Fourier Transform Question - Negative Frequencies?

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SUMMARY

The discussion centers on the interpretation of negative frequencies in the context of the Fourier Transform (FT). It is established that negative frequencies are mathematically valid and represent the complex conjugate of positive frequencies, as demonstrated by the equation e^{-\omega ix} = e^{\omega (-i)x}. Furthermore, the physical interpretation of these frequencies can be related to left and right circularly polarized waves, providing a clearer understanding of their significance in signal processing.

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This discussion is beneficial for signal processing engineers, physicists, and students studying Fourier analysis who seek to deepen their understanding of frequency domain representations.

J$C
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When I see a graph of a Fourier Transform, or something in the frequency domain, say band-limited from -\Omega to \Omega, I'm confused to what the interpretation is of the negative frequencies. Physically it would seem as though something considered in cycles/second for example, should be positive but mathematically the FT will allow for negative frequencies.

Can somebody clarify this for me?
 
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e^{-\omega ix}= e^{\omega (-i)x} so negative frequencies are just the complex conjugate of the positive frequencies.
 
Hadn't really thought about it. Physically positive/negative frequencies could be interpreted as left/right circularly polarized waves.
 

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