The General Scalar Cramer Rao Bound English Language Essay

The Direction of Arrival ( DOA ) appraisal algorithm which may take assorted signifiers by and large follows from the homogenous solution of the wave equation. The theoretical accounts of involvement in this thesis may every bit use to an EM moving ridge every bit good as to an acoustic moving ridge. Assuming that the extension theoretical account is basically the same, we will, for analytical expedience, show that it can follow from the solution of Maxwell ‘s equations, which clearly are merely valid for EM moving ridges. In empty infinite the equation can be written as:

=0 ( 3.1 )

=0 ( 3.2 )

( 3.3 )

( 3.4 )

where “ . ” and “ A- ” , severally, denote the “ divergency ” and “ coil. ” Furthermore, B is the magnetic initiation. Tocopherol denotes the electric field, whereas and are the magnetic and dielectric invariables severally. Raising 3.1 the undermentioned coil belongings consequences as:

( 3.5 )

( 3.6 )

( 3.7 )

The changeless degree Celsius is by and large referred to as the velocity of extension. For EM moving ridges in free infinite, it follows from the derivation c = 1 / = 3 x m / s. The homogenous moving ridge equation ( 3.7 ) constitutes the physical motive for our assumed informations theoretical account, irrespective of the type of moving ridge or medium. In some applications, the implicit in natural philosophies are irrelevant, and it is simply the mathematical construction of the informations theoretical account that counts.

3.2 Plane moving ridge

In theA physicsA ofA wave extension, aA plane moving ridge is a constant-frequency moving ridge whoseA wave frontsA are infinite parallel planes of changeless peak-to-peakA amplitudeA normal to theA stage velocityA vector [ ] .

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Figure 3.1

Actually, it is impossible to hold a rare plane moving ridge in pattern, and merely a plane moving ridge of infinite extent can propagate as a plane moving ridge. Actually, manyA wavesA are about regarded as plane moving ridges in a localised part of infinite, e.g. , a localized beginning such as anA antennaA produces a field which is about a plane moving ridge far plenty from the aerial in itsA far-field part. Likely, we can handle the moving ridges asA light raysA which correspond locally to shave moving ridges, when theA length scalesA are much longer than the moving ridge ‘s wavelength, as is frequently looking of visible radiation in the field ofA optics.

3.2.1 Mathematical definition

Two maps which meet the standards of holding a changeless frequence and changeless amplitude are defined as theA sineA or cosineA maps. One of the simplest ways to utilize such aA sinusoidA involves specifying it along the way of the x axis. As the equation shown below, it uses the cosine map to show a plane moving ridge traveling in the positive ten way.

( 3.8 )

Where A ( x, T ) is the magnitude of the shown moving ridge at a given point in infinite and clip. A is theA amplitudeA of the moving ridge which is the peak magnitude of the oscillation. kA is the wave’sA moving ridge numberA or more specifically theA angularA wave figure and equalsA 2Iˆ/I» , whereA I»A is theA wavelengthA of the wave.A K has the units ofA radiansA per unit distance and is a criterion of how quickly the perturbation alterations over a given distance at a peculiar point in clip.

xA is a point along the x axis.A yA andA zA are non considered in the equation because the moving ridge ‘s magnitude and stage are the same at every point on any givenA y-zA plane. This equation defines what that magnitude and stage are.

A is the wave’sA angular frequencyA which equalsA 2Iˆ/T, andA TA is theA periodA of the moving ridge. In item, A omega , A has the units of radians per unit clip and is besides a criterion of how rapid the perturbation altering in a given length of clip at a peculiar point in infinite.

A is a given peculiar point in clip, and varphi , A is the moving ridge stage shiftA with the units of radians. It must do clear that a positive stage displacement will switch the moving ridge along the negative ten axis way at a given point of clip. A stage displacement ofA 2IˆA radians means switching it one wavelength precisely. Other preparations which straight use the moving ridge ‘s wavelength, period T, A frequencyA fA andA velocityA c , A are shown as follows:

A=A_o cos [ 2pi ( x/lambda- t/T ) + varphi ] , ( 3.9 )

A=A_o cos [ 2pi ( x/lambda- foot ) + varphi ] , ( 3.10 )

A=A_o cos [ ( 2pi/lambda ) ( x- Nutmeg State ) + varphi ] , ( 3.11 )

To appreciate the equality of the above set of equations denote that

f=1/T , !


c=lambda/T=omega/k , !

