The Hilbert-transform

The Hilbert transform

Though it's use is frequent in signal processing, it does have a significance in understanding tomographic image reconstruction, the Hilbert transform.
. The Hilbert transform is defined as:

The definition looks simple, to evaluate the integral looks a lot harder as the denominator harbours a singularity. It means we have to take the integral in a Cauchy principal value, so

Existence of such a limit can easily imagined, as for function g=1 the function 1/x can be integrated in Principa Value, being odd the result is 0, since the range of the integral is symmetric.

To ease the evaluation of the integral

• note that the Hilbet transfrom is a convolution with function 1/x
• Fourier transform then inverse Fourier transform the expression

Let us evaluate the Fourier transform of 1/x:

The function cos is even, divided by the odd "x" function the first term is again zero.
We have the next two terms remaining:

and

Since

The result is, then:

where sgn is the sign function.

This result, regarding the numerical evaluation technique is a lot simpler then the application of the basic definition, since the digital Fourier transform and its implementation technique (FFT) is a routinely applied, accessible and fast.

Our results also shows, that if we apply the Hilbert transform twice on the same function we obtain:

thus the inverse of the Hilbert-transfrom is -apart from a sign- is itself.
As an illustration we have prepared the 2D Hilbert transform of an image: