Fifth International Congress on Sound and Vibration

Harmonic wavelets have simple formulations in the fi-equency domain and they have proved a good basis for the time-frequency mapping of transient signals. Moreover their computational algorithm allows bandwidth to be chosen arbitrarily so that they offer a variable Q transform, where Q is the ratio of centre frequency to bandwidth. In contrast, the short-time Fourier transform and the Wlgner-Ville frequency decomposition method are constant bandwidth transforms, so that Q increases as frequency rises. Although properties of wavelet orthogonality may be used to permit easy retrieval of the input signal, to achieve a high-definition time-frequency map more data is required than that obtained from a single decomposition of the input signal. This is achieved by repeating the frequency decomposition for different, overlapping bandwidths in order to increase the number of points that can be plotted. Because bandwidth can be chosen arbitrarily, a frequency zoom feature can be incorporated into the harmonic wavelet transform algorithm. Also, because harmonic wavelets are complex, with real and imaginary parts, phase variations can be studied. Local changes in the spectral composition of a signal can be recognised, using either wavelet amplitude or phase as the discriminator. Examples of fi-equency zoom and of segmentation by amplitude and phase are given below. They demonstrate that the complex harmonic wavelet transform offers a computationally-efficient method of signal decomposition. Its principal advantage over the STFT is its variable Q property which becomes important when large amounts of data have to be processed. THE HWT ALGORITHM FOR TIME-FREQUENCY MAPPING Frequency versus time decomposition is possible by the short-time Fourier transform (STFT) and the Wigner-Ville method. The application of the first method is common and it offers a practical and conceptually simple route for time-frequency analysis. Its disadvantage is that it produces frequency coefficients for harmonics uniformly spaced across the whole frequency spectrum. For example, if an N point (real) time series is decomposed by the FFT there will be N/2 complex frequency coefficients covering the range from zero to the NyqzW frequency. Each harmonic is a constant fi-equency step from the last, so that the frequency interval is NyqzM/(N/2). This means that to achieve close frequency resolution at the low frequency end of the spectrum, many high frequency harmonics have to be computed. The Wigner-Ville method of analysis is an alternative method (see, for example, Cohen, 1995) which relies on the result that the spectral density of a signal is the Fourier transform of its autocorrelation finction. For a stationary random process, autocorrelation is a fi.mction only of the separation time T and then spectral density is independent of absolute time t. But for a transient signal, frequency composition changes with time and autocorrelation is a finction of r and t. Then spectral density is a fimction of absolute time t. Its amplitude is a measure of the distribution of energy over frequency and time. In practice there are complications due to the cross-terms that occur when two or more distinct frequencies intetiere with each other because of the squaring operation involved in computing the second-order correlation fimction x(flx(t+ ~ (Mark, 1970). There are various ways of overcoming or at least greatly reducing this effect (Cohen, 1995; Hammond (cd), 1997) but there are two drawbacks. One is the high computational effort needed to compute the smoothed Wigner-Ville distribution. The other is the consequence of using an FFT computation to find the Fourier transform of the correlation ii.mction. As for the STFT this computes frequency components that are uniformly spaced on the frequency scale, so that to achieve acceptable resolution at low frequency, an excessively large number of harmonics may have to be computed at high frequency, The harmonic wavelet transform is illustrated in figure 1 (iYomNewland, 1997a). This makes the calculation a(t) = ‘jf(r)wJ*(r-t) dr.