1

Design of Transmit Beam-Width and Dwell Timeunder K-distributed Clutter and Gaussian Noise

Emanuele Grossi, Senior Member, IEEE, Marco Lops, Senior Member, IEEE,and Luca Venturino, Senior Member, IEEE

DIEI, Università degli Studi di Cassino e del Lazio Meridionale, Italy 03043Email: e.grossi@unicas.it, lops@unicas.it, l.venturino@unicas.it

Abstract—This paper handles the problem of optimizing thescanning policy of a surveillance system in the presence ofRayleigh targets in a mixture of K-distributed clutter andGaussian noise. Adopting the detection rate (DR) and false alarmrate (FAR) as performance measures, we show that the saidoptimization amounts to solving a constrained DR maximization,wherein the free parameters are the transmit beam-width, thedwell time, and the detection threshold, while the constraintsconcern the maximum FAR and time-on-target. A thoroughperformance assessment is offered, aimed at eliciting the influenceof the signal-to-noise and clutter-to-noise ratios, the clutter shapeparameter, and the rate of change of the target response. Theresults indicate that, although a uniformly optimum transmitpolicy does not exist, some trends can be devised, which appearsof particular relevance especially for systems which can availthemselves of some degree of cognition.

Index Terms—Surveillance, Radar detection, Cognitive radar,Radar clutter, Beam-width design.

I. INTRODUCTION

The detection performances of many surveillance systemsare deeply affected by clutter, i.e., the unwanted reverberationfrom the surrounding environment onto the receive antenna,which often completely hides the echo from targets withlow Radar Cross Section (RCS): this is particularly true forcoastal radars where the returns are mostly dominated bythe backscattering from the sea-surface, but also in scenarioswhere land clutter is the primary source of interference [1]–[8].From a system design point of view, the presence of clutter istypically modeled as an additional additive interference, whoseenvelope distribution is ruled by at least two parameters: ascale parameter, which is an increasing function of the signalpower impinging on the reflector and captures the signal-dependent nature of reverberation, and a shape parameter,ruling the tail of the distribution, which may account forthe substantial departures from the customary Rayleigh lawobserved at low grazing angles and/or for high-resolutionsystems. As a consequence, clutter cannot be handled byincreasing or focusing the transmit power, which would resultin a “mirror effect” blinding the receiver; instead, we shouldexploit its temporal variation [2], [7], so that proper low-pass “filtering” represents the most straightforward strategyto get rid of it. This leads to the use of broad-beam—and

A preliminary version of this work has been presented at the 2016 IEEE 9thSensor Array and Multichannel Signal Processing Workshop, Rio de Janerio,Brazil.

hence reduced directional gain—transmission as an effectivestrategy to increase the time-on-target (TOT) for the same scantime and average the clutter out: it is understood that suchtransmit architectures, which may get as far as non-scanning(floodlight) sensors, common in many high-frequency surfacewave radars [9], should be complemented at the receiverby a proper azimuth sectorization by generating multiplenarrow beams (multi-beam on receive, MBR) [10]–[14]. Itis worth underlining at this stage that, in a noise-dominatedenvironment, the above considerations could be somewhatreversed: from an intuitive point of view, in this scenario thepriority tends to be focusing the beam so as to increase thesignal-to-noise ratio (SNR) up to a point where the targetfluctuation model becomes an increasingly important playerin determining the scanning policy.

In spite of the intuitions outlined before, no systematicstudy appears available so far aimed at gauging the achievabledetection performance of surveillance radars as a function ofthe transmit beam-width and of the dwell time under differentoperating conditions concerning both clutter and target models.To the best of our knowledge, previous related works haveonly considered the problem of optimizing the dwell-time(for a fixed transmit beam-width) assuming either a uniformscanning speed with fixed sample size detection [15]–[19]or a variable scanning speed with sequential detection [14],[20]–[24]; also, these studies have mostly assumed absence ofclutter.

Starting from the preliminary results presented in [25], theaim of this paper is to move a step forward to fill this gap.Once time is incorporated as a design parameter, it is shownin [18], [24] that the performance of a surveillance radar isbest described through the detection rate (DR), defined as theaverage number of target detections per unit of time [26], andby the false-alarm rate (FAR), defined as the average numberof false alarms per unit of time. Hence, we adopt DR andFAR as the basic performance measures: this corresponds toweighing the rate at which correct/wrong decisions are maderather than focusing on what happens in a single isolatedcells in an ensemble of realizations; moreover, it allowsdistinguishing between two basic situations, where the scantime can either be fixed at some preassigned value or be itselfa design parameter. From a practical point of view, maximizingDR under a FAR constraint facilitates subsequent track-before-detect [27]–[31] and/or tracking [32]–[35] algorithms, whichfollow the detector in the radar processing chain, since more

2

12

2

1

LΨM

12

N

ΨM

ML

Ψ Ψ

1 2 L

Transmit Receive

M

Figure 1. Transmit and receive scheme of the radar.

frequent hits result in a smaller association gate in the trackestimation process: this appears of great importance in modernmultifunction radar systems, wherein several tasks must beexecuted and completed within a certain period [36]–[41].

