Discrete Fourier transform mirrored?

AI Thread Summary
The discrete Fourier transform (DFT) produces two peaks for a single sine wave due to the presence of both positive and negative frequency components, resulting in a mirrored spectrum. This mirror image appears because of aliasing from sampling, where negative frequencies are represented in the upper half of the DFT output. For real input signals, the second half of the DFT is a mirror image of the first half, allowing for simplifications in analysis. The real part of the DFT output exhibits even symmetry, while the imaginary part shows odd symmetry. It is generally acceptable to discard the second half when extracting magnitude and phase information for real-valued inputs.
lordchaos
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Why does a discrete Fourier transform seems to produce two peaks for a single sine wave? It seems to be the case that the spectrum ends halfway through the transform and then reappears as a mirror image; why is that? And what is the use of this mirror image? If I want to recover the frequency, phase and magnitude of an oscillation, do I need to use any data from this mirror image?
 
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because

\cos(\omega t + \phi) = \frac{1}{2} \left( e^{+i \omega t} + e^{-i \omega t} \right)

so there is a frequency component at +\omega and at -\omega.

because of aliasing due to sampling, negative frequencies are displayed in the upper half of the output of the DFT.
 
Thanks for that. Does this affect how I should extract the magnitude & phase from the transform? Or is it OK to throw the second half away for that purpose?
 
if your input to the DFT is real (i.e. they are complex numbers, but the imaginary part is zero), then yes, the second half is a mirror image of the first half. the real part (or the magnitude) of the DFT output has even symmetry and the imaginary part (or the phase) has odd symmetry.
 
Thanks for your help!
 
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