deterministic asymptotic cramér–rao bound for the multidimensional harmonic model

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Signal Processing

Signal Processing 88 (2008) 2869– 2877

0165-16

doi:10.1

� Tel.

E-m

journal homepage: www.elsevier.com/locate/sigpro

Deterministic asymptotic Cramer–Rao bound for themultidimensional harmonic model

Remy Boyer �

Laboratoire des Signaux et Systemes, CNRS - Universite Paris-Sud - Supelec, 3, rue Joliot-Curie, 91190 Gif-sur-Yvette, France

a r t i c l e i n f o

Article history:

Received 3 October 2007

Received in revised form

27 May 2008

Accepted 5 June 2008Available online 21 June 2008

Keywords:

Parameter estimation

Multidimensional signal processing

Harmonic model

Cramer–Rao bound

84/$ - see front matter & 2008 Elsevier B.V. A

016/j.sigpro.2008.06.011

: +33169 8517 12; fax: +33169 8517 69.

ail address: [email protected]

a b s t r a c t

The harmonic model sampled on a P-dimensional grid contaminated by an additive

white Gaussian noise has attracted considerable attention with a variety of applications.

This model has a natural interpretation in a P-order tensorial framework and an

important question is to evaluate the theoretical lowest variance on the model

parameter (angular-frequency, real amplitude and initial phase) estimation. A standard

Mathematical tool to tackle this question is the Cramer–Rao bound (CRB) which is a

lower bound on the variance of an unbiased estimator, based on Fisher information.

So, the aim of this work is to derive and analyze closed-form expressions of the

deterministic asymptotic CRB associated with the M-order harmonic model of

dimension P with P41. In particular, we analyze this bound with respect to the

variation of parameter P.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The one-dimensional harmonic model is very useful inmany fields such as in signal processing, audio compres-sion, digital communications, biomedical signal proces-sing, electromagnetic analysis and others. A generalizationof this model to P41 dimensions can be encountered inseveral domains such as in MIMO channel modeling fromchannel sounder measurements [1,2], wireless commu-nications [3], passive localization and radar processing,etc. In particular, we can find in [4] a tensorial-basedESPRIT algorithm adapted to the multidimensional har-monic model. In addition, we can find in [5,6] an analysisof the identification problem associated with this model.

For many practical estimation problems, optimalestimators such as the maximum likelihood (ML) estima-tor, the maximum a posteriori (MAP) estimator or theminimum mean squared error (MMSE) estimator areinfeasible. Therefore, one often needs to resort to

ll rights reserved.

suboptimal techniques such as expectation maximization,gradient-based algorithms, Markov chain Monte Carlomethods, particle filters, or combinations of those meth-ods. These techniques are usually evaluated by computingthe mean square error (MSE) through extensive MonteCarlo simulations and compare it to theoretical perfor-mance bounds.

In signal processing, a popular lower bound is thedeterministic Cramer–Rao bound (CRB) [7]. In spite ofthe fact that this bound is optimistic for low and moderatesignal-to-noise ratio (SNR) [8], the predominance of thisbound can be probably explained by its relative simplealgebraic derivation in comparison to other lower bounds.

More precisely, in this work, we propose closed-form(nonmatrix) expressions of the deterministic CRB forthe M-order harmonic model (sum of M waveforms) ofdimension P, viewed as an N1 � � � � � NP tensor, contami-nated by an additive white Gaussian noise. This work is anextension of the seminal work of Stoica and Nehorai [7]for the one-dimensional (P ¼ 1) harmonic model. Ob-viously, many works have been done on the determinationof the deterministic CRB for small P, i.e., for P ¼ 2 (two-dimensional harmonic model) [9,10] or for P ¼ 3 and 4 in

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the context of sensor array [11]. Other contributionsprovide matrix-based expressions of the CRB for any P

[12], but at our best knowledge, we cannot find closed-form expressions of the deterministic CRB for anydimension P. Closed-form expressions [13] are importantfor at least two reasons: (i) they provide useful insightinto the behavior of the bound and (ii) for large analysisduration ðNpb1; 8p 2 ½1 : P�Þ and/or dimension P, comput-ing the CRB in a brute force manner becomes animpracticable task.1

This article is organized as follows. Section 2 presentsthe multidimensional harmonic model and the associatedCanDecomp/Parafac decomposition. Section 3 introducesand analyzes a closed-form expressions of the determi-nistic CRB for asymptotic analysis duration. Section 4presents the analysis of the ACRB for a constant amountof data. Next, Section 5 is dedicated to the conclusion. Thederivation of the asymptotic CRB is given in Appendix Aand we present in Appendix B, the exact (nonasymptotic)CRB for a first-order harmonic model of dimension P.

