Convolution of a Rectangular ”Pulse” With Itself

1 Convolution of a Rectangular Pulse With ItselfMike Wilkes10/3/2013 After failing in my attempts to locate online a derivation of the Convolution of a general rectangularpulse with Itself , and not having available a textbook on communications or signal processing theory, Idecided to write up my attempt at computing it. I expect, however, that it is the first example one wouldfind in any textbook that discusses the general definition of the convolutionf gof two real-valued functions:(f g)(t)= f(u)g(t u)du= f(t u)g(u)du.(1)We apply this to the problem wherefandgare both given byf(t)=g(t)= 0,t<a,A,a t b,0,t>b,(2)where [a,b]isatimeintervalontherealline,witha< of amplitudeAand width or durationT=b function(f f)(t)= f(u)f(t u)du.(3)How isf(t u)relatedtof(u)? Defineg(u)=f(u+t), which fort>0representsahorizontaltranslation off(u) (u)=g( u)=f( u+t)=f(t u)isareflectionofg(u)across the vertical axisu=0.

2 Thus,asafunctionofu,f(t u)isareplicaoff(u)which,fort>0, istranslated to the leftadistancet,thenreflectedacross the vertical axisu=0. Thus,f(t u)isafunctionwhose values depend on the real line byintegrating over the real line (with respect tou)theproductof the two functionsf(u)andf(t u)atthatvalue (2),graphsofthefunctionsf(u),f(u+t), andf(t u)fort>0areillustratedbelow:Aabua tb tt at bf(u+t)f(t u)f(u) tReferring to the Figure, observe that as the distancetfromaincreases, the translated and reflected pulsef(t u)movestotherighttoward + .Ontheotherhand,asthedistancetto the left ofadecreases,1thetranslatedpulsef(u+t)mo vestowardtheright,andthetranslatedandref lectedpulsef(t u)movestoward first on theleftedgeu=t boff(t u)(representedinthenexttwoFiguresbythere ctanglewith dashed line sides), we see that foru=t b>b,theoriginalpulsehasvaluef(u)=0,sothe Convolution will be zero fort>2b,correspondingtou> <t b b,ora+b<t 2b,bothf(u)=Aandf(t u)= (t u)=0foru t b,so the integrand is nonzero only fort b u b,asshowninthisFigure:f(t u)abuf(u)At bt aThe two pulses coincide exactly whent b=a,andt a= ,whent=a+ >a+b,wefocusontherightedgeu=t aoff(t u)asitmovesthruf(u)totheleft.

3 Fora t a<b,or2a t<a+b,bothf(u)andf(t u)haveamplitudeA,butf(t u)=0foru>t a,hence the integrand is nonzero only fora u t :f(t u)abuAt bt af(u)Finally, foru=t a<a,ort<2a,theoriginalpulsef(u)=0,sotheconvolutionisagainzeroforu< results are summarized in the following calculations of the Convolution for any value oft,listedin the opposite order of the above discussion: (f f)(t)=0,fort<2a,(f f)(t)= t aaA Adu=A2(t 2a),for2a t a+b,(f f)(t)= bt bA Adu=A2(2b t),fora+b t 2b,(f f)(t)=0,fort>2b.(4)The graph of this piecewise-defined function is anisosceles triangleof heightA2(b a)atthevertexpoint((a+b),A2(b a)),andbaseofwidth2(b a)withverticesatthepoints(2a,0) and (2b,0) on thet details are illustrated in the next (b a)2aa+b2b(f f)(t)=A2(t 2a)(f f)(t)=A2(2b t)t