# 3.10 Estimation theory: problems  (Page 3/3)

In this section , we questioned the existence of an efficient estimator forsignal parameters. We found in the succeeding example that an unbiased efficient estimator exists for the signalamplitude. Can a nonlinearly represented parameter, such as time delay, have an efficient estimator?

Simplify the condition for the existence of an efficient estimator by assuming it to be unbiased. Note carefullythe dimensions of the matrices involved.

Show that the only solution in this case occurs when the signal depends "linearly" on the parameter vector.

In Poission problems, the number of events $n$ occurring in the interval $\left[0 , T\right)$ is governed by the probability distribution (see The Poission Process ) $(n)=\frac{(T)^{n}}{n!}e^{-(T)}$ where  is the average rate at which events occur.

What is the maximum likelihood estimate of average rate?

Does this estimate satisfy the Cramr-Rao bound?

In the "classic" radar problem, not only is the time of arrival of the radar pulse unknown but also the amplitude.In this problem, we seek methods of simultaneously estimating these parameters. The received signal $r(l)$ is of the form $r(l)={}_{1}s(l-{}_{2})+n(l)$ where ${}_{1}$ is Gaussian with zero mean and variance ${}_{1}^{2}$ and ${}_{2}$ is uniformly distributed over the observation interval. Find the receiver that computes the maximum a posteriori estimates of ${}_{1}$ and ${}_{2}$ jointly. Draw a block diagram of this receiver and interpret its structure.

We state without derivation the Cramr-Rao bound for estimates of signal delay (see this equation ).

The parameter  is the delay of the signal $s()$ observed in additive, white Gaussian noise: $r(l)=s(l-)+n(l)$ , $l\in \{0, , L-1\}$ . Derive the Cramr-Rao bound for this problem.

In Time-delay Estimation , this bound is claimed to be given by $\frac{{}_{n}^{2}}{E^{2}}$ , where $^{2}$ is the mean-squared bandwidth. Derive this result from your general formula. Does the bound make sense for allvalues of signal-to-noise ratio $\frac{E}{{}_{n}^{2}}$ ?

Using optimal detection theory, derive the expression (see Time-Delay Estimation ) for the probability of error incurred when trying todistinguish between a delay of  and a delay of $+$ . Consistent with the problem pposed for the Cramr-Rao bound, assume the delayed signals are observed in additive, white Gaussian noise.

In formulating detection problems, the signal as well as the noise are sometimes modeled as Gaussian processes. Let'sexplore what differences arise in the Cramr-Rao bound derived when the signal is deterministic. Assume thatthe signal contains unknown parameters  , that it is statistically independent of the noise, and that the noise covariancematrix is known.

What forms do the conditional densities of the observations take under the two assumptions? What are thetwo covariance matrices?

Assuming the stochastic signal model, show that each element of the Fisher information matrix has the form $F_{i, j}=\frac{1}{2}\mathrm{tr}(K^{(-1)}\frac{\partial^{1}K}{\partial {}_{i}}K^{(-1)}\frac{\partial^{1}K}{\partial {}_{j}})$ where $K$ denotes the covariance matrix of the observations. Make this expression more complex by assuming the noisecomplement has no unknown parameters.

Compare the stochastic and deterministic bounds, the latter is given by this equation , when the unknown signal parameters are amplitude and delay. Assume thenoise covariance matrix equals ${}_{n}^{2}I$ . Do these bounds have similar dependence on signal-to-noise ratio?

The histogram probability density estimator is a special case of a more general class of estimators known as kernel estimators . $(p(r, x))=\frac{1}{L}\sum_{l=0}^{L-1} k(x-r(l))$ Here, the kernel $k()$ is usually taken to be a density itself.

What is the kernel for the histogram estimator.

Interpret the kernel estimator in signal processing terminology. Predict what the most time consumingcomputation of this estimate might be. Why?

Show that the sample average equals the expected value of a random variable having the density $(p(r, x))$ regardless of the choice of kernel.

Random variables can be generated quite easily if the probability distribution function is "nice." Let $X$ be a random variable having distribution function $P(X, )$ .

Show that the random variable $U=P(X, X)$ is uniformly distributed over $\left(0 , 1\right)$ .

Based on this result, how would you generate a random variable having a specific density with a uniform randomvariable generator, which is commonly supplied with most computer and calculator systems?

How would you generate random variables having the hyperbolic secant density $p(X, x)=\frac{1}{2}\mathrm{sech\,}\left(\frac{\pi x}{2}\right)$ ?

Why is the Gaussian not in the class of "nice" probability distribution functions? Despite this fact, the Gaussianand other similarly unfriendly random variables can be generated using tabulated rather than analytic forms forthe distribution function.

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
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