introduction to signals and systems lecture #4 - input...

Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems

Guillaume Drion Academic year 2017-2018

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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.

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Outline

Systems modeling: input/output approach of LTI systems.




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Systems modeling

Modeling and analysis of systems: open loop. “Observing and analyzing the environment”

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?

SYSTEMInput Output

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Systems modeling: state-space representation

Last lecture, we saw that the state-space representation of a model can describe its behavior.

Example: RLC circuit.

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.

R

LV

i

vL(t)vR(t)

C

vC(t)

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Example: the Windkessel model for variations in blood pressure

r

RCaP(t)

u(t)

PCa(t)Pr(t)

Model Simulation

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Response of N-dimensional continuous systems

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Systems with more than one variable:

x = Ax

! x(t) =?

is a square matrix of parameters. The response will be a sum of exponentials whose coefficients are the eigenvalues of the matrix .A

A

Eigenvalues can be real or complex conjugate pairs. In general:

where is the imaginary unit.j

xi(t) = xi,0e�i = xi,0e

(�i+j!i)t

The complex exponential

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Condition for stability of continuous linear systems

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STABLE UNSTABLE

Response of N-dimensional discrete systems

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Systems with more than one variable:

is a square matrix of parameters. The response will be a sum of exponentials whose coefficients are the eigenvalues of the matrix .A

A

Eigenvalues can be real or complex conjugate pairs. In general:

where is the imaginary unit.j

x[n+ 1] = Ax[n]

! x[n] =?

xi[n] = xi,0�ni = xi,0⇢

ni e

j!in

The discrete time complex exponential

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u[n] = zn, z = ⇢ej!

⇢ < 1 ⇢ = 1 ⇢ > 1

Condition for stability of continuous linear systems

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u[n] = zn, z = ⇢ej!

⇢ < 1 ⇢ = 1 ⇢ > 1

STABLE UNSTABLE


But what if the system is like this? How many states/equations would you need?

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or like this?

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or like this?

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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!

or like this?

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Input-output representation in time domain

Any system S can be represented using an input-output representation:

u yS

Can we mathematically describe the system using the following relationship?

What kind of system can be described this way?

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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?

400 ms20 mV

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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!

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Linear, Time-Invariant (LTI) systems


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.

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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)

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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

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Input-output representation of LTI systems

Can we mathematically describe a LTI system using the following relationship?

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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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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.

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Outline

Systems modeling: input/output approach and LTI systems.




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Definition of a pulse in discrete-time

The pulse in discrete time is defined by

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1

0 1 2 3 4-4 -3 -2 -1n

δ[n]

Role of a pulse in discrete-time


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If we multiply any signal by , we retrieve a signal that only contains the value of the input signal at : .

u[0]

0 1 2 3 4-4 -3 -2 -1n

u[n]δ[n]



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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[2]

0 1 2 3 4-4 -3 -2 -1n

u[n]δ[n-2]


If we retrieve signals that only contain the value of the input signal at for all and sum them, we retrieve the initial signal:

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A signal can therefore be decomposed into an infinite sum of unit impulse signals.

Example: decomposition of the unit step function:


A signal can therefore be decomposed into an infinite sum of unit impulse signals.

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0 1 2 3 4-4 -3

-2

-1n

u[n]

0 1 2 3 4-4 -3

-2

-1n

u[-2]δ[n+2]

0 1 2 3 4-4 -3

-2

-1n

u[-1]δ[n+1]

0 1 2 3 4-4 -3

-2

-1n

u[0]δ[n+0]

=

+

+

...

...

!!!!!!

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

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where is the impulse response of the system .


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

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where is the impulse response of the system .

is called “convolution of the signals and “and writes .


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 .

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If the impulse response has a “finite window”, the size of this windows defines the memory of the system.

Examples of convolutions.

Cascade of systems

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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 yh1[n] h2[n]x

Outline





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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 such thatfor all signal continuous at the origin, and

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(step function)

is “the derivative” of the step function, which is discontinuous at the origin!

Definition of a pulse in continuous-time: the Dirac delta function

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The Dirac delta function can be defined as a square of width and height with (defined by its integral equal to 1).

t

δ(t)

0-ε/2 ε/2

1/ε

ε 0

Definition of a pulse in continuous-time: the Dirac delta function

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The Dirac delta function can be defined as a square of width and height with (defined by its integral equal to 1).

t

δ(t)

0-ε/2 ε/2

1/ε

ε 0

Convolution in continuous-time

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Any continuous signal can be expressed as a sum (integral) of delta functions:

Therefore, the output of a continuous LTI system can be expressed aswhere is the impulse response of the LTI system.

In continuous time, the convolution is

Convolution in continuous-time

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Properties of convolution

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Convolutions are commutative (show it):

Convolutions are associative:

Convolutions are distributive (show it):

Outline





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Properties of LTI systems

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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

Memory of LTI systems

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The memory of a system is defined by the window of its impulse response. The larger the impulse response window, the bigger the memory.

0 1 2 3 4-4 -3 -2 -1n

h[n]

0 1

2

3 4-4 -3 -2 -1n

h[n]

Output depends on the present and the previous input values

Output depends on the present and the 4 previous input values


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Static systems: depends on only: . This gives where is the static gain of the system.

Input 0

1

Output

0

K


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Static systems: depends on only: . This gives where is the static gain of the system.

Dynamical systems: the response of the system is limited by the window of its impulse response! (reaches steady-state after some time).

Input 0

1

Output

0

K

Input 0

1

Output

0

K

Response time of LTI systems

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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 .

Time-constant of LTI systems

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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 .


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If the impulse response is the exponential decay:

Then

This is the typical response of a first order system. First order systems are characterized by a static gain and a time-constant .


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Example: high energy photon detector.


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Example: high energy photon detector.


The time-constant is important for filtering: with = cutoff frequency. Example: the cardiovascular system modeled in lecture #1 (low-pass filter).

Atherosclerosis: loss of arterial compliance => Ca decreases => τ=RCa decreases

τ = RCa

Low pass filter

H(s) =R

RCas + 1+ r

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Other useful response: the step response.

Highlights of the day

Input/output approach.

Linear, Time-Invariant systems.

Superposition principle.

Impulse response and step response.

Dirac delta function in continuous-time.

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Delta function in discrete-time.

Convolution (+properties).

Causality, memory response-time.

Time-constant and cutoff frequency.



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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?

The complex exponential

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The discrete time complex exponential

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u[n] = zn, z = ⇢ej!

⇢ < 1 ⇢ = 1 ⇢ > 1

Transmission of complex exponentials through LTI systems

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where is the transfer function of the LTI system.

LTI system

Continuous case:

Transmission of complex exponentials through LTI systems

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where is the transfer function of the LTI system.

Discrete case:

LTI system



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?

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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.

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introduction to signals and systems lecture #4 - input...

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