3.2.2 Application

Airplane moving ridges are solutions for aA scalarA wave equationA in the homogenous medium. As forA vectorA moving ridge equations, e.g. , moving ridges in an elastic solid or the 1s describingA electromagnetic radiation, the solution for the homogenous medium is similar. In vector moving ridge equations, theA scalarA amplitudeA A is replaced by a constantA vector. e.g. , in electromagnetismA A is the vector of theA electric field, A magnetic field, orA vector potency. TheA cross waveA is a sort of moving ridge in which the amplitude vector isA perpendicularA toA K, which is the instance for electromagnetic moving ridges in anA isotropicA infinite. On the contrast, theA longitudinal waveA is a sort of moving ridge in which the amplitude vector is parallel toA K, typically, such as for acoustic moving ridges in a gas or fluid.

The plane moving ridge equation is true for arbitrary combinations ofA I‰A andA k. However, all existent physical mediums will merely let such moving ridges to propagate for these combinations ofA I‰A andA kA that satisfy theA scattering relationA of the mediums. The scattering relation is frequently demonstrated as a map, A I‰ ( K ) , where ratioA I‰/|k| gives the magnitude of theA stage velocityA andA dI‰/dkA denotes theA group speed. As for electromagnetism in an isotropic instance with index of refractionA coefficient N, the stage speed isA c/n, which equals the group speed on status that the index is frequency independent.

In additive unvarying instance, a wave equation solution can be demonstrated as a superposition of plane moving ridges. This method is known as theA Angular Spectrum method. Actually, the solution signifier of the plane moving ridge is the general effect ofA translational symmetricalness. And in the more general instance, for periodic constructions with distinct translational symmetricalness, the solution takes the signifier ofA Bloch moving ridges, which is most celebrated inA crystallineA atomic stuffs, inA the photonic crystalsA and other periodic moving ridge equations.

3.3 Propagation

Many physical phenomena are either a consequence of moving ridges propagating through a medium or exhibit a moving ridge like physical manifestation. Though 3.7 is a vector equation, we merely consider one of its constituents, say E ( R, T ) where R is the radius vector. It will subsequently be assumed that the measured detector end products are relative to E ( R, T ) . Interestingly plenty, any field of the signifier E ( R, T ) = , which satisfies 3.7, provided with T denoting heterotaxy. Through its dependance on merely, the solution can be interpreted as a moving ridge going in the way, with the velocity of extension. For the latter ground, I± is referred to as the awkwardness vector. The main involvement herein is in narrowband forcing maps. The inside informations of bring forthing such a forcing map can be found in the authoritative book by Jordan [ 59 ] . In complex notation [ 63 ] and taking the beginning as a mention, a narrowband transmitted wave form can be expressed as:

( 3.12 )

where s ( T ) is slowly clip changing compared to the bearer. For, where B is the bandwidth of s ( T ) , we can compose:

( 3.13 )

In the last equation 3.13, the alleged moving ridge vector was introduced, and its magnitude is the wavenumber. One can besides compose, where is the wavelength. Make certain that K besides points in the way of extension, e.g. , in the x-y plane we can acquire:

( 3.14 )

where is the way of extension, defined counter clockwise relative the x axis. It should be noted that 3.12 implicitly assumed far-field conditions, since an isotropic, which refers to uniform propagation/transmission in all waies, point beginning gives rise to a spherical travelling moving ridge whose amplitude is reciprocally relative to the distance to the beginning. All points lying on the surface of a domain of radius R will so portion a common stage and are referred to as a moving ridge forepart. This indicates that the distance between the emitters and the receiving antenna array determines whether the spherical grade of the moving ridge should be taken into history. The reader is referred to e.g. , [ 10, 24 ] for interventions of close field response. Far field having conditions imply that the radius of extension is so big that a level plane of changeless stage can be considered, therefore ensuing in a plane moving ridge as indicated in Eq. 8. Though non necessary, the latter will be our false on the job theoretical account for convenience of expounding.

Note that a additive medium implies the cogency of the superposition rule, and therefore allows for more than one going moving ridge. Equation 8 carries both spacial and temporal information and represents an equal theoretical account for separating signals with distinguishable spatial-temporal parametric quantities. These may come in assorted signifiers, such as DOA, in general AZ and lift, signal polarisation, transmitted wave forms, temporal frequence etc. Each emitter is by and large associated with a set of such features. The involvement in blossoming the signal parametric quantities forms the kernel of detector array signal processing as presented herein, and continues to be an of import and active subject of research.

