Hilbert transform is an operator in mathematics, which is widely used in signal processing, image processing, communication, audio and video coding and other fields. It was first introduced into the study of quantum mechanics in physics, which was put forward by German mathematician and physicist David Hilbert.

Hilbert transform can transform real function f(t) into complex function H(f(t)). This complex function has some special properties. It is called analytic signal and contains all the information of the original signal f(t). Therefore, Hilbert transform is often used to extract some specific information from the original signal or to establish some mathematical models.

One of the main functions of Hilbert transform is its application in modulation and demodulation. Taking a modem as an example, in mobile communication, analog signals will be converted into high-frequency signals, and then broadcast or sent to the receiver through wireless channels.

In the receiver, the high frequency signal needs to be demodulated back to the original analog signal. In this process, the phase and amplitude of the signal can be restored to the original state by using Hilbert transform, so as to correctly process digital signals and analog signals.

Another important application field is signal analysis. In fact, Hilbert transform realizes the transformation from time domain to frequency domain, which is closely related to Fourier transform. Hilbert transform can improve the resolution of a signal in a specific part of the frequency domain, especially for high-energy and low-frequency information that tends to appear near the baseband, such as complex sound or music signals.

In addition, Hilbert transform can also be used to accurately calculate the instantaneous characteristics, spectrum and other statistical values of the original signal. By introducing phase angle to measure the relative contribution of each frequency component, more detailed characteristic lines can be obtained and signal information can be further understood.

Generally speaking, Hilbert transform plays an important role in communication, signal analysis, image processing, audio and video coding and other fields, and its application scope has wide applicability and expansibility.