Chapter 2 Week 3 EE 341 Clark Lecture Notes - 15 april

# Chapter 2 Week 3 EE 341 Clark Lecture Notes - 15 april - EE...

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EE 341 Discrete-Time Linear Systems – Week 2 Spring 2011 Page 38 3.  0 [] c o s [] y nn x n  Not linear, and not time-invariant. 4. [] [ 2 1 ] yn x n  Linear, and not time-invariant. 5. [1 ] , [ ] 0 0, [ ] 0 xn  Not linear, and time-invariant. 6. [ ], 0 0 n n Linear, and not time-invariant.
EE 341 Discrete-Time Linear Systems – Week 3 Spring 2011 1 Text reading assigment for this week: Chapter 2, Sec 0, 1, 3, 4 (emphasis on discrete time) Chapter 3, Sec 6 Chapter 2 – Discrete-Time Linear Time-Invariant Systems We will study a very important case: discrete-time systems that are both linear and time-invariant and will see that their input/output relationship is described by discrete-time convolution. Sequence Representation of Discrete-Time Signals We can describe any discrete-time sequence [] x n as scaled and summed unit impulse sequences: [ ] [ 1 ] [1 ][ 0 ] [ 1 ] 2 ] [2 ] x n…x n x nx n x n     which is equivalent to the more succinct notation [] [][ ] k x nx k n k  This equation expresses [] x n as a series of impulse functions shifted in time, all scaled with weights [] x k . We will see this again when we show that the input/output relationship of a DT LTI system is DT convolution.

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EE 341 Discrete-Time Linear Systems – Week 3 Spring 2011 2 2.1 Convolution for Discrete-Time Linear Time-Invariant Systems Using the result from the last page [] [][ ] k x nx k n k  Now add that we are assuming the system is linear and has unit pulse response [ ] [ ] k yn h n to arbitrarily (integer) delayed or advanced unit pulse input [] [ ] x nn k . ] [] [] k k k x k n k xkh n Due to time-invariance , we get [ ] [ ] k hn hn k ] k k k xkhn k DISCRETE-TIME CONVOLUTION!! Note that [] hn is the impulse response.
EE 341 Discrete-Time Linear Systems – Week 3 Spring 2011 3 The Convolution Equation Definition [] [] [] [][ ] k yn xn hn xkhn k   Which means something important: The output of an LTI system is the input convolved with the impulse response where [] hn is the impulse response.

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Chapter 2 Week 3 EE 341 Clark Lecture Notes - 15 april - EE...

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