Computationally efficient PCRLB for tracking in cluttered environments: measurement existence conditioning approach

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Published in IET Signal ProcessingReceived on 12th March 2008Revised on 29th July 2008doi: 10.1049/iet-spr:20080040

ISSN 1751-9675

Computationally efficient PCRLB for trackingin cluttered environments: measurementexistence conditioning approachH. Meng1 M.L. Hernandez2 Y. Liu3 X. Wang1

1Department of Electronic Engineering, Tsinghua University, Beijing 100084, People’s Republic of China2Intelligence, Communications and Security (ICS) Department, QinetiQ Ltd., St Andrew’s Road, Malvern WR14 3PS, UK3SVA Communication Technology Co., LTD, People’s Republic of ChinaE-mail: [email protected]

Abstract: In this paper, we consider the problem of calculating the posterior Cramer–Rao lower bound (PCRLB)for tracking in cluttered domains in which there can be both missed detections and false alarms. We introduce anovel approach, whereby we condition on the ‘existence sequence’, which is a sequence of zeros and onesdepending on whether at least one measurement exists at each sampling time. An existing Riccati-likerecursion then provides a PCRLB conditional on each existence sequence, and an unconditional PCRLB iscalculated as a weighted average of these conditional bounds. This new approach is referred to as‘measurement existence sequence conditioning’ (MESC). The MESC approach is compared with both theinformation reduction factor (IRF) approach and measurement sequence conditioning (MSC) approach. It isproved that the MESC approach provides a less optimistic bound than the IRF approach. This is a desirableproperty, as it shows that the MESC bound is more realistic than the IRF bound. It is also shown that theMESC bound provides a more optimistic bound than the MSC approach. Although this is undesirable, in thesimulations differences between the MESC and MSC bounds are very small (typically less than 5%). Thissuggests that the key reason for the over-optimism of the IRF bound is the fact that it does not take intoaccount the effect of missed detections. Although the MESC approach treats cases with one or moredetection differently to the MSC approach, the similarity between these two bounds suggests thatdiscriminating between such cases is of less importance. However, the greatest value of the new MESCapproach is that the bound can be enumerated precisely, without the need for inefficient and computationallyexpensive sampling. In case studies, we show that the MESC bound can be calculated 10 – 100 times morequickly than the MSC bound. It is concluded that the novel MESC formulation introduced herein representsan exciting development in the determination of the PCRLB in cluttered environments.

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1 IntroductionThe posterior Cramer–Rao lower bound (PCRLB) [1]provides a mean square error bound on the performance ofany unbiased estimator of an unknown and stochasticparameter vector. In the context of target tracking, thePCRLB (often referred to as the ‘Bayesian’ PCRLB [1]provides a powerful tool, enabling one to determine abound on the optimal achievable accuracy of target stateestimation. This provides a mechanism for establishing themaximum degree to which sub-optimal filtering algorithms

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could potentially be improved (e.g. [2]), and thereforeprovides an indication as to where future algorithmicdevelopment should be focused. The PCRLB also providesa means of establishing limitations in current sensortechnology, enabling one to easily determine ‘impossible’scenarios (i.e. scenarios in which operational requirementscannot possibly be met) given current sensor capabilities.

Recent interest in the PCRLB is primarily as a result of theexcellent paper [3], in which an efficient and generalformulation of the bound was derived. This derivation

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exploited the fact that the PCRLB for the estimation of thetarget state at sampling time k could be expressed in terms ofthe PCRLB for the estimation of the target state at samplingtime k 2 1. The computational simplicity of this formulationhas led to the PCRLB being utilised as a predictive measureof system performance in sensor management applications.For example, it has been used in automating thedeployment of passive sonobuoys in anti-submarine warfare[4], and in controlling a large network of bandwidth-limited sensors in multi-target tracking [5]. Moreover, thePCRLB has recently been used as the basis for performingreal-time UAV trajectory planning in bearings-onlytracking [6, 7].

The focus of the current paper is the calculation of thePCRLB for the nonlinear filtering problem [8] in whichthere are both false positives and missed detections. Weconcentrate on the single target case, for which there has beensignificant interest in the last decade. In [9], for the problemwith deterministic (i.e. not randomly evolving) targetdynamics, the measurement origin uncertainty was shown tomanifest itself as a constant information reduction factor(IRF). This IRF scaled the measurement contribution, suchthat the Cramer–Rao bound increased in magnitude (i.e.optimal performance would be degraded) as the environmentbecame more cluttered (i.e. as the number of false positivesincreased). The paper [10] (see also [11]) then derived ageneral expression for the PCRLB for the linear filteringproblem with randomly evolving target dynamical equations.The paper [12] extended [10] to provide a PCRLB for thenonlinear filtering problem. The methodology detailed in thepapers [10, 12] is referred to as the ‘IRF approach’.

However, it has recently been shown that the IRF approachprovides an optimistic bound on tracker performance [13, 14].This is primarily because the IRF approach does not take intoaccount the effect of missed detections, which can have aconsiderable impact on performance, particularly during theearly stages of tracking. In [14], an enumeration approachwas introduced for the case with missed detections (i.e.Pd � 1) but no false alarms (i.e. Pfa ¼ 0). This enumerationapproach created a bound conditional on the sequence ofdetections and missed detections, and then created anunconditional bound as a weighted average. It was thenshown in [15] that the enumeration bound was greater thanthe IRF bound, and therefore less optimistic. The paper [13]generalised the enumeration bound of [14] to the case inwhich there were also false alarms (i.e. Pd � 1 and Pfa � 0)by introducing the ‘measurement sequence conditioning’(MSC) approach. The MSC approach created a boundconditional on the number of measurements at eachsampling time, and then created an unconditional bound as aweighted average (weighted by the probability of eachparticular sequence occurring). It was then shown (in [13])that the MSC bound was also less optimistic than thecorresponding IRF bound, which generalised the finding of[15] to the case in which Pfa � 0. In simulations, it was thenshown that differences between the two bounds are greatest

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initially, when the impact of the actual measurementsequence is most significant.

The one major limitation of the MSC approach is thatcalculating the bound is computationally demanding, whichtherefore limits the usefulness of the technique in time-critical applications (such as in performing real-time sensormanagement). This is because it is typically not possible tocalculate the MSC bound precisely, as the number ofpotential measurement sequences is infinite for Pfa . 0. Oneis then forced to create an approximation by either:(i) excluding sequences that have a low probability ofoccurrence [14], or (ii) sampling a number of potentialmeasurement sequences using the known distribution ofmeasurements [13]. Method (ii) is the easiest to implement,but it is unclear how many sequences should be sampled inorder to accurately estimate the MSC bound. Moreover,method (ii) is inefficient, as sequences that occur withrelatively high probability are sampled on more than oneoccasion. In the case with no false alarms, it has recently beenshown in [16, 17] that a modified Riccati (MR) equationprovides an upper bound on the enumeration bound [14], butwith a much lower computational overhead. The MRapproach has a computational complexity similar to that ofthe IRF approach [10]. In simulations, the MR equation wasshown to bound the enumeration PCRLB reasonably tightly,making it an effective measure of system performance.However, the MR equation cannot be guaranteed to providea bound on tracker performance, and can only be applied inthe case of Pfa ¼ 0.

Motivated by the computational issues regarding the MSCbound, in this paper we again consider the problem ofcalculating a PCRLB for the general discrete-time,nonlinear filtering problem with Pd � 1 and Pfa � 0.We introduce a novel approach in which we firstdetermine an ‘existence sequence’, which specifies whetherat least one measurement exists at each sampling time. Ifno measurements exist, then as in the MSC approach, themeasurement contribution to the PCRLB is zero.However, if at least one measurement exists, themeasurement contribution is equal to that used in the IRFapproach [10, 12] multiplied by a scale factor that is greaterthan or equal to unity. An unconditional PCRLB is thencalculated as a weighted average of the bounds given byeach existence sequence, where the weight factor is theprobability of each sequence occurring. This approach isreferred to herein as ‘measurement existence sequenceconditioning’ (MESC). The beauty of this technique is thatit contains the key feature of the MSC approach – that themeasurement contribution is zero if no measurements areavailable. Furthermore, because each existence sequenceconsists of zeros and ones, the total number of sequences(given a fixed number of time steps) is finite and calculable.

