Chapter 3 Week 4 EE 341 Clark Lecture Notes - 20 april

Chapter 3 Week 4 EE 341 Clark Lecture Notes - 20 april - EE...

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EE 341 Discrete-Time Linear Systems – Week 3 Spring 2011 18 Unit-Step Response [] [] [] [][ ] n kk x nu n sn hkun k hk    Can get [] hn from [] as: [] [] [ 1 ] sn sn  Ex. Given we already determined that for an impulse response n aun , 1 1 1 n n a a un aun un  , show that you can obtain the impulse response back from the step response. You might find it helpful to remember that [1 ][ ] n .
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EE 341 Discrete-Time Linear Systems – Week 3 Spring 2011 19 Summary of DT LTI Systems. 1. System Attributes Memory, linearity, TI, Causality, Stability, Invertibility. We saw how to determine attribute from impulse response (except inverse). 2. We saw how an LTI system has its input/output (I/O) relationship described by convolution. 3. Superposition–Break an input down into basis functions , where for each basis function it is easy to calculate system response. Then use superposition to find the output for that input. Examples of bases functions: Impulses (with all possible delays), step functions (with all possible delays), complex exponentials (parameterized by frequency and, perhaps, by magnitude). 4. Can get step response from impulse response and vice versa. () ( ) [ ] () () [] [] [] [] [ 1 ] [] [] [ 1 ] n t k n t k ut d un k st h d ht ut sn hk hn un d t n un un dt d ht hn sn sn dt         Continuous Time Discrete Time
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EE 341 Discrete-Time Linear Systems – Week 4 Spring 2011 Page 1 Text reading assignment for this week: Chapter 3, Sec 2, 6, 7 Read problems 5.53 and 5.54 on pages 417 – 420. Section 3.6 Fourier Transforms of DT Periodic Sequences A periodic sequence always satisfies[] [ ] x nx nN , where its fundamental period (now just called period) is N . Discrete-time periodic signals can also be described by a Fourier series expansion: 0 [ ] [ ] synthesis equation jk n kk k kN xn a n ae     0 1 [ ] analysis equation jk n k ax n e N  As one would expect, the continuous-time integral in time now is a sum. However, there is one more key difference: We define to be any complete sequence of the N elements of the set   0,1,2,. .., 1  , starting from any value of n . For example, for N =4, the set   Sn N  can be represented by any of these row vectors: Computer Implementation As needed for computer implementation, the sum in the synthesis equation is finite. But the
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Chapter 3 Week 4 EE 341 Clark Lecture Notes - 20 april - EE...

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