Topic 2: LTI Systems and Convolution Response of LTI Systems - PowerPoint PPT Presentation

ELEC361: Signals And Systems Topic 2: LTI Systems and Convolution � Response of LTI Systems � Impulse response and unit response � Characterization of LTI systems � Properties using the system response � Convolution Integral � Convolution Integral Properties � Convolution Sum � Convolution Sum Properties � Systems and Difference/Differential Equations � Block Diagrams of Systems Dr. Aishy Amer � Summary Concordia University Electrical and Computer Engineering Figures and examples in these course slides are taken from the following sources: • A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 • M.J. Roberts, Signals and Systems, McGraw Hill, 2004 • J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

Response of LTI Systems � Linearity ∑ = if x [ n ] a x [ n ] k k k ∑ = then y [ n ] a y [ n ] k k k � Time Invariance → ⇒ − → − x [ n ] y [ n ] x [ n n ] y [ n n ] 0 0 2

Response of LTI Systems � 1 = L If x ( t ) Weighted sum of several signals x ( t ), x ( t ), 1 2 ⇒ = y ( t ) The weighted sum of the responses of the system to each of those indivual signals + ⇒ + ax ( t ) bx ( t ) ay ( t ) by ( t ) � Linear Combination 1 2 1 2 � If we represent x(t) to LTI system as a linear combination of a set of BASIC signals � we can use the superposition to compute y(t) as the response to those basic signals � BASIC signals: δ jn L u [ n ], [ n ], e , cos[ n ], sin[ n ], 3

Response of LTI Systems � Once the response to a unit impulse is known, the response of any discrete-time LTI system to any arbitrary excitation can be found � Any arbitrary excitation is a sequence of amplitude- scaled and time-shifted DT impulses = + − δ + + δ + δ − + L L [ ] [ 1 ] [ 1 ] [ 0 ] [ ] [ 1 ] [ 1 ] x n x n x n x n ∞ ∑ = δ − [ ] [ ] x k n k � Therefore the response is a sequence of amplitude- − ∞ scaled and time-shifted DT impulse responses � This we call convolution The impulse response is conventionally designated by � the symbol, h[ n ] or h(t) 4

Response of LTI Systems: Example 5

Response of LTI Systems: Example 6

Response of LTI Systems = � 1 S { x [ n ]} y [ n ] � 2 = δ If x [ n ] [ n ] ⇒ = = δ = y [ n ] S { x [ n ]} S { [ n ]} h [ n ] h [ n ] : Impulse response of S � Question: if h[n] known, how to find y[n]? 7

Response of LTI System: Identity System = δ if h [ n ] K [ n ] = with K h [ 0 ] a constant = [ ] [ ] y n Kx n = = ⇒ if K 1 then y[n] x[n] Identity System • Output equal to the input & • Unit impulse response is the unit impulse = ∗ δ y [ n ] x [ n ] [ n ] ∑ = δ − = y [ n ] x [ k ] [ n k ] x [ n ] 1 4 4 4 2 4 4 4 3 k Shifting Property 8 = Preserves ONLY the x[k] at k n

Response of LTI System: Example If the system LTI = ∑ − [ ] [ ] [ ] y n h k x n k = − + − [ 0 ] [ 0 ] [ 1 ] [ 1 ] h x n h x n = + − [ ] [ 1 ] x n x n Can be the response of a non- linear system but it is not possible to determine y[n] used 9 on h[n]

Response of LTI System: Example 10

Response of LTI System: Example = ∗ δ y [ n ] x [ n ] [ n ] ∑ = δ − = x [ k ] [ n k ] x [ n ] 1 4 4 4 2 4 4 4 3 k Preserves only the x[n] at (shifts) = k n 11

Outline Response of LTI systems � Impulse response and unit response � Characterization of LTI systems � Properties using the system response � Convolution Integral � Convolution Integral Properties � Convolution Sum � Convolution Sum Properties � Systems and Difference/Differential Equations � Block Diagrams of Systems � Summary � 12

Relationship: DT impulse response and step response � In any discrete-time LTI system let an excitation, x[ n ], produce the response, y[ n ] � Then the excitation x[ n ] - x[ n - 1] will produce the response y[ n ] - y[ n - 1] � It follows then that the unit impulse response is the first backward difference of the unit step response and, conversely that the unit sequence (step) response is the accumulation of the unit impulse response = − − h [ n ] s [ n ] s [ n 1 ] where s [ n ] is the step response n ∑ = s [ n ] h [ k ] = −∞ k 13

