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Unformatted text preview: Lecture 8: More on the CT Unit Impulse Function Tie Liu September 20, 2011 1 Operational Definition of the CT Unit Impulse func- tion The CT unit impulse function ( t ): Idealization of a unit-area pulse that is so short that, for any physical systems of interest to us, the system responds only to the area of the pulse and is insensitive to its duration Operationally, the unit impulse is the signal which when applied to any CT LTI system results in an output equal to the impulse response of the system. That is, ( t ) is defined by what it does under convolution: ( t ) * h ( t ) = h ( t ) 2 Differentiators and Integrators Consider the first-order differentiator system as shown below: x ( t ) y ( t ) = dx ( t ) dt d dt The system is LTI with the unit impulse response h ( t ) = d ( t ) dt Just like ( t ), d ( t ) /dt is a singularity function (usually denoted by u- 1 ( t ) and known as the unit doublet function) which is defined by what it does under convolution: x ( t ) * u- 1 ( t ) = dx ( t ) dt 1 For any positive integer...
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- Fall '09