We also prove that the MESC approach is less optimisticthan the IRF approach. This is a desirable property, as itshows that the MESC bound is more realistic than the

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IRF bound. We also show that the MESC bound is moreoptimistic than the MSC bound. Although this isundesirable, in our simulations differences between theMESC and MSC bounds are very small (typically less than5%). This suggests that the key reason for the over-optimism of the IRF bound is the fact that it does not takeinto account the effect of missed detections. Although theMESC approach treats cases with one or more detectiondifferently to the MSC approach, the similarity betweenthese two bounds suggests that discriminating betweensuch cases is of less importance. However, the greatestvalue of the new MESC approach is that the bound can beenumerated precisely, without the need for inefficient andcomputationally expensive sampling. In case studies, weshow that the MESC bound can be calculated 10–100times more quickly than the MSC bound.

The remainder of this article is structured as follows. InSection 2, the PCRLB is defined, and we present theefficient recursion of [3] that is central to all clutteredenvironment PCRLBs. In Section 3, we then detail theIRF (e.g. [10]) and MSC [13] approaches. In Section 4,we introduce the novel new MESC approach, and inSection 5 we examine the relationship between the threebounds. Following this, in Section 6 we describe thesimulation scenario (which is a variant of a scenario definedin [13]). In Section 7 we present simulation results, whichinclude: (i) comparisons between the various bounds, (ii) aqualitative assessment of the differences between theMESC and MSC bounds, (iii) investigations into theconvergence of the MSC approach and (iv) a comparisonbetween the computational costs of calculating the MESCand MSC bounds. Following this, in Section 8 we discussthe circumstances in which each of the three performancebounds should be preferred. Conclusions are given inSection 9. For completeness, in Appendix 1 we present ageneral expression for the IRF, taken from [13]. Finally, inAppendices 2 and 3 we present proofs of several key resultsregarding the relationship between the various bounds.

2 Posterior Cramer–Raolower bound2.1 Notation

Throughout this paper, ‘N (m, C)’ is used to denote themultivariate Gaussian distribution with mean m andcovariance C; ‘Ef’ will denote mathematical expectationwith respect to f; ‘p(E)’ will denote either: (i) theprobability of the event E, or: (ii) the probability massfunction of the random variable E; a matrix A is positive-

definite, and denoted by A . 0, if and only if aT Aa . 0for all non-zero vectors a.

2.2 Definition

Suppose Xk is the (unknown and random) target state atsampling time k. Let X k denote any unbiased estimator of

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Xk based on the measurements Z1:k ¼ Z1, . . . , Zk

� �available at discrete sampling times 1, . . . , k. The PCRLBfor the estimation of Xk given measurements Z1:k is definedas follows [1]

Ck W E Xk � X k

� �Xk � X k

� �T� �

� J�1k (1)

Jk is the Fisher information matrix (FIM). The inequality in(1) means that the difference Ck � J�1

k is a non-negativedefinite matrix.

2.3 Computationally efficient formulation

Consider the discrete-time, nonlinear filtering problem withadditive Gaussian process and measurement noise. Thetarget state evolution is given by

Xkþ1 ¼ fk(Xk)þ vk (2)

where fk(�) is a nonlinear function, which models the stateevolution in time, and vk � N (0, Sk). A is non-negativedefinite (denoted by A � 0) if and only if aT Aa � 0 for allnon-zero vectors a. The measurement sequence Z1:k will bedescribed in Section 3.1.

The FIM, Jkþ1 at sampling time kþ 1 is given by thefollowing recursion [3]

Jkþ1 ¼ D33k � D12

k

� TJk þD11

k

��1D12

k þ JZ(kþ 1) (3)

where

D11k ¼ EXk,Xkþ1

rXk

ln p(Xkþ1jXk)�h

�rXk

ln p(Xkþ1jXk)�Ti

(4)

D12k ¼ EXk,Xkþ1

rXk

ln p(Xkþ1jXk)�h

�rXkþ1

ln p(Xkþ1jXk)�Ti

(5)

D33k ¼ EXk,Xkþ1

rXkþ1

ln p(Xkþ1jXk)�h

�rXkþ1

ln p(Xkþ1jXk)�Ti

(6)

and

JZ(kþ 1) ¼ EXkþ1,Zkþ1

rXkþ1

ln p(Zkþ1jXkþ1)�h

�rXkþ1

ln p(Zkþ1jXkþ1)�Ti

(7)

rz is a first-order partial derivative operator with respect tothe parameter vector z. The recursive formula is initialisedwith J0 ¼ C�1

0 , where C0 is the target state covariance aftertrack initialisation.

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If the target dynamics are linear (i.e. fk(Xk) ¼ AkXk), thenit is straightforward to show (e.g. [12]) that

Jkþ1 ¼ Sk þ Ak J�1k AT

k

� �1þJZ(kþ 1) (8)

Equation (8) also holds if there is no process noise (i.e.Sk ¼ 0) [18].

3 Established PCRLB approachesin cluttered environments3.1 Measurement equations andfalse alarms

Purely for brevity we will consider a single sensor scenario.The extension to multi-sensor systems is thenstraightforward, provided that, conditional on the targetstate, measurements are independent. Let mk denote thenumber of measurements available at this sensor at time k– which can include both measurements generated by thetarget and false alarms. A target-generated measurement, attime k, has a Gaussian distribution with mean hk(Xk) andvariance Rk. False alarms have a uniform distributionthroughout the hyper-region A, with associated hyper-volume V under surveillance.

Hence, target-generated measurements (indexed with‘TG’) and false alarm measurements (indexed with ‘FA’)are given by

ZTGk ¼ hk(Xk)þ 1k with 1k � N (0, Rk)

ZFAk ¼ uk with p(uk) ¼

1

Vfor uk [ A

(9)

3.2 Information reduction factorapproach

The IRF approach [10, 12] formulates the PCRLB as follows

PCRLB(IRF ; k) ¼ J�1k (10)

where to remind the reader, the FIM, Jk is given by therecursion (3). This then reduces to recursion (8) if thetarget dynamics are linear.

The difference between the various PCRLB approaches isin the way in which the measurement contribution JZ(k) (see(7)) is dealt with. In the IRF approach, the measurementcontribution is given by

JZ(k) ¼ EmkJZ(k : mk) �

(11)

¼Xmk

p(mk)JZ(k : mk) (12)

where p(mk) is the probability mass function of mk, andJZ(k : mk) is the measurement contribution at sampling time

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k given that there are mk measurements at that time. Thisis given as follows

JZ(k : mk) ¼ EXk ,Zk

�rXk

ln p(ZkjXk, mk)�

�rXk

ln p(ZkjXk, mk)�T�

(13)

After some calculations [12, 13], it can be shown that

JZ(k) ¼ qkEXkH T

k R�1k Hk

�ð14Þ

where H Tk ¼ rXk

hk(Xk)T

h iis the Jacobian of the

measurement vector, and

qk ¼X1mk¼1

p(mk)qk(mk)

" #(15)

qk(mk) [ [0, 1] is the IRF conditional, there being mk

measurements at sampling time k [13] and qk [ [0, 1] is theoverall IRF [12]. These IRFs are independent of the targetstate, and quantify the effect of missed detections and falsealarms. If there is no clutter and there are no misseddetections, then qk ¼ 1. In all other cases, qk , 1 and theIRF reduces the measurement contribution (compared withthe no clutter case), and therefore increases the PCRLB.

Of particular note, if the measurement, clutter and detectionmodels are time invariant then so too is the IRF qk(mk), whichneed then only be calculated once. This calculation can beperformed offline, enabling the PCRLB to be calculatedefficiently, and implemented in real-time applications [4]. Forcompleteness, in Appendix 1 we present a general expressionfor the IRF qk(mk) taken from [13] (see (62)). We alsopresent a numerical approximation (see (68)).

