Periodic Signals 31 ExponentialSinusoidal Signals as Building Blocks for Many Signals TimeFrequency Domain Basis Sets Time Domain Frequency Domain ID: 303295
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Slide1
3.0 Fourier Series Representation of
Periodic Signals
3.1 Exponential/Sinusoidal Signals as
Building Blocks for Many SignalsSlide2
Time/Frequency Domain Basis Sets
Time
Domain
Frequency Domain
Slide3
Signal Analysis
(
P.32
of 1.0)Slide4
Response of A Linear Time-invariant
System to An Exponential Signal
Initial Observation
if the input has a single frequency component, the output will be exactly the same single frequency component, except scaled by a constant
time-invariant
scaling property
0
tSlide5
Input/Output Relationship
Time
Domain
Frequency Domain
0
0
matrix
vectors
eigen
value
eigen
vector
Slide6
Response of A Linear Time-invariant System to An Exponential Signal
More Complete Analysis
continuous-timeSlide7
Response of A Linear Time-invariant System to An Exponential Signal
More Complete Analysis
continuous-time
Transfer Function
Frequency Response
: eigenfunction of any linear time-invariant
system
: eigenvalue associated with the eigenfunction
e
stSlide8
Response of A Linear Time-invariant System to An Exponential Signal
More Complete Analysis
discrete-time
Transfer Function
Frequency Response
eigenfunction, eigenvalueSlide9
System Characterization
Superposition Property
continuous-time
discrete-time
each frequency component never split to other frequency components, no convolution involved
desirable to decompose signals in terms of such
eigenfunctionsSlide10
3.2 Fourier Series Representation of
Continuous-time Periodic Signals
Fourier Series Representation
Harmonically related complex exponentials
all with period
T
T: fundamental periodSlide11
Harmonically Related Exponentials for Periodic Signals
[n]
[n]
(N)
(N)
integer multiples of
ω
0
‧
Discrete in frequency domainSlide12
Fourier Series Representation
Fourier Series
: j-th harmonic components
realSlide13
Real Signals
For orthogonal basis:
(unique representation)Slide14
Fourier Series Representation
Determination of
ak
Fourier series coefficients
dc componentSlide15
Not unit vector
orthogonal
(
分析
)Slide16
Fourier Series Representation
Vector Space Interpretation
vector space
could be a vector space
some special signals (not concerned here) may need to be excludedSlide17
Fourier Series Representation
Vector Space Interpretation
orthonormal basis
is a set of orthonormal basis expanding a vector space of periodic signals with period TSlide18
Fourier Series Representation
Vector Space Interpretation
Fourier SeriesSlide19
Fourier Series Representation
Completeness
IssueQuestion: Can all signals with period T be represented this way? Almost all signals concerned here can, with exceptions very often not importantSlide20
Fourier Series Representation
Convergence
Issueconsider a finite series
It can be shown
a
k
obtained above is exactly the value needed even for a finite seriesSlide21
Truncated DimensionsSlide22
Fourier Series Representation
Convergence
IssueIt can be shown
Slide23
Fourier Series Representation
Gibbs Phenomenon
the partial sum in the vicinity of the discontinuity exhibit ripples whose amplitude does not seem to decrease with increasing NSee Fig. 3.9, p.201 of textSlide24Slide25
Discontinuities in Signals
All basis signals are continuous, so converge to average valuesSlide26
Fourier Series Representation
Convergence
Issuex(t) has no discontinuitiesFourier series converges to x(t
) at every tx
(t) has finite number of discontinuities in each periodFourier series converges to
x
(t) at every
t except at the discontinuity points, at which the series converges to the average value for both sidesSlide27
