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.