Recover signal from its spectral amplitude and envelope I am given an amplitude spectrum (no phase) and the envelope of a signal. What I know about the original signal is, that it can be thought of as wavepacket like structure (sinusoids under a bell curve) and the spectrum is narrow banded.
To be able to further analyse the original signal I would like to recover the input signal from those two given functions.

My questions are: Can I recover the signal from its amplitude spectrum and envelope or would you advice me to do otherwise.



To clarify what is given:

The signal is thought of wavepacket structure, which is

$$x(t) \approx \sum_n\frac{1}{\sigma_n\sqrt{2\pi}}e^{-\dfrac{1}{2}\left(\dfrac{x-\mu_n}{\sigma_n}\right)^2} \cdot \sin(2\pi f_n x)$$

with $n \gt 1$, $\lvert\mu_k-\mu_l\rvert \lt \sigma_k, \sigma_l$ and if $\mu_k \gt \mu_l$ then $f_k \lt f_l$. 

The amplitude spectrum is the absolute of the Fourier transform of the signal, which is

$$a(\omega) = \lvert\mathcal{F}\{x(t)\}\rvert$$

The envelope can be thought of as absolute value of the analytic signal, which is

$$e(t) = \lvert x(t) + j \cdot \mathcal{H}\{x(t)\}\rvert$$

I'm not sure if it is possible or even that beneficial to recover the signal and I am pleased for any answer or every hint.
One counterexample is two wave packets of identical spread and amplitude, well separated in time so that their envelopes do not overlap, and of high enough (differing) well-separated frequencies such that their analytic signals can be considered Gaussian-enveloped complex exponentials. The Hilbert envelope will thus consist of two identical well-separated Gaussians. The amplitude spectrum will consist of well-separated Gaussians of identical magnitude. There is not enough information to assign the frequencies to the wave packets.



So in general you cannot resolve the original signal, not even up to an ambiguity of a constant phase shift over all frequencies.

With sufficiently large time spreads of the two wave packets their frequencies need not be far apart to still be well separated in the spectrum, satisfying your constraint for a narrow-banded signal.