Even and Odd Functions

Ηλεκτρονική - Συσκευές

10 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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Even and Odd Functions
A function,f,is even (or symmetric) when
f (x) = f (x):
A function,f,is odd (or antisymmetric) when
f (x) =f (x):
Even and Odd Functions (contd.)
Theorem 5.1 Any function can be written as a
sumof even and odd functions.
f (t) =
1
2
2
4
f (t) + f (t)  f (t)
|
{z
}
0
+f (t)
3
5
=
1
2
[ f (t) + f (t)]
|
{z
}
f
e
+
1
2
[ f (t)  f (t)]
|
{z
}
f
o
f
e
is even because f
e
(t) = f
e
(t):
f
e
(t) = f (t) + f (t)
= f (t) + f (t)
= f
e
(t):
f
o
is odd because f
o
(t) =f
o
(t):
f
o
(t) = f (t)  f (t)
= [ f (t)  f (t)]
= f
o
(t):
Even and Odd Functions (contd.)
Theorem5.2 The integral of the product of odd
and even functions is zero.
Z


f
e
(x) f
o
(x)dx =
Z
0

f
e
(x) f
o
(x)dx+
Z

0
f
e
(x) f
o
(x)dx:
Substituting x for x and dx for dx in the ﬁrst
termyields:
Z
0

f
e
(x) f
o
(x)dx+
Z

0
f
e
(x) f
o
(x)dx
=
Z

0
f
e
(x) f
o
(x)dx+
Z

0
f
e
(x) f
o
(x)dx
=
Z

0
[ f
e
(x) f
o
(x) + f
e
(x) f
o
(x)] dx:
Substituting f
e
(x) for f
e
(x) and f
o
(x) for
f
o
(x) yields:
Z

0
[ f
e
(x) f
o
(x)  f
e
(x) f
o
(x)]
|
{z
}
0
dx:
Fourier TransformSymmetry
The Fourier transformof f (t) is deﬁned to be:
F(s) =
Z


f (t)e
j2st
dt:
This can be rewritten as follows:
F(s) =
Z


f (t)cos(2st)dt  j
Z


f (t)sin(2st)dt:
Substituting f
e
(t) + f
o
(t) for f (t) yields:
F(s) =
Z


f
e
(t)cos(2st)dt +
Z


f
o
(t)cos(2st)dt
j
Z


f
e
(t)sin(2st)dt  j
Z


f
o
(t)sin(2st)dt:
Fourier TransformSymmetry (contd.)
However,the second and third terms are zero
(Theorem5.2):
F(s) =
Z


f
e
(t)cos(2st)dt  j
Z


f
o
(t)sin(2st)dt:
It follows that:
F(s) =F
e
(s) +F
o
(s):
Even Functions
Theorem 5.3 The Fourier transform of a real
even function is real.
F(s) =
Z


f (t)e
j2st
dt
=
Z


f (t)[cos(2st)  j sin(2st)] dt
=
Z


f (t)cos(2st)dt
which is real.
Odd Functions
Theorem 5.4 The Fourier transform of a real
odd function is imaginary.
F(s) =
Z


f (t)e
j2st
dt
=
Z


f (t)[cos(2st)  j sin(2st)] dt
= j
Z


f (t)sin(2st)dt
which is imaginary.
Even Functions (contd.)
Theorem5.5 The Fourier transformof an even
function is even.
F(s) =
Z


f (t)e
j2st
dt
Substituting f (t) for f (t) yields:
F(s) =
Z
t=
t=
f (t)e
j2st
dt:
Substituting u for t and du for dt yields:
=
Z
u=
u=
f (u)e
j2s(u)
du
=
Z


f (u)e
j2(s)u
du
= F(s):
Odd Functions (contd.)
Theorem 5.6 The Fourier transform of an odd
function is odd.
F(s) =
Z


f (t)e
j2st
dt
Substituting f (t) for f (t) yields:
F(s) =
Z
t=
t=
f (t)e
j2st
dt:
Substituting u for t and du for dt yields:
=
Z
u=
u=
f (u)e
j2s(u)
du
=
Z


f (u)e
j2(s)u
du
= F(s):
Fourier TransformSymmetry (contd.)
 The Fourier transform of the even part (of a
real function) is real (Theorem5.3):
F ff
e
g(s) =F
e
(s) =Re(F
e
(s)):
 The Fourier transformof the even part is even
(Theorem5.5):
F ff
e
g(s) =F
e
(s) =F
e
(s):
 The Fourier transform of the odd part (of a
real function) is imaginary (Theorem5.4):
F ff
o
g(s) =F
o
(s) =Im(F
o
(s)):
 The Fourier transformof the odd part is odd
(Theorem5.6):
F ff
o
g(s) =F
o
(s) =F
o
(s):
Hermitian Symmetry
We can summarize all four symmetries possessed
by the Fourier transform of a real function as
follows:
F(s) = F
e
(s) +F
o
(s)
= F
e
(s) F
o
(s)
=
F
e
(s) +F
o
(s)
=
F(s):
Hermitian Symmetry (contd.)
This symmetry matches the symmetry of the
functions which comprise the Fourier basis:
e
j2st
=
e
j2s(t)
: