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Statistical resolution limit for the multidimensional harmonic retrieval model: hypothesis test and CramérRao Bound approaches
EURASIP Journal on Advances in Signal Processing volume 2011, Article number: 12 (2011)
Abstract
The statistical resolution limit (SRL), which is defined as the minimal separation between parameters to allow a correct resolvability, is an important statistical tool to quantify the ultimate performance for parametric estimation problems. In this article, we generalize the concept of the SRL to the multidimensional SRL (MSRL) applied to the multidimensional harmonic retrieval model. In this article, we derive the SRL for the socalled multidimensional harmonic retrieval model using a generalization of the previously introduced SRL concepts that we call multidimensional SRL (MSRL). We first derive the MSRL using an hypothesis test approach. This statistical test is shown to be asymptotically an uniformly most powerful test which is the strongest optimality statement that one could expect to obtain. Second, we link the proposed asymptotic MSRL based on the hypothesis test approach to a new extension of the SRL based on the CramérRao Bound approach. Thus, a closedform expression of the asymptotic MSRL is given and analyzed in the framework of the multidimensional harmonic retrieval model. Particularly, it is proved that the optimal MSRL is obtained for equipowered sources and/or an equidistributed number of sensors on each multiway array.
Introduction
The multidimensional harmonic retrieval problem is an important topic which arises in several applications [1]. The main reason is that the multidimensional harmonic retrieval model is able to handle a large class of applications. For instance, the joint angle and carrier estimation in surveillance radar system [2, 3], the underwater acoustic multisource azimuth and elevation direction finding [4], the 3D harmonic retrieval problem for wireless channel sounding [5, 6] or the detection and localization of multiple targets in a MIMO radar system [7, 8].
One can find many estimation schemes adapted to the multidimensional harmonic retrieval estimation problem, see, e.g., [1, 2, 4–7, 9, 10]. However, to the best of our knowledge, no work has been done on the resolvability of such a multidimensional model.
The resolvability of closely spaced signals, in terms of parameter of interest, for a given scenario (e.g., for a given signaltonoise ratio (SNR), for a given number of snapshots and/or for a given number of sensors) is a former and challenging problem which was recently updated by Smith [11], Shahram and Milanfar [12], Liu and Nehorai [13], and Amar and Weiss [14]. More precisely, the concept of statistical resolution limit (SRL), i.e., the minimum distance between two closely spaced signals^{a} embedded in an additive noise that allows a correct resolvability/parameter estimation, is rising in several applications (especially in problems such as radar, sonar, and spectral analysis [15].)
The concept of the SRL was defined/used in several manners [11–14, 16–24], which could turn in it to a confusing concept. There exist essentially three approaches to define/obtain the SRL. (i) The first is based on the concept of mean null spectrum: assuming, e.g., that two signals are parameterized by the frequencies f_{1} and f_{2}, the Cox criterion [16] states that these sources are resolved, w.r.t. a given highresolution estimation algorithm, if the mean null spectrum at each frequency f_{1}and f_{2}is lower than the mean of the null spectrum at the midpoint. Another commonly used criterion, also based on the concept of the mean null spectrum, is the Sharman and Durrani criterion [17], which states that two sources are resolved if the second derivative of the mean of the null spectrum at the midpointis negative. It is clear that the SRL based on the mean null spectrum is relevant to a specific highresolution algorithm (for some applications of these criteria one can see [16–19] and references therein.) (ii) The second approach is based on detection theory: the main idea is to use a hypothesis test to decide if one or two closely spaced signals are present in the set of the observations. Then, the challenge herein is to link the minimum separation, between two sources (e.g., in terms of frequencies) that is detectable at a given SNR, to the probability of false alarm, P_{fa} and/or to the probability of detection P_{d}. In this spirit, Sharman and Milanfar [12] have considered the problem of distinguishing whether the observed signal contains one or two frequencies at a given SNR using the generalized likelihood ratio test (GLRT). The authors have derived the SRL expressions w.r.t. P_{fa} and P_{d} in the case of real received signals, and unequal and unknown amplitudes and phases. In [13], Liu and Nehorai have defined a statistical angular resolution limit using the asymptotic equivalence (in terms of number of observations) of the GLRT. The challenge was to determine the minimum