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Fourier Coefficients Analysis: Real and Odd vs Real and Even Signals, Exercises of Signals and Systems Theory
An in-depth analysis of fourier coefficients for real and odd signals versus real and even signals. Topics covered include the relationship between time domain functions and fourier coefficients, time reversal, and time scaling. The document also includes various plots and equations to illustrate the concepts.
Typology: Exercises
2011/2012
1 / 13
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(^
1
1 T
t
T
T
t
t x^
7/20/
3
T T
dt
T
dt
e t x
T
a
T T
T
t
jkw
1
0
0
1 1
0
^
The interval for integration is from –T/
≤
t
≤
T/
For k= 0
1 1
0
1 1
0
0 1
T T
t jkw
T T
t jkw
k^
e T
jkw
dt
e
T
a
^
Otherwise
k
T
kw
T
T
k
T
kw
T
kw
T
kw
e j
e T
kw
a
T jkw
T jkw
k
sin(
sin( 2
)
sin( 2
1 0
1 0
0
1 0
0
1 0
1 0
^
Assuming T = 4T
1
k k
a
k
sin(
3
3
1
1
0
^
a
a
a
a
a
docsity.com
k
k
FS
t
jk
T
t
jk
T
FS
k
t
jk
T
FS
k
t
jk
T
FS
k
t
jk
T
FS
k
t
jk
T
FS
b a t y t x
dt
e t y
T
dt
e t x T t y t x
b
dt
e t y T t y
a
dt
e t x T t x
b
dt
e t y T t y
a
dt
e t x T t x
o
o
o o o o
k
t
jk
jk
T
t
jk
t
jk
T
t
jk
T
t
jk
T
o
FS
o
a
e
d
e
x
T
e
d
e
x
T
dt
e
x
T
dt
e t t x T t t x
o o
o
o o
o
o
o
o
o
) (
)^1 (
1
) (
1
)
(
1
)
(
)
(
)
(
Replace t–t
o^
by
τ
Time reversal applied to a continuous time signal results in a
time reversal of the corresponding sequence of Fourier Coefficients
Consequence of this property:
If x(t) is even i.e. x(-t)=x(t) – then its FS Coeffs are also even a
=a-k
k
If x(t) is odd i.e. x(-t)=-x(t) – then its FS Coeffs are also odd a
=-a-k
k
k
m
jk
T
t
jk
T
t
jk
T
FS
k
t
jk
T
FS
a
dm
e
m
x
T
dt
e
t
x
T
dt
e t x T t x
a
dt
e t x T t x
o o
o
o
) (
)
(
Taking complex conjugate of a signal the FS coefficients obtained
are time reversed and complex conjugate
Case: x(t) is real.In this case x(t)=x*(t) Then a
=a*-k
k^ i.e. FS coeffs are Conjugate Symmetric