Noise properties of Hilbert transform and fast white-light interferometry

Hilbert transform is a standard method for the calculation of the envelope and phase of a modulated signal. An advantage of Hilbert transform is that it can be used for the arbitrary size of sampling step. Because each measured signal is affected by a noise, the envelope and phase calculated from the signal by Hilbert transform are also affected by the noise. The demodulation by means of Hilbert transform (and even each demodulation method) transforms the measured signal. Thus one can expect that the noise is also transformed. We calculate the variance and spectral density of both the envelope and phase noise. Because Hilbert transform is suitable for the arbitrary size of sampling step, it can be used also for a large sampling step where the modulated signal is undersampled. The results are demonstrated on the example of fast white-light interferometry.