2. CanDecomp/Parafac decomposition of themultidimensional harmonic model

The multidimensional harmonic model assumes thatthe observation can be modeled as the superposition of M

undamped exponentials sampled on a P-dimensional grid.More specifically, we define a noisy M-order harmonicmodel of dimension P according to

½Y�n1 ...nP¼ ½X�n1 ...nP

þ s½E�n1 ...nP(1)

where ½Y�n1 ...nPdenotes the ðn1; . . . ;nPÞ-th entry of the

ðN1 � � � � � NPÞ tensor (multiway array) Y associated withthe noisy M-order harmonic model of dimension P. LetNp41 be the analysis duration along the p-th dimensionand define np 2 ½0 : Np � 1�. Tensor X in model (1) isthe ðN1 � � � � � NPÞ tensor associated with the noise-freeM-order harmonic model of dimension P defined by

½X�n1 ...nP¼XMm¼1

am

YP

p¼1

eioðpÞm np (2)

in which the m-th complex amplitude is denoted by am ¼

ameifm where am40 is the m-th real amplitude, fm is them-th initial phase and oðpÞm is the m-th angular-frequencyalong the p-th dimension. Let dðoðpÞm Þ ¼ ½1 eioðpÞm . . .

eioðpÞm ðNp�1Þ�T be the Vandermonde vector containing theangular-frequency parameters. As

½dðoð1Þm Þ � dðoð2Þm Þ � � � � � dðoðPÞm Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Dm

�n1 ...nP¼YP

p¼1

eioðpÞm np (3)

in definition (2) and in which � denotes the outerproduct, it is straightforward to see that the tensor, X,associated with the noise-free M-order harmonic model ofdimension P can be expressed as the linear combining ofM rank-1 tensors: D1; . . . ;DM , each of size N1 � � � � � NP ,

1 The computation of the CRB for the considered model is of

OðN1N2 . . .NP Þ.

according to

X ¼XMm¼1

amDm 2 CN1��NP . (4)

Consequently, the noise-free M-order harmonic model ofdimension P follows a CanDecomp/Parafac model [5,6,14]and its vectorized expression is

x ¼ vecðXÞ

¼ ½½X�000 . . . ½X�N1�1 N2�1 0 ½X�001 . . .

. . . ½X�N1�1 N2�1 N3�2 ½X�00 N3�1 . . . �T

¼XMm¼1

am vecðDmÞ (5)

where

vecðDmÞ ¼ dðoð1Þm Þ � dðoð2Þm Þ � � � � � dðoðPÞm Þ (6)

in which � denotes the Kronecker product. Tensor sE inmodel (1) is the noise tensor where s is a positive realscalar and each entry ½E�n1 ...nP

follows a Gaussian distribu-tion Nð0;1Þ. In addition, we assume the decorrelation of(i) the noise-free signal and the noise and (ii) the noise ineach dimension, i.e.,

Ef½X�n1 ...nP½E��n0

1...n0

Pg ¼ 0, (7)

Ef½E�n1 ...nP½E��n0

1...n0

Pg ¼

YP

p¼1

dnpn0p0

(8)

where Ef:g is the mathematical expectation and dij is theKronecker delta. So, based on expressions (5), (7) and (8),the final expression of the vectorized noisy model is

y ¼ vecðYÞ ¼ xþ se (9)

where e ¼ vecðEÞNð0; IN1 ...NPÞ.

3. CRB for the multidimensional harmonic model

The noisy observation y in expression (9) follows aGaussian distribution, i.e., yNðx;s2IN1 ...NP

Þ and is afunction of the real parameter vector y given by

y ¼ ½y0T s2�T

in which

y0 ¼ ½oT aT fT�T

where

a ¼ ½a1 . . . aM�T, (10)

f ¼ ½f1 . . .fM �T, (11)

o ¼ ½oð1ÞT . . .oðPÞT�T

with oðpÞ ¼ ½oðpÞ1 . . .oðpÞM �T. (12)

3.1. Deterministic CRB

3.1.1. Covariance inequality principle

A fundamental result [15,16] is the following. Let G ¼Efðy� yÞðy� yÞTg be the covariance matrix of an unbiasedestimate of y, denoted by y and define the CRB associated

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R. Boyer / Signal Processing 88 (2008) 2869–2877 2871

with the M-order harmonic model of dimension P,denoted by CRBðPÞ. The covariance inequality principlestates that under quite general/weak conditions, G�CRBðPÞðyÞ is a positive semidefinite matrix or equivalentlyin terms of the MSE, we have

MSEð½y�iÞ ¼ Efð½y�i � ½y�iÞ2gXCRBðPÞð½y�iÞ (13)

In other words, the variance of any unbiased estimate isalways bounded below by the CRB. In addition, if the MSEfor a given unbiased estimator is equal to the CRB, we saythat the considered estimator is statistically efficient.

