Why does FFT produce complex numbers instead of real numbers?

algorithm math audio signal-processing fft
steve landiss picture steve landiss · Apr 24, 2012 · Viewed 69.6k times · Source

All the FFT implementations we have come across result in complex values (with real and imaginary parts), even if the input to the algorithm was a discrete set of real numbers (integers).

Is it not possible to represent frequency domain in terms of real numbers only?

Answer

zmccord picture zmccord · Apr 24, 2012

The FFT is fundamentally a change of basis. The basis into which the FFT changes your original signal is a set of sine waves instead. In order for that basis to describe all the possible inputs it needs to be able to represent phase as well as amplitude; the phase is represented using complex numbers.

For example, suppose you FFT a signal containing only a single sine wave. Depending on phase you might well get an entirely real FFT result. But if you shift the phase of your input a few degrees, how else can the FFT output represent that input?

edit: This is a somewhat loose explanation, but I'm just trying to motivate the intuition.

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