3 Fourier Series Representation of Periodic Signals
3.Fourier Series Representation of Periodic Signal
Jean Baptiste Joseph
Fourier,born in 1768,
in France.
1807,periodic signal
could be represented
by sinusoidal series.
1829,Dirichlet provided
precise conditions.
1960s,Cooley and Tukey
discovered fast Fourier
transform,
3 Fourier Series Representation of Periodic Signals
3.2 The Response of LTI Systems to Complex
Exponentials
(1) Continuous time LTI system
h(t)x(t)=e
st y(t)=H(s)est
)(
)()(
)()()(*)()(
)(
sHe
dehedhe
dhtxthtxty
st
sstts












dehsH s)()( ( system function )
3 Fourier Series Representation of Periodic Signals
(2) Discrete time LTI system
h[n]x[n]=z
n y[n]=H(z)zn
)(
][][
][][][*][][
)(
zHz
khzzkhz
khknxnhnxny
n
k
kn
k
kn
k









k
k
zkhzH?


][)( ( system function )
3 Fourier Series Representation of Periodic Signals
(3) Input as a combination of Complex Exponentials
Continuous time LTI system:

N
k
ts
kk
N
k
ts
k
k
k
esHaty
eatx
1
1
)()(
)(
Discrete time LTI system:

N
k
n
kkk
N
k
n
kk
zzHany
zanx
1
1
)(][
][
Example 3.1
3 Fourier Series Representation of Periodic Signals
3.3 Fourier Series Representation of Continuous-time
Periodic Signals
(1) General Form
2,1,0,)( )/2(0 keet tTjktjkk
3.3.1 Linear Combinations of Harmonically Related
Complex Exponentials
The set of harmonically related complex
exponentials:
Fundamental period,T ( common period )
3 Fourier Series Representation of Periodic Signals
So,arbitrary periodic signal can be represented as
tjtj ee 00,
,Fundamental components
tjtj ee 00 22,
,Second harmonic components
tjNtjN ee 00,
,Nth harmonic components


k
tjk
k eatx
0)(?
( Fourier series )
Example 3.2
3 Fourier Series Representation of Periodic Signals
(2) Representation for Real Signal
Real periodic signal,x(t)=x*(t)
So a*k=a-k


k
tjk
k eatx 0)(




1
0
1
0
]R e [2
][)(
0
00
k
tjk
k
tjk
k
k
tjk
k
eaa
eaeaatx

Let (A) )(
00,kk tkjktjkkjkk eAeaeAa


1
00 )c o s (2)(
k
kk tkAatx
3 Fourier Series Representation of Periodic Signals
Let (A) )(
00,kk tkjktjkkjkk eAeaeAa


1
00 )c o s (2)(
k
kk tkAatx
(B)
kkk jCBa
]s inc o s[2)( 0
1
00 tkCtkBatx
k
kk


3 Fourier Series Representation of Periodic Signals
2,1,0,)( )/2( ket tTjkk?
3.3.2 Determination of the Fourier Series
Representation of a Continuous-time Periodic
Signal


k
tjk
k eatx 0)(
( Orthogonal function set )
Determining the coefficient by orthogonality:
( Multiply two sides by )



k
tnkj
k
tjn eaetx 00 )()(
tjne 0
3 Fourier Series Representation of Periodic Signals
Fourier Series Representation:


nk
nkTdte
T
tnkj
,0
,
0)(?
Tadteadtetx k
k T
tnkj
kT
tjn

00 )()(
T tjnn dtetxTa 0)(1?



