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Estimation theory - Example: DC gain in white Gaussian noise | | Estimation theory - Example: DC gain in white Gaussian noise: Encyclopedia II - Estimation theory - Example: DC gain in white Gaussian noise | | Consider a received discrete signal, x[n], of N independent samples that consists of a DC gain A with Additive white Gaussian noise w[n] with known variance σ2 (i.e., ). Since the variance is known then the only unknown parameter is A.
The model for the signal is then
Two possible (of ...
See also:Estimation theory, Estimation theory - Fields that use estimation theory, Estimation theory - Estimation process, Estimation theory - Basics, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Maximum likelihood, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Books | | | Estimation theory, Estimation theory - Basics, Estimation theory - Books, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Estimation process, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Fields that use estimation theory, Estimation theory - Maximum likelihood, Bias (statistics), Completeness (statistics), Detection theory, Efficiency (statistics), Expectation-maximization algorithm (EM algorithm), Information theory, Rao-Blackwell theorem, Sufficiency (statistics), Maximum likelihood, Method of moments, generalized method of moments, Cramér-Rao inequality, Minimum mean squared error (MMSE), Maximum a posteriori (MAP), Minimum variance unbiased estimator (MVUE), Best linear unbiased estimator (BLUE), Unbiased estimators — see bias (statistics)., Particle filter, Markov chain Monte Carlo (MCMC), Kalman filter, Wiener filter | | |
| | | Estimation theory: Encyclopedia II - Estimation theory - Example: DC gain in white Gaussian noise
Estimation theory - Example: DC gain in white Gaussian noise
Consider a received discrete signal, x[n], of N independent samples that consists of a DC gain A with Additive white Gaussian noise w[n] with known variance σ2 (i.e., ). Since the variance is known then the only unknown parameter is A.
The model for the signal is then
Two possible (of many) estimators are:
Both of these estimators have a mean of A, which can be shown through taking the expected value of each estimator
and
At this point, these two estimators would appear to perform the same. However, the difference between them becomes apparent when comparing the variances.
and
It would seem that the sample mean is a better estimator since, as , the variance goes to zero.
Estimation theory - Maximum likelihood
Continuing the example using the maximum likelihood estimator, the probability density function (pdf) of the noise for one sample w[n] is
and the probability of x[n] becomes (x[n] can be thought of a )
By independence, the probability of becomes
Taking the natural logarithm of the pdf
and the maximum likelihood estimator is
Taking the first derivative of the log-likelihood function
and setting it to zero
This results in the maximum likelihood estimator
which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for N samples of AWGN with a fixed, unknown DC gain.
Estimation theory - Cramér-Rao lower bounds
To find the Cramér-Rao lower bounds (CRLB) of the sample mean estimator, it is first necessary to find the Fisher information number
and copying from above
Taking the second derivative
and finding the negative expected value is trivial since it is now a deterministic constant
Finally, putting the Fisher information into
results in
Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is equal to the Cramér-Rao lower bounds for all values of N and A. The sample mean is the minimum variance unbiased estimator (MVUE) in addition to being the maximum likelihood estimator.
This example of DC gain + WGN is a recurring example in Kay's Fundamentals of Statistical Signal Processing.
Other related archivesActuator, Additive white Gaussian noise, Articles to be merged, Bayesian statistics, Best linear unbiased estimator, Bias (statistics), CAT, Channel, Clinical trials, Completeness (statistics), Control theory, Cramér-Rao inequality, Cramér-Rao lower bounds, DC, Detection theory, Digital image processing, Digital signal processing, EEG, EKG/ECG, ESPRIT, Efficiency (statistics), Estimation theory, Expectation-maximization algorithm, Fisher information, Information theory, Kalman filter, MRI, Markov chain Monte Carlo, Maximum a posteriori, Maximum likelihood, Medical ultrasonography, Medicine, Method of moments, Minimum mean squared error, Minimum variance unbiased estimator, Network intrusion detection system, Opinion polls, Particle filter, Quality control, Radar, Rao-Blackwell theorem, Seismology, Signal processing, Statistics, Sufficiency (statistics), Telecommunications, Wiener filter, bias (statistics), computer vision, derivative, discrete signal, epistemic probability, estimator, estimators, expected value, generalized method of moments, independent, information, maximum likelihood, mean, minimum mean squared error, minimum variance unbiased estimator, natural logarithm, noise, noisy, optimal, optimality, periodogram, probability, probability density function, probability mass function, radar, random vector, sample mean, samples, signal, signal processing, sonar, statistical samples, statistics, variance, vector
 Adapted from the Wikipedia article "Example: DC gain in white Gaussian noise", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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