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23/04/18System Arch 2008 (Fire Tom
Wada) 1
OUTLINE
Periodic Signal Fourier series introduction Sinusoids Orthogonality Integration vs inner product
23/04/18System Arch 2008 (Fire Tom
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
23/04/18System Arch 2008 (Fire Tom
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
23/04/18System Arch 2008 (Fire Tom
Wada) 4
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!
23/04/18System Arch 2008 (Fire Tom
Wada) 8
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.
23/04/18System Arch 2008 (Fire Tom
Wada) 9
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
)(
23/04/18System Arch 2008 (Fire Tom
Wada) 10
Anyway, when you see the periodic signal,Please think it is just sum of sinusoids!!!
23/04/18System Arch 2008 (Fire Tom
Wada) 11
How we can divide f(t) into sinusoids?
FilterPass
nω (Hz)
)sin()cos( tnbtna nn
•Filter is used
•an and bn
23/04/18System Arch 2008 (Fire Tom
Wada) 12
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
23/04/18System Arch 2008 (Fire Tom
Wada) 13
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
23/04/18System Arch 2008 (Fire Tom
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
23/04/18System Arch 2008 (Fire Tom
Wada) 15
By changing multiplier, each coefficient computedTc
)sin(
)cos(
tn
or
tn
cT
dt0
One coefficient
23/04/18System Arch 2008 (Fire Tom
Wada) 16
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
23/04/18System Arch 2008 (Fire Tom
Wada) 17
Another Orthogonality (1)
Vector inner product
)2,5( B
)5,2(Α
cos
0255)2(
BA
BΑ
Orthogonal
Θ = 90 degree
23/04/18System Arch 2008 (Fire Tom
Wada) 18
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.
23/04/18System Arch 2008 (Fire Tom
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)
23/04/18System Arch 2008 (Fire Tom
Wada) 20
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(()(
23/04/18System Arch 2008 (Fire Tom
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