signal and systems chapter 3: fourier series representation...
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Signal and SystemsChapter 3: Fourier Series Representation of Periodic Signals
• Complex Exponentials as Eigenfunctions of LTI Systems• Fourier Series representation of CT periodic signals• How do we calculate the Fourier coefficients?• Convergence and Gibbs’ Phenomenon• CT Fourier series reprise, properties, and examples• DT Fourier series • DT Fourier series examples and differences with CTFS• Fourier Series and LTI Systems• Frequency Response and Filtering• Examples and Demos
Portrait of Jean Baptiste Joseph Fourier
Image removed due to copyright considerations.
Signals & Systems, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997, p. 179.
Book Chapter 3 : Section 1
Computer Engineering Department, Signal and Systems 2
Desirable Characteristics of a Set of “Basic” Signals
a. We can represent large and useful classes of signals using these building blocks
b. The response of LTI systems to these basic signals is particularly simple, useful, and insightful
Previous focus: Unit samples and impulses
Focus now: Eigenfunctions of all LTI systems
Computer Engineering Department, Signal and Systems 3
Book Chapter 3 : Section 1
The eigenfunctions and their properties
Eigenfunction in →same function out with a “gain”
Computer Engineering Department, Signal and Systems 4
(Focus on CT systems now, but results apply to DT systems as well.)
( )k t
From the superposition property of LTI systems:
Now the task of finding response of LTI systems is to determine λk.
Book Chapter 3 : Section 1
Complex Exponentials as the Eigenfunctions of any LTI Systems
Computer Engineering Department, Signal and Systems 5
stetx )(
dehty ts )()()(
sts edeh
)(
eigenvalue eigenfunction
)(sHste
eigenvalue eigenfunction
m
mnzmhny ][][
n
m
m zzmh
][
)(zHnz
Book Chapter 3 : Section 1
DT:
dtethsH st)()(
k k
ts
kk
ts
kkk easHtyeatx )()()(
nznhzH ][)(
k
n
kkk
k
n
kk zazHnyzanx )(][][
Book Chapter 3 : Section 1
What kinds of signals can we represent as “sums” of complex exponentials?
For Now: Focus on restricted sets of complex exponentials
CT:
DT:
s = jω – purely imaginaly,i.e., signals of the form ejωt
i.e., signals of the form ejωn
Magnitude 1
CT & DT Fourier Series and Transforms
Periodic Signals
jZ e
Book Chapter 3 : Section 1
for all t
Computer Engineering Department, Signal and Systems 8
Fourier Series Representation of CT Periodic Signals
- smallest such T is the fundamental period- is the fundamental frequency
Periodic with period T
-periodic with period T-{ak} are the Fourier (series) coefficients-k= 0 DC -k= 1 first harmonic-k= 2 second harmonic
0 2 /( )jk t jk t T
k kx t a e a e
( ) ( )x t x t T
Book Chapter 3 : Section 1
Computer Engineering Department, Signal and Systems 9
Question #1: How do we find the Fourier coefficients?First, for simple periodic signals consisting of a few sinusoidal terms
tttx 8sin24cos)(
][2
2][
2
1 8844 tjtjtjtj eej
ee
0 4 0
2 2 1
4 2T
Book Chapter 3 : Section 1
For real periodic signals, there are two other commonly used forms for CT Fourier series:
Because of the eigenfunction property of e jωt , we will usually use the complex exponential form.
