![Page 1: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/1.jpg)
CT Systems:
Impulse response
Convolution integral
Block diagram of systems
Properties using the impulse response
Systems characterized by Differential Equations
oDT Systems
Impulse response
Convolution sum
Block diagram of systems
Properties using the impulse response
Systems characterized by Difference Equations
Summary
ELEC264: Signals And Systems
Topic 2: LTI Systems and Convolution
Aishy Amer
Concordia University
Electrical and Computer Engineering
Figures and examples in these course slides are taken from the following sources:
•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997
•M.J. Roberts, Signals and Systems, McGraw Hill, 2004
•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003
![Page 2: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/2.jpg)
2
Properties of the Unit Impulse
The area under the unit impulse
Sampling (Sifting) Property:
Scaling Property
Equivalence Property
Relation to u(t):
1)( dtt
)/()( ;||
1 abtbata
![Page 3: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/3.jpg)
3
Impulse Response of Linear
Time-Invariant (LTI) Systems
Linearity
Time Invariance
)()()()(
)()(then
)()( if
00 ttyttxtytx
tyaty
txatx
k
kk
k
kk
![Page 4: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/4.jpg)
4
Impulse Response of LTI
systems The response of a Linear Time-Invariant (LTI) system
to the unit impulse δ(t) is called "The impulse
response" h(t)
h(t) = LTI-SYS { δ(t) }
The impulse response h(t) completely characterizes an
LTI system
y(t) = FUNCTION { x(t),h(t) }
CONVOLUTION
h(t)
![Page 5: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/5.jpg)
5
Impulse response of Basic
Systems
???????)(;)]([)(:LTI)(not Squarer
)()( ;
)()( :ator(LTI)Differenti
)()( ;)(y(t) :(LTI) Integrator
)()(x(y(t) :(shifter)delay timeIdeal
)()( x(t);y(t) :systemIdentity
2
'
0 );0
thtxty
tdt
tdth
dt
tdxty
tuthdx
ttthtt
tth
t
![Page 6: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/6.jpg)
6
Basic Systems: Differentiators &
Integrators
Differentiators are:
Difficult to implement
Sensitive to noise and errors
Alternatives : Integrators
Integrators : amplifiersfinite) (to )(2)()()(
)(2)()(
)(2)()()()(
0)( assume
)(2)()(
0
t
t
tt
dyxtyty
dyxty
dyxtytydtdt
tdy
y
tytxdt
tdy
0;)()(
txt
txttx
dtd
![Page 7: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/7.jpg)
7
Relationship: CT impulse response and
step response
In any CT LTI system let an excitation, x(t), produce the response, y(t). Then the excitation
will produce the response
It follows then that the unit impulse response is the first derivative of the unit step response and, conversely that the unit step response is the integral of the unit impulse response
))(( txdt
d
))(( tydt
d
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8
![Page 9: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/9.jpg)
9
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Difference Equations
Summary
![Page 10: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/10.jpg)
10
The Convolution Integral
1) An arbitrary input x(t) can be expressed as a weighted
sum of time-shifted impulses
2) An LTI is described by an impulse response h(t)
![Page 11: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/11.jpg)
11
The Convolution Integral
The output y(t) must be a weighted sum of time-shifted
impulse responses
![Page 12: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/12.jpg)
12
The Convolution Integral:
Proof
dtSxtxSty
S
dtxStxSty
tSthdtxtx
)}({)()}({)(
mswitch thecan We
opeartorlinear a also is
operator linear a is
} )()( {} )( { )(
} )( {)( )()()(
Assuming S is LTI
![Page 13: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/13.jpg)
13
The Convolution Integral:
Proof
)()}({
invariant- timeis
thtS
S
tth parameter with ofFunction :)(
over on Intergrati:Important
)(*)()()()( thtxdthxty
} )( {)( tSth
CONVOLUTION
![Page 14: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/14.jpg)
14
The Convolution Integral:
Interpretation
Interpretation:
replacing each signal amplitude at time t by a
weighted sum of its neighbors
dhtxtxthty
dthxthtxty
)()()(*)()(
)()()(*)()(
![Page 15: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/15.jpg)
15
Convolution Integral:
Graphical Illustration
Let the excitation, x(t), and the impulse
response, h(t), be
![Page 16: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/16.jpg)
16
Convolution Integral:
Graphical Illustration
The convolution integral:
In the convolution integral there is a factor
We visualize this quantity
dthxty )()()(
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17
Convolution Integral:
Graphical Illustration
The functional transformation in going
from to is
![Page 18: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/18.jpg)
18
Convolution Integral:
Graphical Illustration
The convolution value is the area under the
product of x(t) and
This area depends on what t is
First, as an example, let t = 5
For this choice of t the area under the product is
zero
So 0y(5) ),()()(with thtxty
![Page 19: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/19.jpg)
19
Convolution Integral:
Graphical Illustration
Now let t=0
Therefore the area under the product is 2, i.e., y(0) = 2
![Page 20: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/20.jpg)
20
Convolution Integral:
Graphical Illustration
The process of convolving to find y(t) is
illustrated below
![Page 21: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/21.jpg)
21
Convolution Integral: Graphical
Illustration
Interpretation: replacing each signal amplitude
at time t by a weighted sum of its neighbors
Smoothing of sharp transitions of x(t)
Filtering out some content from x(t)
Removing some content
...
