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Ve216LectureNotesChapter7Part2

# Ve216LectureNotesChapter7Part2 - Ve216 Lecture Notes...

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Ve216 Lecture Notes Dianguang Ma Spring 2009

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Chapter 7 (Part II) Representing Signals by Using Discrete-Time Complex Exponentials: The z-transform
7.6 The Transfer Function Having defined the transfer function as the z- transform of the impulse response, we examine the relationship between the transfer function and input- output descriptions of discrete-time LTI systems. = - = - = - = - = = = = = - = - = = = M k k k M k k k M k k k N k k k M k k N k k z a z b z X z Y z H z X z b z Y z a k n x b k n y a z X z Y z H z X z H z Y n x n h n y 0 0 0 0 0 0 ) ( ) ( ) ( ) ( ) ( ] [ ] [ ) ( ) ( ) ( ) ( ) ( ) ( ] [ ] [ ] [

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7.6 The Transfer Function Example 7.13 Find the transfer function and impulse response of a causal LTI system if ( 29 1 1 1 1 1 [ ] [ ], [ ] 3 1 [ ] [ ]. 3 3 [ANSWER] 1 4(1 ) 3 ( ) 1 (1 )(1 ) 3 1 [ ] 2( 1) [ ] 2 [ ] 3 n n n n n x n u n y n u n u n z H z z z h n u n u n - - - = - = - + + = + - = - +
7.6 The Transfer Function Example 7.14 Find the transfer function and impulse response of a causal LTI system described by ] [ 4 3 ] [ ) 2 1 ( 2 ] [ 8 3 4 1 1 2 1 ) ( [ANSWER] ]. 1 [ 2 ] [ ] 2 [ 8 3 ] 1 [ 4 1 ] [ 2 1 1 n u n u n h z z z z H n x n x n y n y n y n n + - - = - - + - = - + - = - - - - - - -

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7.7 Causality and Stability A discrete-time system is causal if and only if the ROC of its transfer function is the exterior of a circle, including infinity (H(z) does not contain any positive power of z). < + + = = - = - ) ( lim ] 1 [ ] 0 [ ] [ ) ( 1 0 z H z h h z n h z H z n n
7.7 Causality and Stability A discrete-time LTI system is causal if and only if the ROC of its transfer function is the exterior of a circle outside the outermost pole. With H(z) expressed as a ratio of polynomials in z, the order of the numerator cannot be greater than the order of the denominator. + - + = + + + + - = + + + - = ] [ 4 9 ] 1 [ ] [ 4 9 2 ) ( 8 1 4 1 2 32 9 16 23 8 1 4 1 2 2 3 n n n h z z z z z z z z z z H δ δ

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7.7 Causality and Stability A discrete-time LTI system is stable if and only if the ROC of its transfer function includes the unit circle. A discrete-time LTI system that is both stable and causal must have all their poles inside the unit circle.
7.7 Causality and Stability Example Consider a stable and causal system with impulse response h[n] and rational transfer function H(z). Suppose it is known that H(z) contains a pole at z=1/2 and a zero somewhere on the unit circle. The precise number and locations of all of the other poles and zeros are unknown. For each of the following statement, let us determine whether we can definitely say that it is true, whether we can definitely say that it is false, or whether there is insufficient information given to determine if it is true or not:

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7.7 Causality and Stability Example (continued) ( 29 { } ( 29 system. stable a of response impulse the is ] [ ] [ ] [ (e) real. is ] [ (d) duration. finite has ] [ (c) . some for 0 ) ( (b) converges. ] [ (a) 2 1 n h n h n n g n h n h e H n h F j n = =
7.7 Causality and Stability Example (continued) ( 29 { } { } . 2 / 1 at pole a contains ) ( But . and/or 0 possibly except plane, - entire the be will ROC The duration finite has ] [ (c) circle. unit on the somewhere zero a contains ) ( . ) ( ) ( (b) . 1 includes ) ( of ROC The ) ( ] [ 2 ] [ (a) [HINTS] 1 2 2 1 = = = = = = = = - z z H z z z n h z H z H e H z z H z H n

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Ve216LectureNotesChapter7Part2 - Ve216 Lecture Notes...

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