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One extension of parametric estimation theory necessary for its application to array processing is the estimation of signalparameters. We assume that we observe a signal s l , whose characteristics are known save a few parameters , in the presence of noise. Signal parameters, such as amplitude, time origin, and frequencyif the signal is sinusoidal, must be determined in some way. In many cases of interest, we would find it difficult to justify aparticular form for the unknown parameters' a priori density. Because of such uncertainties, the minimum mean-squared error and maximum a posteriori estimators cannot be used in many cases. The minimum mean-squared error linear estimator does not require this density, but it is most fruitfully used when the unknownparameter appears in the problem in a linear fashion (such as signal amplitude as we shall see).

Linear minimum mean-squared error estimator

The only parameter that is linearly related to a signal is the amplitude. Consider, therefore, the problem where theobservations at an array's output are modeled as

l l 0 L 1 r l s l n l
The signal waveform s l is known and its energy normalized to be unity ( l s l 2 1 ). The linear estimate of the signal's amplitude is assumed to be of the form l h l r l , where h l minimizes the mean-squared error. To use the Orthogonality Principle expressed by this equation , an inner product must be defined for scalars. Little choice avails itself butmultiplication as the inner product of two scalars. The Orthogonality Principle states that the estimation error mustbe orthogonal to all linear transformations defining the kind of estimator being sought. h l 0 L 1 h LIN l r l k 0 L 1 h k r k 0 Manipulating this equation to make the universality constraint more transparent results in h k 0 L 1 h k l 0 L 1 h LIN l r l r k 0 Written in this way, the expected value must be 0 for each value of k to satisfy the constraint. Thus, the quantity h LIN of the estimator of the signal's amplitude must satisfy k l 0 L 1 h LIN l r l r k r k Assuming that the signal's amplitude has zero mean and is statistically independent of the zero-mean noise, the expectedvalues in this equation are given by r l r k 2 s l s k K n k l r k 2 s k where K n k l is the covariance function of the noise. The equation that must be solved for the unit-sample response h LIN of the optimal linear MMSE estimator of signal amplitude becomes
k l 0 L 1 h LIN l K n k l 2 s k 1 l 0 L 1 h LIN l s l
This equation is easily solved once phrased in matrix notation. Letting K n denote the covariance matrix of the noise, s the signal vector, and h LIN the vector of coefficients, this equation becomes K n h LIN 2 1 s h LIN s The matched filter for colored-noise problems consisted of the dot product between the vector of observations and K n s (see the detector result ). Assume that the solution to the linear estimation problem is proportional to the detectiontheoretical one: h LIN c K n s , where c is a scalar constant. This proposed solution satisfies the equation; the MMSE estimate ofsignal amplitude corresponds to applying a matched filter to the observations with
h LIN 2 1 2 s K n s K n s
The mean-squared estimation error of signal amplitude is given by 2 2 l 0 L 1 h LIN l r l Substituting the vector expression for h LIN yields the result that the mean-squared estimation error equals the proportionality constant c defined earlier. 2 2 1 2 s K n s

Thus, the linear filter that produces the optimal estimate of signal amplitude is equivalent to the matched filter used todetect the signal's presence. We have found this situation to occur when estimates of unknown parameters are needed to solvethe detection problem (see Detection in the Presence of Uncertainties ). If we had not assumed the noise to be Gaussian, however, thisdetection-theoretic result would be different, but the estimator would be unchanged. To repeat, this invarianceoccurs because the linear MMSE estimator requires no assumptions on the noise's amplitude characteristics.

Let the noise be white so that its covariance matrix is proportional to the identity matrix ( K n n 2 I ). The weighting factor in the minimum mean-squared error linear estimator is proportional to thesignal waveform. h LIN l 2 n 2 2 s l LIN 2 n 2 2 l 0 L 1 s l r l This proportionality constant depends only on the relative variances of the noise and the parameter. If the noise variance can be considered to be much smaller than the a priori variance of the amplitude, then this constant does not depend on these variances and equals unity. Otherwise, thevariances must be known.

We find the mean-squared estimation error to be 2 2 1 2 n 2 This error is significantly reduced from its nominal value 2 only when the variance of the noise is small compared with the a priori variance of the amplitude. Otherwise, this admittedly optimum amplitude estimateperforms poorly, and we might as well as have ignored the data and "guessed" that the amplitude was zero

In other words, the problem is difficult in this case.
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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Estimation theory. OpenStax CNX. May 14, 2006 Download for free at http://cnx.org/content/col10352/1.2
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