Zoom out Search Issue

ManualsBrandsContents Manualsaudio & home theatreZoom in

161

162

163

164

165

166

167

168

169

170

[

lecture

NOTES

]

continued

IEEE SIGNAL PROCESSING MAGAZINE [166] MARCH 2015

By applying the Schur determinant

formula [8], [11]

JJ JJJ

ab ba

ba ab

along with | | | | ,JJ

to (5)–(7), we

can now state the CRB analogs of the

chain rule (3),

,|CRBCRBCRB

ab a b b=

(8)

and of Bayes’ rule (4),

|.CRB

CRB

(9)

The results are, of course, symmetric, i.e.,

one can interchange a and .b

From (8) we see that the joint error

bound for

a and b equals the error bound

for ,

a when b is known, multiplied by the

error bound for .

b More interestingly, (9)

tells us that the error bound for

a is equal

to the bound for

a when b is known,

multiplied by a factor, viz. /CRB

|,1CRB $ba

that quantifies the influ-

ence of

b on one’s ability to estimate .a

REMARK 1

The rules can be applied to cases with any

number of additional parameters, besides

a and .b Consider, for instance, the case

of ,

and ,c where c is an unknown

nuisance parameter. Then applying the

chain rule twice yields

,, |,

CRBCRB

CRB

CRBCRB

abc c ab

ab b

cab

ba a

(10)

where the factors without c signify that

the nuisance parameter is unknown.

Combining the two expressions in (10)

yields the analog of Bayes’ rule (9) for any

number of additional parameters.

The joint error bound for a set of param-

eters

,,,

123

faaa can be similarly decom-

posed by a recursive application of the chain

rule to analyze their interdependency and

its impact on estimation.

REMARK 2

The CRB analog of Bayes’ rule (9) general-

izes the result in [7], which concerns only

scalar parameters

and

amid a vector

of nuisance parameters .c Our proof of (9)

is also more direct than in [7].

REMARK 3

These results are also applicable to the pos-

terior, or Bayesian, CRB (PCRB), in which

i is modeled as a random variable with a

prior distribution. The PCRB is valid for the

entire class of estimators

whether

biased or not [2]. The posterior Cramér–

Rao inequality is then ,

where

,plE nJy

2_ i-

is the Bayesian

Fisher information matrix, ,p y

is the

joint pdf and the expectation is with respect

to this pdf. Letting

[],

iab=

<<<

the

matrix can be partitioned correspondingly,

and thereby the results (8)–(10) can be

applied to the PCRB as well.

EXAMPLES

Next, we illustrate via two examples how a

decomposition like (9) can be used for anal-

ysis. The examples show that, by quantify-

ing the impact of nuisance parameters, it is

possible to study the tradeoff between the

gain of obtaining them through indepen-

dent side information versus estimating

them jointly with the parameters of interest.

LINEAR MIXED MODEL

Consider a linear model

,y AxBzw R

!=++

where w is Gaussian noise with covariance

matrix ,vI and x R

! and z R

! are

unknown parameters. The matrices are

known and

([ ]) ,kk nrank AB

1=+

which implies that the parameters x and z

are embedded into two distinct range

spaces,

()AR and ( ),BR respectively.

Here ()AR denotes the linear subspace

spanned by the columns of .A Under these

conditions the joint Fisher information

matrix equals [9]

From this expression, we see that the

bound for v is independent of that

for x and .z That is, ( , )vCRB,xz =

() ().vCRB,CRBxz This is a CRB analog

of the independence for random variables.

Furthermore, we obtain

()CRB|zx=

|||( )| | |vvJBB BB

and

() |( ) | | (vCRB zJJJJ BB

zzx

=- = -

() )| | |,vBAAA AB B B

k11 1

<< < <

-- -

where

is the projector onto the

orthogonal complement of ().AR

The increase in the error bound for x

due to the lack of information about z can

now be quantified using (9)

()

(),CRBCRB|x

(11)

where the factor ||BB

measures the

alignment of ()AR and ( ) .BR When the

range spaces are orthogonal we have that

||||,BBBB

P =

and by (11) the

bound for x is unaffected by one’s igno-

rance about .z In scenarios where it is pos-

sible to obtain z through additional

side-information or calibration instead of

estimation, the cost can be weighed against

the reduction of the error bound for

x by

the given factor | |/| |.BBBB

This example has illustrated the inter-

dependencies between the unknown

parameters

,x,z and .v Next we consider

an example where the unknown parame-

ters become asympotically independent as

the number of samples

n grows large.

SINE-WAVE FITTING

Sine-wave fitting is a problem that arises

in system testing, e.g., of waveform

recorders, and IEEE Standard 1057 for-

malizes procedures to do so (see [12] and

references therein).

Consider

n uniform samples of a sinu-

soid in noise

() ( ) (),sinyk k C wka~z=+++

where ( )wk is a Gaussian white noise pro-

cess with variance v and , , .kn01f=-

The amplitude

a and phase z of the sinu-

soidal signal, along with the offset ,C are of

interest. In certain cases, the frequency

of the test signal may be obtained separately

from the estimation of

az and .C For

simplicity, we first consider an alternative

parameterization of the sinusoid:

( ) () (),sin cos sinkAkBka~z ~ ~+= +

where ()sinA

az= and ( ).cosB az=

The parameters are [].ABC v

i ~=

THE WORLD’S NEWSSTAND

THE WORLD’S NEWSSTAND