elec 264 review
Post on 04-Jun-2018
224 Views
Preview:
TRANSCRIPT
-
8/13/2019 Elec 264 Review
1/114
-
8/13/2019 Elec 264 Review
2/114
2
Mathematical Review
sincos
:relation'
.,|z|r
z,ofphaseorangletheisand
z,ofmagnitudetheis0rwhere
-:formPolar
numbers,realareyandxand1
,:forrrectangulaor
je
sEuler
z
rez
jwhere
jyxzCartesian
j
j
+=
=
>=
=
+=
-
8/13/2019 Elec 264 Review
3/114
3
Periodic Complex Exponential
& Sinusoidal Signal
tjte ootj o sincos
:signalssinusoidalofterms
inwrittenbecanperiod,lfundamentathe
withsignal,lexponentiacomplexThe
+=
-
8/13/2019 Elec 264 Review
4/114
4
tjjtjj
ooo ee
Aee
AtA
+=+22
)cos(
:lexponentiacomplex
periodicofin termswritten
becanperiod,lfundamentawithsignal,sinusoidalThe
Periodic Complex Exponential& Sinusoidal Signal
-
8/13/2019 Elec 264 Review
5/114
5
ondcycles/secorHertzinfrequencylfundamentatheisof
condradians/seinfrequencyangularlfundamentaisof2o
periodlfundamentatheis
})(
{.)sin(
})(
Re{.)cos(:followsIt
=
+=+
+=+
oT
with
tojeIMAtoAor
tojeAt
oA
Periodic Complex Exponential& Sinusoidal Signal
-
8/13/2019 Elec 264 Review
6/114
6
increasing ,
increases the rate of oscillation
hzfo 1000=
hzfo 2000=
)cos()( += tAtx o
o
o
-
8/13/2019 Elec 264 Review
7/114
7
The phase of the signal is the offset in the
displacement from a specified reference point at time t = 0 represents a "shift" from zero phase A change in is also referred to as a phase-shift For infinitely long sinusoids, a change in is the same as
a shift in time, such as a time-delay If x(t) is delayed (time-shifted) by 1/4 of its cycle T
whose "phase" is now -pi/2it has been shifted by pi/2
Phase shift
)cos( +tA o
Signals with different phase shift
compared to the bottom signal
-
8/13/2019 Elec 264 Review
8/114
8
Case: x(t) is a constant
x(t) is periodic with period T
for any positive value of TThus fundamental period is
undefinedOn other hand can define
fundamental frequency to bezero, i.e., constant signal (d.c)
has zero rate of oscillation
0=o
-
8/13/2019 Elec 264 Review
9/114
9
Periodic Complex Exponential
Play a central role in signals &
systems
Serve as building block for many
other signalsSets of harmonically related
complex exponential are periodicwith a common period To
-
8/13/2019 Elec 264 Review
10/114
10
Periodic Complex Exponential
A necessary condition for a complex
exponentialtj oe
to be periodic with
period To is 1=oTje
,...2,1,0,2Ti.e..2ofmultipleaisthatimplieswhich
o == kkTo
-
8/13/2019 Elec 264 Review
11/114
11
Harmonically Complex
Exponential Signals
.frequencypositive
singleaofmultiplesarethatsfrequencielfundamentawithlsexponentiaperiodic
ofsetaislexponentiacomplex
ofsetrelatedlyharmonical
.k...of
multipleintegeranbemustthen
,2
defineweIf
o
oo
o
A
ei
To
=
=
-
8/13/2019 Elec 264 Review
12/114
12
Harmonically ComplexExponential Signals
|||k|
2
periodlfundamentaand|k|sfrequencie
lfundamentawithperiodicis)(0,k
constant.ais)(0,kFor2,...1,0,k,)(
o
o
k
T
t
tet
o
k
k
tjk
k o
=
=
==
-
8/13/2019 Elec 264 Review
13/114
13
Harmonically Complex
Exponential Signals
.Tlengthofintervalany time
duringperiodslfundamentaitsof|k|exactly
throughgoesitaswell,asTperiodwith
periodicstillis)(harmonickthThe
o
o
tk
-
8/13/2019 Elec 264 Review
14/114
14
General Complex Exponential
Signals
.|||C|
)(Then
.jraand|C|C
:formcartesianina
andformpolarinexpressedisCIf
numbers.complexarea''andC''bothwhere,)(
)()(
o
++==
=
+==
=
tjrttjrj
at
j
at
oo eeCee
Cetx
e
Cetx
-
8/13/2019 Elec 264 Review
15/114
15
General Complex Exponential
Signals
l.exponentiadecayingbymultipledsignalssinusoidalisx(t)0,rIf
l.exponentiagrowingbymultipledsignalssinusoidalisx(t)
0,rIf
.sinusoidalarepartsimaginary&realthe
0,rIf
)sin(||)cos(||)(
relation,sEuler'Using
=
+++== teCjteCCetx ort
o
rtat
-
8/13/2019 Elec 264 Review
16/114
16
r>0
r
-
8/13/2019 Elec 264 Review
17/114
17
pi=3.142;t=-10:.1:10;
f=2000;
w=2*pi*f;
r=0.1;x=zeros(size(t));
x=exp((r+w*i)*t);
theta=pi/4;
c=1*exp(i*theta);
y=c*x;
subplot(2,1,1);
plot(t,y);grid;
r=-0.1;x=zeros(size(t));
x=exp((r+w*i)*t);
theta=pi/4;
c=1*exp(i*theta);
y=c*x;
subplot(2,1,2);
plot(t,y);grid;
end;
Matlab Program for
Growing & Decaying Sinusoids
-
8/13/2019 Elec 264 Review
18/114
18
Decaying or
Damped Sinusoids
Examples:Response of RLC circuits.
