Lecture 4: Linear Time Invariant (LTI) systems LTI systems ...DT LTI systems: the convolution sum...
Transcript of Lecture 4: Linear Time Invariant (LTI) systems LTI systems ...DT LTI systems: the convolution sum...
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Lecture 4: Linear Time Invariant (LTI) systems
2. Linear systems, Convolution (3 lectures): Impulse
response, input signals as continuum of impulses.
Convolution, discrete-time and continuous-time. LTI
systems and convolution
Specific objectives for today:
We’re looking at discrete time signals and systems
• Understand a system’s impulse response properties
• Show how any input signal can be decomposed into
a continuum of impulses
• DT Convolution for time varying and time invariant
systems
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LTI systems
• Two important basic Properties of systems:
• Linearity.
• Time-invariance (superposition property).
• Plays an important role in signals and systems
analysis.
• Many of physical processes possess these properties.
• They are modeled as LTI systems
• Any system possess these two properties is called
linear time- invariant (LTI) system.
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LTI systems Properties
• Develop a complete characterization of LTI system in terms
to its impulse response using convolution sum for DTS and
convolution integral for CTS. 4/17
Representation of DTS in Terms of Impulses
• A DTS x[n] can be viewed as sequence of individual
impulses or as a linear combination of time-shifted
impulses:
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Representation of DTS in Terms of Impulses
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Representation of DTS in Terms of Impulses
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Discrete Impulses & Time Shifts
Basic idea: use a (infinite) set of of discrete time impulses to represent any signal.
Consider any discrete input signal x[n]. This can be written as the linear sum of a set of unit impulse signals:
Therefore, the signal can be expressed as:
In general, any discrete signal can be represented as:
k
knkxnx ][][][
101]1[
]1[]1[
000]0[
][]0[
101]1[
]1[]1[
nnx
nx
nnx
nx
nnx
nx
]1[]1[ nx
actual value Impulse, time
shifted signal
The sifting property
]1[]1[][]0[]1[]1[]2[]2[][ nxnxnxnxnx
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Representation of DTS in Terms of Impulses
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Example
The discrete signal x[n]
Is decomposed into the
following additive
components
x[-4][n+4] +
x[-3][n+3] + x[-2][n+2] + x[-1][n+1] + …
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DT LTI systems: the convolution sum
• Is the mathematical relationship that links the input and
output signals in any LTI discrete-time system.
• Given an:
• LTI system,
• Input signal x[n].
• Impulse Response H[n]: the response to one of the
basic signals such as impulse signal;
• The convolution sum will allow us to compute the
corresponding output signal y[n] of the system.
• Compute y[n] ??????????
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Introduction to Convolution
Definition Convolution is an operator that takes an input signal and
returns an output signal, based on knowledge about the system’s unit
impulse response h[n].
The basic idea behind convolution is to use the system’s response to a
simple input signal to calculate the response to more complex signals
This is possible for LTI systems because they possess the
superposition property (lecture 3):
k kk nxanxanxanxanx ][][][][][ 332211
k kk nyanyanyanyany ][][][][][ 332211
System y[n] = h[n]x[n] = [n]
System: h[n] y[n]x[n]
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Response of LTI as a linear combination of
impulse response
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Response of LTI as a linear combination of
impulse response
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Response of LTI as a linear combination of impulse response
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Response of LTI as a linear combination of
impulse response
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Convolution Sum
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Convolution Sum
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Response of LTI as a combination of H[n]
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Convolution sum
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Convolution sum
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Example : LTI Convolution
A LTI system with the following
unit impulse response:
h[n] = [0 0 1 1 1 0 0]
For the input sequence:
x[n] = [0 0 0.5 2 0 0 0]
The result is:
y[n] = … + x[0]h[n] + x[1]h[n-1] +
…
= 0 +
0.5*[0 0 1 1 1 0 0] +
2.0*[0 0 0 1 1 1 0] +
0
= [0 0 0.5 2.5 2.5 2 0]
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Example 2: LTI Convolution
Consider the problem
described for example 1
Sketch x[k] and h[n-k] for any
particular value of n, then
multiply the two signals and
sum over all values of k.
