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$$s(t) = \cos(wt)\cdot u(t)$$

with $u(t)$ being the unit step.

Suppose we can represent such a signal as the sum of a transient and of a "steady state". A transient is a short-time wide-band spectral component of part of the signal. In this case the transient occurs at $t = 0$.

My question is, given $s(t)$ above, what are $s_t(t)$ and $s_s(t)$ in the decomposition

$$s(t) = s_t(t) + s_s(t)$$

where $s_t(t)$ is localized around t = 0 with minimum support in time and $s_s(t)$ has minimum support in frequency. Essentially we are trying to maximize the time-locality for $s_t(t)$ and maximize the frequency-locality for $s_s(t)$.

Note that solutions such as $s_t(t) = -\cos(wt)\cdot u(-t)$ and $s_s(t) = \cos(wt)$ are not allowed as $s_t(t)$ is not localized around t. In fact I think we should require the transient to be causal in nature. Another possibility is $s_t(t) = \cos(wt)\cdot u(t) \cdot u(1 + \alpha - t)$ and $s_s(t) = \cos(wt)\cdot u(t - 1 - \alpha)$ but $s_s(t)$ does not have minimal bandwidth.

I would expect the above problem to have a unique solution. I imagine $s_s(t)$ have a Gaussian like attack. Any ideas?

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How about $$s_t(t)= u(t) \cos(\omega t) (1-e^{t_0/t})$$ and $$s_s(t)= u(t) \cos(\omega t) e^{-t_0/t}?$$ The constant $t_0$ you can still adjust to fit your needs.

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I'm not looking for just any solution but one that is optimal. There are many ways to express the slowly rising slope of the steady state solution and the transient part is easily found from that. But not all functions produce the properties I'm after. – JonSlaughter Oct 2 '12 at 21:51
@JonSlaughter: can you phrase the "optimality" in mathematical terms? – Fabian Oct 3 '12 at 12:25
I did, re-read my post. – JonSlaughter Oct 9 '12 at 15:17
@JonSlaughter: there is no mathematical definition there. I would assume that you have to define the support as the the size of the interval where the function is nonzero? Then if $\sigma_t$ is the (time)support of the transient and $\sigma_\omega$ the (frequency)support of the steady state. Would you want to minimize $\sigma_t \sigma_\omega$ or $\sigma_t + \sigma_\omega$ or $\text{max}(\sigma_t, \sigma_\omega)$ or ...? Furthermore, why you care about the support and not of a weaker (but maybe more suitable measure) like standard deviation? – Fabian Oct 9 '12 at 17:19

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