This thread seems to be blowing out of proportion. There seem to be two main differences of opinion between us:-

(1) The use of the words, "negative frequency".

(2) Whether or not using complex numbers to represent AC signals (and AC impedance) actually means there is another, un-measurable, dimension somehow orthogonal to reality.

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Regarding point (1), I've found some internet resources that use these words, and some that don't. I think we just basically have different opinions on whether or not we like these words. Either way, regardless of the words, the mathematics of the DFT/FFT remains EXACTLY the same for both of us.

I have now been educated about what people are referring to when they say, "negative frequency" in this context, and for this I'm both grateful and interested! I have learned something new, thanks.

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Regarding point (2), it seems like a belief/ faith situation. I can't, and neither can anyone else, disprove the existence of a physical manifestation of an imaginary component of current in some other dimension (for example).

I have quite a scientific brain and tend to only believe statements backed by physical evidence/ experiment. None of the systems, or thought experiments, that you have described prove its existence to me (see * below). I view the use of complex number REPRESENTATION in the mathematics of AC analysis as just a very clever, back compatible, convenient, way to represent a system where two real world quantities are represented in one complex variable (like amplitude and phase for example).

* - Your example of a negative frequency radio in post#8 requires two real-world signals that are both sent with the same POSITIVE PHYSICAL frequency - and the two signals are simply just out of phase with each other. For me, there's no physical manifestation of complex signals nor real world negative frequency reception.

Obviously a real-only signal cannot exhibit a direction of rotation, though strictly speaking it consists of equal amounts of positive and negative power at the same frequency such that their imaginary components cancel.

...I'm not sure what you're saying here. Are you saying that adding together the corresponding positive frequency and negative frequency bins in the output generated by the FFT results in the actual, real, signal level? My understanding is that the imaginary and real components of a single output bin supply information about the overall amplitude

**and phase** at that frequency (as far as I'm aware). To remove the imaginary part, by adding two conjugate bins, seems to remove information and give an incorrect amplitude at that frequency. For example:-

fft([0 1 0 -1 0 1 0 -1]) = 0 0 0-4i 0 0 0 0+4i 0

You're probably not saying that this real signal has zero amplitude after adding (0-4i) + (0+4i) = 0?