Book Chapter 3 : Section 1
Computer Engineering Department, Signal and Systems 10
or
- A consequence of this is that we need to include terms for both positive and negative frequencies:
0 0 0
1
( ) [ cos sin ]k k
k
x t a k t k t
0 0
1
( ) [ cos( )]k k
k
x t a k t
Computer Engineering Department, Signal and Systems 11
Now, the complete answer to Question #1
denotes integral over any interval of lengthHere
Next, note that
Orthogonality
0( )jk t
k
k
x t a e
0 0 0( )jn t jk t jn t
k
kT T
x t e dt a e e dt
0( )j k n t
k
k T
a e dt
0( )j k n t
T
e dt
Book Chapter 3 : Section 1
Given x(t), how to find 𝑎𝑘
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CT Fourier Series Pair
(Synthesis equation)
(Analysis equation)
0 0( )( ) . [ ]
jn t j k n t
k k
T T
x t e dt a e dt a T k n
0( )jn t
n
T
x t e dt a T
0
2( )
T
0( )jk t
kx t a e
01
( )jk t
k
T
a x t e dtT
Book Chapter 3 : Section 1
Computer Engineering Department, Signal and Systems 13
Ex: Periodic Square Wave
DC component is just the average
2
2
10
2)(
1T
TT
Tdttx
Ta
10 0
1
2
2
1 1( )
TT
jk t jk t
TkT
a x t e dt e dtT T
0 1
1
0 1
0
sin1|
jk t T
T
k Te
jk T k
0
2( )
T
Book Chapter 3 : Section 1
How can the Fourier series for the square wave possibly make sense?
The key is: What do we mean by
One useful notion for engineers: there is no energy in the difference
Book Chapter 3 : Section 1
Computer Engineering Department, Signal and Systems 14
Convergence of CT Fourier Series
(just need x(t) to have finite energy per period)
0( )jk t
kx t a e
0( ) ( )jk t
ke t x t a e
T
dtte 0|)(| 2
?
T
dttx 2|)(|
Computer Engineering Department, Signal and Systems 15
Under a different, but reasonable set of conditions (the Dirichlet conditions)
Condition 1. x(t) is absolutely integrable over one period, i. e.
AndCondition 2. In a finite time interval,
x(t) has a finite number of maxima and minima.
Ex. An example that violates Condition 2.
AndCondition 3. In a finite time interval, x(t) has only a
finite number of discontinuities.Ex. An example that violates
Condition 3.
T
dttx |)(|
10)2
sin()( tt
tx
Book Chapter 3 : Section 1
Dirichlet conditions are met for the signals we will encounter in the real world. Then
Still, convergence has some interesting characteristics:
Book Chapter 3 : Section 1
Computer Engineering Department, Signal and Systems 16
- The Fourier series = x(t) at points where x(t) is continuous
- The Fourier series = “midpoint” at points of discontinuity
- As N→ ∞, xN(t) exhibits Gibbs’ phenomenon at points of discontinuity
Demo: Fourier Series for CT square wave (Gibbs phenomenon).
0( )N
jk t
N k
k N
x t a e
CT Fourier Series Pairs
Book Chapter3: Section2
Computer Engineering Department, Signals and Systems 17
𝜔0 =2𝜋
𝑇
Review:
𝑥(𝑡) =
𝑘=−∞
∞
𝑎𝑘𝑒𝑗𝑘𝜔0𝑡 =
𝑘=−∞
∞
𝑎𝑘𝑒𝑗2𝜋𝑘 Τ𝑡 𝑇
𝑎𝑘 =1
𝑇න
𝑇
𝑥(𝑡)𝑒−𝑗𝑘𝜔0𝑡𝑑𝑡
Skip it in future
for shorthand
𝑥(𝑡)𝐹𝑆𝑎𝑘
Another (important!) example: Periodic Impulse Train
𝑥 𝑡 = 𝑛=−∞
∞)𝛿(𝑡 − 𝑛𝑇 sampling function
important for sampling
𝑎𝑘 =1
𝑇න−𝑇2
𝑇2𝑥(𝑡)𝑒−𝑗𝑘𝜔0𝑡 𝑑𝑡 =
1
𝑇න−𝑇2
𝑇2𝛿(𝑡)𝑒−𝑗𝑘𝜔0𝑡𝑑𝑡 =
1
𝑇𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘!
𝑥(𝑡) = 𝑘=−∞
∞𝑎𝑘𝑒
𝑗𝑘𝜔0𝑡
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 18
─ All components have:(1) the same amplitude,
&(2) the same phase.
