强大的时频分析工具ITD附原文Intrinsic timescale decomposition.pdf

强大的时频分析工具ITD附原文Intrinsic timescale decomposition.pdf 据说克服了傅里叶、加窗傅里叶、小波、EMD等所有时频分析方法所存在的缺点。。。值得一看。。。
Downloaded from rspa. royalsocietypublishing org on December 13, 2011 322 M. G. Frei and. Osorio Figure 1.(Opposite ) Comparison between TFE information obtained using windowedFourier wavelet and intrinsic timescale decomposition(ITD) methods. (a) A sample brain wave signal with a seizure beginning at t=5S.(b, c tFe distribution of this signal obtained with windowed Fourier analysis and the tradeoffs between(b) good temporal but poor frequency localization and (c) poor temporal but good frequency localization. Brighter regions(blue and, primarily, red) correspond to those with higher power throughout this figure. The simultaneous improvement of both temporal and frequency localizations of energy is restricted by the Heisenberg uncertainty windowed Fourier analysis approach uses sequential time windows wIth is transform. The principle. Note the rectangular grid of TFE information naturally derived from this transform. The segmentation of the signal that fails to use any information about signal changes(such as loca extrema).(d) The TFE distribution for the same input signal, but obtained using the fast wavelet transform. Note the dyadic grid of TFE information naturally derived from this transform. The FWT provides an improvement over Fourier analysis in that it has the ability to localize higher frequency information on shorter timescales and lower frequency information on longer time scales. However it still suffers from predetermined temporal segmentation and inaccuracies in both temporal and frequency localizations. These are due in part to the bleeding of signal energy across different levels of resolution in the wavelet transform and are attributed in large part to the temporal and frequency bins that are predetermined by the choice of basis and not by the sign under study. (e The ITDbased TFE distribution for the same signal, illustrating the important gain in precision of timefrequency localization that the ITD method provides Each displayed line segment's start and end points correspond exactly to the start and end of an actual wave in an itd proper rotation component and the line segment shading is determined by the amplitude of the corresponding wave(red indicates greater amplitude (ii)avoid a priori assumptions about the content/ morphology of the signal being analysed (e.g. make the decomposition 'basis free), and (iii)perform the analysis in an efficient and rapid manner (ideally, online and in realtime) The Fourier transform has long been the dominant TfE analysis tool for all ypes of signals and data, despite its inherent assumption of signal stationarity. Attempts to address this limitation in analysis of nonstationary signals typically involve the application of the Fourier transform in moving windows of demeaned and or tapered data or the application of linear, finite(FIR or infinite impulse response filters in order to extract the desired TFE information(e. g Oppenheim Schafer 1989 ). However, it is well known that these approaches force a difficult choice between:(i) good frequency resolution in the TFE information(by using long windows/filters)at the expense of temporal resolution, or(ii) good temporal resolution(by using short windows/filters)at the expense of frequency resolution In addition, although efficient, the fast Fourier transform uses a rectangular partitioning of the timefrequency plane on which the information is obtained, so that the temporal information regarding highfrequency fluctuations in the signal s blurred over the entire window. Additional drawbacks of Fourierbased analysis include the introduction of spurious harmonics and the necessity to use a wide frequency spectrum in order to represent brief transient changes(e. g impulses or teps) or other data that are nonstationary in time Over the past couple of decades, alternative approaches for obtaining tFE information based upon the wavelet transform have become popular(e.g. Strang 1989; Haar 1910; Daubechies 1992 Strang Nguyen 1996), owing to this transform's dyadic partitioning of the time frequency plane, which allows a better temporal localization of highfrequency signal changes and better frequency information for Proc. R. Soc. A(2007) Downloaded from rspa. royalsocietypublishing org on December 13, 2011 Intrinsic timescale decomposition 323 sample signal 一~ windowed Fourierbased spectrogram(16 pts/win 120 90 60 windowed Fourierbased spectrogram(256 pts/win) 120 timefreqenergy distribution from the fast wavelet transform 120 30 u 0 timefreq energy distribution from the Itd 120 60 30 6 Figure 1.