Linear Time-Invariant Systems

Transcription

1 9/9/00 LIEAR TIE-IVARIAT SYSTES Uit, d Part Liear Time-Ivariat Sstems A importat class of discrete-time sstem cosists of those that are Liear Priciple of superpositio Time-ivariat dela of the iput sequece causes a correspodig shift i the output sequece This tpe of sstems ca be completel characterized b its impulse respose

2 9/9/00 Iput-Output Relatioship A sequece ca be epressed i terms of a uit impulse sequece as follows [ ] [ ] δ [ ] The, the output sequece ca be obtaied as follows [ ] T [ ] δ [ ] Iput-Output Relatioship Usig the priciple of superpositio (liear sstems) [ ] [ ] T{ δ [ ] } [ ] h [ ] where h [] is the respose of the sstem to δ[-]. Addig the time-ivariat restrictio, the output sequece become [ ] [ ] h[ ]

3 9/9/00 Covolutio Sum The last equatio represets the covolutio sum of sequeces [] ad h[] [ ] [ ] h[ ] [ ] h[ ] * The importace of this equatio is: Full-characterized a Liear Time-Ivariat (LTI) Sstems Provide a wa to implemet a LTI sstem Covolutio Sum [ ] [ ] h[ ] [] h[] h[-] 3

4 9/9/00 Covolutio Sum [ 0] [ ] h[ ] [ 0] h[ 0] [ ] [ 0] h[ ] [ ] h[ 0] [ ] [ 0] h[ ] [ ] h[ ] [ ] h[ 0] [ 7] [ 4] h[ 3] Properties of LTI Sstems The covolutio is commutative Cascade Coectio [ ] * h[ ] h[ ] [ ] * [ ] *( h [ ] * h [ ] ) [ ] * h [ ] h [ ] * [] h []*h [] [] [] h [] h [] [] [] h [] h [] [] 4

5 9/9/00 Properties of LTI Sstems Covolutio is distributive over the additio [ ] *( h [ ] h [ ] ) [ ] * h [ ] [ ] h [ ] * [] h []h [] [] [] h [] h [] [] Stabilit Coditio for LTI Sstems A LTI sstem is BIBO stable if ad ol if its impulse respose is absolutel summable, i.e. [ ] h[ ] [ ] [ h[ ] B h[ ] < ] The stabilit coditio is s h [ ] < 5

6 9/9/00 Causalit Coditio A LTI sstem is causal if ad ol if the impulse respose satisfies the followig coditio h [ ] 0 for < 0 A o-causal sstem ca ofte be realized as a causal sstem b delaig the output sequece b a appropriate amout. Classificatio of LTI sstems LTI Sstems Impulse Respose Legth Output Calculatio Process Coefficiets Fiite Impulse Respose (FIR) Recursive Real discretetime sstems Ifiite Impulse Respose (IIR) o-recursive Comple discrete-time sstems 6

7 9/9/00 Fiite Impulse Respose Sstems If h[] is a fiite impulse respose [ ] 0 h For > ad < with < The, the sstem is ow as Fiite Impulse Respose (FIR) Sstem, for which the covolutio sum reduces to h [ ] [ ] [ ] If h[] is a ifiite impulse respose, the it is ow as a Ifiite Impulse Respose (IIR) Sstems Output Calculatio Process If the output sample ca be calculated sequetiall, owig ol the preset ad past iput samples, the sstem is called orecursive Discrete-time Sstem If the computatio of the output ivolves past output samples as well as preset ad past iput samples, the the sstem is ow as Recursive Discrete-time Sstem 7

8 9/9/00 8 Eample Cosider a sstem with impulse respose The, the output sequece ca be obtaied from the covolutio sum [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] h [ ] otherwise h 0, 0,,...,5, Eample To develop the recursive versio of this sstem, we ote That ca be rewritte as Substitutig the above equatio i the origial equatio [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 6 [ ] [ ] [ ] [ ] [ ] [ ] [ ] 5 4 3

9 9/9/00 9 Differece Equatio The output of a fiite-order LTI sstem at time ca be epressed as a liear combiatio of the iputs ad outputs i the followig form: For a causal sstem f 0 The differece equatio provide a wa to implemet the discrete-time filters [ ] [ ] p f b a ] [ Eample The geeral movig average sstem is defied as Cosider 0 so that the sstem is causal [ ] [ ] [ ] [ ] 0

10 9/9/00 h Eample The sstem correspods to this class of sstem where a 0, b /( ) for 0,.., [ ] [ ] 0 Its impulse respose ca be epressed as [ ] δ [ ] ( µ [ ] µ [ ] ) 0 Eample The accumulator sstem is defied as [ ] [ ] The impulse respose is obtaied as h [ ] δ [ ] µ [ ] 0

11 9/9/00 Eample 3 The causal movig average sstem impulse respose is h [ ] ( µ [ ] µ [ ] ) The impulse respose ca be rewritte as h µ [ ] ( δ [ ] δ [ ] )* [ ] Eample 3 The causal movig average sstem ca be see as a cascade of two sstems [ ] ( [ ] [ ] ) [ ] [ ] [ ] The differece equatio ca be rewritte as [ ] [ ] ( [ ] [ ] )

12 9/9/00 Implemetig Discrete-Time Sstems The discrete-time sstem ca be full describe b usig a differece equatio. This equatios use the followig operatios [ ] a [] p [ ] b [ ] f w[][][] [] [] []a [] [] Z - [][-] Direct Structure b -f Z Z b - [ ] a p [ ] b [ ] f [] Z Z - b 0 b a 0 Z - [] Z - Z - Z - b p a Z -

13 9/9/00 Direct Structure for Causal sstems [ ] a p [ ] b [ ] 0 [] Z - b 0 b a 0 Z - [] Z - Z - Z - b p a Z - Program _7 % Program _7 % Computatio of cross-correlatio sequece % iput('tpe i the referece sequece '); iput('tpe i the secod sequece '); % Compute the correlatio sequece legth()-; legth()-; r cov(,fliplr()); (-):'; stem(,r); label('lag ide'); label('amplitude'); v ais; ais([- v(3:ed)]); 3

14 9/9/00 Program _8 % Program _8 % Computatio of Autocorrelatio of a % oise Corrupted Siusoidal Sequece % 96; :; cos(pi*0.5*); % Geerate the siusoidal sequece d rad(,) - 0.5; % Geerate the oise sequece d; % Geerate the oise-corrupted siusoidal r cov(, fliplr()); % Compute the correlatio seq -8:8; stem(, r(68:4)); label('lag ide'); label('amplitude'); 4