(A few of the) Properties of CT Fourier Series
Linearity
Conjugate Symmetry
𝑥(𝑡)𝑖𝑠 𝑟𝑒𝑎𝑙 ⇒ 𝑎−𝑘 = 𝑎𝑘∗
⇓𝑎𝑘 = Re{𝑎𝑘} + 𝑗Im{𝑎𝑘} = |𝑎𝑘|𝑒
𝑗∠𝑎𝑘
Re{𝑎𝑘} 𝑖𝑠 𝑒𝑣𝑒𝑛, Im{𝑎𝑘} 𝑖𝑠 𝑜𝑑𝑑𝑜𝑟
|𝑎𝑘| 𝑖𝑠 𝑒𝑣𝑒𝑛, ∠𝑎𝑘 𝑖𝑠 𝑜𝑑𝑑
Time shift 𝑥 𝑡 → 𝑎𝑘𝑥(𝑡 − 𝑡0) → 𝑎𝑘𝑒
−𝑗𝑘𝜔0𝑡0 = 𝑎𝑘𝑒−𝑗𝑘2𝜋 Τ𝑡0 𝑇
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 19
𝑥(𝑡) ↔ 𝑎𝑘 , 𝑦(𝑡) ↔ 𝑏𝑘 ⇒ 𝛼𝑥(𝑡) + 𝛽𝑦(𝑡) ↔ 𝛼𝑎𝑘 + 𝛽𝑏𝑘
Introduces a linear phase shift of to
Example: Shift by half period
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 20
using
𝑦(𝑡) = 𝑥(𝑡 − Τ𝑇 2) ↔ 𝑎𝑘𝑒−𝑗𝑘𝜋 = −1 𝑘𝑎𝑘
𝑒−𝑗𝑘𝜔0 Τ𝑇 2 = 𝑒−𝑗𝑘𝜋
𝑦(𝑡) ↔ −1 𝑘𝑎𝑘 𝑎𝑘 =1
𝑇= 𝐹. 𝐶. 𝑜𝑓
−∞
∞
)𝛿(𝑡 − 𝑛𝑇
||
−1 𝑘
𝑇
Parseval’s Relation
Multiplication Property
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 21
1
𝑇
𝑇
|𝑥(𝑡)|2𝑑𝑡 =
−∞
∞
|𝑎𝑘|2
Power in the kth harmonic Average signal power
kk btyatx )(,)( (Both x(t) and y(t) are periodic with the same period T)
l
kklklk babactytx *)().(
Energy is the same whether measured in the time-domain or the frequency-domain
Proof:
0 0 0 0( )
,
.jl t jm t j l m t jk tl m k
l m l m l k l
l m l m k l
a e b e a b e a b e
X(t) y(t) ck
Periodic Convolutionx(t), y(t) periodic with period T
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 22
dtyxtytx )()()(*)( - Not very meaningful
E.g. If both x(t) and y(t) are positive, then
)(*)( tytx
Periodic Convolution (continued)Periodic convolution : Integrate over any one period (e.g. -T/2 to T/2)
z(t) is periodic with period T
𝑥𝑇 𝑡 = ቐ𝑥 𝑡 −𝑇
2< 𝑡 <
𝑇
20 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 23
/ 2
/ 2
( ) ( ) ( ) ( ) ( )
T
T
T
z t x y t d x y t d
where
Periodic Convolution (continued) Facts
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 24
1) z(t) is periodic with period T (why?)From Lecture #2, x(t) = x(t + T) → y(t) = y(t + T) for LTI systems.In the convolution, treat y(t) as the input and xT(t) as h(t)
2) Doesn’t matter which period we choose to integrate over:
Periodic3) Convolution
in time
)()()()()( tytxdtyxtzT
kkk ctzbtyatx )(,)(,)(
0 01 1
( ) ( ) ( )jk t jk t
k
T T T
c z t e dt x y t d e dtT T
0 0( )1( ) ( )
jk t jk
T T
y t e dt x e dT
0( )jk
k k k
T
b x e d Ta b
MultiplicationIn frequency!
