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?
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.



So how are these helpful?

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.



Other examples of usage include:


Sampling of narrowband signals in telecommunications (mostly using Hilbert filters).
Medical imaging.
Array processing for Direction of Arrival.
System response analysis.