More specifically, the CRB with respect to (wrt) thesignal parameters is given by

CRBðPÞð½y0�iÞ ¼

s2

2½F�1

y0y0 �ii for i 2 ½1 : ðP þ 2ÞM� (14)

where

Fy0y0 ¼

Joo Jo a Jof

JTo a Ja a Jaf

JTof JT

af Jff

26664

37775 (15)

is the Fisher information matrix (FIM) wrt the signalparameter y0. In addition, in (15), we have defined eachblock of the FIM by

Jpq ¼ Rqx

qp

� �H qx

qq

( )(16)

with Rf:g being the real part of a complex number and x isthe noise-free M-order harmonic model of dimension P

introduced in expression (5). Note that to obtain (14),we have exploited the property that the signal and thenuisance (noise) parameters are decoupled. So, the CRBfor the i-th signal parameter, denoted by ½y0�i, is given bythe ði; iÞ-th term of the FIM inverse weighed by s2=2.

3.1.2. Deterministic asymptotic CRB for the M-order

harmonic model of dimension P

In the sequel, we consider large analysis duration(Npb1; 8p) where analytic inversion of the FIM is feasibleand thus closed-form expressions of the deterministicCRBðPÞ can be obtained.

Theorem 1. The deterministic asymptotic CRBðPÞ (ACRBðPÞ)for the M-order harmonic model of dimension P defined in

(1) wrt the model parameter y0, i.e., ACRBðPÞðy0Þ, is given by

ACRBðPÞðoðpÞm Þ ¼6

N2pðQP

p¼1 NpÞSNRm

, (17)

ACRBðPÞðamÞ ¼a2

m

2ðQP

p¼1 NpÞSNRm

, (18)

ACRBðPÞðfmÞ ¼3P þ 1

2ðQP

p¼1 NpÞSNRm

, (19)

where SNRm ¼ a2m=s2 is the local SNR.

Proof. See Appendix A.

The deterministic ACRBðPÞ is fully characterized by thetensor size (and thus dimension P) and the local SNR. Inthe sequel, we list some important properties of the ACRB.

P1.
The deterministic ACRBðPÞ is invariant to the specificvalue of the initial phase.
P2.
The deterministic ACRBðPÞ is invariant to the specificvalue of the angular-frequency.
P3.
According to expression (17), the ACRB for the p-thangular-frequency depends on the cube of thecorresponding dimension, Np, and is only linear inthe other ones.
P4.
As expected at an intuitive level, the ACRB for the realamplitude and for the initial phase are invariant to thespecific dimension p.
3.2. Convergence with respect to (wrt) dimension P

In this part, the ACRBðPþ1Þ is associated with the tensorof size N1 � � � � � NP � NPþ1, i.e., the first P dimensions, i.e.,N1; . . . ;NP , remains identical as for the tensor associatedwith the ACRBðPÞ and the last one, i.e., the ðP þ 1Þ-th, isadded. In addition, it makes sense to consider the ACRB forthe same waveform m and dimension p. In this case, westudy the behavior of the ACRBðPÞ wrt dimension P.

Theorem 2. The ACRBðPÞ is a strictly monotonically decreas-

ing sequence wrt dimension P, i.e., ACRBðPÞð½y0�iÞo

ACRBðP�1Þð½y0�iÞo � � �oACRBð1Þð½y

0�iÞ.

Proof. Using (17)–(19), the quotient of two consecutiveACRB is given by

ACRBðPþ1ÞðoðpÞm Þ

ACRBðPÞðoðpÞm Þ¼

ACRBðPþ1ÞðamÞ

ACRBðPÞðamÞ¼

1

NPþ1, (20)

ACRBðPþ1ÞðfmÞ

ACRBðPÞðfmÞ¼

3P þ 4

3P þ 1

� �1

NPþ1. (21)

As NPþ1 in expressions (20) and (21) is large, meaning1=NPþ151, and as 1oð3P þ 4Þ=ð3P þ 1Þo2 in (21), wehave ACRBðPþ1Þð½y

0�mÞoACRBðPÞð½y

0�mÞ. Consequently, the

ACRBðPÞ is a strictly monotonically decreasing sequencewrt dimension P.