T
tjk
k
k
tjk
k
e qu at i onA n al y s i sdtetx
T
a
e qu at i onSy n t he s i seatx
)()(
1
)()(
0
0
3 Fourier Series Representation of Periodic Signals
Example 3.3 3.5
x(t)
ak
3 Fourier Series Representation of Periodic Signals
3.4 Convergence of the Fourier Series


N
Nk
tjk
kN eatx 0)(
Approximation error:
(1) Finite series






Nk
tjk
k
N
Nk
tjk
k
k
tjk
k
NN
eaeaea
txtxte
||
000
)()()(

If 0|)(|1lim 2
T NN dtteT
,then the series is
convergent,( xN(t)? x(t) )
3 Fourier Series Representation of Periodic Signals
Condition 1,Absolutely Integrable
(2) Dirichlet condition
T dttx |)(|
ttx
1)(?
3 Fourier Series Representation of Periodic Signals
Condition 2,Finite number of maxima and minima
during a single period
)2sin()( ttx
3 Fourier Series Representation of Periodic Signals
Condition 3,Finite number of discontinuity
3 Fourier Series Representation of Periodic Signals
1898,Albert Michelson,An American physicist
Constructed a harmonic analyzer
Observed truncated Fourier series
xN(t) looked very much like x(t)
Found a strange phenomenon
Josiah Gibbs,An American mathematical physicist
Given out a mathematical Explanation
(3) Gibbs phenomenon
3 Fourier Series Representation of Periodic Signals
3 Fourier Series Representation of Periodic Signals
Any continuity,
xN(t1)? x(t1)
Vicinity of discontinuity,
ripples
peak amplitude does not seem to decrease
Discontinuity,
overshoot 9%
Gibbs’s conclusion:
3 Fourier Series Representation of Periodic Signals
3.6.1 Linear Combination of Harmonically Related
Complex Exponentials
Periodic signal x[n] with period N:
x[n]=x[n+N]
3.6 Fourier Series Representation of Discrete-time
Periodic Signals
2,1,0,][ )/2(0 keen nNjknjkk
Discrete-time complex exponential orthogonal
signal set:
3 Fourier Series Representation of Periodic Signals



rkN
rk
enn
Nn
nrkj
Nn
rk
,
,0
][][)2( 0
)(*?
Property of orthogonal signal set:
][][)1( nn rNkk
3 Fourier Series Representation of Periodic Signals
3.6.2 Determination of the Fourier Series
Representation of Periodic Signals
Fourier series of periodic signal x[n]:
Determine the coefficients ak by orthogonality:



Nk
njk
k
Nk
nNjk
k eaeanx 0
)/2(][


Nk
nNrkj
k
nNjr eaenx )/2)(()/2(][
Na
ea
eaenx
r
Nk Nn
nNrkj
k
Nn Nk
nNrkj
k
Nn
nNjr






)/2)((
)/2)(()/2(
][

3 Fourier Series Representation of Periodic Signals
The equations of Fourier series:


Nn
nNjr
r enxNa
)/2(][1?


Nn
nNjk
k
Nk
nNjk
k
enx
N
a
eanx
)/2(
)/2(
][
1
][
( ak is periodic )
3 Fourier Series Representation of Periodic Signals
Example 3.10 3.12
x[n]
ak
3 Fourier Series Representation of Periodic Signals
(1) System function
3.8 Fourier Series and LTI System
Continuous time system:
Discrete-time system:
dehsH s)()(
k
k
zkhzH?


][)(
(2) Frequency response
Continuous time system:
Discrete-time system:
dtethjH tj )()(
nj
n
j enheH

][)(
3 Fourier Series Representation of Periodic Signals
(3) System response
Continuous time system:
Discrete-time system:



k
tjk
k
k
tjk
k
ejkHaty
eatx
0
0
)()(
)(
0


Nk
N
knj
N
kj
k
Nk
N
knj
k
eeHany
eanx

22
2
)(][
][
Example 3.16 3.17
3 Fourier Series Representation of Periodic Signals
3.9 Filtering
3.9.1 Frequency-shaping filters
Example1:
Equalizer
3 Fourier Series Representation of Periodic Signals
Example2,Image Filtering (Edge enhancement)
3 Fourier Series Representation of Periodic Signals
3.9.2 Frequency-selective filters
Several type of filter,
(1) Lowpass filter
(2) Highpass filter
(3) Bandpass filter
3 Fourier Series Representation of Periodic Signals
3 types of filter ( Continuous time )
3 Fourier Series Representation of Periodic Signals
3 types of filter ( Discrete time )
3 Fourier Series Representation of Periodic Signals
Problems:
3.1 3.13 3.15 3.34 3.35 3.43