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}

 

Example

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.

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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.

Example

 

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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(t)=|s_a(t)|

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

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

Example

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Resources:

  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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