We can say:
Increasing the dimension of the harmonic modeldecreases the ACRBðPÞ. We explain this, at an intuitivelevel, according to the following argumentation. Whendimension P increases, i.e., P! P þ 1, we have toestimate more parameters so the degree of freedomdecreases but in the same time the ACRBðPþ1Þ bene-ficiates from NPþ1 additional samples. This two facttogether explains why the ACRBðPÞ is decreased by afactor 1=NPþ1. We have
ACRBðPÞðoðpÞm Þ ¼1QP

p¼2 NP

ACRBð1ÞðomÞ, (22)

ACRBðPÞðamÞ ¼1QP

p¼2 NP

ACRBð1ÞðamÞ, (23)


4QP

p¼2 NP

ACRBð1ÞðfmÞ, (24)

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0 5 10 15 20 25 30 35 4010−4

10−3

10−2

SNR [linear scale]

CR

B [l

og s

cale

]

Angular frequency

10−2

10−1

[log

sca

le]

Real amplitude

Numerical CRB(3)(ω)ACRB(3)(ω)

Exact CRB(3)(ω)

Numerical CRB(3)(a)ACRB(3)(a)

Exact CRB(3)(a)


where ACRBð1Þ is the bound derived by Stoica andNehorai [7] for P ¼ 1. &

3.2.1. The cubic tensor case

A cubic or balanced tensor is a tensor with identicalsizes, i.e., Np ¼ N; 8p. According to the previous theoremthe ACRBðPÞ are


NPþ2SNRm

¼1

NP�1ACRBð1ÞðomÞ, (25)


m

2NPSNRm

¼1

NP�1ACRBð1ÞðamÞ, (26)


2NPSNRm

¼3P þ 1

4NP�1ACRBð1ÞðfmÞ. (27)

For cubic tensors, we can say:
C
RB
The ‘‘magnitude of order’’ of the ACRBðPÞ for the realamplitude and the initial phase is OðN�P
Þ and OðN�Pþ2Þ

for the angular-frequency.

−3

0 5 10 15 20 25 30 35 40SNR [linear scale]

10

10−2

10−1

100

CR

B [l

og s

cale

]

Initial phase

Numerical CRB(3)(φ)ACRB(3)(φ)

Exact CRB(3)(φ)

The rate of convergence [17], i.e., the ‘‘speed’’ at whichthe ACRBðPÞ approaches its limit, for the angular-frequency and for the real amplitude is geometric.For the initial phase parameter, the convergence is alsogeometric for large dimension P.

3.3. Illustration of the ACRB

In this part, we choose to illustrate the derived boundsfor small and large cubic tensors of size N ¼ 3 and 1000,respectively. The dimension of the multidimensionalharmonic model is P ¼ 3 and its ‘‘vectorized’’ form isx ¼ 2eip=3 ðdð1Þ � dð0:5Þ � dð0:2ÞÞ. We illustrate in Fig. 1,the behavior of the derived bounds wrt the SNR in linearscale in range ½1;40�. More precisely, we have reported:

0 5 10 15 20 25 30 35 4010−3
SNR [linear scale]
Fig. 1. CRB Vs. SNR for a first-order harmonic model of dimension three

The numerical CRB which is the bound based on bruteforce computation of expression (14). Its complexity isOðN3

Þ.

for very short analysis duration (N ¼ 3).
The asymptotic CRB defined in expressions (17)–(19). The exact CRB defined in expressions (45)–(47) for a
first-order multidimensional harmonic model of di-mension three.

As expected for very short duration, the ACRB is notaccurate for the angular-frequency and in particular for thereal amplitude parameter. In addition, we can observe thatthe exact CRB and the numerical CRB are merged. In Fig. 2,we have drawn the ACRBðPÞ for P in range ½1 : 5� and for longanalysis duration (N ¼ 1000). Note that the complexity isvery high (Oð109

Þ) then the numerical CRB or matrix-basedderivation of this bound are impracticable. As we can seeincreasing the dimension P decreases the ACRB.

