Another useful application of the Hilbert transform is through the Kramers-Kronig relations. Basically, for a causal signal (e.g. an impulse that turns on), the

Another useful application of the Hilbert transform is through the Kramers-Kronig relations. Basically, for a causal signal (e.g. an impulse that turns on), the real and imaginary parts of the Fourier transform are related by a Hilbert transform. This is very useful, for example, when trying to extract a time-domain impulse from imperfect frequency-domain measurements, for example with a network analyzer. Rather than attempting to inverse Fourier transform the frequency domain measurements, which will tend to produce an unphysical mess due to measurement errors, it's often better to take either just the real or imaginary part and synthesize the other one. Even better would be to try to find the nearest causal waveform to what is measured, but that's a bit more difficult... Another application is finding the minimum phase response given an amplitude response.