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EE313_cont_time_convolution

# EE313_cont_time_convolution - EE 313 Linear Systems and...

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Continuous-Time Convolution EE 313 Linear Systems and Signals Fall 2008 Courtesy of Prof. Brian Evans Prof. Adnan Kavak Dept. of Electrical and Computer Engineering The University of Texas at Austin

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4 - 2 Useful Functions Unit gate function (a.k.a. unit pulse function) What does rect(x / a) look like? Unit triangle function ( 29 < = = 2 1 1 2 1 2 1 2 1 0 rect x x x x ( 29 < - = 2 1 2 1 2 1 0 x x x x 0 1 1/2 -1/2 x rect( x ) 0 1 1/2 -1/2 x ( x )
4 - 3 ( 29 ( 29 ( 29 ( 29 t Y T t y c T t f c t F m i i i m i i i = - - = = = 1 1 LTI Systems Recall for LTI system with input f and output y Homogeneity: c f ( t ) c y ( t ) Time-invariance: c f ( t - T ) c y ( t - T ) Adding additivity: If a signal ( F ( t ) ) can be expressed as a sum of shifted ( t - T i ) and weighted ( c i ) copies of a simpler signal ( f ( t ) ), we easily find a system’s output ( Y ( t ) ) to that signal if we only know system’s output ( y ( t ) ) to that simpler signal A common choice for f ( t ) is the impulse

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4 - 4 Impulse Response Impulse response of a system is response of the system to an input that is a unit impulse (i.e., a Dirac delta functional in continuous time) Linear constant coefficient differential equation When initial conditions are zero, this differential
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EE313_cont_time_convolution - EE 313 Linear Systems and...

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