Transcription of The Dirac Delta Function and Convolution 1 The Dirac Delta ...
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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.(3)The Dirac Delta Function (also known as the impulse Function ) can be defined as the limiting formof the unit pulse T(t) as the durationTapproaches zero. As the durationTof T(t) decreases,the amplitude of the pulse increases to maintain the requirement of unit area under the Function ,and (t) = limT 0 T(t).
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Sysytems The Dirac Delta Function and Convolution
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