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Mathematics of the DFT
    Fourier Theorems for the DFT
       Fourier Theorems
          Symmetry

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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 [67]).

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Previous: Conjugation and Reversal
Next: Shift Theorem
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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