Fourier Series Representation
Convergence
IssueDirichlet’s condition for Fourier series expansion(1) absolutely integrable, (2) finite number of maxima & minima in a period(3) finite number of discontinuities in a periodSlide28
3.3 Properties of Fourier Series
Linearity
Time Shift
phase shift linear in frequency with amplitude unchangedSlide29
Linearity
Slide30
Time ShiftSlide31
Time Reversal
the effect of sign change for
x
(
t
) and
a
k
are identical
Time Scaling
positive real number
periodic with period
T
/
α
and fundamental frequency
αω
0
a
k
unchanged, but
x
(
α
t
) and each harmonic component are differentSlide32
Time Reversal
unique representation for orthogonal basisSlide33
Time ScalingSlide34
Multiplication
ConjugationSlide35
Multiplication
Slide36
Conjugation
unique representationSlide37
Differentiation
Parseval’s
Relation
average power in the
k
-
th
harmonic component in a period
T
total average power in a period
T
but
Slide38
DifferentiationSlide39
3.4 Fourier Series Representation of
Discrete-time Periodic Signals
Fourier Series Representation
Harmonically related signal sets
all with period
only N distinct signals in the setSlide40
Harmonically Related Exponentials for Periodic Signals
[n]
[n]
(N)
(N)
integer multiples of
ω
0
‧
Discrete in frequency domain
(P. 11 of 3.0)Slide41
Continuous/Discrete Sinusoidals
(
P.36
of 1.0)Slide42
Exponential/Sinusoidal Signals
Harmonically related discrete-time signal sets
all with common period NThis is different from continuous case. Only
N distinct signals in this set.
(
P.42
of 1.0)Slide43
Orthogonal Basis
N different values in
N-dimensional vector space
(
合成
)
(
分析
)Slide44
repeat with period
N
Note: both
x
[
n
] and
a
k
are discrete, and periodic with period
N
, therefore summed over a period of
N
Fourier Series Representation
Fourier SeriesSlide45
Fourier Series Representation
Vector Space Interpretation
is a vector spaceSlide46
Fourier Series Representation
Vector Space Interpretation
a set of orthonormal basesSlide47
Fourier Series Representation
No
Convergence Issue, No Gibbs Phenomenon, No Discontinuityx[n] has only N parameters, represented by
N coefficientssum of N
terms gives the exact valueN
odd
– N even
See Fig. 3.18, P.220 of textSlide48Slide49
Properties
Primarily Parallel with those for continuous-time
Multiplication
periodic convolutionSlide50
Time Shift
Periodic Convolution
Slide51
Properties
First Difference
Parseval’s Relation
average power in a period
average power in a period for each harmonic componentSlide52
3.5 Application Example
System Characterization
y
[
n
],
y
(
t
)
h
[
n
],
h
(t)
x[
n], x(t)δ[n], δ
(t)
n
n
z
n
,
e
st
H(z)
z
n
, H(s)
e
st
, z=e
j
ω
, s=j
e
j
ω
n
,
e
j
ω
t
H(
e
j
ω
)
e
j
ω
n
, H(j
)e
j
ω
tSlide53
Superposition Property
Continuous-time
Discrete-time
H
(
j
), H(ej
ω) frequency response, or transfer functionSlide54
Filtering
modifying the amplitude/ phase of the different frequency components in a signal, including eliminating some frequency components entirely
frequency shaping, frequency selective Example 1
See Fig. 3.34, P.246 of textSlide55Slide56
Example 2
Filtering
See Fig. 3.36, P.248 of textSlide57Slide58
Examples
Example 3.5, p.193 of textSlide59
Examples
Example 3.5, p.193 of textSlide60
Examples
Example 3.5, p.193 of text
(a)
(b)
(c)Slide61
Examples
Example 3.8, p.208 of text
(a)
(b)
(c)Slide62
Examples
Example 3.8, p.208 of textSlide63
Example 3.17, p.230 of text
Examples
y
[
n
],
y
(
t
)
h
[
n
],
h
(t)
x
[n], x(t)
δ[n],
δ(t)
x[n]
y[n]
h[n]Slide64
.
.
.
Problem 3.66
, p.275 of textSlide65
Problem 3.70
, p.281 of text
2-dimensional signalsSlide66
Problem 3.70
, p.281 of text
2-dimensional signalsSlide67
Problem 3.70
, p.281 of text
2-dimensional signals