DT impulse response and step response:Example � Suppose that the step response is given by ⎡ ⎤ n ⎛ ⎞ 4 = − ⎜ ⎟ ⎢ ⎥ y [ n ] 5 4 u [ n ] ⎝ ⎠ ⎢ ⎥ 5 ⎣ ⎦ � What is the impulse response h[n] ? = − − h [ n ] y [ n ] y [ n 1 ] ⎡ ⎤ ⎡ − ⎤ n n 1 ⎛ ⎞ ⎛ ⎞ 4 4 = − − − − ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ 5 4 u [ n ] 5 4 u [ n 1 ] ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎢ ⎥ 5 5 ⎣ ⎦ ⎣ ⎦ δ [ n ] 6 4 4 4 7 4 4 4 8 n n ⎛ ⎞ ⎛ ⎞ 4 4 = δ − δ + − ⎜ ⎟ ⎜ ⎟ 5 [ n ] 4 [ n ] 4 u [ n 1 ] ⎝ ⎠ ⎝ ⎠ 5 5 14

Relationship: CT impulse response and step response � In any continuous-time LTI system let an excitation, x( t ) , produce the response, y( t ). Then the excitation d ( x ( t )) dt will produce the response d ( y ( t )) dt � It follows then that the unit impulse response is the first derivative of the unit step response and, conversely that the unit step response is the integral of the unit impulse response 15

Outline Response of LTI systems � Impulse response and unit response � Characterization of LTI systems � Properties using the system response � Convolution Integral � Convolution Integral Properties � Convolution Sum � Convolution Sum Properties � Systems and Difference/Differential Equations � Block Diagrams of Systems � Summary � 16

Characterization of LTI System � The impulse response h[n] completely characterizes an LTI system DT LTI Systems: � Use the unit impulse to construct any signal � A DT signal is a sequence of individual weighted impulses � The response of the system is the sum of delayed h[n] 17

Characterization of LTI System � DT LTI systems are described mathematically by difference equations � CT LTI systems are described mathematically by differential equations 18

Characterization of LTI System For any excitation, x[ n ] or x(t) � the response, y[ n ] or y(t) can be found by � finding the response to x[ n ] or x(t) as the only forcing 1. function on the right-hand side and then adding scaled and time-shifted versions of that 2. response to form y[ n ] or y(t) If x[ n ] or x(t) is a unit impulse, � the response to it as the only forcing function is � simply the homogeneous solution of the difference or differential equation with initial conditions applied 19

Outline Response of LTI systems � Impulse response and unit response � Characterization of LTI systems � Properties using the system response � Convolution Integral � Convolution Integral Properties � Convolution Sum � Convolution Sum Properties � Systems and Difference/Differential Equations � Block Diagrams of Systems � Summary � 20

Properties using the system response: system Stability � It can be shown that a BIBO-stable DT system has an impulse response that is absolutely summable ∞ ∑ < ∞ h [ n ] = −∞ � Proof n ∞ ∑ = − y [ n ] x [ n k ] h [ k ] = −∞ k ∞ ∑ ≤ − [ ] [ ] x n k h k = −∞ k ∞ ∑ ≤ B h [ n ] = −∞ n 21

Properties using the system response: system Stability � A CT system is BIBO stable if its impulse response is absolutely integrable ∞ ∫ < ∞ h ( t ) − ∞ 22

Properties using the system response : Invertible Systems = ∑ n � 1 The accumulato r : y [ n ] x [ k ] = −∞ k � 1 = − − The backward difference : w [ n ] y [ n ] y [ n 1 ] are inverse systems 23

Properties using the system response: Differentiators & Integrators � Differentiators are: � Difficult to implement � Sensitive to noise and errors � Alternatives : Integrators dy ( t ) = − x ( t ) 2 y ( t ) dt −∞ = assume y ( ) 0 t t dy ( t ) ∫ ∫ ⇒ = τ − τ τ dt y ( t ) y ( t ) x ( ) 2 y ( ) d dt − ∞ − ∞ t ∫ = τ − τ τ y ( t ) x ( ) 2 y ( ) d − ∞ t ∫ = + τ − τ τ y ( t ) y ( t ) x ( ) 2 y ( ) d (to finite) 0 � Integrators : amplifiers − ∞ 24

25

Outline Response of LTI systems � Impulse response and unit response � Characterization of LTI systems � Properties using the system response � Convolution Integral � Convolution Integral Properties � Convolution Sum � Convolution Sum Properties � Systems and Difference/Differential Equations � Block Diagrams of Systems � Summary � 26