# 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?

## 1 Answer

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.