angular separation, in the case of complex received signals, which allows to resolve two sources knowing the direction of arrivals (DOAs) of one of them for a given P_{fa} and a given P_{d}. Recently, Amar and Weiss [14] have proposed to determine the SRL of complex sinusoids with nearby frequencies using the Bayesian approach for a given correct decision probability. (iii) The third approach is based on a estimation accuracy criteria independent of the estimation algorithm. Since the CramérRao Bound (CRB) expresses a lower bound on the covariance matrix of any unbiased estimator, then it expresses also the ultimate estimation accuracy [25, 26]. Consequently, it could be used to describe/obtain the SRL. In this context, one distinguishes two main criteria for the SRL based on the CRB: (1) the first one was introduced by Lee [20] and states that: two signals are said to be resolvable w.r.t. the frequencies if the maximum standard deviation is less than twice the difference between f_{1}and f_{2}. Assuming that the CRB is a tight bound (under mild/weak conditions), the standard deviation, and , of an unbiased estimator is given by and , respectively. Consequently, the SRL is defined, in the Lee criterion sense, as 2max . One can find some results and applications in [20, 21] where this criterion is used to derive a matrixbased expression (i.e., without analytic inversion of the Fisher information matrix) of the SRL for the frequency estimates in the case of the conditional and unconditional signal source models. On the other hand, Dilaveroglu [22] has derived a closedform expression of the frequency resolution for the real and complex conditional signal source models. However, one can note that the coupling between the parameters, CRB(f_{1}, f_{2}) (i.e., the CRB for the cross parameters f_{1} and f_{2}), is ignored by this latter criterion. (2) To extend this, Smith [11] has proposed the following criterion: two signals are resolvable w.r.t. the frequencies if the difference between the frequencies, δ_{ f }, is greater than the standard deviation of the DOA difference estimation. Since, the standard deviation can be approximated by the CRB, then, the SRL, in the Smith criterion sense, is defined as the limit of δ_{ f } for which is achieved. This means that, the SRL is obtained by solving the following implicit equation
In [11, 23], Smith has derived the SRL for two closely spaced sources in terms of DOA, each one modeled by one complex pole. In [24], Delmas and Abeida have derived the SRL based on the Smith criterion for DOA of discrete sources under QPSK, BPSK, and MSK model assumptions. More recently, Kusuma and Goyal [27] have derived the SRL based on the Smith criterion in sampling estimation problems involving a powersum series.
It is important to note that all the criteria listed before take into account only one parameter of interest per signal. Consequently, all the criteria listed before cannot be applied to the aforementioned the multidimensional harmonic model. To the best of our knowledge, no results are available on the SRL for multiple parameters of interest per signal. The goal of this article is to fill this lack by proposing and deriving the socalled MSRL for the multidimensional harmonic retrieval model.
More precisely, in this article, the MSRL for multiple parameters of interest per signal using a hypothesis test is derived. This choice is motivated by the following arguments: (i) the hypothesis test approach is not specific to a certain highresolution algorithm (unlike the mean null spectrum approach), (ii) in this article, we link the asymptotic MSRL based on the hypothesis test approach to a new extension of the MSRL based on the CRB approach. Furthermore, we show that the MSRL based on the CRB approach is equivalent to the MSRL based on the hypothesis test approach for a fixed couple (P_{fa}, P_{d}), and (iii) the hypothesis test is shown to be asymptotically an uniformly most powerful test which is the strongest statement of optimality that one could expect to obtain [28].
The article is organized as follows. We first begin by introducing the multidimensional harmonic model, in section "Model setup". Then, based on this model, we obtain the MSRL based on the hypothesis test and on the CRB approach. The link between theses two MSRLs is also described in section "Determination of the MSRL for two sources" followed by the derivation of the MSRL closedform expression, where, as a by product the exact closedform expressions of the CRB for the multidimensional retrieval model is derived (note that to the best of our knowledge, no exact closedform expressions of the CRB for such model is available in the literature). Furthermore, theoretical and numerical analyses are given in the same section. Finally, conclusions are given.
Glossary of notation
The following notations are used through the article. Column vectors, matrices, and multiway arrays are represented by lowercase bold letters (a, ...), uppercase bold letters (A, ...) and bold calligraphic letters , whereas

