NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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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 
Hz to
Hz.
On the other hand, the spectrum of a complex signal must be
shown, in general, from 
to (or from to 
),
since the positive and negative frequency components of a complex
signal are independent.
Theorem: If 
, 
is even and 
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 few 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
Theorem: A real even signal has a real even
transform:
Proof: This follows immediately from setting
in the preceding proof and seeing that the DFT of a real and
even
function reduces to a type of cosine transform8.5,
or we can show it directly:
Definition: A signal with a real spectrum (such as a 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
.
Nevertheless, it is common to call such signals ''zero phase, ''
even though the phase switches between and
at
the zero-crossings of the spectrum. Such zero-crossings typically
occur at low amplitude
in practice, such as in the ''sidelobes'' of the DTFT of an FFT
window.
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