![Page 22: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/22.jpg)
22
Steps for graphical convolution
1. Sketch the waveform for input x(τ) by changing the independent variable
from t to τ and keep the waveform for x(τ) fixed during convolution.
2. Sketch the waveform for the impulse response h(τ) by changing the
independent variable t to τ.
3. Reflect h(τ) about the vertical axis to obtain the time-inverted impulse
response h(-τ).
4. Shift the time-inverted impulse function h(-τ) by a selected value of "t".
The resulting function represents h(t-τ).
5. Multiply function x(τ) by h(t- τ) and plot the product function x(τ)h(t-τ).
6. Calculate the total area under the product function x(τ)h(t-τ) by integrating
it over τ =[-∞,∞].
7. Repeat steps 4-6 for different values of t to obtain y(t) for all times, -
∞≤t≤∞.
)(*)()()()( thtxdthxty
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23
Steps for graphical convolution
)(*)()()()( thtxdthxty
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24
Steps for graphical convolution
01&0:case3
01&0:case2
01&0:1 case
)()()(
tt
tt
tt
dthxty
![Page 25: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/25.jpg)
25
Convolution: Example 1 infinite-duration signals
![Page 26: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/26.jpg)
26
Convolution: example 2 finite & infinite-duration signals
![Page 27: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/27.jpg)
27
Convolution: example 3 with finite-duration signals
![Page 28: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/28.jpg)
28
Convolution: example 4
Exam question
only. itiesdiscontinu threehas )(
if ? (b)
?)( (a)
10)(
10
:0
:1)(
t
ty
ty
txth
else
ttx
![Page 29: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/29.jpg)
29
Solution: Step 1
(a) Write down x(t) and h(t) functionally and graphically
Note that h(t) is a scaled version of x(t)
dthxthtxty
else
t
else
tt
xth
else
ttx
)()()()()(
0
:0
:110
:0
:1)(
10
:0
:1)(
![Page 30: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/30.jpg)
30
Solution: Step 2
Sketch h(-τ) and h(t-τ)
h(-τ)
Rreflection around y-axis
Chage t to τ
h(t-τ) = h(-τ+t)
Add t to all axis points
Move the graph away to the left
![Page 31: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/31.jpg)
31
Solution: Step 3
Slide h(t-τ) to the right and collect the overlap
As you go, find
Limits for y(t)
Limits for integration
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32
Solution: Step 3a
tdd
tORtandtfor
ty
t
t
0
0-
][1)-)h(tx(
graph) from (findn integratiofor Limits
000
)(for Limits
![Page 33: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/33.jpg)
33
Solution: Step 3b
t
t
t
dd
tORtandtfor
ty
][1)-)h(tx(
graph) from (findn integratiofor Limits
110
)(for Limits
-t-
![Page 34: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/34.jpg)
34
Solution: Step 3c
tdd
tORtandtfor
ty
t 1][1)-)h(tx(
graph) from (findn integratiofor Limits
1111
)(for Limits
1
1
-t
1
-t
![Page 35: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/35.jpg)
35
Solution: Step 4
Put the limits together to make y(t)
else
t
t
t
t
t
ty11
1
0
:
:
:
:
0
1)(
![Page 36: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/36.jpg)
36
Solution: Step 5
(b) find the first derivative of y with respect to t both
functionally and graphically
This function has 4 discontinuities
Only when = 1, it has 3 discontinuities
• (two discontinuities become one)
Note that we know 0< 1
else
t
t
t
t
ty
11
1
0
:
:
:
:
0
1
0
1
)(
![Page 37: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/37.jpg)
37
Outline
CT Systems
Impulse response
Convolution integral
Block Diagram of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum (DT)
DT properties using the impulse response
DT Systems characterized by Difference Equations
Summary
![Page 38: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/38.jpg)
38
System Block Diagrams
LTI Systems can be described using the impulse
response h(t) which completely characterizes an LTI system
LTI systems can also be described
mathematically by differential equations
)()()()( 0012 txbtyatyatya
h(t)
)(*)()()()( thtxdthxty
![Page 39: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/39.jpg)
39
Block Diagram ElementsBlock diagram: A very useful method for describing and analyzing systems is the block diagram
![Page 40: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/40.jpg)
40
System Block Diagrams
A block diagram can be drawn directly from the differential equation which describes the system
For example, if the system is described by
It can also be described by this block diagram …
)()()()( 0012 txbtyatyatya
![Page 41: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/41.jpg)
41
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum (DT)
DT properties using the impulse response
DT Systems characterized by Difference Equations
Summary
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42
System properties via the
convolution properties Convolution properties help to solve convolution of
complex signals in term of operations on another
signal for which the convolution is known
Example:
)(*)()(*)(
)(*)]()([)(*)()(y
:property edistibutiv theUsing
)()(
)()()()()(
21
21
212
