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- Xref: sparky sci.physics:22559 alt.sci.physics.new-theories:2732
- Newsgroups: sci.physics,alt.sci.physics.new-theories
- Path: sparky!uunet!well!sarfatti
- From: sarfatti@well.sf.ca.us (Jack Sarfatti)
- Subject: Wavelets, squeezed states 12: Wavelet Transform.
- Message-ID: <C0q8qL.MqK@well.sf.ca.us>
- Sender: news@well.sf.ca.us
- Organization: Whole Earth 'Lectronic Link
- Date: Tue, 12 Jan 1993 06:01:32 GMT
- Lines: 73
-
-
- 12. The window called "the basic wavelet" is scaled for waves of different
- frequencies. Introduce a =/ 0 and define
-
- h[a,s](t) = |a|^-1/2 h((t-s)/a) (85)
-
- *Note -earlier I did not distinguish "||...||" from "|..|" - the integrals
- of |...| are ||..|| - so some of earlier formulae have to be corrected. I
- will use correct notation from now on as those earlier formulae recur.
-
- ||h[a,s]||^2 = Integral(-+inf)[dt|h[a,s](t)|^2]= ||h||^2 (86)
-
- * a can be, and must be, under some conditions, negative real.
-
- Classically h and f are real - take them as complex and take real part as
- in classical wave theory. In QM h and f are really complex - do not take
- real part only.
-
- Define the wavelet transform as
-
- F(a,s) = <h[a,s]|f> = Int(-+inf)[dt|a|^-1/2 h((t-s)/a)* f(t)] (87)
-
- where * means complex conjugate (cc).
-
- This wavelet transform also localizes signals (i.e., time-series of complex
- systems - possibly chaotic with fractal attractors in phase space) in the
- time-frequency plane (i.e. phase space).
-
- First look at localization in time. If the window h(t) is concentrated near
- t = 0 ( drop requirement of window with compact support), then F(a,s) is a
- weighted average of the signal f(t) around t = s.
-
- *The weight function is not generally positive so you cannot think of it as
- a classical probability density - wavelets demand a kind of "negative
- probability density" in a naive sense (e.g. Wigner phase - space function).
-
- Next look at the localization in frequency. Like the earlier windowed
- Fourier transform, the wavelet transform has rigid time-translations of the
- window looking like a convolution. For example, let f(t) be the Dirac delta
- function. The Green's function impulse response is then
-
- ga(s) = |a|^-1/2 h(-s/a)* (88)
-
- F(a,s) = (ga*f)(s) (89)
-
- Now the * is a convolution in eq. 89 but it is cc in eq. 88. A limit on
- text files. The context should make clear which is which.
-
- Ga(v) = Int(-+inf)[dt e^i2pivt |a|^-1/2 h(-t/a)*] = |a|^1/2 H(av)* (90)
-
- A window h(t) with compact support in a finite band vmin < v < vmax have
- convolutions called "bandpass filters" that pass only frequency components
- of the signal in the band.
-
- The wavelet transform F(a,s) only depends on those frequency components of
- the original signal f(t) in the band vmin/a < v < vmax/a if a > 0, or
- vmax/a < v < vmin/a if a < 0. Thus frequency localizations, unlike the
- windowed Fourier transform, is achieved by "dilations" rather than
- translations in frequency space. Remember, the wavelet transform still uses
- rigid time translations of the window, but it does not use translations in
- frequency space.
-
- Consider the example of audio signals. We perceive "meaning" (logarithmic
- response) frequency ratios rather than simple linear frequency differences.
- Hearing an octave difference is achieved by doubling the frequency. We also
- "hear" linear frequency differences as "beats" and there is also a
- phenomenon of non-linear "difference tones" (p.46).
-
- to be continued.
-
-
-
-
-