View
220
Download
0
Category
Preview:
Citation preview
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
1/41
Fourier Series Expansion
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
2/41
Fourier Series Expansion
x t=k=
cke
jk 0
t Fourier Series
(Complex ExponentialForm)
x t=A0
k=1
Ak
cos k0
tk
Fourier Series
(Trigonometric Form)
These are 3
different
forms for the
same
expression.
x t=a0k=1
[ akcos k0 tbksin k0 t ]
Fourier Series(Trigonometric Form)
T = 2/0
0= fundamental frequency (rad/sec)
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
3/41
=k=
ck
ejk
0t
x t=a0k=1
[ akcos k0 tbksin k0 t ]
How to find cand c! from aand "#
akcos(k0 t)+bksin(k0 t)=ak[1
$e
jk 0 t+1
$ejk 0 t]+bk[
1
$%e
jk 0 t1
$%ejk 0 t]
=[a
k$+
bk
$% ]ejk 0 t+ [
ak
$b
k$% ]e
jk 0 t
ck ck
ck=1
$(akj bk)
ck=1
$(ak+ j bk)
ak=ck+ck
bk=j (ckck)
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
4/41
&imilarly "y expanding out t'e t'term a"oe using Eulers
Formula you can see t'at
ck=A
k
$ e j k
ck=Ak
$ e
jk
Ak=$ck
k=ck
x t=A0
k=1
Ak
cos k0
tk =
k=
ck
ejk
0t
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
5/41
c0=A0
ck=1
$Ake
jk k=1* $* +*
ck=1$
Akejk k=1* $* +*
c0=a0
ck=1
$akjbk k=1* $* +*
ck=1
$ akjbk k=1* $* +*
Exponential Form
x t=k=
cke
jk 0t
Trig Form: Amplitude & Phase
x t=A0
k=1
Akcos k
0t
k
Trig Form: Sine-Cosinex t=a
0
k=1
[ akcos k0 tbksin k0 t ]
a0=c
0
ak=ckck, k=1*$*+*
bk=jckck, k=1*$*+*
A0=c0
Ak=$ck k=1* $* +*
k, ck k=1* $* +*
A0
=a0
Ak=ak$bk
$
k=tan1bka
k
a0=A0
ak=Akcosk
bk=A
ksin
k
Three (Eui!alent" Forms of FS and Their #elationships
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
6/41
x t=k=
cke
jk 0
t
ck= 1
Tt0t0Tx tejk 0 tdt
Complex Exponential Form
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
7/41
Fourier Series Trigonometric Form
f(t)=a0+k=1 [akcos(k0 t)+bksin (k0 t)]
ak=$
T
Tf(t)cos(k0 t)dt (k0)
bk= $TT f(t)sin(k0 t)dt (k0)
a0
=1
TT
ftdt
-e can derie t'ese results in t'e same way as t'e
complex exponential case using ort'ogonal functions.
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
8/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
9/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
10/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
11/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
12/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
13/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
14/41
f(t) = 1 for t = [-T/2, T/2]
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
15/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
16/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
17/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
18/41
Fourier Series Expansion
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
19/41
Fourier Series Properties
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
20/41
Fourier Series Expansion
x t=k=
cke
jk 0
t Fourier Series
(Complex ExponentialForm)
x t=A0
k=1
Ak
cos k0
tk
Fourier Series
(Trigonometric Form)
These are 3
different
forms for the
sameexpression.
x t=a0k=1
[ akcos k0 tbksin k0 t ]
Fourier Series(Trigonometric Form)
T = 2/0
0= fundamental frequency (rad/sec)
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
21/41
=
k=
ck
ejk
0t
x t=a0k=1
[ akcos k0 tbksin k0 t ]
How to find cand c
!from a
and "
#
akcos(k0 t)+bksin(k0 t)=ak[1
$
ejk 0 t+
1
$
ejk 0 t]+bk[
1
$%
ejk 0 t
1
$%
ejk 0 t]
=[ak
$+
bk
$%]e jk 0 t+ [
ak
$
bk
$%]ejk 0 t
ck ck
ck=1
$(akj bk)
ck=1
$(ak+ j bk)
ak=ck+ck
bk=j (ckck)
ecall from last time
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
22/41
Fourier &eries roperties
if x (t)=a0+k=1
[ akcos (k0 t)+bksin (k0 t) ] is real*t'en "ot' akand bkare real 2nd since
ck=akj bk
$
ck=ak+ j bk
$
ck=ck3
t'en
ck=ck ck=ckie*
4Hermitian &ymmetry5
6agnitude is an een function 'ase is an odd function
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
23/41
ecall t'e full wae rectifier example t'at we did -e deried
and plotted t'e complex exponential F& coefficients for t'e
rectified sinusoid
The t$o-sided spe%trum after the re%tifier:
A$
+
$A
0
0$ 0+ 07 0800$0+0708
0
0$ 0+ 07 08
0
0$
0+
07
08
ck
kc
x (t)=Asin ($9
T1t)
:ote t'e Hermitian symmetry.
t
x(t)
A
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
24/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
25/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
26/41
Fourier Series
x t=k=
cke
jk 0
t
ck=
1
T
t0
t0Tx te
jk 0
tdt
T = 2/0
0= fundamental frequency (rad/sec)
T'is is true for a
ery wide class of
periodic signals.
(see ;iric'let
Conditions)
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
27/41
Time s'ifting a periodic signal creates a new periodic signal. How
are t'e Fourier series coefficients of t'e two signals related#
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
28/41
hat has the time shift
done to the FS %oeffi%ents'
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
29/41
Fourier Series
x t=k=
cke
jk 0
t
ck=
1
T
t0
t0Tx te
jk 0
tdt
T = 2/0
0= fundamental frequency (rad/sec)
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
30/41
ftcn
ftt0cn ejn0 t0
Fourier &eries roperties
Time &'ifting
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
31/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
32/41
Fourier Series Expansion
x t=k=
cke
jk 0
t Fourier Series
(Complex ExponentialForm)
x t=A0
k=1
Ak
cos k0
tk
Fourier Series
(Trigonometric Form)
These are 3
different
forms for the
sameexpression.
x t=a0k=1
[ akcos k0 tbksin k0 t ]
Fourier Series(Trigonometric Form)
T = 2/0
0= fundamental frequency (rad/sec)
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
33/41
Can we use t'e
differentiation property
to 'elp find t'e F&coefficients off(t) ?
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
34/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
35/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
36/41
6atla" example
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
37/41
ftcn
ftt0cn ejn0 t0
Fourier &eries roperties
Time &'ifting
Time ;ifferentiation
ftcndft
dt jn0cn
E l
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
38/41
Find t'e Fourier series coefficients* dn* of !(t)
Example
dn=1
TT/$
T/$!tejn0 tdt
dn=1TT/7
T/7tT
7ejn0 tdt
-e 'ae to integrate t'is "y parts. =nstead* we can use
t'e differentiation t'eorem.
E l
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
39/41
Example
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
40/41
8/10/2019 EE3TP4 13b FourierSeriesProperties v3
41/41
Recommended