thtxthtx
thtxtxthtxt
tuth
txtxtuetuetx tt
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43
System properties via the
convolution properties
)(*)()(*)(
)(*)]()([)(y
:Example
2
2
thtuethtue
thtuetuet
tt
tt
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44
System properties via the
convolution properties
systems two thecascade
i.e., ity,associativ theusecannot we
LTInot )(
)()( );(2)(:Example
2
221
ty
txtytxty
![Page 45: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/45.jpg)
45
System properties via the convolution
properties: System Interconnections
Example:
o Since the integrator and differentiator are both LTI system operations, when
used in combination with another system having impulse response h(t), we find
that the cascade property holds
Performing differentiation or integration before a signal enters an LTI system,
gives the same result as performing the differentiation or integration after the
signal passes through the system
![Page 46: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/46.jpg)
46
System properties via the
convolution properties
• “Convolution” property:
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47
System properties using the
impulse response
000
000
for causal-non is)()(
for causal is )()(
)integratoran is system (this causal is )()(
:Examples
tttth
tttth
tuth
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48
System properties using the
impulse response
System stability: A CT system is BIBO stable if its impulse response is
absolutely integrable
)(th
stable is )(h(t) )2
|)(| since unstable is system
)()()()(*)()(
summer)or or (accumulat integratoran is )()( )1
:Examples
0tt
dtth
dxdtuxthtxty
tuth
t
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49
System properties using the
impulse response
)(2
1)( with invertable is )(2)()(2)()2
)()(*)(
)((t)h with invertable is )(h(t)
)t-x(ty(t):delay Ideal )1
:Examples
0i0
0
tthtthtxty
tthth
tttt
i
i
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50
System properties using the
impulse response
A CT LTI system is memory less if and only if
)constant );()( (
)()(
0for 0)(
KtKxty
tKth
tth
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51
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum (DT)
DT properties using the impulse response
DT Systems characterized by Difference Equations
Summary
![Page 52: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/52.jpg)
52
LTI Systems: Differential
Equations LTI Systems can be described using the impulse
response h(t)
LTI systems can also be described
mathematically by a differential equation
A linear combination of a function and its derivatives
)()()()( 0012 txbtyatyatya
)(*)()()()( thtxdthxty
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53
LTI Systems: Differential Equations
General Nth-order linear constant-coefficient differential equation
Differential equations play a central role in describing input-output relationships in (electrical) systems
The general solution is given by: y(t) = yp(t) + yh(t)
yp(t) is a particular solution
yh(t) is the homogeneous solution satisfying
• To get yh(t), N auxiliary conditions are required
• Auxiliary conditions are the values of:
at some point in time
constants real , ;)()(
00
kkk
kN
k
kk
kN
k
k badt
txdb
dt
tyda
0)(
0k
kN
k
kdt
tyda
1
1 )(,,
)(),(
N
N
dt
tyd
dt
tdyty
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54
LTI Systems: Differential
Equation: Example
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55
LTI Systems: Differential
Equation: Example
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56
LTI Systems: Differential
Equation: Example
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57
LTI Systems: Differential
Equation: Example
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58
Outline
CT Systems
Impulse response
Convolution integral
Block Diagram of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum (DT)
Block Diagram of Systems
DT properties using the impulse response
DT Systems characterized by Difference Equations
Summary
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59
Impulse response of DT LTI
Systems
Linearity
Time Invariance
][][][][
][][then
][][ if
00 nnynnxnynx
nyany
nxanx
k
kk
k
kk
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60
Impulse Response of LTI
Systems
Once the response to a unit impulse is known, the response of any discrete-time LTI system to any arbitrary excitation can be found
Any arbitrary excitation is a sequence of amplitude-scaled and time-shifted DT impulses
Therefore the response is a sequence of amplitude-scaled and time-shifted DT impulse responses
][][
]1[]1[][]0[]1[]1[][
knkx
nxnxnxnx
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61
Impulse Response of LTI
systems
The impulse response h[n] completely
characterizes an LTI system
DT LTI Systems: Use the unit impulse to construct any signal
A DT signal is a sequence of individual weighted impulses
The response of the system is the sum of delayed h[n]
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62
Response of LTI Systems
][]}[{ nynxS
Snh
nhnSnxSny
nnx
of response Impulse :][
][]}[{]}[{][
][][ If
1
2
Question: if h[n] known, how to find y[n]?