Throughout the paper, we consider a Rayleigh-distributedtarget in a mixture of compound-Gaussian K-distributed clut-ter and Gaussian noise; notice that the Rayleigh model wellfits the response of a target reflector composed of manyindependent scatters, while the K model is widely used todescribe the amplitude statistics of both sea clutter [7], [42]–[45] and ground clutter [1], [4]. We assume that the receiverundertakes coherent integration on a given number of pulses(a batch) and then further processes these batches. Since weare interested in outlining trends more than introducing newdetection strategies, we consider a canonical energy detectorwith perfect knowledge of the local (i.e., for a given realizationof the clutter texture) interference power: this corresponds toassuming a locally homogeneous clutter and defining a tileof secondary data large enough that any consistent powerestimator allows implementing, at least approximately, theideal CFAR detector of [42]. Under these conditions, theoptimum scanning policy is thus dictated by the solution ofa constrained DR optimization, wherein the free parametersare the transmit beam-width, the number of processed batches,and the detection threshold, while the constraints concern FARand the TOT (to avoid target cell migration).1 A thoroughperformance assessment is offered, and the optimum scanningpolicies are determined under various conditions concerningthe SNR and the clutter-to-noise ratio (CNR), the clutter shapeparameter (i.e., the impulsiveness of its envelope distribution)and the rate of change of the target response.

The reminder of the paper is organized as follows. Inthe next section, the radar model is presented. Section IIIis devoted to system optimization, while the performanceanalysis and the numerical results are provided in Section. IV.Finally, concluding remarks are given in Section V.

II. SYSTEM DESCRIPTION

We consider a radar equipped with an antenna array whichmonitors an azimuth sector of width Ψ. The emitted signal isa pulse train with repetition time T and bandwidth W , so that

1A similar approach has been followed in [25], but the analysis in thispreliminary study is limited to batch-to-batch target fluctuation (more on thisin the next section) and based on a different detector.

N = bTW c range bins are defined. The sector Ψ is dividedin M azimuth bins of width Ψ/M , as shown in Fig. 1. At thetransmit side, a beam of width LΨ/M , where L is a divisor ofM , is formed to illuminate L adjacent azimuth bins; this beamhas a loss L in the radiated power with respect to a full-gainbeam of width Ψ/M . At the receive side, L beams of widthΨ/M are formed towards the azimuth bins illuminated bythe transmitter. The transmit beam is rotated at steps of Ψ/Lto cover the whole area in M/L consecutive illuminations.At each illumination, BP pulses, parsed in B batches of Pelements, are emitted, which gives a dwell time of BPT anda scan time of2

Ts = MBPT/L. (1)

For each batch, the continuous-time signal received in eachazimuth bin undergoes a standard processing chain, whichinvolves passband-to-baseband conversion, pulse-matched fil-tering, range gating, and coherent integration through a bankof P Doppler filters, each corresponding to a different Dopplerbin. Finally, the test statistics from the B batches are combinedto make a decision.

A. Signal model

For the k-th batch, k = 1, . . . , B, we model the return fromthe range-azimuth-Doppler bin under inspection as

rk =

√PEsL

sk +

√PEcL

ck + wk (2)

if a target is present (hypothesis H1), and as

rk =

√PEcL

ck + wk (3)

otherwise (hypothesis H0). In the previous equations, sk andck models the target and clutter returns at the k-th batch,respectively. We assume that {sk}Bk=1 are complex circularly-symmetric Gaussian random variables with zero mean andunit variance; the phases of the target response in differentbatches are mutually independent, while the correspondingamplitudes (which are Rayleigh distributed) are either indepen-dent or identical: we refer to these two cases as batch-to-batch(B2B) and scan-to-scan (S2S) fluctuation, respectively. As tothe clutter, we adopt a compound-Gaussian model, so thatck =

√τxk, where τ is the texture, a unit-mean non-negative

random variable with probability density function f( · ) ac-counting for the local mean power, and xk is the speckle,a complex circularly-symmetric Gaussian random variablewith zero mean and unit variance accounting for the localbackscattering [7]; {xk}Bk=1 are independent, as the speckletypically decorrelates on a short timescale (on the order oftens of milliseconds) [42]. Es and Ec are the received signaland clutter energy per pulse, respectively, when a full gain (i.e.,L = 1) transmit beam is employed.

