# Representing stereo signal with complex numbers as input to DFT

Usually, when we have a stereo signal, we process each channel separately in order to extract the frequencies using Fourier transform.

However, Fourier transform can also be applied to complex numbers as well. So what if I represent the original stereo signal with complex numbers, do a Fourier transform, and back?

I know that in this case, for DTF, I would need to keep all N frequency numbers as opposed to N/2 for DFT on the real data. Should I then be able to recover my original signal from the frequencies? (My guess would be "yes".)

Now, what if I want to detect if a specific frequency is present in the signal? I could use the formula for DFT:

So, what if instead of real $x_n$, I pass a complex number containing values of the stereo channels as real and imaginary parts? Would that make any sense? What would that detect?