y[n] = x[n]*x[n]
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63
Response of LTI Systems:
Example
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64
Response of LTI Systems:
Example
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65
Relationship: DT impulse response
and step response
In any DT LTI system let an excitation, x[n], produce the response, y[n]
Then the excitation x[n] - x[n - 1] will produce the response y[n] - y[n - 1]
It follows then that the unit impulse response is the first backward
difference of the unit step response and, conversely that the unit
sequence (step) response is the accumulation of the unit impulse
response
n
k
khns
nsnsnsnh
][][
response step theis ][ where]1[][][
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66
DT impulse response and step
response:Example
Suppose that the step response is given by
What is the impulse response h[n] ?
][5
445][ nuns
n
]1[5
44][
5
44][5
]1[5
445][
5
445
]1[][][
][
1
nunn
nunu
nsnsnh
n
n
n
nn
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67
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum
System properties using the impulse response
Systems characterized by Difference Equations
Summary
![Page 68: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/68.jpg)
68
Convolution of Two Signals
A signal x[n] can be represented as linear
combination of DELAYED Impulses
If the system is LINEAR
k
k
k
k
k
k
k
nhkxny
knykh
knny
nykxny
knkxnx
][][][
][ toresponse [n]][
][ tosystem theof response ][with
][][][
][][][
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69
Convolution of Two Signals
If the system is Time Invariant
n Convolutio ][][][
][][][
][][][
with
0 Omit the ]][]0[][ OR[
][][then
][ toresponse ][ if
00
0
nhnxny
knhkxny
nhkxny
nhnhnh
knhnh
knnh
k
k
k
k
k
![Page 70: Signals & Systems - Concordia Universityamer/teach/elec264/notes/topic2... · •A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 •M.J](https://reader031.vdocuments.site/reader031/viewer/2022012403/5ae630bd7f8b9ae1578d4c29/html5/thumbnails/70.jpg)
70
The Convolution Sum
The response, y[n], to an arbitrary excitation, x[n], is of
the form
h[n] is the impulse response
This can be written in a more compact form,
called the convolution sum
)1()1()()0()1()1()( nhxnhxnhxny
k
knhkxny ][][][
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71
Obtain the sequence h[n-k]
Reflecting h[k] about the origin to get h[-k]
Shifting the origin of the reflected sequence to k=n
Multiply x[k] and h[n-k] for
Sum the products to compute the output
sample y[n]
Computation of the convolution
sumk
knhkxnhnxny ][][][*][][
k
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7272
Convolution sum: graphical
steps
1) Sketch the waveform for input x[k] by changing the independent variable of
x[n] from n to k and keep the waveform for x[k] fixed during steps (2)-(7).
2) Sketch the waveform for the impulse response h[k] by changing the
independent variable from n to k.
3) Reflect h[k] about the vertical axis to obtain the time-inverted impulse
response h[-k].
4) Shift the sequence h[-k] by a selected value of n. the resulting function
represents h[n-k].
5) Multiply the input sequence x[k] by h[n-k] and plot the product function
x[k]h[n-k].
6) Calculate the summation .
7) Repeat steps (4)-(6) for -∞≤n≤+∞ to obtain the output response y[n] over all
time n.
