Re: a question about the UnitStep function

• To: mathgroup at smc.vnet.net
• Subject: [mg58438] Re: [mg58411] a question about the UnitStep function
• From: Pratik Desai <pdesai1 at umbc.edu>
• Date: Sat, 2 Jul 2005 04:06:57 -0400 (EDT)
• References: <200507010602.CAA09193@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Zhou Jiang wrote:

>Dear Mathgroup,
>I want to let Mathematica compute the convolution of two sqare waves. I did as follows
>
>f[x_]:=(UnitStep[x+1]-UnitStep[x-1])/2;
>
>integrand=f[z] f[x-z];
>
>Assuming[Element[x, Reals], Integrate[integrand, {z, -Infinity, Infinity}]]
>
>Mathematica gave me the result as follows,
>((-1 + x) UnitStep[-1 + x] - x UnitStep[x] + (2 + x) UnitStep[2 + x])/4
>
>I plot the result to check
>
>Plot[%,{x,-10,10}, PlotRange->All];
>
>It is clear wrong since the convolution of two square waves should be convergent. Can anyone give me some help with the subtlties about the UnitStep function? Any thoughts are appriciable.
>
>
>
>
Try this,
f[x_]:=(UnitStep[x+1]-UnitStep[x-1])/2;
integrand1=f[x] f[x-1]
d[\[Omega]_]=FourierTransform[integrand1,x,\[Omega]]//ExpToTrig//Simplify
g[x_]=InverseFourierTransform[Evaluate[d[\[Omega]]],\[Omega],x]
DisplayTogether[Plot[f[x],{x,-10,10}],Plot[f[x-1],{x,-10,10}]]
Plot[g[x],{x,-10,10}]

Hope this is what your are looking for

--
Pratik Desai