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Fourier Analysis
S. Awad, Ph.D.
M. Corless, M.S.E.E.
D. Cinpinski
E.C.E. Department
University of Michigan-Dearborn
Math Review with Matlab:
Fourier Series
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
2
Periodic Signal Definition
Parseval’s Theorem
Fourier Series
Complex Exponential Representation
Magnitude and Phase Spectra of Fourier Series
Fourier Series Representation of Periodic Signals Fourier Series Coefficients Orthogonal Signals
Example: Full Wave Rectifier
Example: Finding Complex Coefficients
Example: Orthogonal Signals
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
3
For example, the normal U.S. AC from wall outlet has a sine wave with a peak voltage of 170 V (110 Vrms)
The Period of a signal is the amount of time it takes for a given signal to complete one cycle.
What is a Periodic Signal ?
A Periodic Signal is a signal that repeats itself every period
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
4
General Sinusoid A general cosine wave, v(t), has the form:
)cos()( tMtv
= Phase Shift, angular offset in radians
F = Frequency in Hz
T = Period in seconds (T=1/F)
t = Time in seconds
M = Magnitude, amplitude, maximum value
= Angular Frequency in radians/sec (=2F)
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
5
General Sinusoid
Plot in Blue:
))60(2sin(5 t
Plot in Red:
)2
)60(2sin(5 t
1 Period = 1/60 sec.
= 16.67 ms.
/2 Phase Shift
Amplitude = 5
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
6
AC Wall Voltage Sine Wave
1 Period
1 Period
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
7
Represent Periodic Signals For a general periodic signal x(t) shown to the right:
x(t+nT) = x(t) for all
t
where n is any integer, i.e. n = 0, ± 1, ± 2,…
T
x(t)
-T/2 T/2 t......
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
8
Frequency of Periodic Signals The frequency of a signal is defined as the
inverse of the period and has the unit “number of cycles/sec.” T
fo1
is the fundamental frequency.of
The frequency of a US standard outlet is 1/T = 60 Hz
T is the period and
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
9
What is Fourier Series ? Fourier Series is a technique developed by J. Fourier.
This technique (studied by Fourier) allows us to represent periodic signals as a summation of sine functions of different frequency, amplitude, and phase shift.
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
10
Represent a Square Wave
Represent the Square Wave at the right using Fourier Series
Notice that as more and more terms are summed, the approximation becomes better
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
11
Fourier Series Representationof Periodic Signals
Any periodic function can be represented in terms of sine and cosine functions:
...2sinsin
...2coscos)(
21
210
tbtb
tataatx
oo
oo
This can also be written as:
1
0 )sincos()(n
onon tnbtnaatx
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
12
Fourier Series Coefficients
The above a0, an, and bn are known as the Fourier Series
Coefficients. These coefficients are calculated as follows.
1
0 )sincos()(n
onon tnbtnaatx
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
13
Calculating the a0 Coefficient
ao, the coefficient outside the summation, is known as the average value or the dc component
ao is calculated as follows:
2
2
)(1
T
T
o dttxT
a
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
14
Calculating the an and bn Coefficients
2
2
,)cos()(2
T
T
on dttntxT
a n = 1, 2,…
2
2
,)sin()(2
T
T
on dttntxT
b n = 1, 2,…
The an and bn coefficients are calculated as follows:
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
15
Orthogonal Signals
Two periodic signals g1(t) and g2(t) are said to be “Orthogonal” if the the integral of their product over one period is equal to zero.
2/
2/
0)(2)(1T
T
dttgtg
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
16
Example: Orthogonal Signals
2/
2/
)()( 21
T
T
dttgtg
2/
2/
2/
2/
)cos()sin(22
1
)cos()sin(
T
T
T
T
dttt
dttt
Show that the following signals are orthogonal:
cos(t) (t)g
sin(t) (t)g
2
1
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
17
Orthogonal Signals
0
)cos()cos(4
1
)2cos(4
1)2sin(
2
1
)cos()sin(22
1
2/
2/
2/
2/
2/
2/
TT
tdtt
dttt
T
T
T
T
T
T
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
18
Note that the rectified wave has a period equal to one-half of the source wave period.