Mechanical systems having bothdamping & restoring forces e.g.
automotive suspension system.
-
8/13/2019 Elec 264 Review
19/114
19
Impulse Function
time
volt
-
8/13/2019 Elec 264 Review
20/114
20
Delayed Impulse
-
8/13/2019 Elec 264 Review
21/114
-
8/13/2019 Elec 264 Review
22/114
22
Summation Operation of x[n]
-
8/13/2019 Elec 264 Review
23/114
23
Discrete-Time Complex
Exponential Sequence
sequence.time-discrete
with thedealingwhenperferredisformformer
the,previouslydescribedhavewesignallexponentia
time-continuousthesimilar toisformhisAlthough t
.where,][-:formfollowing
in thesequencetheexpresscanely weAlternativ
numbers.complexgeneralinareandCwhere
,][
eCenx
Cnx
n
n
==
=
-
8/13/2019 Elec 264 Review
24/114
24
Real Exponential Signalnnxge 1.1*2][.. =1where*][ >= nCnx
-
8/13/2019 Elec 264 Review
25/114
25
Real Exponential Signal10where*][
-
8/13/2019 Elec 264 Review
26/114
26
Real Exponential Signal01-where*][
-
8/13/2019 Elec 264 Review
27/114
27
Real Exponential Signal1where*][
-
8/13/2019 Elec 264 Review
28/114
-
8/13/2019 Elec 264 Review
29/114
29
Real Exponential Signal
Real-valued discrete exponentials
are used to describe:1) Population growth as function of
generation2) Total return on investment as a
function of day, month or
quarter
-
8/13/2019 Elec 264 Review
30/114
30
Sinusoidal Signals
njjnjj
o
oo
nj
o
o
nj
o
n
oo
o
o
ee
A
ee
A
nA
njne
nAnx
enx
Cenx
+=+
+=
+=
=
===
22)cos(
sincos-:relationsEuler'From
radians.ofunitshaveand
boththeness,dimensionlasnTaking
).cos(][
:signalsinusoidaltorelatedcloselyissignalThis
.][
number.imaginaryanpurelybej&1Clet,][
-
8/13/2019 Elec 264 Review
31/114
31
Discrete-time SinusoidalSignals
)12/2cos(][ nnx =
)31/8cos(][ nnx =
)6/cos(][ nnx =
-
8/13/2019 Elec 264 Review
32/114
32
Discrete-time Sinusoidal
Signals
These discrete-time signals possessed:
1) Infinite total energy2) Finite average power
-
8/13/2019 Elec 264 Review
33/114
33
General Complex Exponential
Signals
lly.exponentiagrowingsinusoidal1,|a|
lly,exponentiadecayingsinusoidal1,|a|
.sinusoidalareparts&,1||
)sin(||||)cos(||||][
.e||,e|C|C
:formpolarinandCWriting
signals.sinusoidaland
lsexponentiarealofin termsdinterpretebecan
lexponentiacomplextime-discretegeneralThe
ojj
>
1
+
-
8/13/2019 Elec 264 Review
41/114
41
Discrete-time Sinusoidal
Signals
12
1
2
,12/2becauseperiodic
)12/2cos(][
o
o
=
=
=
nnx
31
4
2,31/8becauseperiodic
)31/8cos(][o
o ==
=
nnx
numberrational2
,6/1becauseperiodicnot)6/cos(][
o
o
==
nnx
-
8/13/2019 Elec 264 Review
42/114
42
Fundamental Period & Frequency of
discrete-time complex exponential
)2