(Caption opposite) lowfrequency changes. The primary strength of wavelets over Fourier analysis lies in their ability, via dilation and translation of basis elements with compact support, to provide tFe information that more closely match the timescales at which the quantified signal changes occur. Figure lad illustrates the performance and some Proc. R. Soc. A(2007) Downloaded from rspa. royalsocietypublishing org on December 13, 2011 324 M. G. Frei and. Osorio limitations of these popular approaches in extracting TFE information from a sample brain signal, which serves as model realworld signal that may exhibit nonlinear, nonstationary, highdimensional and noisy characteristics a drawback inherent to both the Fourier and the wavelet analyses as well as most other approaches to tfe analysis, such as those based on th WignerVille distribution (e.g. Cohen 1995) or the evolutionary spectrum (Priestly 1965), is that these methods represent the signal in terms of basis functions that are defined a priori and the performance of the analysis method is closely tied to how well the morphology of signals under study is represented by the selected basis elements. For example, sound waves produced by an orchestra and consisting of superimposed pure tones with characteristic harmonics and fixed durations(quarter notes, sixteenth notes, etc. are well represented via Fourier analysis(e. g. sheet music), whereas more complex nonstationary signals, such as ambient room noise, meteorologic signals, financial signals or brain waves, make a priori construction of suitable basis elements difficult, if not impossible, and also preclude accurate temporal segmentation with prespecified windows, as the precise times and timescales of signal changes are typically unknown in advance An alternative approach to TFE analysis that made progress towards addressing these difficulties is known as the empirical mode decomposition (EMD; Huang et al. 1998, 2003). The EMD is an algorithm for decomposing physical signals, including those that may be nonlinear or nonstationary, into a collection of intrinsic mode functions(IMFs), which the method,s developers claim are indicative of intrinsic oscillatory modes in the physical signals. The EMD method obtains this decomposition by defining a baseline trend signal for a given window of an input signal, such that when this baseline is subtracted from the input, the resulting residual signal (which they call an IMF)is representative of the highest relative frequency riding waves present in the input. The baseline is constructed as the mean value of the upper and lower envelopes of the input signal, with these upper and lower envelopes defined by cubic spline interpolation of the input signal' s local maxima and local minima, respectivel The eMD procedure for obtaining the baseline is intended to produce a residual imf that is well behaved, with welldefined instantaneous amplitude and frequency. More precisely, the EMD's objective is to produce an IMF that is a 'proper rotation,, which is a signal that has strictly positive values at all local maxima and strictly negative values at all local minima. Once the first residual imf function has been obtained. it is subtracted from the nput signal and the process is repeated on the resulting lower frequency baseline signal. This process is iterated, producing a multilevel decom position of the original or raw input signal into IMFS of successively lower relative frequency, until all the riding waves are removed and the decomposition is completed, leaving a trend signal. The Hilbert transform can then be applied to the IMF components to extract instantaneous amplitude and frequency information While the introduction of the EMD constitutes a conceptual advance in TFE analysis for nonstationary and nonlinear signals, it has several practical limitations that reduce its practical utility and may cause inaccuracies in depicting signal dynamics. The EMDdefined baseline often fails to produce Proc. R. Soc. A(2007) Downloaded from rspa. royalsocietypublishing org on December 13, 2011 Intrinsic timescale decomposition 325 a residual that has the important property of being a proper rotation. In an effort to work around this inability to consistently generate proper rotation components, Huang et al.(1998) developed a ',process that is applied in an iterative manner to generate a sequence of signal baseline candidates until one is found with a proper rotation residual. This sifting process has two stated goals:(i) to separate out highfrequency, smallamplitude waves, which are riding,atop, or superimposed on, larger amplitude, lower frequency waves, and (ii) to smooth out uneven amplitudes in the IMf being extracted However, these goals are often conflicting for nonstationary signals, since riding waves may be transient in nature and or highly variable in amplitude and smoothing out the uneven amplitudes via sifting can prevent faithful extraction of these waves. Moreover, repetitive sifting causes smearing of TFE information across different decomposition levels and an intralevel smoothing of TFE information, which is unlikely to reflect the intrinsic characteristics of the signal under analysis. Despite this effort, the sifting process still often fails to produce proper rotations and thus requires a stopping criterion in the EMD procedure to keep the process bounded and limit the amount of computational energy expended The requisite sifting also forces the emd to be applied in a windowbased manner, with datadependent(and thus uncertain) computational expense and loss of TFE information on timescales longer than the window length. Each sifting then causes an inward propagation of window boundary or edge effects hindering its ability to extract meaningful TFE information. In addition, the