What does it mean that an input data stream is "real". I'm thinking like a stream of A/D conversions. How could it be anything but real? What would it mean if it were complex?

#FAQ

First, I suppose that you understand complex numbers and it appears that you do.

Your question is a good one that comes up fairly often.  The simple answer is that the samples ARE "real" in every sense.

The confusion arises when you are dealing with quadrature sampling and are generating "in phase" and "quadrature" samples or "channels".  They are both "real".

Example:  Suppose that you compute a Discrete Fourier Transform of the two channels of real sample values.  The output of the transform is generally complex.  Now what do we call them?  Imaginary part of the Real part and Imaginary part of the Imaginary part?  No.  We have the Re and Im parts of the in-phase part and Re and Im parts of the quadrature part.

Even so, these channels are often called Re and Im .... and there may be any number of good reasons for that. Perhaps someone else can illuminate if warranted.

I simply wanted here to get the definitions clear. 

Called "Re" and "Im" (or "complex" when you're talking about the pair of 'em) because a quadrature pair of signals acts like a complex signal, with, for the most part, the same arithmetic, etc.

Being pedantic, I would call out the signals as "quadrature pairs" in the circuit documentation and the point where they were brought into the digital realm, but being practical I'd have no problem with using math packages that called the numbers involved "complex".

Normally we do complex sampling where the incoming data is splitted in two channels, each channel is multiplied with 2 sinusoids(which are 90 degree out of phase) for down conversion, and then fed to 2 ADC chips. By doing so we sample a signal(with occupied bandwidth B) at B MHz, instead of 2B MHz. 

So effectively you get samples from two channels, one of them is termed real part of complex sample and another is imaginary part of complex sample. 

If you receive just one channel, its a real part.  

Thanks! It sounds like maybe the advantage of complex sampling is that you trade off one fast ADC for two ADCs operating at half the speed. Is this correct? Are there other advantages to complex sampling? Also, is complex sampling also called quadrature sampling?

YES

Are there other advantages to complex sampling?

No side bands are created hence no filtering required in contrast to direct RF sampling which uses single high speed ADC. 

Also, is complex sampling also called quadrature sampling?

YES. Its called Quadrature demodulation as well. 

Thanks for your kind response. Now I understand enough to start learning about it!

See my reply to Fred.  It almost always means that just one channel is being sampled (or that the speaker is talking about the inphase part of a quadrature pair), and that the speaker calls quadrature pairs "complex signals" even though (I claim) they're not.

If it means "real" as opposed to "surreal" then you may want to reconsider working with that group of people.