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.

Recall from §7.3 that a signal $ x(n)$ is said to be even if $ x(-n)=x(n)$, and odd if $ x(-n)=-x(n)$. Below are are Fourier theorems pertaining to even and odd signals and/or spectra.


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:

$\displaystyle \zbox {x\;\mbox{even} \;\longleftrightarrow\;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

$\displaystyle X(k)$ $\displaystyle \isdef$ $\displaystyle \sum_{n=0}^{N-1}x(n) e^{-j\omega_k n}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^{N-1}[x_r(n)+jx_i(n)] \cos(\omega_k n) - j [x_r(n)+jx_i(n)] \sin(\omega_k n)$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^{N-1}[x_r(n)\cos(\omega_k n) + x_i(n)\sin(\omega_k n)]$  
    $\displaystyle \;\,\mathop{+} j [x_i(n)\cos(\omega_k n) - x_r(n)\sin(\omega_k n)].
\protect$ (7.5)

Let even$ _n$ denote a function that is even in $ n$, such as $ f(n)=n^2$, and let odd$ _n$ denote a function that is odd in $ n$, such as $ f(n)=n^3$, Similarly, let even$ _{nk}$ denote a function of $ n$ and $ k$ that is even in both $ n$ and $ k$, such as $ f(n,k)=n^2k^2$, and odd$ _{nk}$ mean odd in both $ n$ and $ k$. Then appropriately labeling each term in the last formula above gives

\begin{eqnarray*}
X(k)&=&\sum_{n=0}^{N-1}\mbox{even}_n\cdot\mbox{even}_{nk}
+ ...
...10pt]
&=& \mbox{even}_k + j \cdot \mbox{even}_k = \mbox{even}_k.
\end{eqnarray*}


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

$\displaystyle \zbox {x\;\mbox{real and even} \;\longleftrightarrow\;X\;\mbox{real and even}}$ (7.6)


Proof: This follows immediately from setting $ x_i(n)=0$ in the preceding proof. From Eq.$ \,$(7.5), we are left with

$\displaystyle X(k) = \sum_{n=0}^{N-1}x_r(n)\cos(\omega_k n).
$

Thus, the DFT of a real and even function reduces to a type of cosine transform,7.12

Instead of adapting the previous proof, we can show it directly:

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


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 $ \pm\pi$, not 0. When a real spectrum is positive at dc (i.e., $ X(0)>0$), 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 $ \pi $ 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 $ \pi $ (or $ -\pi$, which is the same phase as $ +\pi$). 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]).


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Shift Theorem
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Conjugation and Reversal