Convolution of a Rectangular ”Pulse” With Itself

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.

Convolution of a Rectangular ”Pulse” With Itself Mike Wilkes 10/3/2013 After failing in my attempts to locate online a derivation of the convolution of a general rectangular pulse with itself, and not having available a textbook on communications or signal processing theory, I decided to write up my attempt at computing it.

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  Convolutions, Rectangular

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