3.4 Smart aerial

Smart aerials are devices which adapt their radiation form to accomplish improved public presentation – either scope or capacity or some combination of these [ 1 ] .

The rapid growing in demand for nomadic communications services has encouraged research into the design of wireless systems to better spectrum efficiency, and increase nexus quality [ 7 ] . Using bing methods more effectual, the smart aerial engineering has the possible to significantly increase the radio. With intelligent control of signal transmittal and response, capacity and coverage of the nomadic radio web, communications applications can be significantly improved [ 2 ] .

In the communicating system, the ability to separate different users is indispensable. The smart aerial can be used to add increased spacial diverseness, which is referred to as Space Division Multiple Access ( SDMA ) . Conventionally, employment of the most common multiple entree strategy is a frequence division multiple entree ( FDMA ) , Time Division Multiple Access ( TDMA ) , and Code Division Multiple Access ( CDMA ) . These independent users of the plan, frequence, clip and codification sphere were given three different degrees of diverseness.

Potential benefits of the smart aerial show in many ways, such as anti-multipath attenuation, cut downing the hold extended to back up smart antenna keeping high informations rate, intervention suppression, cut downing the distance consequence, cut downing the outage chance, to better the BER ( Bit Error Rate ) public presentation, increasing system capacity, to better spectral efficiency, back uping flexible and efficient handoff to spread out cell coverage, flexible direction of the territory, to widen the battery life of nomadic station, every bit good as lower care and operating costs.

3.4.1 Types of Smart Antennas

The environment and the system ‘s demands make up one’s mind the type of Smart Antennas. There are two chief types of Smart Antennas. They are as follows:

Phased Array Antenna

In this type of smart aerial, there will be a figure of fixed beams between which the beam will be turned on or steered to the mark signal. This can be done, merely in the first phase of accommodation to assist. In other words, as wanted by the traveling mark, the beam will be the Steering [ 2 ] .

Adaptive Array Antenna

Integrated with adaptative digital signal processing engineering, the smart aerial uses digital signal processing algorithm to mensurate the signal strength of the beam, so that the aerial can dynamically alter the beam which transmit power concentrated, as figure 3.2 shows. The application of spacial processing can heighten the signal capacity, so that multiple users portion a channel.

Adaptive aerial array is a closed-loop feedback control system dwelling of an antenna array and real-time adaptative signal receiving system processor, which uses the feedback control method for automatic alliance of the antenna array form. It formed nulling intervention signal beginning in the way of the intervention, and can beef up a utile signal, so as to accomplish the intent of anti-jamming [ 3 ] .

Figure 2 – chink for text version

Figure 3.2

3.4.2 Advantages and disadvantages of smart aerial


First of wholly, a high degree of efficiency and power are provided by the smart aerial for the mark signal. Smart antennas generate narrow pencil beams, when a large figure of antenna elements are used in a high frequence status. Therefore, in the way of the mark signal, the efficiency is significantly high. With the aid of adaptative array aerial, the same sum times the power addition will be produce, on status that a fixed figure of antenna elements are used.

Another betterment is in the sum of intervention which is suppressed. Phased array aerial suppress the intervention with the narrow beam and adaptative array aerials suppress by seting the beam form [ 2 ] .


The chief disadvantage is the cost. Actually, the cost of such devices will be more than earlier, non merely in the electronics subdivision, but in the energy. That is to state the device is excessively expensive, and will besides diminish the life of other devices. The receiving system ironss which are used must be decreased in order to cut down the cost. Besides, because of the usage of the RF electronics and A/D convertor for each aerial, the costs are increasing.

Furthermore, the size of the aerial is another job. Large base Stationss are needed to do this method to be efficient and it will increase the size, apart from this multiple external aerial needed on each terminus.

Then, when the diverseness is concerned, disadvantages are occurred. When extenuation is needed, diverseness becomes a serious job. The terminuss and base Stationss must fit with multiple aerials.

3.5 White noiseA

White noiseA is a randomA signal with a flatA power spectral denseness [ ] . In another word, the signal contains the equal power within a particularA bandwidthA at the Centre frequence. White noise draws its name fromA white lightA where the power spectral denseness of the visible radiation is distributed in the seeable set. In this manner, the oculus ‘s three coloring material receptors are about every bit stirred [ ] .

In statistical instance, a clip series can be characterized as holding weak white noise on status that { } is a sequence of serially uncorrelated random quivers with zero mean and finite discrepancy. Particularly, strong white noise has the quality to be independent and identically distributed, which means no autocorrelation. In peculiar, the series is called the Gaussian white noise [ 1 ] , if A is usually distributed and it has zero mean and standard divergence.

