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# 5-23 to 5-25 - EE 341 Discrete-Time Linear Systems Week 9...

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EE 341 Discrete-Time Linear Systems – Week 9 Spring 2011 1 Guest Lecturer, Prof. Les Atlas, [email protected] Atlas office hrs this week, MTW 10:30-11:20 in EE 410. Text reading assignment for this week: Sections 6.2, 10.0-10.5. Example: Given an input signal   3 4 [] n x nu n , and an LTI impulse response [] (5 ) [] n hn un  , find the steady-state output 33 44 n ss yn H    which is the forced response to this input, after the initial conditions die out. YOU FINISH by finding 3 4 H :

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EE 341 Discrete-Time Linear Systems – Week 9 Spring 2011 2 Spectral Response of a System: Relation between the upcoming z - Transform (Ch. 10), to the DTFT (Ch. 5), to the magnitude response of a system (Sec. 6.2). If we evaluate () H z for j ze , as we are about to see, we get the z -transform on the unit circle ( 1 z ), which is identical to the DTFT     j j He Hz , as we have defined previously. () j is periodic with period 2 . Example: Find the magnitude, magnitude–squared, and dB spectrum of the transfer function j with impulse response [] n hn aun , 01 a   Plot the magnitude response, for the frequency range   , carefully labeling magnitudes at critical frequencies ,0,   . 2 ()() , jj j j j j He H e   

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EE 341 Discrete-Time Linear Systems – Week 9 Spring 2011 3 Chapter 10 – The z -transform But what if we want more than just the steady state response?? If we can evaluate () H z at j zr e , we get the output sequence (response) for exponentially decreasing or increasing input sequences. This transform, called “the z -transform” is the discrete-time counterpart of the Laplace transform. Relative definitions: Laplace transform ( ) ( ) -transform ( ) [ ] st n n X sx t e d t zX z x n z    The z -transform is: Used heavily in all aspects of digital signal processing
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5-23 to 5-25 - EE 341 Discrete-Time Linear Systems Week 9...

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