Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems Guillaume Drion Academic year

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1 Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems Guillaume Drion Academic year

2 Outline Systems modeling: input/output approach of LTI systems. Convolution in discrete-time. Convolution in continuous-time: the Dirac delta function. Causality, memory, responsiveness of LTI systems. 2

3 Outline Systems modeling: input/output approach of LTI systems. Convolution in discrete-time. Convolution in continuous-time: the Dirac delta function. Causality, memory, responsiveness of LTI systems. 3

4 Systems modeling Modeling and analysis of systems: open loop. Observing and analyzing the environment Input SYSTEM Output Can be used to understand/analyze the behavior of a dynamical system. A good model can predict the future evolution of a system. How can we use systems modeling to predict the future? What is a good model or a good system to model? 4

Systems modeling: state-space representation Last lecture, we saw that the state-space representation of a model can describe its behavior. V Example: RLC circuit.

5 Systems modeling: state-space representation Last lecture, we saw that the state-space representation of a model can describe its behavior. V Example: RLC circuit. i R v R (t) C v C (t) v L (t) Such representation can be used to predict the future behavior of the system when subjected to a specific input, providing that we know its current state. L 5

6 Systems modeling: state-space representation Example: the Windkessel model for variations in blood pressure Model Simulation u(t) r P(t) Pr(t) PCa(t) Ca R 6

7 Response of N-dimensional continuous systems Systems with more than one variable: ẋ = Ax! x(t) =? A is a square matrix of parameters. The response will be a sum of exponentials whose coefficients are the eigenvalues of the matrix A. Eigenvalues can be real or complex conjugate pairs. In general: x i (t) =x i,0 e i = x i,0 e ( i+j! i )t where j is the imaginary unit. 7

8 The complex exponential 8

9 Condition for stability of continuous linear systems STABLE UNSTABLE 9

10 Response of N-dimensional discrete systems Systems with more than one variable: x[n + 1] = Ax[n]! x[n] =? A is a square matrix of parameters. The response will be a sum of exponentials whose coefficients are the eigenvalues of the matrix A. Eigenvalues can be real or complex conjugate pairs. In general: x i [n] =x i,0 n i = x i,0 n i e j! in where j is the imaginary unit. 10

11 The discrete time complex exponential u[n] =z n,z = e j! < 1 =1 > 1 11

12 Condition for stability of continuous linear systems u[n] =z n,z = e j! < 1 =1 > 1 STABLE UNSTABLE 12

13 Systems modeling: state-space representation But what if the system is like this? How many states/equations would you need? 13

14 or like this? Systems modeling: state-space representation 14

15 or like this? Systems modeling: state-space representation 15

16 or like this? Systems modeling: state-space representation Sometimes, you want to know how the system reacts to inputs, but you do not care about all the details of the internal dynamics. Input-output representation! 16

17 Input-output representation in time domain Any system S can be represented using an input-output representation: u S y Can we mathematically describe the system using the following relationship? What kind of system can be described this way? 17

In particular, can we find a function that will predict the output of

18 Input-output representation in time domain Can we mathematically describe a time-invariant system using the following relationship? In particular, can we find a function that will predict the output of the system for any input, whatever the complexity of the input? 20 mv 400 ms 18

19 Input-output representation in time domain Can we mathematically describe a time-invariant system using the following relationship? It is possible if the system obeys the superposition principle: if and then Superposition principle = additivity + homogeneity. If a system obeys the superposition principle, we can express the (possibly complex) input signal as a sum of simple input signals! 19

20 Linear, Time-Invariant (LTI) systems Can we mathematically describe a time-invariant system using the following relationship? It is possible if the system obeys the superposition principle: if and then The superposition principle is valid for linear systems. For all these reasons, this course will focus on Linear, Time-Invariant (LTI) systems. 20

21 Does it make sense to study linear systems? Is a physical/biological system linear? Linearity implies homogeneity:. All physical/biological systems saturate none of them are totally linear. Examples: I/V curve of a diode (left) and force/travel curve of a suspension (right) 21

