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lecture23 - EE313 Linear Systems and Signals Fall 2010...

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EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Frequency Response of Discrete-Time Systems
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23 - 2 Complex Exponentials Complex exponentials have special property when they are input into LTI systems. Output will be same complex exponential weighted by H [ z ] When we specialize the z -domain to frequency domain, the magnitude of H [ z ] will control which frequencies are attenuated or passed [ ] [ ] [ ] [ ] [ ] [ ] z H z z m h z z m h z n h n f n h n y n m m n m m n n = = = = = -∞ = - -∞ = - ] [
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23 - 3 Frequency Response Response to sinusoids and complex sinusoids In continuous time In discrete time For real-valued impulse response, H ( e -j ) = H * ( e j ) ( 29 ( 29 ( 29 ( 29 ( 29 ϖ ϖ ϖ ϖ ϖ ϖ ϖ j H t j H t e j H e t j t j + cos cos ( 29 ( 29 [ ] [ ] ( 29 + cos cos j j k j j k j e H k e H k e e H e frequency response frequency response [ ] [ ] k j j k j j k j k j e e H e e H e e - - - - + + [ ] ) cos( 2 ) cos( 2 k e H k j
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23 - 4 Response to Sampled Sinusoids Start with a continuous-time sinusoid Sample it every T s seconds ( see slide 7-6 ) We show discrete-time sinusoid with Resulting in Discrete-time frequency is equal to continuous- time frequency multiplied by sampling period ( 29 t t x cos ) ( ϖ = ( 29 s nT t T n t x n x s cos ) ( ] [ ϖ = = = ( 29 ( 29 n T n s cos cos ϖ = s T ϖ =
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23 - 5 Example Calculate the frequency response of the system given as a difference equation as
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