Re: a question about the UnitStep function

*To*: mathgroup at smc.vnet.net*Subject*: [mg58431] Re: a question about the UnitStep function*From*: Torsten Coym <torsten.coym at eas.iis.fraunhofer.de>*Date*: Sat, 2 Jul 2005 04:06:37 -0400 (EDT)*Organization*: Fraunhofer Gesellschaft (http://www.fraunhofer.de/)*References*: <da2msl$944$1@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. > > Your square wave vanishes for all values outside the interval [-1,1]. Adapting the integration interval leads to the desired result: Assuming[x \[Element] Reals, Integrate[integrand, {z, -1, 1}]] (1/4)*((-(2 + x))*UnitStep[-2 - x] - (-2 + x)*UnitStep[2 - x] + 2*x*UnitStep[-x]) Plot[%, {x, -5, 5}, PlotRange -> All]; Torsten