3.3 Measurement sequence conditioningapproach

In the MSC approach, the PCRLB is formulated as (see[13])

PCRLB(MSC ; k) W Em1:kJ�1

k (m1:k) �

(16)

¼Xm1:k

p(m1:k)� J�1k (m1:k)

�(17)

where Jk(m1:k) is the FIM conditional on the measurementsequence m1:k ¼ m1, . . . , mk

� �and p(m1:k) is the

probability of the measurement sequence m1:k occurring.

Repeating the derivation of the recursive formula (3),conditional on m1:k, it is easily shown that

Jk(m1:k) ¼ D33k�1 � D12

k�1

� T[Jk�1(m1:k�1)þD11

k�1]�1

�D12k�1 þ JZ(k : mk) (18)



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If the target dynamics are linear this reduces to

Jk(m1:k) ¼ Sk�1 þ Ak�1J�1k�1(m1:k�1)AT

k�1

� �1

þ JZ(k : mk) (19)

After some calculations (again, see [13]) it can then be shownthat

JZ(k : mk) ¼ qk(mk)EXkH T

k R�1k Hk

�(20)

where qk(mk) is again the IRF conditional there being mk

measurements at sampling time k. The recursive formula(18) is initialised with J0(:) ¼ C�1

0 , where C0 is the targetstate covariance after track initialisation.

4 Conditioning on the existenceof at least one measurementWe now introduce a new PCRLB in cluttered environments(with Pd � 1 and Pfa � 0) based on conditioning on theexistence of at least one measurement at each samplingtime. We will call this approach the MESC approach.

First, let us define ek as follows

ek W0 if mk ¼ 01 if mk � 1

�(21)

and let e1:k W e1, . . . , ek

� �.

Clearly, in the same way as the PCRLB was conditionedon m1:k in [13], we now condition on e1:k

E (Xk � Xk)(Xk � Xk)T

h i¼ Ee1:k

hE[(Xk � Xk)

� (Xk � Xk)Tje1:k]

i(22)

� Ee1:kJ�1

k (e1:k) �

(23)

where p(e1:k) is the probability of the existence sequence e1:k

occurring, and Jk(e1:k) is the FIM conditional on theexistence sequence e1:k.

We will then denote this new PCRLB as follows

PCRLB(MESC ; k) W Ee1:kJ�1

k (e1:k) �

(24)

¼Xe1:k

p(e1:k)� J�1k (e1:k)

�(25)

Repeating the derivation of the recursive formula (3),conditional this time on e1:k, it can easily be shown that

Jk(e1:k) ¼ D33k�1 � D12

k�1

� T[Jk�1(e1:k�1)þD11

k�1]�1

�D12k�1 þ JZ(k : ek) (26)



If the target dynamics are linear this reduces to

Jk(e1:k) ¼ Sk�1 þ Ak�1J�1k�1(e1:k�1)AT

k�1

� �1þ JZ(k : ek) (27)

Clearly, the measurement contribution is zero if there are nomeasurements:

JZ(k : ek ¼ 0) ¼ 0 (28)

In the case ek ¼ 1, we then have

JZ(k : ek ¼ 1) ¼ EXkþ1,Zkþ1

rXkþ1

ln p(Zkþ1jXkþ1)�h

�rXkþ1

ln p(Zkþ1jXkþ1)�Tjek ¼ 1

i(29)

¼X1mk¼1

p(mkjek ¼ 1)

� EXkþ1,Zkþ1

rXkþ1

ln p(Zkþ1jXkþ1, mk)�h

�rXkþ1

ln p(Zkþ1jXkþ1, mk)�Ti

(30)

¼X1mk¼1

p(mkjek ¼ 1)JZ(k : mk) (by (13)) (31)

¼ Emkjek¼1 JZ(k : mk) �

(32)

This can then be written as

JZ(k : ek ¼ 1)¼1

p(ek ¼ 1)

X1mk¼1

p(mk)JZ(k : mk) (33)

(using Bayes rule: p(mkjek ¼ 1)

¼p(ek ¼ 1jmk)p(mk)

p(ek ¼ 1)¼

p(mk)

p(ek ¼ 1)for mk � 1)

¼JZ(k)

p(ek ¼ 1)(using (12)) (34)

¼qk

p(ek ¼ 1)EXk

H Tk R�1

k Hk

�(using (14)) (35)

¼ ~qkEXkH T

k R�1k Hk

�(36)

where

~qk ¼qk

p(ek ¼ 1)(37)

Therefore if there are no missed detections (i.e. Pd ¼ 1), thenek ¼ 1 for all k; and p(ek ¼ 1)¼ 1. Clearly, in such cases theMESC approach reduces to the IRF approach.

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5 Relationships between thevarious performance bounds5.1 Relationship between IRF andMESC bounds

As we can see from the following result, the MESC bound isalso less optimistic than the IRF bound, therefore making it amore realistic bound on estimator performance.

Proposition 1: For the general case with the potential forboth missed detections (i.e. Pd � 1) and false alarms (i.e.Pfa � 0), provided mild regularity conditions hold, thefollowing result is true

J�1k � PCRLB(MESC; k) (38)

for all k, where J�1k ¼ PCRLB(IRF ; k) (see (10)).

Proof: This can be found in Appendix 2. A

5.2 Relationship between MESC andMSC bounds

The following result shows us that the MESC bound has theunappealing feature that it is more optimistic than the MSCbound. However, as we will see in later simulations (seeSection 7.1), differences between the MESC and MSCbounds are very small (typically less than 5%).

Proposition 2: For the general case with the potential forboth missed detections (i.e. Pd � 1) and false alarms (i.e.Pfa � 0), provided mild regularity conditions hold, thefollowing result is true

PCRLB(MESC ; k) � PCRLB(MSC; k) (39)

for all k.

Proof: Quite simply

PCRLB(MSC; k) ¼Xm1:k

p(m1:k)J�1k (m1:k) (40)

¼Xe1:k

p(e1:k)Xm1:k

p(m1:kje1:k)J�1k (m1:k) (41)

(using the total probability theorem)

�Xe1:k

p(e1:k)J�1k (e1:k) (42)

(using Proposition 4 given in

Appendix 3)

¼ PCRLB(MESC ; k) (43)

This completes the proof. A



5.3 Special cases

Again, we note that if there are no missed detections (i.e.Pd ¼ 1) then the IRF and MESC bounds are identical,that is,

PCRLB(IRF ; k) ; PCRLB(MESC; k) if Pd ¼ 1 (44)

and we have equality in Proposition 1.

Moreover, for Pd � 1, and Pfa ¼ 0, there can be amaximum of one measurement per sampling time (i.e.mk � 1). In this case, it is clear that ek ¼ mk for all k, andthe MESC and MSC approaches are identical. Theseapproaches are also then identical to the enumerationapproach of [14]. Therefore

PCRLB(MESC ; k) ; PCRLB(MSC; k) if Pfa ¼ 0 (45)

and we have equality in Proposition 2.

6 Simulation scenario: tracking anearly constant velocity target ina high clutter environmentThis scenario is a one-dimensional variant of the scenariopresented in Section X of [13]. The key feature of thisscenario is that the probability of detection is low comparedwith the false alarm rate, and was constructed in [13] inorder to show that large differences could exist between theMSC and the original IRF bounds [10]. This is becausethe IRF method does not take into account the fact thatwhen no measurements are received early in the scenario,this can have a huge detrimental effect on trackingperformance.

The target moves with nearly-constant velocity (NCV) in aone-dimensional space. The target state is expressed asXk ¼ (xk _xk)

T , where xk is the location of the target in theone-dimensional space and _xk is its velocity. The NCVtarget motion is then prescribed by

Xkþ1 ¼ AkXk þ vk (46)

where vk � N (0, Sk). Ak and Sk are given as follows

Ak ¼1 T0 1

�Sk ¼ ‘�

T 3

3

T 2

2T 2

2T

0BB@

1CCA (47)

where l is the power spectral density and T is the timebetween measurements (i.e. the scan duration).