ℝ and ℂ denote the body of real and complex values, respectively,

and denote the real and complex multiway arrays (also called tensors) body of dimension D_{1} × D_{2} × ⋯ ×D_{ I }, respectively,

j = the complex number .

I_{ Q }= the identity matrix of dimension Q,

= the Q_{1} × Q_{2} matrix filled by zeros,

[a]_{ i }= the i th element of the vector a,

= the i_{1}th row and the i_{2}th column element of the matrix A,

= the (i_{1}, i_{2}, ..., i_{ N })th entry of the multiway array ,

[A]_{i,p:q}= the row vector containing the (q  p + 1) elements [A]_{i,k}, where k = p, ..., q,

[A]_{p:q,k}= the column vector containing the (q  p + 1) elements [A]_{i,k}, where i = p, ..., q,

the derivative of vector a w.r.t. to vector b is defined as follows: ,

A^{T} = the transpose of the matrix A,

A* = the complex conjugate of the matrix A,

A^{H} = (A*)^{T},

tr {A} = the trace of the matrix A,

det {A} = the determinant of the matrix A,

ℜ{a} = the real part of the complex number a,

= the expectation of the random variable a,

denotes the normalized norm of the vector a(in which L is the size of a),

sgn (a) = 1 if a ≥ 0 and 1 otherwise.

diag(a) is the diagonal operator which forms a diagonal matrix containing the vector a on its diagonal,