For n<0, we see that x[k]h[n-k]
= 0 for all k, since the non-
zero values of the two
signals do not overlap.
y[0] = Skx[k]h[0-k] = 0.5
y[1] = Skx[k]h[1-k] = 0.5+2
y[2] = Skx[k]h[2-k] = 0.5+2
y[3] = Skx[k]h[3-k] = 2
As found in Example 1
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Convolution sumExample 5: Compute y[0] for the input signal and impulse response of
an LTI system shown in the following Figure.
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Convolution sumExample 5: Compute y[1] for the input signal and impulse response of
an LTI system shown in the following Figure.
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Convolution sum
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Convolution sum
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Example 3: LTI Convolution
Consider a LTI system that has a step
response h[n] = u[n] to the unit
impulse input signal
What is the response when an input
signal of the form
x[n] = anu[n]
where 0<a<1, is applied?
For n0:
Therefore,
a
a
a
1
1
][
1
0
n
n
k
kny
][1
1][
1
nunyn
a
a
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Convolution sum
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Convolution sum
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Convolution sum
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Convolution sum
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Discrete, Unit Impulse System Response
A very important way to analyse a system is to study the
output signal when a unit impulse signal is used as
an input
Loosely speaking, this corresponds to giving the system
a kick at n=0, and then seeing what happens
This is so common, a specific notation, h[n], is used to
denote the output signal, rather than the more
general y[n].
The output signal can be used to infer properties about
the system’s structure and its parameters q.
System: q h[n][n]
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Types of Unit Impulse Response
Looking at unit impulse
responses, allows you to
determine certain system
properties
Causal, stable, finite impulse response
y[n] = x[n] + 0.5x[n-1] + 0.25x[n-2]
Causal, stable, infinite impulse response
y[n] = x[n] + 0.7y[n-1]
Causal, unstable, infinite impulse response
y[n] = x[n] + 1.3y[n-1] EE-2027 SaS, L4: 34/17
Linear, Time Varying Systems
If the system is time varying, let hk[n] denote the response
to the impulse signal [n-k] (because it is time varying,
the impulse responses at different times will change).
Then from the superposition property (Lecture 3) of linear
systems, the system’s response to a more general input
signal x[n] can be written as:
Input signal
System output signal is given by the convolution sum
i.e. it is the scaled sum of impulse responses
k
k nhkxny ][][][
k
knkxnx ][][][
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Example: Time Varying Convolution
x[n] = [0 0 –1 1.5 0 0 0]
h-1[n] = [0 0 –1.5 –0.7 .4 0 0]
h0[n] = [0 0 0 0.5 0.8 1.7 0]
y[n] = [0 0 1.4 1.4 0.7 2.6 0]
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Linear Time Invariant Systems
When system is linear, time invariant, the unit impulse
responses are all time-shifted versions of each other:
It is usual to drop the 0 subscript and simply define the
unit impulse response h[n] as:
In this case, the convolution sum for LTI systems is:
It is called the convolution sum (or superposition sum)
because it involves the convolution of two signals x[n]
and h[n], and is sometimes written as:
knhnhk 0][
nhnh 0][
k
knhkxny ][][][
][*][][ nhnxny
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System Identification and Prediction
Note that the system’s response to an arbitrary input signal is
completely determined by its response to the unit impulse.
Therefore, if we need to identify a particular LTI system, we
can apply a unit impulse signal and measure the system’s
response.