Measurements are of the target location, therefore eachtarget-generated measurement is given by:

Zk ¼ xk þ ek with ek � N (0, s2x) (48)



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It is then relatively straightforward to show that

JZ(k : mk) ¼ qk(mk)

1

s2x

0

0 0

0@

1A (49)

As in previous papers (e.g. [10, 13]) it is assumed that (i)there is a maximum of one target-generated measurementper sampling time, which occurs with constant probabilityPd ; and (ii) at each sampling time the number of falsealarms has a Poisson distribution with mean l per unit areaof the surveillance region. Each IRF calculation considersonly the number of false alarms in a small gated region ofvolume Vg around the target. Hence, the average numberof false alarms considered is lVg per sampling time. TheIRF qk(mk) is then given by (62) in Appendix 1 (withn ¼ 1 and s1 ¼ sx), and approximated using (68).

Unless stated otherwise, the parameter settings used arethose given in Table 1.

7 Simulation results7.1 Calculating the various performancebounds

As an aid to the reader, in this section we summarise the stepsin the calculation of the three performance bounds. The IRFbound is the simplest of the three performance bounds, and iscalculated using the recursion (3). The IRF, qk used in the

Table 1 Parameter settings in the simulation scenario. Theunit of distance is m and unit of time is s. The unit of velocityis m/s

Parameter Value

sensor location at origin

initial (mean) target state, X0 (1000 210)T

initial target state covariance, C0 diag(10002, 102)

probability of detection, Pd 0.2 – unless statedotherwise

number of standard deviations ofmeasurement gate, g

4

average number of false alarms insurveillance region (¼lVg)

0.4 – unless statedotherwise

power spectral density (of NCVmotion), l

1.0 � 1028

measurement error standarddeviation, sx

2.5

sampling time, T 5

number of samples, Ni, in theapproximation of each IRF (see(68))

10 000



measurement contribution is then calculated using (15);qk(mk) is calculated using (68) with Ni ¼ 10 000 samples.The MSC bound is given by (16), with the measurement-dependent FIM, Jk(m1:k) calculated using the recursion(18). We approximate the MSC bound by using thesample-based technique introduced in Section 6 of [13].The basis of this technique is as follows.

Step 1: generate NS measurement sequences mi1:k,

i ¼ 1, . . . , NS by sampling from the distribution p(mk).

Step 2: calculate Jk(mi1:k), k ¼ 1, . . . using the recursive

formula (18).

Step 3: approximate the MSC bound as follows:

PCRLB(MSC; k) �1

NS

XNS

i¼1

J�1k (mi

1:k) (50)

The rate of convergence of this approach is investigated inSection 7.4.

In order to calculate the MESC bound, we could also use asample-based approach, whereby we first determine existencesequences e1:k, i ¼ 1, . . . , NS using the measurement

sequences mi1:k, i ¼ 1, . . . , NS (i.e. ei

k ¼ 1 if and only ifmi

k . 0). We can then approximate the bound bycalculating J�1

k (ei1:k), for k � 1 by using the recursive

formula (26) and then creating an (unweighted) average (asin (50)). The convergence of the approach compared to the(sample-based) MSC approach is investigated in Section 7.2.

Alternatively, we can determine the exact value of theMESC bound by: (i) enumerating all possible existencesequences e1:k, then (ii) determining the conditional bound

J�1k (e1:k) for each sequence, and finally (iii) creating a

weighted average – weighted by the probability of eachsequence occurring. There are a total of 2k possibleexistence sequences (at scan number k), and thecomputational expense of this approach is also discussed, inSection 7.5. Calculating the exact MESC bound byenumeration is also useful in establishing the rate ofconvergence of the sample-based approximations.

In Sections 7.2 to 7.5, we present both mean square error(MSE) and root mean square error (RMSE) location bounds.These are given as follows:

location MSE bound ¼ PCRLB(:; k)11 (51)

location RMSE bound ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilocation MSE boundp

(52)

where PCRLB(.;k)11 denotes the (1, 1) element ofPCRLB(.;k) (which gives the PCRLB on xk).

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7.2 Comparisons between the bounds

For the scenario detailed in Section 6, in Fig. 1a we show thefollowing.

† The IRF location RMSE bound (in grey).

† The exact MESC location RMSE bound (calculated byenumeration – in black).

Figure 1 Comparison between the location RMSE boundscalculated using the three PCRLB approaches. In eachcase, Pd ¼ 0.2 and lVg ¼ 0.4

a Black line indicates the exact MESC bound determined byenumerating all possible existence sequencesb We calculate the ratio of the approximate RMSE bound(calculated by sampling potential evolutions of themeasurement sequence) divided by the exact MESC bound



† Sample-based approximations of the MESC locationRMSE bound and the MSC location RMSE bound, eachwith NS ¼ 1 000, 10 000, 50 000 and 100 000 (see dashedand dot-dashed lines).

The first thing we note is that the relationship between thethree bounds is in accordance with Proposition 1 andProposition 2 (i.e. PCRLB(IRF; k) � PCRLB(MESC; k) �PCRLB(MSC; k) for all k). Moreover, by construction, theIRF bound is extremely optimistic, and is orders ofmagnitude lower than the MESC and MSC bounds (seeFig. 1a).

Furthermore, in the cases considered, differences betweenthe sample-based MESC and MSC bounds are small for allsample sizes NS . This is clear when one looks at the values ofthe two bounds in Table 2 (for Pd ¼ 0:2 and lVg ¼ 0:4),and when one looks at the percentage difference between thebounds (for a range of values of Pd and lVg – see Table 3).This suggests that the key reason for the over-optimism of theIRF bound is the fact that it does not take into account theeffect of missed detections. Although the MSC approachtreats cases with one or more detections differently to theMESC approach, the similarity between these two boundssuggests that discriminating between such cases is of lessimportance. Percentage differences between the MESC andMSC bounds are greatest when Pd ¼ 1, and we note that insuch cases the MESC bound is identical to the IRF boundwhich, by construction, is extremely optimistic.

Because the sample-based MESC and MSC bounds arevirtually identical for all samples sizes NS, they have thesame rate of convergence. Hence, somewhat surprisingly,calculating an accurate approximation of the MESC boundusing the sample-based technique is not computationallysimpler than approximating the MSC bound usingsamples. However, we can calculate the MESC boundexactly (because the number of potential existencesequences is finite (¼2k)). This exact bound is then usefulin showing the convergence of sample-based techniques(see Fig. 1b and Section 7.4). As we will then show inSection 7.5, the computational expense required in order tocalculate the MESC bound by enumeration is many timesless then the computational expense required in determiningan accurate sample-based estimate of the MSC bound.

Table 2 Overall location RMSE bounds (in m), using the three approaches. In each case, the overall location RMSE bound is

given byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=20

P20k¼1 PCRLBð:; kÞ11

q. The exact value of the MESC bound, calculated by enumerating all potential existence

sequences, is 244.806272

# Sequences, NS IRF approach MSC approach MESC approach

1000 6.375090 242.678020 242.677085

10 000 as above� 242.764859 242.764280

100 000 as above� 243.839153 243.838681

�The IRF bound is calculated by using the recursive formula (3) and does not require the use of samples



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Table 3 Largest percentage difference between the MSC and MESC bounds, given by: maxk¼1, . . . ,20[PCRLB(MSC; k)11 2 PCRLB(MESC; k)11/PCRLB(MESC; k)11], for different values of Pd and different values of the average number of false alarms per samplingtime (¼lVg). The figures in brackets are the sampling time at which the greatest difference occurred. In each case, the MESC andMSC bounds are each calculated by sampling NS ¼ 10 000 sequences

# False alarms, lVg

Probability of detection, Pd

0.2 0.4 0.6 0.8 1.0

0.4 0.24% (20) 0.17% (14) 0.78% (11) 1.81% (7) 9.16% (1)

0.8 1.11% (13) 0.10% (13) 0.83% (7) 2.71% (5) 14.97% (1)

1.6 4.57% (7) 0.90% (5) 0.56% (5) 3.28% (5) 21.93% (1)

3.2 7.46% (4) 1.93% (4) 0.56% (4) 4.04% (3) 24.39% (1)

7.3 Qualitative assessment of thedifferences between the MESC andMSC bounds

In this section, we investigate the mechanisms responsible fordifferences between the MESC and MSC bounds. As we cansee from Table 3, differences between the two bounds aretypically less than 5% in the cases considered in this paper.Furthermore, in other simulations not included in thisarticle (for scenarios taken from [13, 14]), differencesbetween the the MESC and MSC bounds were extremelysmall. We note that this is not unexpected, as Propositions1 and 2 tell us that differences between the MESC andMSC bounds can be no greater than differences betweenthe IRF and MSC bounds. Herein we have considered ascenario deliberately constructed (in [13]) to maximisedifferences between the IRF and MSC bounds. Such ascenario also then allows differences between the MESCand MSC bounds to be at their greatest.

