FFTs of a complex signal - separating the real and imaginary parts

1. What are the real and imaginary parts of FFT?
2. Is the FFT of a real signal real?
3. What is the FFT of a complex number?

What are the real and imaginary parts of FFT?

The real portion of an FFT result is how much each frequency component resembles a cosine wave, the imaginary component, how much each component resembles a sine wave.

Is the FFT of a real signal real?

Most real-world signals are real-valued. Therefore, you can use the real fast Fourier transform (FFT) for most applications. You also can use the complex FFT by setting the imaginary part of the signal to zero.

What is the FFT of a complex number?

The FFT provides you with amplitude and phase. The amplitude is encoded as the magnitude of the complex number (sqrt(x^2+y^2)) while the phase is encoded as the angle (atan2(y,x)). To have a strictly real result from the FFT, the incoming signal must have even symmetry (i.e. x[n]=conj(x[N-n])).