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Dear All,
I'm dealing with sort of laplace problem and perhaps someone could help
me (i'm not mathemathician, rather electronics guy).
I'd like to compute response of the signal to the high pass filter. The
problem is, that the signal is a periodic series of Gauss-shaped
impulses, thus each impulse is defined from -infnty..infnty.
Normally one would proceed with invlaplace (laplace (myfunc(t)) * H(s)),
where H(s) is the laplace of the response of the filter, to get the
filters' response on that signal. However, when using gaussian
this is rather complicated task.
So I told to myself, that a good approximation of this gaussian pulse
could be something as cos(a*t)^4, providing that integral of the two
functions gives the same result and FWHM of both functions is the same.
This function approximation gives great results, however, it is periodic
with 2Pi.
In order to get laplace transform of piecewise function replacing 2Pi
periodicity with the periode I want (say X), I've substituted
this function:
f(t) = a*cos(b*t)^4 when 0 < t < h (1)
0 when h <= t <= X
- which is well defined, easy to integrate over the specified interval
when substituted into laplace integral - into laplace integral, and I
got the result which is more-or-less like:
F(s) = 1/(1-exp(-s*t)) * int (exp(-s*t)*f(t), t=0..X) (2)
where s is the laplace operator, t is time (say so), X is the periode of
the function.
So far it is ok, but now I need to multiply F(s) by transfer function of
the filter:
G(s) = tau*s / (tau*s + 1) * F(s), thus entire function becomes quite
complex stuff hard to deal (at least for me)
---------->
and here is the problem. My mathematical knowledge doesn't allow me to
calculate the inverse laplace transform of G(s).
Could someone help me with this calculation? Or - does anyone know how
to get the output signal of the filter by any other way?
I'm aware of one other possibility:
If I make Fourier series of my function f(t) to make it periodic with
the period of my interest, then I could make Fourier transform of filter
function and Fourier-series-approximated-signal in order to get the
result back by inverse fourier.
I'm not sure whether I want this. It is not purely analytic solution,
and more - (AFAIK) the fourier transform decomposes the signal into
spectra which is identified by harmonic number, thus it is a discrete
spectra. I would prefer continuous spectra as I'd like to deal with
those signals analytically.
ANY HELP VERY KINDLY APPRECIATED.
thanks
david
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