introduction to signals and systems lecture #8 - …guilldrion/files/syst0002-2019...the region of...

Introduction to Signals and Systems Lecture #8 - Frequency-Domain Representation of LTI Systems

Guillaume Drion Academic year 2019-2020

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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

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Simple signals we considered: - pulses (time-domain): the system is characterized by its impulse response. - complex exponential (frequency domain): response?

The complex exponential

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Transmission of complex exponentials through LTI systems

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where is the Fourier transform of the impulse response of the system.

LTI system

u(t) = ej!t

y(t) = h(t) ⇤ u(t)

=

Z +1

�1h(⌧)ej!(t�⌧)d⌧

= ej!t

Z +1

�1h(⌧)e�j!⌧d⌧

= ej!tH(j!)

H(j!) =

Z +1

�1h(⌧)e�j!⌧d⌧

u(t) = ej!t

y(t) = ej!tH(j!)


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LTI system

u(t) = ej!t

y(t) = ej!tH(j!)

The response of an LTI system to a complex exponential at a specific frequency is a complex exponential at the same frequency whose amplitude and phase have been modified.

This modification is fully characterized by the Fourier transform of the system impulse response.


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LTI system

u(t) = ej!t

y(t) = ej!tH(j!)

The response of an LTI system to a complex exponential at a specific frequency is a complex exponential at the same frequency whose amplitude and phase have been modified.

This modification is fully characterized by the Fourier transform of the system impulse response.

However, the Fourier transform of a system impulse response does not always converge. In particular, it does not converge for unstable systems.

Generalization of Fourier transforms to all complex exponentials

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Fourier transform General case


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where is the transfer function of the LTI system.

LTI system

Continuous case:

Transmission of complex exponentials through LTI systems (continuous time)

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Transfer function Impulse response

The transfer function is a transformation of the impulse response.

This transformation is called the Laplace transform: the transfer function of a LTI system is the Laplace transform of its impulse response (continuous time).

There always exists a region where the Laplace transform of an LTI system converges, even for unstable systems.

The (two-sided/bilateral) Laplace transform

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The Laplace transform of a signal is given by

It can be written or or .

The (continuous time) Fourier transform of a signal is the Laplace transform on the imaginary axis ( ):

The inverse Laplace transform

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The inverse Laplace transform of a signal is given by

It is an integration in the complex plane on the axis.

The inverse Laplace transform: interpretation

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The inverse Laplace transform of a signal is given by

weight frequency

A signal can be represented as an infinite sum of exponentially growing/decaying sine waves.

The specific weight of a frequency is given by .

Transform tables

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Region of convergence of a transform

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The Laplace transform only exists if, and if yes where, the integral converges!

Region of convergence of Laplace transform

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Example #1: Laplace transform of the decaying exponential

It writes

The integral only converges for .


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The transform is therefore given by


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It writes

The integral only converges for .


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Example: Laplace transform of the decaying exponential


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The region of convergence (ROC) of the Laplace transform is the set of complex values for where the integralexists. We can derive a condition by writing

The condition of existence therefore writes


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If is the interval containing the signal , i.e. the condition writes

(I) The Laplace transform of a signal with finite support ( )exists on the whole complex plane.

(II) The Laplace transform of a signal with left finite support ( )and bounded by an exponential ( and s.t. ) exists in the half plane .

(III) The Laplace transform of a signal with right finite support ( )and bounded by an exponential ( and s.t. ) exists in the half plane .


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Left finite support:

Note that the Fourier transform does not exist in this case!

Time-shift and frequency-shift

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Continuous:

Duality convolution/multiplication

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Continuous:

Differentiation and integration

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Continuous:

Outline

Transfer function

Response of LTI systems: zero-state (bilateral Laplace transform)

Response of LTI systems: non-zero-state (unilateral Laplace transform)

Stability of LTI systems

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The transfer function

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LTI system

Transfer function:

In practice, analysis and design of LTI systems is done using the transfer function.

Transfer function of LTI systems (continuous case)

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Let’s consider the discrete LTI system described by the ODE the Laplace transform gives

The transfer function is therefore given by

Transfer function of LTI systems

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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.

Transfer function of LTI systems: relationship with state-space representation

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Transfer function from state-space representation: which givesand therefore

(1)

(1) (2)

Relationship between ROC of transfer function and causality

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Recall the example of last lecture:

If was the impulse response of a system, it would correspond to a causal system, and to its anti-causal equivalent.

The ROC of the transfer function of a causal system is an half-plane bounded on the left by the pole that has the biggest real part.

Relationship between ROC of transfer function and causality

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ROC of transfer function of causal systems (continuous and discrete):

Block diagrams and transfer function

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The duality convolution/multiplication makes it easy to connect LTI systems using the transfer function.

H1

H2

H

U Y

H1 H2

H

U Y

H1

H2

H

U Y-

Parallel: H = H1 + H2

Series: H = H1H2

Feedback: H = H1/(1+ H1 H2)

Outline

Transfer function




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Response of LTI systems: zero-state

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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).


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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:


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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)!


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The response of a LTI system is determined by the poles of the transfer function.


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Problem?

Outline

Transfer function




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The transfer function

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LTI system

Transfer function:

In real-life, the input is time-limited and the system has often a non-zero state!

The unilateral Laplace transform

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To cope with non-zero initial conditions, we define the unilateral Laplace transform:where 0- means that we include the effect of a pulse at t=0.

The unilateral Laplace transform of a signal is the bilateral Laplace transform of a signal .

The ROC is always a half-plane bounded on the left!

For a causal LTI system, the unilateral and bilateral Laplace transforms of the impulse response both give the transfer function.

Unilateral Laplace transform and initial conditions

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Why do we define the unilateral Laplace transform?

The main difference between the bilateral and unilateral Laplace transforms concerns the derivation:

Initial state!

Similarly,

Unilateral z-transform and initial conditions

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Response of LTI systems: non-zero-state

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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 ODEThe 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).


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The problem writes and the response can be derived using the unilateral Laplace transform: which gives


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The problem writes and the response can be derived using the unilateral Laplace transform: which gives

zero-state responsezero-input response

Response of LTI systems: modes

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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?

Outline

Transfer function




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Response of LTI systems

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Stability?

Bounded input bounded output (BIBO) stability

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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?

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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?

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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!

Relationship between ROC of transfer function and stability

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ROC of transfer function of an unstable causal systems:

K>0 K>1

Relationship between transfer function and systems response

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introduction to signals and systems lecture #8 - …guilldrion/files/syst0002-2019...the region of...

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