Multistatic Cloud Radar Systems: Joint Sensing and Communication Design

0.42 MB·English

by  Seongah Jeong

1
Multistatic Cloud Radar Systems: Joint Sensing and
Communication Design
Seongah Jeong, Shahrouz Khalili, Osvaldo Simeone, Alexander Haimovich and Joonhyuk Kang
Abstract—In a multistaticcloud radar system, receive sensors system(GPS),andarecapableofcommunicatingwithafusion
measure signals sent by a transmit element and reflected from center in the “cloud” through a backhaul wireless or wired
a target and possiblyclutter, in thepresenceof interference and
6 network. For example, the receive sensors could be mounted
noise.Thereceive sensorscommunicateover non-idealbackhaul
1 on light poles, trucks or unmanned aerial vehicles (UAV’s)
linkswithafusioncenter,orcloudprocessor,wherethepresence
0 orabsenceofthetargetisdetermined.Thebackhaularchitecture and could be connected to a wireless access point via Wi-
2
canbecharacterizedeitherbyanorthogonal-accesschannelorby Fi or dedicated mmWave links. As this architecture can be

n a non-orthogonal multiple-access channel. Two backhaul trans- implemented by means of cloud computing technology, we
a mission strategies areconsidered,namelycompress-and-forward refer to it as “cloud radar.”
J (CF), which is well suited for the orthogonal-access backhaul,
2 and amplify-and-forward (AF), which leverages the superposi- The main purpose of this work is to study the interaction
1 tion property of the non-orthogonal multiple-access channel. In between the sensing and backhaul communication functions
this paper, the joint optimization of the sensing and backhaul in a cloudradar architecture,andto developan understanding
] communication functions of the cloud radar system is studied. of the performance gain to be expected by means of a joint
T Specifically, the transmitted waveform is jointly optimized with
optimization of these two functions, namely of waveform
I backhaul quantization in the case of CF backhaul transmission
design for sensing and of backhaul transmission.
. andwiththeamplifyinggainsofthesensorsfortheAFbackhaul
s
c strategy. Inboth cases, theinformation-theoreticcriterion of the
[ Bhattacharyya distance is adopted as a metric for the detection A. Background
performance. Algorithmic solutions based on successive convex
2 approximation are developed under different assumptions on The separate design of radar waveforms, under the as-
v the available channel state information (CSI). Numerical results sumption of an ideal backhaul, has long been a problem
1 demonstratethattheproposedschemesoutperformconventional of great interest [2], [3]. For monostatic radar systems, i.e.,
7 solutions that perform separate optimizations of the waveform
radars with single transmit and receive elements, optimal
3 and backhaul operation, as well as the standard distributed
5 detection approach. waveforms for detection in the Neyman-Pearson sense were
0 studiedin [4]. In a multistatic radar system, wherethe signals
. IndexTerms—CloudRadioAccessNetworks(C-RANs),quan- received by a set of distributed sensors are processed jointly,
1 tization, localization, Crame´r-Rao bound (CRB).
0 the performance of the Neyman-Pearson optimal detector is
5 in general too complex to be suitable as a design metric.
1 I. INTRODUCTION As a result, various information-theoretic criteria such as the
v: This paper1 addresses a distributed radar system that in- Bhattacharyya distance, the Kullback Leibler divergence, the
i volvessensingandcommunication:atransmitelementillumi- J-divergenceandthemutualinformation,whichcanbeshown
X
natesanareaofinterest,inwhichatargetmaybepresent,and to provide various bounds to the probability of error (missed
r
a the signals returned from the target are observed by sensors. detection,falsealarmandBayesianrisk),havebeenconsidered
The sensors have a minimal processing capabilities, but com- as alternative design metrics [5]–[7].
municate over a backhaul network with a processing center, Instead, the separate design of backhaul communication
referredtohenceforthasafusioncenter,wheretargetdetection functions, for fixed radar waveforms was studied in [8]–
takes place (see Fig. 1). Such architecture is different from a [11] under a compress-and-forward (CF) strategy, for which
classical multistatic radar system in which each constituent backhaul quantization was optimized, and in [12], [13] under
radar performs the full array of radar functions, including an amplify-and-forward (AF) scheme, for which the power
target detection and tracking. The considered architecture is allocation at the sensors was investigated using the minimum
motivated by the proliferation of low-cost, mobile or fixed mean square error (MMSE) as the performance criterion.
sensors in the “Internet of Things,” which are supported by
globalsynchronizationservices such as the globalpositioning
B. Main Contributions
Seongah JeongandJoonhyukKangarewiththe Department ofElectrical Unlike prior work, in this paper, we tackle the problem
Engineering, KoreaAdvancedInstituteofScienceandTechnology(KAIST), of jointly designing the waveform, or code vector, and the
