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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.

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    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 $E[n^2(t)] = R_n(0) = \sigma^2$?2011-10-08
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    @DilipSarwate, Fixed. Thanks.2011-10-08
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    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.2011-10-09
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    @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.2011-10-09
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    "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.2011-10-09
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    Could you assist me to state the question correctly and than I'll post it at the places you suggested? Basically it is the model of a Radar Signal (Or for that mater received bit in a communication system). Thanks.2011-10-09
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    @DilipSarwate, Will you help me phrase it correctly and transfer it? Thanks.2011-10-10

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