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73
Forming the sequence h[n-k]
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74
Computing a discrete convolution:
Example 1
]1[][
]1[]1[]0[]0[
][][][
nxnx
nxhnxh
knxkhny
If the system LTI
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75
Computing a discrete convolution:
Example 2
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76
Convolution sum:
Example 3: infinite-duration signals
-„k‟ refers here to the “n” of the other
examples
-Compare to the CT equivalent example
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77
Convolution sum:
Example 4: finite-duration signals
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78
Convolution sum : Example 5
.1 ),1
1(
,10 ,1
1
,0 ,0
][
1
1
nNa
aa
Nna
a
n
ny
NNn
n
][][
otherwise. ,0
,10 ,1
][][][
nuanx
Nn
Nnununh
n
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79
Convolution Sum:
Example 6
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80
Convolution Sum:
Example 6
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81
Convolution Sum:
Example 6
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82
Convolution Sum:
Example 6
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Convolution Sum:
Example 7 Consider an LTI system with input x[n] and unit impulse h[n] response
shown. Find the output of this system
Solution: the output of the system
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84
Convolution Sum:
Analytical Example 8
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85
Convolution Sum:
Analytical Example 9
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Convolution Sum:
Analytical Example 9
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87
Convolution Sum:
Analytical Example 9
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88
Convolution Sum:
Analytical Example 9
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89
Convolution Sum:
Analytical Example 9
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90
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Difference Equations
Summary
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91
System Block Diagrams
LTI Systems can be described using the impulse
response h[n] which completely characterizes an LTI system
LTI systems can also be described
mathematically by difference equation
k
knhkxnhnxny ][][][*][][
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92
System Block Diagrams
A block diagram can be drawn directly from the difference equation which describes the system
For example, if the system is described by
It can also be described by
the block diagram below in
which “D” represents a delay
of one in discrete time
][]2[2]1[3][ nxnynyny
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93
Block Diagram Elements
Discrete-Time
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94
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Difference Equations
Summary
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95
System properties via
Convolution properties Commutative
Distributive or Linear
The distributive property implies that the
following two LTI systems are equivalent
][*][][*][ nxnhnhnx
][*][][*][])[][(*][ 2121 nhnxnhnxnhnhnx
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96
System properties via Convolution
properties: System Interconnections
Direct consequence of the distributivity property:
If two systems are excited by the same signal and their responses are added they are said to be parallel connected.
The parallel connection of two systems can be viewed as a single system whose impulse response is the sum of the two individual system impulse responses
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97
System properties via
Convolution properties
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98
System properties via Convolution
properties: System Interconnections
Direct consequence of the associativity property:
If the response of one system is the excitation of another system the two systems are said to be cascade connected
The cascade connection of two systems can be viewed as a single system whose impulse response is the convolution of the two individual system impulse responses
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99
System properties via
Convolution properties
The following properties can be proven from the
convolution definition:
3
])1[][(][]1[][
:property difference Backward
][][][][][
:propertyDelay
][][][
:property n"Convolutio"
000
00
nhnhnxnyny
nnAhnxnhnnAxnny
nnAxnnAnx
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100
Delay property: Example
][][][][][ 000 nhnnAxnnAhnxnny
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101
System properties via Convolution
properties: Example 1
))5(exp()5()5()(
:have weproperty,n convolutio By the
ANSWER
).5()( Compute
)exp()(Let
5 nanxnnx
nnx
nanx
n
n
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102
System properties via Convolution
properties: Example 2
Problem: a discrete-time LTI system has impulse response
Find the output y[n] due to input
x[n] = u[n + 1] – u[n - 1] + 2δ[n - 2],
where u[n] is the discrete time unit step function
Suggestions: Use convolution properties
Plot the functions of h[n] and x[n]
In other problems: you may be • Given y(t) = integral (..); find h(t) analytically or graphically
• Given x(t) and h(t) ; find y(t) analytically or graphically
• Pay attention that you may need to do variable substitution, e.g.,
integral(e^(t-p) h(p-5) dp) –inf to tp' = p-5 p=p'+5integral(e^(t-p'-5) h(p') dp') -inf to t-5
Solution: the simplest way to solve for the output y[n] would be to first plot the functions of h[n] and x[n]
]1[2][3][ nnnh
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103
System properties via Convolution sum
properties: Example 2
The sequence h[n] consists of two samples. Therefore, convolving x[n] and h[n] can be simplified by convolving
x[n] with h[n] one sample at a time.
For example, we can convolved x[n] first with and then with
Finally, the convolution sum (y[n]) can be then obtained by adding the two sequences (adding sample by corresponding sample).
