# 3.7 Maximum likelihood estimators of signal parameters

 Page 1 / 1

Many situations are either not well suited to linear estimation procedures, or the parameter is not well described as a randomvariable. For example, signal delay is observed nonlinearly and usually no a priori density can be assigned. In such cases, maximum likelihood estimators are more frequentlyused. Because of the Cramr-Rao bound , fundamental limits on parameter estimation performance can be derived for any signal parameter estimation problem where the parameter is not random.

Assume that the data are expressed as a signal observed in the presence of additive Gaussian noise.

$\forall l, l\in \{0, , L-1\}\colon r(l)=s(l, )+n(l)$
The vector of observations $r$ is formed from the data in the obvious way. Evaluating the logarithm of the observationvector's joint density, $\ln p(r, , r)=-1/2\ln \det (2\pi {K}_{n})-1/2(r-s())^T{K}_{n}^{(-1)}(r-s())$ where $s()$ is the signal vector having $P$ unknown parameters, and ${K}_{n}$ is the covariance matrix of the noise. The partial derivative of this likelihood function with respect to the ${i}^{\mathrm{th}}$ parameter ${}_{i}$ , for real-valued signals, $\frac{\partial^{1}\ln p(r, , r)}{\partial {}_{i}}=(r-s())^T{K}_{n}^{(-1)}\frac{\partial^{1}s()}{\partial {}_{i}}$ and, for complex-valued ones, $\frac{\partial^{1}\ln p(r, , r)}{\partial {}_{i}}=\Re ((r-s()){K}_{n}^{(-1)}\frac{\partial^{1}s()}{\partial {}_{i}})$ If the maximum of the likelihood function can be found by setting its gradient to $0$ , the maximum likelihood estimate of the parameter vector is the solution of the set of equations
$\forall i, i\in \{1, , P\}\colon (, ({}_{\mathrm{ML}}), (r-s())^T{K}_{n}^{(-1)}\frac{\partial^{1}s()}{\partial {}_{i}})=0$

The Cramr-Rao bound depends on the evaluation of the Fisher information matrix $F$ . The elements of this matrix are found to be

$\forall i, j, (i\land j)\in \{1, , P\}\colon F_{i, j}=\frac{\partial^{1}s()^T}{\partial {}_{i}}{K}_{n}^{(-1)}\frac{\partial^{1}s()}{\partial {}_{j}}$
Further computation of the Cramr-Rao bound's components is problem dependent if more than one parameter is involved, andthe off-diagonal terms of $F$ are nonzero. If only one parameter is unknown, the Cramr-Rao bound is given by
$(^{2})\ge b()^{2}+\frac{(1+\frac{d b()}{d }})^{2}}{\frac{\partial^{1}s()^T}{\partial }{K}_{n}^{(-1)}\frac{\partial^{1}s()}{\partial }}$
When the signal depends on the parameter nonlinearly (which constitute the interesting cases), the maximum likelihoodestimate is usually biased. Thus, the numerator of the expression for the bound cannot be ignored. One interestingspecial case occurs when the noise is white. The Cramr-Rao bound becomes $(^{2})\ge b()^{2}+\frac{{}_{n}^{2}(1+\frac{d b()}{d }})^{2}}{\sum_{l=0}^{L-1} \frac{\partial^{1}s(l, )}{\partial }^{2}}$ The derivative of the signal with respect to the parameter can be interpreted as the sensitivity of the signal to theparameter. The mean-squared estimation error depends on the"integrated" squared sensitivity: The greater this sensitivity, the smaller the bound.

For an efficient estimate of a signal parameter to exist, the estimate must satisfy the condition we derived earlier . $(\frac{d s()}{d }}^T{K}_{n}^{(-1)}(r-s()), I+\frac{d b}{d }}^T^{(-1)}\frac{d s()}{d }}^T{K}_{n}^{(-1)}\frac{d s()}{d }}(((r))--b))$ Because of the complexity of this requirement, we quite rightly question the existence of any efficient estimator,especially when the signal depends nonlinearly on the parameter (see this problem ).

Let the unknown parameter be the signal's amplitude; the signal is expressed as $s(l)$ and is observed in an array's output in the presence of additive noise. The maximum likelihood estimate of theamplitude is the solution of the equation $(r-({}_{\mathrm{ML}})s)^T{K}_{n}^{(-1)}s=0$ The form of this equation suggests that the maximum likelihood estimate is efficient. The amplitude estimate isgiven by $({}_{\mathrm{ML}})=\frac{r^T{K}_{n}^{(-1)}s}{s^T{K}_{n}^{(-1)}s}$ The form of this estimator is precisely that of the matched filter derived in the colored-noise situation (see equation ). The expected value of the estimate equals the actual amplitude.Thus the bias is zero and the Cramr-Rao bound is given by $(^{2})\ge s^T{K}_{n}^{(-1)}s^{(-1)}$ The condition for an efficient estimate becomes $(s^T{K}_{n}^{(-1)}(r-s), s^T{K}_{n}^{(-1)}s(({}_{\mathrm{ML}})-))$ whose veracity we can easily verify.

In the special case where the noise is white, the estimator has the form $({}_{\mathrm{ML}})=r^Ts$ , and the Cramr-Rao bound equals ${}_{n}^{2}$ (the nominal signal is assumed to have unit energy). The maximum likelihood estimate of the amplitude has fixed error characteristics that do not depend on the actual signal amplitude. A signal-to-noiseratio for the estimate, defined to be $\frac{^{2}}{(^{2})}$ , equals the signal-to-noise ratio of the observed signal.

When the amplitude is well described as a random variable, its linear minimum mean-squared error estimator has the form $({}_{\mathrm{LIN}})=\frac{{}_{}^{2}r^T{K}_{n}^{(-1)}s}{1+{}_{}^{2}s^T{K}_{n}^{(-1)}s}$ which we found in the white-noise case becomes a weighted version of the maximum likelihood estimate (see example ). $({}_{\mathrm{LIN}})=\frac{{}_{}^{2}}{{}_{}^{2}+{}_{n}^{2}}r^Ts$ Seemingly, these two estimators are being used to solve the same problem: Estimating the amplitude of a signalwhose waveform is known. They make very different assumptions, however, about the nature of the unknownparameter; in one it is a random variable (and thus it has a variance), whereas in the other it is not (and variance makesno sense). Despite this fundamental difference, the computations for each estimator are equivalent. It isreassuring that different approaches to solving similar problems yield similar procedures.

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
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!