1
Chapter 3
Fourier Series Representations
of Periodic Signals
2
Chapter 3 Fourier Series
§ 3.2 The Response of LTI Systems to Complex Exponentials
LTI 系统对复指数信号的响应
tyste
th
1,Continuous-time system
stH s h t e d t
Eigenfunction
特征函数
—— Eigenvalue (特征值）
nh
nynz
2,Discrete-time system
Eigenfunction
特征函数
—— Eigenvalue (特征值） n
n
znhzH?



ste stesH
nz nzzH
3
Chapter 3 Fourier Series
Example 3.1
Consider an LTI system,3 tth?
3y t x t
2 1 jtx t e?
23 3jty t e x t
2 c o s 4 c o s 7x t t t
c o s 4 3 c o s 7 3 3y t t t x t
4
Chapter 3 Fourier Series
§ 3.3 Fourier Series Representation（傅立叶级数）
of Continuous-time Periodic Signals
§ 3.3.1 Linear Combinations （线性组合）
of Harmonically Related Complex Exponentials
tjkk
k
eatx 0


—— Fourier Series
ka —— Fourier Series Coefficients
Spectral Coefficients （频谱系数）
5
Chapter 3 Fourier Series
Example 3.2
tjkk
k
eatx 2
3
3






3/1,2/1
4/1,1
32
10
aa
aa
Example,
Consider an LTI system for which the input
and the impulse response determine the output
ttx?2c o s211
tueth tty
22
11
441
1 2 1 2
j t j ty t e e
jj




6
Chapter 3 Fourier Series
§ 3.3.2 Determination of Fourier Series Representation
tjkk
k
eatx 0


0
00
1 jk t
k Ta x t e d tT



Synthesis equation
综合公式
Analysis equation
分析公式
ka —— Fourier Series Coefficients
Spectral Coefficients
7
Chapter 3 Fourier Series
Example 3.5 Periodic square wave defined over one period as




2/ t T 0
t 1
1
1
T
T
tx
1
tx
-T -T/2 –T1 0 T1 T/2 T t

dttxTa TT 2/ /2- 0 1
Defining
x
xxc s i ns i n?
101 s i n2 TkcTTa k
01s in 0
k
kTak
k

T
T12?
8
Chapter 3 Fourier Series
1 1s in?TTc
21 1s in?TTc
31 1s in?TTc
T1固定,的包络 固定kTa11 s in2 TcT?
9
Chapter 3 Fourier Series
TT /20 谱线变密
Figure 4.2
14 a TT?
18 b TT?
116 c TT?
10
Chapter 3 Fourier Series
Example Periodic Impulse Trains (周期冲激串 )
tx
tT? 0 T T2T2?
1

21 j k tT
k
x t eT



ka
T
1

-ω0 0 ω0 2ω0
2 2/ T/T,2,1,0 1 k
Ta k
212 j k tT
k
y t H j k eTT






k
x t t k T



11
Chapter 3 Fourier Series
§ 3.4 Convergence（收敛） of the Fourier Series
1,Approximation(近似性 )
tjkkN
Nk
N eatx
0

txtxte NN —— Error
1 NEN
dtetxTaa tjkTkk 01? 2
dtteE NTN 2
EN最小
0 N NE
dtteteT NN
12
ka
Chapter 3 Fourier Series
2,Dirichlet Conditions:
Condition 1
dttxT
1T,1t0,/1 ttx
13
Chapter 3 Fourier Series
Condition 2.
In any finite interval,is of bounded variation.tx
1T,1t0,/2s i n ttx?
14
Chapter 3 Fourier Series
Condition 3.
In any finite interval,there are only a finite number
of discontinuities.
15
Chapter 3 Fourier Series
Gibbs Phenomenon:
Figure 3.9
16
Chapter 3 Fourier Series
§ 3.5 Properties of Continuous-Time Fourier Series
§ 3.5.1 Linearity
kkx t a y t bFS FS
k k kz t A x t B y t c A a B bFS
§ 3.5.2 Time Shifting
kx t aFS
000 j k tkx t t a eFS
17
txtx
§ 3.5.3 Conjugation and Conjugate Symmetry
（共轭及共轭对称性）
kk atxatx FSFS
Chapter 3 Fourier Series
kk aatxtx

kk aao r
kkaa kkaa
00
1
a 2 c o skk
k
x t a A k t


0 0 0
1
b 2 c o s s i nkk
k
x t a B k t C k t


18
Chapter 3 Fourier Series
§ 3.5.4 Time Reversal
kx t aFS kx t aFS
real eventx
ka
real even
x t x t kkaa
real oddtx
ka
Purely imaginary
odd
x t x t kkaa
19
Chapter 3 Fourier Series
§ 3.5.5 Time Scaling
tjkk
k
eatx 0


0j k a tk
k
x a t a e?



