fourier series analysis
TRANSCRIPT
Fourier Series Analysis
Trig and Exponential Series
MODIFIED BY TLH
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 2
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 3
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 4
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 5
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 6
Harmonic Signal->Periodic
tFkj
k
keatx 02)(
0
0
0
00
1or
22
FT
TF
Sums of Harmonic
complex exponentials
are Periodic signals
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 8
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 9
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LECTURE OBJECTIVES
Work with the Fourier Series Integral
ANALYSIS via Fourier Series
For PERIODIC signals: x(t+T0) = x(t)
Draw spectrum from the Fourier Series coeffs
0
0
0
0
)/2(1 )(T
dtetxatTkj
Tk
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 11
0 100 250–100–250f (in Hz)
3/7 je3/7 je
2/4 je 2/4 je
10
SPECTRUM DIAGRAM
Recall Complex Amplitude vs. Freq
kk aX 21
Xk Akejk
12 Xk
*
N
k
tfjk
tfjk
kk eXeXXtx1
2
212
21
0)(
ka{ *ka0a
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 12
Harmonic Signal->Periodic
tFkj
k
keatx 02)(
0
0
0
00
1or
22
FT
TF
Sums of Harmonic
complex exponentials
are Periodic signals
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 13
Notation for Fundamental
Frequency in Fourier Series
The k-th frequency is fk = kF0
Thus, f0 = 0 is DC
This is why we use upper case F0 for the
Fundamental Frequency
tfj
k
k
tFkj
k
kkeaeatx
22 0)(
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 15
STRATEGY: x(t) ak
ANALYSIS
Get representation from the signal
Works for PERIODIC Signals
Measure similarity between signal & harmonic
Fourier Series
Answer is: an INTEGRAL over one period
0
0
0
0)(1
T
dtetxatkj
Tk
CALCULUS for complex exp
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 16
ttjtt ejedt
dee
dt
d
ab
b
a
t
b
a
t eeedte
11
ajbj
b
a
tj
b
a
tj eej
ej
dte
11
component) (DC)(
)(
0
0
0
0
0
0
10
0
)/2(1
T
T
T
tkTj
Tk
dttxa
dtetxa
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 18
00 /1
Freq.lFundamenta
TF
Fourier Series Integral
Use orthogonality to determine ak from x(t)
real is )( when* txaa kk
THIS IS THE AVERAGE!
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 19
Fourier Series: x(t) ak
ANALYSIS
Given a PERIODIC Signal
Fourier Series coefficients are obtained via
an INTEGRAL over one period
Next, consider a specific signal, the FWRS
Full Wave Rectified Sine
0
0
0
0)(1
T
dtetxatkj
Tk
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 20
Full-Wave Rectified Sine
121
01 is Period)/2sin()( TTTttx
Absolute value flips the
negative lobes of a sine wave
Frequency
Doubles
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 21
Full-Wave Rectified Sine {ak}
0
0
0
)/2(
0
)(1
T
ktTj
k dtetxT
a
0
00
00
00
0
0
0
0
0
0
0
0
0
00
0
0
0
00
0
))12)(/((2
)12)(/(
0
))12)(/((2
)12)(/(
0
)12)(/(
21
0
)12)(/(
21
0
)/2()/()/(
1
0
)/2(1
2
)sin(
T
kTjTj
tkTjT
kTjTj
tkTj
T
tkTj
Tj
T
tkTj
Tj
T
ktTjtTjtTj
T
T
ktTj
TTk
ee
dtedte
dtej
ee
dteta
Full-Wave Rectified Sine
)/sin()(
:
)/2sin()(
0
121
0
1
Tttx
TTPeriod
Tttx
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)14(
21)1(
11
11
2
2
)14(
)12(12
)12(
)12(1)12(
)12(1
)12)(/(
)12(21)12)(/(
)12(21
0
))12)(/((2
)12)(/(
0
))12)(/((2
)12)(/(
2
0000
0
00
00
00
0
k
ee
ee
eea
k
k
kk
kj
k
kj
k
TkTj
k
TkTj
k
T
kTjTj
tkTjT
kTjTj
tkTj
k
Full-Wave Rectified Sine {ak}
Fourier Coefficients: ak
ak is a function of k
Complex Amplitude for k-th Harmonic
NOTE: 1
𝑘2𝑓𝑜𝑟 𝑙𝑎𝑟𝑔𝑒 𝑘
Does not depend on the period, T0
DC value is
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 23
)14(
22
kak
6336.0/20 a
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 24
Spectrum from Fourier Series
Plot a for Full-Wave Rectified Sinusoid
0000 2and/1 FTF
)14(
22
kak
ka
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 25
Reconstruct From Finite Number
of Harmonic Components
Full-Wave Rectified Sinusoid
Hz100
ms10
0
0
F
T
)/sin()( 0Tttx
6336.0/20 a
)14(
22
kak
N
k
tFkj
k
tFkj
kN eaeaatx1
22
000)(
?)/sin()( to)( is closeHow 0TttxtxN
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 26
Reconstruct From Finite Number
of Spectrum Components
Full-Wave Rectified Sinusoid )/sin()( 0Tttx
Hz100
ms10
0
0
F
T
6336.0/20 a
)14(
22
kak
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 32
Synthesis: up to 7th Harmonic)350sin(
7
2)250sin(
5
2)150sin(
3
2)50cos(
2
2
1)(
2ttttty