m(N
-:aswrittenisperiodlfundamentaThe
,N
2isfrequencylfundamentaIts
N,periodlfundamentawithperiodicisex[n]If
o
jo
=
=
=
m
o
n
-
8/13/2019 Elec 264 Review
43/114
43
nt oo jj eandesignaltheofComparison
.ofvalues
distinctforsignalsDistinct
ee
o
jj oo
nt
.ofchoiceanyforPeriodic o
ofrequencylFundamenta
o
undefined
2
:0
:0
periodlFundamenta
o
o
=
2ofmultiplesbyseparated
offor valuessignalsIdentical o
mand0Nintegerssomefor
,
2
ifonlyPeriodic o
>
= N
m
m
frequencylFundamenta 0
)
2
(:0
:0
periodlFundamenta
o
o
om
undefined
=
Harmonically related
-
8/13/2019 Elec 264 Review
44/114
44
Harmonically related
periodic exponential sequence
Considering periodic exponentialswith common period N samples:
1,...0,kfor,][ )/2( == nNjkk en
This set of signals possessfrequencies which are multiples of
N/2
Harmonically related
-
8/13/2019 Elec 264 Review
45/114
45
Harmonically related
periodic exponential sequence
In continuous-time case
][e][
2/N)njk(2
)/2)((
neen
k
nj
nNNkj
Nk
==
=
+
+
tTjk
e)/2(
are all distinct signals for ,....2,1,0 =k
Harmonically related
-
8/13/2019 Elec 264 Review
46/114
46
Harmonically related
periodic exponential sequence
Therefore , there are only N distinct
periodic exponentials in the discrete
harmonic sequences.
)][][(e.g.above.theof
onetoidenticalis][otherAny
][.......
,][,][,1][
0N
k
/)1(2
1
/42
/21
nn
n
en
enenn
NnNj
N
NnjNnjo
=
=
===
-
8/13/2019 Elec 264 Review
47/114
47
o Mathematical review
o Transformation of the independent variable
o LTI Signals & Systems
o Convolutiono Frequency representation of signals
o Fourier Series
o Fourier Transform
ELEC264: Signals And Systems
Review
Dr. Aishy AmerConcordia 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
http://metalab.uniten.edu.my/~zainul/images/Signals&Systems/
T f i f
-
8/13/2019 Elec 264 Review
48/114
48
Transformation ofIndependent Variable
Aircraft control system:
Input correspond to pilot action theseaction are transformed by electrical &mechanical system of the aircraft tochanges to aircraft trust or positioncontrol surfaces such as the rudder& ailerons
finally these changes affect thedynamics & kinematics such as the
aircraft velocity and heading
M difi i f i d d
-
8/13/2019 Elec 264 Review
49/114
49
Modification of independentvariable (time axes)
Introducing several basic properties of signals& systems through elementarytransformations.
Examples of elementary transformation:- time shift, x(t-t0), x[n-n0]
time reversal, x(-t), x[-n].
time scaling, x(0.5t), x[2n].
and combinations of these. x(at+b), x[an-b],where a & b are signed constants*.
