DIGITAL SIGNAL PROCESSING LECTURE 3

Transcription

1 DIGITAL SIGNAL PROCESSING LECTURE 3 Fall 2 2K8-5 th Semester Tahir Muhammad tmuhammad_7@yahoo.com Cotet ad Figures are from Discrete-Time Sigal Processig, 2e by Oppeheim, Shafer, ad Buc, Pretice Hall Ic.

2 Itroductio to LTI System Impulse ad Step Respose If the iput to the DTS system is Uit Impulse (δ), the output of the system will be Impulse Respose (h). If the iput to the DTS system is Uit Step (µ), the output of the system will be Step Respose (s). Digital Sigal Processig 2

3 Impulse Respose Digital Sigal Processig 3

4 Iput-Output Relatioship A Liear time-ivariat ivariat system satisfied both the liearity ad time ivariace properties. A LTI discrete-time system is characterized by its impulse respose Eample:.5δ+2 +.5δ- - δ-4 will result i y.5h+2 +.5h- - h-4 Digital Sigal Processig 4

5 Iput-Output Relatioship ca be epressed i the form δ where deotes the th sample of sequece {} The respose to the LTI system is y or represeted as h y h h Digital Sigal Processig 5

6 Computatio of Discrete Covolutio Digital Sigal Processig 6

7 Eample 2.3 N N u u h u α otherwise N u u h for i) < for iii) > N ( ) N ( ) N y for ii) N + + a a a a a y N N N ) ( ( ) N + a a a N Digital Sigal Processig 7 a a a y +

8 Fiite Legth Eample 5 otherwise h α otherwise h Digital Sigal Processig 8

9 <.8 - h Digital Sigal Processig 9

10 < < y + α α α Digital Sigal Processig

11 6< < y y α α 5 5 α + ( 5) 5 6 α 5 α α α 5 Digital Sigal Processig

12 < < α y α y α ( 5) 6 α 5 α α α Digital Sigal Processig 2

13 > y Digital Sigal Processig 3

14 Digital Sigal Processig 4

15 Iput-output Relatioship Properties of covolutio Properties of covolutio Commutative Associative 2 2 Di t ib ti ) ( Distributive ) ( ) ( Digital Sigal Processig 5

16 Properties of LTI Systems Stability if ad oly if, sum of magitude of Impulse Respose, h is fiite S h h < Digital Sigal Processig 6

17 Properties of LTI Systems Causality if ad oly if Impulse Respose,h for all < h, < Digital Sigal Processig 7

18 Properties of LTI Systems Digital Sigal Processig 8

19 Liear Costat-Coefficiet Differece Equatios Fiite Impulse Respose (FIR) For causal, FIR systems Covolutio reduces to h, for < ad M > y M h Ifiite Impulse Respose (IIR) y h Digital Sigal Processig 9

20 Liear Costat-Coefficiet Differece Equatios A importat class of LTI systems of the form N a M b y The output is ot uiquely specified for a give iput The iitial coditios are required Liearity, time ivariace, ad causality deped o the iitial coditios If iitial coditios are assumed to be zero system is liear, time ivariat, ad causal Eample Movig Average Differece eq. y b where a b a y Digital Sigal Processig 2

21 Liear Costat-Coefficiet Differece Equatios y output : iput : N M m m m b y a Eample Accumulator : y y y y + + Digital Sigal Processig 2 y y X is the differece of y

22 Bloc Diagram Digital Sigal Processig 22

23 Stability, Causality, ad Time ivariace of systems describe by LCCD LCCD: iput : output : y N a y M m b m m Particular cases M b i. y m m ii. m N a y Digital Sigal Processig 23

24 Stability Case i: May be stable if y Whe M b < m m bm m M m m <, ad bm < < Digital Sigal Processig 24

25 Stability Case ii: ii. N a y Evethough a <, the system is ustable E.g. For a > this system is ustable Digital Sigal Processig 25

26 Liearity ad Causality Suppose that for a give iput we have foud oe particular output sequece y P so that a LCCD equatio is satisfied. The the same equatio with the same iput is satisfied by ay output of the form y yp + y Where y H is ay solutio to the LCCD equatio with zero iput. Remar: y P ad y H are referred to as the particular ad homogeeous solutios respectively. Digital Sigal Processig 26 H

27 Liearity ad Causality A LCCD equatio does ot provide a uique specificatio of output for a give iput Auiliary iformatio or coditios are required to specify uiquely the output for a give iput Eample Let auiliary iformatio be i the form of N sequetial output values. The, Later values ca be obtaied by rearragig LCCD equatio as a recursive relatio ruig forward i. Prior values ca be obtaied by rearragig LCCD equatio as a recursive relatio ruig bacward i. Digital Sigal Processig 27

28 LCCD equatios as recursive procedures Digital Sigal Processig 28

29 Digital Sigal Processig 29

30 Digital Sigal Processig 3

31 LTI ad causal Digital Sigal Processig 3

32 Liearity ad causality Digital Sigal Processig 32

33 Time ivariace Digital Sigal Processig 33

34 Liear, Time ivariat, ad Causal system Digital Sigal Processig 34

35 Digital Sigal Processig 35