06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1

Transcription

1 IV. Continuous-Time Signals & LTI Systems [p. 3] Analog signal definition [p. 4] Periodic signal [p. 5] One-sided signal [p. 6] Finite length signal [p. 7] Impulse function [p. 9] Sampling property [p.11] Impulse properties [p.17] Continuous time system building blocks [p. 18] Delay block [p. 20] Integrator block [p. 25] Differentiator block [p. 26] LTI system [p. 26] Definition [p. 27] Testing for linearity [p. 30] Testing for time invariance [p. 32] Evaluating convolution [p.37] Convolving unit steps [p.38] Convolving impulses [p. 39] Convolution properties [p. 44] Combining LTI systems [p. 47] System stability [p. 51] System causality [p. 53] Examples 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1

2 Processing of continuous time Signals x ( t ) ANALOG y ( t ) System Analog signal x(t) INFINITE LENGTH SINUSOIDS: (t = time in secs) PERIODIC SIGNALS ONE-SIDED, e.g., for t>0 UNIT STEP: u(t) FINITE LENGTH SQUARE PULSE 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 3

3 CT Signals: PERIODIC Signals of INFINITE DURATION x (t) = 10 cos( 200 π t) Sinusoidal signal Square Wave 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 4

4 CT Signals: ONE-SIDED ut () 1 t 0 = 0 t < 0 Unit step signal One-Sided Sinusoid v(t ) = e t u(t ) Suddenly applied Exponential 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 5

5 CT Signals: FINITE LENGTH p(t) =u(t 2) u(t 4) Square Pulse signal Sinusoid multiplied by a square pulse 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 6

6 What is an Impulse? A signal that is concentrated at one point. δ () t = lim () t δ Δ Δ 0 where δ Δ ( t ) dt = 1 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 7

7 Defining the Impulse δ(t) =0, t 0 One INTUITIVE definition is: Concentrated at t=0 with unit area δ ( τ ) d τ = 1 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 8

8 δ(t) Sampling Property f (t)δ Δ (t) f (0)δ Δ (t) f ( t )δ ( t ) = f (0)δ ( t ) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 9

9 General Sampling Property f (t)δ (t t 0 ) = f (t 0 )δ (t t 0 ) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 10

10 Properties of the Impulse δ (t t 0 ) = 0, t t 0 δ ( t t 0 )dt = 1 f (t)δ(t t 0 ) = f(t 0 )δ(t t 0 ) f ( t ) δ ( t t 0 ) dt = f ( t 0 ) 1 δ ( at) = δ ( t) a du ( t) dt = δ (t) Concentrated at one time Of Unit area Scaling Property Sampling Property Extract one value of f(t) sifting property Derivative of unit step 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 11

11 Example: xt e ut 2( t 1) () = ( 1) Compute and plot dx( t) / dt 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 12

12 Example: Compute: t 1 τ A= δτ ( + 3) e dτ d B= e u t dt C t+ 1 t { 3t ( 1) } = δτ ( + 3) dτ jt D= sin( t) e δ ( τ + 3) dτ 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 13

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15 Example: Assume x(t)=u(t)-u(t-5), sketch: 1) dx(t)/dt, 2) x(2-t) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 16

16 CT Building Blocks DELAY by t o INTEGRATOR (CIRCUITS) DIFFERENTIATOR MULTIPLIER & ADDER Others 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 17

17 Ideal Delay: System S x(t) Mathematical Definition: y(t) y(t) = x(t t d ) To find the IMPULSE RESPONSE of a system, h(t), let x(t) be an impulse, so x(t) y(t) System S x(t)=δ(t) y(t)=h(t) ht () =? 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 18

18 Output of Ideal Delay of 1 sec y(t) = x(t 1) = e (t 1) u(t 1) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 19

19 Integrator: x(t) System S y(t) Mathematical Definition: t y ( t ) = x (τ )d τ Running Integral To find the IMPULSE RESPONSE, h(t), let x(t) be an impulse, so t h(t) = δ (τ )dτ = u(t) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 20

