Multitarget range-azimuth tracker - IEEE Xplore

I. INTRODUCTION

Multitarget Range-Azimuth Tracker

FRANCESCO BANDIERA, Member, IEEE Università del Salento Lecce, Italy MARCO DEL COCO INO, Consiglio Nazionale delle Ricerche Arnesano (LE), Italy GIUSEPPE RICCI Università del Salento Lecce, Italy

We focus on the design of a Kalman filter-based algorithm to track multiple targets in range and azimuth. The detection stage implements a multitarget strategy aimed at collecting and eventually processing all target measurements. Existing tracks are updated using a joint probabilistic data association (JPDA) algorithm to handle the data association problem. We model target echoes as coherent pulse trains in white Gaussian noise with unknown power. The performance analysis assumes a maritime scenario and shows that the proposed tracker outperforms more conventional approaches.

Manuscript received June 30, 2013; revised July 16, 2014; released for publication January 16, 2015. DOI. No. 10.1109/TAES.2015.130426. Refereeing of this contribution was handled by C. Jauffret. This work has been partly supported by CNIT within the “HABITAT” project (PON-011936) funded by MIUR. Authors’ addresses: F. Bandiera and G. Ricci, Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Via Monteroni, 73100 Lecce, Italy. E-mail: ([email protected]). M. Del Coco, INO, Consiglio Nazionale delle Ricerche, Viale della Libertà 3, 73010 Arnesano (LE), Italy. C 2015 IEEE 0018-9251/15/$26.00

Tracking algorithms are designed assuming that the sensor provides a set of point measurements at each scan. In a radar system such measurements could be obtained by thresholding the output of a matched filter fed by a baseband version of collected data. Then, the tracking algorithm estimates the trajectories of the targets with the help of the a priori information on their motion; in a real scenario it is also necessary to take into account false alarms and, more generally, to adequately solve the data association problem [1–3]. As to the detection stage, most detectors considered so far assume that the target is located exactly “where the matched filter is sampled” and, hence, that there is no spillover of target energy to adjacent matched filter samples. However, the spillover is taken into account in [4] where the case of several closely spaced targets, ruled by a Swerling 2 model, that fall within the same beam of a monopulse radar and among three or more adjacent matched filter samples in range, is considered; therein, a maximum likelihood (ML) extractor is developed. The idea is further investigated in order to estimate the angles and ranges of multiple unresolved extended targets in [5]. In [6] an adaptive space-time detector is derived and assessed: it takes advantage of the possible spillover of target energy for the case of coherent target echoes. In [7] this idea has been extended to address localization of multiple point-like targets within the same range cell or range cells in spatial proximity; therein, it is shown that the multitarget approach outperforms the corresponding single target one (that is also one of the instances considered in [6]) in the localization of a weak target for the case of targets with different strength. In [8], a Kalman filter-based algorithm, to track a single target in range, has been designed and assessed. The proposed approach exploits spillover of target energy between consecutive matched filter samples; to this end, at the kth scan, a detector based on the generalized likelihood ratio test (GLRT) is fed by the predicted target position (computed by the tracking stage using measurements up to the (k–1)th scan) and returns an estimate of the current target position (measurement) obtained by taking into account target strength over consecutive samples. The target echo is modeled as a coherent pulse train; moreover, the target is embedded in white Gaussian noise with unknown power. In this work, we propose and assess an algorithm to track multiple targets in range and azimuth. It is the natural, though not trivial, extension of the Kalman filter-based algorithm, proposed in [8], to track a single target in range, to multitarget scenarios and to range-azimuth tracking. The main novelty of the proposed algorithm is the potential to deal with multiple targets in spatial proximity resorting to “adaptive selection” of the detection stage, i.e., either a single target detector or a multiple target one. It is important to stress that the multiple target detector is new, though sharing with the localization scheme proposed in [7] the same rationale,

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 2 APRIL 2015

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while the single target detector is an extension of the one proposed in [6] that, in fact, has been modified to collect all target measurements in both range and azimuth. As a final remark, observe that use of the Kalman filter (actually in conjunction with a joint probabilistic data association (JPDA) algorithm) requires conversion of polar measurements to a Cartesian frame. It is well known that the classical conversion is biased and that computation of the covariance matrix of the errors associated with converted measurements would require knowledge of the true range and azimuth that is obviously unavailable [9–12]. We resort to the solution proposed in [13], a point better explained in Section III, that, under certain conditions, would compensate the bias of the conventional conversion and handle the correlation between the estimate of the measurement error covariance and the measurement error. The paper is organized as follows. Section II is devoted to the problem formulation while Section III focuses on the derivation of the proposed tracker. Section IV deals with the performance assessment of the new algorithm, also in comparison to a more conventional one. Finally, Section V contains some concluding remarks and hints for future work. II. PROBLEM FORMULATION

The aim of this section is twofold. First, we describe the radar scenario; then, we introduce the discrete-time signal and noise models for such a scenario, following the lead of [6, 7, 14, 15]. A mechanically rotating reflector is assumed as radar antenna. It is mounted at a certain height above the sea level and illuminates the surveillance area transmitting a bandpass modulated signal whose lowpass equivalent is a train of rectangular pulses of duration Tp . We neglect the dependence of the antenna radiation pattern on the elevation angle within the surveillance area. The rate at which observations are made in azimuth depends on the time (in seconds) for the antenna to make one rotation, namely 2π/|ωa | where ωa denotes the angular velocity of the antenna, in radians per seconds (rad/s), that we assume to be positive in case of counterclockwise rotation. For subsequent developments, tracking is done open loop, in that the antenna position is not controlled by the processed tracking data; the proposed algorithm can be classified as an automatic detection and track (ADT) strategy. For more details on ADT strategies, the interested reader is referred to [14]. Notice also that the number of consecutive pulses on a stationary target N is (approximately) equal to π θ3 N= (1) 180 |ωa |T where θ 3 is the (−3 dB) beamwidth (in degrees), namely the beamwidth measured between the half-power points (in azimuth) of the antenna radiation pattern, and T is the pulse repetition time1 (PRT). In practice, the pulses of 1

Remember also that T is the inverse of the pulse repetition frequency (PRF).

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TABLE I An Example of Radar Parameters fc Tp PRF ωa θ 3

9.375 GHz 0.2 μs 3 kHz π rad/s 0.36◦

each train are subject to an amplitude modulation process (that at first we neglect in order not to burden too much the introduction of the main ideas involved in the derivation of the system). For instance, assuming the radar parameters of Table I, it turns out that N = 6. Now consider a Cartesian coordinate frame with the z axis coincident with the rotation axis of the antenna and the x – y plane locally coincident with the sea surface. Moreover, suppose that the (azimuthal) pointing direction of the antenna at time instant t is θa (t) = θ0a + ωa t

(2)

with θ 0a ∈ [0, 2π) the angular position of the antenna at t = 0. In addition, assume a prospective (point-like and slowly fluctuating) target moving within the (nonambiguous) surveillance area (and in the antenna far field) with approximately constant radial velocity v (v > 0 if the target is approaching the radar) and approximately constant azimuth θ(t) over the coherent processing interval. Finally, denote by r(t) the range of the target at time instant t. Typically, a discrete form for the signal corresponding to each range-azimuth cell is obtained by properly sampling the output of a filter matched to the transmitted pulse (and fed by the received signal). To be more quantitative, the signal transmitted by the radar antenna is given by +∞ Iϕ I 2πfc t e Ae (3) p(t − nT )e n=−∞

