02_linear time invariant system

# 02_linear time invariant system - ELEC211: Signals and...

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1 Chapter 2 AKW Fall 2009 ELEC211: Signals and Systems Chapter 2: Linear Time-Invariant systems and Impulse Response Discrete-time LTI systems: convolution sum Continuous-time LTI systems: convolution integral Properties of LTI systems Causal LTI systems described by differential and difference equations

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2 Chapter 2 AKW Fall 2009 Characterizing Discrete-Time (DT) LTI systems System (DT) ] [ n x What is the response ??? Given an input to an LTI system, what can we say about the output or response of the system? How can we characterize an LTI system? Let’s start with the Discrete-Time case Given an input [] yn
3 Chapter 2 AKW Fall 2009 Unit Impulse Response of (DT) LTI systems The simplest DT input signal is the unit impulse signal δ [ n ]. Let h [ n ] be the response to [ n ]. We call h [ n ] the unit impulse/sample response , or simply the impulse response . h [ n ] completely characterizes a DT LTI system. That is, if we know what h [ n ] is, we can determine the response of this system under any input. The key lies in the fact that all DT input signals can be viewed as a superposition of shifted unit impulse signals . n 0 1 -1 2 -2 ] [ n ] [ n h n unit impulse Impulse response

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4 Chapter 2 AKW Fall 2009 Representation of DT Signals as Impulses −∞ = = k k n k x n x ] [ ] [ ] [ δ Mathematically: n 0 1 -1 2 -2 ] [ n x The signal x [ n ] on the left below can be regarded as a superposition of the shifted impulses on the right shifted impulse n 0 1 -1 2 -2 n 0 1 -1 2 -2 n 0 1 -1 2 -2 ] 1 [ ] 1 [ + n x ] [ ] 0 [ n x ] 1 [ ] 1 [ n x 1 = k 0 = k 1 = k = + +
5 Chapter 2 AKW Fall 2009 Output of LTI (DT) systems (cont.) Linear System −∞ = = k k n k x n x ] [ ] [ ] [ δ −∞ = = k k n h k x n y ] [ ] [ ] [ system ] [ n x What is the response ??? Let be the system’s responses the shifted impulse function . That is, ] [ ] [ n h k n k ] [ n h k ] [ k n So, if the system is linear , the output is the sum of the responses to the weighted and shifted impulse functions! Now, we return to the question: Eq. (2.3)

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6 Chapter 2 AKW Fall 2009 Eq. (2.6) is called the convolution sum . It means that y [ n ], the output at any time n , can be founded by summing up the responses of the system due to the input at time k for all values of k . Output of LTI (DT) systems (cont.) Linear Time-Invariant (LTI) System −∞ = = k k n k x n x ] [ ] [ ] [ δ −∞ = = k k n h k x n y ] [ ] [ ] [ ] [ ] [ 0 k n h n h k = Next, if the system is Time Invariant , then where which is the unit impulse response ] [ ] [ 0 n h n h = Replacing h k [ n ] by h [ n – k ] in Eq. (2.3) , we have: Eq. (2.6)
7 Chapter 2 AKW Fall 2009 −∞ = = = k k n h k x n h n x n y ] [ ] [ ] [ ] [ ] [ The Convolution Sum We also say that the output of an LTI system is the convolution of the input with the system’s impulse response. We notate the convolution of x [ n ] and h [ n ] as x [ n ]* h [ n ] Meaning : The meaning of the convolution sum is extremely simple: The output at time n is the sum of all of the responses due to the individual x [ k ]’s. The response at time n due to the input at time k is x [ k ] h [ n - k ].

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8 Chapter 2 AKW Fall 2009 Meaning of The Convolution Sum Example:
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## This note was uploaded on 09/24/2010 for the course ELEC 211 taught by Professor Albertk.wong during the Fall '09 term at HKUST.

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02_linear time invariant system - ELEC211: Signals and...

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