2015/11/28 system arch 2008 (fire tom wada) 1 outline periodic signal fourier series introduction...
TRANSCRIPT
23/04/18System Arch 2008 (Fire Tom
Wada) 1
OUTLINE
Periodic Signal Fourier series introduction Sinusoids Orthogonality Integration vs inner product
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Wada) 2
Consider any wave is sum of simple sin and cosine Periodic Tc
)(tf10 a
)cos(31 ta )sin(11 tb )2cos(22 ta )2sin(22 tb )3cos(13 ta )3sin(13 tb
)4cos(24 ta )4sin(24 tb
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Wada) 3
Periodic signal is composed of DC + same frequency sinusoid + multiple frequency sinusoids
)4sin(2)4cos(2
)3sin(1)3cos(1
)2sin(2)2cos(2
)sin(1)cos(3
1
)(
tt
tt
tt
tt
tf
Frequency = 0 Hz
cc Tf
122
Basic frequency fc=1/Tc
2 x fc
3 x fc
4 x fc
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Spectrum of periodic signal
frequency f (Hz)
0 fc 2 ・fc
3 ・fc
4 ・fc
5 ・fc
-fc-2 ・fc
-3 ・fc
-4 ・fc
-5 ・fc
There are only n * fc (n=integer) frequencies!
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Another example (even rectangular pulse)
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Increase the number of sum (1)
N=1 N=2
N=3 N=10
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Increase the number of sum (2)
N=20 N=50
N=100 N=200
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Fourier
Jean Baptiste Joseph, Baron de FrourierFrance, 1778/Mar/21 – 1830/May/16
Fourier Series paper is written in 1807
Even discontinue function (such asrectangular pulse) can be composedof many sinusoids.
Nobody believed the paper at that time.
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Fourier Series
If f(t) ‘s period is Tc…c
c Tf
122
10
10
))2
sin()2
cos(()(
))sin()cos(()(
n cn
cn
nnn
tT
nbtT
naatf
tnbtnaatf
If we use complex exponential…,
tjn
nn ectf
)(
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Anyway, when you see the periodic signal,Please think it is just sum of sinusoids!!!
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How we can divide f(t) into sinusoids?
FilterPass
nω (Hz)
)sin()cos( tnbtna nn
•Filter is used
•an and bn
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If we integrate in [ 0 to Tc] Tc
dttfcT
0
)(
0
0
0 aTdta c
Tc
0)1cos(
0
1 dttaTc
0)1sin(
0
1 dttbTc
0)2cos(
0
2 dttaTc
0)1sin(
0
1 dttbTc
0
0
)( aTdttf c
Tc
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If we integrate in [ 0 to Tc] (2)
Tc
)1cos( t cT
dt0
dtttfcT
)1cos()(0
0
21cTa
00
0
0
0
00
•a1 can be computed
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Wada) 14
If we integrate in [ 0 to Tc] (3)
Tc
)1sin( t cT
dt0
dtttfcT
)1cos()(0
0
21cTb
0
0
0
0
0
00
•b1 can be computed
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Wada) 15
By changing multiplier, each coefficient computedTc
)sin(
)cos(
tn
or
tn
cT
dt0
One coefficient
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Sinusoidal Orthogonality
m,n: integer, Tc=1/f0
0)2sin()2cos(
)(0
)(2)2sin()2sin(
)(0
)(2)2cos()2cos(
0 00
0 00
0 00
c
c
c
T
T c
T c
dttnftmf
nm
nmT
dttnftmf
nm
nmT
dttnftmf
Orthogonal
Orthogonal
Orthogonal
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Another Orthogonality (1)
Vector inner product
)2,5( B
)5,2(Α
cos
0255)2(
BA
BΑ
Orthogonal
Θ ＝ 90 degree
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Another Orthogonality (2) n dimensional vector
nn
n
n
bababa
abb
aaa
2211
21
21
),,,(
),,,(
BΑ
B
Α
0BΑIF
THEN A and B are Orthogonal.
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Wada) 19
is same as the N dim inner product
111100
1
0
0
1
00
0 00
)2cos()2cos(
)2cos()2cos(
)2cos()2cos(
,
NN
N
i
N
ic
T
yxyxyx
N
in
N
im
TN
infT
N
imf
dttnftmf
sampledissignalIfc
cT
dt0
Freq=nω(Hz) sinusoids are Orthogonal each other (n=integer)
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Fourier Series Summary
c
c
c
T
cn
T
cn
T
c
n cn
cn
nnn
dttntfT
b
dttntfT
a
dttfT
a
tT
nbtT
naatf
tnbtnaatf
0
0
0
0
10
10
)sin()(2
)cos()(2
)(1
))2
sin()2
cos(()(
))sin()cos(()(
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Wada) 21
Complex form Fourier Series
dtetfT
c
ectf
tjnT
cn
tjn
nn
c
0)(
1
)(
)(0
)(0
)(
0
nm
nmTdte
dtee
cT tmnj
tjmT tjn
c
c
Orthogonal
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HW2
[2-1]Compute the complex form Fourier Series coefficient cn for f(x).
[2-2]Draw the Spectrum of f(t) when T0=0.04sec.
2.30
00
0
2
11
2
101
)(TtT
Tttf