# Maximum likelihood estimator of SNR for a known signal superimposed in AWGN

Assuming the following model (Fir instant, received signal of a communication / RADAR system):

$$x[n] = \alpha s[n] + w[n]$$

Where:

• $s[n]$ is a known signal of length $L$ and known energy $p = \sum_{n=0}^{L-1} {s}^{2}[n]$.
• $\alpha$ is unknown attenuation factor. Though it is known $\alpha \leq 1$.
• $w[n]$ is additive white gaussian noise (AWGN) where $E[w[n]] = 0$ and $E[w[m]w[n] = \delta[n-m] {\sigma}^{2}$. The noise variance ${\sigma}^{2}$ is unknown parameter.
• There are $N$ samples, namely, $n \in [0, \ldots, N-1]$ where $N > L$.

The question is, what is the Maximum Likelihood Estimator of the SNR of the received signal - $\frac{{\alpha}^{2}p}{{\sigma}^{2}}$?

Using Matched Filter / Correlation one could calculate the ML estimator of $\alpha$ and ${\sigma}^{2}$. Yet since the SNR isn't one to one function of the parameters, it cannot be calculated using them.

Hence the question is: what's the MLE of the SNR of the received signal?

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"... since the SNR isn't one to one function of the parameters, it cannot be calculated using them" isn't always true. In this case, the MLE of the SNR is exactly what you'd hope, just plug in the MLEs of $\alpha$ and $\sigma^2$.
To see this heuristically, assume the MLE of the SNR, label it $\widehat{SNR}$, is something else. Then the likelihood calculated using $\widehat{SNR}$ and the MLE of, say, $\alpha$ and using the relationship $SNR = \alpha^2p / \sigma^2$ will be lower than it would be if the MLEs of both $\alpha$ and $\sigma^2$ were used, because the estimate of $\sigma^2$ will be different from the MLE of $\sigma^2$. Therefore $\widehat{SNR}$ isn't maximizing the likelihood. Therefore it isn't the MLE of the SNR.