20 Integrator: y ( t ) = x (τ ) d τ Integrate the impulse t δ (τ )d τ = u (t ) t IF t<0, we get zero IF t>0, we get one Thus we have h(t) = u(t) for the integrator 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 21

21 Graphical Representation δ (t ) = du (t) dt t 1 t 0 ut () = δτ ( ) dτ= 0 t < 0 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 22

22 Output of Integrator x(t) System S y(t) t y ( t) = x( τ ) dτ y(t)= 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 23

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24 Differentiator: Differentiator Output: Example: x(t) y ( t ) = System S dx (t ) dt y(t) y(t)= 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 25

25 Linear and Time-Invariant (LTI) Systems If a continuous-time system is both linear and timeinvariant, then the output y(t) is related to the input x(t) by a convolution integral where h(t) is the impulse response of the system. x(t) x(t)=δ(t) System S y(t) h(t)=y(t) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 26

26 Testing for Linearity Identical process to discrete signal case x(t) System S y(t) S x () t + x () t = { α β } 1 2 αs x () t + S x () t Examples: y(t)=2x(t) y(t)=2x(t)+1 y(t)=x(t 2 ) y(t)=x(t-d) { } β { } 1 2 t y () t = x ( τ ) dτ 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 27

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29 Testing Time-Invariance A system is time-invariant if a shift in the input produces the same shift in the output x(t) System S y(t) Examples: y(t)=2x(t) y(t)=x(2t) y(t)=x(t-d) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 30

30 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 31

31 Evaluating a Convolution x( t) = u( t 1) t h( t) = e u( t) y( t) = h( τ ) x( t τ ) dτ = h( t) x( t) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 32

32 Solution y(t)= 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 33

33 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 34

34 Example at bt x() t e = u(), t h() t = e u() t y(t)= 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 35

35 Special Case: h(t)=u(t) at x() t = e u(), t a 0, h() t = u() t y(t)= 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 36

36 Convolving Unit Steps x() t = u(), t h() t = u() t y(t)= 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 37

37 Convolution Properties 1) Commutative 2) Associative 3) Distributive 4) Derivative of convolution d dt d x() t y() t = x() t y() t dt d = x() t () dt [ ] [ ] [ y t ] 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 39

38 Convolution Properties 5) Time invariance yt () = xt () ht () xt ( t)* ht ( ) = yt ( t) /12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 40

39 Proofs: 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 41

40 Examples: Compute t A = δ( t 1)*( δ( t+ 2) + 2 e ) d B= sin(5 t) u( t 1/ 2) dt ( ) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 42

41 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 43

42 Cascade of LTI Systems h(t) = h 1 (t) h 2 (t) = h 2 (t) h 1 (t) 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 44

43 Stability A system is stable if every bounded input produces a bounded output. A continuous-time LTI system is stable if and only if h ( t ) dt < 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 47

44 Stability Examples: yt () = 3 xt (), yt () = x( τ ) dτ dx() t yt () =, yt () = txt () dt xt () yt () = e t 0 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 48

45 Note: 1) Finding a counter-example is OK. 2) You can t prove a property with an example 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 49

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47 Causal Systems A system is causal if and only if y(t 0 ) depends only on x(τ) for τ< t 0. An LTI system is causal if and only if h( t) = 0 for t < 0 Example: Are the LTI systems with the following imput/output relationships or impulse responses causal? y () t = x( t) 1 y t x t 2 2 () = ( ) h () t = u( t + 3) 2 u( t 3) 3 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 51

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50 Example: LTI system h 1 (t)=δ(t+2) LTI system h 2 (t)=δ(t-2) + - LTI system h 3 (t)=u(t Τ) 1) Compute the impulse response to the overall system and plot it 2) How do you need to pick T so that the overall system is causal 3) Are systems 1, 2, 3 causal, is the overall system causal? 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 55

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52 Example: 06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 57

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