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Math Help - Exponential averaging with 2 time constants

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    Exponential averaging with 2 time constants

    Hello,

    let f(t) = \sin^2(a \cdot t) with a > 0 some constant.

    How do I calculate x(t) = \int_{-\infty}^{t} f(t') e^{-\frac{t-t'}{\tau(t')}} \text{d}t' where \tau(t) = \begin{cases} \tau_0, & f(t) \ge x(t)\\ \tau_1, & \text{else} \end{cases} with \tau_0 \ge 0 and \tau_1 \ge 0 constant? I am particularly interested in \lim_{t \to \infty} \{ \max x(t) \} and in the respective minimum.

    In words, we are exponentially averaging a function f(t) with different smoothing constants for increasing and decreasing function values relative to the smoothing result. E.g., whenever f(t_0) \ge x(t_0) for some t_0, the function is considered rising and the smoothing constant \tau(t_0) = \tau_0 is used.

    Thank you for any hint.
    Last edited by Golan; May 31st 2012 at 12:22 PM.
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