introduction to signals and systems lecture #4 - input...
TRANSCRIPT
![Page 1: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/1.jpg)
Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems
Guillaume Drion Academic year 2017-2018
1
![Page 2: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/2.jpg)
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.
2
![Page 3: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/3.jpg)
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.
3
![Page 4: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/4.jpg)
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
4
![Page 5: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/5.jpg)
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)
5
![Page 6: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/6.jpg)
Systems modeling: state-space representation
Example: the Windkessel model for variations in blood pressure
r
RCaP(t)
u(t)
PCa(t)Pr(t)
Model Simulation
6
![Page 7: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/7.jpg)
Response of N-dimensional continuous systems
7
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
![Page 8: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/8.jpg)
The complex exponential
8
![Page 9: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/9.jpg)
Condition for stability of continuous linear systems
9
STABLE UNSTABLE
![Page 10: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/10.jpg)
Response of N-dimensional discrete systems
10
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
![Page 11: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/11.jpg)
The discrete time complex exponential
11
u[n] = zn, z = ⇢ej!
⇢ < 1 ⇢ = 1 ⇢ > 1
![Page 12: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/12.jpg)
Condition for stability of continuous linear systems
12
u[n] = zn, z = ⇢ej!
⇢ < 1 ⇢ = 1 ⇢ > 1
STABLE UNSTABLE
![Page 13: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/13.jpg)
Systems modeling: state-space representation
But what if the system is like this? How many states/equations would you need?
13
![Page 14: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/14.jpg)
Systems modeling: state-space representation
or like this?
14
![Page 15: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/15.jpg)
Systems modeling: state-space representation
or like this?
15
![Page 16: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/16.jpg)
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?
16
![Page 17: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/17.jpg)
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?
17
![Page 18: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/18.jpg)
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
18
![Page 19: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/19.jpg)
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!
19
![Page 20: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/20.jpg)
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.
20
![Page 21: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/21.jpg)
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)
21
![Page 22: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/22.jpg)
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
22
![Page 23: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/23.jpg)
Input-output representation of LTI systems
Can we mathematically describe a LTI system using the following relationship?
23
![Page 24: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/24.jpg)
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
24
![Page 25: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/25.jpg)
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.
25
![Page 26: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/26.jpg)
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.
26
![Page 27: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/27.jpg)
Definition of a pulse in discrete-time
The pulse in discrete time is defined by
27
1
0 1 2 3 4-4 -3 -2 -1n
δ[n]
![Page 28: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/28.jpg)
Role of a pulse in discrete-time
The pulse in discrete time is defined by
28
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]
![Page 29: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/29.jpg)
Role of a pulse in discrete-time
The pulse in discrete time is defined by
29
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]
![Page 30: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/30.jpg)
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:
30
A signal can therefore be decomposed into an infinite sum of unit impulse signals.
Example: decomposition of the unit step function:
![Page 31: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/31.jpg)
Role of a pulse in discrete-time
A signal can therefore be decomposed into an infinite sum of unit impulse signals.
31
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]
=
+
+
...
...
!!!!!!
![Page 32: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/32.jpg)
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
32
where is the impulse response of the system .
![Page 33: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/33.jpg)
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
33
where is the impulse response of the system .
is called “convolution of the signals and “and writes .
![Page 34: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/34.jpg)
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 .
34
If the impulse response has a “finite window”, the size of this windows defines the memory of the system.
Examples of convolutions.
![Page 35: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/35.jpg)
Cascade of systems
35
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
![Page 36: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/36.jpg)
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.
36
![Page 37: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/37.jpg)
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
37
(step function)
is “the derivative” of the step function, which is discontinuous at the origin!
![Page 38: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/38.jpg)
Definition of a pulse in continuous-time: the Dirac delta function
38
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
![Page 39: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/39.jpg)
Definition of a pulse in continuous-time: the Dirac delta function
39
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
![Page 40: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/40.jpg)
Convolution in continuous-time
40
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
![Page 41: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/41.jpg)
Convolution in continuous-time
41
![Page 42: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/42.jpg)
Properties of convolution
42
Convolutions are commutative (show it):
Convolutions are associative:
Convolutions are distributive (show it):
![Page 43: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/43.jpg)
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.
43
![Page 44: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/44.jpg)
Properties of LTI systems
44
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
![Page 45: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/45.jpg)
Memory of LTI systems
45
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
![Page 46: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/46.jpg)
Memory of LTI systems
46
Static systems: depends on only: . This gives where is the static gain of the system.
Input 0
1
Output
0
K
![Page 47: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/47.jpg)
Memory of LTI systems
47
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
![Page 48: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/48.jpg)
Response time of LTI systems
48
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 .
![Page 49: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/49.jpg)
Time-constant of LTI systems
49
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 .
![Page 50: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/50.jpg)
Time-constant of LTI systems
50
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 .
![Page 51: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/51.jpg)
Time-constant of LTI systems
51
![Page 52: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/52.jpg)
Time-constant of LTI systems
52
Example: high energy photon detector.
![Page 53: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/53.jpg)
Time-constant of LTI systems
53
Example: high energy photon detector.
![Page 54: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/54.jpg)
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
54
![Page 55: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/55.jpg)
Time-constant of LTI systems
55
Other useful response: the step response.
![Page 56: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/56.jpg)
Highlights of the day
Input/output approach.
Linear, Time-Invariant systems.
Superposition principle.
Impulse response and step response.
Dirac delta function in continuous-time.
56
Delta function in discrete-time.
Convolution (+properties).
Causality, memory response-time.
Time-constant and cutoff frequency.
![Page 57: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/57.jpg)
Input-output representation of LTI systems
Can we mathematically describe a LTI system using the following relationship?
57
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?
![Page 58: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/58.jpg)
The complex exponential
58
![Page 59: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/59.jpg)
The discrete time complex exponential
59
u[n] = zn, z = ⇢ej!
⇢ < 1 ⇢ = 1 ⇢ > 1
![Page 60: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/60.jpg)
Transmission of complex exponentials through LTI systems
60
where is the transfer function of the LTI system.
LTI system
Continuous case:
![Page 61: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/61.jpg)
Transmission of complex exponentials through LTI systems
61
where is the transfer function of the LTI system.
Discrete case:
LTI system
![Page 62: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/62.jpg)
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
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?
62
![Page 63: Introduction to Signals and Systems Lecture #4 - Input ...guilldrion/Files/SYST0002-2017-18-Lecture4.pdf · Introduction to Signals and Systems Lecture #4 - Input-output Representation](https://reader030.vdocuments.mx/reader030/viewer/2022040422/5e157b97ed060f7d434ac52a/html5/thumbnails/63.jpg)
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
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.
63