Hilbert transform and the curious case of enveloping [My notes]

Hilbert transform in frequency domain

-It is a peculiar sort of filter that changes the phase of the spectral components depending on the sign of their frequency.
-It only effects the phase of the signal. It has no effect on the amplitude at all.For any signal g(t), its Hilbert Transform has the following property:
\mathbf{\hat{G}}(f)=\begin{cases} -j & \text{for} f>0 \\ j & \text{for} f<0\end{cases}



The role of Hilbert transform is to take the carrier for example: a cosine wave and create a sine wave out of it. In case of multi component signal, each frequency is transformed and added accordingly.


Image resource: http://complextoreal.com/wp-content/uploads/2013/01/tcomplex.pdf

Hilbert transform in time domain

In time domain the Hilbert transform \hat{g(t)} of a function g(t) is its convolution with \frac{1}{\pi t} . The choice of the function \frac{1}{\pi t} is because the Fourier transform of the function gives:
\mathbf{F}(\frac{1}{\pi t})=-j sgn(f)=\begin{cases} -j & \text{for} f>0 \\ j & \text{for} f<0\end{cases}
-The signal and its Hilbert Transform are orthogonal. This is because by rotating the signal 90° we have now made it orthogonal to the original signal, that being the definition of orthogonality.
-The signal and its Hilbert Transform have identical energy because phase shift do not change the energy of the signal only amplitude changes can do that.




So how does this all fit into enveloping functions?

Analytical signal

Analytical signal is s_a (t): the signal s(t)as the real part+ the Hilbert transform of the signal \hat{s}(t) as the imaginary part.

s_a (t) = s(t)+j  \hat{s}(t)

s_a (t) = A(t)e^{j \phi t}


A 3-D representation of the analytical signal from previous example

The instantaneous amplitude : A(t) gives the envelope of the function.





  1. http://complextoreal.com/wp-content/uploads/2013/01/tcomplex.pdf
  2. https://www.tutorialspoint.com/signals_and_systems/hilbert_transform.htm
  3. https://dsp.stackexchange.com/questions/25845/meaning-of-hilbert-transform

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