signal analysis - Meaning of Hilbert Transform

I understand the Fourier Transform which is a mathematical operation that lets you see the frequency content of a given signal. But now, in my comm. course, the professor introduced the Hilbert Transform.

I understand that it is somewhat linked to the frequency content given the fact that the Hilbert Transform is multiplying a FFT by $-j\operatorname{sign}(W(f))$ or convolving the time function with $1/\pi t$.

What is the meaning of the Hilbert transform? What information do we get by applying that transform to a given signal?

Answer

One application of the Hilbert Transform is to obtain a so-called Analytic Signal. For signal $s(t)$, its Hilbert Transform $\hat{s}(t)$ is defined as a composition:

$$s_A(t)=s(t)+j\hat{s}(t) $$

The Analytic Signal that we obtain is complex valued, therefore we can express it in exponential notation:

$$s_A(t)=A(t)e^{j\psi(t)}$$

where:

$A(t)$ is the instantaneous amplitude (envelope)

$\psi(t)$ is the instantaneous phase.

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The instantaneous amplitude can be useful in many cases (it is widely used for finding the envelope of simple harmonic signals). Here is an example for an impulse response:

Secondly, based on the phase, we can calculate the instantaneous frequency:

$$f(t)=\dfrac{1}{2\pi}\dfrac{d\psi}{dt}(t)$$

Which is again helpful in many applications, such as frequency detection of a sweeping tone, rotating engines, etc.

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Other examples of usage include:

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  Sampling of narrowband signals in telecommunications (mostly using Hilbert filters).

  
  


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  Medical imaging.

  
  



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  Array processing for Direction of Arrival.

  
  


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  System response analysis.