Introduction to Signals and Systems Lecture #9 - Frequency Response. Guillaume Drion Academic year

Transcription

1 Introduction to Signals and Systems Lecture #9 - Frequency Response Guillaume Drion Academic year

2 Transmission of complex exponentials through LTI systems Continuous case: LTI system where is the transfer function of the LTI system. 2

3 Transmission of complex exponentials through LTI systems Discrete case: LTI system where is the transfer function of the LTI system. 3

4 Transfer function of LTI systems The transfer function of LTI systems has a specific form: it is rational. The roots of The roots of are called the zeros of the transfer function. are called the poles of the transfer function. 4

5 Response of LTI systems: non-zero-state Can we evaluate the response of a LTI system by simply looking at its transfer function? Example: consider the continuous-time LTI system described by the ODE The transfer function writes We want to compute the response of this system in non-zero-state to a step of amplitude a that starts at t=0 (step response). 5

6 Response of LTI systems: non-zero-state The problem writes and the response can be derived using the unilateral Laplace transform: which gives 6

7 Response of LTI systems: non-zero-state The problem writes and the response can be derived using the unilateral Laplace transform: which gives zero-input response zero-state response 7

8 Response of LTI systems: modes zero-input response zero-state response The zero-input (i.e. autonomous) response of a LTI system is composed of (complex) exponentials determined by the poles of the transfer function. The poles of the transfer function define the modes of the systems response (i.e. natural response). If the transfer function possesses a positive real pole, the modes contain a growing exponential! Stability of the system? 8

9 Bounded input bounded output (BIBO) stability A system is BIBO stable of all input-output pairs where is often referred as the gain of the system. satisfy In practice, it means that in a stable system, a bounded input will always give a bounded output. Stability is critical in engineering! 9

10 How do we characterize BIBO stability? A LTI system is stable if the poles of the transfer function all have negative real parts, i.e. the imaginary axis is included in its ROC. 10

11 How do we characterize BIBO stability? In general, stability is ensured if continuous:. discrete:. In other words: continuous: the imaginary axis is included in the ROC of the transfer function. discrete: the unit circle is included in the ROC of the transfer function. Note that stability conditions imply that the Fourier transform exists! 11

12 Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 12

13 How do we characterize the response of a LTI systems to Continuous case: an oscillatory signal at a specific frequency? Using the polar representation, we have 13

14 How do we characterize the response of a LTI systems to an oscillatory signal at a specific frequency? Using the polar representation, we have Change in amplitude Change in phase When an oscillatory signal goes through a LTI systems, his amplitude (amplification/attenuation) and phase (advance, delay) are affected. Not his frequency! 14

15 Frequency response of LTI systems The frequency response of a LTI system can be fully characterize by, and in particular: : GAIN (change in amplitude) : PHASE (change in phase) A change in phase in the frequency domain corresponds to a time delay in the time domain: which gives The slope of the phase curve corresponds to a delay in the time domain. 15

16 Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 16

17 Frequency response of LTI systems The frequency response of a LTI system can be fully characterize by, and in particular: : GAIN (change in amplitude) : PHASE (change in phase) A plot of and for all frequencies gives all the informations about the frequency response of a LTI system: the BODE plots. In practice, we use a logarithmic scale for such that becomes 17

18 The Bode plots The Bode plots graphically represent the frequency response of a LTI system. They are composed of two plots: The amplitude plot (in db):. The phase plot:. For discrete time systems, we use a linear scale for the frequencies, ranging from to. 18

19 The Bode plots Examples of Bode plots of continuous (left) and discrete (right) LTI systems. 19

20 Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 20

21 Bandwidth The bandwidth of a system is the range of frequencies that transmits faithfully through the system. We can define the bandwidth of a system the same way we defined its timeconstant: 21

22 Bandwidth We define the bandwidth of a system the same way we defined its time-constant: There is a tradeoff between the bandwidth of a system and its time-constant. Indeed, let s consider, which gives If we now consider and, we have 22

23 Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 23

24 Time and frequency responses of 1st order systems We consider the general 1st order system of the form The transfer function of the system is given by The frequency response ( ), impulse response ( ) and step response ( ) writes H(j!) = 1 j! +1, h(t) = 1 e t/ I(t), s(t) =(1 e t/ )I(t) 24

25 Time and frequency responses of 1st order systems 25

26 Bode plots of 1st order systems: amplitude Amplitude plot: If : If : Low frequencies: constant frequency response ( ) High frequencies: frequency response linear decays by -20dB/dec. Cutoff frequency:. 26

27 Bode plots of 1st order systems: amplitude Amplitude plot: first order systems are low-pass filters! (but the slope at HF might be too low to achieve good filtering properties...). 27

28 Bode plots of 1st order systems: phase Phase plot: Low frequencies: no phase shift. Mid frequencies: phase response decays linearly (slope = time-delay = ). High frequencies: phase-delay of. 28

29 Bode plots of 1st order systems: amplitude and phase 29

30 Time and frequency responses of 2nd order systems We consider the general 2nd order system of the form = natural frequency = damping factor The transfer function of the system is given by The frequency response writes 30

31 Time and frequency responses of 2nd order systems The transfer function of a 2nd order system is given by The transfer function has two poles: Case 1 ( ) : two real poles cascade of two first order systems. Case 2 ( ) : two complex conjugates poles. New behaviors (oscillations, overshoot, etc.) 31

32 Time and frequency responses of 2nd order systems 32

33 Bode plots of 2nd order systems Amplitude plot: Phase plot: 33

34 Bode plots of 2nd order systems Amplitude plot: second order systems are low-pass filters! (higher slope, possible resonant frequency with overshoot in the frequency response). 34

35 Bode plots of 2nd order systems: resonant frequency If, there is an overshoot in the frequency response at a resonant frequency. The amplitude of the peak is given by For, there is no peak in the frequency response. 35

36 Time and frequency responses of 2nd order systems 36

37 Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 37

38 Frequency response of LTI systems Bode plots of first and second order systems are building blocks for the construction of Bode plots of any LTI systems. Indeed, the transfer function of LTI systems is rational, and the denominator terms can all be expressed as or In other terms, the Bode plots of LTI systems can be sketched from the poles and zeros of the transfer function! 38

39 Frequency response of LTI systems: poles and zeros The Bode plots of LTI systems can be sketched from the poles and zeros of the transfer function! Each real pole induce a first order system response where. Each pair of complex conjugate poles system response where induce a second order Zeros induce the opposite behavior. 39

40 Frequency response of LTI systems: poles and zeros 40

41 Amplitude: Frequency response of LTI systems: Bode plots any real pole induces a decrease in the slope of -20dB/dec. any real zero induces an increase in the slope of 20dB/dec. any pair of complex conjugate poles induces a decrease in the slope of -40dB/dec. Phase: any real pole induces a decrease in the phase of. any real zero induces an increase in the phase of. any pair of complex conjugate poles induces a decrease in the phase of. 41

42 Frequency response of LTI systems: Bode plots Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz. 42

43 Frequency response of LTI systems: Bode plots Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz. 43

44 Frequency response of LTI systems: poles and zeros 44