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 signal satisfies
In other terms, if a signal is real, then its spectrum is Hermitian (``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:
Real Even (or Odd) Signals
If a signal is even in addition to being real, then its DTFT is 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.,
If 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
Shift Theorem for the DTFT