EE200_Weber_10-30

EE200_Weber_10-30 - EE 200 Frequency and Impulse Responses For a continuous-time system with impulse response h(t the output is given by the

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26 EE 200 Frequency and Impulse Responses For a continuous-time system with impulse response h(t) , the output is given by the convolution integral If the input is a complex exponential this becomes We also know that for an input of a complex exponential, the output is given by y ( t ) = h ( " ) e i # ( t $ ) d $% % = e i t h ( ) e $ i #" d $% % y ( t ) = H ( ) e i t y ( t ) = x ( ) h ( t #" ) #$ $ % d = h ( ) x ( t #" ) #$ $ % d
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27 EE 200 Frequency and Impulse Responses This allows us to relate the impulse response to the frequency response. This is the continuous-time Fourier transform (CTFT) or just Fourier transform . The CTFT of the impulse response of a system its frequency response. H ( " ) = h ( # ) $% % e $ i "# d y ( t ) = e i t h ( ) e $ i d $% % = H ( ) e i t
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28 EE 200 Frequency and Impulse Responses Example: A T-second delay system. The impulse response of a system that implements a T-second delay is The CTFT of this is the frequency response H ( " ) = h ( t ) #$ $ % e # i t dt = ( t # T ) e # i t dt #$ $ % = e # i T h ( t ) = ( t # T )
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29 EE 200 Frequency and Impulse Responses Example: A pulse of duration T H ( " ) = h ( t ) #$ $ % e # i t dt = e # i t dt 0 T % = 1 # i e # i t 0 T = i e # i T # 1 ( ) = i e # i T / 2 e # i T / 2 # e i T / 2 ( ) = e # i T / 2 sin( T /2) /2 h ( t ) = 1 0 " t " T 0 otherwise # $ % T h(t)
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30 EE 200 Frequency and Impulse Responses The magnitude of the response is given by H ( " ) = e # i T / 2 sin( T /2) /2 $ H ( ) = sin( T /2) /2 2 π /T 4 π /T 6 π /T -2 π /T -4 π /T -6 π /T ω |H( ω )|
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31 EE 200 Causality
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This note was uploaded on 12/03/2008 for the course EE 200 taught by Professor Zadeh during the Fall '08 term at USC.

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EE200_Weber_10-30 - EE 200 Frequency and Impulse Responses For a continuous-time system with impulse response h(t the output is given by the

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