In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both deterministic and random parameters. Lately computationally tractable hybrid Cramér-Rao lower bounds for discrete-time Markovian dynamic systems depending on unknown time invariant deterministic parameters has been released. However in many applications (radar, sonar, telecoms, ...) the unknown deterministic parameters of the measurement model are time variant which prevents from using the aforementioned bounds. It is therefore the aim of this communication to tackle this issue by introducing new computationally tractable hybrid Cramér-Rao lower bounds

Chaumette, Eric

Galy, Jerome

Larzabal, Pascal

Renaux, Alexandre

Vincent, François

International audienceIn statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both deterministic and random parameters. Lately computationally tractable hybrid Cramér-Rao lower bounds for discrete-time Markovian dynamic systems depending on unknown time invariant deterministic parameters has been released. However in many applications (radar, sonar, telecoms, ...) the unknown deterministic parameters of the measurement model are time variant which prevents from using the aforementioned bounds. It is therefore the aim of this communication to tackle this issue by introducing new computationally tractable hybrid Cramér-Rao lower bounds

Galy, Jérôme

Hal-Diderot

Recursive hybrid CRB for Markovian systems with time-variant measurement parameters

HAL-CentraleSupelec

Open Archive Toulouse Archive Ouverte

 Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr  Open Archive TOULOUSE Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.  This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 16660 To cite this version: Galy, Jerome and Renaux, Alexandre and Chaumette, Eric and Vincent, François and Larzabal, Pascal Recursive hybrid CRB for Markovian systems with time-variant measurement parameters. (2016) In: CAMSAP 2015, 13 December 2015 - 16 December 2015 (Cancun, Mexico).  Official URL: http://dx.doi.org/10.1109/CAMSAP.2015.7383839 Recursive Hybrid CRB for Markovian Systems withTime-Variant Measurement ParametersJerome Galy(1), Alexandre Renaux(2), Eric Chaumette(3), Francois Vincent(3), Pascal Larzabal(4)(1) Universite´ de Montpellier 2/LIRMM, 161 rue Ada, Montpellier, France (galy@lirmm.fr)(2) Universite´ Paris-Sud/LSS, 3 Rue Joliot-Curie, Gif-sur-Yvette, France (renaux@lss.supelec.fr)(3) ISAE-SUPAERO, Universite´ de Toulouse, 31055 Toulouse Cedex 4, France(eric.chaumette@isae.fr, francois.vincent@isae.fr)(4) Universite´ Paris-Sud/SATIE, 61 av. du President Wilson, Cachan, France (pascal.larzabal@satie.ens-cachan.fr)Abstract—In statistical signal processing, hybrid parameterestimation refers to the case where the parameters vector to esti-mate contains both deterministic and random parameters. Latelycomputationally tractable hybrid Crame´r-Rao lower bounds fordiscrete-time Markovian dynamic systems depending on un-known time invariant deterministic parameters has been released.However in many applications (radar, sonar, telecoms, ...) theunknown deterministic parameters of the measurement modelare time variant which prevents