transform [1,2]. The idea of STFT is to emphasize the samples f (i)(x) at a distinct spatial position x = χ , whose local properties are to be studied, and to suppress them at positions farther away from χ . This is achieved by multiplying the samples with a window w ( x − χ ) of finite width T centered at the position of interest χ . The window is then moved along the spatial axis at n equally-spaced positions x = χ j , j = 1 ... n , and the Fourier transform is applied at each χ j . This strategy results in ho~ mogeneous spectrum components S j(ω,x) for each χ j ⎡ ~ 1 S j (ω , x) = E ⎢ ⋅ ⎢ 2π T ⎣
χ +T / 2
χ −T / 2
( x) ⋅ w x − χ
⎤ dx ⎥ ⎥ ⎦ 2
−I ω x
The complete evolutionary spectrum estimate, also known as the spectrogram, can finally be obtained by combining all components j=1,2,..,N. The basic limitation of the method is its inability to achieve simultaneous localization in both frequency and space, which is a consequence of the uncertainty principle [1,2,16]. If the spatial localization is increased by shortening the window width T , the frequency resolution of the spectrum deteriorates. In turn, if the width T is increased, frequencies are resolved better, but the spatial localization is reduced. The result of the STFT based estimation for the Kanai-Tajimi benchmark spectrum is shown in Fig. 4 and is obtained with a simple non-overlapping rectangular window ⎧ 1 w( x) = ⎨ ⎩ 0
−T /2 ≤ x ≤ T /2 elsewhere
centered at 16 equally-spaced positions χ j . A compromise in the spatial localization of the window function is chosen, that allows for a fair localization in space without distorting the frequency localization too severely.
Fig. 5. Harmonic wavelet based estimate of the benchmark spectrum
Fig. 6. Wigner-Ville based estimate of the Kanai-Tajimi spectrum
The harmonic wavelet transform For the joint space-frequency representation of evolutionary power spectra, harmonic wavelets developed by Newland  have proved to be especially suitable due to their exact box-like Fourier spectrum. Wavelet functions of different frequency scales (m, n) and positions k can be interpreted as a collection of window functions that are mutually orthogonal, so that a generalized wavelet transform of a series of samples f (i ) ( x) can be constructed, resulting in wavelet coefficients a(i)(n,m),k. Because of the strongly localized amplitudes of harmonic wavelets of the same frequency scale (m, n) at different positions x − k /(n − m) , corresponding wavelet coefficients a(i)(n,m),k can be used to distinguish local events of samples f (i ) ( x) at the same frequency. On this basis, Spanos and co-workers  have recently shown that an evolutionary power spectrum can be estimated from the wavelet coefficients as ~ S ( m,n ),k
2 4 E ⎡ a((mi ),n ),k ⎤ ⎢⎣ ⎥⎦ n−m
This formula defines the localized spectral density in the discrete space-frequency regions
m 2π n 2π ≤ ω < L L
kL (k + 1) L ≤ x < n−m n−m
where integers k = 0 K (n − m − 1) determine the coupling between space and frequency localization. The major drawback of the harmonic wavelet based method is again the limitation of simultaneous space-frequency localization by Eqs. (12). They allow either for a fine resolution in frequency, if differences in scales (m, n ) are small, or for a fine resolution in space, if differences in scales (m, n ) are large. The wavelet based estimate for the Kanai-Tajimi benchmark is shown in Fig. 5, which shows reasonable localization properties due to a compromise in space-frequency resolution.
The Wigner-Ville transform A qualitatively different approach is provided by the Wigner-Ville transform , which yields the following evolutionary spectrum expression from a series of samples f (i)(x) ~ ⎡ 1 L S (ω , x) = E ⎢ ⋅∫ f ⎣ 2π 0
τ ⎞ (i ) ⎛ τ ⎞ − I ω τ ⎤ ⎛ dτ ⎥ ⎜x + ⎟⋅ f ⎜x − ⎟⋅e 2⎠ 2⎠ ⎝ ⎝ ⎦
where τ is a shifting parameter. Eq. (13) can be interpreted as a folding of the left over the right part of the sample to determine possible overlaps, which will be then present in the spectrum at the current position x . In a more formal sense, Eq. (13) can be interpreted as the Fourier transform of an autocorrelation estimate . The major strength of the Wigner-Ville based spectrum estimate is the exact representation of the marginal densities Eqs. (1) and (2), which guarantees an exact representation of the incremental energy content of the spectrum. Its major drawback is its potential ability to yield negative spectral values, which contradicts the mathematical definition of the power spectrum and its physical energy interpretation [1,2]. The Wigner-Ville estimate of the Kanai-Tajimi benchmark spectrum is shown in Fig. 6.