√P/L is a normalization

factor which accounts for the coherent integration of the Ppulses in a batch and for the reduced antenna gain when Lazimuth bins are simultaneously illuminated. wk is a complex

2If L = M , a single illumination covers the whole sector Ψ [13], [14]: inthis case, dwell and scan time become coincident.

3

circularly-symmetric Gaussian random variable with varianceσ2, which accounts for the thermal noise contribution. Target,clutter, and noise components in each batch are mutuallyindependent; also, the noise components in different batchesare mutually independent.

We assume that no cell migration occurs in B batches3.Notice that the receiver typically has little or no prior informa-tion as to the clutter-plus-noise statistics, whereby one shouldin principle resort to Constant False Alarm Rate (CFAR)detectors: a number of detectors can be used (see, e.g., [46]–[48]) depending on the amount of prior knowledge available atthe receiver. A discussion on this point is however outside thescope of the present contribution and would overshadow themajor object of interest we are focused upon. For this reason,to combine the returns {rk}Bk=1, we rather use here the idealCFAR detector

1

µ(τ)

B∑k=1

|rk|2{≥ η, ⇒ accept H1

< η, ⇒ accept H0

(4)

where η ≥ 0 is the detection threshold and

µ(τ) = σ2 + τPEcL

(5)

is the local power of the overall disturbance (i.e., noise plusclutter), assumed known. Notice that the above detector is akinto the one considered in [42] to analyze the effect of sea-clutterspikyness: the test statistics is clearly parameter-free under H0

(see next section), and the threshold takes on one and the samevalue independent of the clutter scale and shape parametersand of the noise power. In practice, µ(τ) is not known, and,under locally homogeneous4 clutter, should be replaced by anestimate, µ̂Q(τ) say, from Q secondary data, whereby theCFAR property may not be rigorously ensured, and somemismatch loss as compared to the performance achievablethrough (4) is to be expected. However, if Q is large enoughand a strongly consistent power estimator is applied thereupon(like, e.g., an elementary sample mean square value), then theperformance of an implementable version of the ideal CFARcan be made to closely follow that of the test (4).

B. Probabilities of false alarm and detection

The derivation of the probability of false alarm, Pfa =P(accept H1 under H0), and the probability of detection,Pd = P(accept H1 under H1), for the test (4) is quitestraightforward; we report here the main results which arerelevant in the following development, while the interestedreader may refer to standard textbooks for more details [7],[9].

Under H0, 2µ(τ)

∑Bk=1 |rk|2 is a chi-squared random vari-

able with 2B degrees of freedom, whereby the probability offalse alarm is

Pfa = 1− γ (B, η)

(B − 1)!(6)

3This condition can be relaxed by resorting to track-before-detect algo-rithms.

4This implies that a number of cells surrounding the one under test sharethe same noise and speckle average power of the cell under test and the samerealization of the texture τ [49], [50].

where γ( · , · ) is the lower incomplete gamma function.Under H1, let

SDR(τ) =PEsLµ(τ)

(7)

be the local signal-to-disturbance ratio per batch. For S2Sfluctuation, we have that s = |s1|2 = · · · = |sB |2, whereby

2µ(τ)

∑Bk=1 |rk|2 | {τ, s} is a non-central chi-squared random

variable with 2B degrees of freedom and non-centrality pa-rameter 2Bs SDR(τ) and the probability of detection is

Pd =

∫ ∞0

∫ ∞0

QB

(√2Bs SDR(τ),

√2ζ)f(τ)dτe−sds

(8)

where QB( · , · ) is the Marcum Q-function of order B. ForB2B fluctuation, the random variables {|sk|2}Bk=1 are inde-pendent, whereby the statistic 2

µ(τ)(1+SDR(τ))

∑Bk=1 |rk|2 | {τ}

is a chi-squared random variable with 2B degrees of freedomand the probability of detection is

Pd = 1− 1

(B − 1)!

∫ ∞0

γ(B, η

1+SDR(τ)

)f(τ)dτ. (9)

C. Texture distribution

While the expressions in (6), (8), and (9) hold for anycompound-Gaussian distribution, in the following analysis,we assume a K-distributed clutter [6], whereby f(τ) =νν

Γ(ν)τν−1e−ντu(τ), with u(τ) being the unit step function and

ν the shape parameter.

III. SYSTEM OPTIMIZATION

The radar continuously scans the azimuth sector Ψ tomonitor the scene, whereby the same test is repeated in eachresolution cell after a period Ts. The available degrees offreedom for system optimization are the detection thresholds,the number of integrated batches, and the transmit beam-width, which together determine the radar scanning policy:5

this allows different trade-offs in terms of probability of falsealarm, probability of detection, and scan time.

