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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?

Thank you in advance.

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    $\begingroup$ There are a lot of questions here. My advice is to take things one step at a time. Step 1: theory of AM using continuous-time signals. Step 2: Sampling. Step 3: re-formulate AM using discrete-time signals. Step 4: How SDRs work. Step 5: Implement Step 3 in SDR. $\endgroup$
    – MBaz
    Commented Feb 9, 2023 at 14:36
  • $\begingroup$ Thank you @MBaz for your precious advice. I believe that I am confused at Step 3. Can you recommend any resource for reformulation of AM that use discrete-time signals? $\endgroup$
    – unique
    Commented Feb 9, 2023 at 17:10
  • $\begingroup$ I'd suggest trying it on your own. It's not very difficult. $\endgroup$
    – MBaz
    Commented Feb 9, 2023 at 18:11
  • $\begingroup$ Welcome to DSP.SE! There are great questions but should be narrowed significantly for this shorter Q&A forum. I recommend following MBaz’s advise; think it through and then within the details of one of the steps, show us where you are stuck and why (one question for one answer even if you do several different posts as you proceed with your project). (Also do some basic research first so that we don't repeat answers readily available). Good luck with your project! $\endgroup$ Commented Feb 10, 2023 at 3:49

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