Symmetry
In the previous section, we found

when

is
real. This fact is of high practical importance. It says that the
spectrum of every real
signal is
Hermitian.
Due to this symmetry, we may discard all
negative-frequency spectral
samples of a real signal and regenerate them later if needed from the
positive-frequency samples. Also, spectral plots of real signals are
normally displayed only for positive frequencies;
e.g.,
spectra of
sampled signals are normally plotted over the range 0 Hz to

Hz. On the other hand, the
spectrum of a
complex signal must
be shown, in general, from

to

(or from 0 to

),
since the positive and negative frequency components of a complex
signal are independent.

Recall from §
7.3 that a signal

is said to be
even if

, and
odd if

. Below
are are
Fourier theorems pertaining to even and odd signals and/or
spectra.
Theorem: If

, then
re

is
even and
im

is
odd.
Proof: This follows immediately from the conjugate symmetry of

for real signals

.
Theorem: If

,

is
even and

is
odd.
Proof: This follows immediately from the conjugate symmetry of

expressed
in polar form

.
The conjugate symmetry of spectra of real signals is perhaps the most
important symmetry theorem. However, there are a couple more we can readily
show:
Theorem: An even signal has an even transform:
Proof:
Express

in terms of its real and imaginary parts by

. Note that for a complex signal

to be even, both its real and
imaginary parts must be even. Then
Let
even

denote a function that is even in

, such as

, and let
odd

denote a function that is odd in

, such as

, Similarly, let
even

denote a
function of

and

that is even in both

and

, such as

, and
odd

mean odd in both

and

.
Then appropriately labeling each term in the last formula above gives
Theorem: A real even signal has a real even transform:
 |
(7.6) |
Proof: This follows immediately from setting

in the preceding
proof. From Eq.

(
7.5), we are left with
Thus, the
DFT of a real and
even function reduces to a type of
cosine transform,
7.12
Instead of adapting the previous proof, we can show it directly:
Definition: A signal with a real
spectrum (such as any real, even signal)
is often called a
zero phase signal. However, note that when
the spectrum goes
negative (which it can), the phase is really

, not 0. When a real spectrum is positive at
dc (
i.e.,

), it is then truly zero-phase over at least some band
containing dc (up to the first zero-crossing in frequency). When the
phase switches between 0 and

at the zero-crossings of the
(real) spectrum, the spectrum oscillates between being zero phase and
``constant phase''. We can say that all real spectra are
piecewise constant-phase spectra, where the two constant values
are 0 and

(or

, which is the same phase as

). In
practice, such zero-crossings typically occur at low magnitude, such
as in the ``
side-lobes'' of the
DTFT of a ``zero-centered symmetric
window'' used for
spectrum analysis (see Chapter
8 and Book IV
[
70]).
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