برآورد طیف فرآیندهای تصادفی تکاملی از طریق موج ضربه ای هارمونیک

The problem of estimating the power spectrum of non-stationary stochastic processes using dyadic, generalized, and filtered harmonic wavelets is addressed. Explicit relationships between the statistical moments of the coefficients of the wavelets based representation of the process and of its evolving spectral values are given. It is shown that the popular concept of a separable evolutionary spectrum involving a deterministic modulating envelope is subject to interpretation. Further, mathematical expressions elucidating the analogy between the wavelets based spectral representation and the traditional one involving trigonometric functions are derived. Finally, the applicability and the physical soundness of the developed procedure are demonstrated by applying it to records of the Kocaeli, Turkey earthquake (8/17/1999).

The wavelet transform yields a useful representation of a function in the time–frequency domain [3] and [15]. Recent applications of the wavelet transform to engineering problems can be found in several studies that refer to dynamic analysis of structures, system identification, damage detection, and a plethora of other themes. This study applies the wavelet transform to the problem of estimating the power spectrum of a non-stationary stochastic process; Refs. [11], [12] and [14] provide examples of previous engineering approaches for addressing this important problem. The notion of the spectrum of a stochastic process can be associated with a trigonometric representation, which involves a decomposition of the process in sines and cosines. In this manner, it is easy to identify the contribution of parts of the process with specific frequency content to the total energy [9]. Other forms of oscillatory functions can be used to represent non-stationary processes and to capture the change in time of the probabilistic characteristics of the process [10]. For instance, the Wigner–Ville time–frequency analysis can be used [4]. However, the Wigner–Ville representation of the spectrum lacks, to a certain degree, physical meaning since it yields negative values for the spectrum in several cases. Similar difficulties are encountered in using other joint time–frequency analysis schemes. Nevertheless, wavelets, which are oscillatory functions of zero mean and of finite energy, can be used to obtain a rigorously defined and physics-compatible time–frequency representation of a stochastic process [1] and [2]. In this paper, harmonic wavelets which possess the appealing property of non-overlapping Fourier transforms are used for capturing the evolving spectral content of non-stationary processes. Explicit mathematical expressions elucidating the spectral representation and efficiency aspects of the various members of the harmonic wavelets family are given. Numerical examples involving an analytical model of a non-stationary process derived as the product of a modulating envelope and of a stationary process, and recorded data from a seismic event in Turkey are given.

The problem of estimating the power spectrum of a non-stationary stochastic process via the wavelet transform has been addressed. For this purpose the family of the harmonic wavelets has been chosen due to the non-overlapping, in the frequency domain, feature of wavelets belonging to adjacent scales. Dyadic, generalized, and filtered harmonic wavelets have been used. Explicit expressions have been derived relating the statistical moments of the coefficients of the wavelets based representation of the process and of its evolving in time spectral values. In this regard, the filtered harmonic wavelet scheme has been found to be the most effective one. Mathematical expressions have been derived elucidating this observation and the analogy between the wavelets-based and the traditional (involving trigonometric functions) representations of the spectral content of stochastic processes. The derivation of these expressions has been included as an Appendix A to the main body of the paper. The theoretical results have been used first to study the spectrum of a separable non-stationary process represented by the product of a stationary process with a “slowly” varying deterministic modulating envelope. In this regard, it has been found that the degree of slow variation is subject to quantification, and the classical formula for the spectral estimation involving the product of the spectrum of the stationary process with the square of the modulating envelope does not necessarily yield results identical to those derived by a wavelets-based analysis. Thus, caution should be exercised in using this formula in refined local features capturing signal processing procedures. The proposed method has also been found to yield physically sound and potentially useful for design purposes results when applied to seismic events such as the Kocaeli, Turkey (08/17/1999) earthquake. Future work could extend this analysis to be applicable for conceptual interpretation and numerical implementation of cross-spectrum estimation procedures of non-stationary processes.