# 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.

### 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.

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

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

$A(t)=|s_a(t)|$

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

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