That data can then be used to predict the system’s response
to any input signal
Note that describing an LTI system using h[n], is equivalent to
a description using a difference equation. There is a direct
mapping between h[n] and the parameters/order of a
difference equation such as:y[n] = x[n] + 0.5x[n-1] + 0.25x[n-2]
System: h[n]
y[n]x[n]
Discrete LTI Convolution in Matlab
In Matlab to find out about a command, you can search the help files or type:
>> lookfor convolution
at the Matlab command line. This returns all Matlab functions that contain the term “convolution” in the basic description
These include:
conv()
To see how this works and other functions that may be appropriate, type:
>> help conv
at the Matlab command line
Example:
>> h = [0 0 1 1 1 0 0];
>> x = [0 0 0.5 2 0 0 0];
>> y = conv(x, h)
>> y = [0 0 0 0 0.5 2.5 2.5 2 0 0 0 0 0]
Consider the DT SISO system:
If the input signal is and the system has no energy at
, the output is called the impulse response
of the system
DT Unit-Impulse Response
[ ]y n[ ]x n
[ ]h n[ ]n
[ ] [ ]x n n[ ] [ ]y n h n
System
System
0n
General Response
Impulse Response
Consider the DT system described by
Its impulse response can be found to be
Example
[ ] [ 1] [ ]y n ay n bx n
( ) , 0,1,2,[ ]
0, 1, 2, 3,
na b nh n
n
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Let x[n] be an arbitrary input signal to a DT LTI system
Suppose that for
This signal can be represented as
Representing Signals in Terms ofShifted and Scaled Impulses
0
[ ] [0] [ ] [1] [ 1] [2] [ 2]
[ ] [ ], 0,1,2,i
x n x n x n x n
x i n i n
1, 2,n [ ] 0x n
Exploiting Time-Invariance and Linearity
0
[ ] [ ] [ ], 0i
y n x i h n i n
This particular summation is called the convolution sum
Equation is called the convolution representation of the system
Remark: a DT LTI system is completely described by its impulse
response h[n]
0
[ ] [ ] [ ]i
y n x i h n i
The Convolution Sum
[ ] [ ]x n h n
[ ] [ ] [ ]y n x n h n
Since the impulse response h[n] provides the complete
description of a DT LTI system, we write
Block Diagram Representation of DT LTI Systems
[ ]y n[ ]x n [ ]h n
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Example:
Suppose that both x[n] and v[n] are equal
Plot of [ ] [ ]x n v n
Associativity
Commutativity
Distributivity w.r.t. addition
Properties of the Convolution Sum
[ ] ( [ ] [ ]) ( [ ] [ ]) [ ]x n v n w n x n v n w n
[ ] [ ] [ ] [ ]x n v n v n x n
[ ] ( [ ] [ ]) [ ] [ ] [ ] [ ]x n v n w n x n v n x n w n
Shift property: define
Convolution with the unit impulse
Convolution with the shifted unit impulse
Properties of the Convolution Sum -Cont’d
[ ] [ ] [ ]w n x n v n
[ ] [ ] [ ] [ ] [ ]q qw n q x n v n x n v n
[ ] [ ]qx n x n q
[ ] [ ]qv n v n q
then
[ ] [ ] [ ]x n n x n
[ ] [ ] [ ]qx n n x n q
Example: Computing Convolution with Matlab
Consider the DT LTI system
impulse response:
input signal:
[ ]y n[ ]x n [ ]h n
[ ] sin(0.5 ), 0h n n n [ ] sin(0.2 ), 0x n n n
n=0:40;
x=sin(0.2*n);
h=sin(0.5*n);
y=conv(x,h);
stem(n,y(1:length(n)))
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Convolution sum
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Convolution sum
Consider the CT SISO system:
If the input signal is and the system has no
energy at , the output
is called the impulse response of the system
CT Unit-Impulse Response