22 Does it make sense to study linear systems? Is a physical/biological system linear? However, many systems are almost linear in their functional range, and/or can be decomposed in linear subsystems! Examples: I/V curve of a diode (left) and force/travel curve of a suspension (right) High g High g Low g 22

23 Input-output representation of LTI systems Can we mathematically describe a LTI system using the following relationship? 23

24 Input-output representation of LTI systems Can we mathematically describe a LTI system using the following relationship? Using the superposition principle, we can analyze the input/output properties by expressing the input signal into the sum of simple signals: if then 24

25 Input-output representation of LTI systems Using the superposition principle, we can analyze the input/output properties by expressing the input signal into the sum of simple signals: if then What is the simplest signal: a pulse! The response of a system to a pulse is called the impulse response. Therefore, any LTI system can be fully characterized by its impulse response. 25

26 Outline Systems modeling: input/output approach and LTI systems. Convolution in discrete-time. Convolution in continuous-time: the Dirac delta function. Causality, memory, responsiveness of LTI systems. 26

27 Definition of a pulse in discrete-time The pulse in discrete time is defined by 1 δ[n] n 27

28 Role of a pulse in discrete-time The pulse in discrete time is defined by If we multiply any signal by, we retrieve a signal that only contains the value of the input signal at :. u[0] u[n]δ[n] n 28

29 Role of a pulse in discrete-time The pulse in discrete time is defined by Similarly, if we want to retrieve a signal that only contains the value of the input signal at any value, we need to multiply the signal by : u[n]δ[n-2] u[2] n

30 Role of a pulse in discrete-time If we retrieve signals that only contain the value of the input signal at for all and sum them, we retrieve the initial signal: A signal can therefore be decomposed into an infinite sum of unit impulse signals. Example: decomposition of the unit step function: 30

31 Role of a pulse in discrete-time A signal can therefore be decomposed into an infinite sum of unit impulse signals.... u[-2]δ[n+2] n u[n] u[-1]δ[n+1] n = n u[0]δ[n+0]!!!!!! n 31...

32 Impulse response of discrete systems Can we use this decomposition to analyze the input/output properties of a discrete LTI system? Yes, we can use the superposition principle which gives where is the impulse response of the system. 32

33 Impulse response of discrete systems Can we use this decomposition to analyze the input/output properties of a discrete LTI system? Yes, we can use the superposition principle which gives where is the impulse response of the system. and writes. is called convolution of the signals and 33

34 Impulse response of discrete systems A LTI system is fully characterized by its impulse response. What does it mean? It means that the response of a LTI system at an instant depends on all the past, present and future values of the input, each of them having a gain equal to. If the impulse response has a finite window, the size of this windows defines the memory of the system. Examples of convolutions. 34

35 Cascade of systems Show that the impulse response of a cascade of LTI systems is the equal to the convolution of the impulse response of each subsystem. u h 1 [n] x h 2 [n] y 35

36 Outline Systems modeling: input/output approach and LTI systems. Convolution in discrete-time. Convolution in continuous-time: the Dirac delta function. Causality, memory, responsiveness of LTI systems. 36

37 Definition of a pulse in continuous-time How can we define a pulse in continuous-time? Similarly to the discrete case, we have to define a signal for all signal continuous at the origin, and such that (step function) is the derivative of the step function, which is discontinuous at the origin! 37

38 Definition of a pulse in continuous-time: the Dirac delta function The Dirac delta function can be defined as a square of width and height with (defined by its integral equal to 1). ε 0 δ(t) -ε/2 ε/2 0 1/ε t 38

39 Definition of a pulse in continuous-time: the Dirac delta function The Dirac delta function can be defined as a square of width and height with (defined by its integral equal to 1). ε 0 δ(t) -ε/2 ε/2 0 1/ε t 39

40 Convolution in continuous-time Any continuous signal can be expressed as a sum (integral) of delta functions: Therefore, the output of a continuous LTI system can be expressed as where is the impulse response of the LTI system. In continuous time, the convolution is 40