0?ka k0?aka k?
§ 3.5.6 Multiplication（相乘）
kk btyatx FSFS
tytx FS
k m k m
m
h a b


Convolution
Su
20
Average Power
of kth harmonic
Average Power
oftx
Chapter 3 Fourier Series
§ 3.5.7 Parseval’s Relation（帕兹瓦尔关系式）
221 k
kT
adttxT


§ 3.5.8 Differential Property
katx FS
knn ajktx 0FS
kajktx 0FS
21
Chapter 3 Fourier Series
Example 3.6tg
-2 -1 0 1 2 t
2
1?
2
1
1 kctx
FS
-4 -2 0 2 4 t

Example 3.7
22
Chapter 3 Fourier Series
§ 3.6 Fourier Series Representation of
Discrete-Time Periodic Signals
§ 3.6.1 Linear Combinations of Harmonically
Related Complex Exponentials
nNjkk
Nk
njk
k
Nk
eaeanx
2
0


Fourier
Series
ka —— Fourier Series Coefficients
Spectral Coefficients
njkk
Nk
eanx 0

njkjkk
Nk
eeHa 00

23
Chapter 3 Fourier Series
§ 3.6.2 Determination of the Fourier Series Representations
nNjk
Nn
njk
Nn
k enxNenxNa
211
0



nNjkk
Nk
njk
k
Nk
eaeanx
2
0


Synthesis
Equation
Analysis
Equation
Discrete-time Fourier Series
k N kaa
The discrete-time series is a finite series with N terms.
24
Chapter 3 Fourier Series
Example 3.10 nnx
0s i n
0 1a / 2 N
0b / 2 MN
-6 -1 4
-4 1 6 k
21j
21j?
5N?ka

-3 2 7
-2 3 8 k
21j
21j?
5N,3Mka

0
2
N

0 2
M
N
25
Chapter 3 Fourier Series
Example 3.12 Discrete-time periodic square wave
n1N? 0 N
1
1N

N?
nNjkN
Nn
k eNa
21
1
1



mNk
e
ee
N
N
jk
N
N
jkN
N
jk

1
1
2
1
22
11




N
jk
N
jk
N
jk
N
N
jkN
N
jk
N
jk
eee
eee
N

2/1
2
2/1
2
11
1


mNkNN 12 1


ka
mNkNN 12 1

N /s i n
/2/12s i n1 1 mk
Nk
NNk
N?
26
Chapter 3 Fourier Series
§ 3.7 Properties of Discrete-time Fourier Series（不要求）
§ 3.8 Fourier Series and LTI Systems
tjkk
k
eatx 0


Linear Combinations
of Eigenfunctions
Frequency Response
of LTI System dtethjH tj
tjkk
k
ejkHaty 00


Periodic Signal
Continuous-time LTI System
27
Chapter 3 Fourier Series
njkk
Nk
eanx 0

Linear Combinations
of Eigenfunctions
Frequency Response
of LTI System nj
n
j enheH



Discrete-time LTI System
njkjkk
Nk
eeHany 00

Periodic Signal
28
Chapter 3 Fourier Series
Example
Consider an LTI system with input
the unit impulse response,determine the
Fourier Series Representation of output
nttx n
n


1
ty
tueth t4
j k t j k tkk
kk
y t b e a H j k e





kb
0 k is even
4
1
jk
k is odd
29
Chapter 3 Fourier Series
§ 3.9 Filtering（滤波）
th tjk
k
k
ejkHaty 00


tjkk
k
eatx 0


dtethjH tj
Filter
Frequency-Shaping Filter
频率成形滤波器
Frequency-Selective Filter
频率选择性滤波器
30
1,Equalizer（均衡器）
Chapter 3 Fourier Series
§ 3.9.1 Frequency-Shaping Filter
31
Chapter 3 Fourier Series
2,Differentiator （微分器）
Dtx
dt
tdx
tuth 1?
H j j
0
jH s g n
2 jH
2

