ece 2c, notes set 1: fourier seriesfourier series mark rodwell university of california, santa...
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class notes, M. Rodwell, copyrighted 2013
ECE 2C, notes set 1: Fourier Series
Mark Rodwell
University of California, Santa Barbara
[email protected] 805-893-3244, 805-893-3262 fax
class notes, M. Rodwell, copyrighted 2013
Goals of this note set:
response.ansient circuit tr find toansformsFourier tr use tohow Review
they workhow ...and
from come they ...where
.*is* ansformFourier tr a what Understand
.*is* seriesFourier a what Understand
phasors as wavessinerepresent tohowRemember tje
class notes, M. Rodwell, copyrighted 2013
Fourier Transforms and Fourier Series
intuition. and ingunderstandbetter want We
this.do totoolsefficient need We
equations. aldifferenti partial andordinary solveoften must We
?Why
class notes, M. Rodwell, copyrighted 2013
Solving Simple Differential Equations
class notes, M. Rodwell, copyrighted 2013
Solving Simple Differential Equations
class notes, M. Rodwell, copyrighted 2013
Phasors
class notes, M. Rodwell, copyrighted 2013
Differential Equations: Easy to Solve with Sinewaves
class notes, M. Rodwell, copyrighted 2013
Sinewaves and Complex Exponentials
2)()sin(
)sin(2
____________________
)sin()cos(
)sin()cos(
2)()cos(
)cos(2
____________________
)sin()cos(
)sin()cos(
)sin()cos( :identityEuler
?it is What ? ith Why work w
jeet
tjee
tjte
tjte
eet
tee
tjte
tjte
tjte
e
tjtj
tjtj
tj
tj
tjtj
tjtj
tj
tj
tj
tj
class notes, M. Rodwell, copyrighted 2013
Sinewaves and Complex Exponentials
ls.exponentia ofpair ait with driving
as same theis wavesineor cosine ath circuit wi a Driving
2)()sin( and 2)()cos( jeeteet tjtjtjtj
)cos(0 tVVin
)exp()2/(
)exp()2/(
02
01
jtjVV
jtjVV
in
in
applied. with and applied with for solving
separately ion,superpositby problem theSolve
11 VVVV outout
class notes, M. Rodwell, copyrighted 2013
Complex Exponentials: Ignoring the Real Part
)exp()exp(||||
)exp()(
)exp()exp(||||
)exp()(
tjjV
tjVtV
tjjV
tjVtV
out
in
Vin
outout
Vin
inin
notes. of page previous from steps though Working:2 Proof
ion.superposit of Principle :1 Proof
output. theofpart imaginary theignore input,complex Apply the
)cos()}sin()Re{cos(}Re{ ofpart Real ttjtee tjtj
)cos(||||
)}exp(Re{)(
)cos(||||
)}exp(Re{)(
out
in
Vout
outout
Vin
inin
jtV
tjVtV
jtV
tjVtV
class notes, M. Rodwell, copyrighted 2013
Representing signals as sets of sinewaves
http://en.wikipedia.org/wiki/File:Voice_waveform_and_spectrum.png
? thisdo wedo How
sinewaves. of sum a asit represent can ...but we
.sinewave.. anot is signalinput This
class notes, M. Rodwell, copyrighted 2013
Understanding Fourier Transforms and Series
? thisdo wedo How
ls).exponentiacomplex (or sinewaves of
sum a as signal a writing: transformand seriesFourier
done. isit or why isit what of sense little having while
memorized, TransformFourier thehaveoften Students
class notes, M. Rodwell, copyrighted 2013
Understanding Fourier: Signals are Vectors
signalinput an Take
steps. discrete of
set aby eApproximat
small) very makecan we:nintegratio as method (same t
t
ninnin VtV ,)(
1t 2t nt
vector.a is )( So, numbers. oflist a isA vector tVin
)(
)(
)(
2
1
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in
in
in
tV
tV
tV
V
.)(),...,(),(
numbers of
list a becomes
21 Nininin
in
tVtVtV
V
class notes, M. Rodwell, copyrighted 2013
Writing Vectors as Sums of Basis Vectors
332211
11131211
1331231111
1
11
332311
321
so 0, and , 0 ,1but
productdot the take, find To
direction.- in the of projection simply the is
,, vectorsbasis theof sum a is vector The
xxVxxVxxVV
VxVxxxxxx
xxVxxVxxVxV
V
xVV
xVxVxVV
xxxV
x
xxx
x
x
xxx
class notes, M. Rodwell, copyrighted 2013
Vector Projection
11
1
11