vec(.) is the vecoperator stacking the columns of a matrix on top of each other,

⊙ stands for the Hadamard product,

⊗ stands for the Kronecker product,

○ denotes the multiway array outerproduct (recall that for a given multiway arrays and , the result of the outerproduct of and denoted by is given by .
Model setup
In this section, we introduce the multidimensional harmonic retrieval model in the multiway array form (also known as tensor form[29]). Then, we use the PARAFAC (PARallel FACtor) decomposition to obtain a vector form of the observation model. This vector form will be used to derive the closedform expression of the MSRL.
Let us consider a multidimensional harmonic model consisting of the superposition of two harmonics each one of dimension P contaminated by an additive noise. Thus, the observation model is given as follows [8, 9, 26, 30–32]:
where , , and denote the noisy observation, the noiseless observation, and the noise multiway array at the t th snapshot, respectively. The number of snapshots and the number of sensors on each array are denoted by L and (N_{1}, ...,N_{ P } ), respectively. The noiseless observation multiway array can be written as follows^{b}[26, 30–32]:
where and s_{ m } (t) denote the m th frequency viewed along the p th dimension or array and the m th complex signal source, respectively. Furthermore, the signal source is given by where α_{ m } (t) and ϕ_{ m } (t) denote the real positive amplitude and the phase for the m th signal source at the t th snapshot, respectively.
Since,
where a(.) is a Vandermonde vector defined as
then, the multiway array follows a PARAFAC decomposition [7, 33]. Consequently, the noiseless observation multiway array can be rewritten as follows:
First, let us vectorize the noiseless observation as follows:
Thus, the full noisefree observation vector is given by
Second, and in the same way, we define y, the noisy observation vector, and n, the noise vector, by the concatenation of the proper multiway array's entries, i.e.,
Consequently, in the following, we will consider the observation model in (5). Furthermore, the unknown parameter vector is given by
where ω denotes the unknown parameter vector of interest, i.e., containing all the unknown frequencies
in which
whereas ρ contains the unknown nuisance/unwanted parameters vector, i.e., characterizing the noise covariance matrix and/or amplitude and phase of each source (e.g., in the case of a covariance noise matrix equal to and unknown deterministic amplitudes and phases, the unknown nuisance/unwanted parameters vector ρ is given by ρ= [α_{1}(1) ⋯ α_{2}(L)ϕ_{1}(1) ⋯ ϕ_{2}(L)σ^{2}]^{T}.
In the following, we conduct a hypothesis test formulation on the observation model (5) to derive our MSRL expression in the case of two sources.
Determination of the MSRL for two sources
Hypothesis test formulation
Resolving two closely spaced sources, with respect to their parameters of interest, can be formulated as a binary hypothesis test [12–14] (for the special case of P = 1). To determine the MSRL (i.e., P ≥ 1), let us consider the hypothesis which represents the case where the two emitted signal sources are combined into one signal, i.e., the two sources have the same parameters (this hypothesis is described by , whereas the hypothesis embodies the situation where the two signals are resolvable (the latter hypothesis is described by ∃p ∈ [1 ... P], such that ). Consequently, one can formulate the hypothesis test, as a simple onesided binary hypothesis test as follows:
where the parameter δ is the socalled MSRL which indicates us in which hypothesis our observation model belongs. Thus, the question addressed below is how can we define the MSRL δ such that all the P parameters of interest are taken into account? A natural idea is that δ reflects a distance between the P parameters of interest. Let the MSRL denotes the l_{1} norm^{c} between two sets containing the parameters of interest of each source (which is the naturally used norm, since in the monoparameter frequency case that we extend here, the SRL is defined as δ = f_{1}  f_{2}[13, 14, 34]). Meaning that, if we denote these sets as C_{1} and C_{2} where , m = 1,2, thus, δ can be defined as
First, note that the proposed MSRL describes well the hypothesis test (8) (i.e., δ = 0 means that the two emitted signal sources are combined into one signal and δ ≠ 0 the two signals are resolvable). Second, since the MSRL δ is unknown, it is impossible to design an optimal detector in the NeymanPearson sense. Alternatively, the GLRT [28, 35] is a wellknown approach appropriate to solve such a problem. To conduct the GLRT on (8), one has to express the probability density function (pdf) of (5) w.r.t. δ. Assuming (without loss of generality) that , one can notice that ξ is known if and only if δ and are fixed (i.e., there is a one to one mapping between δ, ϑ, and ξ). Consequently, the pdf of (5) can be described as p(yδ,ϑ). Now, we are ready to conduct the GLRT for this problem:
where , , and denote the maximum likelihood estimates (MLE) of δ under , the MLE of ϑ under and the MLE of ϑ under , respectively, and where ς' denotes the test threshold. From (10), one obtains
in which Ln denotes the natural logarithm.
Asymptotic equivalence of the MSRL
Finding the analytical expression of T_{ G } (y) in (11) is not tractable. This is mainly due to the fact that the derivation of is impossible since from (2) one obtains a multimodal likelihood function [36]. Consequently, in the following, and as in^{d}[13], we consider the asymptotic case (in terms of the number of snapshots). In [35, eq (6C.1)], it has been proven that, for a large number of snapshots, the statistic T_{ G } (y) follows a chisquare pdf under and given by
where and denote the central chisquare and the noncentral chisquare pdf with one degree of freedom, respectively. P_{fa} and P_{d} are, respectively, the probability of false alarm and the probability of detection of the test (8). In the following, CRB(δ) denotes the CRB for the parameter δ where the unknown vector parameter is given by [δ ϑ^{T}]^{T}. Consequently, assuming that CRB(δ) exists (under and ), is well defined (see section "MSRL closedform expression" for the necessary^{e} and sufficient conditions) and is a tight bound (i.e., achievable under quite general/weak conditions [36, 37]), thus the noncentral parameter κ'(P_{fa}, P_{d}) is given by [[35], p. 239]
On the other hand, one can notice that the noncentral parameter κ'(P_{fa}, P_{d}) can be determined numerically by the choice of P_{fa} and P_{d}[13, 28] as the solution of
in which and are the inverse of the right tail of the and pdf starting at the value ϖ. Finally, from (13) and (14) one obtains^{f}
where is the socalled translation factor [13] which is determined for a given probability of false alarm and probability of detection (see Figure 1 for the behavior of the translation factor versus P_{fa} and P_{d}).
Result 1: The asymptotic MSRL for model (5) in the case of P parameters of interest per signal (P ≥ 1) is given by δ which is the solution of the following equation:
where A_{direct} denotes the contribution of the parameters of interest belonging to the same dimension as follows
and where A_{cross} is the contribution of the cross terms between distinct dimension given by
in which .
Proof see Appendix 1.
Remark 1: It is worth noting that the hypothesis test (8) is a binary onesided test and that the MLE used is an unconstrained estimator. Thus, one can deduce that the GLRT, used to derive the asymptotic MSRL [13, 35]: (i) is the asymptotically uniformly most powerful test among all invariant statistical tests, and (ii) has an asymptotic constant falsealarm rate (CFAR). Which is, in the asymptotic case, considered as the strongest statement of optimality that one could expect to obtain [28].

Existence of the MSRL: It is natural to assume that the CRB is a nonincreasing (i.e., decreasing or constant) function on ℝ^{+} w.r.t. δ since it is more difficult to estimate two closely spaced signals than two largelyspaced ones. In the same time the left hand side of (15) is a monotonically increasing function w.r.t. δ on ℝ^{+}. Thus for a fixed couple (P_{fa}, P_{d}), the solution of the implicit equation given by (15) always exists. However, theoretically, there is no assurance that the solution of equation (15) is unique.

