Why do we read that Hilbert transform can be used for envelope detection? We can often read that Hilbert transform is useful for envelope detection (e.g.
Hilbert transform to compute signal envelope?)

I have done some tests with various soundfiles, and

x[n] -&gt; absolue value -&gt; 1-pole low pass filter -&gt; envelope


or

x[n] -&gt; Hilbert transform -&gt; absolute value -&gt; 1-pole low pass filter -&gt; envelope


gives the same kind of result, it's not better with a Hilbert transform (top : envelope computed with hilbert, bottom : envelope computed without hilbert) :



Moreover, I know that computing the Hilbert transform is very time-consuming (big FIR filters involved).

So it is really a good method for envelope detection ?
Using a Hilbert transform with a sufficiently narrow-band signal, the low pass filter may not be needed even for continuous envelope estimation.