Shifti i ht l i
-
8/13/2019 Elec 264 Review
50/114
50
Shifting right or laggingsignal x(t)
X(t)
tX(t-t0)
0
0 t
t0
is a positive value
Shifti l ft l di i l
-
8/13/2019 Elec 264 Review
51/114
51
Shifting left or leading signalx(t)
X(t)
tX(t+t1)
0
0 t
-t1
Folded or Flipped x(t)
-
8/13/2019 Elec 264 Review
52/114
52
Folded or Flipped x(t)=x(-t), time reversal
Signal Flip about y axes
-
8/13/2019 Elec 264 Review
53/114
53
Signal Flip about y- axes
x[-n], time reversal
-
8/13/2019 Elec 264 Review
54/114
-
8/13/2019 Elec 264 Review
55/114
55
Examples
x[n]
x[n-5]
-
8/13/2019 Elec 264 Review
56/114
56
Examples
x[n]
x[n+5]
-
8/13/2019 Elec 264 Review
57/114
-
8/13/2019 Elec 264 Review
58/114
58
Example 1.1x(t)
x(t+1), x(t) shifted left by 1sect
t
1
1
0
0
12
1 2-1
Tables of x(t) & x(t+1) & x(
-
8/13/2019 Elec 264 Review
59/114
59
Tables of x(t) & x(t+1) & x(-
t+1)
t x(t) x(t+1) x(-t+1)-2 0 0 0
-1 0 1 0
0 1 1 11 1 0 1
2 0 0 0
3 0 0 0
-
8/13/2019 Elec 264 Review
60/114
60
Example 1.1x(t+1) is x(t) shifted left by 1
x(-t+1) is x(t+1) flipped about t=0t
t
1
1
0
0
12
1 2-1
-1
-
8/13/2019 Elec 264 Review
61/114
61
Example 1.1 Alternative 1x(t-1) is x(t) shifted right by 1se
x(-t+1)=x(-1(t-1))
Flip about axis t=1
t
t
1
1
0
0
12
1 2-1
-
8/13/2019 Elec 264 Review
62/114
62
Example 1.1, Method 2x(-t), flip about axis t=0
x(-t+1), shift right (because -t) by 1t
t
1
1
0
0
1 2
1 2-1
-1
-
8/13/2019 Elec 264 Review
63/114
63
Example 1.1x(t)
x(t+1), x(t) shifted left by 1sect
t
1
1
0
0
12
1 2-1
-
8/13/2019 Elec 264 Review
64/114
-
8/13/2019 Elec 264 Review
65/114
65
o Mathematical review
o Transformation of the independent variable
o LTI Signals & Systems
o Convolutiono Frequency representation of signals
o Fourier Series
o Fourier Transform
ELEC264: Signals And Systems
Review
Dr. Aishy AmerConcordia 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
http://metalab.uniten.edu.my/~zainul/images/Signals&Systems/
Topic 1:
-
8/13/2019 Elec 264 Review
66/114
66
Topic 1:
LTI Signals & Systems
Topic 1:
-
8/13/2019 Elec 264 Review
67/114
67
Topic 1:
LTI Signals & Systems
Topic 1:
-
8/13/2019 Elec 264 Review
68/114
68
op c
LTI Signals & Systems
Topic 1:
-
8/13/2019 Elec 264 Review
69/114
69
p
LTI Signals & Systems
-
8/13/2019 Elec 264 Review
70/114
70
o Mathematical review
o Transformation of the independent variable
o LTI Signals & Systems
o Convolutiono Frequency representation of signals
o Fourier Series
o Fourier Transform
ELEC264: Signals And Systems
Review
Dr. Aishy AmerConcordia 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
http://metalab.uniten.edu.my/~zainul/images/Signals&Systems/
-
8/13/2019 Elec 264 Review
71/114
71
Topic 2: Convolution
-
8/13/2019 Elec 264 Review
72/114
72
Topic 2: Convolution
Types of Systems
Time-invariant, linear, causal, memory, stable,invertible Need to be able to prove whether a system is linear
and time-invariant
Convolution Sum Know how to compute impulse response h[n]
Convolution Integral Graphical Convolution
Convolution Properties Stability and Impulse Response
-
8/13/2019 Elec 264 Review
73/114
73
Topic 2: Convolution
-
8/13/2019 Elec 264 Review
74/114
74
Topic 2: Convolution
Convolution:
1. Write down equations for both signals2. Reflect and shift one of the two signals (preferably the finite
duration signal)3. Label the edge(s) of the reflected/shifted signal with
t and/or t-a appropriately4. Shift signal to the left until there is no overlap between the two
signals5. Identify number of different cases in which there is overlap
between the two signals
6. For each of the above cases, evaluate the convolution integralwith the limits of integration determined by the region ofoverlap expressed as a function of t ( see step 3)
-
8/13/2019 Elec 264 Review
75/114
75
Topic 2: Convolution
-
8/13/2019 Elec 264 Review
76/114
76
Topic 2: Convolution
-
8/13/2019 Elec 264 Review
77/114
77
o Mathematical review
o Transformation of the independent variable
o LTI Signals & Systems
o
Convolutiono Frequency representation of signals
o Fourier Series
o Fourier Transform
ELEC264: Signals And Systems
Review
Dr. Aishy AmerConcordia 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
http://metalab.uniten.edu.my/~zainul/images/Signals&Systems/
-
8/13/2019 Elec 264 Review
78/114
-
8/13/2019 Elec 264 Review
79/114
79
Waveform Representation
Waveform representation
Plot of the signal value vs. time
Sound amplitude, temperaturereading, stock price, ..
