Re: convolution involving UnitStep

• To: mathgroup at smc.vnet.net
• Subject: [mg126172] Re: convolution involving UnitStep
• From: Dana DeLouis <dana01 at me.com>
• Date: Mon, 23 Apr 2012 05:39:45 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```> results in a conditional expression requiring t>1, but I want to
> evaluate and plot t values from [0,1] as well as t>1.

Hi.  I'm not an expert, but I believe the problem is with the edges of UnitStep.
I can never do these correctly in my head.  Therefore, I let the program do it by
switching to the frequency domain, and then back...

equ = 2*UnitStep[t] - UnitStep[t - 1];

FullSimplify[FourierTransform[equ, t, w]]

(-Sqrt[Pi/2])*DiracDelta[w] +  Sqrt[2*Pi]*DiracDelta[w] +
(2*I - I*Cos[w] + Sin[w])/(Sqrt[2*Pi]*w)

Then, invert it back...

FullSimplify[InverseFourierTransform[%, w, t]]

(1/2)*(1 + Sign[1 - t]) + Sign[t]

Now, you have a condition for t > 0.

h[t_]=Sin[t];
g[t_]=(1/2)*(1 + Sign[1 - t]) + Sign[t];
y[t_]=Integrate[h[t-s]g[s],{s,0,t}]//FullSimplify

ConditionalExpression[1+Cos[1-t]-2 Cos[t]-(-1+Cos[1-t]) HeavisideTheta[1-t], t > 0]

= = = = = = = = = = = =
Good Luck!  :>)
Dana DeLouis
Mac & Math 8
= = = = = = = = = = = =

On Apr 19, 3:56 am, J Davis <texasauti... at gmail.com> wrote:
> h[t_] = Sin[t];
> g[t_] = 2 UnitStep[t] - UnitStep[t - 1];
> y[t_] = Integrate[h[t-s]g[s],{s,0,t}]
>
> results in a conditional expression requiring t>1, but I want to
> evaluate and plot t values from [0,1] as well as t>1.
>
> I tried HeavisideTheta as well as := in the definition of y to no
> avail. Thanks for any help...

```

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