Lecture 2 Linear and Time Invariant Systems

Transcription

1 EE3054 Sigals ad Systems Lecture 2 Liear ad Time Ivariat Systems Yao Wag Polytechic Uiversity Most of the slides icluded are extracted from lecture presetatios prepared by McClella ad Schafer

2 Licese Ifo for SPFirst Slides This work released uder a Creative Commos Licese with the followig terms: Attributio The licesor permits others to copy, distribute, display, ad perform the work. I retur, licesees must give the origial authors credit. No-Commercial The licesor permits others to copy, distribute, display, ad perform the work. I retur, licesees may ot use the work for commercial purposes uless they get the licesor's permissio. Share Alike The licesor permits others to distribute derivative works oly uder a licese idetical to the oe that govers the licesor's work. Full Text of the Licese This (hidde) page should be kept with the presetatio 1/30/ , JH McClella & RW Schafer 2

3 Review of Last Lecture Geeral FIR System IMPULSE RESPONSE FIR case: h[]=b_ CONVOLUTION { b k } h[] y[ ] = h[ ] x[ ] For ay FIR system: y[] = x[] * h[] 1/30/ , JH McClella & RW Schafer 3

4 1/30/ , JH McClella & RW Schafer 4 GENERAL FIR FILTER FILTER COEFFICIENTS {b k } DEFINE THE FILTER For example, = = M k k k x b y 0 ] [ ] [ 3] [ 2] [ 2 1] [ ] [ 3 ] [ ] [ = = = x x x x k x b y k k 1, 2,1} 3, = { b k Feedforward Differece Equatio

5 GENERAL FIR FILTER SLIDE a Legth-L WINDOW over x[] Whe h[] is ot symmetric, eeds to flip h() first! x[-m] x[] 1/30/ , JH McClella & RW Schafer 5

6 7-pt AVG EXAMPLE Iput : x[ ] = (1.02) + cos(2π /8 + π / 4) for 0 40 CAUSAL: Use Previous 1/30/ , JH McClella & RW Schafer 6 LONGER OUTPUT

7 Uit Impulse Sigal x[] has oly oe NON-ZERO VALUE δ [ ] = 1 0 = 0 0 UNIT-IMPULSE 1 1/30/ , JH McClella & RW Schafer 7

8 4-pt Avg Impulse Respose δ[] READS OUT the FILTER COEFFICIENTS h i h[] deotes Impulse Respose y[ ] = 1 ( x[ ] + x[ 1] + x[ 2] + x[ 4 h[ ] = {, 0, 0, 1, 1, 1, 1, 0, 0, } 4 = 1 =0 =1 4 = /30/ , JH McClella & RW Schafer 8 4 =4 =5 NON-ZERO Whe widow overlaps δ[] 3])

9 What is Impulse Respose? Impulse respose is the output sigal whe the iput is a impulse Fiite Impulse Respose (FIR) system: Systems for which the impulse respose has fiite duratio For FIR system, impulse respose = Filter coefficiets h[k] = b_k Output = h[k]* iput 1/30/ , JH McClella & RW Schafer 9

10 1/30/ , JH McClella & RW Schafer 10 FIR IMPULSE RESPONSE Covolutio = Filter Defiitio Filter Coeffs = Impulse Respose = = M k k x k h y 0 ] [ ] [ ] [ CONVOLUTION = = M k k k x b y 0 ] [ ] [

11 Covolutio Operatio Flip h[] SLIDE a Legth-L WINDOW over x[] y [ ] = M k = 0 h [ k ] x [ k ] CONVOLUTION x[-m] x[] 1/30/ , JH McClella & RW Schafer 11

12 More o sigal rages ad legths after filterig Iput sigal from 0 to N-1, legth=l1=n Filter from 0 to M, legth = L2= M+1 Output sigal? From 0 to N+M-1, legth L3=N+M=L1+L2-1 1/30/ , JH McClella & RW Schafer 12

13 DCONVDEMO: MATLAB GUI 1/30/ , JH McClella & RW Schafer 13

14 Go through the demo program for differet types of sigals Do a example by had Rectagular * rectagular Step fuctio * rectagular 1/30/ , JH McClella & RW Schafer 14

15 CONVOLUTION via Sythetic Polyomial Multiplicatio 1/30/ , JH McClella & RW Schafer 15

16 Covolutio via Sythetic Polyomial Multiplicatio More example 1/30/ , JH McClella & RW Schafer 16

17 MATLAB for FIR FILTER yy = cov(bb,xx) VECTOR bb cotais Filter Coefficiets DSP-First: yy = firfilt(bb,xx) FILTER COEFFICIENTS {b k } y[ ] = M k= 0 b k x[ k] cov2() for images 1/30/ , JH McClella & RW Schafer 17

18 POP QUIZ FIR Filter is FIRST DIFFERENCE y[] = x[] - x[-1] INPUT is UNIT STEP 1 u[ ] = 0 < 0 0 Fid y[] y [ ] = u [ ] u [ 1] = δ [ ] 1/30/ , JH McClella & RW Schafer 18