√ where I = −1 is the imaginary unit and e {z} indicates the real part of the complex number z; A > 0 is an amplitude factor related to the transmitted power; ϕ ∈ [0, 2π) is the initial phase of the carrier signal; p(t) is a unit-energy rectangular pulse waveform of duration Tp (and one-sided bandwidth Wp ≈ 1/Tp ), namely, ⎧ 1 ⎨ √T , 0 < t < T p p (4) p(t) = ⎩ 0, elsewhere and fc = c/λ is the carrier frequency with c the velocity of propagation in the medium and λ the carrier wavelength. Neglecting compression or stretching of the time scale, the complex envelope of the received signal, containing N consecutive useful echoes backscattered by the target (approximately corresponding to the main beam of the


range gate, l = 1, . . . , L, and to the sth azimuth cell, s = 1, . . . , H, are those at time instants

antenna), is given by [15] zr (t) = αeI 2πfd t

nθ (h)+N−1

p(t − nT − τ ) + wr (t),

tl,n = tmin + (l − 1)Tp + nT ,

n=nθ (h)

n = nθs (h), . . . , nθs (h) + N − 1

(5)

t ∈ (nθ (h)T , (nθ (h) + N)T )

where α ∈ C is a factor which accounts for AeIϕ , the effects of the transmitting (and receiving) antenna gain (remember that, for the time being, we neglect the amplitude modulation process), the two-way path loss, the radar cross section of the target, etc.; fd Wp is the Doppler frequency shift of the signal backscattered by the target (i.e.,fd = 2vc fc ); nθ (h) denotes the first pulse of the hth train of N pulses that “illuminates” the target: nθ (h) is approximately given by (we assume that N is even) θ(t) + 2hπ sgn(ωa ) − θ0a nθ (h) = round − N/2 (6) ωa T

with θ s = θ 0a + sθ 3 and nθs (h) obtained by (6) with θ s in place of θ(t); in addition, the time samples can be grouped to form an N-dimensional vector as follows z l (nθs ) = z(tl,nθs (h) ) · · · z(tl,nθs (h)+N−1 ) = sl + nl (12) where sl is the signal component, nl the noise component, and denotes transpose. As to tmin , it is the beginning of the sampling process. Remember also that Tp is the pulse duration and T is the PRT. However, assuming that the round-trip delay τ of the received signal is different from (l – 1)Tp and is such that τ ∈ [tmin , tmin + (L − 1)Tp ]

with h any integer such that θ(t) = θ0a + ωa t − 2hπ sgn(ωa )

(7)

sgn(x) the signum of x and round(x) the integer closest to x; τ is the round-trip delay of the received signal, namely τ = 2r(tm )/c, tm ∈ (nθ (h)T, (nθ (h) + N)T); and wr (t) is the complex envelope of the overall disturbance. The discrete form for the received signal is obtained by sampling the output of a filter matched to p(t) and fed by zr (t) in (5). As a matter of fact, the matched filter output is given by [15] z(t) = α

nθ (h)+N−1

χp (t − nT − τ, fd )eI 2π nν + w(t),

n=nθ (h)

t ∈ (nθ (h)T , (nθ (h) + N)T )

(8)

where α is a modified version of the previous constant, χ p (·, ·) is the (complex) ambiguity function of the pulse waveform p(t) [16], i.e.,

+∞ χp (τ, f ) = p(t)p∗ (t − τ )eI 2πf t dt (9) −∞

with ∗ denoting in turn complex conjugate, w(t) is the noise component, and ν = fd T

(13)

It follows that ∃ l0 ∈ {1, . . . , L − 1} : τ = tmin + (l0 − 1)Tp + , (14) 0 ≤ ≤ Tp where (or Tp – , if > Tp /2) represents a residual delay that gives rise to the spillover of target energy. In other words, the target contributes to the l0 th and to the (l0 + 1)th range cells. In addition, the actual samples that contain the signal backscattered by a target with azimuthal position θ, when it is illuminated through the (−3 dB) antenna beamwidth, are those corresponding to the vectors z l (nθ ) = z(tl,nθ ) · · · z(tl,nθ +N−1 ) , l = l0 , l0 + 1 (15) with nθ obtained by (6) with θ in place of θ(t) (we neglect here the dependence of nθ on h). Thus, it follows that [6] ⎧ l = l0 αχ (− , ν/T )v(ν), ⎪ ⎪ ⎨ p (16) sl = αχp (Tp − , ν/T )v(ν), l = l0 + 1 ⎪ ⎪ ⎩ 0, l = l , l + 1 0

(10)

(11)

0

where

is the normalized target Doppler frequency. In order to generate the vector of the noisy returns, it is customary to assume that the target is located exactly “where the matched filter is sampled”; in other words, the matched filter output z(t) is sampled to obtain returns corresponding to a certain number of range-azimuth cells. For the sake of clarity, assume that 2π/|ωa | is a multiple integer of T and define the sth azimuth cell as the one corresponding to the values of θ that belong to the set s = {θ : θ0a + sθ3 − θ3 /2 < θ ≤ θ0a + sθ3 + θ3 /2}. It follows that samples corresponding to the lth

is the temporal steering vector. Equation (16) highlights the relationship between the amplitude of the target in the range cell where it is located and that in the adjacent range cell affected by spillover. As a final remark, notice that generalization of the above model to the case that several targets are present, within a given set of range-azimuth cells, is possible by using the superposition principle.

BANDIERA ET AL.: MULTITARGET RANGE-AZIMUTH TRACKER

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1 v(ν) = √ [1 eI 2π ν · · · eI 2π (N−1)ν ] N

(17)

III. DESIGN OF THE DETECTION AND TRACKING STAGES

In this section, we illustrate how measurements, to be fed to the tracking stage, are obtained and processed. We assume a multitarget scenario. However, even in a multitarget scenario, it seems reasonable to attack detection and localization (but also the tracking stage) resorting to either single target or multitarget strategies depending on whether targets are well separated (in range and/or in azimuth) or there are two or more closely spaced targets, respectively. The actual scenario being in force impacts the design approach:

the matched filter at the time instants tl,n = tmin + (l − 1)Tp + nT ,

n = nθ , . . . , nθ + N − 1 (18) We have to choose between the H0 hypothesis that zl contains disturbance only and the H1 hypothesis that it also contains a useful target signal, proportional to the (temporal) steering vector v(ν). Now, under the H1 hypothesis, there is spillover of target energy to either zl–1 or zl+1 depending on the value of the target delay τl, = tmin + (l − 1)Tp + ∈ [tmin + (l − 1)Tp − Tp /2, tmin + (l − 1)Tp + Tp /2]

1) for “noninterfering targets” the detector processes data resorting to the single target approach, namely using an algorithm designed under the assumption that a single target is present; 2) in case of closely spaced targets, instead, it seems reasonable to take into account, at the detection stage, the presence of mutual interference; for this reason, we intend to resort to a decision scheme designed to address detection and localization of multiple targets (the number of targets is assumed known at the detection stage).

with –Tp /2 ≤ ≤ Tp /2. It seems reasonable to use the above additional vectors, in addition to zl , to discriminate between H0 and H1 and to localize the possible target [6]. Finally, we assume that the corresponding noise vectors ni , i = l – 1, l, l + 1, are (N-dimensional) independent and identically distributed (IID) complex normal random vectors [6]; in particular, ni ∼ CNN (0, σ 2 I N ), with IN the N × N identity matrix, and σ 2 > 0 an unknown deterministic quantity. Following the lead of [6], the corresponding hypothesis test is

⎧ ⎪ ⎪ H0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

: z i (nθ ) = ni , i = l − 1, l, l + 1 ⎧ z l−1 (nθ ) = αχp −Tp − , Tν v l−1 (ν, θ) + nl−1 , ⎪ ⎪ ⎪ ⎪ ⎪ z l (nθ ) = αχp − , Tν v l (ν, θ) + nl , ⎪ ⎪ ⎪ ⎪ ⎪ z (n ) = nl+1 , ⎪ ⎪ ⎨ l+1 θ ⎪ H1 : or ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z l−1 (nθ ) = nl−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z l (nθ ) = αχp − , Tν v l (ν, θ) + nl , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z l+1 (nθ ) = αχp Tp − , ν v l+1 (ν, θ) + nl+1 , ⎩ T

The way targets are clustered (and eventually the type of detector used to process each cluster) depends on the available information on each target location (and possibly on how much such information is reliable). To this end, we can use the predicted positions of the targets returned by the tracking stage. The actual clustering criteria are described in Section III-B.

