Magnitude of the analytic signal

Question

It's well known that magnitude of the analytic signal for narrowband signals gives the envelope. For example we can demodulate AM signal by abs(hilbert(s_AM)) in MATLAB. I'm looking for the exact mathematical conditions which specify when we get the envelope by computing $|s(t)|$. By analytic signal, I mean $$s(t) = x(t) + j\hat{x}(t)$$ Where $\hat{x}(t)$ is the Hilbert transform of $x(t)$, $$\hat{x}(t) = \frac{1}{\pi t}*x(t) = \frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{x(\tau)}{t-\tau}d\tau$$Magnitude of $s(t)$ is $\sqrt{x^2(t) + \hat{x}^2(t)}$ and I don't know how to proceed further.

Answer 1

By definition, the magnitude of the analytic signal $s_a(t)$ can always be considered the envelope of the signal in a purely mathematical sense, simply because the real-valued signal is given by

$$s(t)=\textrm{Re}\big\{s_a(t)\big\}=|s_a(t)|\cos\big(\arg\{s_a(t)\}\big)\tag{1}$$

However, this does not mean that $|s_a(t)|$ has necessarily a physical significance as an actual envelope of $s(t)$. It does have physical significance if the term $\cos(\arg\{s_a(t)\})$ in $(1)$ has the function of a carrier. This carrier doesn't need to be sinusoidal but it must be a narrowband signal, i.e., its center frequency must be large compared to its bandwidth.

If you want a strict mathematical definition of the conditions under which $|s_a(t)|$ is an envelope in a physically significant way, then you must define what you mean by 'envelope' and by 'physical significance'. This is not trivial and probably also not very useful.

If your question is just about the feasibility of envelope detection, then that's a very different question, and one that's much easier to answer. If the modulated signal is given by

$$s(t)=m(t)\cos(\omega_ct)\tag{2}$$

then the magnitude of the corresponding analytic signal equals $m(t)$ if

1. $m(t)$ has a maximum frequency smaller than the carrier frequency $\omega_c$
2. $m(t)\ge 0$

These conditions are met for standard AM, but not for DSB-SC.

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EDIT: For the signal $(2)$ it's straightforward to show that $|s_a(t)|=m(t)$ if above conditions are met. If $m(t)$ is a low pass signal with a maximum frequency smaller than the carrier frequency $\omega_c$, then the analytic signal is given by

$$s_a(t)=m(t)e^{j\omega_ct}\tag{3}$$

with magnitude

$$|s_a(t)|=|m(t)|\tag{4}$$

Clearly, if $m(t)\ge 0$ then $|s_a(t)|=m(t)$.

Answer 2

Using your notation for the (complex valued) analytic signal;

$$s(t) = x(t) + j \hat{x}(t) = |s(t)| e^{j \phi(t)} \tag{1}$$ where $x(t)$ is the message signal and $\hat{x}(t)$ is its continuous-time Hilbert transform.

You can see that the following signal $$ z(t) = \mathcal{Re}\{ s(t) \cdot e^{j \omega_c t} \} \tag{2}$$

is $$ z(t) =\mathcal{Re}\{ |s(t)| \cdot e^{j \phi(t)} e^{j \omega_c t} \} $$

$$ z(t) = |s(t)| \cos(\omega_c t+\phi(t)) \tag{3}$$

And the carrier $\cos(\omega_c t)$ has the envelope of $|s(t)|$; the magnitude of the analytic signal $s(t)$.

This can also be seen in the following equivalent expression of the Narrow-Band modulated signal in Eq.(3):

$$ z(t) = x(t) \cos(\omega_c t) - \hat{x}(t) \sin(\omega_c t) \tag{4}$$

which defines a QAM (quadrature amplitude modulated) signal. If you are familiar with AM demdulation with coherent receiver , then you know that you can recover the original mesage signal $x(t)$ by lowpass filtering the following signal :

$$ x(t) = LPF\{ z(t) \cos(\omega_c t) \} \tag{5}$$

To see the extend of Narrow-Band assumption, note the relation between carrier frequency $\omega_c$, the angle $\phi(t)$ and the instantaneous frequency of the complex exponential (using FM modulation reasoning). Then it can be seen that for $$\phi(t) \approx 0 $$ Eq.3 becomes:

$$ z(t) = |s(t)| \cos(\phi(t)) \cos(\omega_c t) - |s(t)| \sin(\phi(t)) \sin(\omega_c t) \tag{6} $$

$$z(t) \approx |s(t)| \cos(\omega_c) \tag{7}$$

which outlines more clearly that complex-modulated analytic signal can be considered as a narrowband sinusoidal signal with envelope $|s(t)|$.