Fisher information matrix for models of systems based

Cybernetics atui Systems Analysis, Voi. 35, No. 4, 1999

FISHER

INFORMATION

M A T R I X F O R MODELS OF S Y S T E M S

BASED ON FACE-SPLITTING

MATRIX

PRODUCTS

UDC 621.391

V. I. Slyusar

Expressions for blocks of the information Fisher matrix are presented based on factorization of the Neudecker derivative of a transposed face-splitting matrix product. Keywords: matrix product, Neudecker derivative, Fisher matrix, face-splitting matrix product.

As is generally known, the measurement accuracy of the parameters of signals can be analyzed using different approaches [1]. Among a variety of factors used in this case, the low Cramer-Rao bound of the variance of an unbiased estimate is most popular. Calculation of this bound involves inversion of the Fisher information matrix; therefore, the formation of this matrix can be considered as the key problem in the entire process of determination of the low bound. The inversion of an information matrix is a trivial operation and can be automatically executed even by the spreadsheet processor MS Excel, let alone specialized software packages such as Mathcad 6.0 (7.0). This paper presents a detailed review of the technique of Fisher-matrix formation for analytical models of radar-tracking systems (RTS). These models have been formalized on the basis of transposed face-splitting matrix product [2]. Recall that the transposed face-splitting product (TFP) of a g x p matrix A =[aij] and an s x p matrix B, represented in the form of a block-matrix of columns [Bj](B=[Bj], j = l ..... p), is a g s x p matrix AaB defined by the equality

AaB =[aijBj]

(1)

(i is the symbol of face-splitting product [2].) F~ example' tbr A = [ al 3 1] al a2 a 2 al a2 13

andB=rbllb12b13]|,.b21 b22 b23.~|

we

have

AaB=

As was mentioned in [3], the Fisher information matrix can be obtained based on the the Neudecker matrix derivative [4]. The general form of the Fisher matrix according [2, 3] is Institute of Land Forces, Kiev, Ukraine. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 141-149, July-August, 1999. Original article submitted October 19, 1998.

636

1060-0396/99/3504-0636522.00 9

Kluwer Academic/Plenum Publishers

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(2)

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OP is the Neudecker derivative of the matrix P with respect to the vector Y, which is composed of unknown OY parameters of the signals of M sources, 1R8 is a unit (RxR)-matrix, and | is the symbol of Kronecker product. The Here,

Fisher matrix in general form can be used to form the Cramer-Rao low bound, achievable by a multicoordinate singleor multiposition radar-tracking system in a multsignal situation. As was mentioned in [2], it is sufficient to use in expression (2) instead of the matrix P its value represented by a face-splitting matrix product, and to appropriately adjust the dimension of the matrix 1RR. A further simplification of the procedure of the Fisher matrix formation in the class of problems being considerect relates to the property of factorization of the Neudecker derivative of TFP. To write it in the general form, we first turn to the Neudecker differentiation of a matrix product

P=SuF,

(3)

where S is a T x M matrix, and F is an R xM matrix. By analogy with [3], the matrix S can be treated as a T x M matrix of responses T of synthesized frequency filters on the frequencies of 3,/signals, and F as an R x M matrix of the directional characteristics R of the reception channels of a digital antenna array (DAA) in the directions of M sources. In this case, if noise is absent, the set of response voltages of the DAA reception channels can be writen as U = PA, where A =[a 1 ,ti 2 "--aM ]T is the vector of the complex amplitudes of the signals. When two signal sources are acting, we can write

s~ (%) s~ (co2)]

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LFR(x,)FR(h)J t, (x2)]

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P=SuF=

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(4)

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637

Let us consider Y = [v I x2 601 dO2 ]T as a vector of unknowns. Then the Neudecker derivative of P will take the form

oP oY

u

i

I ,.,I. ,,

1

f

(~', J f

s(,,,)

as(..,)

': as(%)

'.~m/

2

% J

m

Analyzing expression (5) one can easily arrive at the idea of factorization of 0 P. Indeed, on closer look, it turns out OY

that (5) can be obtained by transposing the block face-splitting product (BFP) [2] of the following two matrices:

S l(~ l)

0

~ a~

0

Ox

1 9

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]

9

1

oe(~)

I

, OS1(("I)

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o

,r(~)

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I ST(COl) OS = Y

0 9

0

, 0~T(CL)I) ~ l 0 i 0~ol 1 a s I(02) Sl(a~2), 0 0w 2 9

I

I

Denoting

638

Sr(~

~

[

0

.OF=

Ox

I

o

,

~(~,)

Y

(6)

---------I

Ox

as(%) 2

I

1

9

9

] 0

I

%%)

1

9

Ox

I

F(~)

S1

OSy = S -

I OS1

OFf

I 0W1

0X1 I

' OF =

I 1 0S2 I Oto2

I

(7)

OF2 Ox2

I i

where

osl (%) 0col S1 =

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,

S2 =

9

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OSr(O,~)

90mI

Ow2

"0'

aS 1(~2) Ow2

aS 2 ~ =

0

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OSr(o~2)

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~

F 2

~

~

[FR(Xl)

0

OFl (~ )

~

FRx2)

OF1 (x2)]

-

0F, ax~

OFR(x~)

OFR(~2)

'~

N

j

according to the definition of t r a n s p o s e d BFP [2], we obtain

I OS1

s~

OP = OS ~ OF = OY Y Y

OF!.,

I oco~ I___

~

os 2

ox a ___

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I F~ I

oF2 I F-~ i

(8) OF1 I 0S1 ,,

S1 9 OXl

I O~ 1 9 F1 I .

OF.2 !I 0S2 Sz 9 ox 2 I OoJ2"F2

It is n o t e w o r t h y that not o n l y the result of N e u d e c k e r differentiation can be subject to factorization but also the product

(__~OP)lT Ooy P" Thus, for relation (5) it is easy to obtain -~-

~ - ~ - =[Pl I P2 ],

(9)

639

where

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2}

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os%)l ,,

1

~: s,(o~ ,o t--I

.j Z ;

{

2}

%%) .

r--I

1

/ml

rml

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