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

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?

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.