# Amplitude modulation in SDR [closed]

I understand the theory of amplitude modulation. However, I couldn't understand it's implementation in SDR, which use IQ data, and I would like to learn end to end amplitude modulation system in detail as mathematical model in discrete time. Let say we want to transmit wav file and assume that each sample of the wav file data (discrete sound) is represented with 8 bits which is directly associated with amplitude. To be clear continious amplitude modulation formula is given as

$$y(t) = (1+m(t))\cos(w_ct)$$

where the $$m(t)$$ is the message signal and $$-1 < m(t) < 1$$. In discrete quadrature form how do we transmit $$m[n]$$? Do we need upsampling to transmit sound file which is sampled at 44 kHz if carrier frequency is 10 MHz? Also which steps should we follow to demodulate transmitted signal. As far as I know, received data can be decoded by simply using the formula below,

$$\sqrt{I^2+Q^2}$$

but I don't understand the reason of simplicity of this formula. I know that the formula is equal to amplitude in quadrature form. But do we need downsampling before decoding as the formula above. Why don't we need phase or frequency synchronization? How should we select sample per symbol in the receiver? Can anyone explain end to end amplitude modulation with a mathematical and/or block model?