Why do signal have mirror signal after pure frequency shift?

1. Why are FFTS mirrored?
2. Why is FFT two sided?
3. Why does FFT have negative frequency?
4. Why does spectral leakage occurs?

Why are FFTS mirrored?

Because both the positive and negative frequency sinusoids are 90 degrees out of phase and have the same magnitude, they will both respond to real signals in the same way.

Why is FFT two sided?

The two-sided results from the analysis functions include the positive half of the spectrum followed by the negative half of the spectrum, as shown in Figure 1. is the root mean square (rms) amplitude of the sinusoidal component at frequency k.

Why does FFT have negative frequency?

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.

Why does spectral leakage occurs?

Spectral leakage occurs when a non-integer number of periods of a signal is sent to the DFT. Spectral leakage lets a single-tone signal be spread among several frequencies after the DFT operation. This makes it hard to find the actual frequency of the signal.