from using the aforementionedbounds. It is therefore the aim of this communication to tacklethis issue by introducing new computationally tractable hybridCrame´r-Rao lower bounds.I. INTRODUCTIONIn the Bayesian estimation framework, discrete-time Marko-vian dynamic systems (MDS) arises in various applicationssuch as adaptive control, analysis and prediction of non-stationary time series [1], or signal source tracking (radar,sonar, telecoms) [2]. As is well known, the optimal estimatorfor this problem cannot be built in general, and it is necessaryto turn to one of the large number of existing suboptimal fil-tering techniques [1][2]. Assessing the achievable performancemay be difficult, and we have to resort to simulations and com-paring proximity to Bayesian lower bounds corresponding tooptimum performance [2]. Actually, most discrete-time MDSincorporate some deterministic parameters which can be eitherknown or unknown according to the experimental conditions[2]. Even when the deterministic parameters are known, someof the true values may originate from a prior calibrationprocess which accuracy impacts on the optimum performanceof random parameter estimates. In both cases, there is a needfor computationally tractable hybrid lower bounds for discrete-time MDS depending on unknown deterministic parameters.The first hybrid lower bound, the so-called hybrid Crame´r-Rao bound (HCRB), has been introduced in [3], extended in[4] to the case where the prior probability density function(pdf) of the random parameters depends on deterministicparameters, and its asymptotic tightness has been furtheranalyzed in [5]. This characterization of hybrid estimation hasThis work has been partially supported by the European Networkof excellence NEWCOM#, by the iCODE institute, research projectof the IDEX Paris-Saclay, by the DGA/DGCIS, by the DGA/MRIS(2015.60.0090.00.470.75.01) and by the Display-Mastodons project of CNRS.been generalized by Reuven and Messer [6] who introducedthe first ”large-error” hybrid bound, the so-called hybridBarankin Bound (HBB), in order to handle the threshold phe-nomena and of which one limiting form yields the HCRB. Thisseminal work [6] has been lately extended to new ”large-error”hybrid bounds [7][8][9] in order to improve the estimation ofthe transition region where the threshold phenomena occurs.Unfortunately, the computational cost of hybrid ”large-error”bounds is prohibitive in most applications when the numberof unknown parameters increases. All these works have shownthat, like the deterministic CRB and Bayesian CRB (BCRB),the HCRB is valid in the asymptotic region only, i.e., whensignal to noise ratio is high or the number of observations(measurements) is large.Therefore, provided that one keeps in mind its limitations,the HCRB is a lower bound of great interest for systemanalysis and design in the asymptotic region. For that pur-pose, computationally tractable HCRB for discrete-time MDSdepending on unknown time invariant deterministic parametershas been released only lately in [10] and extended in [11].However in many applications (radar, sonar, telecoms, ...) theunknown deterministic parameters of the measurement