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In a wireless communication scenario, received signal $y$ is governed by the following equation:

$$y=hx+n$$

where $h=|h|\exp(j\theta)$ and $j=\sqrt{-1}$ is the random channel coefficient with distinct distributions for magnitude $|h|$ and phase $\theta$. $x$ is the complex valued symbol to be sent over wireless channel and $n$ is random Gaussian noise (again complex).

To estimate we take help of a known signal $\bar{x}$ and jointly estimate phase and magnitude of channel, i.e., $|h|, \theta$, in the presence of noisy reception $y$. I use Bayesian estimate for the same.

My problem: Is it possible to separately estimate magnitude $|h|$ and phase $\theta$ of the channel given the equation
$$y=hx+n$$ where $x$ is again known as $\bar{x}$. Again, I want to use Bayesian estimate for separately estimating phase and magnitude of $h$.

I believe we can't because the channel itself is defined as a complex whole and it is not possible to estimate its magnitude and phase separately. But I am not sure. Please help

PS:- As per the suggested answer by Martin my scenario is (1) wherein I am performing Bayesian estimation of $h$ for a single pair of $y$ and $\bar{x}$ and I have used priors for phase and magnitude distribution of $h$. The data do not let me disentangle phase and magnitude (and noise) at all. However, just to be sure is there some more mathematical explanation for the given scenario?

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    $\begingroup$ dsp.stackexchange.com would likely be a better place for this question since digital communication theory is covered in detail there $\endgroup$ Commented 3 hours ago
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    $\begingroup$ The use of $j$ to denote $\sqrt{-1}$ instead of $i$ (universal in more mathematical circles) reinforces @Mahmoud's suggestion that dsp.SE would be a better place for this question. They use $j$ over there all the time. $\endgroup$ Commented 3 hours ago

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Major edits as I previously misunderstood the question.

If you only observe one $y$, $\bar x$ pair and don't know the variance of the noise term, then the data do not let you disentangle phase and magnitude and noise. Any inferences about them individually will be driven by your prior. The technical term here is non-identifiability - any single $y$ observation would be equally consistent with a) $hx$ being close to $y$ and the noise magnitude being small and b) $hx$ being far from $y$ and the noise magnitude being large and c) anything in between (more precisely only the likelihood is not identified, when you add a strong prior, you may get an identifiable model overall, but all the identifying information comes from the prior not the data.

If you know the variance of the noise term than this in fact should work.

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  • $\begingroup$ Thanks. I am performing Bayesian estimation of $h$ for a single pair of $y$ and $\bar{x}$ and I have used priors for phase and magnitude distribution of $h$. So I think my case is (1) and the data do not let me disentangle phase and magnitude (and noise) at all. So, I believe I am correct then no separate estimates exit? However, is there some more mathematical justification for scenario (1)? $\endgroup$ Commented 8 hours ago
  • $\begingroup$ Furthermore, by $j_i$ you mean $\theta_i$ (Phase), right? Because $j=\sqrt{-1}$ in my question, I have edited it. $\endgroup$ Commented 8 hours ago
  • $\begingroup$ @Userhanu oh, that didn't occur to me - thought that that's something analogous for time. Yeah, than you know it and I will edit the answer to reflect that. $\endgroup$ Commented 3 hours ago

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