The Hilbert Transform

The Hilbert transform is a time-domain to time-domain transformation which shifts the phase of a signal by 90 degrees. Positive frequency components are shifted by +90 degrees, and negative frequency components are shifted by – 90 degrees. Applying a Hilbert transform to a signal twice in succession shifts the phases of all of the components by 180 degrees, and so produces the negative of the original signal. IDL's HILBERT function accepts both real and complex valued signals as inputs; the imaginary part of the result is zero for real inputs.

In optics and signal analysis, the Hilbert transform of the time signal r(t) is known as the quadrature function of r(t), which is used to form a complex function known as the analytic signal. The analytic signal is defined as:

where j is the square root of –1 and H is the Hilbert function.

The projection of the analytic signal onto the plane defined by the real axis and the time axis is the original signal. The projection onto the plane defined by the imaginary axis and the time axis is the Hilbert transform of the original signal.

The following example plots the complex analytic signal of a periodic time signal with a slowly varying amplitude.

Example Code
Type @sigprc09 at the IDL prompt to run the batch file that creates this display. The source code is located in sigprc09, in the examples/doc/signal directory. See Running the Example Code if IDL does not find the batch file.