modelare time variant which prevents from using the bounds derivedin [11]. It is therefore the aim of this communication totackle this issue by introducing new computationally tractableHCRB.II. BACKGROUND ON HCRB FOR MDS WITH UNKNOWNDETERMINISTIC PARAMETERSIn hybrid parameter estimation one wishes to estimatean unknown hybrid parameter vector (x;θ) from a randomobservation vector y ∈ RN ′ , where for L column vectorsal, (a1; a2; . . . ;aL) ! (aT1 , aT2 , . . . , aTL)T denotes the verticalconcatenation. Some prior knowledge is available on randomparameter x ∈ RP ′ that is incorporated by an a prioripdf p (x) which support is a subset Πr of RP′. No suchknowledge is available on θ ∈ Πd ⊂ RD′ and thus itis considered deterministic. In the general case, p (x) maydepend on the unknown parameter θ, and it is denoted p (x|θ).The conditional pdf of y given x parameterized by θ isp (y|x,θ) and their joint pdf parameterized by θ is given byp (y,x|θ) = p (y|x,θ) p (x|θ). Then, for any estimators θ̂ (y)of θ and x̂ (y) of x, the HCRB is given by [3][4][5][11]:Ey,x|θ[e (y) e (y)T]≽ HCRB(x;θ) = J−1(x;θ), (1)J(x;θ) = Ey,x|θ[∂ ln p (y,x|θ)∂ (x;θ)∂ ln p (y,x|θ)∂ (x;θ)T], (2)where e (y) = (x̂ (y) − x; θ̂ (y) − θ), Ey,x|θ [g (y,x)] isthe statistical expectation of the vector of functions g ( )with respect to y and x parameterized by θ, and for twomatrices, A ≽ B means that A−B is positive semi-definite.The regularity conditions for the hybrid Fisher informationmatrix (HFIM) J(x;θ) to be of the usual form (2) are [11]:(R1): Πr = RP′,(R2): Ey,x|θ[∂ ln p(y,x|θ)∂xp′2], Ey,x|θ[∂ ln p(y,x|θ)∂θd′2]<∞.Moreover, under its usual form (1), the HCRB is a lower boundfor the class of estimates satisfying (R3):Ey,x|θ[(x̂ (y)− x; θ̂ (y)− θ)]= (µ;0).Our main concern is the derivation of a computationallytractable HFIM (2) for hybrid discrete-time MDS representedwith the state and measurement equations:xk = fk−1 (xk−1,wk−1,α) , yk = hk (xk,vk,λk) (3)where k ≥ 1 is a time index, xk is the P -dimensionalstate vector, yk is the N -dimensional measurement vector,fk ( , ,α) and hk ( , ,λk) are known parametric vector func-tions depending on an unknown deterministic parameter vector(α and λk respectively). The process noise sequence {wk}and the measurement noise sequence {vk} are mutually inde-pendent white sequences described by known pdfs p (wk|β)and p (vk|µk), respectively, depending on an unknown deter-ministic parameter vector (β and µk respectively). The noisesare independent of the initial state x0 described by the knownpdf p (x0|α).Let adopt the notational convention: ∀l ≤ k, δl:k =(δl; . . . ; δk) where all vectors are of same dimension. At timeindexes 1, 2, . . . , k the dependency of the state and measure-ment on deterministic unknown parameters is as follows:∣∣∣∣ x1 ! x1 (α,β) = f0 (x0,w0 (β) ,α)y1 ! y1 (λ1,µ1,α,β) = h1 (x1 (α,β) ,v1 (µ1) ,λ1) ,∣∣∣∣ x2 ! x2 (α,β) = f1 (x1 (α,β) ,w1 (β) ,α)y2 ! y2 (λ2,µ2,α,β) = h2 (x2 (α,β) ,v2 (µ2) ,λ2) ,...∣∣∣∣ xk ! xk (α,β) = fk−1 (xk−1 (α,β) ,wk−1 (β) ,α)yk ! yk (λk,µk,α,β) = hk (xk (α,β) ,vk (µk) ,λk) .