We now provide a qualitative assessment of whendifferences between the MESC and MSC bounds will beat their greatest. Referring to Table 3, the most interestingrow is the final one (i.e. with lVg ¼ 3:2), which has thelargest variation in performance, with differences of 0.56–24.39%. To understand the mechanisms at work, let uslook at the IRFs. In Fig. 2a we show the MSC approachIRFs: qk(mk); and the overall IRF: ~qk used in the MESCapproach. If we then divide qk(mk) by ~qk (see Fig. 2b),values close to unity for all likely values of mk show that theMESC and MSC approaches scale the measurementcontribution to the same degree for all likely measurementsequences, and in such cases one would expect the twobounds to be very similar.

Given that the average number of measurements at eachtime step is Pdþ lVg ¼ 3.4, 3.6, 3.8, 4.0 and 4.2 inFig. 2b, the likely values of mk are in the range of 2–7.Hence, it is clear from Fig. 2b that Pd ¼ 0:6 will result inthe smallest difference in the bounds, and Pd ¼ 0:2, 1:0will result in the largest difference. As we can see fromTable 3, this is indeed the case. Hence, we conclude thatthe ratio qk(mk)=~qk provides an effective means of



determining when differences between the MESC andMSC bounds will be at their greatest (and smallest). Thisis useful in deciding whether to use the MESC approach,with its lower computational burden (see Section 7.5), oruse the less optimistic MSC bound.

Figure 2 The IRF qk(mk) and the ratio qk(mk)/qk for fivedifferent probabilities of detection. In each case, the averagenumber of false alarms per time step is lVg ¼ 3.2, and theIRFs are calculated using (58) from [13] (with 10 000 samples)

a The IRF qk(mk). The dashed horizontal lines show the value of qk

for each value of Pd

b The ratio qk(mk)/qk, each plotted against mk for five differentprobabilities of detection

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7.4 Convergence of the MSC approach

The convergence of the Monte Carlo sampling method usedin calculating the MSC bound depends on the distribution ofm1:k and the subsequent variation in Jk(m1:k) for differentvalues of m1:k. However, the variation in Jk(m1:k) is hard toquantify theoretically. Therefore in investigating theconvergence of the MSC approach, we calculate the MSCbound by sampling NS measurement sequences mi

1:k, andrepeat each calculation NR times. Throughout this article,we take NS ¼ 100. We then propose a rule of convergencegiven by

hRk

W

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiV Rk

h ir

E Rk

h i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1= NR � 1� � PNR

i¼1 Rik � E Rk

h i� �2r

E Rk

h i� h0 (53)



where

Rik W location RMSE bound at sampling time k, under

the MSC approach and using the ith measurement

sequence (W mi1:k) (54)

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi[ J�1

k (mi1:k)]11

q(where [ J ]11 denotes the (1, 1)

element of J ) (55)

and

Rk W Rik: i ¼ 1, . . . NR

� �(56)

E Rk

h i¼

1

NR

XNR

i¼1

Rik (57)

hRk

is the variance–mean ratio of samples, which iscompared with a threshold h0 given a typical value of 0.05.

Figure 3 One hundred approximations of the MSC location RMSE bound (given by black dots), calculated by sampling NS

measurement sequences. Four different values of NS are considered. Also shown are the exact MESC bound (in black –calculated by enumerating all potential existence sequences), and the IRF bound (in grey)



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In Fig. 3, we show NR ¼ 100 approximations of the MSClocation RMSE bound (with Pd ¼ 0:2 and lVg ¼ 0:4).Also shown is the exact MESC location RMSE bound(which was calculated by enumerating all 2k potentialexistence sequences e1:k). The convergence of the MSCapproach for different values of NS is then shown inTable 4. Using the convergence rule given by (53), it canbe seen that for k ¼ 5, 10, 15, 20, convergence is achievedwith NS ¼ 2000, 20 000, 200 000, 1 000 000 measurementsequences, respectively. We note that for k ¼ 5, 10 and 15the value of h

Rkis extremely close to the threshold value of

h0( ¼ 0:05, see the bold values in Table 4), but a veryslight reduction in the number of measurement sequenceswould still satisfy the convergence criterion. When k ¼ 20,the test statistic is equal to the threshold (i.e. h

Rk¼ h0).

Clearly, in this case no reduction in the number ofmeasurement sequences would still result in convergence ofthe MSC approximation.

7.5 Comparison between thecomputational costs of the MESCand MSC approaches

Using the rule of convergence given in (53), we can easilycompare the computational costs of the MSC and MESCapproaches. In calculating the MSC bound at each timestep k, the total number of times we have to perform oneiteration of the recursion (18) is

total number of calculations (18) ¼ NSk (58)

In calculating the MESC bound at each time step k,we enumerate all potential existence sequences e1:k.Hence, in calculating the MESC bound, the total numberof times we have to perform one iteration of the recursion(26) is:

total number of calculations (26) ¼Xk

r¼1

2r¼ 2kþ1

� 2

(59)

Therefore

total number of calculations in the MSC approach

¼ UcNSk (60)

and

total number of calculations in the MESC approach

¼ Uc 2kþ1� 2

� (61)

where Uc denotes the number of calculations required inperforming one iteration of the recursive formulae (18) or(26).


Authorized licensed use limited to: IEEE Xplore. Downloaded on April 14, 2009 at 11:21

Tab

le4

Var

ian

ce–

mea

nra

tio

shR

ko

f10

0M

SCb

ou

nd

app

roxi

mat

ion

s.Th

eva

lues

inb

old

sho

wth

esm

alle

stn

um

ber

ofs

equ

ence

sN

S(o

fth

ose

eval

uat

edb

elo

w)

for

wh

ich

the

MSC

app

roac

hh

asco

nver

ged

(i.e

.at

wh

ichhR

k�

0.05

)at

sam

plin

gti

mes

k¼

5,10

,15

and

20

k1

23

45

67

89

1011

1213

1415

1617

1819

20

NS¼

1000

0.01

520.

0239

0.03

450.

0467

0.06

630.

0929

0.12

200.

1566

0.20

490.

2456

0.31

210.

3698

0.49

000.

5749

0.63

870.

6549

0.80

580.

7942

0.82

960.

8281

NS¼

2000

0.01

010.

0169

0.02

310.

0337

0.04

840.

0633

0.07

910.

1117

0.12

340.

1515

0.21

140.

2837

0.32

980.

4294

0.53

490.

5947

0.63

290.

7359

0.76

860.

7552

NS¼

1000

00.

0046

0.00

820.

0117

0.01

710.

0209

0.02

800.

0357

0.04

520.

0594

0.06

850.

0871

0.10

820.

1453

0.20

960.

2410

0.29

090.

3334

0.36

300.

3774

0.40

40

NS¼

2000

00.

0031

0.00

530.

0075

0.01

120.

0133

0.01

740.

0261

0.03

80.

0442

0.04

860.

0592

0.07

230.

1103

0.12

540.

1719

0.19

680.

2578

0.27

180.

3126

0.31

52

NS¼

100

000

0.00

150.

0026

0.00

350.

0052

0.00

680.

0090

0.01

190.

0146

0.01

790.

0227

0.02

870.

0338

0.04

700.

0560

0.07

410.

0861

0.11

640.

1261

0.14

690.

1685

NS¼

200

000

0.00

110.

0018

0.00

270.

0042

0.00

470.

0063

0.00

820.

0115

0.01

410.

0179

0.02

130.

026

0.03

630.

0477

0.04

980.

0639

0.06

70.

0882

0.11

310.

1102

NS¼

300

000

0.00

090.

0015

0.00

240.

0031

0.00

350.

0055

0.00

660.

0080

0.01

000.

0135

0.01

620.

0209

0.02

900.

0403

0.04

030.