Note that, in practical situation, the case where CRB(δ) is not a function of δ is important since in this case, CRB(δ) is constant w.r.t. δ and thus the solution of (15) exists and is unique (see section "MSRL closedform expression").
In the following section, we study the explicit effect of this socalled translation factor.
The relationship between the MSRL based on the CRB and the hypothesis test approaches
In this section, we link the asymptotic MSRL (derived using the hypothesis test approach, see Result 1) to a new proposed extension of the SRL based on the Smith criterion [11]. First, we recall that the Smith criterion defines the SRL in the case of P = 1 only. Then, we extend this criterion to P ≥ 1 (i.e., the case of the multidimensional harmonic model). Finally, we link the MSRL based on the hypothesis test approach (see Result 1) to the MSRL based on the CRB approach (i.e., the extended SRL based on the Smith criterion).
The Smith criterion: Since the CRB expresses a lower bound on the covariance matrix of any unbiased estimator, then it expresses also the ultimate estimation accuracy. In this context, Smith proposed the following criterion for the case of two source signals parameterized each one by only one frequency [11]: two signals are resolvable if the difference between their frequency, , is greater than the standard deviation of the frequency difference estimation. Since, the standard deviation can be approximated by the CRB, then, the SRL, in the Smith criterion sense, is defined as the limit of for which is achieved. This means that, the SRL is the solution of the following implicit equation
The extension of the Smith criterion to the case of P ≥ 1: Based on the above framework, a straightforward extension of the Smith criterion to the case of P ≥ 1 for the multidimensional harmonic model is as follows: two multidimensional harmonic retrieval signals are resolvable if the distance between C_{1}and C_{2}, is greater than the standard deviation of the δ_{CRB}estimation. Consequently, assuming that the CRB exists and is well defined, the MSRL δ_{CRB} is given as the solution of the following implicit equation
Comparison and link between the MSRL based on the CRB approach and the MSRL based on the hypothesis test approach: The MSRL based on the hypothesis test approach is given as the solution of
whereas the MSRL based on the CRB approach is given as the solution of (17). Consequently, one has the following result:
Result 2: Upon to a translation factor, the asymptotic MSRL based on the hypothesis test approach (i.e., using the binary onesided hypothesis test given in (8)) is equivalent to the proposed MSRL based on the CRB approach (i.e., using the extension of the Smith criterion). Consequently, the criterion given in (17) is equivalent to an asymptotically uniformly most powerful test among all invariant statistical tests for κ(P_{fa}, P_{d}) = 1 (see Figure 2 for the values of (P_{fa}, P_{d}) such that κ (P_{fa}, P_{d}) = 1).
The following section is dedicated to the analytical computation of closedform expression of the MSRL. In section "Assumptions," we introduce the assumptions used to compute the MSRL in the case of a Gaussian random noise and orthogonal waveforms. Then, we derive non matrix closedform expressions of the CRB (note that to the best of our knowledge, no closedform expressions of the CRB for such model is available in the literature). In "MSRL derivation" and thanks to these expressions, the MSRL wil be deduced using (16). Finally, the MSRL analysis is given.
MSRL closedform expression
in section "Determination of the MSRL for two sources" we have defined the general model of the multidimensional harmonic model. To derive a closedform expression of the MSRL, we need more assumptions on the covariance noise matrix and/or on the signal sources.
Assumptions

The noise is assumed to be a complex circular white Gaussian random process i.i.d. with zeromean and unknown variance .

We consider a multidimensional harmonic model due to the superposition of two harmonics each of them of dimension P ≥ 1. Furthermore, for sake of simplicity and clarity, the sources have been assumed known and orthogonal (e.g., [7, 38]). In this case, the unknown parameter vector is fixed and does not grow with the number of snapshots. Consequently, the CRB is an achievable bound [36].