4. Asymptotic CRB for a constant amount of data

In the given contexts/applications, we have a constantamount of data, D for every dimensions, P. In this section,we investigate the derived bound which integrates thisconstraint. Let NðPÞp be the number of samples in the p-thdimension among P. So, we have

YP

p¼1

NðPÞp ¼ D: (28)

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10−15

10−10

10−5

CR

B [l

og s

cale

]

Angular frequency

P=1

P=2

P=3

P=4

P=5

10−12

10−10

10−8

10−6

10−4

10−2

100

Real amplitude

P=1

P=2

P=3

P=4

P=5

0 5 10 15 20 25 30 35 40SNR [linear scale]

CR

B [l

og s

cale

]

10−12

10−10

10−8

10−6

10−4

10−2

100

0 5 10 15 20 25 30 35 40SNR [linear scale]

0 5 10 15 20 25 30 35 40SNR [linear scale]

CR

B [l

og s

cale

]

Initial phase

P=1

P=2

P=3

P=4

P=5

Fig. 2. ACRB Vs. SNR for a first-order harmonic model of dimension 3 for

very short analysis duration (N ¼ 1000).


Constraint (28) implies that we have no assurance that theACRBðPÞ exists for fixed N and P. In other terms, it is notalways possible to find integers, NðPÞp , which satisfyconstraint (28) for all P and N. For instance, assume thatD ¼ 9, the ACRBð3Þ does not exist since the integer 9cannot be decomposed into the product of three integersstrictly greater than one. But, if the ACRB exists, then itsexpression is


NðPÞp

2DSNRm

, (29)


m

2DSNRm, (30)


2DSNRm. (31)

The main difference to the ACRB without constraint (28) isthat the dependance wrt dimension P is only through thesquare of term NðPÞp for the angular-frequency parameterand term 3P þ 1 for the initial phase. Remark theimportant point that the ACRB for the real amplitudeparameter becomes invariant to parameter P.

4.1. Real amplitude and initial phase

The ACRBðPÞ for the real amplitude is constant wrtdimension P and is equal to the ACRBð1Þ, i.e.,

ACRBðPÞðamÞ ¼ ACRBðP�1ÞðamÞ

¼ � � � ¼ ACRBð1ÞðamÞ. (32)

So, the accuracy for real amplitude is not affected byconsidering multidimensional harmonic model. Contraryto the real amplitude parameters, the rate for the initialphase is not constant (wrt P) and is given by

ACRBðPþ1ÞðfmÞ

ACRBðPÞðfmÞ¼

P þ 43

P þ 13

¼ lðPÞ.

As the rate is higher than one, the ACRBðPÞðfmÞ strictlymonotonically increases with P and we have

ACRBðPÞðfmÞ4ACRBðP�1ÞðfmÞ

4 � � �4ACRBð1ÞðfmÞ. (33)

As rate lðPÞ is a strictly monotonically decreasingsequence in the following interval:

lð1Þ ¼7

4plðPÞo1 ¼ lim

P!1lðPÞ (34)

the increasing of the bound remains relatively low forsmall and moderate P and becomes almost constant forlarge P. In conclusion, the estimation accuracy of thisparameter is degraded but not seriously.

4.2. Angular-frequency parameter

For the p-th angular-frequency parameter, the numberof samples into the p-th dimension, NðPÞp , plays animportant role since the quotient of two consecutiveACRB is given by

ACRBðPþ1ÞðoðpÞm Þ

ACRBðPÞðoðpÞm Þ¼

NðPÞp

NðPþ1Þp

!2

for p 2 ½1 : P�. (35)
For cubic tensors, we have NðPÞ4NðPþ1Þ and thus theACRB is a monotonically increasing sequence. For unbalanced tensors, the ACRB can be locally, i.e.,
for a given dimension p, a constant, a strictlymonotonically increasing or decreasing sequences,depending on the specific distribution of the tensorsizes. But, if the accuracy is improved in a givendimension, this means that the accuracy decreases inanother one.

