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Fourier phase (unwrapping problem)

  1. May 5, 2010 #1
    given an integrable function f(x), and its Fourier transform

    [tex]\mathcal{F}\{f\}(\omega)=\int_{\mathbb{R}}f(x)e^{-i\omega x}dx[/tex],

    we consider the phase [tex]\mathrm{Ph}_f : \mathbb{R}\rightarrow [-\pi,\pi)[/tex] which is given by:

    [tex]\mathrm{Ph}_f (\omega) = \mathrm{arg}(\mathcal{F}\{f\}(\omega))[/tex]

    In general the phase function will have discontinuities (when it wraps from [itex]-\pi[/itex] to [itex]\pi[/itex], and there are algorithms that attempts to recover a continuous phase function.
    My question is: why should the phase be a continuous function? What is the condition/theorem that guarantees that the phase is always continuous?
  2. jcsd
  3. May 5, 2010 #2
    Last edited by a moderator: May 4, 2017
  4. May 5, 2010 #3
    I am afraid that chapter doesn't answer my question which was:

    "...Is there a condition on f (or a theorem) that guarantees that the phase is certainly continuous? ..."
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