Nonstationary signals

Flicker-noise spectroscopy — The spectral density of - flicker noise (also known as 1// noise, excess noise, semiconductor noise, low-frequency noise, contact noise, and pink noise) increases with frequency. Flicker noise spectroscopy (FNS) is a relatively new method based on the representation of a nonstationary chaotic signal as a sequence of irregularities (such as spikes, jumps, and discontinuities of derivatives of various orders) that conveys information about the time dynamics of the signal [i—iii]. This is accomplished by analysis of the power spectra and the moments of different orders of the signal. The FNS approach is based on the ideas of deterministic chaos and maybe used to identify any chaotic nonstationary signal. Thus, FNS has application to electrochemical systems (-> noise analysis). 

Transformation — Several approaches are available for transformation of time domain data into the - frequency domain, including - Fourier transformation, the maximum entropy method (MEM) [i], and wavelet analysis [ii]. The latter two methods are particularly useful for nonstationary signals whose spectral composition vary over long periods of time or that exhibit transient or intermittent behavior or for time records with unevenly sampled data. In contrast to Fourier transformation which looks for perfect sine... 

Fourier analysis is very useful to describe the signals frequency content. However, it has a serious drawback. In a typical FT of a signal, it is impossible to tell when an event has happened because time information is lost. For all nonstationary signals that contain drift, trends, or abrupt changes, it is important to keep the time information in the transformation. In principle, time information can be found in the imaginary part of a complex FT. However, since only the real part is usually considered, the time information will not be available. 

The following sections arc organized as follows. Section 18.3 deals with an ysis of deterministic and stationary stochastic signals. Analysis of nonstationary signals is discussed in Sec. 18.4. Sub-... 

Given the limitations of the FT, some approximations are needed to handle nonstationary signals. The discrete FT (DFT) and the short-time FT (STFT, a.k.a. the Gabor transform) are two alternative transformation methods that address this issue. 3 jn the mid-20th century, Jean Ville pointed out that... 

Figure 3 A nonstationary signal (solid line) is fitted with an FT basis function (dashed line). The basis function is fitted well against a single signal peak, but outside of the immediate region of the peak, the signal approximation suffers. It illustrates the ability of a Fourier basis function to isolate frequency information but not time-domain information.

In STFT, a nonstationary signal is divided into small windows in an attempt to achieve a locally stationary signal, as depicted in Figure 4. Each... 

Figure 4 (a)-(c) The windowing of a nonstationary signal (solid line) in STFT analysis gives some locality of time information to the FT (dashed line is Fourier basis function). Even so, this method still suffers from a tradeoff of knowledge between time-domain and frequency-domain information. 

Figure 6 Shown here are (a)-(c) three wavelet windows over a nonstationary signal that illustrate a partial wavelet scaling analysis and how wavelets simultaneously identify component frequency and position information within the signal. The CWT performs an exhaustive fitting of the different features of the signal at different scales and positions of the wavelet function.

Since the results presented in this chapter require that a series be stationary, it is necessary to consider the procedure for obtaining a stationary series. A nonstationary signal can be made stationary by taking the difference between two adjacent values. This procedure is called differencing. If the differences themselves are not stationary, then they can be differenced until a stationary differenced signal is obtained. However, it should be noted that differencing will lead to a loss of information in the signal and can introduce correlations where there are none. 

An alternative approach has been proposed by Cacciola (2010) the author s contribution allows a straightforward evaluation of a non-separable power-spectral density function compatible with a target response spectrum. In the model proposed by Cacciola (2010), it is assumed that the nonstationary spectrum-compatible evolutionary ground motion process is given by the superposition of two independent contributions the first one is a fully nonstationary known counterpart which accounts for the time variability of both intensity and frequency content the second one is a corrective term represented by a quasi-stationary zero-mean Gaussian process that adjusts the nonstationary signal in order to make it spectrum compatible. Therefore the grotmd motion can be split in two contributions ... 

Note that the stationarity assumption results in a spectrum consisting of identically shaped (window) pulses placed at the sine-wave frequencies. Most signals of interest however are generally nonstationary (e.g., the frequencies may change over the window extent due to amplitude and frequency modulation), and so the window transform may deviate from this fixed shape. Naylor and Porter [Naylor and Boll, 1986] have developed an extension of the approach of this section that accounts for the deviation from the ideal case. 

Wavelet analysis is a rather new mathematical tool for the frequency analysis of nonstationary time series signals, such as ECN data. This approach simulates a complex time series by breaking up the ECN data into different frequency components or wave packets, yielding information on the amplitude of any periodic signals within the time series data and how this amplitude varies with time. This approach has been applied to the analysis of ECN data [v, vi]. Since electrochemical noise is 1/f (or flicker) noise, the new technique of -> flicker noise spectroscopy may also find increasing application. 

Ganesan R, Das T, Sikder A, Kumar A. Wavelet-based identification of delamination defect in CMP (Cu-low k) using nonstationary acoustic emission signal. IEEE Trans Semicond Mannf 2003 16(4) 677-685. 

The quantity in square brackets in (3.5) resembles the overlap kernel in the time-dependent theory of the continuous-wave absorption spectrum [34], but here involves the nonstationary ground state wave packet vibrational wave function. The interference signal in the impulsive limit directly measures the overlap between pseudo-rotating wave packets propagated in the ground and excited states for a time... 

Abstract We review the basic theoretical formulation for pulsed X-ray scattering on nonstationary molecular states. Relevant time scales are discussed for coherent as well as incoherent X-ray pulses. The general formalism is applied to a nonstationary diatomic molecule in order to highlight the relation between the signal and the time-dependent quantum distribution of intemuclear positions. Finally, a few experimental results are briefly discussed. 

It is instructive to consider the time-resolved X-ray scattering for a nonstationary diatomic molecule. Our aim here is to present a simplified analysis which highlights the relation between the key features in the dynamics and the time-dependent diffraction signals. We assume in the following that the nuclear density created by the pump pulse, pon(R, 0. is distributed over the electronic ground state and an excited electronic state, that is, pon(R, 0 = Pgr(R, 0 + Pex(R> 0-... 

Next, we consider the scattering signal when the molecule is in a nonstationary excited state obtained by excitation out of the initial stationary vibrational-rotational ground state. 

For instance, if the formation resistivity and its thickness are 100 ohm-m and 6 m, respectively, then the maximal time when the influence of the surrounding medium is still negligible can not exceed 2 sec (pi/p2 < 16). Taking into account that at such times currents induced in a moderately conductive medium create relatively weak signals we can expect serious technical problems of measuring in inductive sensors due to intrinsic processes. For this reason it is appropriate to consider some features in a behavior of the nonstationary field of the electric dipole. 

A stochastic process is termed stationary, if the signal generating process is time invariant. All distributions and statistical parameters of a stationary process are independent of time. A process varying with time is called nonstationary. 

The spectral density estimator is Wishart distributed regardless of the distribution of the orig-inai time-domain signal. Therefore, the expressions for the likelihood function in Chapter 3 are valid even for nonlinear systems. The only challenge is on the computation of the mean spectrum but this may be accommodated by simulations. However, the Bayesian spectral density approach is not applicable for nonstationary response measurements. 

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