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lecture

NOTES

]

continued

IEEE SIGNAL PROCESSING MAGAZINE [168] MARCH 2015

As shown in [12], the Fisher informa-

tion matrix can be decomposed into

,JJJ=+

where J

, shown in the box

at the bottom of the page, contains the

dominant terms and

contains the

remainder, so that JJ

for large .n

Using this approximation we now analyze

the bounds for

,A ,B and C by applica-

tion of (9).

First, let

[]ABCv

i =

be the

parameter vector without .

~ Then

() | |

()

ABn

212

CRB JJJ

223

32 2

~ =-

~~i

Second, let []BCv

i =

be the parame-

ter vector without

~ and .A Then

(|) | |

()

A J

ABn

A n

AB B

CRB JJJ

223

32 22

~ =-

~~i

Thus ()/ ( |) /AB13CRB CRB

~~=+

()AB

+ [,].14! Note that the domi-

nant terms of J

and J

are diagonal,

making their inverses particularly easy to

compute. Applying (9), we obtain

()

(| )

(|)

(|),

CRB

where the bounds for B and C are

derived in a similar manner as for .A This

shows that the bound for the offset C

becomes independent of the knowledge of

the frequency

~ as n increases, while the

bounds for A and B are inflated by fac-

tors ranging between one and four due to

one’s ignorance about

When considering the original parame-

terization

[]C v

j az ~=

there exists an

invertible relation, ABg

ji== +

/.arctan AB C v~

Therefore we have

that ggJJ

22ii=

^^hh

[2], where

denotes the first-order differential or

gradient with respect to

i and

sin cos

cos sin

2 i

Exploiting the approximation

once again, one obtains [12]

()

(|)

(|).4

CRB

This shows that, in large samples, the error

bound for the amplitude

a also becomes

independent of knowledge about the fre-

quency

~ whereas not knowing ~ inflates

the bound for the phase

z by a factor of four.

For large data records, the cost of pre-

calibrating the frequency can be weighed

against a reduction of the error bound for

the phase, while the error bounds for the

amplitude and offset will not be improved.

WHAT WE HAVE LEARNED

An analog of Bayes’ rule for the CRB has

been derived. This analogous rule enables

a formalized decomposition and quantifi-

cation of the mutual dependencies

between multiple unknown parameters.

The use of the rule was illustrated in two

estimation problems.

AUTHORS

Dave Zachariah (dave.zachariah@it.uu.se)

received the M.S. degree in electrical engi-

neering from Royal Institute of Technolo-

gy (KTH), Stockholm, Sweden, in 2007.

He received the Tech. Lic. and Ph.D. de-

grees in signal processing from KTH in

2011 and 2013, respectively. He is current-

ly a postdoctoral researcher at Uppsala

University in Sweden.

Petre Stoica (ps@it.uu.se) is a research-

er and educator in the field of signal pro-

cessing and its applications to radar/sonar,

communications and biomedicine. He is a

professor of signal and system modeling at

Uppsala University in Sweden and a mem-

ber of the Royal Swedish Academy of

Engineering Sciences, the Romanian

Academy (honorary), the European

Academy of Sciences, and the Royal

Society of Sciences in Uppsala.

REFERENCES

[1] H. Cramér, “A contribution to the theory of statis-

tical estimation,” Scand. Actuarial J., vol. 1946, no.

1, pp. 85–94,1946.

[2] H. Van Trees and K. Bell, Detection Estimation

and Modulation Theory, Part I. Detection Estima-

tion and Modulation Theory, 2nd ed. Hoboken, NJ:

Wiley, 2013.

[3] P. Stoica and J. Li, “Study of the Cramér-Rao

bound as the numbers of observations and unknown

parameters increase,” IEEE Signal Process. Lett.,

vol. 3, no. 11, pp. 299–300, 1996.

[4] P. Stoica and P. Babu, “The Gaussian data as-

sumption leads to the largest Cramér-Rao bound,”

IEEE Signal Process. Mag., vol. 28, no. 3, pp. 132–

133, 2011.

[5] S. Park, E. Serpedin, and K. Qaraqe, “Gaussian as-

sumption: The least favorable but the most useful,” IEEE

Signal Process. Mag., vol. 30, no. 3, pp. 183–186, 2013.

[6] M. Stein, A. Mezghani, and J. Nossek, “A lower

bound for the Fisher information measure,” IEEE

Signal Process. Lett., vol. 21, pp. 796–799, July 2014.

[7] A. D’Amico, “A “reciprocity” property of the un-

biased Cramér-Rao bound for vector parameter esti-

mation,” IEEE Signal Process. Lett., vol. 21, no. 5,

pp. 615–619, 2014.

[8] T. Söderström and P. Stoica, System Identifi-

cation.Englewood Cliffs, NJ:Prentice Hall,1988.

[9] S. M. Kay, Fundamentals of Statistical Signal

Processing: Estimation Theory. Englewood Cliffs,

NJ: Prentice Hall, 1993.

[10] T. Anderson, An Introduction to Multivariate

Statistical Analysis, 2nd ed. (Wiley Series in Prob-

ability and Statistics). Hoboken, NJ: Wiley, 2003.

[11] R. Horn and C. Johnson, Matrix Analysis.Cam-

bridge, U.K.: Cambridge Univ. Press, 1990.

[12] T. Andersson and P. Händel, “IEEE standard

1057, Cramér–Rao bound and the parsimony prin-

ciple,” IEEE Trans. Instrum. Measure., vol. 55,

no. 1, pp. 44 –53, 2006.

[SP]

()

ABn

223

~~~

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