Symmetry of the DTFT for Real Signals
Most (if not all) of the signals we deal with in practice are real
signals. Here we note some spectral symmetries associated with real signals.
The previous section established that the spectrum
of every real
In other terms, if a signal
is real, then its spectrum
(``conjugate symmetric''). Hermitian spectra
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
is antisymmetric about argument zero.
Real Even (or Odd) Signals
If a signal is even
in addition to being real, then its DTFT
also real and even. This follows immediately from the Hermitian
symmetry of real signals, and the fact that the DTFT of any even
signal is real:
This is true since cosine is even, sine is odd, even times even is
even, even times odd is odd, and the sum over all samples of an odd
signal is zero. I.e.
is real and even, the following are true:
Similarly, if a signal is odd
and real, then its DTFT is odd
and purely imaginary
. This follows from Hermitian symmetry for
real signals, and the fact that the DTFT of any odd signal is imaginary.
where we used the fact that
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