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...