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Hilbert Transform
Hilbert transform of a signal x(t) is defined as the transform in which phase angle of all components of the signal is shifted by $\pm \text{90}^o $.
Hilbert transform of x(t) is represented with $\hat{x}(t)$,and it is given by
$$ \hat{x}(t) = { 1 \over \pi } \int_{-\infty}^{\infty} {x(k) \over t-k } dk $$
The inverse Hilbert transform is given by
$$ \hat{x}(t) = { 1 \over \pi } \int_{-\infty}^{\infty} {x(k) \over t-k } dk $$
x(t), $\hat{x}$(t) is called a Hilbert transform pair.
Properties of the Hilbert Transform
A signal x(t) and its Hilbert transform $\hat{x}$(t) have
The same amplitude spectrum.
The same autocorrelation function.
The energy spectral density is same for both x(t) and $\hat{x}$(t).
x(t) and $\hat{x}$(t) are orthogonal.
The Hilbert transform of $\hat{x}$(t) is -x(t)
If Fourier transform exist then Hilbert transform also exists for energy and power signals.