998 [6] G. Strang, zyxwvutsrqponml Linear Algebra and Its Applications. 1976. New York: Academic, zyxwvutsr Localization of the Complex Spectrum: The zyxwvut S Transform R. G. Stockwell, L. Mansinha, and R. P. Lowe Abstract-The zyxwvutsrqpon S transform, which is introduced in this correspondence, is an extension of the ideas of the continuous wavelet transform (CWT) and is based on a moving and scalable localizing Gaussian window. It is shown here to have some desirable characteristics that are absent in the continuous wavelet transform. The S transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 4, APRIL 1996 I. INTRODUCTION In geophysical data analysis and in many other disciplines, the concept of a stationary time series is a mathematical idealization that is never realized and is not particularly useful in the detection of signal arrivals. Although the Fourier transform of the entire time series does contain information about the spectral components in a time series, for a large class of practical applications, this informa- tion is inadequate. An example from seismology is an earthquake seismogram. The first signal to amve from an earthquake is the P (primary) wave followed by other P waves traveling along different paths. The P arrivals are followed by the S (secondary) wave and by higher amplitude dispersive surface waves. The amplitude of these oscillations can increase by more than two orders of magnitude within a few minutes of the arrival of the P. The spectral components of such a time series clearly have a strong dependence on time. It would be desirable to have a joint time- frequency representation (TFR). This correspondence proposes a new transform (called the zyxwvutsr S transform) that provides a TFR with frequency-dependent resolution while, at the same time, maintaining the direct relationship, through time-averaging, with the Fourier spectrum. Several techniques of examining the time-varying nature of the spectrum have been proposed in the past; among them are the Gabor transform [7], the related short-time Fourier transforms, the continuous wavelet transform (CWT) 181, and the bilinear class of time-frequency distributions known as Cohen’s class [4], of which the Wigner distribution 191 is a member. Manuscript received November 28, 1993; revised September 22. 1995. This work was supported by the Canadian Network for Space Research, one of fifteen Networks of Centres of Excellence supported by the Govemment of Canada, by the Natural Sciences and Engineering Research Council of Canada, and by Imperial Oil Resources Ltd. The associate editor coordinating the review of this paper and approving it for publication was Dr. Boualem Boashash. R. G. Stockwell and R. P. Lowe are with the Canadian Network for Space Research and Department of Physics, The University of Western Ontario, London, Ont., Canada N6A 5B7 (e-mail: stockwell@uwo.ca). L. Mansinha is with the Department of Earth Sciences, The University of Westem Ontario, London, Ont., Canada N6A 5B7. Publisher Item Identifier S 1053-587X(96)02790-9. zyxwvutsrq 11. THE S TRANSFORM There are several methods of arriving at the S transform. We consider it illuminating to derive the S transform as the “phase correction” of the CWT. The CWT W(T,~) of a function h(t) is defined by 00 Tl-(r. d) zyxwvu = h(t)w(t - T, d) dt (1) lo where ~(t. d) is a scaled replica of the fundamental mother wavelet. The dilation d determines the “width’ of the wavelet w(t. d) and thus controls the resolution. Along with (1), there exists an admissibility condition on the mother wavelet zyxw w (t, d) 151 that w(t, d) must have zero mean. Refer to Rioul and Vetterli [lo] and Young 1111 for reviews of the literature. The S transform of a function h(t) is defined as a CWT with a specific mother wavelet multiplied by the phase factor S(r, f) = er2+W(T, d) (2) where the mother wavelet is defined as Note that the dilation factor d is the inverse of the frequency f. The wavelet in (3) does not satisfy the condition of zero mean for an admissible wavelet; therefore, (2) is not strictly a CWT. Written out explicitly, the S transform is If the S transform is indeed a representation of the local spectrum, one would expect a simple operation of averaging the local spectra over time to give the Fourier spectrum. It is easy to show that cc 1, S(T. f)dT = H(f) (5) (where H(f) is the Fourier transform of h(t)). It follows that h(t) is exactly recoverable from S(T. f). Thus This is clearly distinct from the concepts of the CWT. The S transform provides an extension of instantaneous frequency (IF) 121 to broadband signals. The 1-D function of the variable T and fixed parameter fl defined by S(T, fl) is called a voice (as with the CWT). The voice for a particular frequency fl can be written as S(T. fl) = A(T, fl)ezQ(T’fl). (7) Since a voice isolates a specific component, one may use the phase in (7) to determine the IF as defined by Bracewell 121. (8) Thus, the absolutely referenced phase information leads to a gener- alization of the IF of Bracewell to broadband signals. The validity of (8) can easily be seen for the simple case of h(t) = cos(2~wt), where the function @(T, f) = 2n(w - f)~. The linear property of the S transform ensures that for the case of additive noise (where one can model the data as data(t) = signal(t) + noise(t)), the S transform gives S{data} = S{signal) + S{noise}. l a IF(T.fl) = %%{27rflT+ @(T,fl)l. 1053-587X/96$05.00 0 1996 IEEE Authorized licensed use limited to: UNIVERSITY OF WESTERN ONTARIO. Downloaded on November 15, 2008 at 09:07 from IEEE Xplore. Restrictions apply.