1) Single Target Detector: First assume that θ and v (and hence ν) are known parameters and that we are looking for the presence of an “isolated target” in the lth range cell. Also remember that z l (nθ ) ∈ CN×1 , l = 1, . . . , L, is the vector of returns from the lth range cell, obtained by stacking the N samples of the output of 1518

−Tp /2 ≤ ≤ 0

(20)

0 < ≤ +Tp /2

where, taking into account the amplitude modulation process due to the antenna gain, we have that 1 v l (ν, θ) = √ [cl,θ (nθ ) cl,θ (nθ + 1)eI 2π ν N · · · cl,θ (nθ + N − 1)eI 2π (N−1)ν ] with cl,θ (n) =

A. Detection Stage

(19)

G(θa (nT ) − θ)G(θa (tl,n ) − θ),

n = nθ , . . . , nθ + N − 1

(21)

(22)

and G(·) the antenna gain. More explicitly, θ a (nT) is the antenna pointing direction at the time instant the nth pulse is transmitted and θ a (tl,n ) is the antenna pointing direction at the time instant the signal backscattered from the target is sampled [given by (18)]. As to θ a (t), it is given by (2).


The previous expression for the cl, θ (n)s can be approximated as cl,θ (n) ≈ G(θa (nT ) − θ) ≈ G(θa (nT ) − θ ) ≡ cθ (n), n = nθ , . . . , nθ + N − 1 (23) by neglecting dependence on the rotation of the antenna over the time interval tl,n – nT = tmin + (l – 1)Tp and replacing θ with θ , given by N ωa T + θ0a − 2hπ sgn(ωa ) (24) θ = nθ + 2 It turns out that 1 v l (ν, θ) ≈ v(ν, θ) = √ [cθ (nθ ) cθ (nθ + 1)eI 2π ν N · · · cθ (nθ + N − 1)eI 2π (N−1)ν ] (25) Solving the above problem will lead to an adaptive decision scheme that takes advantage of the possible spillover of target energy and that could return more accurate measurements (to be fed to the tracking stage) than a conventional detector. In order to formulate and eventually solve the above problem we define an augmented received vector z˜ (nθ ) = [z l−1 (nθ ) z l (nθ ) z l+1 (nθ )] ∈ C3N×1

(26)

Accordingly, we also define an augmented steering vector; as a matter of fact, we have that

As to maximization over , it cannot be conducted in closed form and, hence, we resort to a grid search. Maximization over is performed assuming that it takes on values in the finite set 2N n − N − 1/2 E= Tp (30) 2N n=1 with N a preassigned positive integer. It is now a good moment to recall that ν and θ are unknown parameters, and, hence, that we also have to face with such a priori uncertainty. As to θ, we assume an uncertainty sector approximately given by = {θ : θ¯ − θ3 < θ < θ¯ + θ3 } where θ¯ is the predicted azimuthal position of the target returned by the tracking stage. More precisely, we assume that nθ ∈ N = {nθ¯ − N, . . . , nθ¯ + N}. A straightforward approach would be to maximize the above statistic, given by (29), also with respect to nθ ∈ N and ν ∈ {0, 1/M, . . . , (M – 1)/M} with M an integer greater than or equal to N. Notice also that, strictly speaking, the resulting detector is no longer a GLRT because it uses different data for each value of nθ . More important, we have now to implement a three-dimensional search. In order to avoid such a computationally intensive optimization, we propose the following heuristic procedure: 1) first, estimate ν ∈ {0, 1/M, . . . , (M – 1)/M}, given nθ ∈ N , resorting to the ML estimator of ν (under the H1

˜ ν, θ) = v( ,

χp −Tp − , Tν v (ν, θ) χp − , Tν v (ν, θ) 0 0 χp − , Tν v (ν, θ) χp Tp − , Tν v (ν, θ)

and the hypothesis test becomes H0 : z˜ (nθ ) = n˜ ˜ ν, θ) + n˜ H1 : z˜ (nθ ) = α v( ,

(28)

Obviously, n˜ ∼ CN3N (0, σ 2 I 3N ). It is not difficult to derive the corresponding GLRT. To this end, it is important to observe that the unknown parameters (leaving aside, for the moment, ν and θ) are σ 2 , α, and under H1 and σ 2 under H0 . Moreover, maximization of the probability density function (pdf) of the observables under the H1 hypothesis is performed in closed form with respect to σ 2 and α, given . It is not difficult to show that the corresponding decision rule is given by [8] † 2 z˜ (nθ )v( , H1 ˜ ν, θ) max † ≷γ (29)

˜ ν, θ) H0 z˜ (nθ )˜z (nθ ) v˜ † ( , ν, θ)v( , where ∈ (–Tp /2, Tp /2); † and | · | denote conjugate transpose and the modulus of a complex number, respectively, while γ is the detection threshold to be set according to the desired probability of false alarm (Pfa ). BANDIERA ET AL.: MULTITARGET RANGE-AZIMUTH TRACKER

−Tp /2 ≤ ≤ 0

(27)

0 < ≤ +Tp /2

hypothesis) for the case of a target located at the center of the lth range cell and known θ, νˆ (nθ ) for example; it is given by †

νˆ (nθ ) = arg max |z l (nθ )v(ν, θ)|2

(31)

ν

and can also be computed via fast Fourier transform (FFT) over NFFT ≥ N points; 2) then, resort to the following decision rule

max max nθ ∈N

∈E

† 2 z˜ (nθ )v( , ˜ νˆ (nθ ), θ)

H1

≷γ ˜ νˆ (nθ ), θ) H0 z˜ † (nθ )˜z (nθ ) v˜ † ( , νˆ (nθ ), θ)v( , (32)

Notice that this decision statistic is obtained by that for known ν and θ by maximizing with respect to nθ after replacing ν with νˆ (nθ ). The output of the detection stage is a decision on the hypothesis actually in force or, equivalently, a value of the binary variable D that takes on values zero and one (D = i is tantamount to choosing Hi , i = 0, 1); in addition, if D = 1519

1, the detector also returns the estimated value of and θ,

ˆ and θˆ for example, and of ν, νˆ (that is not forwarded to the tracking stage). The tracking stage will use ˆ and θˆ to compute the measurement. It is worth stressing that the above detection strategy can be extended to attack detection of a single target within a larger area, namely considering a certain number of adjacent range-azimuth cells. Such a generalized approach might be effective when the information on the range cell l the target belongs to and/or the uncertainty sector we assume (i.e., the value of θ¯ ) is the result of a not too reliable prediction on the actual position of a (previously detected) target; in fact, such a prediction might be affected by an estimation error, due, for instance, to a target that has initiated a maneuver or to an adverse scenario in terms of signal-to-noise ratio (SNR) value. Thus, for future reference, we also consider the following hypotheses testing problem ⎧ H0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

and θ i (t) range and azimuth of the ith target, respectively, and suppose that the azimuth is approximately constant, θ i for example, over the coherent processing interval. We leave aside, for the moment, the fact that the v i s (and, hence, the normalized Doppler frequency shifts νi = 2vc i fc · T ) are unknown parameters. As to the θ i s, they are unknown parameters too; however, we suppose that θm =

