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30Wavelet(1)

# 30Wavelet(1) - Time-frequency Map Short-time Fourier...

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Time-frequency Map Short-time Fourier Transform (STFT) : Spectrogram, Sonogram A short data window centered at time t with time duration T has the frequency bandwidth of approximately B T 1 . Thus STFT features that all spectral estimates have the same(constant) resolution bandwidth. Wigner-Ville Method The spectral density function of a nonstationary process x t ( ) is defined as: S f t R t e d E x t x t e d xx xx j f j f ( , ) ( , ) ( ) ( ) = = + −∞ −∞ τ τ τ τ τ π τ π τ 2 2 2 2 But, in practice, it is never possible to compute an ensemble average. Instead, we get, for a single sample function x t ( ) , the Wigner distribution given by S R xx xx j f j f f t t e d x t x t e d ( , ) ( , ) ( ) ( ) = = + − ∞ − ∞ τ τ τ τ τ π τ π τ 2 2 2 2 A disadvantage is that, although the above integral is centered at time t , it covers an infinite range of and so depends on the character of τ x t ( ) far away from the local time t . Therefore it does not describe the truly local behavior of x t ( ) at time t. Because of the continuing nature of harmonic waves, it is impossible to have a local spectral density. In other words, Wigner- Ville method tries to break down a signal into its harmonics, which are global functions that go on for ever. This is a fundamental uncertainty principle for time-dependent spectra. In addition, high resolution cannot be obtained simultaneously in time and frequency. Wavelet Analysis A transient signal is broken down into a series of local basis functions called wavelets . Each wavelet is located at a different position on the time axis and is local in the sense that it decays to zero when sufficiently far from its center. Any particular local features of a signal can be identified from the scale and position of the wavelets in which it is decomposed. Wavelets are a powerful tool for presenting local features of a signal. When the size and shape of a wavelet are exactly the same as a section of the signal, the wavelet transform gives a maximum absolute value, a property which can be used to detect transients in a signal.

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30Wavelet(1) - Time-frequency Map Short-time Fourier...

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