# My notes on Hilbert transform

Hilbert transform in frequency domain
-It is a peculiar sort of filter that changes the phase of the spectral components depending on the sign of their frequency.
-It only effects the phase of the signal. It has no effect on the amplitude at all.For any signal g(t), its Hilbert Transform has the following property:
$\mathbf{\hat{G}}(f)=\begin{cases} -j & \text{for} f>0 \\ j & \text{for} f<0\end{cases}$

Hilbert transform in time domain

In time domain the Hilbert transform $\hat{g(t)}$ of a function $g(t)$ is its convolution with $\frac{1}{\pi t}$. The choice of the function $\frac{1}{\pi t}$ is because the Fourier transform of the function gives:
$\mathbf{F}(\frac{1}{\pi t})=-j sgn(f)=\begin{cases} -j & \text{for} f>0 \\ j & \text{for} f<0\end{cases}$
-The signal and its Hilbert Transform are orthogonal. This is because by rotating the signal 90° we have now made it orthogonal to the original signal, that being the definition of orthogonality.
-The signal and its Hilbert Transform have identical energy because phase shift do not change the energy of the signal only amplitude changes can do that.
2. https://www.tutorialspoint.com/signals_and_systems/hilbert_transform.htm
3. https://dsp.stackexchange.com/questions/25845/meaning-of-hilbert-transform