Discrete Fourier transform mirrored?

[lordchaos]

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?

 

[rbj]

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.

 

[lordchaos]

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?

 

[rbj]

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.

 

[lordchaos]

Thanks for your help!