0474

0.06

090.

0700

0.09

440.

0924

NS¼

100

000

00.

0005

0.00

080.

0012

0.00

170.

0020

0.00

30.

0039

0.00

490.

0060

0.00

730.

0088

0.01

250.

0148

0.01

920.

0223

0.02

720.

0348

0.04

070.

0482

0.05

00

143


from IEEE Xplore. Restrictions apply.

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Table 5 Comparison of total calculations and computer run times under the MESC and MSC approaches. Uc denotes the totalcalculations required for one iteration of the recursion (18) or (26). In the final column we calculate the ratio: (run time of MSCapproach)/(run time of MESC approach)

MSC approach MESC approach Ratio of run times

k ¼ 5NS ¼ 2000

total computations ¼ 10 000 � Uc

run time ¼ 0.69 stotal computations ¼ 30 � Uc

run time ¼ 0.031 s22.3

k ¼ 10NS ¼ 20 000

total computations ¼ 200 000 � Uc

run time ¼ 13.1 stotal computations ¼ 2046 � Uc

run time ¼ 0.14 s93.6

k ¼ 15NS ¼ 200 000

total computations ¼ 3 000 000 � Uc

run time ¼ 194 stotal computations ¼ 65 534 � Uc

run time ¼ 3.47 s55.9

k ¼ 20NS ¼ 1 000 000

total computations ¼ 20 000 000 � Uc

run time ¼ 1241 stotal computations ¼ 2 097 150 � Uc

run time ¼ 109 s11.4

4

The computational costs of the MSC and MESCapproaches are shown in Table 5, with all simulations run ona PC with an IntelwCoreTM2 Duo CPU [email protected] GHzprocessor. In each case, the results show that the run timeis almost proportional to the total number of calculations of(18) or (26), which gives us a convenient way of makingfuture comparisons between the computational costs of thetwo algorithms.

In calculating the MSC bound, we used NS ¼ 2000measurement sequences for k ¼ 5; NS ¼ 20 000 sequencesfor k ¼ 10; NS ¼ 200 000 sequences for k ¼ 15 andNS ¼ 1 000 000 sequences for k ¼ 20. These were thevalues identified in Section 7.4 for which the MSCapproach converged. The total number of calculations,and required run time of calculating the MESC bound isthen 10–100 times less than that required in calculatingthe MSC bound. As noted in Section 7.4, for k ¼ 5, 10,15 (but not for k ¼ 20) it is possible to reduce NS veryslightly, and still generate an MSC bound that satisfiesthe convergence criterion (53). However, given that inthese cases the run time in calculating the MSC bound is22–94 times greater than that required in calculating theMESC bound, it is clear that even if we can reduce NS

slightly, calculating the MESC bound will still besignificantly more efficient.

Given that differences between the MSC and MESCbounds are small in these and other cases, we conclude thatthe MESC approach is a powerful and computationallyefficient means of calculating RMSE performance boundsin cluttered environments.

8 The circumstances under whichwe should favour each approachIf computational expense is not an issue (e.g. we areconducting a retrospective assessment of the performanceof a particular filtering methodology), then the MSCapproach should be preferred because it provides the least

The Institution of Engineering and Technology 2009


optimistic bound. The bound should then be calculated byusing the sample-based technique outlined in Section 7.1(and detailed in Section 6 in [13]), with the number ofsamples chosen according to the convergence criterion ofSection 7.4.

However, in time critical scenarios, such as in performingonline sensor resource management (in which there is a needto quickly evaluate and compare the predicted futureperformance of different system architectures), thecomputational expense of the MSC approach can limit itsusefulness. In this case there are three potential scenarios,given as follows.

† Scenario I: the probability of detection is close to unity and thefalse alarm rate is low. In this case, differences between thethree bounds are small. The IRF approach should then beutilised as it provides a computationally efficient andrealistic performance bound.

† Scenario II: the probability of detection is lower and/or thefalse alarm rate is higher. It was shown in [13] that in thiscase, differences between the IRF and MSC bounds aregreatest if the measurement errors are small in relation tothe initial target state uncertainty. This scenario makes thesequence of detections of critical importance. In such cases,we should then use the MESC approach, and calculate thebound by enumerating all of the potential existencesequences. This provides a bound that is typically almostidentical to that of the MSC approach, but with a reducedcomputational overhead.

† Scenario III: enumerating the MESC bound is stillcomputationally prohibitive. One should then calculate theMSC bound using the sample-based approach, but thenumber of samples used should be the maximum numberthat allows decision-making to be made in the availabletime. Although convergence of the estimate will notbe achieved, this approach provides the best compromisebetween the accuracy of the estimate and the need to make‘just-in-time’ decisions.



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The boundary between scenarios I and II depends in acomplex way on the probability of detection, false alarmrate, measurement accuracy and the target trajectory inrelation to the sensors. Therefore a degree of intuition isrequired in deciding whether the IRF approach providesa reasonable estimate of the more complex MESC andMSC bounds.

9 ConclusionsIn this paper, we have considered the problem of calculatingthe PCRLB for the nonlinear filtering problem in whichthere can be both missed detections and false alarms. Wehave introduced a novel new approach, which we havecalled MESC. The basis of the approach is to condition onwhether at least one measurement is acquired at eachsampling time, and then create an unconditional PCRLBas a weighted average of these conditional bounds.

The key findings were as follows.

† We rigourously proved that the new MESC bound willalways be less optimistic than the IRF bound (e.g. [10, 12]).

† We also proved the less desirable property that the MESCbound will always be more optimistic than the MSC bound[13]. However, in simulations, differences between the twobounds were small (typically less than 5%), suggesting thatthe MESC approach is a good approximation to the MSCbound.

† The greatest value of the new MESC approach is that thebound can be enumerated precisely, without the need forinefficient and computationally expensive sampling. In oursimulations, the MESC bound was calculated 10–100times more quickly than the MSC bound, which is forcedto employ an inefficient sample-based approach.

† Qualitatively, we showed the the IRF ratio: qk(mk)=~qk

provided an effective means of determining scenarios inwhich differences between the MESC and MSC boundswill be greatest. This is useful in deciding whether to usethe MESC approach, with its lower computational burden,or employ the less optimistic MSC bound.

Furthermore, we have discussed the circumstancesin which each of the three performance bounds shouldbe preferred. We conclude that the MESC approach is apowerful and computationally efficient means of calculatingRMSE performance bounds in cluttered environments.

10 AcknowledgmentThe contribution of Dr Hernandez was sponsored by theUnited Kingdom MOD Data and Information FusionDefence Technology Centres (DIF DTC) ResearchProgramme. The authors would also like to thank two



anonymous reviewers whose comments have improved thisarticle.

11 References

[1] VAN TREES H.: ‘Detection, estimation, and modulationtheory’ (Wiley, New York, USA, 1968)

[2] BERGMAN N.: ‘Posterior Cramer– Rao bounds forsequential estimation’, in DOUCET A., DE FREITAS N., GORDON

N. (EDS.): Sequential Monte Carlo methods in Practice(Springer-Verlag, New York, USA, 2001)

[3] TICHAVSKY P., MURAVCHIK C., NEHORAI A.: ‘Posterior Cramer–Rao lower bounds for discrete-time nonlinear filtering’,IEEE Trans. Signal Process., 1998, 46, (5), pp. 1386–1396

[4] HERNANDEZ M.L., KIRUBARAJAN T., BAR-SHALOM Y.: ‘Multisensorresource deployment using posterior Cramer–Rao bounds’,IEEE Trans. Aerosp. Electr. Syst., 2004, 40, (2), pp. 399–416

[5] THARMARASA R., KIRUBARAJAN T., HERNANDEZ M.L., SINHA A.:‘PCRLB based multisensor array management formultitarget tracking’, IEEE Trans. Aerosp. Electr. Syst.,2007, 43, (2), pp. 539–555

[6] HERNANDEZ M.L.: ‘Optimal sensor trajectories in bearings-only tracking’. Proc. Seventh Int. Conf. on InformationFusion, Stockholm, Sweden, 2004