Fig. 7. Wigner-Ville Based Estimate of the Kanai-Tajimi Spectrum
Fig. 8. Convergence behaviour of the examined estimation techniques for the Kanai-Tajimi benchmark
The Method of Separation: Evolutionary Spectrum Estimation of Separable Random Fields The method of separation is generally valid for the estimation of separable power spectra, but in particular designed to cope with the challenge of accurate simultaneous space-frequency localization. For the estimation of non-separable power spectra, it can be complemented by a joint strategy, which is based on the partitioning of the space-frequency domain into several sub-spectra that have to be separable only within themselves. These sub-spectra can then be estimated consecutively by the method of separation and finally be recomposed to the full non-separable spectrum. For the introduction of the principles of the method, the present text confines itself to the separable case, details on the estimation of non-separable spectra with the method of separation can be found in Schillinger and Papadopoulos .
Theory and Derivation The method of separation assumes that input samples f (i ) ( x) represent a separable or at least approximately separable random field. The essential advantage of this assumption is the breakdown of the
combined evolutionary spectrum estimation into a frequency and a spatial part, which can be dealt with separately. The definition of spectral separability in Eq. (3) allows that its multiplicative components can be chosen arbitrarily from a group of pairs [ S ′; g ′ ] that satisfy with respect to the original components [ S; g ] S ′(ω ) = λ ⋅ S (ω ) 1 g ′( x) = ⋅ g ( x)
where λ is an arbitrary positive number. Eqs. (14) constitute sets of geometrically similar functions, whose energy content is varied by λ , but whose energy distribution over frequency or space, i.e. the relative shapes of the curves, remain the same. The product of Eqs. (14) yields always the correct evolutionary spectrum S(ω,x) in the sense of Eq. (3). In the method of separation, the spectrum component S ′ of Eq. (14) is chosen as the homogeneous Fourier power spectrum S h (ω ) =
1 L ⋅ ∫ S (ω , x) dx = λ h ⋅ S (ω ) L 0
corresponding to the frequency content of the evolutionary spectrum S (ω , x) averaged over the signal length L. Its counterpart g h (x) can be obtained from the first expression of Eqs. (14) by factor λh
1 ⋅ g (x) dx L ∫
The evolutionary spectrum S (ω, x) in the method of separation is thus decomposed into S (ω , x) = S h (ω ) ⋅ g h ( x)
An estimate S h (ω ) of Eq. (15) can be readily obtained from the periodogram Eq. (8), where the frequency content of the separable input samples are averaged over their length L by the Fourier transform [1,2], which corresponds to the definition of S h (ω ) in Eq. (15).
