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The HILBERT function returns a series that has all periodic terms phase-shifted by 90 degrees. The output is a complex-valued vector with the same size as the input vector. This transform has the interesting property that the correlation between a series and its own Hilbert transform is mathematically zero.
HILBERT generates the fast Fourier transform using the FFT function, and shifts the first half of the transform products by +90 degrees and the second half by -90 degrees. The constant elements in the transform are not changed. Angle shifting is accomplished by multiplying or dividing by the complex number, i = (0.0000, 1.0000). The shifted vector is then submitted to FFT for transformation back to the "time" domain and the output is divided by the number elements in the vector to correct for multiplication effect peculiar to the FFT algorithm.
Note: Because HILBERT uses FFT, it exhibits the same side effects with respect to input arguments as that function.
This routine is written in the IDL language. Its source code can be found in the file
hilbert.pro
in the
lib
subdirectory of the IDL distribution.