Each parameter of interest w.r.t. to the first signal, , can be as close as possible to the parameter of interest w.r.t. to the second signal , but not equal. This is not really a restrictive assumption, since in most applications, having two or more identical parameters of interest is a zero probability event [[9], p. 53].
Under these assumptions, the joint probability density function of the noisy observations y for a given unknown deterministic parameter vector ξ is as follows:
where . The multidimensional harmonic retrieval model with known sources is considered herein, and thus, the parameter vector is given by
where
in which
CRB for the multidimensional harmonic model with orthogonal known signal sources
The Fisher information matrix (FIM) of the noisy observations y w.r.t. a parameter vector ξ is given by [39]
For a complex circular Gaussian observation model, the (i th, k th) element of the FIM for the parameter vector ξ is given by [34]
Consequently, one can state the following lemma.
Lemma 1: The FIM for the sum of two Porder harmonic models with orthogonal known sources, has a block diagonal structure and is given by
where, the (2P) × (2P) matrix F_{ ω }is also a block diagonal matrix given by
in which Δ = diag {α_{1}^{2} ,α_{2}^{2}} where
and
Proof see Appendix 2.
After some calculation and using Lemma 1, one can state the following result.
Result 3: The closedform expressions of the CRB for the sum of two Porder harmonic models with orthogonal known signal sources are given by
where denotes the SNR of the m th source and where
Furthermore, the crossterms are given by
where
Proof see Appendix 3.
MSRL derivation
Using the previous result, one obtains the unique solution of (16), thus, the MSRL for model (1) is given by the following result:
Result 4: The MSRL for the sum of Porder harmonic models with orthogonal known signal sources, is given by
where the socalled extended SNR is given by .
Proof see Appendix 4.
Numerical analysis
Taking advantage of the latter result, one can analyze the MSRL given by (26):

First, from Figure 3 note that the numerical solution of the MSRL based on (12) is in good agreement with the analytical expression of the MSRL (23), which validate the closedform expression given in (23). On the other hand, one can notice that, for P_{d} = 0.37 and P_{fa} = 0.1 the MSRL based on the CRB is exactly equal to the MSRL based on hypothesis test approach derived in the asymptotic case. From the case P_{d} = 0.49 and P_{fa} = 0.3 or/and P_{d} = 0.32 and P_{fa} = 0.1, one can notice the influence of the translation factor κ(P_{fa}, P_{d}) on the MSRL.

The MSRL^{g} is which is consistent with some previous results for the case P = 1 (e.g., [12, 14, 24]).

From (26) and for a large number of sensors N_{1} = N_{2} = ⋯ = N_{ P }= N ≫ 1, one obtains a simple expression
meaning that, the SRL is .

Furthermore, since P ≥ 1, one has
and consequently, the ratio between the MSRL of a multidimensional harmonic retrieval with P parameters of interest, denoted by δ _{ P } and the MSRL of a multidimensional harmonic retrieval with P + 1 parameters of interest, denoted by δ_{P+1}, is given by
meaning that the MSRL for P + 1 parameters of interest is less than the one for P parameters of interest (see Figure 4). This, can be explained by the estimation additional parameter and also by an increase of the received noisy data thanks to the additional dimension. One should note that this property is proved theoretically thanks to (27) using the assumption of an equal and large number of sensors. However, from Figure 4 we notice that, in practice, this can be verified even for a small number of sensors (e.g., in Figure 4 one has 3 ≤ N_{ p } ≤ 5 for p = 3, ..., 6).

Furthermore, since
one can note that, the SRL is lower bounded by .

One can address the problem of finding the optimal distribution of power sources making the SRL the smallest as possible (s.t. the constraint of constant total source power). In this issue, one can state the following corollary: Corollary 1: The optimal power's source distribution that ensures the smallest MSRL is obtained only for the equipowered sources case.
Proof see Appendix 5.
This result was observed numerically for P = 1 in [12] (see Figure 5 for the multidimensional harmonic model). Moreover, it has been shown also by simulation for the case P = 1 that the socalled maximum likelihood breakdown (i.e., when the mean square error of the MLE increases rapidly) occurs at higher SNR in the case of different power signal sources than in the case of equipowered signal sources [40]. The authors explained it by the fact that one source grabs most of the total power, then, this latter will be estimated more accurately, whereas the second one, will take an arbitrary parameter estimation which represents an outlier.

