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The Dirac Delta Function and Convolution 1 The Dirac Delta ...

Dirac Delta Function and Convolution1 The Dirac Delta (Impulse) FunctionThe Dirac Delta Function is a non-physical, singularity Function with the following definition (x)= 0forx =0undefinedatx=0(1)but with the requirement that (x)dx=1,(2)that is, the Function has unit ) Unit pulses of different extentsb) The impulse function1/T11/T21/T31/T4td(t)0t d (x)XFigure 1: Unit pulses and the Dirac Delta 1 shows aunit pulsefunction T(t), that is a brief rectangular pulse Function of durationT, defined to have a constant amplitude 1/Tover its extent, so that the areaT 1/Tunder thepulse is unity: T(t)= 0fort 01/T0<t T0fort>0.

In Fig. 3 an arbitrary continuous input function u(t) has been approximated by a staircase function ˜uT(t) ≈ u(t), consisting of a series of piecewise constant sections each of an arbitrary fixedduration,T,where u˜T(t)=u(nT)fornT ≤ t<(n+1)T (7) foralln. ItcanbeseenfromFig.3thatastheintervalT isreduced,theapproximationbecomes moreexact ...

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