Symmetry

In the previous section, we found $\hbox{\sc Flip}(X) = \overline{X}$ when $x$ 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 $f_s/2$ Hz. On the other hand, the spectrum of a complex signal must be shown, in general, from $-f_s/2$ to $f_s/2$ (or from 0 to $f_s$), since the positive and negative frequency components of a complex signal are independent.

Theorem: If $x\in{\bf R}^N$, then re$\left\{X\right\}$ is even and im$\left\{X\right\}$ is odd.

Proof: This follows immediately from the conjugate symmetry of $X$ for real signals $x$.

Theorem: If $x\in{\bf R}^N$, $\left\vert X\right\vert$ is even and $\angle{X}$ is odd.

Proof: This follows immediately from the conjugate symmetry of $X$ expressed in polar form $X(k)= \left\vert X(k)\right\vert e^{j\angle{X(k)}}$.

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:

$$\zbox {x\,\mbox{even} \leftrightarrow X\,\mbox{even}}$$

Proof: Express $x$ in terms of its real and imaginary parts by $x\isdeftext x_r + j x_i$. Note that for a complex signal $x$ to be even, both its real and imaginary parts must be even. Then

$$\begin{eqnarray*} X(k) &\isdef & \sum_{n=0}^{N-1}x(n) e^{-j\omega_k n} \\ &=&\s... ...,\mathop{+} j [x_i(n)\cos(\omega_k n) - x_r(n)\sin(\omega_k n)]. \end{eqnarray*}$$

Substituting the label ``odd'' or ``even'' for each signal above gives

$$\begin{eqnarray*} &=&\sum_{n=0}^{N-1}\mbox{even}\cdot\mbox{even} + \underbrace... ...t\mbox{even} = \mbox{even} + j \cdot \mbox{even} = \mbox{even}. \end{eqnarray*}$$

Theorem: A real even signal has a real even transform:

$$\zbox {x\,\mbox{real and even} \leftrightarrow X\,\mbox{real and even}}$$

Proof: This follows immediately from setting $x_i(n)=0$ in the preceding proof and seeing that the DFT of a real and even function reduces to a type of cosine transform,^7.7

$$X(k) = \sum_{n=0}^{N-1}x(n)\cos(\omega_k n),$$

or we can show it directly:

$$\begin{eqnarray*} X(k) &\isdef & \sum_{n=0}^{N-1}x(n) e^{-j\omega_k n} = \sum_{... ...ven}\cdot\mbox{even} = \sum_{n=0}^{N-1}\mbox{even} = \mbox{even} \end{eqnarray*}$$

Definition: A signal with a real spectrum (such as a 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 $\pm\pi$, not 0. Nevertheless, it is common to call such signals ``zero phase, '' even though the phase switches between 0 and $\pi$ at the zero-crossings of the spectrum. Such zero-crossings typically occur at low amplitude in practice, such as in the ``sidelobes'' of the DTFT of an FFT window (see Chapter 8).