# Understanding MMSE equalizer equation

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?

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

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.
• 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. Jun 29, 2021 at 8:41