For example, for the same Ts, we can trade a smaller gainof the transmit beam for a longer dwell time to exploit thevariation of the clutter and/or target response over consecutivebatches (if any): this might help improving Pd for a fixedPfa. Similarly, Pd might be improved by increasing Ts for afixed Pfa: however, a series of tests with a larger Pd and alonger Ts is not necessarily preferable to another one with asmaller Pd and a shorter Ts when constantly monitoring thesame region [18], [24], [52], [53].

In order to compare scanning policies involving a differentscan time, it is meaningful to evaluate the radar performancein terms of detection rate, defined as DR = Pd/Ts, and falsealarm rate, defined as FAR = Pfa/Ts. The rational behind thismethodology, firstly introduced in [18], [24], is that the samelevel of Pd is more valuable if achieved with a shorter Ts, asquicker target detections occur; for the same reason, the same

5The number of pulses in each batch could also be considered as adesign parameter; however, changing P would require modifying the receiverprocessing chain and the Doppler resolution [51]. In this work, we assumethat the value of P is given and fixed.

4

level of Pfa becomes less valuable if achieved with a shorter Ts:accordingly, the optimal scanning policy is the one maximizingDR under a constraint on FAR. Let L, Bmax, and FAR0 be theset containing the divisors of M , the maximum number ofprocessed batches (which is tied to the target mobility and,possibly, to the minimum measurement update rate requiredby the radar system), and the maximum tolerable FAR level,respectively. The optimization problem becomes

maxL∈L,B∈Nη≥0

LPd

MBPT

s.t.LPfa

MBPT≤ FAR0

B ≤ Bmax

(10)

where Pfa is given by (6), while Pd is given by (9) and (8) forB2B and S2S target fluctuation, respectively. Notice that, foreach feasible pair (L,B), DR is maximized by choosing thelargest η satisfying the FAR constraint (since Pd is increasingwith Pfa); the optimum pair (L,B) must be found throughexhaustive search over the finite set defined by the constraints.

We hasten to underline that the solution to (10) depends onEs, Ec, σ2, and f(·), which we assume known so as to elicitthe achievable system performance and trade-offs: this may befor example the case of a system which has some degree ofcognition of the surrounding environment. If these quantitiesare instead unknown, as typical in many applications, then theycan be set at some design or estimated value, and losses areaccepted in case of mismatch. Also, notice that (10) reducesto the standard Neyman-Pearson design criterion if we includethe additional constraint that the scan time (i.e., the ratio B/L)is kept fixed at some desired value.

IV. ANALYSIS

In this section, we discuss some examples to illustrate theachievable performance tradeoffs. Specifically, we consider along-range radar where the pulse repetition time is T = 1 ms,the batch length is P = 20, the maximum number of integratedbatch is Bmax = 50, the monitored area is Ψ = 60◦, and thenumber of range bins is N = 1000, the number of azimuthbins is M = 10, so that L = {1, 2, 5, 10}. Unless otherwisestated, DR and FAR are expressed in detections per second(det/s) and false alarms per second (fa/s), respectively.

In the following, we first examine the effect of varying Lon Pd (the numerator of DR) when both Pfa (the numerator ofFAR) and Ts (the denominator of DR and FAR) are kept fixed:in this case, a larger L allows processing more of batches at theprice of a smaller antenna gain. Then, we examine the effect ofvarying L and B on DR for a fixed FAR: in this case, a largerL results into a shorter scan time at the price of a smallerantenna gain, while a larger B results into a larger number ofintegrated bacthes at the price of a longer scan time. Beforeproceeding with this analysis, we point out that the receivedsignal-to-noise and clutter-to-noise ratios per pulse, given by

SNR =EsLσ2

(11)

CNR =EcLσ2

(12)

-20 -10 0 10 20 30SNR [dB]

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0.3

0.5

0.7

0.9

0.99

Pd

!1 dB0 dB

15 dB

30 dB

Figure 2. Pd vs SNR for CNR = − inf, 0, 15, 30 dB and L = 1, 2, 5, 10, whenB = 5L, Pfa = 8.3× 10−8, and ν = 0.5. Scan-to-scan target fluctuation isassumed. The direction of the dashed arrow indicates an increasing value ofL, while its label the value of CNR.

respectively, are decreasing with L for a fixed transmit power,as a consequence of the reduced antenna gain. Hence, toallow a fair comparison among system configurations with adifferent antenna beam-width, Pd and DR are analyzed here asa function of the received signal-to-noise and clutter-to-noiseratios per pulse normalized by antenna gain loss 1/L, i.e.,

SNR = Es/σ2 (13)

CNR = Ec/σ2 (14)

respectively.

