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 signal above gives

**Theorem: **A real even signal has a real even transform:

*Proof: *This follows immediately from setting in the preceding
proof. From Eq. (7.7), we are left with

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 an FFT window (see
Chapter 8).

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