I am aware that complex discrete-time signals are of the form:

$$ x(n) = a(n) \cdot e^{j\omega n T} $$

However, I have not yet worked with complex signals, only real ones. What are some practical examples of complex signals? Answers with plenty of detail are appreciated.

  • $\begingroup$ no, your signal is a very special case: it's a complex sinusoid with amplitude modulated over time. Overall, your question is very broad, and doesn't make very much sense: Shift any real-valued signal in frequency and you end up with a complex one. You should probably read a few more pages in a textbook that introduces complex equivalent baseband signals, see if that clarifies things to you. Asking for a list of example signals doesn't fulfill the criterion of "questions should allow for a correct answer". $\endgroup$ – Marcus Müller Feb 14 at 14:59
  • $\begingroup$ The purpose of my question was to motivate the use of complex signals. So far, I have not seen them used in practice, only on paper. Similarly, I have not seen non-symmetric spectra in practice. Does this mean that all complex signals are just frequency shifted real signals? Should I re-word my question to reflect this? $\endgroup$ – mhdadk Feb 14 at 15:15
  • $\begingroup$ Have you heard about analytic (Hilbert transform) or IQ signals? Apart from communications, you can find complex signals in radar processing or medicine (MRI). $\endgroup$ – jojek Feb 14 at 15:27
  • $\begingroup$ motivate the use: I've given you arecommendation what to read, and you'll find what you're looking for. No, not all complex signals are just frequency-shifted real signals. $\endgroup$ – Marcus Müller Feb 14 at 15:28
  • $\begingroup$ Thanks for the references $\endgroup$ – mhdadk Feb 14 at 15:32

Here is one aspect that often creates confusion: In many cases, complex signals are used solely as a mathematical convenience, not because the complex representation is required by anything physical.

A good example is circuit analysis in electronics: all physical quantities (voltages, currents, impedances) are real and you can do the analysis with these quantities and the associated physical equations directly.

However, it's a lot easier if you convert the real quantities to complex ones, run the analysis in the complex domain and convert the results back to real. Hence, that's the way everyone does it.

So you don't HAVE to use complex signals, it just gets a lot easier if you do.

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    $\begingroup$ I don’t think it is accurate to suggest that “complex” numbers are some mathematical description that is somehow different than “real” numbers. Both are our way of describing things with math and it is a very unfortunate naming given by “real” and “imaginary”. We can represent complex quantities with real numbers by using pairs of real numbers; and for the physical quantities that we describe with real numbers, we can equally create complex physical quantities as pairs (I and Q in a receiver implementation). $\endgroup$ – Dan Boschen Feb 14 at 18:55

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