Fourier transform why can I convert one of the axes into an imaginary number?

1. Can a Fourier transform be complex?
2. What do the real and imaginary parts of a Fourier transform represent?
3. Why do we need complex Fourier series?
4. Why does the Fourier transform have imaginary numbers?

Can a Fourier transform be complex?

The complex Fourier transform is important in itself, but also as a stepping stone to more powerful complex techniques, such as the Laplace and z-transforms. These complex transforms are the foundation of theoretical DSP.

What do the real and imaginary parts of a Fourier transform represent?

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.

Why do we need complex Fourier series?

The complex Fourier series obeys Parseval's Theorem, one of the most important results in signal analysis. This general mathematical result says you can calculate a signal's power in either the time domain or the frequency domain.

Why does the Fourier transform have imaginary numbers?

Since Fourier Transforms are used to analyze real-world signals, why is it useful to have complex (or imaginary) numbers involved at all? It turns out the complex form of the equations makes things a lot simpler and more elegant. As such, everyone uses complex numbers, from physicists, to engineers, and mathematicians.