A. Fixed scan-time

Fig. 2 shows Pd vs SNR for S2S target fluctuation, whenL = 1, 2, 5, 10, B = 5L (which keeps the scan time fixedto Tup = 1 s), Pfa = 8.3 × 10−8, and ν = 0.5. Noticethat the choice of Tup and Pfa results in one false alarm perminute from the whole inspected area, which is a reasonablerequirement for surveillance radars. Different values of CNR

are considered, namely, −∞ (i.e., no clutter), 0, 15, and 30dB. It is seen by inspection that the effect of changing L(i.e., the beamwidth of the transmit antenna) on Pd dependson the specific operating regime. In the absence of clutter,Pd is a decreasing function of L at any SNR, and the optimalscanning policy requires maximizing the transmit antenna gain(i.e., L = 1): here, increasing the transmit beamwidth isdetrimental, as the antenna gain loss is not payed off by theincoherent integration gain granted by the larger number ofprocessed batches. In the presence of clutter, Pd is decreasingwith L at sufficiently small SNR’s and increasing at sufficientlylarge SNR’s: overall, the optimal value of L increases from 1 to

5

Table IOPTIMUM L WHEN B = 5L, PFA = 8.3× 10−8 , AND ν = 0.5.

SCAN-TO-SCAN TARGET FLUCTUATION IS ASSUMED.

Pd

CNR [dB] 0.01 0.1 0.3 0.5 0.9 0.99

−∞ 1 1 1 1 1 1−20 1 1 1 1 1 1−15 1 1 1 1 2 2−12 1 1 2 2 2 2−10 1 2 2 2 5 50−8 1 2 5 5 5 50−6 1 5 5 5 10 100−4 2 5 10 10 10 100−2 2 5 10 10 10 10−00 2 10 10 10 10 10−02 5 10 10 10 10 10−04 5 10 10 10 10 10−08 5 10 10 10 10 10−15 10 10 10 10 10 10−30 10 10 10 10 10 10

Table IIOPTIMUM L WHEN B = 5L, PFA = 8.3× 10−8 . SCAN-TO-SCAN TARGET

FLUCTUATION IS ASSUMED.

Pd = 0.1 Pd = 0.5ν ν

CNR [dB] 0.1 0.5 1 5 0.1 0.5 1 5

−20 1 1 1 1 1 1 1 1−15 1 1 1 2 1 1 1 2−12 1 1 2 2 1 2 2 2−10 1 2 5 5 1 2 2 50−8 1 2 5 5 1 5 5 50−6 1 5 5 10 1 5 5 100−4 1 5 10 10 1 10 10 100−2 1 5 10 10 1 10 10 10−00 1 10 10 10 1 10 10 10−02 1 10 10 10 1 10 10 10−04 1 10 10 10 2 10 10 10−08 1 10 10 10 2 10 10 10−015 1 10 10 10 5 10 10 10−030 1 10 10 10 10 10 10 10

10 as SNR gets larger. These results are explained by noticingthat, while keeping the received signal-to-clutter ratio fixed,increasing L produces two opposite effects. On the one hand,the received SNR is reduced for the same transmit power, as aconsequence of the reduced transmit antenna gain; on the otherhand, more batches are processed in the same scan time, asa consequence of the increased transmit antenna beamwidth,which allows to take advantage of the temporal variation of theclutter over multiple observations. To provide more insightson this latter point, we report in Table I the optimal valueof L as6 a function of Pd and CNR. Notice that, for a givenPd, the optimal L increases with CNR, as the clutter becomesmore severe and exploiting its variation over multiple batchesgets more rewarding; similarly, for a given CNR, the optimal Lincreases with Pd (or equivalently SNR), since a larger antennagain becomes less rewarding as the target signal gets stronger.

Next, we study the effect of the clutter shape on the

6Here, the optimal L should be understood as the one providing the givenvalue of Pd with the smallest SNR.

-20 -10 0 10 20 30SNR [dB]

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0.3

0.5

0.7

0.9

0.99

Pd

0:1

0:5

1

5

Figure 3. Pd vs SNR for scan-to-scan target fluctuation for ν = 0.1, 0.5, 1, 5and L = 1, 2, 5, 10, when B = 5L, Pfa = 8.3 × 10−8, and CNR = 15 dB.The direction of the dashed arrow indicates an increasing value of L, whileits label the value of ν.

system design. Fig. 3 shows Pd vs SNR when CNR = 15dB and ν = 0.1, 0.5, 1, 5; also, Table II shows the optimalL as function of ν and CNR when Pd = 0.1, 0.5. All otherparameters are set as in Fig. 2. In the inspected SNR range,the best performance is achieved in the spikiest clutter case(i.e., the lowest value of ν): this is a well known result,since the probability density function of the clutter amplitudeweights more the small (with respect to the mean) values asν gets smaller, and this results into longer quiescence periods[6], [45]. More interestingly, for a fixed CNR, the optimal Lgets smaller as the clutter becomes spikier. Again, this is aconsequence of the fact that a spikier clutter presents a verylow local mean level for a larger fraction of time: in theseregions, the additive noise is the major impairment, wherebya target is more easily detected if a larger antenna gain isadopted.