An estimate g~h ( x) for the spatial envelope can be derived from the mean square of samples f (i ) ( x) . The basic analytical expression is found from Eq. (1) as ⎡ E ⎢ f ( x) ⎣
∞ ⎤ 2 = ∫ S h (ω ) g h ( x) dω ⎥⎦ 0
g h ( x) ⋅ 2 ∫ S h (ω ) dω 0
The estimate g~h ( x) is obtained by replacing in Eq. (18) the analytical homogeneous Fourier spectrum S h (ω ) and the mean square 2 E [ f (x) ] by corresponding estimates, which yields
g~h ( x) =
2 E ⎡ f (i ) ( x) ⎤ ⎢⎣ ⎥⎦ ∞ ~ 2 ∫ S h (ω ) dω
The final estimate of the spectrum can now be established by replac~ ing the analytical expressions in Eq. (18) by their estimates S h (ω ) of Eqn. (9) and g~h ( x) of Eq. (18), which results in 2 ~ S (ω , x) = E ⎡ f (i ) ( x) ⎤ ⋅ ⎥⎦ ⎢⎣
~ S h (ω ) ∞ ~ 2 ∫ S h (ω ) dω
Factor 1/2 in the right-hand side fraction is necessary, because Eq. (20) takes into account only one side of the symmetric two-sided power spectrum. The validity of Eq. (20) can be succinctly illustrated by energy considerations. The right-hand side fraction contains the estimate ~ S h (ω ) in the numerator, which is normalized by the denominator in the sense that its total energy content, i.e. the area under the two~ sided curve S h (ω ) , is a constant of 1. This can be trivially verified by integration of the right-hand side fraction in Eq. (20) over frequency as
~ S h (ω )
~ ∫ S h (ω ) dω 0
The left-hand side of Eq. (20) represents the mean square of the samples f (i ) ( x) , which on the basis of Eq. (2) is an estimation of the incremental energy in space. Hence, Eq. (21) can be interpreted as the distribution of the mean square over the frequency domain by a normalized homogeneous spectrum for each position x . This makes the method of separation a direct implementation of the initial intuitive concept of the power spectrum.
Performance Test with Kanai-Tajimi Benchmark Estimation The complete Kanai-Tajimi spectrum estimate obtained with the method of separation is shown in Fig. 7. The method of separation captures exact spectrum gradients and peak values as seen in Fig. 2 considerably better than STFT and wavelet transforms of Figs. 4 and 5 and leads to a more regular spectrum surface than the Wigner– Ville transform of Fig. 6. Stochastic convergence in a Monte-Carlo sense  is tested by the squared difference between exact analytical and estimated Ka~ nai–Tajimi spectra S ex and S m , respectively, integrated over the space-frequency domain in the form e ( m) =
∫0 ∫0 (S ex − S m )dxdω ∞ L
The error e of Eq. (22), which depends on the number m of input ~ samples f (i)(x), i = 1,. . . ,m used for the computation of S m , is plotted in Fig. 8 for each estimation method. Whereas STFT and wavelet estimates do not converge due to the systematic error of the uncertainty principle, the method of separation achieves monotonic convergence, approaching the exact solution faster than the Wigner– Ville method.
(a) Estimate of frequency distribution
(b) Estimate of the mean square
Figure 9. Method of separation based estimates of the frequency and spatial component of the Kanai-Tajimi spectrum with different sample sizes
(a) Marginal density in space (mean square)
(b) Marginal density in frequency
Fig. 10. Marginal densities (incremental energies) of the Kanai-Tajimi spectrum estimates
The two principal components the method of separation, i.e. the estimated mean square (left-hand side of Eq. (20)) and the estimated normalized frequency distribution (right-hand side of Eq. (20)), are shown in Figs. 9a and 9b, respectively. The geometric similarity of the component estimates in Fig. 9 with respect to the analytical components in Fig. 1 illustrates the validity of the fundamental expressions in Eq. (4.1). The marginal densities Eqs. (1) and (2), which are an excellent indicator for the quality of space and frequency localization, respectively , are plotted in Fig. 10. The method of separation and the Wigner–Ville transform are able to reproduce the
analytical incremental energies exactly due to their accurate localization properties in both space and frequency directions, whereas STFT and harmonic wavelet based methods exhibit observable deviations from the exact solutions as a consequence of the uncertainty principle. In summary, it can be stated that the method of separation based estimate of the Kanai–Tajimi spectrum is more accurate in terms of regularity of the spectrum surface, space-frequency localization and convergence towards the exact spectrum than the results of the established estimation techniques.
Stochastic Modeling of Imperfections in Structures: Estimation of Strongly Narrow-Band Power Spectra The introduced space-frequency analysis techniques are now applied to a practical problem from stochastic imperfection modeling in structures. The target power spectrum represents the stochastic variation of geometric imperfections in an I-section flange and has to be estimated from corresponding experimental measurements obtained from 6 nominally equal I-section flanges. The measured geometric imperfections represent the deviation of the true flange edge position from perfect plate geometry along the beam axis as illustrated in Fig. 11. The zero-mean parts of the imperfection measurements, which have been previously separated from their deterministic mean μ(x), represent 6 input samples f (i)(x), i=1,2,..,6, which are plotted in Fig. 12(a). Before the presentation of the estimation results, some key issues related to the space-frequency analysis of strongly narrow-band random functions are discussed on the basis of the imperfection example.
