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The image displays techniques for construction of complex envelopes from bandpass signals and vice versa.

From what little I've been able to gather, the complex envelope preserves all information required for the reconstruction of the original signal and that it basically sends a low pass version of a bandpass signal.(Correct me if I'm wrong).I need an explanation for how the complex envelope is being created as per this text book? And also of how the original signal is obtained at the end?

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    $\begingroup$ In the system you show, the complex envelope is not used. This recent answer may help you: dsp.stackexchange.com/a/25835/11256 $\endgroup$ – MBaz Sep 17 '15 at 23:39
  • $\begingroup$ Votes or best answer validation are required $\endgroup$ – Laurent Duval Jul 28 at 11:58
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Let’s get one thing straight. You cannot recover the original real and imaginary parts of a complex signal based only on the magnitude values of the complex signal. That’s equivalent to saying, “I have the magnitude of a complex number. How do I compute that complex number’s real and imaginary parts?” There are an infinite number of possible correct answers to that question!

However, as MBaz rightly said, you don’t have any “envelope” signal in your block diagrams. And by the way, there are no such things as “complex envelope” signals. All “envelope” signals are real-only. Having said all of that, you can learn everything about your two block diagrams by writing down the algebraic equations for all their various signals. (Your textbook should have done that for you.) Then pay attention to all the various trigonometric identities in your math reference book. It’s a mild pain to do all of that, but it’s terrifically educational.

And another thing, your two block diagrams are not the normal “complex modulator” and “complex demodulator” used in modern digital communications systems. In a normal digital comms system the real-valued gI(t)/2 and gQ(t)/2 would be added (or subtracted, I forget) to produce a new real-valued signal that’s transmitted via an antenna. Have a look at Figure 1 of the following PDF file: http://www.testequity.com/documents/pdf/keysight/complex-modulation-generation-wp.pdf

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The complex envelope seems a not-so-well-defined concept to me.

Some consider it an equivalent of the analytic signal (see What exactly is complex envelope?), from a real input. In other words, a complex signal whose real part is the real signal $f(t)$, and the imaginary part the Hilbert transform $\mathcal{H}$: $g(t) = f(t) + \imath\mathcal{H}(f(t))$. What I understand from your diagram is that $g(t)$ is already analytic. For others (see Analytic signal/Complex envelope/baseband), it denotes a modulation of the analytic signal, generally to shift the center frequency: $g_c(t) = g(t)e^{-\imath \omega_c t}$. This definition is not unique, as it depends on $\omega_c$. If this center frequency is shifted toward $0$, then a low-pass filtering can extract the baseband low-frequency component of the "complex signal".

Forgetting the $\frac{1}{2}$ factor, this would correspond to the upper part and the lower part of your diagram. You can find a graphical explanation in Figure 2-1 from Complex band-bass filters for analytic signal generation and their application, where the left subfigure relates to your diagram, and the right one to the above formula for $g_c(t)$, with $w_c = 2\pi f_c$.

To me, an envelope can be complex if one defines it properly, althought this is counter-intuitive as the traditional envelope is usually real and non-negative, or somehow symmetric about the time-axis.

Does this transform preserve all the information from the input? In some cases, possibly. This could be stated as "In general, a real-valued bandpass signal/system has a complex-valued lowpass equivalent" (Mikko Valkama, Complex-valued signals and systems). This, of course, depends on the properties of the signal and an appropriate choice of $\omega_c$ and the low-pass filter.

I do not see where you talk about magnitude, so I do not think the question is about "recovering a complex signal from magnitude only". However, this is an important topic in signal processing, and can be performed in some cases with additional conditions on signals.

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