Finally, we analyze the system performance in the presenceof B2B target fluctuation, when all other parameters are set asin Fig. 2. Fig. 4 shows Pd vs SNR, while Table III reportsthe optimal value of L as a function of Pd and CNR. Bycomparing Figs. 2 and 4 and Tables I and III, similar trends areobserved in the presence of clutter under both target fluctuationmodels. A major difference is instead observed in the absenceof clutter: indeed, for B2B target fluctuation the optimal Lnow increases from 1 to 10 as SNR (or equivalently Pd) getslarger. This is a consequence of the fact that, at lower SNR’s,a larger antenna gain is preferable to signal diversity and viceversa at higher SNR’s.7

7Recall that, for a fixed scan time, a larger L results here into the processingof a larger number of independent batches (granting signal diversity) at theprice of a smaller antenna gain.

6

-20 -10 0 10 20 30SNR [dB]

1e-08 1e-07 1e-06 1e-05

0.0001

0.001

0.01

0.1

0.3 0.5 0.7

0.9

0.99

0.999

0.9999

0.999990.999999

Pd

!1 dB0 dB

15 dB

30 dB

Figure 4. Pd vs SNR for CNR = − inf, 0, 15, 30 dB and L = 1, 2, 5, 10, whenB = 5L, Pfa = 8.3× 10−8, and ν = 0.5. Batch-to-batch target fluctuationis assumed. The direction of the dashed arrow indicates an increasing valueof L, while its label the value of CNR.

Table IIIOPTIMUM L WHEN B = 5L, PFA = 8.3× 10−8 , AND ν = 0.5.

BATCH-TO-BATCH TARGET FLUCTUATION IS ASSUMED.

Pd

CNR [dB] 0.1 0.3 0.5 0.7 0.9 0.99

−∞ 1 1 1 1 2 5−20 1 1 1 1 2 5−15 1 1 1 2 5 5−12 1 1 2 2 5 10−10 1 1 2 5 10 100−8 1 2 2 5 10 100−6 1 2 5 10 10 100−4 1 2 5 10 10 100−2 1 5 10 10 10 10−00 1 5 10 10 10 10−02 2 5 10 10 10 10−04 2 10 10 10 10 10−08 2 10 10 10 10 10−015 10 10 10 10 10 10−030 10 10 10 10 10 10

We hasten to underline that, while radar engineers maybe familiar with the trade-offs outlined here for a fixed scantime, to the best of authors’ knowledge, no previous work haspresented them in this form.

B. Optimized scan-time

In Figs. 5 and 6, we report the optimal DR—i.e., the solutionto the problem (10)—vs SNR for different values of CNR andν = 0.5 (in the former figure) and for different values ofν and CNR = 15 dB (in the latter figure). The per-cell FARconstraint is set to one false alarm per minute from the wholeinspected area, as in the examples discussed in Section IV-A.

-30 -20 -10 0 10 20 30 40SNR [dB]

10-8

10-6

10-4

10-2

100

102

DR

[det

/s]

!1 dB

0 dB

15 dB30 dB

S2SB2B

Figure 5. Optimal DR vs SNR when CNR = − inf, 0, 15, 30 dB, FAR =8.3 × 10−8 fa/s, and ν = 0.5. Both scan-to-scan and batch-to-batch targetfluctuations are considered.

-30 -20 -10 0 10 20 30 40SNR [dB]

10-8

10-6

10-4

10-2

100

102

DR

[det

/s]

0:10:5

1

5

S2SB2B

Figure 6. Optimal DR vs SNR when ν = 0.1, 0.5, 1, 5, FAR = 8.3 ×10−8 fa/s, and CNR = 15 dB. Both scan-to-scan and batch-to-batch targetfluctuations are considered.

7

0

20

40

B

0

5

10

L

0

5

10

Ts

S2SB2B

0

0.5

1

Pd

-20 0 20SNR [dB]

10-10

10-8

10-6

Pfa

-20 0 20SNR [dB]

Figure 7. Optimal values of B, L, Ts, Pd, and Pfa vs SNR when FAR =8.3× 10−8 fa/s. Both scan-to-scan and batch-to-batch target fluctuations areconsidered. In the left plots we have CNR = −∞ dB, while in the right plotsCNR = 15 dB and ν = 0.5.