Mathematical representation: x(t)
x: variable value
T: independent variable
-
8/13/2019 Elec 264 Review
80/114
80
Sample Speech Waveform
-
8/13/2019 Elec 264 Review
81/114
81
Sample Music Waveform
-
8/13/2019 Elec 264 Review
82/114
82
Sinusoidal Signals
Sinusoidal signals: important because they can be used to synthesize any signal An arbitrary signal can be expressed as a sum of many sinusoidal signals with
different frequencies, amplitudes and phases
Phase shift: how much the max. of the sinusoidal signal is shifted away from t=0 Music notes are essentially sinusoids at different frequencies
Complex Exponential
-
8/13/2019 Elec 264 Review
83/114
83
Signals
)sin(2)cos(2
:ForumlaEuler
)()(:Note:ConjugateComplex
phase)(initialshiftphasetheis
)sin()cos(||)(
:SignallExponentiaComplex
sincos:tionrepresentaPolar
taniszofPhase
||iszofMagnitude
:tionrepresentaCartesian
:NumberComplex
***
00
)(
1
22
0
jeeee
realarezzandzzjbaz
tzjtzeztx
zjzezz
a
bz
baz
jbaz
tj
j
==+
+=
+++==
+==
==
+=
+=
+
o
o
o
o
What is frequency of an
-
8/13/2019 Elec 264 Review
84/114
84
arbitrary signal?
Sinusoidal signals have a distinct (unique) frequency
An arbitrary signal x(t) does not have a unique frequency x(t) can be decomposed into many sinusoidal signals
with different frequencies, each with different magnitudeand phase
Spectrumof x(t): the plot of the magnitudes and phasesof different frequency components
Fourier analysis: find spectrum for signals
Bandwidthof x(t): the spread of the frequencycomponents with significant energy existing in a signal
-
8/13/2019 Elec 264 Review
85/114
85
Frequency content in signals
-
8/13/2019 Elec 264 Review
86/114
86
Frequency content in signals A constant : only zero frequency component (DC
component)
A sinusoid : Contain only a single frequency component Periodic signals : Contain the fundamental frequency and
harmonics : Line spectrum Slowly varying : contain low frequency only
Fast varying : contain very high frequency Sharp transition : contain from low to high frequency Music: :
contain both slowly varying and fast varyingcomponents, wide bandwidth
: increasing , increases
h f ill i
)cos( +tA o o
-
8/13/2019 Elec 264 Review
87/114
87
the rate of oscillation
hzfo 1000=
hzfo 2000=
o
S l
-
8/13/2019 Elec 264 Review
88/114
88
Spectrum analyzer
x1(t) & x2(t) look similar; A spectrum analyzer reveals the difference:x2(t) contains a sinusoid that causes the two large spikes
Fourier Analysis: Types of signals
-
8/13/2019 Elec 264 Review
89/114
89
Fourier Analysis: Types of signals
-
8/13/2019 Elec 264 Review
90/114
90
Fourier Analysis: Types of
i l
-
8/13/2019 Elec 264 Review
91/114
91
signals
All continuous signals are
CT but not all CT signalsare continuous
Fo rier Anal sis
-
8/13/2019 Elec 264 Review
92/114
92
Fourier Analysis
Discrete inTime
Periodic inFrequency
Continuousin Time
Aperiodic in
Frequency
Aperiodic in Time
Continuous in Frequency
Periodic in TimeDiscrete in Frequency
=
=
=
k
tjkk
T
tjk
k
eatx
dtetxT
a
0
0
)(
P-CTDT:SeriesFourierInverseCT
)(1
DTP-CT:SeriesFourierCT
T
0
T
=
+
=
+
=
2
2
2
)(2
1][
DTPCT:TransformFourierDTInverse
][)(
PCTDT:TransformFourierDT
deeXnx
enxeX
njj
n
njj
=
=
=
=
1
0
NN
1
0
NN
0
0
][1
][
P-DTP-DTSeriesFourierDTInverse
][][
P-DTP-DTSeriesFourierDT
N
k
knj
N
n
knj
ekXN
nx
enxkX
=
=
dejXtx
dtetxjX
tj
tj
)(2
1)(
CTCT:TransformFourierCTInverse
)()(
CTCT:TransformFourierCT
Relations Among Fourier
Methods
-
8/13/2019 Elec 264 Review
93/114
93
Methods
ELEC264: Signals And Systems
-
8/13/2019 Elec 264 Review
94/114
94
o Mathematical review