fact that the EMD's baseline is exclusively determined by extrema, and ignores all other critical signal information, contributes to mislocalization of temporal information and produces spurious phase shifts and distortion in the components it extracts. Its use of splines to obtain the baseline also causes the emd to inherit all the wellknown difficulties associated with this form of interpolation. Fc example, overshoots and undershoots of the interpolating cubic splines generate spurious extrema, and may shift or exaggerate the existing ones. This results in the EMD's production of IMFS with critical points, which typically differ from those of the original signal and do not retain the important TFE information that these points naturally possess In this paper, we present a new method for TFE analysis, called the intrinsic timescale decomposition(ITD), which is specifically formulated for application to nonlinear or nonstationary signals of arbitrary origin and obtained from complex systems with underlying dynamics that change on multiple timescales simultaneously The ITD overcomes the limitations of EMD listed earlier, as well as those previously mentioned and associated with more classical approaches such as Fourier and wavelets. In particular, the ITD provides (i) efficient signal decomposition into proper rotation'components, for which instantaneous frequency and amplitude are well defined, along with the underlying monotonic signal trend, without the need for laborious and ineffective sifting or splines These criteria may be, for example, (i)when the resulting IMF becomes a proper rotation, (ii) when a Cauchy criteria is satisfied, or (iii) when some predetermined number of iterations has been exhausted without obtaining a proper rotation Proc. R. Soc. A(2007) Downloaded from rspa. royalsocietypublishing org on December 13, 2011 326 M. G. Frei and. Osorio (ii)precise temporal information regarding instantaneous frequency and amplitude of component signals with a temporal resolution equal to the timescale of occurrence of extrema in the input signal, ane (iii)a new class of realtime signal filters that utilize the newly availabl instantaneous amplitude and frequency/phase information together with additional features and morphology information obtained via singlewave analysis. Moreover, the resulting feature extraction and featurebased filtering can be performed in realtime and is easily adapted to extract feature signals of interest at the timescales on which they naturally occur while preserving their morphology and relative phases 2. The intrinsic timescale decomposition Given a signal, Xt, we define an operator, L, which extracts a baseline signal from Xt in a manner that causes the residual to be a proper rotation. More specifically Xt can be decomposed as Xt=LX +(1CX=Lt Ht, (21) where Lt=LX is the baseline signal and Ht=(1C)Xt is a proper rotation Suppose Xt, t20 is a realvalued signal, and let Tk, k=1, 2, .. denote the local extrema of Xt, and for convenience define To=0. In the case of intervals on which Xt is constant, but which contain extrema due to neighbouring signal fuctuations, Tk is chosen as the right endpoint of the interval. To simplify notation, let Xk and Lk denote X(Tk) and L(Tk), respectively Suppose that Lt and Ht have been defined on [0, Tk and that Xt is available for E0, TK+2]. We can then define a(piecewise linear) baselineextracting operator, L, on the interval(Tk, Tk+1 between successive extrema as follows LXt=L=Lk+ k+1 L )(X1X),t∈(r, 2.2 whe k+1=a+(7+1 )(Xk+2X)+(1a k+1, (23) k+2Tk and 0<a<l is typically fixed with a=1/2. We construct the baseline signal in this manner in order to maintain the monotonicity o of xt between extrema while at the same time remaining inside an envelope' generated by some wave riding atop this baseline. The extrema are interpreted as evidence of some proper rotation, riding wave to be extracted The baseline is constructed as a linearly transformed contraction of the original signal in order to make the residual function monotonic between extrema, a necessity for proper rotations. The approach also enables information intrinsic' to the original signal to be passed down to the baseline and residual components. We found that other attempts at baseline construction that were not based on use of the input signal, inevitably failed to produce proper rotation residuals Proc. R. Soc. A(2007) Downloaded from rspa. royalsocietypublishing org on December 13, 2011 Intrinsic timescale decomposition 327 After defining the baseline signal according to equations(2.2)and(2.3), we are able to define the residual, properrotationextracting operator, 7, as HX=(1CX=H,= XtLt R emarks (i) Since Xt is monotonic on (Tk, Tk+1, and Lt and Ht are linear transformations of X, on this interval, it follows that l and ht are also monotonic on(Tk, Tk+1. The extrema of H, i.e. Tk, k21, coincide with extrema of the input signal, X+. The extrema of Xt are either extrema for Lt or inflection points (ii)The manner of defining these components of Xt on interextrema intervals allows the operation to be performed in a recursive manner, interval b interval, with the arrival of each new local extrema. The process is illustrated in figure 2 and allows the method to