# Maximum Likelihood Estimator of SNR for a Known Signal Superimposed in AWGN

I would like to evaluate the Maximum Likelihood Estimator for the SNR of a given signal:

$x(t) = as(t-\tau) + n(t)$

Under the following assumptions (This is the model of Radar Signal):

• The input signal is the sum of the attenuated and delayed reference signal and AWGN. This is the model of a Radar signal.
• The signal $s(t)$ is known. The signal has finite Support and Energy s.t. $\int_{0}^{T}{s(t)}^{2}dt = p < \infty$.
• The attenuation factor, $a$, is unknown.
• The noise, $n(t)$ is Additive White Gaussian Noise with $E[n(t)] = 0$ and $E[{n(t)}^{2}] = \delta(t){\sigma}^{2}$ where ${\sigma}^{2}$ is unknown.

How would you estimate the SNR of the given signal $x(t)$? How would you change the answer given $a$, ${\sigma}^{2}$ or both?

Thanks.

-
Shouldn't the autocorrelation function of the white noise process be something like $R_n(t) = A\delta(t)$ (where $\delta(t)$ is an impulse or Dirac delta) which does not quite match up with $E[n^2(t)] = R_n(0) = \sigma^2$? –  Dilip Sarwate Oct 8 '11 at 16:44
@DilipSarwate, Fixed. Thanks. –  Drazick Oct 8 '11 at 17:02
This question is almost the same as another one posted by the same author a few days ago. I had suggested that it be moved to stat.stackexchange or dsp.stackexchange but the author voluntarily withdrew it instead. For this version, since $s(t)$ is known and not a random signal, $E[s(t)^2] = p$ simply means that $s(t) = \pm \sqrt{p}$ for all real numbers $t$. Thus, the energy $\int_{-\infty}^{\infty} s^2(t) dt$ of $s(t)$ cannot be finite unless $s(t)$ has finite support: if it does, estimating $\sigma^2$ is easy. What is meant by SNR is left unspecified. So this question is unanswerable. –  Dilip Sarwate Oct 9 '11 at 1:11
@DilipSarwate, I deleted the previous question since you said it was badly written. I will add the assumption of finite support. If you think somewhere else people will be able to answer Estimation problems better, just direct me. Thanks. –  Drazick Oct 9 '11 at 6:25
"If you think somewhere else people will be able to answer Estimation problems better, just direct me." I did direct you, twice, to dsp.stackexchange and stat.stackexchange but the last time you said you wanted mathematicians to answer the question. You still have not said what you mean by SNR; if you don't say that the finite support of $s(t)$ is $(-0, T]$, whatever $-0$ is (typo?), then the energy of $s(t)$ might be larger than $p$, and so on. –  Dilip Sarwate Oct 9 '11 at 12:01