iHz

Shinsukke, check out Michael Ossman’s Hackrf / SDR tutorials on youtube (also linked via his Great Scott gadgets site). Really good explanations of quadrature and related concepts assuming HS math concepts only (FYI, although it might seem counterintuitive, using complex exponentials is 10x easier than trig). AFTER viewing that course, you will understand this short answer to your question: “because negative frequencies exist.”

A sin(x) + B cos(x)

trigonometry tells us that cos(x) = sin(x + 90°)

Therefore:

A sin(x) + B sin(x + 90°) can express any arbitrary sine wave, both in amplitude and phase

I/Q is just that, the readings of those 2 coefficients, A and B. With those, you can very easily represent the exact same sine signal, with the exact same phase and exact same amplitude

Otherwise, you could only represent the same amplitude, but your signal wont be on the same phase

Read: https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinations

One way of looking at it is this: we all know that you need to sample at 2*f if you were sampling a frequency up to f (the Nyquist criterion). This is because if your sampling frequency is too low you just subsample and get a low-frequency aliased result. To see that, imagine sampling a signal with frequency ‘f’ with samples also at ‘f’: you would just get a constant value, and if your signal would increase in frequency to ‘f+delta’ you would measure something with a low frequency ‘delta’ because your sampling point would creep slowly along the high frequency ‘f+delta’ signal (this is actually used to sample very-high-frequency signals by so called ‘_sampling oscilloscopes_’, but it assumes that the measured signal repeats over and over, so it can’t be used to recover arbitrary signals)

Now, imagine what happens if you measure a frequency ‘f’ with samples at 2*f. You would just be hitting the signal at the same points over and over; if you’re ‘very lucky’, you could hit it at the maximum and minimum and get a correct measurement, but all the other phases would give you a fake reduced amplitude, and in particular If you’re ‘very unlucky’, you could be sampling the signal at zero-crossings, and get a constant zero output.

Note that ‘very lucky’ and ‘very unlucky’ measurements are just sampling the signal with the phases 90 degrees off each other. So the idea is to always measure both of them, and it’s called ‘quadrature sampling‘ or I+Q sampling. It turns out that you can correctly reconstruct all signals up to frequency ‘f’ from I+Q samples at frequency 2f. It turns out they are easy to capture because shifting the phase is just a fixed time delay of 1/4f, so the I channel measures samples at t0 and t0+1/2f, and the Q channel measures at t0+1/4f and t0+3/4*f.

We discovered that depending on how fast you spin that wheel, is correlated to what phenomena you get.

You can go from nothing, up to microwaves, visible light, UV, xrays.

I just keep thinking how interesting is to see the wheel, even as a concept in logical spaces.

How something as simple as 90° makes stuff such a sine appear as cosine, real numbers appear as imaginary, orthogonal frequencies appear that wont affect your signal, and such.

There is a huge universe there, 90° right ahead of you.