(4)Therefore, the dependency of the state and measurement ondeterministic unknown parameters on a horizon of k pointsfrom the first measurement is as follows:k ≥ 1 :∣∣∣∣∣ x0:k ! x0:k (α,β) ! x0:k (θ0)y1:k ! y1:k (λ1:k,µ1:k,α,β) ! y1:k (θk) , (5)θ0 = (α;β) , k ≥ 1 : θk =(λ1:k;µ1:k;θ0), (6)where θk is the vector gathering all the unknown deterministicparameters. The state transition and the measurement pdfsdepend on unknown deterministic parameters:p (xk|xk−1) ! p (xk|xk−1,θ0) , p (yk|xk) ! p (yk|xk,λk,µk,θ0) ,(7)and we suppose that both p (xk|xk−1,θ0) andp (yk|xk,λk,µk,θ0) are twice differentiable with respect toall their arguments. Since (4) is a MDS:p (y1:k,x0:k|θk) = p (x0|α)k∏l=1p (yl|xl,λl,µl,θ0) p (xl|xl−1,θ0)(8)From a theoretical point of view, we are primarily interestedin the HFIM (2) on (x;θ) ! (xk;θk) associated to themeasurement vector y ! y1:k resulting from the set of kmeasurements y1,...,yk:J(xk;θk) = Ey1:k,xk|θk[∂ ln p (y1:k,xk|θk)∂ (xk;θk)∂ ln p (y1:k,xk|θk)∂ (xk;θk)T]which alternative formula is [3][4][5][6]:J(xk;θk) = Ey1:k,xk|θk[−∂2 ln p (y1:k,xk|θk)∂ (xk;θk) ∂ (xk;θk)T](9)Unfortunately the computation of (9) requires the deriva-tion of the marginal pdf p (y1:k,xk|θk) from (8) which isgenerally mathematically intractable [2][5][12]. However anupper bound of (9) can be derived from the HFIM (2) on(x;θk) ! (x0:k;θk) associated to the measurement vectory ! y1:k [11]:J(x0:k;θk) = Ey1:k,x0:k|θk[−∂2 ln p (y1:k,x0:k|θk)∂ (x0:k;θk) ∂ (x0:k;θk)T].(10)Indeed, if we set J(x0:k;θk) =[J11k BkBTk Ck]where:J11k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂x0:k−1∂xT0:k−1]Bk = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂x0:k−1∂(xk;θk)T]Ck = Ey1:k,x0:k|θk[− ∂2 ln p(y1:k,xk|θk)∂(xk;θk)∂(xk;θk)T] ,then [11]:J˜(xk;θk) ≽ J(xk;θk) ⇔ HCRB(xk;θk) ≽ J˜−1(xk;θk), (11)J˜(xk;θk) = Ck −BTk(J11k)−1Bk. (12)J˜−1(xk;θk) defines a looser (in comparison with HCRB(xk;θk))but general computable hybrid bound for discrete-time MDSpdf (8). However the computation of J˜−1(xk;θk) (11) maybecome rapidly computationally prohibitive and unstable sinceJ11k and J˜(xk;θk) are matrices with unbounded dimension asthe time index k increases (a (PK × PK) matrix and a((P + dimθk)× (P + dimθk)) matrix, respectively). Hencethe need of a recursive form involving matrices with boundeddimensions.III. A NEW RECURSIVE HCRB FOR MDS WITHTIME-VARIANT MEASUREMENT PARAMETERSFrom a practical point of view, in estimation problem such(3)(4), at each time index k, the vector of parameters of interestis rather (xk;ωk), ωk = (λk;µk;α;β), than (xk;θk). In thatperspective, we can reparameterize (x0:k;θk) as:(z0:kθ0),∣∣∣∣ k = 0 : z0 = x0k ≥ 1 : zk = (xk;λk;µk) , (13)and rearrange (xk;θk) as:vk =(γk−1; zk;θ0), γk−1 =(λ1:k−1;µ1:k−1). (14)Then:HCRBvk =[HCRBγk−1vk HCRBγk−1,(zk;θ0)vkHCRB(zk;θ0),γk−1vk HCRB(zk;θ0)vk]and:HCRB(xk;ωk)(xk;θk) = HCRB(zk;θ0)vk ≽ J˜−1(zk;θ0) (15)where J˜(zk;θ0) can be decomposed into block matrices:J˜(zk;θ0) =[J˜zk,zkk J˜zk,θ0kJ˜θ0,zk J˜θ0,θ0k](16)which obey the recursion:∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣J˜zk,zkk = D22k−1 −(D12k−1)T (D11k−1 + J˜zk−1,zk−1k−1)−1D12k−1J˜zk,θ0k = D23k−1 −(D12k−1)T (D11k−1 + J˜zk−1,zk−1k−1)−1×(D13k−1 + J˜zk−1,θ0k−1)J˜θ0,θ0k = D33k−1 + J˜θ0,θ0k−1 −(D13k−1 + J˜zk−1,θ0k−1)T×(D11k−1 + J˜zk−1,zk−1k−1)−1 (D13k−1 + J˜zk−1,θ0k−1)(17)where:∣∣∣∣∣∣∣∣∣∣∣∣∣D11k−1 = Exk,xk−1|θ0[−∂2 ln p(xk|xk−1,θ0)∂zk−1∂zTk−1]D12k−1 = Exk,xk−1|θ0[−∂2 ln p(xk|xk−1,θ0)∂zk−1∂zTk]D22k−1 = Eyk,xk|λk,µk,θ0[−∂2 ln p(yk|xk,λk,µk,θ0)∂zk∂zTk]+ Exk,xk−1|θ0[−∂2 ln p(xk|xk−1,θ0)∂zk∂zTk](18)∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣D13k−1 = Exk,xk−1|θ0[−∂2 ln p(xk|xk−1,θ0)∂zk−1∂θT0]D23k−1 = Eyk,xk|λk,µk,θ0[−∂2 ln p(yk|xk,λk,µk,θ0)∂zk∂θT0]+ Exk,xk−1|θ0[−∂2 ln p(xk|xk−1,θ0)∂zk∂θT0]D33k−1 = Eyk,xk|λk,µk,θ0[−∂2 ln p(yk|xk,λk,µk,θ0)∂θ0∂θT0]+ Exk,xk−1|θ0[−∂2 ln p(xk|xk−1,θ0)∂θ0∂θT0](19)J˜(z0;θ0) = J(z0;θ0) = Ex0|θ0[−∂2 ln p (x0|α)∂ (x0;θ0) ∂ (x0;θ0)T](20)Proof:The first step consists in reformulating (11)(12) after therearrangement (14) leading to:J˜vk ≽ Jvk , J˜vk = Ck −BTk(J11k)−1Bk, (21)J11k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂x0:k−1∂xT0:k−1]Bk = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂x0:k−1∂vTk]Ck = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,xk|θk)∂vk∂vTk]Let uk =(x0:k−1;γk−1; zk;θ0)! (x0:k−1;vk), that is(x0:k;θk) after rearrangement, then:Juk = Ey1:k,x0:k|θk[−∂2 ln p (y1:k,x0:k|θk)∂uk∂uTk]=[J11k BkBTk Ck].Therefore, using block matrix inversion:HCRBuk = J−1uk =[HCRBx0:k−1uk HCRBx0:k−1,vkukHCRBvk,x0:k−1uk HCRBvkuk],HCRBvkuk =(Ck −BTk(J11k)−1Bk)−1= J˜−1vk .Additionally, according to (13)(14):z0:k−1 = P0:k−1(x0:k−1γk−1)⇒⎛⎝ z0:k−1( zkθ0) ⎞⎠ = [ P0:k−1 00 I]ukwhere P0:k is a permutation matrix; therefore:HCRBuk =[PT0:k−1 00 I]HCRB(z0:k;θ0)[P0:k−1 00 I],leading to the key identity:HCRB(zk;θ0)uk = HCRB(zk;θ0)(z0:k;θ0). (22)Indeed, (21) yields:J−1vk ≽ J˜−1vk ⇔ HCRBvk ≽ HCRBvkuk ,that is, in particular:HCRB(zk;θ0)vk ≽ HCRB(zk;θ0)uk = HCRB(zk;θ0)(z0:k;θ0). (23)The second step consists in showing thatJ˜(zk;θ0) =(HCRB(zk;θ0)(z0:k;θ0))−1=[J˜zk,zkk J˜zk,θ0kJ˜θ0,zkk J˜θ0,θ0k]obey the recursion defined by (17-20). First J(z0:k;θ0) can bebroken down as:J(z0:k;θ0) =⎡⎢⎣ J11k J12k J13k(J12k)TJ22k J23k(J13k)T (J23k)TJ33k⎤⎥⎦ ,J11k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂z0:k−1∂zT0:k−1],J12k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂z0:k−1∂zTk],J13k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂z0:k−1∂θT0],J22k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂zk∂zTk],J23k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂zk∂θT0],J33k = Ey1:k,x0:k|θk[−∂2 ln p(y1:k,x0:k|θk)∂θT0 ∂θT0].Therefore, using block matrix inversion:J˜(zk;θ0) =[J22k J23kJ32k J33k]−[J21kJ31k] [J11k]−1 [J12k J13k]that is: ∣∣∣∣∣∣∣∣J˜zk,zkk = J22k − J21k(J11k−1)−1J12k ,J˜θ0,zk = J23k − J21k(J11k)−1J13k ,J˜θ0,θ0k = J33k − J31k(J11k)−1J13k .(24)Moreover, for MDS, (8) leads to:p (y1:k,x0:k|θk) = p (yk|xk,λk,µk,θ0) p (xk|xk−1,θ0)× p (y1:k−1,x0:k−1|θk−1)yielding:J11k =[J11k−1 J12k−1J21k−1 J22k−1 +D11k−1], J12k =[0D12k−1],J22k = D22k−1, J13k =[J13k−1J23k−1 +D13k−1],J23k = D23k−1, J33k = J33k−1 +D33k−1,where Dijk−1 are given by (18-19). Last, using once againblock matrix inversion:(J11k)−1=[Φk−1 −Γk−1∆−1k−1−∆−1k−1ΓTk−1 ∆−1k−1],∆k−1 = D11k−1 + J22k−1 − J21k−1(J11k−1)−1J12k−1,Γk−1 =(J11k−1)−1J12k−1,Φk−1 =(J11k−1)−1+ Γk−1∆−1k−1ΓTk−1.Then noting that∆k−1 = D11k−1+J˜zk−1,zk−1k−1 , a few additionallines of calculus allows to show that equivalent