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0 5 10 15 20 25 30 35 4010−35

10−30

10−25

10−20

10−15

10−10

SNR [linear scale]

CR

B [l

og s

cale

]

Angular frequency

0 5 10 15 20 25 30 35 4010−17

10−16

10−15

10−14

SNR [linear scale]

0 5 10 15 20 25 30 35 40SNR [linear scale]

0 5 10 15 20 25 30 35 40SNR [linear scale]

CR

B [l

og s

cale

]

Angular frequency

10−12

10−11

10−10

10−9 10−9

CR

B [l

og s

cale

]

10−12

10−11

10−10

CR

B [l

og s

cale

]

Real amplitude

ACRB1(a)ACRB2(a)ACRB3(a)ACRB4(a)

ACRB1(φ)ACRB2(φ)ACRB3(φ)ACRB4(φ)

ACRB2(ω(2))ACRB3(ω(2))ACRB4(ω(2))

Initial phase

ACRB3(ω(1))

ACRB1(ω(1))ACRB2(ω(1))

ACRB4(ω(1))

Fig. 3. ACRB Vs. SNR for a first-order harmonic model of dimension 3 under constraint (28).


4.3. Illustration of the ACRB for constant amount of data

Consider a total amount of data equals toD ¼ 1:2� 1010. For instance, there exits tensors of sizes:

Dim. 1 1:2� 1010

Dim. 2 ð4� 107Þ � ð3� 102

Þ

Dim. 3 105� ð4� 102

Þ � ð3� 102Þ

Dim. 4 ð2:5� 102Þ � ð4� 102

Þ � ð3� 102Þ � ð1:5� 102

Þ

Obviously, other distributions of parameter NðPÞp arepossible. In Fig. 3, we have illustrated the ACRB for a first-order harmonic model of dimension 3 under constraint(28). These figures confirm the conclusions of the previoussection. In particular,
In Fig. 3a, we can see that for increasing para-meter P, the bound increases. But locally, we canhave other behaviors as we can see in Fig. 3bsince for parameter oð2Þ, the ACRB for dimension

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three and four are lower than the one in dimensiontwo.
In Fig. 3c, we can check that the ACRB for the real
amplitude is invariant to parameter P. As expected inSection 3.4.1.
Finally, Fig. 3d indicates that the ACRB for the initial
phase increases with parameter P.

So, constraint (28) modifies drastically the behavior of theACRB.

5. Conclusion

This paper deals with the asymptotic estimationperformance on the model parameters (angular-frequency,initial phase and real amplitude) for an M-order multi-dimensional harmonic model of dimension P. We haveshown that increasing the dimension of the harmonicmodel decreases the asymptotic CRB and thus improvesthe minimal theoretical variance of the estimation of themodel parameters. For P-order cubic tensors of sizeN � � � � � N, the ‘‘magnitude of order’’ of the asymptoticCRB for the real amplitude and the initial phase is OðN�P

Þ

and OðN�Pþ2Þ for the angular-frequency. Finally, the last

conclusion is if the amount of data is constant for alldimension (i.e.,

QPp¼1Np ¼ cst), the asymptotic CRB for the

angular-frequency is a strictly monotonically increasingsequence for cubic tensors but can be locally (for a specificdimension) a constant or a strictly monotonically decreas-ing sequence for unbalanced tensors. Regarding the realamplitude parameter, the asymptotic CRB becomes invar-iant to parameter P. Finally, we show that the estimationaccuracy for the initial phase is degraded for increasing P

but not seriously.

Acknowledgment

The author would like to thank the reviewers and theeditor for their valuable comments that led to theimprovement of this paper.

Appendix A. Proof of Theorem 1

The partial derivatives of the noise-free signal wrt theangular-frequency, the real amplitude and the initialphase are given by

qx

qoðpÞm

¼ iamðdðoð1Þm Þ � � � � � d0ðoðpÞm Þ � � � � � dðoðPÞm ÞÞ

qx

qam¼ eifm ðdðoð1Þm Þ � � � � � dðoðPÞm ÞÞ

qx

qfm

¼ iamðdðoð1Þm Þ � � � � � dðoðPÞm ÞÞ

for m 2 ½1 : M�, j 2 ½1 : P� and d0ðoðpÞm Þ ¼ ½0 eioðpÞm 2e2ioðpÞm . . .ðNp � 1ÞeðNp�1ÞioðpÞm �T.

Using the asymptotic properties of the harmonic

model [7], ð1=N3pÞd0ðoðpÞk Þ

Hd0ðoðpÞm Þ �!Npb1

13 dk�m; ð1=N2

pÞd0

ðoðpÞk ÞHdðoðpÞm Þ �!