Spectral Separability The assumption of spectral separability in the method of separation introduces an error in case of non-separable input samples. Nonseparability implies that the energy distribution over the frequency
(a) Perfect geometry
(b) True geometry (150x enlarged imperfections)
(c) Measured flange edge imperfections δ
Figure 11. The I-section beam experimentally investigated by Hasham and Rasmussen 
Imperfection Amplitude [mm]
1000 Beam Length x [mm]
1 0 -1 1 0 -1
Fig. 12. Imperfection samples: a) Measurements; b) Method of separation based simulation; c) STFT based simulation
domain, i.e. the spectrum shape in frequency direction, differs along the length x . In the case of the strongly narrow-band imperfection measurements, possible variations of energy distribution in frequency are considerably limited, since the main lobe, which contains the largest part of the energy, is located only within a very small fixed bandwidth between 0 and 0.005 rad / mm . This is illustrated by a STFT based spectrum estimate according to Eqn. (9), obtained at 6 equally spaced positions χ j by a non-overlapping Hamming window [1,2,10] of length L / 6 . The 6 individual STFT components plotted in Fig. 13 are normalized with respect to the total energy
Fig. 13. Normalized STFT based spectrum components
represented by the area under their curves. The discrepancy between the plotted curves with respect to shape and amplitude thus indicates the degree of non-separability. Within the narrow-band main lobe containing about 95 % of the total energy, approximate spectral separability can be readily verified, since the 6 curves exhibit the same qualitative shape and diverge only slightly.
Number of Samples and Spectral Smoothing In most practical applications, only a very limited number of measurements are available. The resulting insufficient ensemble averaging in the evaluation of the operator E[ ] in Eq. (20) leads to spurious oscillations in the evolutionary spectrum estimate, due to localized under- and overestimation of the true spectrum. The spurious oscillations are stronger in spatial direction x, because the evaluation of the homogeneous spectrum component Eq. (8) inherently implies an additional averaging of frequency content over length L. An effective damping of the spurious oscillations in spatial direction can be accomplished by spectral smoothing Sˆ (ω , xk ) =
1 2n + 1
~ ∑ S (ω, xk + m )
where x k denote sample points in space of the discrete spectrum representation and Sˆ is the smoothed spectrum counterpart. Eq. (23) can be imagined as a window, which is moved in small steps along the spatial axis, successively replacing the central spectrum values by the arithmetic average of all visible values. The empirical window size (2 n + 1) has to be chosen small enough not to distort the
(a) Increase in Localization in the Mean Square of Measurements f (i)(x) about x=L/2
(b) Decrease in Localization in the Two-Sided Homogeneous Fourier Spectrum Estimate S(ω) about ω=0 Fig. 14. The Uncertainty Principle: Basic Mechanism for the Imperfection Example
evolutionary trend, but large enough to effectively smoothen the spurious local oscillations.
The Uncertainty Principle Originally motivated from a physical point of view by Heisenberg’s observations in quantum mechanics, the uncertainty principle has turned out to be a general property of Fourier analysis [1,2,16]. It states that the two components of a Fourier transform pair, such as a geometric imperfection sample f (x) and its Fourier transform F (ω ) , cannot be completely localized at the same time. The implication of the uncertainty principle can be illustrated by an increase of localization in the mean square E [ f (i ) (x) 2 ] as illustrated in Fig. 14(a), which is obtained by applying Hamming windows [1,2,10] of decreasing widths to measurements f (i ) ( x) . Shortening the window width T , however, necessitates a spread of the corresponding ho-
mogeneous spectrum estimate S h (ω ) Eqn. (8) as shown in Fig. 14(b). Due to their small frequency bandwidths, strongly narrowband spectra are especially sensitive to this phenomenon.