forms ofJ˜zk,zkk , J˜θ0,θ0k and J˜θ0,zk in (24) are given by (17).IV. FURTHER CONSIDERATIONSFirst, some special cases of interest can be easily derivedby updating the definitions of θk, p (xk|xk−1), p (yk|xk) andp (x0|α) accordingly.A first case of interest is θk = θ0, i.e. the measurementsdo not depend on deterministic parameters, then zk = xk,p (xk|xk−1) ! p (xk|xk−1,θ0), p (yk|xk) ! p (yk|xk,θ0),and (15-16) reduce to:HCRB(xk;θ0) ≽ J˜−1(xk;θ0), J˜(xk;θ0) =[J˜xk,xkk J˜xk,θ0kJ˜θ0,xkk J˜θ0,θ0k],which are [11, (7)(10)] where θ ! (α;β), and (17-20) reduceto [11, (11-14)].A second case of interest is θk = ∅, i.e. there is no un-known deterministic parameter, then zk = xk, p (xk|xk−1) !p (xk|xk−1), p (yk|xk) ! p (yk|x), and (15-16) and (17-20)reduce to:HCRBxk ≽ J˜−1xk , J˜xk = J˜xk,xkk ,J˜xk,xkk = D22k−1 −(D12k−1)T (D11k−1 + J˜xk−1,xk−1k−1)−1D12k−1,D11k−1 = Exk,xk−1[−∂2 ln p(xk|xk−1)∂xk−1∂xTk−1],D12k−1 = Exk,xk−1[−∂2 ln p(xk|xk−1)∂xk−1∂xTk],D22k−1 = Eyk,xk[−∂2 ln p(yk|xk)∂xk∂xTk]+ Exk,xk−1[−∂2 ln p(xk|xk−1)∂xk∂xTk]and J˜(z0;θ0) ! Jx0 , which are (21-25) in [12].A third case of interest is θ0 = ∅, i.e. the state doesnot depend on deterministic parameters, then p (xk|xk−1) !p (xk|xk−1), p (yk|xk) ! p (yk|zk), and (15-16) and (17-20)reduce to:HCRBzk ≽ J˜−1zk , J˜zk = J˜zk,zkk ,J˜zk,zkk = D22k−1 −(D12k−1)T (D11k−1 + J˜zk−1,zk−1k−1)−1D12k−1,D11k−1 = Exk,xk−1[−∂2 ln p(xk|xk−1)∂zk−1∂zTk−1],D12k−1 = Exk,xk−1[−∂2 ln p(xk|xk−1)∂zk−1∂zTk],D22k−1 = Eyk,xk|λk,µk[−∂2 ln p(yk|zk)∂zk∂zTk]+Eyk,xk|λk,µk[−∂2 ln p(xk|xk−1)∂zk∂zTk]and J˜z0 ! Jx0 which is a generalization of [12].Second, it is worth noting that, at each time index k, there isno particular requirements on the dimension of λk or µk; inparticular it is possible to have λl = ∅ and/or µl = ∅ for asubset of time index l ∈ {l1,l2, . . . , lJ ≤ k}, what means thatthe new HCRB (15) allows to take into account a mixture ofmeasurements where the deterministic parameters are eitherknown or unknown (which is not possible with the HCRBderived in [11]).REFERENCES[1] D. 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Tabrikian, “Hybrid lower bound via compression ofthe sampled CLR function,” Proc. of IEEE Workshop on SSP 2009.[8] C. Ren, J. Galy, E. Chaumette, P. Larzabal, and A. Renaux, “Hybridlower bound on the MSE based on the Barankin and Weiss-Weinsteinbounds,” in Proc. of IEEE ICASSP 2013.[9] C. Ren, J. Galy, E. Chaumette, P. Larzabal, and A. Renaux, “A Ziv-Zakaı¨ type bound for hybrid parameter estimation,” in Proc. of IEEEICASSP 2014.[10] L. Xu and X. Rong Li, “Hybrid Cramer–Rao Lower Bound on TrackingGround Moving Extended Target,” in Proc. of IEEE ISIF 2009.[11] C. Ren, J. Galy, E. Chaumette, F. Vincent, P. Larzabal, A. Renaux,”Recursive hybrid Crame´r-Rao bound for Discrete-Time MarkovianDynamic System”, IEEE SP Letters, 22(10): 1543-1547, 2015.[12] P. Tichavsky, C. Muravchik and A. Nehorai, ”Posterior Crame´r-Raobounds for discrete-time nonlinear Filtering”, IEEE Trans. on SP,46(5):1386-1396, 1998.

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Recursive hybrid CRB for Markovian systems with time-variant measurement parameters

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