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I am new to signal processing.

I want to find the envelope of a given speech signal.

When I searched about it, I found about Mean Hilbert envelope function. I want to know what other functions/methods can do the same and why Mean Hilbert envelope is considered the best?

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The notion of envelope is an ill-defined concept that describes the smooth upper and lower boundaries of more or less oscillating signals.

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The most classical technique rectifies the signal (take the absolute value) and low-pass filters the result. Of course, results are not unique.

Other techniques identify local extrema and fit them with low-order functions like polynomials or splines. This is used for instance in some implementations of the Empirical Mode Decomposition.

The Hilbert transform sometimes offers a more solid alternative. Roughly speaking, suppose the signal can be written as the product of an envelope times an oscillation: $s(t) = e(t).o(t)$. Then, with other technical conditions, if the spectra of $e$ and $o$ are well separated, $H(s(t)) =e(t).H(o(t))$, where $H$ is the Hilbert transform. This is a loose transcription of the Bedrosian identity.

If $o(t)$ is a fine pass-band oscillations (the purest being a sine or a cosine), and $e(t)$ is lower in frequency, the Hilbert transform can separate them. This is why for speech, the signal is often decomposed first through a multirate filter bank.

All these techniques are often post-filtered in practice.

More recent approaches try to better define the regularity or smoothness of the envelope, using optimization techniques.

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