In the inspected operating regimes, the optimized DR is adecreasing function of both CNR and ν, in keeping with theresults presented in Sec. IV-A; more interestingly, the solutionsfor B2B and S2S target fluctuation are quite close over a wideSNR range: this result is explained by noticing that, as SNR getslarger, DR is greatly improved by reducing the scan time (seenext, Fig. 7), while a small Pd variation due to the differenttarget fluctuation model has a marginal effect.

In Fig. 7, we report the values of B, L, Ts, Pd, and Pfagiving the optimal DR in Figs. 5 and 6 in the absence of

10-5

100

norm

.D

R

SNR = !15 dB

10-4

10-2

100SNR = !5 dB

10-2

10-1

100

norm

.D

R

SNR = 0 dB

10-2

10-1

100SNR = 2 dB

100 101

B

10-2

10-1

100

norm

.D

R

SNR = 5 dB

100 101

B

10-2

100SNR = 15 dB

Figure 8. Normalized DR vs B for L = 1, 2, 5, 10 (the direction of thedashed arrow indicates an increasing value of L), when FAR = 8.3× 10−8

fa/s, CNR = −∞ dB, and SNR = −15,−5, 0, 2, 5, 15 dB. Batch-to-batchtarget fluctuation is considered. In each subplot, the star marker indicates thepair (B,L) achieving the maximum DR used for normalization.

10-4

10-2

100

norm

.D

R

SNR = !15 dB

10-4

10-2

100SNR = !5 dB

10-2

10-1

100

norm

.D

R

SNR = 0 dB

10-2

10-1

100SNR = 2 dB

100 101

B

10-2

10-1

100

norm

.D

R

SNR = 5 dB

100 101

B

10-2

100SNR = 15 dB

Figure 9. Normalized DR vs B for L = 1, 2, 5, 10 (the direction of the dashedarrow indicates an increasing value of L), when FAR = 8.3 × 10−8 fa/s,CNR = 15 dB, ν = 0.5, and SNR = −15,−5, 0, 2, 5, 15 dB. Batch-to-batchtarget fluctuation is considered. In each subplot, the star marker indicates thepair (B,L) achieving the maximum DR used for normalization.

8

clutter and when CNR = 15 dB and ν = 0.5. It is seen thatthe optimal scan time, which results from the combined effectof the number of azimuth bins simultaneously illuminated (L)and the number of processed batches (B), has a decreasingtrend8 for an increasing SNR, in keeping with the basic intuitionthat the dwell time in each angular sector can be reducedfor stronger targets. Indeed, as SNR gets larger, the marginalincrements of Pd obtained by using a narrower transmit beamwith a larger gain (i.e., a lower L) and/or by processing alarger number of batches may not be enough to justify thelonger scan-time. Notice that the same trend is observed underboth target models and with or without clutter. Clearly, if thescan time is reduced, Pfa must also be reduced to maintainthe same FAR. To better understand the effect of varyingB and L, Figs. 8 and 9 show DR as a function of B foreach possible L: the reported curves are normalized by themaximum DR value given in Fig. 5. In the former figure, weassume no clutter, while in the latter we assume CNR = 15dB and ν = 0.5. For brevity, only batch-to-batch fluctuationis considered. In each figure, six subplots are present, eachcorresponding to a different value of SNR; in each subplot, thestar marker indicates the pair (B,L) providing the maximumDR in Fig. 5. Notice that, when SNR is extremely small, DRis increasing with B and decreasing with L: indeed, in thisregime the detection probability is so small that the mainconcern is increasings both the number of processed batchesand the transmit antenna gain; on the other hand, when SNR

is extremely large, DR is decreasing with B and increasingwith L: indeed, in this regime the detection probability is solarge that the main concern is minimizing the scan time. Inbetween, the contrasting needs of a large detection probabilityand small scan time must be balanced; interestingly, in thisintermediate regime several pairs of (B,L) may provide quasi-optimal performance. Finally, observe that DR may be quitesensitivity to a variation of L for a fixed B, while it is generallyquite robust to a small variation of B for a fixed L.

V. CONCLUSIONS

In this paper, we have considered the problem of optimizingthe scanning policy, expressed in terms of transmit beam-width and dwell time, for surveillance systems in the presenceof Rayleigh targets in a mixture of Gaussian noise and K-distributed clutter. At the design stage, we have adopted thedetection rate (DR) achievable for a fixed false alarm rate(FAR) as the key performance measure: this is perfectlyequivalent to a Neyman-Pearson optimization framework if thescan time is constrained to be fixed, while being substantiallydifferent if such a scan time is itself a degree of freedom.