19 SYSTEM PROPERTIES x[] SYSTEM y[] MATHEMATICAL DESCRIPTION TIME-INVARIANCE INVARIANCE LINEARITY CAUSALITY No output prior to iput 1/30/ , JH McClella & RW Schafer 19

20 TIME-INVARIANCE IDEA: Time-Shiftig the iput will cause the same time-shift i the output EQUIVALENTLY, We ca prove that The time origi (=0) is picked arbitrary 1/30/ , JH McClella & RW Schafer 20

21 TESTING Time-Ivariace 1/30/ , JH McClella & RW Schafer 21

22 Examples of systems that are time ivariat ad o-time ivariat 1/30/ , JH McClella & RW Schafer 22

23 LINEAR SYSTEM LINEARITY = Two Properties SCALING Doublig x[] will double y[] SUPERPOSITION: Addig two iputs gives a output that is the sum of the idividual outputs 1/30/ , JH McClella & RW Schafer 23

24 TESTING LINEARITY 1/30/ , JH McClella & RW Schafer 24

25 Examples systems that are liear ad o-liear 1/30/ , JH McClella & RW Schafer 25

26 LTI SYSTEMS LTI: Liear & Time-Ivariat Ay FIR system is LTI Proof! 1/30/ , JH McClella & RW Schafer 26

27 LTI SYSTEMS COMPLETELY CHARACTERIZED by: IMPULSE RESPONSE h[] CONVOLUTION: y[] = x[]*h[] The rule defiig the system ca ALWAYS be rewritte as covolutio FIR Example: h[] is same as b k 1/30/ , JH McClella & RW Schafer 27

28 Proof of the covolutio sum relatio by represetig x() as sum of delta(-k), ad use LTI property!

29 Properties of Covolutio Covolutio with a Impulse Commutative Property Associative Property

30 Covolutio with Impulse x[]*δ[]=x[] x[]*δ[-k]=x[-k] Proof

31 Commutative Property x[]*h[]=h[]*x[] Proof

32 Associative Property (x[]* y[])* z[]=x[]*(y[]*z[]) Proof

33 HARDWARE STRUCTURES x[] FILTER y[] y[ ] = M k= 0 b k x[ k] INTERNAL STRUCTURE of FILTER WHAT COMPONENTS ARE NEEDED? HOW DO WE HOOK THEM TOGETHER? SIGNAL FLOW GRAPH NOTATION 1/30/ , JH McClella & RW Schafer 33

34 1/30/ , JH McClella & RW Schafer 34 HARDWARE ATOMS Add, Multiply & Store = = M k k k x b y 0 ] [ ] [ ] [ ] [ x y β = 1] [ ] [ = x y ] [ ] [ ] [ 2 1 x x y + =

35 FIR STRUCTURE Direct Form SIGNAL FLOW GRAPH y[ ] = M k= 0 b k x[ k ] 1/30/ , JH McClella & RW Schafer 35

36 FILTER AS BUILDING BLOCKS x[] FILTER OUTPUT INPUT FILTER + + y[] FILTER BUILD UP COMPLICATED FILTERS FROM SIMPLE MODULES Ex: FILTER MODULE MIGHT BE 3-pt FIR Is the overall system still LTI? What is its impulse respose? 1/30/ , JH McClella & RW Schafer 36

37 CASCADE SYSTEMS Does the order of S 1 & S 2 matter? NO, LTI SYSTEMS ca be rearraged!!! WHAT ARE THE FILTER COEFFS? {b k } S 1 S 2 1/30/ , JH McClella & RW Schafer 37

38 proof 1/30/ , JH McClella & RW Schafer 38

39 CASCADE SYSTEMS x[] h 1 [] h 2 [] y[] Is the cascaded system LTI? What is the impulse respose of the overall system? 1/30/ , JH McClella & RW Schafer 39

40 CASCADE SYSTEMS h[] = h 1 []*h2[]! Proof o board

41 Does the order matter? S 1 S 2 S 2 S 1 1/30/ , JH McClella & RW Schafer 41

42 proof

43 Example Give impulse resposes of two systems, determie the overall impulse respose

44 Parallel Coectios h 1 [] x[] + y[] h 2 [] h[]=?

45 Parallel ad Cascade h 1 [] h 2 [] x[] + y[] h 3 [] h[]=?

46 Summary of This Lecture Properties of liear ad time ivariat systems Ay LTI system ca be characterized by its impulse respose h[], ad output is related to iput by covolutio sum: y[]= ]=x[]*h[] Properties of covolutio Computatio of covolutio revisited Slidig widow Sythetic polyomial multiplicatio Block diagram represetatio Hardware implemetatio of oe FIR Coectio of multiple FIR Kow how to compute overall impulse respose

47 READING ASSIGNMENTS This Lecture: Chapter 5, Sectios