Wang Yun Maigo
Links: https://www.zhihu.com/question/30372795/answer/47876447
Source: Know
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The physical meaning of the Hilbert transform is simple:
delay the phase of all frequency components of the signal by 90 degrees.
In other words, if the original signal can be represented,
The signal after the Hilbert transform is.
This is easily seen through the frequency domain of the Hilbert transform:
, where
Of course, I know the main interest of the topic is: What is the use of delaying the phase 90 degrees?
The answer is:
Hilbert transform can be used to do demodulator, amplitude modulation, frequency modulation can be solved.
, Blue is a modulated signal whose amplitude and frequency are modulated.
Green is the Hilbert transform of the blue signal. Because the amplitude and instantaneous frequency of the modulated wave are very slow (compared to the carrier frequency), the frequency components are relatively single (all concentrated around the carrier frequency), so the effect of the Hilbert transform-the phase delay of 90 degrees-is obvious.
Now construct the signal and we'll figure out how to draw this signal in three-dimensional space.
The following figure has three axes: the timeline, the real axis, and the imaginary axis.
The timeline and real axis are drawn on a plane (blue),
The time axis and the imaginary axis are drawn on a plane (green),
The three-dimensional space is drawn (red).
As you can see, it looks like a spring with a change in weight and density.
At any one time, we can read the instantaneous amplitude and instantaneous phase:
The instantaneous amplitude is the tangent of the instantaneous phase.
The instantaneous phase-to-time derivative is the instantaneous frequency.
In this way, we use Hilbert transform to demodulated amplitude and frequency from a modulated wave modulated by amplitude and frequency.
Of course, the actual demodulator does not do this, an important reason is
Hilbert transforms are not causal.and cannot be demodulated in real time.
The physical meaning of Hilbert transform