In the same way, let us consider the problem of the optimal placement of the sensors^{h}N_{1}, ...,N_{P} , making the minimum MSRL s.t. the constraint that the total number of sensors is constant (i.e., in which we suppose that N_{total} is a multiple of P).
Corollary 2: If the total number of sensors N_{total}, is a multiple of P, then an optimal placement of the sensors that ensure the lowest MSRL is (see Figure 6 and 7)
Proof see Appendix 6.
Remark 3: Note that, in the case where N_{total} is not a multiple of P, one expects that the optimal MSRL is given in the case where the sensors distribution approaches the equisensors distribution situation given in corollary 3. Figure 7 confirms that (in the case of P = 3, N_{1} = 8 and a total number of sensors N = 22). From Figure 7, one can notice that the optimal distribution of the number of sensors corresponds to N_{2} = N_{3} = 7 and N_{1} = 8 which is the nearest situation to the equisensors distribution.
Conclusion
In this article, we have derived the MSRL for the multidimensional harmonic retrieval model. Toward this end, we have extended the concept of SRL to multiple parameters of interest per signal. First, we have used a hypothesis test approach. The applied test is shown to be asymptotically an uniformly most powerful test which is the strongest statement of optimality that one could hope to obtain. Second, we have linked the asymptotic MSRL based on the hypothesis test approach to a new extension of the SRL based on the CramérRao bound approach. Using the CramérRao bound and a proper change of variable formula, closedform expression of the MSRL are given.
Finally, note that the concept of the MSRL can be used to optimize, for example, the waveform and/or the array geometry for a specific problem.
Appendix 1
The proof of Result 1
Appendix 1.1: In this appendix, we derive the MSRL using the l_{1} norm.
From CRB(ξ) where ξ= [ω^{T}ρ^{T}]T in which , one can deduce where in which . Thanks to the Jacobian matrix given by
where h = [g_{1}g_{2} ... g_{ P } ]^{T} ⊗ [1  1]^{T}, in which and A = [0 I]. Using the change of variable formula
one has
Consequently, after some calculus, one obtains
where and where Finally using (30) one obtains (16)
Appendix 1.2: In this part, we derive the MSRL using the l_{k} norm for a given integer k ≥ 1. The aim of this part is to support the endnote a, which stays that using the l_{1} norm computing the MSRL using the l_{1} norm is for the calculation convenience.
Once again, from CRB(ξ), one can deduce where in which the distance between C_{1} and C_{2} using the l_{ k } norm is given by δ(k) ≜ knorm and where . The Jacobian matrix is given by
where h_{ k } = [1  1]^{T} ⊗ [g_{1}(k)g_{2}(k) ... g_{ P } (k)]^{T}, in which and A = [0I]. Since x ^{k} can be written as . Thus, for × ≠ 0, one has
Again, using the change of variable formula (29), one has
Consequently, after some calculus, one obtains
where and where .
Consequently, note that resolving analytically the implicit equation (32) w.r.t. δ(k) is intractable (aside from some special cases). Whereas, resolving analytically the implicit equation (30) can be tedious but feasible (see section "MSRL closed form expression").
Furthermore, denoting g_{ p } (1) = g_{ p } , A_{cross}(1) ≜ A_{cross} and A_{direct}(1) ≜ A_{direct} and using (32) one obtains (16).
Appendix 2
Proof of Lemma 1
From (20) one can note the wellknown property that the model signal parameters are decoupled from the noise variance [42]. Consequently, the blockdiagonal structure in (21) is selfevident.
Now, let us prove (22). From (4), one obtains
where
Thus,
where s_{ m } = [s_{ m } (1) ... s_{ m } (L)]^{T}. Using the distributivity of the Hermitian operator over the Kronecker product and the mixedproduct property of the Kronecker product [43] and assuming, without loss of generality that p' < p, one obtains
On the other hand, one has
whereas
Finally, assuming known orthogonal wavefronts [38] (i.e., ) and replacing (35) and (34) into (33), one obtains
where α_{ m }= [α_{ m } (1) ... α_{ m } (L)] for m ∈ {1, 2}: Consequently, using (36), F_{ ω }can be expressed as a block diagonal matrix
where each P × P block J_{ m } is defined by
where
Consequently, from (37) and (38) one obtains (22).
Appendix 3
Proof of Result 3
Using (22) one obtains
where . In the following, we give a closedform expression of G^{1}. One can notice that the matrix G has a particular structure such that it can be rewritten as the sum of a diagonal matrix and of a rankone matrix: G = Q + γγ^{T} where and Thanks to this particular structure, an analytical inverse of G can easily be obtained. Indeed, using the matrix inversion lemma
A straightforward calculus leads to the following results,
and
Consequently, replacing (41) and (42) into (40), one obtains
where . Finally, replacing (43) into (39) one finishes the proof.
Appendix 4
Proof of Result 4
Using Results 1 and 3, one has
and
Consequently, replacing (44) and (45) into (16), one finishes the proof.