Actually, an infinite bandwidth white noise signal is merely a theoretical building which can non be reached. In pattern, the bandwidth of white noise is restricted by the transmittal medium, the mechanism of noise coevals, and finite observation capablenesss. If a random signal is observed with a level spectrum in a medium ‘s widest possible bandwidth, we will mention it as “ white noise ” .

3.5.1 Mathematical definition

White random vector

A random vectorA WA is a white random vector merely if itsA mean vectorA andA autocorrelationA matrix are matching to the follows:

mu_w = mathbb { E } { mathbf { tungsten } } = 0 ( 3.15 )

R_ { ww } = mathbb { E } { mathbf { tungsten } mathbf { tungsten } ^T } = sigma^2 mathbf { I } . ( 3.16 )

That is to state, the white random vector is a nothing mean vector, and its autocorrelation matrix is equal to the multiple of theA individuality matrix. When the autocorrelation matrix is the multiple of the individuality, so we can see it as spherical correlativity.

White random procedure

A clip uninterrupted random processA A whereA A is a white noise signal merely if its mean map and autocorrelation map are satisfied with the undermentioned equation:

mu_w ( T ) = mathbb { E } { tungsten ( T ) } = 0 ( 3.17 )

R_ { ww } ( t_1, t_2 ) = mathbb { E } { tungsten ( t_1 ) tungsten ( t_2 ) } = ( N_ { 0 } /2 ) delta ( t_1 – t_2 ) . ( 3.18 )

Since its autocorrelation map is theA Dirac delta map, it is certain that the procedure is zero mean for all clip and has infinite power at zero clip displacement.

The autocorrelation map 3.18 implies the followingA power spectral denseness map 3.19.

S_ { ww } ( omega ) = N_ { 0 } /2, ! ( 3.19 )

TheA Fourier transformA of theA delta functionA is equal to one. Because this power spectral denseness map 3.19 is the same with each other at all frequences, we can specify it whiteA as an analogy to theA frequence spectrumA ofA white visible radiation. The generalisation toA random elementsA on the infinite dimensional infinites, e.g.A random Fieldss, is theA white noise step.

3.6 Normal Distribution

Harmonizing to theA chance theory, theA normalA distribution ( orA Gaussian distribution ) A is aA uninterrupted chance distribution which has a bell-shapedA chance denseness map, known as theA Gaussian functionA or informally as the bell curve [ 1 ] .

degree Fahrenheit ( ten ; mu, sigma^2 ) = frac { 1 } { sigmasqrt { 2pi } } e^ { -frac { 1 } { 2 } left ( frac { x-mu } { sigma }
ight ) ^2 } ( 3.20 )

Where the parameterA I?A is theA meanA orA expectationA ( location of the extremum ) andA I?aˆ‰2A is theA discrepancy. And I?A is known as theA standard divergence. The distribution, as figure 3.3 shown, withA I?A = 0A andA I?aˆ‰2A = 1A is defined as theA criterion normal distributionA or theA unit normal distribution. The normal distribution is frequently regarded as a first estimate to show existent valuedA random vibrationA that bunch around a individual ‘s average value.

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Figure 3.3

The normal distribution is considered the most outstanding chance distribution in statistics. There are several grounds for this [ 1 ] . First, the normal distribution comes from theA cardinal bound theorem, which declares that the mean of a big series ofA random variablesA drawn from the same distribution is distributed about usually, irrespective of the signifier of the original distribution in mild conditions. Then we can connote it exceptionally broad application in, e.g. , trying. Second, the standard distribution is greatly manipulable analytically. That is to state, a big series of consequences including this distribution can be derived in expressed signifier.

In this instance, the standard distribution is normally appeared in pattern, and is used throughout statistics, A natural scientific disciplines, andA even societal scientific disciplines [ 2 ] .A As for this thesis, theA experimental errorA in the way of arrival appraisal is normally assumed to follow a normal distribution, and the extension of uncertaintyA is computed by this premise. It is certain that a criterion distributed information has a symmetric distribution about its average axis.

