Conditions for analytical signal reconstruction with the Hilbert Transform (HT)

Question

I'm new to the Hilbert Transform, I was reading the paper: "SISAR Imaging - Radio Holography Signal Reconstruction Based on Receiver-Transmitter Motion", and the next sentence was written in the paper:

The HT states that an analytical complex signal can be reconstructed from the signal’s real part under the condition that the real amplitude and frequency modulated signal in the form: $$ S(t) = I(t)\cos(\theta(t)) $$ has the spectrum of $\cos(\theta(t))$ outside the spectrum of its envelope $I(t) \in [−f_0, f_0]$.

I have checked some documentation on the Hilbert Transform and I have not been able to find that condition, so, why is this condition needed? What is the reason behind it?

Answer 1

Let me use the notation from Bedrosian's research report "The analytic signal representation of modulated waveforms" in a slightly simplified version:

$$s(t)=c(t)m(t)\tag{1}$$

where $s(t)$ is the modulated signal, $c(t)$ is a (not necessarily sinusoidal) carrier function, and $m(t)$ is the message signal. The carrier is assumed to be a relatively narrow-band band pass signal, and the message signal has a low pass character.

Let $c_a(t)$ be the analytic signal corresponding to $c(t)$. By definition, $c_a(t)$ has only positive frequency components. The analytic signal $s_a(t)$ corresponding to $s(t)$ can then be written as

$$s_a(t)=c_a(t)m(t)\tag{2}$$

if and only if the spectra of $c(t)$ of $m(t)$ do not overlap. Because only then does $s_a(t)$ as defined in $(2)$ contain positive frequency components only. If the spectra overlap, then $s_a(t)$ is not analytic because it will have negative frequency components. This is illustrated by the figure below, taken from Bedrosian's report:

The condition that the spectra of $m(t)$ and $c(t)$ do not overlap is equivalent to the requirement that $s(t)$ be a true band pass signal, i.e., $s(t)$ has no DC component.

I'm sure that this is what is alluded to by the sentence you quoted. I do agree that this is not obvious from the chosen formulation.