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Math Help - Waveforms, instantaneous frequency, and oscillation [soft question]

  1. #1
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    Waveforms, instantaneous frequency, and oscillation [soft question]

    (Partially inspired by the recent question about wave length, and also by thinking about the waveforms for sounds.)

    (wall of text follows)

    The sinusoids A\sin (\omega t + \varphi) have certain global properties -- amplitude, frequency, and phase -- which we think of as "wave properties" and are reflected on local scales. We also think of sinusoids as oscillatory, meaning roughly, they go back and forth. Many functions seem to exhibit similar properties. We can think of a wide class of "wavelike" functions where the function hits a minimum, increases, hits a maximum, decreases, hits a minimum, and continues in this manner going up and down. High degree polynomials tend to look like this away from infinity. On the other hand, there are a lot of functions that do not look like this: constants, low degree polynomials, monotonic functions like the exponential and logarithms...

    Thinking about oscillation a little bit: it often seems that oscillation is an obstuction to smoothness. The function x^2\sin(\frac1x) is a standard counterexample for properties of differentiability. I'm also fairly certain that continuous but non-differentiable functions are all highly oscillatory in the sense that they pass through infinitely many extrema in any interval (but perhaps not? I know for sure that Weirstrauss's function does). And if f_n\to f is a sequence of functions converging (even uniformly) to f which oscillate heavily, their derivatives will not necessarily converge to f' (for example, take \frac1n\sin(n^2x)); this is related to the fact that we can squeeze arbitraily much arc length into an arbitrarily small area if we oscillate very heavily. This can all happen in spite of other "good" assumptions on our functions (continuity, C^\infty differentiability), so it feels like "oscillation" is an orthogonal to these (though oscillatory functions will have oscillatory derivatives!) in some sense and is an additional measure of how "nice" a function is (nice functions have small oscillation). I'm aware vaguely of the space BMO, and that some norms take into account "oscillation" but I've never seen any real theorems using them. I also know that decay at infinity of the Fourier transform rules out oscillation in some sense, but I'm not sure to what degree Fourier analysis is what I'm talking about....

    Returning to the global properties of sinusoids--amplitude and frequency--I wonder to what extent we can measure these in non-sinusoidal waves. This is related to the derivative (conceptually). Differentiation looks at a function as being locally polynomial and draws conclusions about its local behaviour from that. For example, an affine function has certain global properties, like slope, that we can assign locally to a differentiable function, and a circle has global properties like curvature that we can assign locally to a twice-differentiable function. Amplitude can be measured locally by the magnitude of the function or its Lp norms (for largish p). But I don't know about "instantaneosu" frequency. Obviously, \sin(\frac1x) would be expected to have large frequency near zero and small frequency near infinity, but I don't know how it would behave at any precise point. Fixing f and x, you could sample f at some points near x and find the interpolating sinusoid, and then call the limiting value of its frequency as the sample points all go to x the instantaneous frequency at x (in analogy to the limiting process for the difference quotient). In know about Fourier analysis, but its not what I'm thinking of, because its is not local (local changes to a function have global changes on its Fourier transform) so it does not measure "infintesimal" frequency.

    So my question, I guess, is: can we talk about frequency and wavelength for non-sinusoids, at least for a large class of "wavelike" functions? Why are infintesimal tools used so heavily in calculus but not in signal processing? How can we control oscillation and to what extent does "badness" arise from high oscillation? Is there any useful notion of instananeous frequency? If not, why is it not useful (does it not control the function as much as I think it does?)? Are there any books developing any of these ideas? Are function spaces like BMO what I'm thinking of? Is Fourier analysis what I'm thinking of? Did I make any sense?

    (I'm posting this in the analysis forum but I don't know where it belongs really...)
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by maddas View Post

    So my question, I guess, is: can we talk about frequency and wavelength for non-sinusoids, at least for a large class of "wavelike" functions? Why are infintesimal tools used so heavily in calculus but not in signal processing? How can we control oscillation and to what extent does "badness" arise from high oscillation? Is there any useful notion of instananeous frequency? If not, why is it not useful (does it not control the function as much as I think it does?)? Are there any books developing any of these ideas? Are function spaces like BMO what I'm thinking of? Is Fourier analysis what I'm thinking of? Did I make any sense?

    (I'm posting this in the analysis forum but I don't know where it belongs really...)
    This looks more complicated than it is in practice but:

    For an a real signal x(t) we define the analytic signal corresponding to it by:

     <br />
x_a(t)=x(t)+i H(x)(t)<br />

    where H(x)(t) denotes the Hilbert transform of x:

    H(x)(t)=-\frac{1}{\pi}\lim_{\varepsilon \to 0_+}\int_{\varepsilon}^{\infty}\frac{x(t+\tau)-x(t-\tau)}{\tau}\;d\tau

    Then the instantaneous (angular) frequency of x(t) is:

     <br />
\omega(t)=\frac{d}{dt}\text{arg}(x_a(t))<br />

    To a handwaving level of exposition we are finding a complex signal of the form:

    u(t)=A(t)e^{i.\phi(t)}

    where A(t) is a real amplitude, and \phi(t) is a real phase such that:

    x(t)=\text{real}(u(t))

    Then the instantaneous frequency is:

    \omega=\frac{d}{dt}\phi(t)

    Often the analytic signal coresponding to a real signal can be written down without going via the Hilbert Transform.

    CB
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