Appendix 5
Proof of Corollary 1
In this appendix, we minimize the MSRL under the constraint SNR_{1} + SNR_{2} = SNR_{total} (where SNR_{total} is a real fixed value). Since, the term is independent from SNR_{1} and SNR_{2}, minimizing δ is equivalent to minimize where
Using the method of Lagrange multipliers, the problem is as follows:
Thus, the Lagrange function is given by where λ denotes the socalled Lagrange multiplier. A simple derivation leads to,
Consequently, from (46) and (47), one obtains SNR_{1} = SNR_{1}. Using (48), one obtains . Using the constraint SNR_{1} + SNR_{2} = SNR_{total} one deduces corollary 1.
Appendix 6
Minimizing δ w.r.t. N_{1}, ..., N_{ P } is equivalent to minimizing the function , where N= [N_{1} ... N_{ P } ]^{T}. However, since the numbers of sensors on each array, N_{1}, ..., N_{ P } , are integers, the derivation of f(N) w.r.t. N is meaningless. Consequently, let us define the function exactly as f (.) where the set of definition is ℝ ^{P} instead of ℕ ^{P} . Consequently,
in which are real (continuous) variables.
Using the method of Lagrange multipliers, the problem is as follows:
where is a real positive constant value. Thus, the Lagrange function is given by where λ denotes the Lagrange multiplier. For a sufficient number of sensors, the Lagrange function can be approximated by
where . A simple derivation leads to,
This system of equations seems hard to solve. However, an obvious solution is given by and in which . Since, , thus the trivial solution is given by . Consequently, if is a multiple of P then, the solution of minimizing the function in ℝ ^{P} coincides the solution of minimizing the function f(N) in ℕ ^{P} . Thus, the optimal placement minimizing the MSRL is . This conclude the proof.
Endnotes
^{a}The notion of distance and closely spaced signals used in the following, is w.r.t. to the metric space (d, C), where d : C × C → ℝ in which d and C denote a metric and the set of the parameters of interest, respectively. ^{b}See [2–9] for some practical examples for the multidimensional harmonic retrieval model. ^{c}This study can be straightforwardly extended to other norms. The choice of the l_{1} is motivated by its calculation convenience (see the derivation of Result 1 and Appendix 1). Furthermore, since the MSRL is considered to be small (this assumption can be argued by the fact that the highresolution algorithms have asymptotically an infinite resolving power [44]), thus all continuous pnorms are similar to (i.e., looks like) the l_{1} norm. More importantly, in a finite dimensional vector space, all continuous pnorms are equivalent [[45], p. 53], thus the choice of a specific norm is free. ^{d}Note that, due to the specific definition of the SRL in [13] (i.e., using the same notation as in [13], and the restrictive assumption in [13] (u_{1} and u_{2} belong to the same plan), the SRL as defined in [13] cannot be used in the multidimensional harmonic context. ^{e}One of the necessary conditions regardless the noise pdf is that . Meaning that each parameter of interest w.r.t. to the first signal can be as close as possible to the parameter of interest w.r.t. to the second signal , but not equal. This is not really a restrictive assumptions, since in most applications, having two or more identical parameters of interest is a zero probability event [[9], p. 53]. ^{f}Note that applying (15) for P = 1 and for κ(P_{fa}, P_{d}) = 1, one obtains the Smith criterion [11]. ^{g}Where O(.) denotes the Landau notation [46]. ^{h}One should note, that we assumed a uniform linear multiarray, and the problem is to find the optimal distribution of the number of sensors on each array. The more general case, i.e., where the optimization problem considers the non linearity of the multiway array, is beyond the scope of the problem addressed herein.
Abbreviations
 CRB:

CramérRao Bound
 DOAs:

direction of arrivals
 FIM:

Fisher information matrix
 GLRT:

generalized likelihood ratio test
 MLE:

maximum likelihood estimates
 MSRL:

multidimensional SRL
 PARAFAC:

PARallel FACtor
 pdf:

probability density function
 SNR:

signaltonoise ratio
 SRL:

statistical resolution limit.
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This project is funded by region Île de France and Digiteo Research Park. This work has been partially presented in communication [41].
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El Korso, M.N., Boyer, R., Renaux, A. et al. Statistical resolution limit for the multidimensional harmonic retrieval model: hypothesis test and CramérRao Bound approaches. EURASIP J. Adv. Signal Process. 2011, 12 (2011). https://doi.org/10.1186/16876180201112
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DOI: https://doi.org/10.1186/16876180201112
Keywords
 Statistical resolution limit
 Multidimensional harmonic retrieval
 Performance analysis
 Hypothesis test
 CramérRao bound
 Parameter estimation
 Multidimensional signal processing