**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## 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 Hz to Hz. On the other hand, the spectrum of acomplexsignal 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 , isevenand isodd.

Proof:This follows immediately from the conjugate symmetry of for real signals.

Theorem:If , isevenand isodd.

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 ofcosine transform^{8.5},

or we can show it directly:

Definition:A signal with a real spectrum (such as a real, even signal) is often called azero phase signal. However, note that when the spectrum goesnegative(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.