mathematica doesn't find periodic response with laplace transform

• To: mathgroup at smc.vnet.net
• Subject: [mg106910] mathematica doesn't find periodic response with laplace transform
• From: nukeymusic <nukeymusic at gmail.com>
• Date: Wed, 27 Jan 2010 06:24:57 -0500 (EST)

```In every course on Laplace transforms you'll find that the Laplace
transform of a periodic signal with period tee is found by multiplying
the Laplace transform of one period by 1/(1-Exp[-tee*s]). That factor
replaces the series expansion  1 + Exp[-tee*s] + Exp[-2*tee*s] + Exp
[-3*tee*s]+ ...
Unfortunately Mathematica doesn't seem to work according to this.
As an example I have taken a square wave with amplitude 3V and period
tee which is applied to a low pass filter with time constant tau.
You'll see that Mathematica doesn't calculate the correct response
using the above mentioned factor.
As a workaround and also to show what the correct result should be
(for three periods) I also calculated the response by multiplying by
(1 + Exp[-tee*s] + Exp[-2*tee*s] + Exp[-3*tee*s])
Here is my code:
hs = 1/(1 + tau*s)
hps = 1/(1 - Exp[-tee*s])
h1Ts = Exp[-tee*s]
tau = 33.0*^-6
tee = 330.0*^-6
ui1t = 3*HeavisideTheta[t]
ui2t = -6*HeavisideTheta[t - tee/2]
ui3t = 1 - HeavisideTheta[t - tee]
uit1 = (ui1t + ui2t)*ui3t
Plot[uit1, {t, 0, 1500*^-6}]
uis = LaplaceTransform[uit1, t, s]
uisT = uis*(1 + Exp[-tee*s] + Exp[-2*tee*s] + Exp[-3*tee*s])
uisTbis = uis*hps
uitT = InverseLaplaceTransform[uisT, s, t]
uitTbis = InverseLaplaceTransform[uisTbis, s, t]
Plot[uitT, {t, 0, 1500*^-6}]
Plot[uitTbis, {t, 0, 1500*^-6}]
uos = uisT*hs
uot = InverseLaplaceTransform[uos, s, t]
Plot[uot, {t, 0, 1500*^-6}]

does anybody here a way to make Mathematica act correctly for this
case?

regards,
nukey

```

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