Shift Property of z-transform. Lecture 16. More z-transform (Lathi 5.2, ) More Properties of z-transform. Convolution property of z-transform

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1 Shift Property of -Transform If Lecture 6 More -Transform (Lathi 5.2, ) then which is delay causal signal by sample period. If we delay x[n] first: Peter Cheung Department of Electrical & Electronic Engineering Imperial College London If we ADVACE x[n] by sample period: URL: p.cheung@imperial.ac.uk L5.2 p508 Lecture 6 Slide Lecture 6 Slide 2 Convolution property of -transform More Properties of -Transform If h[n] is the impulse response of a discrete-time LTI system, then For all these cases, we assume: Scaling Property: then If Multiply by n property: Then Time reversal property: That is: convolution in the time-domain is the same as multiplication in the -domain. Therefore, we can derive the input-output relationship fo any LTI systems in -domain: Initial value property: L5.2 p5 L5.2 p52 Lecture 6 Slide 3 Lecture 6 Slide 4

2 Summary of -transform properties () Summary of -transform properties (2) L5.2 p54 L5.2 p54 Lecture 6 Slide 5 Lecture 6 Slide 6 Discrete LTI System and Difference Equation Realiation of LTI System Direct Form I Consider a discrete time system where the input-output relation is described by: yn [ ] 5 yn [ ] + 6 yn [ 2] = xn [ ] + 3 xn [ ] + 5 xn [ 2] This is known as a difference equation, where current output is dependent on current input x[n], and two previous inputs and outputs x[n-], x[n-2], y[n-] and y[n-2]. Take -transform on both sides and assume ero-state condition: Y Y Y X X X 2 2 () 5 () + 6 () = () + 3 () + 5 () + = + + Y 2 () 3 5 () + + = H = 2 X() ( 5 6 ) Y( ) ( 3 5 ) X( ) The transfer function of a general th order causal discrete LTI system is: b + b H [] = + + a a + a The general transfer function H() can be realised using Direct Form I as follows: b + b Y [] = HX [] [] = X [] + a a + a + = ( b0 b... b b ) X[ ] a a + a = W() + + a a + a Lecture 6 Slide 7 Lecture 6 Slide 8

3 Realiation of LTI System Direct Form II Realiation of LTI System Transposed Direct Form II Or use Canonical Director Form II: b + b Y [] = HX [] [] = X [] + a a + a + = ( b + b ) X[ ] + + a a + a + ( 0 ) = b + b b + b W( ) + 0 = = + a a + a Or the transposed version: b + b b + b Y [] HX [] [] X [] ( ) ( ) + + ( 0 ) ( ) Y [ ] + a a + a Y [ ] = b+ b b + b X [ ] Y [ ] = b+ b b + b X [ ] a a + a Y [ ] ( bx[ ] ay[ ]) + ( b X[ ] a Y[ ]) ( b X[ ] a Y[ ]) Lecture 6 Slide 9 Lecture 6 Slide 0 Examples Frequency Response of Discrete-time Systems 2 2 H [] = = H [] = = + + Direct Form II Transposed Remember that for a continuous-time system, the system response to an input e t is H() e t. The response to an input cos ωt is H() cos (ωt + H()). j t e ω H( ) e t Continuous-time (.) continuous-time cosωt System H() ow, consider a discrete-time system with -domain transfer function H[]. Let = e, the system response to an input e n is H[e ] e n. The response to an input cos Ωn is H[e ] cos (cos Ωn + H[e ]). S where T S is the sampling period H [] = = L5.4 p j n e Ω cos Ωn Discrete-time System H [e ] He [ ] e n He [ ]cos( Ω n+ He [ ]) [.] discrete-time L5.5 p53 Lecture 6 Slide Lecture 6 Slide 2

Frequency Response Example () Frequency Response Example (2) For a system specified by the following difference

8 yn [ ] = xn [ + ] Take -transform on both sides to find the transfer function: Amplitude response Y[] 0.

5 p533 Lecture 6 Slide 3 Lecture 6 Slide 4 Mapping from to -plane Mapping from to -plane = e = e = e e we can

4 Frequency Response Example () Frequency Response Example (2) For a system specified by the following difference equation, find the frequency response of the system. yn [ + ] 0.8 yn [ ] = xn [ + ] Take -transform on both sides to find the transfer function: Amplitude response Y[] 0.8[] Y = X[] Y [] H [] = = = X [ ] Therefore the frequency response is: Phase response L5.5 p533 L5.5 p533 Lecture 6 Slide 3 Lecture 6 Slide 4 Mapping from to -plane Mapping from to -plane = e = e = e e we can map the to the -plane as below: Im( ) -plane = e = e = e e we can map the to the -plane as below: Im( ) -plane s + s + + Re( ) + Re( ) s s Lecture 6 Slide 5 Lecture 6 Slide 6

5 Mapping from to -plane = e = e = e e we can map the to the -plane as below: Im( ) -plane s + + Re( ) s Lecture 6 Slide 7

Lecture 11 FIR Filters

Lecture 11 FIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/4/12 1 The Unit Impulse Sequence Any sequence can be represented in this way. The equation is true if k ranges