At the analysis stage, the big divide is between the noise-dominated (i.e., the target signal is much weaker than the noiselevel, but much stronger than the clutter level) and the clutter-dominated environment (i.e., the target signal is much strongerthan the noise level, but much weaker than the clutter level).In the former case, a directional high-gain beam should be

8Notice that the scan-time is not here a monotonically decreasing functionof SNR as consequence of the fact that B and L are optimized over a discreteset and, hence, a small ripple may occur.

preferred, as increasing the received signal-to-noise ratio is themain concern; in the latter case, a broad low-gain beam shouldbe preferred, which allows either to reduce the scan time bymaintaining nearly the same detection probability (since thesignal-to-clutter-plus-noise ratio is substantially unaffected bythe transmit antenna gain in this regime) or to improve thedetection probability by exploiting the temporal variation ofthe clutter over a longer dwell time. In general, the optimalscanning policy depends on the signal-to-noise and clutter-to-noise ratios, the clutter spikiness, and the rate of change(namely, batch-to-batch or scan-to-scan) of the target response;interestigly, some simple trends can be devised, which appearsof particular relevance for systems which can avail themselvesof some degree of cognition. The optimal scan time is decreas-ing with the signal-to-noise ratio in keeping with the intuitionthat a stronger target can be detected by using a smaller dwelltime and/or a transmit antenna with a larger beam-width (and,hence, a smaller gain); moreover, for the same scan time, theoptimal transmit beam-width is increasing with the clutter-to-noise ratio, as the transmit antenna gain has no impact onthe signal-to-clutter ratio, but the system should rather exploitthe temporal variation of the clutter over a longer dwell time,and decreasing with the clutter spikiness, as a spiky clutteris quiescent for large time intervals. Finally, while the rate ofchange of the target response may have a significant impacton the achievable detection probability for a preassigned scantime, it only has a marginal effect on the achievable detectionrate when the scan time is a design parameter that can beoptimized, thus simplifying to some extend the system design.

A key assumption in our work is that the speckle de-correlates on a short timescale, thus making the elaboration ofmultiple consecutive batches rewarding for clutter mitigation.Our feeling is that the trends found here should also hold forother clutter distributions, provided that the clutter responsede-correlates over multiple batches. However, a more in depthstudy is necessary to verify this intuition and, more generally,to assess the impact of other clutter fluctuation models onthe scanning policy. Moreover, the results in this work arebased on the use of an ideal CFAR detector and the absenceof cell migration over the dwell time. While these simplifyingassumptions have allowed to derive manageable close-formexpressions for DR and FAR, thus facilitating the analysis,future works should study the effect of the CFAR losses andof the target mobility in practical systems.

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Emanuele Grossi (M’08) was born in Sora, Italy,on May 10, 1978. He received the Dr. Eng. degreein Telecommunication Engineering in 2002 and thePh.D. degree in Electrical Engineering in 2006, bothfrom the University of Cassino and Southern Lazio,Italy. In 2005 he was visiting Scholar with theDepartment of Electrical & Computer Engineeringof the University of British Columbia, Canada, andin 2009 he had a visiting appointment in the DigitalTechnology Center, University of Minnesota, MN.Since February 2006, he is assistant professor at the

University of Cassino and Southern Lazio. His research interests concernwireless communication systems, radar detection and tracking, and statisticaldecision problems with emphasis on sequential analysis.

Marco Lops (M’96-SM’01) was born in Naples(Italy) on March 16, 1961. He obtained his “Laurea”and his Ph.D. degrees from “Federico II” University(Naples), where he was assistant (1989-1991) andassociate (1991-2000) professor. Since March 2000he has been a professor at University of Cassinoand Southern Lazio and, in 2009-2011, he was alsowith ENSEEIHT (Toulouse). In fall 2008 he wasa visiting professor with University of Minnesotaand in spring 2009 at Columbia University. Hisresearch interests are in detection and estimation,

with emphasis on communications and radar signal processing.

Luca Venturino (S’03-M’06) was born in Cassino,Italy, on August 26, 1979. He received the Ph.D.degree in Electrical Engineering in 2006 from theUniversity of Cassino and Southern Lazio, Italy.In 2004 and 2009, he was Visiting Researcher atthe Columbia University, New York (NY). Between2006 and 2008, he spent nine months at NECLaboratories America, Princeton (NJ). Currently, heis an associate professor at the University of Cassinoand Southern Lazio. His research interests are indetection, estimation, and resource allocation, with

emphasis on communications and radar signal processing.