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 ext{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 ampptude 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.