Fourier Series Representation of DT Periodic Signals
x[n] -periodic with fundamental period N, fundamental frequency
Only e jωn which are periodic with period N will appear in the FS
There are only N distinct values of signals of this form
So we could just use
However, it is often useful to allow the choice of N
consecutive values of k to be arbitrary.
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 25
][][ nxNnx and0
2
N
02 , 0, 1, 2,...N k k k
0 0 0 0( )j k N n jk n jN n jk ne e e e
2πn
DT Fourier Series Representation
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 26
Nk
nNjk
keanx )/2(][
Sum over any N consecutive values of k
— This is a finite series
𝑘= 𝑁
=
𝑎𝑘 - Fourier (series) coefficients
Questions:
1) What DT periodic signals have such a representation?
2) How do we find ak?
Answer to Question #1:Any DT periodic signal has a Fourier series representation
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 27
0
0
0
0
2
( 1)
[ ]
[0]
[1]
[2]
[ 1]
jk n
k
k N
k
k N
jk
k
k N
j k
k
k N
j N k
k
k N
x n a e
x a
x a e
x a e
x N a e
N equations for N unknowns, a0, a1, …, a N-1
A More Direct Way to Solve for akFinite geometric series
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 28
1,
1,1
1
1
0
NN
n
nN
0jkwe
0 0
1 12 /
0 0
( ) ( )N N
jk n jk n jk N n
n N n n
e e e
( 2 / )
0
, 0, , 2 ,...
10 ,
1
jk N N
jk
N k N N
eotherwise
e
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 29
So, from0[ ]
jk n
k
k N
x n a e
multiply both sides by
and then
0jm ne
Nn
0 0 0[ ]jm n jk n jm n
k
n N n N k N
x n e a e e
0( )
[ ]
j k m n
k
k N n N
N k m
a e
orthogonality
mNa
DT Fourier Series Pair
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 30
𝜔0 =2𝜋
𝑁
0
0
[ ]
1[ ]
jk n
k
k N
jk n
k
n N
x n a e
a x n eN
(Synthesis equation)
(Analysis equation)
Note : It is convenient to think of ak as being defined for all integers k. So:
1) ak+N = ak—Special property of DT Fourier Coefficients.
2) We only use N consecutive values of ak in the synthesis equation. (Since x[n] is periodic, it is specified by N numbers, either in the time or frequency domain)
Example #1: Sum of a pair of sinusoids
Book Chapter3: Section2
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)4/4/cos()8/cos(][ nnnx
- periodic with period N = 16 → ω0 = π/8
0 0 0 02 2/ 4 / 41 1[ ] [ ] [ ]
2 2
j n j n j n j nj jx n e e e e e e
0
0
2/
2/
2/1
2/1
0
3
3
4/
2
4/
2
1
1
0
a
a
ea
ea
a
a
a
j
j
2/
2
1
4/
2164266
116115
jeaaa
aaa
Example #2: DT Square Wave
using n = m - 𝑁1
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 32
1
1
10 6
2 11[ ]
N
N N N
n N
Na x n a a a
N N
𝐹𝑜𝑟 𝑘 ≠ 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓𝑁:
1 1
0 0 1
1
2( )
0
1 1N N
jk n jk m N
k
n N m
a e eN N
0 11
0 1 0 0 1
0
(2 1)2
0
1 1 1( )
1
jk NNjk N jk jk Nm
jkm
ee e e
N N e
1 0 1
0
sin[ ( 1/ 2) sin[2 ( 1/ 2) / ]1 1
sin( / 2) sin( / )
k N k N N
N k N k N
Example #2: DT Square Wave (continued)
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 33
)/sin(
]/)2/1(2sin[1 1
Nk
NNk
Nak
Book Chapter3: Section2
Computer Engineering Department, Signal and Systems 34
Convergence Issues for DT Fourier Series:Not an issue, since all series are finite sums.
