SNR for the magnitude of a complex signal The SNR for a complex valued signal is presented by Dilip Sarwate in: https://dsp.stackexchange.com/a/42359
Following the same steps I would like to estimate the SNR for the magnitude of a complex signal. The received sample signal $y[n]$ is consisted of the transmitted signal $x[n]$ plus noise $w[n]$, such as,
$$y[n]=x[n]+w[n].$$
Also, it is given that, $w$ is AWGN $\sim CN(0,2\sigma^2_w)$.
When the received signal is treated as magnitude,
\begin{align}
E\{|y[n]|\}
&amp;= E\{ |x[n] + w[n]| \} \\
&amp;= E\{ \sqrt{ x^2_I[n] + 2x_I[n]w_I[n] + w^2_I[n] + x^2_Q[n] + 2x_Q[n]w_Q[n] + w^2_Q[n]} \}
\end{align}
And this is were I hit a road block (assuming that my math is correct). How can you extract the SNR out of this equation?
Which I don't know if it is solvable since the square rot expands all over the parameters.
Another solution I had in mind is to transfer the signals to FFT and use integrals to compare the two different magnitudes. Does that make more sense?
The magnitude $|y[n]|$ follows a Rice (aka Rician) PDF. You can find analytical expressions for the average, variance and SNR in Voltage Signal-to-Noise Ratio (SNR) Nonlinearity Resulting From Incoherent Summations and in The Squaring-Loss Paradox. These two references further investigate the SNR of the addition of independent magnitudes.