[7] HERNANDEZ M.L.: ‘Adaptive horizon sensorresource management: validating the core concept’.DRUMMOND, O. (ED.) : Proc. Society of Photo-OpticalInstrument Engineers Conf. on Signal and Data Processingof Small Targets, San Diego, CA, USA, 2007, vol. 6699

[8] BAR-SHALOM Y., LI X.R., KIRUBARAJAN T.: ‘Estimation withapplications to tracking and navigation’ (John Wiley andSons, Inc., New York, USA, 2001)

[9] NIU R., WILLETT P.K., BAR-SHALOM Y.: ‘Matrix CRLB scaling dueto measurements of uncertain origin’, IEEE Trans. SignalProcess., 2001, 49, (7), pp. 1325–1335

[10] ZHANG X., WILLETT P.K.: ‘Cramer-Rao bounds fordiscrete time linear filtering with measurement originuncertainty’. Proc. Workshop on Estimation, Tracking andFusion: A Tribute to Yaakov Bar-Shalom, Monterey, CA,USA, 2001

[11] ZHANG X., WILLETT P., BAR-SHALOM Y.: ‘Dynamic Cramer–Raobound for target tracking in clutter’, IEEE Trans. Aerosp.Electr. Syst., 2005, 41, (4), pp. 1154–1167

[12] HERNANDEZ M.L., MARRS A.D., GORDON N.J., MASKELL S., REED C.M.:‘Cramer– Rao bounds for non-linear filtering withmeasurement origin uncertainty’. Proc. 5th Int. Conf.

145



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Information Fusion, Annapolis, Maryland, USA, 2002,vol. 1, pp. 18–25

[13] HERNANDEZ M.L., FARINA A., RISTIC B.: ‘PCRLB for tracking incluttered environments: measurement sequenceconditioning approach’, IEEE Trans. Aerosp. Electr. Syst.,2006, 42, (2), pp. 680–704

[14] FARINA A., RISTIC B., TIMMONERI L.: ‘Cramer–Rao bounds fornon-linear filtering with Pd , 1 and its application to targettracking’, IEEE Trans. Signal Process., 2002, 50, (8),pp. 1916–1924

[15] HERNANDEZ M.L., RISTIC B., FARINA A., TIMMONERI L.: ‘Acomparison of two Cramer – Rao bounds for non-linearfiltering with Pd , 1’, IEEE Trans. Signal Process., 2004,52, (9), pp. 2361–2370

[16] BOERS Y., DRIESSEN H.: ‘Modified Riccati equation and itsapplication to target tracking’, IEE Proc. Radar, SonarNavig., 2006, 153, (1), pp. 7–12

[17] BOERS Y., DRIESSEN H.: ‘Results on the modified Riccatiequation: target tracking applications’, IEEE Trans. Aerosp.Electr. Syst., 2006, 42, (1), pp. 379–384

[18] TAYLOR J.: ‘The Cramer – Rao estimation error lowerbound computation for deterministic nonlinear systems’,IEEE Trans. Automat. Control, 1979, 24, (2), pp. 343–344

12 Appendix 1: calculation of theinformation reduction factors12.1 General expression

In calculating the IRFs, the following assumptions are made(e.g. see [13]).

† There is a maximum of one target-generated measurementper sampling time, which occurs with constant probabilityPd .

† At each sampling time the number of false alarms has aPoisson distribution with mean l per unit area of thesurveillance region.

† We consider only false alarms that fall in a small gatedregion of volume Vg around the target (e.g. see (39) in[13]). Hence the average number of false alarms consideredis lVg per sampling time.

† We assume that Rk ¼ diag(s21, . . . , s2

n), where to remindthe reader, Rk is the error covariance of target-generatedmeasurements. The parameter n denotes the dimensionalityof each measurement (e.g. with range-only measurements,n ¼ 1; with range and bearing measurements, n ¼ 2).



The IRF, qk(mk) is then given by, for example, (54) in [13].This equation is as follows

qk(m)¼eg(m)2

jRkj(m�2)=2

mV 2m�2g (2p)n

�fðg

Ymn¼�g

. . .

ðg

Ym1¼�g

. . .

ðg

Y1n¼�g

. . .

ðg

Y11¼�g

�Y 2

11 exp{�Pn

i¼1 Y 21i}

[ð(1� eg(m))=V mg Þþ (eg(m)=mV m�1

g

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(2p)n

jRkj)p Pm

r¼1 exp{�(1=2)Pn

j¼1 Y 2rj }]

dY11 . . .dYmng(62)

The volume of the gated observation region is given by

Vg ¼ (2g)nYn

i¼1

si ¼ (2g)nffiffiffiffiffiffiffiffijRkj

p(63)

The parameter pg(mk) is the probability that there are mk

observations in the gated observation region, and eg(mk) isthe probability that one of these mk measurements is targetgenerated. These are given by

pg(mk)¼ (1�Pgd )

(lVg)mk exp �lVg

� �mk!

þPgd

(lVg)mk�1 exp �lVg

� �(mk�1)!

(64)

eg(mk)¼P

gd

pg(mk)

(lVg)mk�1 exp �lVg

� �(mk�1)!

(65)

Pgd is the probability of obtaining a target-generated

measurement in the gated observation region. This is givenby

Pgd ¼ Pd F(g)�F(�g)

�(66)

where F(.) is the cumulative distribution function of astandard normal random variable, that is

F(t)¼1

2p

ðt

�1

exp�x2

2

( )dx (67)

We note that if the measurement, clutter and detectionmodels are time invariant then so too is the IRF, qk(mk),which then need only be calculated once.



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12.2 Numerical approximation

We can create a Monte Carlo approximation of (62) asfollows (see (58) in [13])

qk(mk) �(2g)nmkeg(mk)

2jRkj

(mk�2)=2

mkV2mk�2

g (2p)n

�1

Ni

XNi

l¼1

U11[l]2 exp �Pn

i¼1 U1i[l]2

n o[ð(1� eg(mk))=V mk

g Þ

þ (eg(mk)=mkV mk�1g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(2p)n

jRkj)p

�Pmk

r¼1 exp {�(1=2)Pn

j¼1 Urj[l]2}]

(68)

where Uij[l], i ¼ 1, . . . , mk; j ¼ 1, . . . , n; l ¼ 1, . . . , Ni

are independent and identically distributed randomvariables drawn from a uniform distribution on [2g, g] (i.e.each Ui, j[l] � U [�g, g]).

13 Appendix 2: proof ofProposition 1We prove the result by induction. We will show that if

PCRLB(MESC ; k) � J�1k (69)

Ee1:kJk(e1:k) �

� Jk (70)

for some k � 0, then

PCRLB(MESC ; kþ 1) � J�1kþ1 (71)

Ee1:kþ1Jkþ1(e1:kþ1) �

� Jkþ1 (72)

with equality in (71) and (72) only if Jk(e1:k) is constant as afunction of e1:k. We note that Jk(e1:k) will be constant as afunction of e1:k if Pd ¼ 1, in which case there is only onepossible existence sequence e1:k, with ek ¼ 1 at all samplingtimes k. In this case, the IRF and MESC approaches areidentical.

First, we note that

Ee1:kJZ(k : ek) �

¼ ~qkEXkH T

k R�1k Hk

�� p(ek ¼ 1) (using (36)) (73)

¼ qkEXkH T

k R�1k Hk

�(using (37)) (74)

¼ JZ(k) (75)

Now, it is straightforward to show that D11k � 0 for all k.

