### 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 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

*cosine transform*,

^{7.12}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 [70]).

**Next Section:**

Shift Theorem

**Previous Section:**

Conjugation and Reversal