L6 - L10 Chapter2 LTI systems

L6 - L10 Chapter2 LTI systems - Chapter 2: Linear...

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Chapter 2: Linear time-invariant systems 2.0 Introduction 2.1 Discrete-time LTI systems: The convolution sum 2.2 Continuous-time LTI system: The convolution integral 2.3 Properties of LTI systems 2.4 Causal LTI systems described by differential and difference equations NIB Impulse response of systems described by D.E. NIB Stability of systems described by D.E.
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ECSE303 Chap. 2 MR - 2/1/2010 2
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2.0 Introduction Many physical processes can be represented by, and successfully analyzed with linear and time- invariant (LTI) system models. For example: DC motor Mixing tank RLC filtering circuit Op-amp circuit Controllers (e.g. cruise control) ECSE303 Chap. 2 MR - 2/1/2010 3
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2.1 Discrete-Time LTI Systems: The Convolution Sum Representation of Discrete-Time Signals in Terms of Impulses A discrete-time signal can be viewed as a combination of weighted independent delayed impulses: This summation illustrates the sifting property of the discrete unit impulse. The impulse decomposes x [ n ] point by point. xn xk n k k [] [ ][ ] =− =−∞ δ ECSE303 Chap. 2 MR - 2/1/2010 4
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x [ n ] n 0 1 2 2 1 n 0 1 2 n 0 1 2 n 0 1 2 2 1 2 1 2 1 xn xk n k k [] [][ ] =− =−∞ δ = + + [0] 1 x = [1] 2 x = [2] 1 x = [ 0 ][ 0 ] [1] [ 1] [2] [ 2] x n = + + Example ECSE303 Chap. 2 MR - 2/1/2010 5
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Example An equivalent expression for u [ n ] using the impulse function is Note: Compare with this form seen in chapter 1 xn xk n k k [] [][ ] =− =−∞ δ 0 [ ] k un = = ECSE303 Chap. 2 MR - 2/1/2010 6 [] [ ] n m m =
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Let h k [ n ] be the response of a linear system S to the shifted impulse δ[ n-k ] . for example, Take note that in this example, the system is clearly not time-invariant . [ ] [ ] ( ) k hn S n k δ =− S S h 2 [ n ] n 0 1 2 3 4 5 3 2 1 δ [ n-2 ] n 0 1 2 3 4 5 3 2 1 h 1 [ n ] n 0 1 2 3 4 5 3 2 1 δ [ n-1 ] n 0 1 2 3 4 5 3 2 1 ECSE303 Chap. 2 MR - 2/1/2010 7
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Principle of superposition of linear systems: the response y [ n ] of a discrete-time linear system S is the sum of the responses to the individual shifted impulses making up the input signal x [ n ]. Example ECSE303 Chap. 2 MR - 2/1/2010 8 y [ n ] = h 1 [ n ]- h 2 [ n ] n 0 1 2 3 4 5 3 2 1 x [ n ]= δ [ n-1 ]- δ [ n-2 ] n 0 1 2 3 4 5 3 2 1 h 2 [ n ] n 0 1 2 3 4 5 3 2 1 δ [ n-2 ] n 0 1 2 3 4 5 3 2 1 h 1 [ n ] n 0 1 2 3 4 5 3 2 1 δ [ n-1 ] n 0 1 2 3 4 5 3 2 1 S S S
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Thus, the response to an input x [ n ] can be written as an infinite sum of all the impulse responses: As a general result, if we know the response of S to each delayed impulse δ k [ n ], we will be able to calculate the response of S to any input signal x [ n ]. [] k k y nx k h n =−∞ = ECSE303 Chap. 2 MR - 2/1/2010 9
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Now considering a linear and time-invariant (LTI) system , the impulse responses for various values of k are just shifted versions of the impulse response at n =0. From this we write The consequence of this: an LTI system is entirely known from its impulse response h [ n-k ] [ ] [ ] [ ] [ ] 0 k hn h n h n k = =− ECSE303 Chap. 2 MR - 2/1/2010 10
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h 2 [ n ] = h [ n-2 ] n 0 1 2 3 4 5 3 2 1 h -1 [ n ]= h [ n+1 ] n 0 1 2 3 4 5 3 2 1 h 0 [ n ] = h [ n ] n 0 1 2 3 4 5 3 2 1 ECSE303 Chap. 2 MR - 2/1/2010 11 LTI
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As a result, the response of a Linear and Time- Invariant (LTI) system to any x [ n ] can be written as: This is called the convolution sum [] [ ] k yn xk hn k
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This note was uploaded on 04/07/2010 for the course ELEC ecse 303 taught by Professor Rochette during the Winter '10 term at McGill.

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L6 - L10 Chapter2 LTI systems - Chapter 2: Linear...

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