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I tried to find a simple explanation of what a complex envelope of a signal is, with no luck. I'm looking for an explanation with simple terms, and possibly that is easy to visualize or imagine, so that I can grab the main concept.

To give some context, I'm studying about optical fibers and WDM channels, and the signal that you can send over a WDM channel. So I would like to understand where the complex envelope concept might apply in this case, and what's the meaning of the complex envelope of a channel.

Sorry if I made some mistakes, but I don't know enough about this subject.

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  • $\begingroup$ Not simple but related $\endgroup$
    – OverLordGoldDragon
    Commented Sep 9, 2021 at 14:31
  • $\begingroup$ I read the answer, and I thank you. However, I didn't understand at which point the complex envelope is used in the answer. Is it to get the phase(t) and amplitude(t) functions that characterize the received signal? $\endgroup$
    – Kenna
    Commented Sep 9, 2021 at 14:54
  • $\begingroup$ The black dashed line is the "envelope"; we obtain it by taking absolute value of the complex waveform, hence "complex envelope". The complex is generated in such a way that, its phase, calculated as arctan(imag/real), equals original's, and amplitude, calculated as sqrt(imag^2 + real^2), likewise matches - and real part stays exactly equal to original signal. The imaginary part is called the "analytic part"; this approach has limitations. $\endgroup$
    – OverLordGoldDragon
    Commented Sep 9, 2021 at 14:58
  • $\begingroup$ Ok so the complex envelope is a different function, but has the same evolution in time of phase and amplitude of the signal over which it is calculated? $\endgroup$
    – Kenna
    Commented Sep 9, 2021 at 15:05
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    $\begingroup$ Yes, exactly. The point is, we can't compute $\phi(t)$ or $a(t)$ directly from $x(t) = a(t) \cos(\phi(t))$ (imaginary part is zero), but we can do it from its analytic waveform $a(t) e^{j\phi(t)}$. $\endgroup$
    – OverLordGoldDragon
    Commented Sep 9, 2021 at 15:07

2 Answers 2

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Here's an intuitive explanation. High-frequency signals (like the WDM signals you are studying) contain very large frequencies. This is inconvenient, because you need to carry a frequency term in your equations, and because you need to sample those signals at huge rates to simulate them.

In many cases, though, the large frequency content is not fundamental, in the sense that we could analyze or simulate the same signal or system at lower frequencies. Consider a message $s(t)$ and the signal $$m(t)\cos(2\pi 10^9t).$$ We are interested in the message $m(t)$, and the fact that there is a 1 GHz carrier is incidental. The carrier could be 10 kHz or 100 THz instead and nothing fundamental in the signal has changed.

The complex envelope is the result of removing all of this superfluous high-frequency content from a signal. The purpose is to focus on the fundamental content of the signal (the message $m(t)$) and not on the carrier. The equations become shorter (no need to keep the GHz around), and it also becomes feasible to simulate (the sampling rate depends only on $m(t)$).

This simpler signal with no carrier is called the "complex envelope" of the original signal. It is comples just because of the way the math works.

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  • $\begingroup$ Ok, so according also to the comments on my question, I calculate the complex envelope on the high frequency signal and I can extract the two functions phase(t) and amplitude(t) that both represent how phase and amplitude evolve in time for the orginal message (the one without the carrier). Could be this the meaning? $\endgroup$
    – Kenna
    Commented Sep 9, 2021 at 15:27
  • $\begingroup$ Yeah, that's correct. $\endgroup$
    – MBaz
    Commented Sep 9, 2021 at 15:51
  • $\begingroup$ Thank you. Unfortunately I cannot upvote your answer but I will once I will have more reputation :) $\endgroup$
    – Kenna
    Commented Sep 9, 2021 at 15:56
  • $\begingroup$ @Kenna I upvoted your question; you may have enough reputation now :) $\endgroup$
    – MBaz
    Commented Sep 9, 2021 at 16:09
  • $\begingroup$ Thank you, however the threshold is at 15 points and I have 13 :( $\endgroup$
    – Kenna
    Commented Sep 9, 2021 at 16:36
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A complex envelope is like a pocket spring or pocket coil, like what you can find in mattresses:

enter image description here

The string can be seen as a cisoid or complex exponential. Its projections can yield a sine or a cosine. The fabric around it is an envelope.

Now, imagine a signal, or its analytic version, like a string (less homogeneous of course). The perfect pocket would be its envelope.

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