3.6.1 Mathematical Definition

Standard normal distribution is the simplest instance of a standard distribution, and it can be described mathematically by theA chance denseness map ( PDF ) as 3.21:

phi ( x ) = frac { 1 } { sqrt { 2pi } } , e^ { – frac { scriptscriptstyle 1 } { scriptscriptstyle 2 } x^2 } . ( 3.21 )

Where the factorA scriptstyle 1/sqrt { 2pi } A in this equation expresses that the entire country under the curveA I• ( x ) must be equal to one, and the coefficient in the advocate ensures the “ width ” of the curve besides equal to one. It is conventional in statistics to stand for this map with the Grecian letterA I• . And the chance denseness functionsA for all other distributions are ever denoted with lettersA degree Fahrenheits [ 5 ] .A

Every normal distribution is the consequence of involution result of aA quadratic map as 3.22:

degree Fahrenheit ( x ) = e^ { a x^2 + B x + degree Celsius } . , ( 3.22 )

This draws the authoritative bell curve form. If aA & lt ; 0, the quadratic map will beA concaveA forA xA when it is close to 0 andA degree Fahrenheit ( x ) & gt ; 0A is true everyplace. We can set parametric quantity aA to command the breadth of the bell form, and adjustA bA to alter the cardinal peak axis of the bell form along theA x axis. Besides we can chooseA cA so that the equationscriptstyleint_ { -infty } ^infty degree Fahrenheit ( x ) , dx = 1A , which is merely possible whenA aA & lt ; A 0, is true.

However it is more frequently to depict a standard distribution by its meanA andA varianceA I?2A = , instead than byA a, A B, andA c. If we replace these new parametric quantities to 3.22 we will rewrite the chance denseness map ( PDF ) in a more normal signifier as 3.23:

degree Fahrenheit ( x ) = frac { 1 } { sqrt { 2pisigma^2 } } , e^ { frac { – ( x-mu ) ^2 } { 2sigma^2 } } = frac { 1 } { sigma } , phi ! left ( frac { x-mu } { sigma }
ight ) . ( 3.23 )

As for a standard normal distribution, A I?A = 0A andA I?2A = 1. As the last portion of the equation in 3.23 shows, any other standard distribution can be seen as a version of the criterion normal distribution which has been stretched horizontally by the factorA parametric quantity I?A and so translated rightward by the distanceA parametric quantity I? . Therefore, the parameterA I?A represents the place of the bell curve ‘s cardinal extremum axis, andA parametric quantity I?A specifies the breadth of the bell curve.

Meanwhile, the parameterA I?A is called the mean, theA medianA or theA modeA of the standard distribution. The parameterA I?2A is theA discrepancy, which is a important base in this thesis about DOA appraisal comparing, and in any random variable instance, it demonstrates the concentration grade of the distribution around its mean. Besides, the square root ofA I?2A is called theA criterion deviationA or root average square and describes the breadth of the denseness map.

The standard distribution is normally denoted byA N ( I? , aˆ‰I?2 ) [ 6 ] . Therefore, if a random map ofA XA is distributed usually with meanA I?A and varianceA I?2, we can compose it as 3.24:

X sim mathcal { N } ( mu, , sigma^2 ) . , ( 3.24 )

3.7 Cramer-Rao Bound

InA appraisal theoryA andA statistics feild, theA Cramer-Rao edge ( CRB ) A orA Cramer-Rao lower edge ( CRLB ) , named afterA Harald CramerA andA Calyampudi Radhakrishna RaoA who were the first clip to deduce it [ 1 ] [ 2 ] [ 3 ] . The CRB expresses the lower edge on theA varianceA ofA the estimatorsA about a deterministic parametric quantity. The lower edge is besides refered as theA Cramer-Rao inequalityA or theA information inequality.

In the simplest instance, the CRB provinces that the discrepancy of anyA unbiasedA calculator is at least every bit high as the opposite of theA Fisher information. An indifferent calculator which achieves this lower edge is said to beA efficient [ ] . A solution like this derives the lowest possibleA mean squared errorA value in indifferent appraisal methods, and therefore it is theA minimal discrepancy unbiasedA ( MVU ) calculator. But in pattern, there is no indifferent technique which achieves the lower edge sometimes. And this job may happen even when the MVU calculator exists.

The Cramer-Rao edge can besides be used to adhere the discrepancy ofA biasedA estimatorsA of a given prejudice. Actually, a colored method can connote both a discrepancy and aA mean squared errorA which areA underA the indifferent Cramer-Rao lower edge.

The mathematical definition of Cramer-Rao edge is declared in this subdivision for several progressively general instances. First of wholly, the instance in which the parametric quantity is aA scalarA and its calculator isA indifferent. All instances of the edge require certain regularity conditions that hold for most good behaved distributions. These conditions are listedA subsequently in this subdivision in detial.

