Image Processing Reference
In-Depth Information
5.4 Hermitian Symmetry of the Fourier Coefficients
The Fourier series equations in Eqs. (5.14) and Eq. (5.13) are also valid for functions
f
that are complex-valued, since our scalar product can cope with complex functions
from the beginning. However, the signals of the real world are real, even if humans
put two real signals together and interpret them as the real and the imaginary parts
of a complex-valued signal, e.g., the electromagnetic signals. Yet the Fourier coeffi-
cients of real signals as well as complex signals are complex-valued. Since there is
much more information in a complex signal than in a real signal, there must be some
redundancy in the Fourier coefficients of the real signals. Here we will bring further
precision to this redundancy.
A real signal is the complex conjugate of itself since the imaginary part is zero,
f
=
f
∗
(5.26)
The Fourier series reconstruction of the signal
f
yields
f
(
t
)=
m
F
(
ω
m
) exp(
imω
1
t
)
(5.27)
so that its complex conjugate becomes
f
(
t
)
∗
=
m
F
∗
(
ω
m
) exp(
−
imω
1
t
)
(5.28)
Replacing the index
m
with
−
m
, we obtain
−∞
f
(
t
)
∗
=
F
∗
(
ω
−m
) exp(
imω
1
t
)
(5.29)
m
=
∞
In doing so, notice that the summation order of
m
has been reversed, i.e.,
m
runs
now from positive to negative integers. But because the relative signs of
m
in the
summands are unchanged, i.e.,
m
in
F
(
ω
m
) with respect to the one in exp(
imω
1
t
),
we have the same terms in the sum as before. However, summing in one direction or
the other does not change the total sum, so we are allowed to reverse the direction of
the summation back to the conventional direction:
∞
f
(
t
)
∗
=
F
∗
(
ω
−m
) exp(
imω
1
t
)
(5.30)
m
=
−∞
Accordingly, the Fourier coefficients of
f
∗
will fulfill
f
∗
=
F
∗
(
ω
−m
)
{
}
m
(5.31)
Now using Eq. (5.26) yields
f
=
f
∗
to the effect that
F
(
ω
m
)=
F
∗
(
ω
−m
)
(5.32)
