Introduction to Signals and Systems Lecture #4 - Input...
Transcript of 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 2019-2020
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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.
Convolution in discrete-time.
Convolution in continuous-time: the Dirac delta function.
Causality, memory, responsiveness 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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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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The complex exponential
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Special case: , giving .
Condition for stability of continuous linear systems
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STABLE UNSTABLE
Systems modeling: state-space representation
But what if the system is like this? How many states/equations would you need?
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Systems modeling: state-space representation
or like this?
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Systems modeling: state-space representation
or like this?
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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!
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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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?
400 ms20 mV
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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!
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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.
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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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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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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.
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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.
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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
The pulse in discrete time is defined by
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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]
Role of a pulse in discrete-time
The pulse in discrete time is defined by
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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]
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:
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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:
Role of a pulse in discrete-time
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 .
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 .
is called “convolution of the signals and “and writes .
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 .
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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
Graphical illustration of convolution (from Manolakis and Ingle)
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y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
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y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
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y[n] =X
k2Zx[k]h[n� k]
<latexit sha1_base64="UaxOYJVqQ8yYPJka6rLmTskVH2g=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBR3FgSEXQjFN24rGAvmIQymU7aIZNJmJmIIfQN3Pgqblwo4tatO9/GSZuFtv4w8PGfc5hzfj9hVCrL+jYqC4tLyyvV1dra+sbmlrm905FxKjBp45jFoucjSRjlpK2oYqSXCIIin5GuH14V9e49EZLG/FZlCfEiNOQ0oBgpbfXNw8zhHryArkyjfh66lLsRUiPfz+/G4wcn9EYOPw69vlm3GtZEcB7sEuqgVKtvfrmDGKcR4QozJKVjW4nyciQUxYyMa24qSYJwiIbE0chRRKSXT+4ZwwPtDGAQC/24ghP390SOIimzyNedxbJytlaY/9WcVAXnXk55kirC8fSjIGVQxbAIBw6oIFixTAPCgupdIR4hgbDSEdZ0CPbsyfPQOWnYmm9O683LMo4q2AP74AjY4Aw0wTVogTbA4BE8g1fwZjwZL8a78TFtrRjlzC74I+PzBzjlnMY=</latexit><latexit sha1_base64="UaxOYJVqQ8yYPJka6rLmTskVH2g=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBR3FgSEXQjFN24rGAvmIQymU7aIZNJmJmIIfQN3Pgqblwo4tatO9/GSZuFtv4w8PGfc5hzfj9hVCrL+jYqC4tLyyvV1dra+sbmlrm905FxKjBp45jFoucjSRjlpK2oYqSXCIIin5GuH14V9e49EZLG/FZlCfEiNOQ0oBgpbfXNw8zhHryArkyjfh66lLsRUiPfz+/G4wcn9EYOPw69vlm3GtZEcB7sEuqgVKtvfrmDGKcR4QozJKVjW4nyciQUxYyMa24qSYJwiIbE0chRRKSXT+4ZwwPtDGAQC/24ghP390SOIimzyNedxbJytlaY/9WcVAXnXk55kirC8fSjIGVQxbAIBw6oIFixTAPCgupdIR4hgbDSEdZ0CPbsyfPQOWnYmm9O683LMo4q2AP74AjY4Aw0wTVogTbA4BE8g1fwZjwZL8a78TFtrRjlzC74I+PzBzjlnMY=</latexit><latexit sha1_base64="UaxOYJVqQ8yYPJka6rLmTskVH2g=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBR3FgSEXQjFN24rGAvmIQymU7aIZNJmJmIIfQN3Pgqblwo4tatO9/GSZuFtv4w8PGfc5hzfj9hVCrL+jYqC4tLyyvV1dra+sbmlrm905FxKjBp45jFoucjSRjlpK2oYqSXCIIin5GuH14V9e49EZLG/FZlCfEiNOQ0oBgpbfXNw8zhHryArkyjfh66lLsRUiPfz+/G4wcn9EYOPw69vlm3GtZEcB7sEuqgVKtvfrmDGKcR4QozJKVjW4nyciQUxYyMa24qSYJwiIbE0chRRKSXT+4ZwwPtDGAQC/24ghP390SOIimzyNedxbJytlaY/9WcVAXnXk55kirC8fSjIGVQxbAIBw6oIFixTAPCgupdIR4hgbDSEdZ0CPbsyfPQOWnYmm9O683LMo4q2AP74AjY4Aw0wTVogTbA4BE8g1fwZjwZL8a78TFtrRjlzC74I+PzBzjlnMY=</latexit><latexit sha1_base64="UaxOYJVqQ8yYPJka6rLmTskVH2g=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBR3FgSEXQjFN24rGAvmIQymU7aIZNJmJmIIfQN3Pgqblwo4tatO9/GSZuFtv4w8PGfc5hzfj9hVCrL+jYqC4tLyyvV1dra+sbmlrm905FxKjBp45jFoucjSRjlpK2oYqSXCIIin5GuH14V9e49EZLG/FZlCfEiNOQ0oBgpbfXNw8zhHryArkyjfh66lLsRUiPfz+/G4wcn9EYOPw69vlm3GtZEcB7sEuqgVKtvfrmDGKcR4QozJKVjW4nyciQUxYyMa24qSYJwiIbE0chRRKSXT+4ZwwPtDGAQC/24ghP390SOIimzyNedxbJytlaY/9WcVAXnXk55kirC8fSjIGVQxbAIBw6oIFixTAPCgupdIR4hgbDSEdZ0CPbsyfPQOWnYmm9O683LMo4q2AP74AjY4Aw0wTVogTbA4BE8g1fwZjwZL8a78TFtrRjlzC74I+PzBzjlnMY=</latexit>
Graphical illustration of convolution (from Manolakis and Ingle)
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y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
37
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
38
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
39
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
40
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
41
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
42
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
43
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
44
y[n] =X
k2Zx[k]h[n� k]
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Graphical illustration of convolution (from Manolakis and Ingle)
45
y[n] =X
k2Zx[k]h[n� k]
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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.
46
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
47
(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
48
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
49
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
50
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
51
Properties of convolution
52
Convolutions are commutative (show it):
Convolutions are associative:
Convolutions are distributive (show it):
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.
53
Properties of LTI systems
54
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
55
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
Memory of LTI systems
56
Static systems: depends on only: . This gives where is the static gain of the system.
Input 0
1
Output
0
K
Memory of LTI systems
57
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
58
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
59
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 .
Time-constant of LTI systems
60
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 .
Time-constant of LTI systems
61
Time-constant of LTI systems
62
Example: high energy photon detector.
Time-constant of LTI systems
63
Example: high energy photon detector.
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).
Atherosclerosis: loss of arterial compliance => Ca decreases => τ=RCa decreases
τ = RCa
Low pass filter
H(s) =R
RCas + 1+ r
64
Time-constant of LTI systems
65
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.
66
Delta function in discrete-time.
Convolution (+properties).
Causality, memory response-time.
Time-constant and cutoff frequency.