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I have read many online materials and journal paper where the equation for MMSE equalizer is given as:

$G_{MMSE}$ = $H^H(HH^H+\nu^2I )$

where $H$ is the channel, $H^H$ is its hermitian. $\nu^2$ is noise variance (a statistical property) and $I$ is a identity matrix.

I know how to compute the channel $H$, my question is how to find $\nu^2$ will it be simply a scalar from Gaussian distribution of mean 0 and variance 1 ? If so, how is it taking care of the real channel noise, as the real channel noise can be anything?

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  • $\begingroup$ Could you write out what "MMSE" is for, please? Does that answer the last sentence in your question? $\endgroup$ Commented Jun 25, 2021 at 22:53

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First, regarding $v^2$, yes it's simply a scalar value. For example, you can get it in Matlab using $v^2$ = 1/10.^(SNR/10); where you SNR is a known scalar too.

In real channels, you usually receive real signal and the noise usually becomes ambient specially with difficult environments. It's hard to estimated the $v^2$ in real communication channels. I personally performed some real tests long time ago with high SNR and I used to set it as the simulation value mentioned above. That was working well.

Is that what you are asking about? I hope I got what you mean.

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  • $\begingroup$ Noise variance should be inversely propotional to SNR. Also, be careful that by setting noise variance $\nu^2 = 10^{-\frac{\mathrm{SNR}}{10}}$, you are assuming that $\mathbf{H}$ is normalized. $\endgroup$
    – AlexTP
    Commented Jun 29, 2021 at 8:41

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