332311
321
product)(dot onto project this,find To
". from made is much how" is
,, vectorsbasis of sum a of up made is
xVV
xV
xVV
xVxVxVV
xxxV
x
x
xxx
class notes, M. Rodwell, copyrighted 2013
Change of Basis Vectors
332211
321
,, vectorsbasis theof sum a is vector The
xxVxxVxxVV
xxxV
332211
321
,, vectorsbasis theof sum a also is vector The
yyVyyVyyVV
yyyV
vectors.basis ofset onethan
more using vector arepresent can We
class notes, M. Rodwell, copyrighted 2013
Understanding Fourier: Continuous time
)()(...)()()()(
? isWhat
12121111
1
NN txtVtxtVtxtVxV
xV
T
dttxtVT
txV(t)
t
0
1111 )()(1
)(
integral.an becomes this,0 make weIf
class notes, M. Rodwell, copyrighted 2013
Understanding Fourier: Our Set of Basis Vectors
tT
ntbtT
nta
tT
tbtT
ta
tT
tbtT
ta
ta
nn
2sin2)(
2cos2)(
22sin2)(
22cos2)(
21sin2)(
21cos2)(
1)(
22
11
0
class notes, M. Rodwell, copyrighted 2013
Understanding Fourier: Our Set of Basis Vectors
T
NN
T
NN
T
dttT
nT
tbtb
dttT
nT
tata
dtT
tata
0
2
0
2
0
00
1...2
sin21
)()(
1...2
cos21
)()(
111
)()(
?length unit of they Are
class notes, M. Rodwell, copyrighted 2013
Understanding Fourier: Our Set of Basis Vectors
.0)()()()(
yourself. nscombinatioother out Sketch
zero.clearly is and 0between integral Its
).()(product thesketched is Here
?other each lar toperpendicu functions theseAre
3132
21
etctatbtata
Ttt
tbtb
vectors.basis ofset a are )(),(),( functions ofset The 0 tbtata nn
class notes, M. Rodwell, copyrighted 2013
We Have Just Derived the Fourier Series
tT
ntbtT
nta
tT
tbtT
ta
tT
tbtT
ta
ta
nn
2sin2)(
2cos2)(
22sin2)(
22cos2)(
21sin2)(
21cos2)(
1)(
22
11
0
T
NN
T
NN
dttbtVT
tbtV
dttatVT
tatV
tbtbtVtbtbtV
tatatVtatatVtatatVtV
0
0
2211
221100
)()(1
)()(
)()(1
)()(
...)()()()()()(...
...)()()()()()()()()()(
class notes, M. Rodwell, copyrighted 2013
We Have Just Derived the Fourier Series
T
n
T
n
T
dttT
ntVT
b
dttT
ntVT
a
dttVT
a
tT
btT
btT
b
tT
atT
atT
aatV
0
0
0
0
321
3210
2sin2)(
1
2cos2)(
1
)(1
...2
3sin22
2sin22
1sin2...
...2
3cos22
2cos22
1cos2)(
ook.your textbin thosefromdifferent is correct, though equations, theseof form the:Warning
class notes, M. Rodwell, copyrighted 2013
Fourier Series: What does it mean ?
T
n
T
n
T
dttT
ntVT
b
dttT
ntVT
a
dttVT
a
tT
btT
btT
b
tT
atT
atT
aatV
0
0
0
0
321
3210
2sin2)(
1
2cos2)(
1
)(1
...2
3sin22
2sin22
1sin2...
...2
3cos22
2cos22
1cos2)(
frequency.at that wavecosineor wavesine theand signal the
of integral)n (projectioproduct dot simply the is cosine)or (sine
frequency any at component ssignal' theof magnitude The
waves.cosine and sinewaves DC, of sum a as signal a writeWe
class notes, M. Rodwell, copyrighted 2013
Complex Exponential Fourier Series
T
n
n
n
n
n
n
nn
n
tj
T
dttjntVT
c
tjnctctctVtV
Ttjntc
tte
dttbtaT
tbta
0
0
0
00
0
*
)exp()(1
where
)exp()()()()(
/2 where)exp()(:functions Basis
)sin()cos( :identityEuler
)()(1
)()( :functionscomplex for product Dot
class notes, M. Rodwell, copyrighted 2013
Fourier Integrals / Fourier Transforms
dttjtVtjtVjF
dtjjFdtjtjtVtV
dttbtaT
tbta
T
)exp()()exp()()( where
)exp()(2
1)exp()exp()(
2
1)(
)()(1
)()( integral.an becomes sfrequencie of sum the
infinity, ofunction t theof Tlimit time the take weIf
0
*
class notes, M. Rodwell, copyrighted 2013
Solving Circuits Using Fourier Transforms
class notes, M. Rodwell, copyrighted 2013
Solving Circuits Using Fourier Transforms
class notes, M. Rodwell, copyrighted 2013
Solving Circuits Using Fourier Transforms
class notes, M. Rodwell, copyrighted 2013
Solving Circuits Using Fourier Transforms
class notes, M. Rodwell, copyrighted 2013
Solving Circuits Using Fourier Transforms
class notes, M. Rodwell, copyrighted 2013
Solving Circuits Using Fourier Transforms
class notes, M. Rodwell, copyrighted 2013
Solving Circuits Using Fourier Transforms