Let

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

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.
@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