Computing the output response of LTI Systems.

Transcription

1 Computig the output respose of LTI Systems. By breaig or decomposig ad represetig the iput sigal to the LTI system ito terms of a liear combiatio of a set of basic sigals. Usig the superpositio property of LTI system to compute the output of the system i terms of its respose to these basic sigals.

2 Geeral Sigal Represetatios By Basic Sigal The basic sigal - i particular the uit impulse ca be used to decompose ad represet the geeral form of ay sigal. Liear combiatio of delayed impulses ca represet these geeral sigals.

3 Respose of LTI System to Geeral Iput Sigal Geeral Iput Sigal Delayed Impulse Sigal LTI SYSTEM LTI SYSTEM Output Respose Sigal Delayed Impulse Sigal N LTI SYSTEM Respose to Impulse sigal N 3

4 Represetatio of Discrete-time Sigals i Terms of Impulses. Discrete-time sigals are sequeces of idividual impulses x x δ x δ + 4 4

5 Discrete-time sigals are sequeces of idividual scaled uit impulses x x δ x δ x δ 4 5

6 x Shifted Scaled Impulses x δ x 3 δ + Geerally: x δ + x δ + x3 δ + + x δ x x δ, + + x δ The arbitrary sequece is represeted by a liear combiatio of shifted uit impulses δ-, where the weights i this liear combiatio are x. The above equatio is called the siftig property of discrete-time uit impulse. 6

7 x +. δ u δ +. δ δ +. δ δ + +. δ As Example cosider uit step sigal xu:- Geerally:- u +. δ, The uit step sequece is represeted by a liear combiatio of shifted uit impulses δ-, where the weights i this liear combiatio are oes from right up to This is idetically similar to the expressio we have derived i our previous lecture a few wees bac whe we dealt with uit step. 7

8 The Discrete-time Uit Impulse Resposes ad the Covolutio Sum Represetatio To determie the output respose of a LTI system to a arbitrary iput sigal x, we mae use of the siftig property for iput sigal ad the superpositio ad timeivariat properties of LTI system. 8

9 Covolutio Sum Represetatio The respose of a liear system to x will be the superpositio of the scaled resposes of the system to each of these shifted impulses. From the time-ivariat property, the respose of LTI system to the time-shifted uit impulses are simply time-shifted resposes of oe aother. 9

10 Uit Impulse Respose h δ LTI System h o δ- LTI System h

11 Respose to scaled uit impulse iput xδ- x-.δ+ - LTI System x-.h - - x.δ x.h o LTI System x+.δ- LTI System x+.h +κ

12 Output y of LTI System With the iput x beig expressed as the delayed trai of + - y x.h. scaled impulses we have: - Thus, if we ow the respose of a liear system to the set of shifted uit impulses, we ca costruct the respose y to a arbitrary iput sigal x.

13 h - x h h 3

14 x-δ+ x-h - xδ xh xδ- xh x y 4

15 I geeral, the respose h eed ot be related to each other for differet values of. If the liear system is also time-ivariat system, the these resposes h to time shifted uit impulse are all time-shifted versios of each other. I.e. h h -. For otatioal coveiece we drop the subscript o h h. h is defied as the uit impluse (sample) respose 5

16 Covolutio sum or Superpositio sum. + y x. h Covolutio operatio otatio give by : - y x*h 6

17 Covolutio sum or Superpositio sum. x h y x h.5 7

18 Covolutio sum or Superpositio sum. x h y y x h.5 x h.5 8

19 Covolutio sum or Superpositio sum. x h y y y x h.5 x h.5 x h.5 9

20 Covolutio sum or Superpositio sum. x h y y y y3 x h.5 x h.5 x h.5 x h3. y x h

21 yxh-+xh-.5h+h- x x x x.5 h x.5.5h h y

22 Modified Example.3

23 3 Modified Example.5 u h u x m m - l -. ) ( ) ( ) ( ) ( y l ad m -, Chagig variable l x y, for ozero samples has, / -, let r,, For + + < < < m m l r r h h x if but h x y α α α

24 4 Modified Example.5 u h u x m m - l -. ) ( ) ( ) ( ) ( y l ad m -, Chagig variable l x y, for ozero samples has, / -, let r,, For + + < < < m m l r r h h x if but h x y α α α