However, it is also important to stress that a test involving a larger area in range might require processing additional samples in range (depending on the actual area under consideration). As a final comment, observe that the GLRT and its ad hoc implementations guarantee the constant false alarm rate (CFAR) property with respect to σ 2 . 2) Multiple Target Detector: Assume that closely spaced (point-like and slowly fluctuating) targets are moving with constant radial velocity (and in the antenna far field) within three consecutive range cells, indexed by l – 1, l, l + 1; v i is the velocity of the ith target (v i > 0 if the ith target is approaching the radar). We denote by ri (t) 1520

(35)

with θmin = min θi and θmax = max θi , is approximately i

i

known. Due to the presence of spillover there might be target energy also in the matched filter outputs corresponding to the (l – 2)th and the (l + 2)th range cells. Then, it is reasonable to process samples at the

: z i (nθ ) = ni , i = l − 1, l, l + 1 ⎧ z l−1 (nθ ) = αχp −Tp − , Tν v(ν, θ) + nl−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z l (nθ ) = αχp − , Tν v(ν, θ) + nl , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z (n ) = nl+1 , ⎪ ⎪ ⎪ l+1 θ ⎪ ⎨ ⎪ ⎪ H1 : or ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z (n ) = nl−1 , ⎪ ⎪ ⎪ ⎪ ⎪ l−1 θ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z l (nθ ) = αχp − , ν v(ν, θ) + nl , ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ν ⎪ ⎪ ⎪ ⎩ ⎩ z l+1 (nθ ) = αχp Tp − , T v(ν, θ) + nl+1 ,

assuming that nθ ∈ N = {nθ¯ − N, . . . , nθ¯ + N}. A solution to the above problem is still given by (32), but replacing E with 3N n − N − 1/2 E = Tp (34) 2N n=−N +1

θmin + θmax 2

−Tp ≤ ≤ 0 (33)

0 < ≤ +Tp

output of the matched filter at the instants tj,n = tmin + (j – 1)Tp + nT, j = l − 2, . . . , l + 2, n = nθm − 3N/2, . . . , nθm + 5N/2 − 1, [again, with nθm obtained by (6) with θ m in place of θ(t)] in order to collect most of the energy backscattered by the targets (the actual pulses to be used depend on the actual targets’ azimuthal location). Thus, for the case at hand, time samples are grouped to form 4N-dimensional vectors as follows z j = z(tj,nθm −3N/2 ) · · · z(tj,nθm +5N/2−1 ) = sj + nj , (36) j = l − 2, . . . , l + 2 with sj the signal component and nj the noise component. It is not difficult to show, following the lead of [6, 7], that the contribution si,j ∈ C4N×1 of the ith target to the jth range cell, j = l – 2, . . . , l + 2, up to a complex factor α i taking into account target and channel effects, is given by


si,l−2 (l − 1, i , νi , θi ) si,l−1 (l − 1, i , νi , θi ) si,l (l − 1, i , νi , θi )

=

χp −Tp − i , νTi v(νi , θi )

−

=0

si,l+2 (l − 1, i , νi , θi )

=0

≤ i ≤ 0

0 < i ≤

0 = χp − i , νTi v(νi , θi ) 0 = χp Tp − i , νTi v(νi , θi )

si,l+1 (l − 1, i , νi , θi )

Tp 2

Tp 2 T − 2p

−

≤ i ≤

Tp 2 Tp −2

Tp 2

≤ i ≤ 0

0 < i ≤ −

Tp 2

(37)

Tp 2

≤ i ≤ ≤ i ≤

Tp 2 Tp 2

if the target is in the (l – 1)th range cell. Obviously, i denotes the “residual delay” from the center of the resolution cell the ith target belongs to and v(ν i , θ i ) is the spatio-temporal steering vector. Assuming that 3N θ3 2 (38) θi ∈ d = θm + r N r=− 3N 2

where

θm

is given by (24), with nθm in place of nθ , the spatio-temporal steering vector can be expressed as 1 v(νi , θi ) = √ [0 · · · 0 cθi (nθi ) cθi (nθi + 1)eI 2π νi · · · cθi (nθi + N − 1)eI 2π (N−1)νi 0 · · · 0] N 3N/2+r

(39)

3N/2−r

N

Similarly, the contribution of the ith target to the range cells from (l – 2) to (l + 2) (up to the complex factor α i ) is given by si,l−2 (l, i , νi , θi )

=0

si,l−1 (l, i , νi , θi )

=

si,l (l, i , νi , θi ) si,l+1 (l, i , νi , θi ) si,l+2 (l, i , νi , θi )

χp −Tp − i , νTi v(νi , θi )

Tp 2 Tp −2

−

≤ i ≤

≤ i ≤ 0

0 < i ≤


Tp 2 T − 2p

=0

−

−

Tp 2

≤ i ≤

Tp 2

(40)

≤ i ≤ 0

0 < i ≤ Tp 2

Tp 2

Tp 2

≤ i ≤

Tp 2

or si,l−2 (l + 1, i , νi , θi )

=0

si,l−1 (l + 1, i , νi , θi )

=0

si,l (l + 1, i , νi , θi ) si,l+1 (l + 1, i , νi , θi ) si,l+2 (l + 1, i , νi , θi )

=

Tp 2 Tp −2 T − 2p

−

χp −Tp − i ,

νi T

v(νi , θi )


≤ i ≤ ≤ i ≤

≤ i ≤ 0

0 < i ≤ T − 2p T − 2p

Tp 2 Tp 2

Tp 2

≤ i ≤

(41) Tp 2

≤ i ≤ 0

0 < i ≤

Tp 2

for a target in the lth or (l + 1)th range cell, respectively. Suppose that at most two targets are present. The generalization to an arbitrary, albeit known, number of targets is straightforward. It follows that the signal component sj is given by sj (j1 , j2 , 1 , 2 , ν1 , ν2 , θ1 , θ2 , α1 , α2 ) =

2

αi si,j (ji , i , νi , θi ), j = l − 2, . . . , l + 2

(42)

i=1


1521

where ji ∈ {l – 1, l, l + 1} denotes the range cell the ith target belongs to. We also assume that signal returns are buried in zero mean, complex normal noise with covariance matrix σ 2 I4N , independent from range cell to range cell. Summarizing, the problem at hand is that of detecting the possible presence of two targets and of estimating the parameters of interest, namely j1 , j2 , 1 , 2 , θ 1 , and θ 2 , together with the nuisance parameters α 1 , α 2 , and σ 2 . In order to formulate and eventually solve the estimation problem we can now define an augmented received vector z˜ = [z l−2 z l−1 z l z l+1 z l+2 ] ∈ C20N×1

(43)

Accordingly, we can define an augmented steering vector, corresponding to the ith target, given by ai (ji , i , νi , θi ) = [s i,l−2 (ji , i , νi , θi ) s i,l−1 (ji , i , νi , θi ) s i,l (ji , i , νi , θi )

s i,l+1 (ji , i , νi , θi ) s i,l+2 (ji , i , νi , θi )] ∈ C

20N×1

(44)

In addition, the hypothesis test to be solved is

H0 : z˜ = n˜ H1 : z˜ = 2i=1 αi ai (ji , i , νi , θi ) + n˜ = A(b1 , b2 )b3 + n˜ (45)

where PA (b1 , b2 ) is the projection matrix onto the space spanned by the columns of the matrix A(b1 , b2 ), i.e., −1 P A (b1 , b2 ) = A(b1 , b2 ) A† (b1 , b2 ) A(b1 , b2 ) A† (b1 , b2 ) (51) and γ (from now on) denotes any modification of the original threshold. Equivalently, the GLRT can be rewritten as max b1