3.7.1 Scalar indifferent Cramer-Rao edge

AssumingA is an unknown deterministic parametric quantity that is to be estimated by measurings informations tens, which are distributed harmonizing to a particularA chance denseness map. And theA varianceA of anyA unbiasedA estimatorA A ofA A is hence bounded by theA reciprocalA value of theA Fisher information as 3.25:

mathrm { volt-ampere } ( hat { heta } ) geq frac { 1 } { I ( heta ) } ( 3.25 )

Where the Fisher informationA A is defined by 3.26:

I ( heta ) = mathrm { E } left [ left ( frac { partial ell ( x ; heta ) } { partial heta }
ight ) ^2
ight ] = -mathrm { E } left [ frac { partial^2 ell ( x ; heta ) } { partial heta^2 }
ight ] ( 3.26 )


ell ( x ; heta ) =log degree Fahrenheit ( ten ; heta ) ( 3.27 )

3.27 is theA natural logarithmA of theA likeliness functionA ( LF ) andA EA represents theA expected value. TheA efficiencyA of an indifferent estimatorA A measures the close grade of this calculator ‘s discrepancy to this lower edge. And the calculator efficiency is defined as:

vitamin E ( hat { heta } ) = frac { I ( heta ) ^ { -1 } } { {
m volt-ampere } ( hat { heta } ) } ( 3.28 )

The minimal possible discrepancy for an indifferent calculator is divided by its existent discrepancy. It implies that The Cramer-Rao lower edge satisfies.

3.7.2 General scalar Cramer-Rao edge

A more general instance of the lower edge can be achieved by sing an indifferent estimatorA A of a functionA A with the parametric quantity. Where the unbiasedness is known as the equation as. Therefore, we can acquire the edge as 3.29:

mathrm { volt-ampere } ( T ) geq frac { [ psi ‘ ( heta ) ] ^2 } { I ( heta ) } ( 3.29 )

WhereA A is the derivative ofA A by parametric quantity, andA A is the Fisher information mentioned above.

3.7.3 Multivariate Cramer-Rao edge

In multivariate instance, the Cramer-Rao edge is decided by multiple parametric quantities. Assume a parametric quantity columnA vector with the chance denseness map ( PDF ) A that satisfies the twoA regularity conditions described as follows.

TheA Fisher information matrixA is aA dA-dA matrix with each elementA A which is defined as 3.30:

I_ { m, K } = mathrm { E } left [ frac { vitamin D } { d heta_m } log fleft ( x ; oldsymbol { heta }
ight ) frac { vitamin D } { d heta_k } log fleft ( x ; oldsymbol { heta }
ight )
ight ] . ( 3.30 )

Assume thatA A is an calculator of any vector map of parametric quantities, where, and replace its outlook vectorA A byA .Then, the Cramer-Rao edge demonstrates that theA covariance matrixA ofA A satisfies the status as 3.31:

mathrm { cov } _ { oldsymbol { heta } } left ( oldsymbol { T } ( X )
ight ) geq frac { partial oldsymbol { psi } left ( oldsymbol { heta }
ight ) } { partial oldsymbol { heta } } [ Ileft ( oldsymbol { heta }
ight ) ] ^ { -1 } left ( frac { partial oldsymbol { psi } left ( oldsymbol { heta }
ight ) } { partial oldsymbol { heta } }
ight ) ^T ( 3.31 )

Where the matrix inequalityA A is known to intend that the matrixA A-BA isA positive semi definite, and theA is theA Jacobin matrixA whoseA I, jth component is expressed byA . IfA A is anA unbiasedA calculator of parametric quantity, so the Cramer-Rao edge reduces to 3.32:

mathrm { cov } _ { oldsymbol { heta } } left ( oldsymbol { T } ( X )
ight ) geq Ileft ( oldsymbol { heta }
ight ) ^ { -1 } . ( 3.32 )

If it is complicated to calculate the reciprocal of theA Fisher information matrix, we can merely utilize the reciprocal of the corresponding diagonal component to happen a lower edge as 3.33 [ 11 ] .

mathrm { volt-ampere } _ { oldsymbol { heta } } left ( T_m ( X )
ight ) = left [ mathrm { cov } _ { oldsymbol { heta } } left ( oldsymbol { T } ( X )
ight )
ight ] _ { millimeter } geq left [ Ileft ( oldsymbol { heta }
ight ) ^ { -1 }
ight ] _ { millimeter } geq left ( left [ Ileft ( oldsymbol { heta }
ight )
ight ] _ { millimeter }
ight ) ^ { -1 } . ( 3.33 )

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