The way I understand how SDR works, there is a receiver input, generally connected to an antenna pulling in signals from the ether. That input is connected to two mixers (linear multipliers), being mixed with the same LO frequency in both mixers, however, the two LO signals are 90º out of phase. This results in two output signals, commonly referred to as I and Q, I for the "in phase" signal, and Q for the "quadrature", or 90º out of phase signal.
Then, if there is a modulated (say AM modulated with voice) frequency of interest, we can tune the LO to the carrier frequency, and because of heterodyning principles we are now only having to deal with the baseband frequencies, which is much easier/cheaper to digitize.
Now if we would have only heterodyned with a single mixer, we would get the difference between the carrier frequency and the baseband frequencies, thus the upper sideband would now appear as frequencies from zero to the upper limit of the baseband frequencies, and the lower sideband would appear as a mirror image to that, thus, would be negative frequencies.
How do we deal with negative frequencies? FWIU, this is where the quadrature (Q) heterodyning comes in. Somehow by shifting the LO frequency 90º, the output from the Q mixer contains the information which was present in the lower sideband.
This is what I am having difficulty understanding/visualizing. I am sure Euler's formula comes into this, and could probably follow the math if presented to me (and by all means don't get me wrong, I am interested in seeing the math as well,) but I am having difficulty visualizing how this can be so.
For one thing, we talk about 2 LO signals 90º apart. But what determines which one is which? Ie, if I were to mix the incoming signal with only one LO output or the other, the incoming signal would not know the difference and in either case would give me difference frequency between the carrier and the baseband. It would look the same whether I heterodyned it with a sine wave of a cosine wave, because who knows what phase angle either of those waveforms would be to the carrier? It could be anything. At least in traditional superhet radios, it didn't matter.
So then why is I the "in phase" signal? In phase to what? The carrier? But then, following the reasoning of the previous paragraph, why would this matter?