(46)

is assumed to be a full-column-rank matrix, with b1 = [j1 j2 1 2 θ1 θ2 ] , b2 = [ν1 ν2 ] , and b3 = [α1 α2 ] . Obviously, n˜ ∼ CN20N (0, σ 2 I 20N ) with σ 2 an unknown parameter. Thus, leaving aside for the moment the unknown parameters ν 1 , ν 2 , the above hypothesis test can be solved resorting to the GLRT that is given by maxb1 ,b3 ,σ 2 f1 (˜z ; b1 , b2 , b3 , σ 2 ) H1 ≷γ maxσ 2 f0 (˜z ; σ 2 ) H0

(47)

where fi denotes the pdf of the observables under the Hi hypothesis, i = 0, 1, and γ is the threshold to be set according to the desired Pfa . Notice that f1 (˜z ; b1 , b2 , b3 , σ 2 ) 1 = (πσ 2 )20N 1 × exp − 2 [˜z − A(b1 , b2 )b3 ]† [˜z − A(b1 , b2 )b3 ] σ (48) and

1 1 † exp − 2 z˜ z˜ f0 (˜z ; σ ) = (πσ 2 )20N σ 2

(49)

It is not difficult to come up with the following form for the GLRT max b1

1522

H1 z˜ † z˜ ≷γ z˜ z˜ − z˜ P A (b1 , b2 )˜z H0 †

†

(50)

(52)

It still remains to overcome the uncertainty on ν 1 and ν 2 . To this end, we resort to a suboptimum procedure. In fact, neglecting the actual expression of the spillover term to adjacent range cells, we can model the received signal, corresponding to the jth range cell, j = l – 2, . . . , l + 2, as z j = [v(ν1 , θ1 ) v(ν2 , θ2 )] uj + nj ∈ C4N×1

(53)

with θi ∈ d and v given by (38) and (39), respectively, uj = α1,j α2,j , and nj ∼ CN4N 0, σ 2 I 4N . It turns out that the ML estimate of b2 , bˆ 2 = [ˆν1 , νˆ 2 ] , is given by bˆ 2 = arg max max ν1 ,ν2

θ1 ,θ2

where A(b1 , b2 ) = [a1 (j1 , 1 , ν1 , θ1 ) a2 (j2 , 2 , ν2 , θ2 )]

z˜ † P A (b1 , b2 )˜z H1 ≷γ z˜ † z˜ H0

l+2

†

z j P(ν1 , ν2 , θ1 , θ2 )z j

(54)

j =l−2

where P(ν 1 , ν 2 , θ 1 , θ 2 ) is the projection matrix onto the space spanned by the columns of the matrix [v(ν1 , θ1 ) v(ν2 , θ2 )] , ν1 = ν2 , νi ∈ {0, 1/M, . . . , (M − 1)/M}, θ1 = θ2 , θi ∈ d . Then, we plug the estimated pair of normalized Doppler frequencies bˆ 2 into the above detector (52), in place of b2 , thus obtaining max b1

z˜ † P A (b1 , bˆ 2 )˜z H1 ≷γ z˜ † z˜ H0

(55)

A few final remarks are in order. First, observe that maximization over b1 = [j1 j2 1 2 θ 1 θ 2 ] cannot be conducted in closed form. We resort to a grid search to maximize with respect to 1 and 2 . To this end, we assume that 1 and 2 take on values in the set given by (30). As to the possible values assumed by the θ i s, again we refer to the set obtainable using (38) and use the corresponding steering vector given by (39). Second, the GLRT and its ad hoc implementation are CFAR with respect to σ 2 . Finally, notice that the generalization of the procedure to an arbitrary (but known) number of targets is in principle straightforward. However, the computational complexity of the grid search would increase exponentially with the number of targets. B. Tracking Stage

Assume that radar data are available over [0, + ∞). Moreover, assume that the motion of each target can be modeled according to a nearly constant velocity model along x and y. More precisely, we suppose that the kinematics of the ith target is ruled by the following state


space model

x i (k + 1)

= Fx i (k) + w1i (k)

yi (k)

= H x i (k) + w2i (k)

(56)

with xi (k) = [xi (k) v xi (k) yi (k) v yi (k)] where xi (k) and yi (k) denote the Cartesian coordinates of the target over the time interval2 (tk,i , tk,i + NT); v xi (k) and v yi (k) are the target velocity components along x and y over the same time interval; xi (k + 1) = [xi (k + 1) v xi (k + 1) yi (k + 1) v yi (k + 1)] is the vector of Cartesian coordinates and velocity components over (tk + 1,i , tk + 1,i + NT), with tk + 1,i – tk,i approximately equal to Ta = 2 π/|ωa |; yi (k) is the vector of the noisy Cartesian coordinates of the target over the time interval (tk,i , tk,i + NT) F = diag (F 1 , F 1 ) with

F1 =

and

H=

1

Ta

(58)

1

1

0

0

0

0

0

1

0

(59)

An important point to be stressed here is that the detection stage returns range and azimuth pairs; conversion of the polar measurements to a Cartesian frame raises several issues. One problem is that the classical conversion is biased [9–11]; a second problem is that the computation of the converted measurement error covariance would require true range and bearing. Practical approaches to the estimate of the converted measurement error covariance, i.e., using the measurements, result in correlation between such an estimate and the measurement error [13]. In the attempt to overcome such drawbacks, we resort to the unbiased conversion proposed in [13], namely we adopt the so-called decorrelated unbiased converted measurements (DUCM) for target tracking3 2

xm = eσα /2 rm cos αm 2

ym = eσα /2 rm sin αm

E[w1i (k)w 1i (j )] = Qδkj and E[w2i (k)w 2i (j )] = Rδkj (62) with E denoting statistical expectation and Q = diag Q 1 , Q 1 (63) where

We assume that during the time interval (tk,i , tk,i + NT) the target is within the (−3 dB) antenna beamwidth. Obviously, the tk,i s depend on the azimuthal position of the ith target over time. Remember also that N is the number of processed pulses and T the PRT. 3 We neglect the antenna elevation. BANDIERA ET AL.: MULTITARGET RANGE-AZIMUTH TRACKER

1 2 T 2 a

(64)

Ta

R 22 = E ym2 |rt , αt − E 2 [ym |rt , αt ]

(66)

R 12 = E [xm ym |rt , αt ] − E [xm |rt , αt ] E [ym |rt , αt ] (67) with E [xm |rt , αt ] = rt cos αt e−σαt /2

(68)

E [ym |rt , αt ] = rt sin αt e−σαt /2

(69)

2

2

1 2 E xm2 |rt , αt = eσα rt2 + σr2 + σr2t 2 2 2 (70) × 1 + e−2(σα +σαt ) cos(2αt )

(61)

2

T3 3 a 1 2 T 2 a

is obtained discretizing a continuous white noise acceleration model with power spectral density of level q [2, 11]. As to δ kj , it is the Kronecker symbol. The parameter q can be set based on the maximum tolerable difference between the nominal and the actual target velocity component (along x or y) δv max over the “sampling interval” Ta . Finally, we computed R according to (Rij is the (i, j) th entry of the matrix R) (65) R 11 = E xm2 |rt , αt − E 2 [xm |rt , αt ]

(60)

where rm and α m are polar measurements and xm and ym the corresponding (converted) Cartesian measurements; σα2 is the variance of α m . Notice though that validity of the above formulas relies on the assumption that range and bearing measurement noises are uncorrelated, zero mean, and Gaussian with variances σr2 and (as already stated) σα2 , respectively. This is not actually true in our scenario where