Example: Full Wave Rectifier
y(t)=|sin(ot)|
t
y=|x|
x
y
x(t)
tT/2
one period
one period
T
Consider the output of a full-wave rectifier:
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
19
Function Characteristics The period of y(t) = T/2 and the fundamental
frequency of y(t) is 2o (rad/sec).
1
0 )2cos()(n
on tnaaty
Thus,
Now bn=0 since y(t) is an even function.
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
20
Finding ao
2/
0
2
0
2
2
)sin(2
)sin(2
1
)(1
T
oo
T
oo
T
T
o
dttT
a
dttT
a
dttyT
a
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
21
Finding ao
1)2
)2
(cos(
)2
(
2
)0*cos()2
cos(2
)cos(2 2/
0
TT
TT
a
T
Ta
tT
a
o
oo
oo
T
oo
o
* Use o = 2pi/T
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
22
Finding ao
2
111
1)cos(1
o
o
o
a
a
a
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
23
Finding an, n = 1, 2, ….
4
0
4
4
)2cos()sin()2(4
)2cos()sin(2
2
T
oo
T
T
oon
dttntT
dttntT
a
2
2
,)cos()(2
T
T
on dttntyT
a
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
24
Solution for an, n = 1, 2, ….
4
0
4
0
4
0
)12(
)12(cos
)12(
)12(cos4
)12(sin)12(sin2
18
T
o
oT
o
on
T
oon
n
tn
n
tn
Ta
dttntnT
a
)12(
1
)12(
1
2
4
)12(
1
)12(
14
nn
nnT oo
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
25
Solution for an, n = 1, 2, …. So:
14
4
14
2222
nnan
Thus:
...6cos
35
14cos
15
12cos
3
142)( tttny ooo
Note: We can only obtain an output signal with a nonzero average value by using a nonlinear system with our zero average value input signal
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
26
Euler’s Identity
)sin()cos( jMMMe j We could also say:
)sin()cos( je j
)sin()cos(
)sin()cos()(
je
jej
j
and ...
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
27
Representing Sin and Cos with Complex Exponentials
)sin()cos( je j )sin()cos( je j
2)cos(
)cos(2
jj
jj
ee
ee
j
ee
jeejj
jj
2)sin(
)sin(2
Add the equations: Subtract the equations:
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
28
Complex Exponential Representation
The Sine and Cosine functions can be written in terms of complex exponentials.
tjntjno
oo eetn 2
1cos
tjntjno
oo eej
tn 2
1sin
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
29
Complex Exponential Fourier Series From previous slides…
10 2
1
2
1)(
n
tjntjnn
tjntjnn
oooo eej
beeaatx
t jn t jn
oo o
e e t n
2
1cos
t jn t jno
o oe e
jt n
2
1sin
Using the Complex Exponential representation of Sine and Cosine, the Fourier series can be written as:
1
0 )sincos()(n
onon tnbtnaatx
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
30
Fourier Series with Complex Exponentials
10
10
2
1
2
1)(
22
1)(
n
tjnnn
tjnnn
n
tjntjnn
tjntjnn
oo
oooo
ejbaejbaatx
eej
beeaatx
Noting that 1/j = -j, we can write:
10 2
1
2
1)(
n
tjntjnn
tjntjnn
oooo eej
beeaatx
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
31
Fourier Series with Complex Exponentials
110
10
10
)(
)(
2
1
2
1)(
n
tjnn
n
tjnn
n
tjnn
tjnn
n
tjnnn
tjnnn
oo
oo
oo
ececctx
ececctx
ejbaejbaatx
Make the following substitutions:
n
tjnn
oectx )(
)(2
1),(
2
1,00 nnnnnn jbacjbacac
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
32
Fourier Series with Complex Exponentials
The Complex Fourier series can be written as:
n
tjnn
oectx )(
where:
2
2
0)(1
T
T
tjnn dtetxT
c
Complex cn *Complex conjugate Note: if x(t) is real, c-n = cn
*
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
33
Line Spectra Line Spectra refers the plotting of discrete coefficients
corresponding to their frequencies For a periodic signal x(t), cn, n = 0, ±1, ± 2,… are
uniquely determined from x(t). The set cn uniquely determines x(t)
Because cn appears only at discrete frequencies, n(n = 0, ± 1, ± 2,… the set cn is called the discrete frequency spectrum or line spectrum of x(t).
