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 0 Hz to Hz. On the other hand, the spectrum of a complex signal must be shown, in general, from to (or from 0 to ), since the positive and negative frequency components of a complex signal are independent.
Recall from §7.3 that a signal is said to be even if , and odd if . Below are are Fourier theorems pertaining to even and odd signals and/or spectra.
Theorem: If , then re is even and im 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 couple 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
Let even denote a function that is even in , such as , and let odd denote a function that is odd in , such as , Similarly, let even denote a function of and that is even in both and , such as , and odd mean odd in both and . Then appropriately labeling each term in the last formula above gives
Theorem: A real even signal has a real even transform:
Proof: This follows immediately from setting in the preceding proof. From Eq.(7.5), we are left with
Instead of adapting the previous proof, we can show it directly:
Definition: A signal with a real spectrum (such as any 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 0. When a real spectrum is positive at dc (i.e., ), it is then truly zero-phase over at least some band containing dc (up to the first zero-crossing in frequency). When the phase switches between 0 and at the zero-crossings of the (real) spectrum, the spectrum oscillates between being zero phase and ``constant phase''. We can say that all real spectra are piecewise constant-phase spectra, where the two constant values are 0 and (or , which is the same phase as ). In practice, such zero-crossings typically occur at low magnitude, such as in the ``side-lobes'' of the DTFT of a ``zero-centered symmetric window'' used for spectrum analysis (see Chapter 8 and Book IV ).
Conjugation and Reversal