41 Convolution in continuous-time 41

42 Properties of convolution Convolutions are commutative (show it): Convolutions are associative: Convolutions are distributive (show it): 42

43 Outline Systems modeling: input/output approach and LTI systems. Convolution in discrete-time. Convolution in continuous-time: the Dirac delta function. Causality, memory, responsiveness of LTI systems. 43

44 Properties of LTI systems We can extract informations about a LTI system using the shape of the impulse response. Causality: the output only depends on past values of the input. It means that only depends on if. In terms of the impulse response, it means that Indeed, if, causality implies that 44

45 Memory of LTI systems The memory of a system is defined by the window of its impulse response. The larger the impulse response window, the bigger the memory. h[n] n Output depends on the present and the previous input values h[n] n Output depends on the present and the 4 previous input values 45

46 Memory of LTI systems Static systems: depends on only:. This gives where is the static gain of the system. Input Output K 46

47 Memory of LTI systems Static systems: depends on only:. This gives where is the static gain of the system. Input Output K Dynamical systems: the response of the system is limited by the window of its impulse response! (reaches steady-state after some time). Input Output K 47

48 Response time of LTI systems The response time of a LTI dynamical system is linked to the time-window of its impulse response. Indeed, if is the length of the impulse response and the length of the input signal, the output signal will have a length of. (can be easily shown graphically). The response-time of a system is defined by its time-constant. 48

49 Time-constant of LTI systems The general form of a the impulse response of a LTI system is a decaying exponential infinite window. We usually define the time-constant of a system as. 49

50 Time-constant of LTI systems If the impulse response is the exponential decay: Then and a time- This is the typical response of a first order system. First order systems are characterized by a static gain constant. 50

51 Time-constant of LTI systems 51

52 Time-constant of LTI systems Example: high energy photon detector. 52

53 Time-constant of LTI systems Example: high energy photon detector. 53

54 Time-constant of LTI systems The time-constant is important for filtering: with = cutoff frequency. Example: the cardiovascular system modeled in lecture #1 (low-pass filter). Low pass filter R H(s) = RC a s +1 + r Atherosclerosis: loss of arterial compliance => C a decreases => τ=rc a decreases τ = RC a 54

55 Time-constant of LTI systems Other useful response: the step response. 55

56 Highlights of the day Input/output approach. Delta function in discrete-time. Linear, Time-Invariant systems. Convolution (+properties). Superposition principle. Causality, memory response-time. Impulse response and step response. Time-constant and cutoff frequency. Dirac delta function in continuous-time. 56

57 Input-output representation of LTI systems Can we mathematically describe a LTI system using the following relationship? We exploit the superposition principle (linear systems): Response to a pulse: impulse response time-domain. Is there any other time of signal we could use? 57

58 The complex exponential 58

59 The discrete time complex exponential u[n] =z n,z = e j! < 1 =1 > 1 59

60 Transmission of complex exponentials through LTI systems Continuous case: LTI system where is the transfer function of the LTI system. 60

61 Transmission of complex exponentials through LTI systems Discrete case: LTI system where is the transfer function of the LTI system. 61

62 Input-output representation of LTI systems Using the superposition principle, we can analyze the input/output properties by expressing the input signal into the sum of simple signals: if then If you put a complex exponential at a specific frequency into a LTI system, you get a complex exponential at the same frequency at the output. Can we use the complex exponential as the basic signal? 62

63 Input-output representation of LTI systems Using the superposition principle, we can analyze the input/output properties by expressing the input signal into the sum of simple signals: if then If you put a complex exponential at a specific frequency into a LTI system, you get a complex exponential at the same frequency at the output. Can we use the complex exponential as the basic signal? Yes if and only if a signal can be decomposed into a sum of complex exponentials! => Topic of next course. 63

Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems. Guillaume Drion Academic year

Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems Guillaume Drion Academic year 2015-2016 1 Outline Systems modeling: input/output approach and LTI systems. Convolution