Furthermore, except in specially constructed cases,Jk(e1:k) . 0 for all k and all sequences e1:k. To show this, inthe simplest case with linear target dynamics, provided theprior covariance C0 . 0 it follows by induction on (27)that Jk(e1:k) . 0 for all k and all sequences e1:k. In themore general case with nonlinear dynamics, we have



to refer to the FIM J1:k(e1:k) for the estimation ofX1:k ¼ X1, . . . , Xk

� �given e1:k. Conditioning the FIM [1]

on e1:k, then given by

J1:k(e1:k) ¼ EX1:k ,Z1:kje1:k�D

X1:kX1:k

log p(X1:k, Z1:kje1:k)h i

(where DQC ¼ rCr

TQ) (76)

¼ EX1:k ,Z1:kje1:k

rX1:k

log p(X1:k, Z1:kje1:k)�h

�rX1:k

log p(X1:k, Z1:kje1:k)�Ti

(77)

Hence, for all non-zero vectors a

aT J1:k(e1:k)a ¼ EX1:k ,Z1:kje1:k

�

"�aTrX1:k

log p(X1:k, Z1:kje1:k)n o�

�

�aTrX1:k

log p(X1:k, Z1:kje1:k)n o�T

#

(78)

Clearly then, J1:k(e1:k) � 0 with strict inequality unless

aTrX1:k

log p(X1:k, Z1:kje1:k)n oh i

¼ 0

for all a . 0 and all (X1:k, Z1:k) such that

p(X1:k, Z1:kje1:k) . 0

(79)

Therefore J1:k(e1:k)�1

. 0 in all but specially constructedcases. The matrix Jk(e1:k)

�1 is the bottom right-hand blockof J1:k(e1:k)

�1, which is also positive definite in all butspecially constructed cases. Therefore in conclusionJk(e1:k) . 0 in all but specially constructed cases. Hence

Jk(e1:k)þD11k

��1. 0 for all sequences e1:k (80)

Now, Jensen’s inequality (e.g. see Lemma A.1 in [15]) statesthat if V is a random positive definite matrix (i.e. V . 0), then

E V �1 �

� E V½ �1 (81)

with strict inequality unless V is a constant.

Hence it follows from inequalities (80) and (81) that

Ee1:kJk(e1:k)þD11

k

��1h i

� Ee1:kJk(e1:k) �

þD11k

h i�1

(82)

with equality only if Jk(e1:k) is constant as a function of e1:k.

It then clearly follows from inductive relation (70) that

Ee1:kJk(e1:k) �

þD11k

h i�1

� Jk þD11k

��1(83)

147



14

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Hence it follows from inequalities (82) and (83) that

Ee1:k(D12

k )T Jk(e1:k)þD11k

��1D12

k

h i� (D12

k )T Jk þD11k

��1D12

k (84)

with equality only if Jk(e1:k) is constant as a function of e1:k.

Therefore it follows that

Ee1:kþ1[Jkþ1(e1:kþ1)]

¼ Ee1:kþ1[D33

k � (D12k )T [Jk(e1:k)þD11

k ]�1D12k

þ JZ(kþ 1:ekþ1)] (85)

¼ Ee1:k[D33

k � (D12k )T [Jk(e1:k)þD11

k ]�1D12k ]

þ JZ(kþ 1) (86)

(using (75) and the independence of Jk(e1:k) from ekþ1)

� D33k � (D12

k )T [Jk þD11k ]�1D12

k

þ JZ(kþ 1) (using (84)) (87)

¼ Jkþ1 (using (3)) (88)

Again, we note that there is equality in (87) only if Jk(e1:k) isindependent of e1:k. This proves the inductive statement (72).

To proceed

PCRLB(MESC ; kþ 1) W Ee1:kþ1J�1

kþ1(e1:kþ1) �

(by definition (24)) (89)

� Ee1:kþ1Jkþ1(e1:kþ1) ��1

(again, by Jensen’s inequality) (90)

� J�1kþ1 (by the inductive

statement (72) proven before) (91)

Once again, we note that there is equality in (90) and (91)only if Jk(e1:k) is independent of e1:k. This proves theinductive statement (71). Finally, we note that

PCRLB(MESC ; 0) ¼ J�10 ¼ C0 (92)

Ee1:0J0(e1:0) �

¼ J0 ¼ C�10 (93)

where ‘e1:0’ denotes the (empty) existence sequence availableat initialisation time 0. C0 is the initial target covariance.This completes the proof. A



14 Appendix 3: relationshipbetween the MESC and MSCbounds – conditional on theexistence sequenceProposition 4: For the general case with the potential forboth missed detections (i.e. Pd � 1) and false alarms (i.e.Pfa � 0), provided mild regularity conditions hold, thefollowing result is true

J�1k (e1:k) � Em1:kje1:k

J�1k (m1:k)

�WXm1:k

p(m1:kje1:k)J�1k (m1:k) (94)

for all k, and all existence sequences e1:k.

Proof: We again prove the result by induction. We will showthat if

J�1k (e1:k) � Em1:kje1:k

J�1k (m1:k)

�(95)

Jk(e1:k) � Em1:kje1:kJk(m1:k) �

(96)

for some k � 0, then

J�1kþ1(e1:kþ1) � Em1:kje1:kþ1

J�1kþ1(m1:kþ1)

�(97)

Jkþ1(e1:kþ1) � Em1:kþ1je1:kþ1Jkþ1(m1:kþ1) �

(98)

First, it follows from the inductive relation (96) that

Jk(e1:k)þD11k

��1� [Em1:kje1:k

Jk(m1:k) �

þD11k ]�1 (99)

Furthermore, it can easily be shown that, in all but speciallyconstructed cases, Jk(m1:k) . 0 for all k and allmeasurement sequences m1:k (this proof is almost identicalto the proof that Jk(e1:k) . 0 given in the previous section).

Hence, it now follows from Jensen’s inequality that

Jk(e1:k)þD11k

��1� Em1:kje1:kþ1

hJk(m1:k)þD11

k

��1i

(100)

with equality only if Jk(m1:k) is constant for all measurementsequences m1:k that have common existence sequence e1:k.Except in specially constructed cases, this requires there to beonly one measurement sequence m1:k that is consistent withthe existence sequence e1:k. Therefore we must then havemk ¼ ek for all k, which is only true if mk � 1 for all k. Toconclude, this requires Pfa ¼ 0.

Clearly, we then have

D12k

� TJk(e1:k)þD11

k

��1D12

k

� Em1:kje1:kþ1

hD12

k

� TJk(m1:k)þD11

k

��1D12

k

i(101)



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To proceed

Em1:kþ1je1:kþ1JZ(kþ 1:mkþ1) �

¼ Emkþ1jekþ1JZ(kþ 1:mkþ1) �

(as JZ(kþ 1:mkþ1)

is independent of m1:k and e1:k) (102)

¼ JZ(kþ 1:ekþ1) (using (32) if ekþ1 ¼ 1

(the case ekþ1 ¼ 0 is trivial)) (103)

Hence

Em1:kþ1je1:kþ1[Jkþ1(m1:kþ1)]

¼ Em1:kþ1je1:kþ1[D33

k � (D12k )T [Jk(m1:k)

þD11k ]�1D12

k þ JZ(kþ 1:mkþ1)] (104)

¼ Em1:kje1:k[D33

k � (D12k )T [Jk(m1:k)

þD11k ]�1D12

k ]þ Emkþ1jekþ1[JZ(kþ 1:mkþ1)]

(exploiting independence – see next) (105)

� D33k � (D12

k )T [Jk(e1:k)

þD11k ]�1D12

k þ JZ(kþ 1:ekþ1)

(using inequality (101) and (103))

¼ Jkþ1(e1:kþ1) (using (26)) (106)

Specifically, (105) is derived by using the following facts



† D33k � D12

k

� TJk(m1:k)þD11

k

��1D12

k is independent ofmkþ1 and ekþ1

† JZ(kþ 1:mkþ1) is independent of m1:k and e1:k.

We again note that there is strict inequality in (106) unlessJk(m1:k) is constant.

This proves the inductive statement (98). To continue

J�1kþ1(e1:kþ1) � Em1:kþ1je1:kþ1

Jkþ1(m1:kþ1) ��1

(by the

inductive statement (98) proven in (104) to (106)) (107)

� Em1:kþ1je1:kþ1J�1

kþ1(m1:kþ1) �

(again,

using Jensen’s inequality) (108)

This proves the inductive statement (97).

Finally, again we note that

J�10 (e1:0) ¼ E J�1

0 (m1:0) �

¼ C0 (109)

J0(e1:0) ¼ E J0(m1:0) �

¼ C�10 (110)

where ‘m1:0’ and ‘e1:0’ denote the (empty) measurement andexistence sequences, respectively, available at initialisationtime 0. C0 is again the initial target covariance. Thiscompletes the proof. A

149



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Computationally efficient PCRLB for tracking in cluttered environments: measurement existence conditioning approach