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Continuous Wavelet Transform

Module by: Elizabeth Chan. E-mail the author

Summary: (Blank Abstract)

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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 (see Figure 1), while low freqs require long time windows for freq resolution (see Figure 2).

Thus, desire a tiling somewhat like

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

(ψ⁢aτ⁢t=1a⁢ψ⁢t−τa)⁢,  ⁢(a>0) ψ a τ t 1 a ψ t τ a ,   a 0
(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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