2
32
Chapter 3 Fourier Series
Figure 3.24
33
Chapter 3 Fourier Series
3,Discrete-time system
121 nxnxny
/2 c o s / 2jjH e e
2/ jeH j
2

2

2/c o sjeH

1
0
34
221 k
kT
adttxT


katx FS
knn ajktx 0FS
kajktx 0FS
Chapter 3 Fourier Series
real oddtx
ka
Purely imaginary
odd
real eventx
ka
real even
35
th tjk
k
k
ejkHaty 00


tjkk
k
eatx 0


dtethjH tj
Chapter 3 Fourier Series
Filter
36
Chapter 3 Fourier Series
§ 3.9.2 Frequency-Selective Filter
1,Ideal Lowpass Filter (LPF)


c
c
jH


0
1
0
1
c
jH
c?Passband Stopband Stopband
Continuous-time system
Discrete-time system






k
k
eH
c
cj
2 0
2 1
0
1
c
jeH
c2?2
Passband Stopband Stopband
37
Chapter 3 Fourier Series
2,Ideal Highpass Filter (HPF)


c
c
jH


0
1
Continuous-time system
Discrete-time system
1 0 cj
c
He?





0
1
jeH
2?2
0
1
c
jH
c?
Passband Stopband Passband
2j jH e H e
38
Chapter 3 Fourier Series
3,Ideal Bandpass Filter (BPF)



o t h e r s 0
1 21
jH
Continuous-time system
0
1
jH
1? 2?12
Discrete-time system



o t h e r s 0
1 21
jeH
210
0
1
jeH
2?2
39
Chapter 3 Fourier Series
0
1
jH
1? 2?12
Ideal Bandpass Filter
0
1
c
jH
c?0
1
c
jH
c?
Ideal Lowpass Filter
jH
1? 2?12
Ideal Highpass Filter
Ideal Bandstop Filter
40
Chapter 3 Fourier Series
Example Consider an LTI system with input
the frequency response of this system is
as shown in Figure 1,determine the output of system
kttx
k
2

ty
jH
0
1

2/3?
jH
Figure 1 (a)
2/3 0
1
jH
Figure 1 (b)
2/5?2/32/3 2/5?
1a c os 2y t t
b c o s 2 y t t
41
Chapter 3 Fourier Series
§ 3.10 Examples of continuous-time filters described
by differential equations
§ 3.10.1 A simple RC Lowpass Filter
tvs
R
ti
C
tvc

21
1
RC
jH
RCa r c t gjH
2?RC
1?RC
5.0?RC
2?RC
1?RC
5.0?RC
42
Chapter 3 Fourier Series
§ 3.10.2 A simple RC Highpass Filter
RCj RCjjG 1
tvs
Rti
C
rvt
jG
tvstvr
43
Chapter 3 Fourier Series
Homework:
3.1 3.13 3.15 3.34 3.35
44
Chapter 3 Fourier Series
dtethsH st
ste stesH
Eigenfunction
Eigenvalue
kks t s tk k k
kk
x t a e y t a H s e
nz nzzH Eigenfunction
Eigenvalue
n
n
znhzH?



nnk k k k k
kk
x n a z y n a H z z
45
tjkk
k
eatx 0


dtetxTa tjk
Tk
0
00
1
Synthesis equation
综合公式
Analysis equation
分析公式
Chapter 3 Fourier Series
01 1 01s i n 2 s i n 0k kT Ta c k T kkT
1,Periodic square wave
2,Periodic Impulse Trains
,2,1,0 1 kTa k
46
Chapter 3 Fourier Series
kk btyatx FSFS
kkk BbAactBytAxtz FS
00FS0 tjkk eattx
kk aatxtx

kk aao r
txtx kkaa kkaa
47
221 k
kT
adttxT


katx FS
knn ajktx 0FS
kajktx 0FS
Chapter 3 Fourier Series
real oddtx
ka
Purely imaginary
odd
real eventx
ka
real even