Symmetry of the DTFT for Real Signals

Most (if not all) of the signals we deal with in practice are real signals. The Fourier transform of real signals exhibits conjugate symmetry. That is,

In other terms, if a signal is real, its spectrum is Hermitian, or ``conjugate symmetric.''

Hermitian spectra have the following equivalent characterizations:

• The real part is even, while the imaginary part is odd:
  
  

• The magnitude is even, while the phase is odd:
  
  

Note that an even function is symmetric about argument zero while an odd function is antisymmetric about argument zero.

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Subsections

• Real Even (or Odd) Signals

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Order a Hardcopy of Spectral Audio Signal Processing

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.