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
34
The Cn coefficients are in general complex.
Line Spectra
The standard practice is to make 2 2D plots. Plot 1: Magnitude of Coefficient vs. frequency
The standard practice is to make 2 2D plots. Plot 1: Magnitude of Coefficient vs. frequency Plot 2: Phase of Coefficient vs. frequency
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
35
Magnitude of Cn
Recall that the magnitude for a complex number a+jb is calculated as follows:
22 bajba
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
36
Phase of Cn Recall that the phase for a complex number a+jb depends on the quadrant that the angle lies in.
a
bTan 1
Quadrant 1: Quadrant 2:
Quadrant 3: Quadrant 4:
a
bTan 1
a
bTan 1
a
bTan 1
Angle(a+jb) =
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
37
Amplitude Spectrum of Cn Note: If x(t) is real then |Cn| is
of even symmetry. nn cc
nc
sec)(radoo o2o2
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
38
Phase Spectrum of Cn
nn cc
Note: If x(t) is real then the Phase of Cn is odd
nc
sec)(radoo o2o2
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
39
Example: Finding Complex Coefficients
Consider the periodic signal x(t) with period T = 2 sec. Thus:
secsec2
2
sec
2 radradrad
To
x(t)
t-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.50
1
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
40
Finding Co(avg)
Co(avg) = 0.5
1
1
5.0
5.0
1
5.0
5.0
1
)( )()()(2
1)(
2
1dttxdttxdttxdttxC avgo
0 0
5.0)5.0(5.02
1
2
1
2
1 5.0
5.0
5.0
5.0
tdt
The area under x(t) from -1 to -.5 and from .5 to 1 is zero.
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
41
Calculating Cn
0 , 2
sin
nn
nCn
22
0
5.0
5.0
5.0
5.0
5.0
5.0
5.0
5.0
1
1
2/
2/
00
2
1
02
10
2
1
2
1
2
1
)(2
1
)(1
nj
nj
tjn
tjntjntjn
tjn
T
T
tjnn
eenj
dte
dtedtedte
dtetx
dtetxT
C
o
ooo
o
o
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
42
Now it can be shown that: sin(n/2) = 0 for n = ±2, ±4, … Cn = 0
sin(2/2) = sin() = 0 sin(-4/2) = sin(-2) = 0 etc .
It can be also be shown that: sin(n/2) = -1 for n = 3, 7, 11,… sin(n/2) = 1 for n = 1, 5, 9,…
sin(3/2) = -1 sin(-7/2) = 1 etc .
Factor Evaluation
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
43
,...11,7,3 , nπ
n)signum(
,...9,5,1n , nπ
signum(n)
etc... 4,2, ,0
nC
C
nC
n
n
n
0 , 2
sin
nn
nCn
Recall:
Factor Evaluation
Co(avg) = 0.5
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
44
Note: Cn= if Cn is negative Therefore:
0n&evenn ,
0
oddn ,
0
5.||
1
nn
Cn
and
otherwise ,
... 11 7, 3, n ,
0
nC
Summary of Results
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
45
Plot the Magnitude Response
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
46
Plot the Phase Response
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
47
What is Parseval’s Theorem ? Parseval’s Theorem states that the average power of a
periodic signal x(t) is equal to the sum of the squared amplitudes of all the harmonic components of the signal x(t).
This theorem is excellent for determining the power contribution of each harmonic in terms of its coefficients
U of M-Dearborn ECE DepartmentMath Review with Matlab
Fourier Analysis: Fourier Series
48
Parseval’s Theorem Average power of x(t) is calculated from the time
or frequency domain by:
)(2
1)(
1
1
2222
2
2
n
nno
T
T
avg baadttxT
P
n n
nonavg cccP1
2222
Time Domain:
Frequency Domain:
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