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Hilbert envelope bandwidth

Dear list,

I am struggling with the following question:

Given a signal x(n) with
   X(f) = 0 for |f| < f1 or |f| > f2
   (bandpass filtered signal with bandwidth B = f2-f1)

e(n) is the Hilbert envelope of x(n) which can then be written as:
   x(n) = e(n)y(n),

where y(n) is the "temporally flattened" version of x(n).

The spectrum of e(n) satisfies:
  E(f) = 0 for |f| > f3

(Due to its DC offset, the evelope e(n) contains frequencies down to

Can f3 be expressed as a function of B (the bandwidth of signal x)?

Any comments/suggestions are appreciated. Thanks,
  Christof Faller