o Transformation of the independent variable
o LTI Signals & Systems
o Convolution
o Frequency representation of signals
o Fourier Series
o Fourier Transform
Review
Dr. Aishy AmerConcordia 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
http://metalab.uniten.edu.my/~zainul/images/Signals&Systems/
Topic 3: Fourier Series
-
8/13/2019 Elec 264 Review
95/114
95
A Fourier Series/transform is unique,
i.e., no two same signals in time givethe same function in frequency
Topic 3: Fourier Series
-
8/13/2019 Elec 264 Review
96/114
96
Topic 3: Fourier Series
Topic 3: Fourier Series
-
8/13/2019 Elec 264 Review
97/114
97
Topic 3: Fourier Series Fourier Series:
Exponential Fourier series
Trigonometric Fourier Series
=
=
==
k
tkfj
T
tkfj
ekXtx
TfdtetxT
kX
0
0
2
0
2
)()(
)1(/1)(1)(
=
=
++=
=
==
==
1
0
1
00
0
)2sin()2cos()(
)0(
)(thatnote)}(Im{2
)2()(thatnote)}(Re{2
k
k
k
k
ksk
kck
tkfbtkfaatx
Xa
bkXkXb
akXkXa
Topic 3: Fourier Series
-
8/13/2019 Elec 264 Review
98/114
98
Topic 3: Fourier Series Fourier Series
Symmetries of signal (even and odd) Even signal: X(k) is purely real
Odd signal: X(k) is purely imaginary
Properties of Fourier series Shifting property
)()( 002
0 kXettx tkfj
-
8/13/2019 Elec 264 Review
99/114
-
8/13/2019 Elec 264 Review
100/114
Topic 3:
Fourier Series
-
8/13/2019 Elec 264 Review
101/114
101
Fourier Series
-
8/13/2019 Elec 264 Review
102/114
Topic 4: Fourier Transform
-
8/13/2019 Elec 264 Review
103/114
103
Topic 4: Fourier Transform
Topic 4: Fourier Transform
-
8/13/2019 Elec 264 Review
104/114
104
Topic 4: Fourier Transform Fourier transform for aperiodic signals
Understand the meaning of the inverse Fouriertransform
Sketch the spectrum
Determine the bandwidth of the signal from itsspectrum
Know how to interpret a spectrogram plot
Topic 4: Fourier Transform
-
8/13/2019 Elec 264 Review
105/114
105
Topic 4: Fourier Transform
Topic 4: Fourier Transform
-
8/13/2019 Elec 264 Review
106/114
106
Topic 4: Fourier Transform
Topic 4: Fourier Transform
-
8/13/2019 Elec 264 Review
107/114
107
Topic 4: Fourier Transform
-
8/13/2019 Elec 264 Review
108/114
108
Topic 4: Fourier Transform
-
8/13/2019 Elec 264 Review
109/114
109
Fourier Series versus
Fourier Transform
-
8/13/2019 Elec 264 Review
110/114
110
Fourier Transform Fourier Series (FS): a representation of periodic
signals as a linear combination of complex
exponentials Fourier Transform (FT): apply to signals that are not
periodic
Aperiodic signals can be viewed as a periodic signal
with an infinite period The CT Fourier Series is a good analysis tool for
systems with periodic excitation but the CT FourierSeries cannot represent an aperiodic signal for all time
The CT Fourier transform can represent an aperiodicsignal for all time
DTFT: Summary
-
8/13/2019 Elec 264 Review
111/114
111
DT Fourier Transform represents a discrete
time aperiodic signal as a sum of infinitelymany complex exponentials, with the
frequency varying continuously in (-, )
DTFT is periodic only need to determine it for
-
8/13/2019 Elec 264 Review
112/114
DT-FT Summary: a quiz
-
8/13/2019 Elec 264 Review
113/114
113
A discrete-time LTI system has impulse response
Find the output y[n] due to input
(Suggestion: work with and using the convolution property)
Solution This can be solved using convolution of h[n] and x[n]. However, the point was to use the convolution in time
multiplication in frequency property.
Therefore, It can be readily shown that
Therefore,
][2
1][ nunh
n
=
][7
1][ nunx
n
=
)()()(][*][][ jjj eXeHeYnxnhny ==
( ) 1,1
1)(][][
top related