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Continuous Wavelet Transform

Module by: Phil Schniter. E-mail the author

Summary: This module introduces continuous wavelet transform.

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The STFT provided a means of (joint) time-frequency analysis with the property that spectral/temporal widths (or resolutions) were the same for all basis elements. Let's now take a closer look at the implications of uniform resolution.

Consider two signals composed of sinusoids with frequency 1 Hz and 1.001 Hz, respectively. It may be difficult to distinguish between these two signals in the presence of background noise unless many cycles are observed, implying the need for a many-second observation. Now consider two signals with pure frequencies of 1000 Hz and 1001 Hz-again, a 0.1% difference. Here it should be possible to distinguish the two signals in an interval of much less than one second. In other words, good frequency resolution requires longer observation times as frequency decreases. Thus, it might be more convenient to construct a basis whose elements have larger temporal width at low frequencies.

The previous example motivates a multi-resolution time-frequency tiling of the form (Figure 1):

Figure 1
Figure 1 (uncertainty.png)

The Continuous Wavelet Transform (CWT) accomplishes the above multi-resolution tiling by time-scaling and time-shifting a prototype function ψt ψ t , often called the mother wavelet. The aa-scaled and ττ-shifted basis elements is given by ψ a , τ t=1|a|ψtτa ψ a , τ t 1 a ψ t τ a where aτR a τ ψtdt=0 t ψ t 0 Cψ =|ψΩ|2|Ω|dΩ< Cψ Ω ψ Ω 2 Ω The conditions above imply that ψt ψ t is bandpass and sufficiently smooth. Assuming that ψt=1 ψ t 1 , the definition above ensures that ψ a , τ t=1 ψ a , τ t 1 for all aa and ττ. The CWT is then defined by the transform pair XCWT aτ=xt ψ a , τ t¯dt XCWT a τ t x t ψ a , τ t xt=1 Cψ XCWT aτ ψ a , τ ta2dτda x t 1 Cψ a τ XCWT a τ ψ a , τ t a 2 In basis terms, the CWT says that a waveform can be decomposed into a collection of shifted and stretched versions of the mother wavelet ψt ψ t . As such, it is usually said that wavelets perform a "time-scale" analysis rather than a time-frequency analysis.

The Morlet wavelet is a classic example of the CWT. It employs a windowed complex exponential as the mother wavelet: ψt=12πe(i Ω0 t)et22 ψ t 1 2 Ω0 t t 2 2 ΨΩ=eΩ Ω0 22 Ψ Ω Ω Ω0 2 2 where it is typical to select Ω0 =π2log2 Ω0 2 2 . (See illustration.) While this wavelet does not exactly satisfy the conditions established earlier, since Ψ07×10-70 Ψ 0 7-7 0 , it can be corrected, though in practice the correction is negligible and usually ignored.

Figure 2
Figure 2 (figure2.png)

While the CWT discussed above is an interesting theoretical and pedagogical tool, the discrete wavelet transform (DWT) is much more practical. Before shifting our focus to the DWT, we take a step back and review some of the basic concepts from the branch of mathematics known as Hilbert Space theory (Vector Space, Normed Vector Space, Inner Product Space, Hilbert Space, Projection Theorem). These concepts will be essential in our development of the DWT.

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