Npb112 dk�m; ð1=NpÞdðoðpÞk Þ

HdðoðpÞm Þ �!Npb1

dk�m, a

straightforward derivation leads to

JoðjÞkoðuÞm¼ R

qx

qoðjÞk

!Hqx

qoðuÞm

8<:

9=;

¼

a2k

N3u

3

QPp¼1;paj

Np for j ¼ u

and k ¼ m

a2k

N2u

2

N2j

2

QPp¼1;paj;u

Np for jau

and k ¼ m

0 otherwise

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(36)

So, we have

JoðjÞoðuÞ ¼

JoðjÞ1oðuÞ

1

. . . JoðjÞ1oðuÞ

M

..

. ...

JoðjÞMoðuÞ

1

. . . JoðjÞMoðuÞ

M

2666664

3777775

M�M

¼

N2j

3

QPp¼1

Np

!D2 for j ¼ u

NuNj

4

QPp¼1

Np

!D2 for jau

8>>>>>><>>>>>>:

(37)

where D ¼ diagfa1; . . . ; aMg and thus

Joo ¼

Joð1Þoð1Þ . . . Joð1ÞoðPÞ

..

. ...

JoðPÞoð1Þ . . . JoðPÞoðPÞ

266664

377775

PM�PM

¼YP

p¼1

Np

!ðUP � D2

Þ (38)

where we have defined the following (P � P) symmetricmatrix:

UP ¼

N21

3

N1N2

4. . . . . .

N1NP

4

N1N2

4

N22

3. . . . . .

N2NP

4

..

. ...

N1NP

4

N2NP

4. . . . . .

N2P

3

2666666666664

3777777777775

(39)

Next, we have JoðjÞkfm�!Njb1

Rfa�kamðNj=2ÞðQP

p¼1 NpÞdk�mg and

thus JoðjÞ f �!Njb1

12 Njð

QPp¼1 NpÞD2. Finally, we find the follow-

ing compact expression for the PM �M matrix Jof ¼

ðQP

p¼1 Np=2ÞðgP �D2Þ where gP ¼ ½N1 . . . NP �

T. In addition,

the other blocks of the FIM are

½Ja a�km �!Npb1

R eiðfm�fkÞYP

p¼1

Np

!dk�m

( )

¼

QPp¼1

Np for k ¼ m

0 otherwise

8><>: (40)

ARTICLE IN PRESS


½Jff�km �!Npb1

R i�a�kiam

YP

p¼1

Np

!dk�m

( )

¼a2

k

QPp¼1

Np for k ¼ m

0 otherwise

8><>: (41)

½Jaf�km �!Npb1

0; 8k;m (42)

½Jo a�km �!Npb1

0; 8k;m (43)

For k ¼ m, expressions (42) and (43) are purely imaginarynumbers. This explains why Jaf and Jo a are null matrices.

Consequently, the blocks of the FIM are asymptoticallydiagonal or null and we obtain

Ja a ¼YP

p¼1

Np

!IM ; Jff ¼

YP

p¼1

Np

!D2

Jaf ¼ 0M�M ; Jo a ¼ 0PM�M

Finally, the FIM wrt y0 is given by

Fy0y0 �!Npb1

ðQP

p¼1NpÞðUP � D2Þ 0

QPp¼1Np

2ðgP � D2

Þ

0 ðQP

p¼1NpÞIM 0QPp¼1Np

2ðgT

P � D2Þ 0 ð

QPp¼1NpÞD2

266666664

377777775

Thanks to the standard inverse of a partitioned matrix [7],

analytic expression of F�1y0y0 is possible. It comes

F�1y0y0 �!

Npb1

L 0 �

0 J�1a a 0

� 0 YLYTþ J�1

ff

2664

3775 (44)

where

L ¼ ðJoo � JofJ�1ffJofÞ

�1

¼1QP

p¼1 Np

UP �1

4gPgT

P

� ��1

�D�2

" #

¼12QP

p¼1Np

ðdiagðgPÞ�2�D�2

Þ

and Y ¼ J�1ffJof ¼

12 ðg

TP � IMÞ. So, the ð3;3Þ-block of matrix

F�1y0y0 is given by

YLYTþ J�1

ff ¼3QP

p¼1Np

gTP diagðgPÞ

�2gP|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}P

�D�2

0B@

1CA

þ1QP

p¼1 Np

D�2

¼3P þ 1QP

p¼1Np

D�2

Hence, the inverse of the FIM is

F�1y0y0 �!

Npb1

12QPp¼1 Np

ðdiagðgPÞ�2� D�2

Þ 0 �

01QP

p¼1 Np

IM 0

� 03P þ 1QP

p¼1Np

D�2

26666666664

37777777775

So, the CRB associated with the M-order harmonic modelof dimension P is given by the diagonal terms of the FIMinverse which proves the theorem.