1 Q1 = q

(57)

0

measurements are obtained as the output of the detection stage. As to the overall set of the w1i (k)s and the w2j (k)s, namely the set of the plant and measurement noise components of the targets, they are assumed to be (approximately) zero-mean, independent normal sequences. Moreover, we assume that

1 2 E ym2 |rt , αt = eσα rt2 + σr2 + σr2t 2 2 2 × 1 − e−2(σα +σαt ) cos(2αt ) (71) and E [xm ym |rt , αt ] 1 2 2 2 = eσα rt2 + σr2 + σr2t e−2(σα +σαt ) sin(2αt ) 2

(72)

where rt and α t denote the one-step predicted values of range and azimuth while σr2t and σα2t are the associated variances as indicated and computed in [13]. 1523

In the following, we describe the main ideas underlying the proposed multitarget tracker. To this end, we denote by Nk , k ≥ 2, the set of (positive) integers indexing the tracks that are active4 over the kth antenna scan (i.e., confirmed and to-be-confirmed). The following are the main actions performed by the tracker. 1) Over the kth scan, for the ith track, i ∈ Nk , we compute the one step prediction of the state xˆ i (k + 1|k) and the associated error covariance matrix by implementing the Kalman filter (KF) equations xˆ i (k + 1|k) = F xˆ i (k|k),

yˆ i (k + 1|k) = H xˆ i (k + 1|k) (73)

P i (k + 1|k) = Q + F P i (k|k)F

(74)

in particular, the above recursion returns an estimate of the position of the target at tk + 1,i (actually at an estimate of tk + 1,i ). 2) Based on the predicted positions of targets over the (k + 1)th scan (corresponding to active tracks over the kth scan) we construct a partition of Nk , Nk1 , . . . , Nkmk , in order to determine targets that might be in close spatial proximity and, hence, groups of range-azimuth cells to be jointly processed by the detection stage; the partition is constructed according to a given criterion (to be described in the following). j 3) If the cardinality of Nk , j = 1, . . . , mk , is greater than one, the multitarget decision scheme is used to detect j and localize the targets whose indices are defined by Nk (remember that, although we have described the detector to be used in case of two targets in close spatial proximity, its generalization to an arbitrary, but known, number of targets is in principle straightforward); otherwise, we resort to the single target detector. 4) To manage possible ambiguities due to targets in close spatial proximity and, more generally, the data association problem, we resort to the nonparametric JPDA algorithm. JPDA updates the estimates using possible measurements constructed over the (k + 1)th scan. For the implementation of the JPDA we assume that the number of targets at scan k + 1 is known and equal to the number of tracks confirmed or to-be-confirmed at scan k (see step 5). Accordingly, the sets of indices labeling measurements used by the JPDA, Mik+1 , i ∈ Nk , are constructed based on measurements computed by any single or multiple target detector applied at scan k + 1 (see step 3). Moreover, we calculate the probability β mi that measurement zmi is originated from target i, i ∈ Nk , m ∈ Mik+1 , according to [2, paragraph 9.3]. Finally, JPDA updates the target states as

4

It is important to stress again that the actual value of tk,i depends on the position of the ith target within the surveillance area over the k scan. Moreover, notice that two closely spaced targets return two closely spaced time instants or time instants whose difference is on the order of Ta .

1524

Si (k + 1) = H P i (k + 1|k)H + R −1

W i (k + 1) = P i (k + 1|k)H Si (k + 1) ν mi (k + 1) = z mi (k + 1) − yˆ i (k + 1|k) ν i (k + 1) = βmi (k + 1)ν mi (k + 1)

(75)

m∈Mik+1

xˆ i (k + 1|k + 1) = e−σαt ( xˆ i (k + 1|k) + W i (k + 1)ν i (k + 1)) 2

where the last equation incorporates the scaling factor of [13] and the error covariance matrices (associated with the updated estimate of the states) as β0i (k + 1) = 1 −

βmi (k + 1)

m∈Mik+1

⎡

P i (k + 1) = W i (k + 1) ⎣

βmi ν mi (k + 1)νmi (k + 1)

m∈Mik+1

− ν i (k + 1)ν i (k + 1) Wi (k + 1)

(76)

P Ci (k + 1|k + 1) = [I − W i (k + 1)H] P i (k + 1|k) P i (k + 1|k + 1) = β0i (k + 1) P i (k + 1|k) + [1 − β0i (k + 1)] P Ci (k + 1|k + 1) + P i (k + 1)

5) The state of each track is updated; more precisely, a to-be-confirmed track can be confirmed (otherwise, it is deleted) using an mc /nc rule and, similarly, a confirmed track can be deleted using an md /nd rule. More precisely, we confirm a track the first time the number of detections over the last nc scans is greater than or equal to mc . Similarly, we resort to an md /nd rule for track deletion: a confirmed track is deleted the first time the number of detections over the last nd scans is less than or equal to md . j In addition, tracks in Nk , j = 1, . . . , mk , are compared to recognize the possible presence of ghost tracks: j ∀r, s ∈ Nk , r < s, j = 1, . . . , mk , if the rth track is close in position and velocity to the sth track, the former is assumed to be a ghost of the latter. In formulas, if [(xˆr (k|k) − xˆs (k|k))2 + (yˆr (k|k) − yˆs (k|k))2 ]1/2 is less than a threshold value Thpos and [(vˆxr (k|k) − vˆxs (k|k))2 + (vˆyr (k|k) − vˆys (k|k))2 ]1/2 is less than a threshold value Thvel the rth track is deleted. 6) After existing tracks have been updated, it is possible to run a more conventional detector over remaining range-azimuth cells belonging to the surveillance region. Range-azimuth cells “occupied by targets” associated with active tracks, but also cells surrounding the above estimated target positions, are not scanned by the conventional detector; notice that it is necessary to set guard cells (not fed to the more conventional detector) to avoid that the spillover due to (already detected) strong targets might originate additional false targets. Thus, if lˆ and θˆ are the estimated range cell and azimuthal position of the target corresponding to an active track, respectively, the conventional detector does ˆ lˆ − 1, and lˆ + 1 together with (at not scan range cells l,


least) pulses nθˆ − N/2, . . . , nθˆ + 3N/2 − 1. The more conventional detector is the cascade of two detectors: the observables corresponding to each range-azimuth cell within the area of interest are fed to a first detection stage designed assuming the presence of a target “matched to” the lth range cell and the hth azimuth cell, namely at range c rl = (tmin + (l − 1)Tp ) (77) 2 with azimuth θh = θ0a + hθ3

(78)

If a target is detected, setting the threshold to guarantee Pfa = 10−2 , the observables are fed to detector (32). Again, the threshold is set to guarantee Pfa = 10−2 . Actually, for low SNR values, the area of interest is scanned for the possible presence of targets resorting to the so-called overlapping-window-based search mode. In other words, we apply the test over data corresponding to partially overlapped range-azimuth cells by shifting the selection window of N/3 pulses in azimuth. Such an approach is obviously more complex than a non-overlapping-window-based search mode, but it guarantees a better match to the design assumptions of the first stage detector. 7) The tracker initiates potential tracks over the (k + 1)th scan in case of detections (which come from the detector described in the previous item) over the kth and the (k + 1)th scans corresponding to range-azimuth cells in spatial proximity. More precisely, assume that a new target has been detected over the kth and the (k + 1)th ˜ ˜ ˜ + 1), y(k ˜ + 1)) scan and denote by (x(k), y(k)) and (x(k its estimated position over the kth and the (k + 1)th scan, respectively; moreover, assume that the Euclidean distance between the above points is less than v max Ta , with v max denoting the maximum admissible target velocity; then, such measurements can be interpreted as originated by the same target and give rise to a new (to be confirmed) track:

It still remains to specify how to partition Nk . To this end, denote by d(x, y) the Euclidean distance between x ∈ R2 and y ∈ R2 and by d0 > 0 a given threshold. j A reasonable partition Nk , j = 1, . . . , mk , should be such that 1) ∀j1 , j2 ∈ {1, . . . , mk }, j1 = j2 , it is true that j

j

∀r ∈ Nk 1 , ∀s ∈ Nk 2 d( yˆ r (k + 1|k), yˆ s (k + 1|k)) > d0 (80) j

2) each Nk cannot be partitioned into two (or more) subsets, Aj and Bj say, such that the distance between “any element of Aj and any element of Bj ” is greater than d0 . In practice, we have to construct at the (k + 1)th scan a graph where each vertex represents a target (associated with an active track) and an undirected edge from one vertex to another means that the (estimated) distance between the two targets is not greater than d0 . Any single connected component of the graph corresponds to targets that should be jointly processed by the detector. Notice also that the distance between targets might be computed taking into account the available information (i.e., predicted state covariances) on the accuracy of the predictions; however, we decided not to use such information to be conservative in the presence of undetected maneuvers. In principle, if the overall graph were a single connected component, all of the targets should be fed to the same multiple target detector. It is clear that, although the above way to partition Nk is reasonable to handle mutual interference between targets, it is very demanding from the computational point of view. Thus, we investigate another partition rule that, leading to sets consisting of one or two elements, keeps the computational complexity of the detection stage at a lower level; this latter rule is the following (we assume that the cardinality of Nk is greater than one).

xˆ i (k + 1|k + 1) ˜ + 1) − x(k) ˜ ˜ + 1) − y(k) ˜ x(k y(k ˜ + 1) ˜ + 1) = x(k y(k Ta Ta (79)

1) Set N = Nk and j = 1. 2) Construct the set of distances D between any two elements yˆ r (k + 1|k) and yˆ s (k + 1|k) of the set

More generally, in the presence of multiple detections over the kth scan and/or the (k + 1)th scan, we initiate new tracks for a subset of available candidates. To this end, we compute all distances between detections over consecutive scans and, among those compatible with the maximum admissible target velocity, choose the pair with the smallest distance. Then, after disregarding all pairs having the first or the second point in common with the selected one, we choose the pair with the smallest distance among the survivors, and so on. The same rationale is used to initiate tracks over the second scan based on possible detections over the first two scans (which come from the detector described in the previous item). At this point Nk+1 contains the set of (positive) integers indexing the active tracks over the (k + 1)th scan.

that satisfy the following additional constraints


{ yˆ i (k + 1|k), i ∈ N } |lr − ls | ≤ 2,

|nθr − nθs | ≤ 3N

(81)

(82)

where li and θ i denote the predicted range cell and azimuthal position of the target corresponding to the ith track in N . 3) Compute the minimum among such distances, i.e., dj = min D (remember that the minimum is equal to + ∞ for an empty set): if such a minimum is different from + ∞, let j

Nk = {r1 , r2 }

(83)

where r1 and r2 are such that d( yˆ r1 (k + 1|k), yˆ r2 (k + 1|k)) = dj ;

(84) 1525

otherwise construct a singleton for any element of { yˆ r (k + 1|k), r ∈ N }

(85)

and quit the loop. 4) Compute the new N as N = N \{r1 , r2 }

(86)

if N is an empty set quit the loop; if, instead, the cardinality of N is one, let j +1

Nk

= {r, r ∈ N }

(87)

and quit the loop; otherwise, let j = j + 1 and return to step 2. IV. PERFORMANCE ASSESSMENT

The aim of this section is to assess the performance of the multiple target detector and eventually that of the overall tracker resorting to Monte Carlo simulation. In order to evaluate the thresholds necessary to ensure the preassigned value of Pfa = 10−2 we resort to 10/Pfa independent trials. The assessment of the multiple target detector is the object of the next subsection while Subsection IV-B focuses on the proposed tracker.

Fig. 1. Pd vs SNR1 for proposed detector. Case 1 (SNR2 = SNR1 ): solid curve; case 2 (SNR2 = SNR1 + 6 dB): solid curve with circle at each data point; case 3 (SNR2 = SNR1 + 10 dB): solid curve with star at each data point.

A. Multiple Target Detector

The performance of the multiple target detector (designed assuming two targets) is assessed in terms of probability of detection (Pd ) values and of RMS errors in range and azimuth. In order to evaluate the Pd s we resort to 500 independent trials. In addition, we use all the available trials to estimate the RMS errors (also when the detection statistics do not cross the threshold). For simulation purposes, target positions and Doppler frequency shifts are generated (independent from trial to trial) and uniformly distributed within the search area. All the illustrative examples assume the parameters of Table I and N = 5. Moreover, we define the SNR for the ith target as SNRi = |αi |2 /σ 2 with σ 2 = 1 and consider the following three cases. Case 1: targets have the same SNR (namely SNR1 = SNR2 ). Case 2: targets have different SNR values; the stronger one is 6 dB above the weaker one, i.e., SNR2 = SNR1 + 6 dB. Case 3: targets have different SNR values; the stronger one is 10 dB above the weaker one, i.e., SNR2 = SNR1 + 10 dB. In Fig. 1 we plot the Pd s vs SNR1 : notice that curves with a marker at each data point refer to targets with different SNR values. Thus, the improvement with respect to the solid curve is due to the presence of a stronger target (SNR2 greater than SNR1 ). Figs. 2–3 show that the localization capabilities of the multiple target detector are negatively affected by the presence of a stronger target; even for SNR2 = SNR1 + 6 dB the detector does not satisfactorily localize the weaker target. This is probably 1526

Fig. 2. RMS error in range vs SNR1 for proposed detector. Case 1 (SNR2 = SNR1 ): solid curves; case 2 (SNR2 = SNR1 + 6 dB): solid curves with circle at each data point; case 3 (SNR2 = SNR1 + 10 dB): solid curves with star at each data point.

Fig. 3. RMS error in azimuth vs SNR1 for proposed detector. Case 1 (SNR2 = SNR1 ): solid curves; case 2 (SNR2 = SNR1 + 6 dB): solid curves with circle at each data point; case 3 (SNR2 = SNR1 + 10 dB): solid curves with star at each data point.


Fig. 4. Simulated trajectories.

Fig. 6. Tracking results assuming strong target (SNR = 30 dB) and weaker one (SNR = 20 dB). Actual trajectories in blue, estimated ones in green for proposed tracker and in red for competitor.

Fig. 5. Tracking results assuming targets with equal strength. SNR = 20 dB. Actual trajectories in blue, estimated ones in green for proposed tracker and in red for competitor.

due to the suboptimum implementation of the GLRT that resorts to (54) to estimate the normalized Doppler frequencies.

Fig. 7. Tracking results assuming targets with equal strength. SNR = 20 dB. Actual trajectories in blue, estimated ones in green for proposed tracker and in red for competitor.

