Why are frequencies represented as complex numbers?

5
votes

In a FFT, the resulting frequencies represent both magnitude and phase. Since each frequency element in the output array of an FFT essentially just describes the SIN wave at each frequency interval, shouldn't it just be magnitude that we need? What is the significance of the phase represented in the imaginary part of the complex number?

To clarify my question, to be able to put a meaning to the phase of a wave, I need a reference point or reference wave.

When an FFT reports the phase for each sin wave in the resulting frequency domain output, what is the reference wave with respect to which it is reporting the phase?

3
mathaudiosignal-processingfft
What about moving this question to DSP? dsp.stackexchange.com – petrichor
In this answer I explained how/why FT works and the meaning of the phase. – Alexey Frunze
Consider looking or asking on dsp.stackexchange.com. – ellisbben
possible duplicate of How does a complex number capture phase, amplitude AND frequency in the result of an FFT? – andand

3 Answers

4
votes

Plot showing the effect of phase in summed sine waves

Because the phase of different components affects the total signal. The two functions in the plot are both summed from sine waves with periods of pi and 2pi, but the phase of the p=2pi sine waves are different. As you can see, the outputs are not the same.

5
votes

Well in layman's words: magnitude tells you how much of that frequency is there, and phase tells you where it is.

0
votes

FFTs (there is more than one convention) usually report phase with respect to the zero-th sample. Or if you use FFTShift, with respect to the sample at the center of an FFT window that indexes from 0 to N-1 (e.g. sample number N/2 = sin(0) for a phase of 0). The latter convention, centering phase using FFTShift, is often better, as there can be a big discontinuity at the edges of an FFT aperture, or nearly no data at the edges after using a tapered window function.

If you use FFTShift to center the phase reference, zero phase represents an even function, and a phase of pi or -pi represents an odd function in the window.

Human hearing, in general, can't discriminate the phase of a single sound source. BUT, phase is important when dealing with combined sounds, or multiple sine waves of the same frequency. Sinusoids that are in phase add or sum. Sinusoids of the opposite phase cancel. So if you have the FFT of, say, two loudspeaker responses without phase, you won't know whether they will sound great or horrible together.