Complex representation of a signal, quadrature signals in receivers

[luislandry]

(I posted this in the elctrical engineering forums because it's technically not homework, but it probably belongs here...it got no replies there)

I'm hoping this thread can clear up some confusion I have with complex signals and moving back and forth from physical signals to the mathematical models. I'll probably ask some questions specifically, but if you would like to help me please treat this whole post as a question because I'll try to walk through my understanding of the subject and it may be wrong in places.

A real received signal will have a form such as A*cos(wt + p). This can also be represented by Euler's formula by two complex sinusoids with additive inverse frequencies: A/2*exp(wt + p) + A/2*exp(-(wt + p)).

A receiver splits this into two signals and phase shifts one by 90 degrees, giving two signals A*cos(wt + p) and A*sin(wt + p), which are represented by A*exp(wt + p).

Q1: When a signal is shown in the frequency domain, and is symmetrical about the zero-frequency line, would this be the real signal?

Q2: When it is shown only in the positive frequencies, it's either just to save space on the diagram, or to represent the in-phase AND quadrature signals. Right?

Q3: What is the purpose of the quadrature signals? I didn't fully follow the explanations I've seen so far.

Q4: How do you interpret signals that are shown as not being symmetric about the zero-frequency line? How can these be realized as real signals?

Thanks a ton for your help here, I feel like these are details that I sort of glossed over before and would like a concrete understanding of.

 

[KataKoniK]

Hi there,

I would be happy to help clarify some of the confusion you have with complex signals and moving between physical signals and mathematical models.

To answer your questions:

Q1: When a signal is shown in the frequency domain and is symmetrical about the zero-frequency line, this would represent a real signal. This is because in a real signal, the positive and negative frequencies are symmetrical.

Q2: When a signal is shown only in the positive frequencies, it is either to save space on the diagram or to represent the in-phase and quadrature signals. This is because in-phase and quadrature signals are typically only shown in the positive frequencies, as they are complex signals.

Q3: The purpose of quadrature signals is to allow for a more efficient representation of a complex signal. By splitting the signal into two components, in-phase and quadrature, the overall signal can be represented using complex numbers instead of trigonometric functions. This makes calculations and analysis of the signal easier.

Q4: Signals that are not symmetric about the zero-frequency line can still be realized as real signals. This can happen if the signal has been modulated or filtered in some way. In these cases, the signal may have a non-symmetrical frequency spectrum, but it can still be represented as a real signal in the time domain.

I hope this helps clarify some of your questions. If you have any further questions or need more clarification, please let me know.