B. Tracking Performance

All the illustrative examples assume the parameters of Table I, N = 5, q = 1 m2 /s3 , σα2 = 7.6 × 10−7 rad2 , σr2 = 4m2 , mc = 5, nc = 8, md = 1, nd = 8, v max Ta = 60 m, Thpos = 3 m, and Thvel = 1 m/s. The true trajectories of the targets are depicted in Fig. 4 and assume a constant acceleration model; markers indicate the position of each target at scans multiple of 5. Notice the crossing of three targets around scan 50 and, concerning the two targets moving along parallel trajectories, the overtaking by the left one of the right one between scans 45 and 55. Figs. 5, 7, and 8 refer to the case of targets with the same strength, while in Fig. 6 one of the two targets moving along parallel trajectories is weaker than the other (the one on the right); similarly, in Fig. 9 one of the three targets whose trajectories cross each other is weaker than the others (the one moving from top-right to bottom-left). For BANDIERA ET AL.: MULTITARGET RANGE-AZIMUTH TRACKER

Fig. 8. Tracking results assuming targets with equal strength. SNR = 30 dB. Actual trajectories in blue, estimated ones in green for proposed tracker and in red for competitor. 1527

assumed a maritime scenario: the proposed tracker has been compared with a JPDA fed by measurements obtained via a single target detector; remarkably, the former tends to outperform the latter when target trajectories are very close. Ongoing research activity is aimed at extending the proposed approach to the case disturbance is modeled as a correlated, possibly non-Gaussian, stochastic process.

REFERENCES [1]

Fig. 9. Tracking results assuming two strong targets (SNR = 30 dB) and weaker one (SNR = 20 dB). Actual trajectories in blue, estimated ones in green for proposed tracker and in red for competitor.

[2]

[3]

comparison purposes the performance of a tracker that employs the single target detector even for targets in spatial proximity is considered too; such a tracker still uses the JPDA to solve the data association problem. Figs. 5–9 show the superiority of the proposed tracker with respect to the competitor in case of targets of equal strength: the former is often able to estimate the tracks even when targets are in close spatial proximity while the latter may lose one of them. However, depending on the specific noise realization, the proposed tracker may fail tracking one of the targets; for instance, Fig. 6 shows that it loses the target on the right. This behavior is probably due to the different strength of the targets and, in addition, to the fact that the implemented detectors estimate target Doppler frequencies via simplified solutions. In other cases, not reported for the sake of brevity, the proposed tracker suffers in the case of crossing of two strong targets and a weaker one; notice that the optimal tracker should use a detector designed for three targets and not, as in the case of our implementation, a single target detector and a multiple target detector for the two remaining ones.

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V. CONCLUSIONS

In this paper, we have focused on the design of a Kalman filter-based algorithm to track multiple targets in range and azimuth. The proposed tracker is aimed at collecting and eventually processing all target measurements in both range and azimuth. To this end, it relies on a detection stage formed by a family of detectors, designed to take into account target strength over multiple range-azimuth cells and possible targets in spatial proximity. Existing tracks are updated resorting to a JPDA to handle the data association problem. New targets can give rise to new tracks at any scan resorting to a more conventional detector. We have modeled target echoes as coherent pulse trains embedded in white Gaussian noise with unknown power. The performance analysis has 1528

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Fortmann, T. E., Bar-Shalom, Y., and Scheffe, M. Sonar tracking of multiple targets using joint probabilistic data association. IEEE Journal of Oceanic Engineering, 8, 3 (July 1983), 173–184. Bar-Shalom, Y., and Fortmann, T. E. Tracking and Data Association. New York, NY USA: Academic Press, 1988. Blackman, S. S. Multiple-Target Tracking with Radar Applications. Boston, MA USA: Artech House, 1986. Zhang, X., Willett, P. K., and Bar-Shalom, Y. Monopulse radar detection and localization of multiple unresolved targets via joint bin processing. IEEE Transactions on Signal Processing, 53, 4 (Apr. 2005), 1225–1236. Zhang, X., Willett, P. K., and Bar-Shalom, Y. Detection and localization of multiple unresolved extended targets via monopulse radar signal processing. IEEE Transactions on Aerospace and Electronic Systems, 45, 2 (Apr. 2009), 455–472. Orlando, D., and Ricci, G. Adaptive radar detection and localization of a point-like target. IEEE Transactions on Signal Processing, 59, 9 (Sept. 2011), 4086–4096. Bandiera, F., Mancino, M., and Ricci, G. Localization strategies for multiple point-like radar targets. IEEE Transactions on Signal Processing, 60, 12 (Dec. 2012), 6708–6712. Del Coco, M., Orlando, D., and Ricci, G. A tracking system exploiting interaction between a detector with localization capabilities and the KF. IEEE Transactions on Signal Processing, 60, 11 (Nov. 2011), 6031–6036. Lerro, D., and Bar-Shalom, Y. Tracking with debiased consistent converted measurements versus EKF. IEEE Transactions on Aerospace and Electronic Systems, 29, 3 (July 1993), 1015–1022. Longbin, M., Xiaoquan, S., Yiyu, Z., and Bar-Shalom, Y. Unbiased converted measurements for tracking. IEEE Transactions on Aerospace and Electronic Systems, 34, 3 (July 1998), 1023–1027. Bar-Shalom, Y., Li, X. R., and Kirubarajan, T. Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software. New York, NY USA: Wiley, 2001. Duan, Z., Han, C., and Li, X. R. Comments on unbiased converted measurements for tracking. IEEE Transactions on Aerospace and Electronic Systems, 40, 4 (Oct. 2004), 1374–1377. Bordonaro, S. V., Willett, P., and Bar-Shalom, Y. Unbiased tracking with converted measurements. In Proceedings of 2012 IEEE Radar Conference, Atlanta, GA, USA, May 2012.


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Francesco Bandiera (M’01) was born in Maglie (Lecce), Italy, on March 9, 1974. He received the Dr. Eng. degree in computer engineering and the Ph.D. degree in information engineering from the University of Salento (formerly University of Lecce), Lecce, in 2001 and 2005, respectively. From July 2001 to Feb. 2002, he was with the University of Sannio, Benevento, Italy, where he was engaged in a research project on mobile cellular telephony. Since Dec. 2004, he has been an Assistant Professor with the Dipartimento di Ingegneria dell’Innovazione, University of Salento, where he is engaged in teaching and research. He has held visiting positions with the Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO, USA (Sept. 2003-Mar. 2004), and with the Department of Avionics and Systems, Institut Supérieur de l’Aéronautique et de l’Espace (ISAE, formerly ENSICA), Toulouse, France (Oct. 2006). His main research interests are in the field of statistical signal processing with focus on radar signal processing, multiuser communications, and pollution detection on the sea surface based upon SAR imagery.

Marco Del Coco was born in Lecce, Italy, on June 28, 1984. He received the Dr. degree in communication engineering in 2010 and the Ph.D degree in information engineering in 2014, both from the University of Salento. In 2013 he held a visiting position at the Queen Mary University of London, United Kingdom, where he investigated the application of parallel computing to tracking problems. Currently, he is a Research Fellow with the National Council of Research (CNR) where his main research interests are in computer vision application for demographic estimation. Giuseppe Ricci was born in Naples, Italy, on Feb. 15, 1964. He received the Dr. degree and the Ph.D. degree, both in electronic engineering, from the University of Naples “Federico II” in 1990 and 1994, respectively. Since 1995 he has been with the University of Salento (formerly University of Lecce) first as an Assistant Professor of Telecommunications and, since 2002, as a Professor. His research interests are in the field of statistical signal processing with emphasis on radar processing and CDMA systems. More precisely, he has focused on high-resolution radar clutter modeling, detection of radar signals in Gaussian and non-Gaussian disturbance, oil spill detection from SAR data, track-before-detect algorithms fed by space-time radar data, localization for wireless sensor networks, multiuser detection in overlay CDMA systems, and blind multiuser detection. He has held visiting positions at the University of Colorado at Boulder, CO, USA in 1997-1998 and in April/May 2001, at Colorado State University, CO, USA in July/Sept. 2003, Mar. 2005, Sept. 2009, and Mar. 2011, at Ensica, Toulouse, France in Mar. 2006, and at the University of Connecticut, Storrs, CT, USA in Sept. 2008. BANDIERA ET AL.: MULTITARGET RANGE-AZIMUTH TRACKER

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