# 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]|\} &= E\{ |x[n] + w[n]| \} \\ &= 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?

• I tried to make your derivation slightly more legible. Hope this helps! – Marcus Müller Oct 8 '20 at 8:58
• It certainly helped me: your math isn't correct, sorry. You can't just say "the expectation of the magnitude of a complex value is equal to the square root of the sum of the expectations of the squares of its real and imaginary part", your first equation, when these might be correlated – and usually, $x$ does have a strong correlation between real and imaginary part, and hence, so does your $y$. – Marcus Müller Oct 8 '20 at 9:08
• Thank you very much for making it legible and your feedback. The first line was indeed wrong, I just corrected it. For cases that $x[n]$ is not IQ modulated data, like a carrier wave, the expectations of real and imaginary parts are indeed correlated. The correct approach it would be to say $y[n] = E\{ y[n]y[n]^* \}$, and similarly with $x[n]$ and $w[n]$? – A V Oct 8 '20 at 9:44
• Ok, so I'm confused now. Do you care about $E\{|y|\}$ or $|E\{y\}|$? Because for zero mean $x$ and independent zero mean $w$: $|E\{y\}|\equiv 0$, and I don't think that is what you wanted to know. – Marcus Müller Oct 8 '20 at 9:49
• Other than that, you're very quick at just pulling the expectation operator through operations; you can't do that, unless the operation is actually linear. And neither the square root, nor the square, is. – Marcus Müller Oct 8 '20 at 9:51

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.