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
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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[] 1/30/2008 2003, JH McClella & RW Schafer 3
1/30/2008 2003, 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 ] [ ] [ 3 0 + + = = = x x x x k x b y k k 1, 2,1} 3, = { b k Feedforward Differece Equatio
GENERAL FIR FILTER SLIDE a Legth-L WINDOW over x[] Whe h[] is ot symmetric, eeds to flip h() first! x[-m] x[] 1/30/2008 2003, JH McClella & RW Schafer 5
7-pt AVG EXAMPLE Iput : x[ ] = (1.02) + cos(2π /8 + π / 4) for 0 40 CAUSAL: Use Previous 1/30/2008 2003, JH McClella & RW Schafer 6 LONGER OUTPUT
Uit Impulse Sigal x[] has oly oe NON-ZERO VALUE δ [ ] = 1 0 = 0 0 UNIT-IMPULSE 1 1/30/2008 2003, JH McClella & RW Schafer 7
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 =0 1 4 1/30/2008 2003, JH McClella & RW Schafer 8 4 =4 =5 NON-ZERO Whe widow overlaps δ[] 3])
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 1/30/2008 2003, JH McClella & RW Schafer 9
1/30/2008 2003, 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 ] [ ] [
Covolutio Operatio Flip h[] SLIDE a Legth-L WINDOW over x[] y [ ] = M k = 0 h [ k ] x [ k ] CONVOLUTION x[-m] x[] 1/30/2008 2003, JH McClella & RW Schafer 11
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 1/30/2008 2003, JH McClella & RW Schafer 12
DCONVDEMO: MATLAB GUI 1/30/2008 2003, JH McClella & RW Schafer 13
Go through the demo program for differet types of sigals Do a example by had Rectagular * rectagular Step fuctio * rectagular 1/30/2008 2003, JH McClella & RW Schafer 14
CONVOLUTION via Sythetic Polyomial Multiplicatio 1/30/2008 2003, JH McClella & RW Schafer 15
Covolutio via Sythetic Polyomial Multiplicatio More example 1/30/2008 2003, JH McClella & RW Schafer 16
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 1/30/2008 2003, JH McClella & RW Schafer 17
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 [ ] 1/30/2008 2003, JH McClella & RW Schafer 18
SYSTEM PROPERTIES x[] SYSTEM y[] MATHEMATICAL DESCRIPTION TIME-INVARIANCE INVARIANCE LINEARITY CAUSALITY No output prior to iput 1/30/2008 2003, JH McClella & RW Schafer 19
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 1/30/2008 2003, JH McClella & RW Schafer 20
TESTING Time-Ivariace 1/30/2008 2003, JH McClella & RW Schafer 21
Examples of systems that are time ivariat ad o-time ivariat 1/30/2008 2003, JH McClella & RW Schafer 22
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 1/30/2008 2003, JH McClella & RW Schafer 23
TESTING LINEARITY 1/30/2008 2003, JH McClella & RW Schafer 24
Examples systems that are liear ad o-liear 1/30/2008 2003, JH McClella & RW Schafer 25
LTI SYSTEMS LTI: Liear & Time-Ivariat Ay FIR system is LTI Proof! 1/30/2008 2003, JH McClella & RW Schafer 26
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 1/30/2008 2003, JH McClella & RW Schafer 27
Proof of the covolutio sum relatio by represetig x() as sum of delta(-k), ad use LTI property!
Properties of Covolutio Covolutio with a Impulse Commutative Property Associative Property
Covolutio with Impulse x[]*δ[]=x[] x[]*δ[-k]=x[-k] Proof
Commutative Property x[]*h[]=h[]*x[] Proof
Associative Property (x[]* y[])* z[]=x[]*(y[]*z[]) Proof
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 1/30/2008 2003, JH McClella & RW Schafer 33
1/30/2008 2003, 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 + =
FIR STRUCTURE Direct Form SIGNAL FLOW GRAPH y[ ] = M k= 0 b k x[ k ] 1/30/2008 2003, JH McClella & RW Schafer 35
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? 1/30/2008 2003, JH McClella & RW Schafer 36
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 1/30/2008 2003, JH McClella & RW Schafer 37
proof 1/30/2008 2003, JH McClella & RW Schafer 38
CASCADE SYSTEMS x[] h 1 [] h 2 [] y[] Is the cascaded system LTI? What is the impulse respose of the overall system? 1/30/2008 2003, JH McClella & RW Schafer 39
CASCADE SYSTEMS h[] = h 1 []*h2[]! Proof o board
Does the order matter? S 1 S 2 S 2 S 1 1/30/2008 2003, JH McClella & RW Schafer 41
proof
Example Give impulse resposes of two systems, determie the overall impulse respose
Parallel Coectios h 1 [] x[] + y[] h 2 [] h[]=?
Parallel ad Cascade h 1 [] h 2 [] x[] + y[] h 3 [] h[]=?
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
READING ASSIGNMENTS This Lecture: Chapter 5, Sectios 5-5 --- 5-9