( )h t( )t
( ) ( )x t t
( ) ( )y t h t
( )y t( )x t System
System
0t
Let x[n] be an arbitrary input signal with
for
Using the sifting property of , we may write
Exploiting time-invariance, it is
Exploiting Time-Invariance
( ) 0, 0x t t
( )t
0
( ) ( ) ( ) , 0x t x t d t
( )h t ( )t System
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Exploiting Time-Invariance
Exploiting linearity, it is
If the integrand does not contain an impulse
located at , the lower limit of the integral can be
taken to be 0,i.e.,
Exploiting Linearity
0
( ) ( ) ( ) , 0y t x h t d t
( ) ( )x h t
0
0
( ) ( ) ( ) , 0y t x h t d t
This particular integration is called the convolution integral
Equation is called the convolution representation of the system
Remark: a CT LTI system is completely described by its
impulse response h(t)
The Convolution Integral
( ) ( )x t h t
( ) ( ) ( )y t x t h t
0
( ) ( ) ( ) , 0y t x h t d t
Since the impulse response h(t) provides the complete
description of a CT LTI system, we write
Block Diagram Representation of CT LTI Systems
( )y t( )x t ( )h t
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Suppose that where p(t) is the rectangular
pulse depicted in figure
Example: Analytical Computation of the Convolution Integral
( ) ( ) ( ),x t h t p t
( )p t
tT0
• In order to compute the convolution integral
we have to consider four cases:
0
( ) ( ) ( ) , 0y t x h t d t
Example –Cont’d
Case 1: 0t ( )x
T0
( )h t
t T t
( ) 0y t
0 t T • Case 2:
( )x
T0
( )h t
t T t
0
( )
t
y t d t
• Case 3:0 2t T T T t T
( )x
T0
( )h t
t T t
( ) ( ) 2
T
t T
y t d T t T T t
• Case 4: 2T t T T t ( )x
T0
( )h t
t T t( ) 0y t
( ) ( ) ( )y t x t h t
T0 t2T
Associativity
Commutativity
Distributivity w.r.t. addition
Properties of the Convolution Integral
( ) ( ( ) ( )) ( ( ) ( )) ( )x t v t w t x t v t w t
( ) ( ) ( ) ( )x t v t v t x t
( ) ( ( ) ( )) ( ) ( ) ( ) ( )x t v t w t x t v t x t w t
Shift property: define
Convolution with the unit impulse
Convolution with the shifted unit impulse
Properties of the Convolution Integral - Cont’d
( ) ( ) ( )w t x t v t
( ) ( ) ( ) ( ) ( )q qw t q x t v t x t v t
( ) ( )qx t x t q
( ) ( )qv t v t q
then
( ) ( ) ( )x t t x t
( ) ( ) ( )qx t t x t q
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Convolution Integral - Properties
)](*)([)](*)([)]()([*)(
)](*)([*)()(*)](*)([
)(*)()(*)(
2121
2121
thtxthtxththtx
ththtxththtx
txththtx
• Commutative
• Associative
• Distributive
Example 1
Consider a CT-LTI system. Assume the impulse response of
the system is h(t)=e^(-at) for all a>0 and t>0 and input
x(t)=u(t). Find the output.
h(t)=e^-atu(t) y(t)
)()1(1
)1(1
)()(
)()()(
)()()()()(
0
tuea
ea
de
dtuue
dtuhty
tuthtxthty
at
at
t
a
a
Draw x(), h(), h(t-),etc. next slide
Because t>0
The fact that a>0 is not an issue!
Example 1 – Cont.
y(t)
t>0
t<0
Remember we are plotting it over
and t is the variable
U(-(-t))
U(-(-t))
)()1(1
)1(1
)()(
)()()(
)()()()()(
0
tuea
ea
de
dtuue
dtuhty
tuthtxthty
at
at
t
a
a
t
y(t); for a=3
t
64
t
Example 2.5: Convolution Integral.
Given a RC circuit below (RC=1s). Use convolution to determine the voltage across the capacitor y(t). Input voltage x(t)=u(t)-u(t-2).
Solution:
y(t)=x(t)*h(t)
- capacitor start charging
at t=0 and discharging
at t=2.
a
b
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65
Cont’d…
66 .
Cont’d…