Positive and negative frequencies how to prove they are independent?

1. What is positive frequency and negative frequency?
2. Why does FFT have negative frequencies?
3. What happens when frequency is negative?
4. Do negative frequencies exist?

What is positive frequency and negative frequency?

Real sinusoidal signals have only positive frequency. Negative frequency is an idea associated with complex exponentials. A single sine wave can be broken down into two complex exponentials ('spinning numbers'), one with a positive exponent and one with a negative exponent.

Why does FFT have negative frequencies?

The reason is that the Fourier transform is symmetric about the y-axis, because the Fourier transform is mathematically defined on the interval (-Inf,Inf). The actual Fourier transform therefore has negative frequencies.

What happens when frequency is negative?

The meaning of negative frequencies is just mathematical(not physical) similarly to the imaginary part of a complex signal. In real world, the negative frequency does not exists and the spectral content on negative frequencies must be added to the spectral content at the positive frequencies, to save energy.

Do negative frequencies exist?

The experimental results show that negative frequency radiation, in some sense, exists. But it also shows that these frequencies are only observed by their generation (either directly or indirectly) of positive frequency radiation.