Signal whose Laplace transform contains derived Dirac-deltas: How do I find the inverse transform?

Question

I must reconstruct the input signal to a system, knowing the output signal and the system transfer function. At the end, I found that the Laplace-Transform of the input signal is something like: $$ s^2\sum_{k=0}^\infty c_k e^{-skT} $$ The coefficients $c_k$ are known (from the output signal) and T is a known constant.

The sum $ \sum_{k=0}^\infty c_k e^{-skT} $ corresponds to a series of Dirac-deltas and this is ok. The $s^2$ means that those deltas must be differentiated twice. The Dirac-delta is a distribution and, given a test function f(t), we have that $<\delta'',f(x)> = f''(0)$. But how do I interpret this for my input signal ?

The only idea I have is that I could consider the delta as the limit of a Gaussian function and the delta's 2nd derivative as the limit of the Gaussian function's 2nd derivative. But I've never seen something like this anywhere, so I'm skeptikal.

Answer 1

You are right in that the Laplace antitransform of $F(t)=s^2 a \, e^{-s b}$ is $f(t)=a \delta''(t-b)$, the second derivative of Dirac delta with origin at $b$ and weighted by $a$ (related). I'm not sure how do you expect to "interpret this for my input signal". Even Dirac deltas are difficult to interpret as input signals, how do make sense of them (probably as limits) depends on your scenario.

It's a little easier (and, I think, more common) to interpret these kind of things as transformations of signals than as signals themselves (say, as impulsive response of LTI filters). Then, if we have $y(t) = x(t) \star h(t)$ (convolution) and $h(t)$ (filter impulsive response) consists of a Dirac delta $h(t)=a \delta(t-b)$, that is easy to interpret as "what this filter does is: delay the input by $b$ and weight it by $a$".

In the same vein, if we have $h(t)=a \delta''(t-b)$ we don't try to interpret this by picturing this strange "function" as a graph (good luck with that) or even as a limit of nicer functions, but simply in operational terms: $h(t)\equiv$"differentiate the input twice, delay it by $b$ and weight it by $a$"

Answer 2

Remember Geometric Series. It will turn into Periodic Summation..

$$ \displaystyle \begin{align} R(s) &= s^2\sum_{k=0}^\infty c_k e^{-skT} \\ R(s) &= s^2 \left( \sum_{k=0}^\infty e^{-skT} \right) \left( \sum_{k=0}^\infty c_k \right) \\ R(s) &= s^2 \left( \frac{1}{1-e^{-sT}} \right) \left( \sum_{k=0}^\infty c_k \right) \\ \mathcal{L}^{-1} \left[R(s)\right] &= \mathcal{L}^{-1} \left[ s^2 \left( \frac{1}{1-e^{-sT}} \right) \left( \sum_{k=0}^\infty c_k \right) \right] \\ \mathcal{L}^{-1} \left[R(s)\right] &= \mathcal{L}^{-1} \left[ s^2 \left( \frac{1}{1-e^{-sT}} \right) \right] \cdot \left( \sum_{k=0}^\infty c_k \right) \\ r(t) &= \sum_{k=0}^\infty \delta''(t-Tk) \ u(t-Tk) \left( \sum_{k=0}^\infty c_k \right) \\ r(t) &= \sum_{k=0}^\infty c_k \ \delta''(t-Tk) \ u(t-Tk) \end{align} $$

Hence the result, Periodic Summation of the derivative of weighed doublet.

Causal real-time system could not directly sense any change without accumulation and proportionality to be exists beforehand. Integrator and gain. Which then compared with the present and past, so we could differentiate the changes. Hence causal differentiator. For example, sRC/(1+sRC).

In order to know completely all of the changes and the rates within all of it, requires knowledge of all of the input, past, present, and future. So that we could completely differentiate it. Post-processing digital system could do that if the data completely stores in memory beforehand. Hence, non-causal.

If proportionality of a signal's magnitude with respect to time is a step, then its velocity is its impulse, its acceleration is its doublet and its jerkiness is its doublet's first derivative.

Lets say there's a system capable of sensing change of jerkiness of a thing with period T. Hence, it's analogous to the mathematical expression above.

In nature, G-Force is one example of jerk. But I don't know any system capable of directly sensing G-Force. Even G-Force sensor uses accelerometer to measure the acceleration's rate of change that fundamentally derived from change of position. Not directly change of jerk itself.

Because directly measuring change of acceleration itself means capable of measuring discontinuities in acceleration. And every known mechanics needs acceleration for its dynamics and kinetics to be exist. Newton's F = ma for example. That would mean the system is capable of sensing nonlocality of the nature. Insane enough to imagine.