Properties of DT Fourier Series: Lots, just as with CT Fourier Series
Example: [ ] kx n a
0 [ ] ?jM n
ke x n b
0 0 0
0
[ ]jM n jr n jM n
r
r N
jk n
k M
k N
x n e a e e
k r m a e
Frequency shift
Mkk ab 0 0( )jk j k M
The Eigenfunction Property of Complex Exponentials
CT
CT System Function:
DT
DT System function:
Book Chapter#: Section#
Computer Engineering Department, Signal and Systems 35
dtethsH st)()(
n
nznhzH ][)(
Fourier Series: Periodic Signals and LTI Systems
,
So , or powers of signals get modified through filter/ system.
,
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0( )jk t
k
k
x t a e
0
0( ) ( )jk t
k
k
y t H jk a e
0
" "
( )k k
gain
a H jk a 0( )
0 0( ) | ( ) |j H jk
H jk H jk e
Includes both amplitude & phase
0[ ]jk n
k
k N
x n a e
0 0[ ] ( )jk jk n
k
k N
y n H e a e
0
" "
( )jk
k k
gain
a H e a
00 0 ( )( ) | ( ) |
jkjk jk j H eH e H e e
Includes both amplitude & phase
0| | | ( ) || |k ka H jk a
The Frequency Response of an LTI System
CT Frequency Response:
DT Frequency Response:
Book Chapter#: Section#
Computer Engineering Department, Signal and Systems 37
( ) ( ) j tH j h t e dt
( ) [ ]j j n
n
H e h n e
Frequency Shaping and Filtering
By choice of (or ) as a function of , we can shape the frequency composition of the output
Preferential amplification
Selective filtering of some frequencies
Example #1:
For audio signals, the amplitude is much more important than the phase
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( )H j ( )jH e
Frequency Shaping and Filtering
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Example #2: Frequency Selective Filters
Filter out signals outside of the frequency range of interest
( 2 )( ) ( )j jH e H e
Lowpass Filters:
Only show
amplitude here.
Note for DT:
Highpass Filters
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Remember:
njn e)1(
Bandpass Filters
Demo: Filtering effects on audio signals
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Idealized Filters
CT
DT
Note: |H| = 1 and ∠H = 0 for the ideal filters in the passbands, no need for the phase plot.
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ωc : cutoff frequency
Highpass
CT
DT
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Bandpass
CT
DT
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lower cut-off upper cut-off
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]}1[][]1[{3
1][ nxnxnxny
]}1[][]1[{3
1][ nnnnh
Example #3: DT Averager/Smoother
FIR (Finite Impulse
Response) filters
1 1 2( ) [ ] [ 1 ] cos
3 3 3
j j n j j
n
H e h n e e e
Example #4: Nonrecursive DT (FIR) filters
Book Chapter#: Section#
Computer Engineering Department, Signal and Systems 46
[( ) / 2]
1 1[ ] [ ] [ ] [ ]
1 1
1 1 sin[ ( 1) / 2]( )
1 1 sin( / 2)
M M
k N k N
Mj jk j N M
k N
y n x n k h n n kN M N M
M NH e e e
N M N M
Rolls off at lower
ω as M+N+1
increases
Example #5: Simple DT “Edge” Detector
DT 2-point “differentiator”
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Computer Engineering Department, Signal and Systems 47
]}1[][{2
1][
]}1[][{2
1][
nnnh
nxnxny
/ 2 / 2 / 2 / 2[ ( ) sin( / 2)2
j j j jje e e je
j
/ 21( ) (1 ) sin( / 2)
2
j j jH e e je
| ( ) | | sin( / 2) |jH e
Demo: DT filters, LP, HP, and BP applied to DJ
Industrial average
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Example #6: Edge enhancement using DT
differentiator
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Courtesy of Jason Oppenheim.
Used with permission.
Courtesy of Jason Oppenheim.
Used with permission.
Example #7: A Filter Bank
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Demo: Apply different filters to two-dimensional image signals.
Note: To really understand these examples, we need to understand frequency contents of aperiodic signals ⇒ the Fourier Transform
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Face of a monkey.
Image removed do to copyright
considerations
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