Introduction to Signals and Systems Lecture #9 - Frequency Response

Guillaume Drion Academic year 2019-2020

1

Transmission of complex exponentials through LTI systems

2

where is the transfer function of the LTI system.

LTI system

Continuous case:

Transfer function of LTI systems

3

The transfer function of LTI systems has a specific form: it is rational.

The roots of are called the poles of the transfer function.

The roots of are called the zeros of the transfer function.

Response of LTI systems: zero-state

4

Can we evaluate the response of a LTI system by simply looking at its transfer function?

Example: consider the continous-time LTI system described by the ODEThe transfer function writes

We want to compute the response of this system in zero-state to a step of amplitude a that starts at t=0 (step response).

Response of LTI systems: zero-state

5

The problem writes and the response can be derived using the transfer function:

Y (s) = H(s)U(s) =

✓1

s2 + 3s+ 2

◆a

s, <(s) > 0

The step response in the time domain can be easily obtained using a partial fraction decomposition of the transfer function:

Response of LTI systems: zero-state

6

The step response in the time domain can be easily obtained using a partial fraction decomposition of the transfer function:

Using the Laplace transform of an exponential, it yields

The response of a LTI system is a combination of exponentials (possibly complex)!

Response of LTI systems: modes

7

zero-state responsezero-input 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?

Bounded input bounded output (BIBO) stability

8

A system is BIBO stable of all input-output pairs satisfy where is often referred as the gain of the system.

In practice, it means that in a stable system, a bounded input will always give a bounded output.

Stability is critical in engineering!

How do we characterize BIBO stability?

9

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.

How do we characterize BIBO stability?

10

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!

Outline

Frequency response of LTI systems

Bode plots

Bandwidth and time-constant

1st order and 2nd order systems (continuous)

Rational transfer functions

11

How do we characterize the response of a LTI systems to an oscillatory signal at a specific frequency?

12

Continuous case:

Using the polar representation , we have

How do we characterize the response of a LTI systems to an oscillatory signal at a specific frequency?

13

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!

Frequency response of LTI systems

14

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.

Outline

Frequency response of LTI systems

Bode plots

Bandwidth and time-constant

1st order and 2nd order systems (continuous)

Rational transfer functions

15

Frequency response of LTI systems

16

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

The Bode plots

17

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 .

The Bode plots

18

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

Outline

Frequency response of LTI systems

Bode plots

Bandwidth and time-constant

1st order and 2nd order systems (continuous)

Rational transfer functions

19

Bandwidth

20

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 time-constant:

Bandwidth

21

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

Outline

Frequency response of LTI systems

Bode plots

Bandwidth and time-constant

1st order and 2nd order systems (continuous)

Rational transfer functions

22

Time and frequency responses of 1st order systems

23

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)

Time and frequency responses of 1st order systems

24

Bode plots of 1st order systems: amplitude

25

Amplitude plot:

If :

If :

Low frequencies: constant frequency response ( )

High frequencies: frequency response linear decays by -20dB/dec.

Cutoff frequency: .

Bode plots of 1st order systems: amplitude

26

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

Bode plots of 1st order systems: phase

27

Phase plot:

Low frequencies: no phase shift.

Mid frequencies: phase response decays linearly (slope = time-delay = ).

High frequencies: phase-delay of .

Bode plots of 1st order systems: amplitude and phase

28

Time and frequency responses of 2nd order systems

29

We consider the general 2nd order system of the form

The transfer function of the system is given by

The frequency response writes

= natural frequency = damping factor

Time and frequency responses of 2nd order systems

30

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.)

Time and frequency responses of 2nd order systems

31

Bode plots of 2nd order systems

32

Amplitude plot:

Phase plot:

Bode plots of 2nd order systems

33

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

Bode plots of 2nd order systems: resonant frequency

34

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.

Time and frequency responses of 2nd order systems

35

Outline

Frequency response of LTI systems

Bode plots

Bandwidth and time-constant

1st order and 2nd order systems (continuous)

Rational transfer functions

36

Frequency response of LTI systems

37

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!

Frequency response of LTI systems: poles and zeros

38

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 induce a second order system response where

Zeros induce the opposite behavior.

Frequency response of LTI systems: poles and zeros

39

Frequency response of LTI systems: Bode plots

40

Amplitude:

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 .

Frequency response of LTI systems: Bode plots

41

Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz.

Frequency response of LTI systems: Bode plots

42

Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz.

Frequency response of LTI systems: poles and zeros

43