Performance of Available Estimation Techniques The spectrum estimates of the corresponding imperfection example obtained from the wavelet, Wigner-Ville, and STFT based techniques and the method of separation are shown in Fig. 15. All measurements have been processed by a Hamming window prior to the evaluation of Fourier transforms to minimize spectral leakage . The harmonic wavelet based estimate in Fig. 15(a) exhibits satisfactory localization in frequency direction, which can be obtained by the smallest possible difference in scales n − m = 1 . However, this choice leads to the complete loss of localization in space according to Eqs. (12). The Wigner-Ville estimate in Fig. 14(b) can be expected to give an accurately localized impression of the true power spectrum. However, its major disadvantage is the appearance of negative spectral density over large parts of the space-frequency domain. The harmonic wavelet based method and the Wigner-Ville method thus yield estimates, which are unsuitable to be used in the framework of spectral representation. The STFT based estimate in Fig. 15(c), which has been obtained with 16 overlapping Hamming windows [1,2,10] of width L / 8 , exhibits good localization in space. However, in comparison to the Wigner-Ville and method of separation results, which rely on a Fourier transform with full window width T = L , the spread of the main lobe of the STFT based estimate is dramatically increased due to the uncertainty principle (observe the frequency scale). The accurately localized result of the method of separation is shown in Fig. 15(d). Due to the small number of only 6 available measurements, spectral smoothing Eq. (23) is applied to the initial estimate in spatial direction, which results in the final smoothed spectrum estimate Fig. 15(e). In view of the uncertainty principle, the frequency localization in the Fourier transform of a strongly narrow-band function corresponds to the availability of a sufficiently
(a) Harmonic wavelet based estimate
(c) STFT based estimate
(b) Wigner-Ville based estimate
(d) Method of separation based estimate
(e) Smoothed method of separation based estimate Fig. 15. Evolutionary power spectrum estimates for the narrow-band imperfection example
long signal length L . Since the method of separation Eq. (20) allows the use of the full available length L in the Fourier transforms involved, its spectrum estimate guarantees an optimum frequency localization and a minimum spread of the energy distribution. The accurate simultaneous space-frequency localization in the method of separation based estimate can be further evidenced by its marginal densities Eqs. (1) and (2) in Fig. 16. The qualitative accordance of its energy distribution and peak values with the Wigner-Ville estimate Fig. 15(b) additionally supports its plausibility. Finally, the smoothed spectrum estimate of the method of separation and the STFT based spectrum estimate are used to generate 6 random imperfection samples by spectral representation Eq. (4), which are plotted in Figs. 12(b) and 12(c). As a result of the large frequency spread in the STFT based spectrum, the wave-lengths of the method of separation based samples correlate considerably better with the measured counterparts of Fig. 12(a) than the STFT based samples.
Summary and Conclusions The present paper deals with space-frequency analysis techniques for the estimation of evolutionary power spectra, which are used in engineering practice for the accurate stochastic simulation of random physical phenomena by spectral representation. In addition to established techniques, a method for the estimation of separable power spectra is introduced, called method of separation, which yields accurate simultaneous space-frequency localization. For the estimation of the separable Kanai–Tajimi benchmark spectrum, the method of separation based estimate is demonstrated to be more accurate in terms of surface regularity, space-frequency localization and stochastic convergence than estimates from the established techniques. It is furthermore illustrated by an example from imperfection modeling in structures that the method of separation yields good localization in space and frequency even in for the case of strong narrow-bandedness of the target spectrum, where established space-frequency analysis techniques fail to provide acceptable estimation results.
(a) Marginal density in space (mean square)
(b) Marginal density in frequency
Fig. 16. Marginal densities (incremental energies) of STFT and method of separation based spectrum estimates of the imperfection example
Acknowledgments The imperfection measurements in I-sections have been kindly provided by Prof. Kim Rasmussen from the University of Sydney. During his stay at NTU Athens, the first author has been partially supported by the German Academic Exchange Service (Deutscher Akademischer Austausch Dienst) and the German National Academic Foundation (Studienstiftung des deutschen Volkes). All support is gratefully acknowledged.
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