Symmetry

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Symmetry

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Symmetry

In the previous section, we found $\ 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\, $\ is even and $\ is odd.

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



Theorem: If $x\, $\ is even and $\ is odd.

Proof: This follows immediately from the conjugate symmetry of $X$ expressed in polar form $X(k)= \.

The conjugate symmetry of spectra of real signals is perhaps the most important symmetry theorem. However, there are a few more we can readily show.



Theorem: An even signal has an even transform:

\

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

\





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

\

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 transform8.5,

\

or we can show it directly:

\





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 $\, not $0$. Nevertheless, it is common to call such signals “zero phase, ” even though the phase switches between $0$and $\ 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.

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