Appendix B. Exact CRB for the first-order harmonicmodel of dimension P

Using the same formalism as before, we derive in thefollowing theorem the exact (nonasymptotic) closed-formof the CRB for the first-order harmonic model of dimen-sion P.

Theorem 3. The exact CRBðPÞ for the first-order harmonic

model of dimension P defined in (1) where M ¼ 1 wrt the

model parameter y0 ¼ ½oð1Þ; . . . ;oðPÞ a f�T, i.e., CRBðPÞðy0Þ, is

given by

CRBðPÞðoðpÞÞ ¼6

N1N2 . . .NPðN2p � 1ÞSNR

(45)

CRBðPÞðaÞ ¼a2

2N1N2 . . .NPSNR(46)

CRBðPÞðfÞ ¼3PP

p¼1

Np � 1

Np þ 1þ 1

2N1N2 . . .NPSNR(47)

where SNR ¼ a2=s2.

Proof. To prove this theorem, we consider the first-orderharmonic model of dimension P given by x ¼

aeifðdðoð1ÞÞ � � � � � dðoðPÞÞÞ where the model parametersare the following triplet: foð1Þ; . . . ;oðPÞ; a;fg. Recallingsome standard results on power sums, we have

d0ðoðpÞÞHd0ðoðpÞÞ ¼XNp�1

n¼0

n2

¼1

6ðNp � 1ÞNpð2Np � 1Þ (48)

d0ðoðpÞÞHdðoðpÞÞ ¼XNp�1

n¼0

n ¼1

2ðNp � 1ÞNp (49)

dðoðpÞÞHdðoðpÞÞ ¼XNp�1

n¼0

1 ¼ Np (50)

Using (48)–(50) this can be expressed according to Joo ¼

ða2=2ÞðQP

p¼1NpÞCP where we have defined the

ARTICLE IN PRESS


following (P � P) symmetric matrix:

CP ¼

ðN1 � 1Þð2N1 � 1Þ

3

ðN1 � 1ÞðN2 � 1Þ

2. . .

ðN1 � 1ÞðNP � 1Þ

2

ðN1 � 1ÞðN2 � 1Þ

2

ðN2 � 1Þð2N2 � 1Þ

3. . .


2

..

. ... ..

.


2


2. . .

ðNP � 1Þð2NP � 1Þ

3

2666666666664

3777777777775

(51)

and

Ja a ¼YP

p¼1

Np; Jff ¼ a2YP

p¼1

Np

!

Jaf ¼ Jo a ¼ 0; Jof ¼a2

2

YP

p¼1

Np

!nP

where nP ¼ ½N1 � 1 N2 � 1 . . . NP � 1�T.

Consequently, the FIM wrt the signal parameters for the

first-order harmonic model of dimension P is given by

matrix (44) and its inverse is given by matrix (44) where

L ¼2

a2

1QPp¼1Np

CP �nPnT

P

2

� ��1

¼2

a2

1QPp¼1Np

DP (52)

where DP ¼ diagf6=ðN21 � 1Þ; . . . ;6=ðN2

P � 1Þg with Y ¼nT

P=2 and

YLYTþ J�1

ff ¼1

a2QP

p¼1 Np

3XP

p¼1

Np � 1

Np þ 1þ 1

!

More precisely, the inverse of the FIM is

F�1y0y0 ¼

2

a2QP

p¼1Np

DP 0 �

01QP

p¼1Np

0

� 01

a2QP

p¼1Np

3PP

p¼1

Np � 1

Np þ 1þ 1

� �

266666666664

377777777775

Considering the diagonal terms of the above matrix

weighed by s2=2, we obtain expressions (45)–(47).

For cubic tensors, we have

CRBðPÞðoðpÞÞ ¼6

NPðN2� 1ÞSNR

(53)

CRBðPÞðaÞ ¼a2

2NPSNR(54)

CRBðPÞðfÞ ¼3P

N � 1

N þ 1þ 1

2NPSNR(55)

Note that as expected if Np goes to infinity for all p, theexact CRB becomes the ACRB derived in the previoussection. The exact CRB for a first-order harmonic ofdimension P is quite similar to the asymptotic analysisderived in the previous sections. In particular, the exactCRBðPÞ for the first-order case shares the same propertiesas the ACRBðPÞ. &

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deterministic asymptotic cramér–rao bound for the multidimensional harmonic model

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