So I recently watched this lovely overview of DSP/SDR explaining how demodulation of AM and FM becomes trivial by turning a signal into a complex number. (AM being the magnitude of the number and FM being the phase angle difference between two samples).
But I'm still confused. It's simply transferring the problem to "How do you get the envelope of a signal" to "how do you represent the signal in 2d". Where does this imaginary component come from?
The guy says to get the envelope of an AM waveform you need only one sample. And while you already are in the complex plane, yes. Sure. Get the magnitude of that one complex number. But a sample, as far as I understood, is a real value. For example a power measurement from a sine wave generator. So you get "0.7V". What's the envelope? It could be anything. Could be a rising edge of the sine, falling edge, a peak, even a valley if you have DC offset. So I call BS you need only one sample.
The person in the video simply glosses over how you actually get the imaginary part? Do you just add a sine wave imaginary component? (sin(t)*i)? Do you phase shift your entire signal by 90° (f(t-tau/4) * i) and add that? Or something else entirely?
Heck I don't even know how to tag this question.