operate very efficiently and in realtime. Thus, the method is applicable to both online or offline signal analysis (iii) The components are defined at the same time points as the original signal, whether X, is a function of a continuous variable, t, or a discrete(sampled) variable, and the method does not require the raw data to be evenly sampled in time. We have attempted to present the description using a notation suitable for either a discrete or a continuous time framework (iv) Although a is most naturally and typically chosen to be 0.5(as in figure 2), any choice of a in the interval (0, 1) produces a proper rotation component. As we will show, the parameter a acts as a gain control that linearly scales the amplitude of the proper rotation component extracted at each application (v) It is important to note that the proper rotation components produced by the ITD method are not unique and the decomposition is nonlinear in the sense that the decomposition of a sum of two signals need not produce components equal to the sum of the components obtained fro om decomposing each signal individuall i)The extracted baseline signal, Lt, has the same smoothness/differentia. bility as the input signal, Xt, at values of t between extrema. At the extrema. successive baseline components will be continuous and differentiable, but in general will not be twice differentiable. This introduction of nonsmoothness in successive baselines must be kept in mind in interpreting instantaneous frequency measurements obtained from analysis of the resulting components (vii) To initialize the decomposition on the interval [0, Til, one may consider the first point of the signal as an extremum(i.e. define To=0), and define =(X+ X, 2. Other alternatives, such as those that extend the function to values of t<o via reflections. can also be used. The effect of this initialization on the resulting decomposition is confined to the interval [0, T2. If the ITD is used to analyse a fixed window of data, a similar procedure may be used to construct the decomposition beyond the Proc. R. Soc. A(2007) Downloaded from rspa. royalsocietypublishing org on December 13, 2011 328 M. G. Frei and. Osorio (b) raw signal raw signal lu LF component LF component LF node LF nodes time tme→> raw signal =Ly Impor LF component nent HF component 2 +1 time time→> Figure 2. Illustration of the online ITD method (a) Extrema [T1,.,T2,Ti1,T, in the input signal (solid line)are determined in real time as the signal evolves. The signal baseline determined through time Ti1 is shown (dashed line). The ITD method awaits'the appearance of the next successive local extrema to trigger performance of its next operation (b)Immediately upon detecting a new extrema in the input signal (the local minimum at Ti+1), the monotonic segment of the baseline on the interval Ti, Ti+1 is constructed. First, the right endpoint (node) for this segment of baseline is determined using an(possibly weighted) average of the local maximum value and the value at T, of the line connected the local minima at T1 and T;+.(c) The baseline segment(dashed line) is then obtained on Ti Ti+i as a linear transformation of the input signal itself on this interval, constrained to begin and end at the two prescribed nodes.(d)Subtracting the baseline(dashed line)from the input signal (solid line) produces the proper rotation highfrequency component dotdashed line). The method may then be iterated using the obtained baseline as an input signal to further decompose it. HF, high frequency; LF, low frequency Proc. R. Soc. A(2007) Downloaded from rspa. royalsocietypublishing org on December 13, 2011 Intrinsic timescale decomposition 329 extrema times are unchanged origi level 1 original level 2 all maxima are positive g〈 level3 level 1 level 4 are negative level 5 level 2 relative frequency decreases with each successive level time time T/2 time monotonic segment= portion of signal between adjacent extrema halfwave= portion of signal between adjacent zerocrossings Figure 3. Illustration of the proper rotation components(and their properties) obtained by the ITD, including preservation of temporal location of extrema. The bottom panel shows monotonic segments and halfwaves, features which may be analysed and utilized in construction of unique, featurebased filters derived from applying the itd to the signal of interest last measured extrema in the interval, should this be desired. In practice if windowbased analysis is used, then resulting TFE information obtained from these beginning and ending partial waves'should be interpreted with caution Proposition 1. Ht is a proper rotation on the interval [T1, TN for any N>1 Proof. Without loss of generality, assume that Xt has a local maximum at Tha nd adjacent local minima at Tk1 and Tk+. It suffices to show that ht has a local maximum at Tk and is monotonic on Tk1, Tk and on Tk,T k+1 X(a1Xt+b)fort∈[7k1,T小 H=XtLt x,(@ +X,+b, +1)for tE[Th, Tk+I (25) where ak=((LkLk1/(XkXk)),Vk Xt monotonic on Tk1, T and [Tk, Tk+1=H, monotonic on Tk1, Tk and TksT By construction, Lk1> Xk1, Lk+1> Xk+1 and LK<XK=Hk1=Xk1Lk1<0 Hk=XkLk>0 and Hk+1=Xk+1Lk+1<0. It follows that Ht has a local maximum at t=Tk Proc. R. Soc. A(2007)
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