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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: ω⁢t−τ⁢ei⁢Ω⁢t 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
(uncertainty.png)

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

Possible to do this via basis functions ψ⁢as⁢t ψ a s t

∀a>0:ψ⁢aτ⁢t=1a⁢ψ⁢t−τa a 0 ψ a τ t 1 a ψ t τ a
(1)
s.t. ∫−∞∞ψ⁢td t =0 t ψ t 0 (BANDPASS FUNCTION)

CWT:

W⁢aτ=1a⁢∫−∞∞ ψ* ⁢t−τa⁢f⁢td t W a τ 1 a t ψ* t τ a f t
(2)
f⁢t=1C⁢∫a∫τ1|a|2⁢W⁢aτ⁢ψ⁢t−τad t d t f t 1 C t a t τ 1 a 2 W a τ ψ t τ a
(3)
C=∫−∞∞|φ⁢ω|2ωd ω C ω φ ω 2 ω
(4)

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