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Continuous Wavelet Transform

Module by: Elizabeth Chan. E-mail the author

Summary: (Blank Abstract)

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Continuous Wavelet Transform

The STFT basis functions: ωtτeiΩt ω t τ Ω t give uniform tiling in T/F plane, but we've reasoned before that high freqs provide better info regarding time localizations, while low freqs require long time windows for freq resolution.

Thus, desire a tiling somewhat like

Figure 1

Note that area ΔΩΔt ΔΩ Δt is preserved, as required by uncertainty principle.

Possible to do this via basis functions ψast ψ a s t

(ψaτt=1aψtτa),  (a>0) ψ a τ t 1 a ψ t τ a ,   a 0
s.t. ψtd t =0 t ψ t 0 (BANDPASS FUNCTION)


Waτ=1aψtτaftd t W a τ 1 a t ψ t τ a f t
ft=1Caτ1|a|2Waτψtτad t d t f t 1 C t a t τ 1 a 2 W a τ ψ t τ a
C=|φω|2ωd ω C ω φ ω 2 ω

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