In doing this, the output y[n] is
The same can be achieved graphically
]1[2][3][ nnnh
][3][1 nnh
]1[2][2 nnh
]3[4]2[6]1[][]1[3][ nnnnnny
k
nxhnxhknxkhny ....)1()1()0()0(][][][
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104
System Properties using impulse
response
It can be shown that a BIBO-stable DT system has an
impulse response that is absolutely summable
Proof
nkk
nhBkhknxkhknxny ][][][][][][
n
nh ][
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105
System properties using impulse
response
systems inverse are
]1[][][ : difference backward The
][][ :r accumulato The
:Example
nynynw
kxnyn
k
][][*][][*][ nnhnhnhnh ii
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System properties via
Convolution sum properties
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“Finite/Infinite” Systems: reflected
in h[n]
Depending on h[n], we divide LTI systems into
Finite-duration impulse response (FIR) systems
Infinite-duration impulse response (IIR) systems
]1[][][ nxnxny
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“Finite/Infinite” Systems: reflected
in h[n]
Finite-duration impulse response (FIR) system
The impulse response has only a finite number of nonzero samples
Ideal delay
Forward difference
Backward difference
integer positive a ],[][
],[][
dd
d
nnnnh
nnnxny
][]1[][
][]1[][
nnnh
nxnxny
]1[][][
]1[][][
nnnh
nxnxny
nd
0
-1
0
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109
“Finite/Infinite” Systems: reflected
in h[n]
Infinite-duration impulse response (IIR) system
The impulse response is infinitive in duration
Accumulator
Stability
FIR systems always are stable, if each of h[n] values is
finite in magnitude
IIR systems can be stable, e.g.
][][][
][][
nuknh
kxny
n
k
n
k 0
…
|)|1(1||
1|| with ][][
0aaS
anuanh
n
n
?
|][|n
nhS
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110
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Difference Equations
Summary
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111
LTI Systems: Difference
Equations An important class of LTI systems:
Input & output satisfy an Nth-order Linear Constant Coefficient Difference
Equations (LCCD) equation
An LTI System can be described by a difference
equation
recursive)-(non memoryless is System ,0 and 1 if
]}[{]}[{]}[{]}[{
0,1],[][
0
0 0
0
maa
nbnxnany
namnxbmnya
m
M
m
M
m
mm
M
m
m
N
k
k mnxbknya00
][][
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112
LTI System: Difference Equation
Example 1
]1[][]1[][
]1[][]1[][
101
101
nxbnxbnyany
nxbnxbnyany
A first order LTI system:
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LTI System: Difference Equation
Example 2
Difference equation representation of the accumulator
][]1[][
]1[][][][][
][]1[
][][
1
1
nxnyny
nynxkxnxny
kxny
kxny
n
k
n
k
n
k
M
m
m
N
k
k mnxbknya00
][][
+
One-sample
delay
x[n]
y[n-1]
y[n]
Recursive representation
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LTI Systems: Solving Difference
Equations: Example 3
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115
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Difference Equations
Summary
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116
Summary
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117
Summary
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118
Outline
CT Systems
Impulse response
Convolution integral
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Differential Equations
DT systems
DT Impulse response
Convolution sum
Block Diagrams of Systems
System properties using the impulse response
Systems characterized by Difference Equations
Summary
Appendix
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119
LTI Systems: Differential Equations
& impulse response
Let a CT system be described by
Let the excitation be a unit impulse at time, t = 0
Then the response, y, is the impulse response, h.
Since the impulse occurs at time, t = 0, and nothing has excited the system before that time, the impulse response before time, t = 0, is zero
After time, t = 0, the impulse has occurred and gone away
Therefore there is no excitation and the impulse response is the homogeneous solution of the differential equation
)()()()( 012 txtyatyatya
)()()()( 012 tthathatha
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120
LTI Systems: Differential Equations
& impulse response
What happens at time t = 0?
The equation must be satisfied at all times. So the left sideof the equation must be a unit impulse
We already know that the left side is zero before time, t = 0because the system has never been excited.
We know that the left side is zero after time, t = 0, becauseit is the solution of the homogeneous equation whose rightside is zero.
This is consistent with an impulse. The impulse responsemight have in it an impulse or derivatives of an impulsesince all of these occur only at time, t = 0.
What the impulse response does have in it depends on theequation.
)()()()( 012 tthathatha
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LTI Systems: Differential Equations
& impulse response