Daejeon, SouthKorea.
transmissionofthereceivesensorsoverthebackhaul.Thisap-
Osvaldo Simeone and Alexander Haimovich are with the Center for
WirelessCommunications andSignalProcessingResearch(CWCSPR),ECE proachismotivatedbythestronginterplaybetweenwaveform
Department, NewJerseyInstitute ofTechnology (NJIT),Newark,NJ07102, and backhaul transmission designs. For instance, waveform
USA.
design may allocate more power at frequencies that are less
1This workwas partially presented atIEEERadar Conference, Arlington,
VA,May2015[1]. affected on average by clutter and interference, while the
2
The rest of the paper is organized as follows. In Section
II, we present the signal model and cover the two types of
Clutter
Target
backhaullinks, namely orthogonal-accessand non-orthogonal
multiple-access backhaul. In Section III, after describing the
Receive sensor N
Backhaul CFbackhaultransmissionstrategiesandreviewingtheoptimal
detectors,we presentthe optimizationof the multistatic cloud
Fusion center radar system with CF. In Section IV, we focus on the AF
Transmit element Receive sensor n
backhaul transmission, and optimize the system with both
instantaneousand stochastic CSI under AF. Numericalresults
are providedin Section V, and, finally,conclusionsare drawn
Receive sensor 1
in Section VI.
Fig. 1. Illustration of a multistatic cloud radar system, which consists of
a transmit element, N receive sensors, and a fusion center. All the nodes II. SYSTEMMODEL
are configured with a single antenna. The receive sensors are connected to
Consider a multistatic cloud radar system consisting of a
the fusion center via orthogonal-access or non-orthogonal multiple-access
backhaul links. transmit element, N receive sensors, and a fusion center, or
cloud processor, as illustrated in Fig. 1. The receive sensors
communicatewiththefusioncenteroveranorthogonal-access
backhaul or a non-orthogonal multiple-access backhaul. All
backhaul transmission strategies are adapted accordingly to
the nodes are equipped with a single antenna, and the set of
devotemostbackhaulresources,namelycapacityor power,to
receive sensors is denoted = 1,...,N .
the transmission of such frequencies to the fusion center. N { }
Thesystemaimstodetectthepresenceofasinglestationary
Twobasictypesofbackhaullinksbetweentheradarreceive
target in a clutter field. To this end, each sensor receives a
sensors and the fusion center are considered, namely orthog-
noisyversionofthesignaltransmittedbythetransmitelement
onal and non-orthogonal access backhaul. In the former, no
andreflectedfromthe surveillancearea,which is conveyedto
interferenceexistsbetweenthesensors,asinawiredbackhaul,
the fusion center on the backhaul channels after either quan-
while in the latter, the backhaul forms a multiple-access
tizingoramplifyingthereceivedsignalsasdiscussedbelow.It
channel,wherechannelsare subjectto mutualinterference,as
is assumed that perfect timing information is available at the
in a wireless backhaul. Furthermore, two standard backhaul
fusioncenter,suchthatsamplesofthe receivedsignalmay be
transmission schemes are investigated, namely CF and AF.
associated with specific locations in some coordinate system.
As inthe CloudRadio AccessNetwork(C-RAN) architecture
For such a location, and based on all the signals forwarded
in communication [14], CF is particularly well suited to an
from the different receive sensors, the fusion center makes a
orthogonal backhaul architecture: each sensor satisfies the
decision about the presence of the target (see, e.g. [7]–[13]).
backhaulcapacityconstraintquantizingthereceivedbaseband
Notethat,asarguedin [6],theassumptionofstationarytarget
signalspriortotransmissiontothefusioncenter.AF,instead,is
and scatterers can be regarded as a worst-case scenario for
better matched to a non-orthogonalmultiple-access backhaul:
more general set-ups with non-zero Doppler.
eachreceivesensoramplifiesandforwardsthereceivedsignal
We consider a pulse compression radar in which the trans-
to the fusion center so that the signals transmitted by the
mitted signal given in baseband form is
receive sensors are superimposed at the fusion center (see,
e.g., [12], [13]). K
s(t)= x φ(t (k 1)T ), (1)
Our specific contributions are as follows: k − − c
k=1
CF:Thejointoptimizationofthewaveformandthequantiza- X
• where φ(t) is, for example, a square root Nyquist with chip
tionstrategyisinvestigatedforCF,withafocusonorthogonal-
rate1/T , sothat φ(t (k 1)T ) K areorthonormal;and
access backhaul.To reflect practicalconstraints, onlystochas- c { − − c }k=1
x K is a sequenceof(deterministic)complexcoefficients
tic channelstate information(CSI) is assumed on the channel { k}k=1
thatmodulatethe waveform.The vectorxxx=[x x ]T is
gains between target or clutter and the receive sensors. For 1 K
···
referred to as waveform or code vector, on which we impose
an optimization objective, we adopt the information-theoretic
the transmit power constraint xxxHxxx P . The design of the
criterionofthe Bhattacharyyadistance in orderto accountfor T
≤
waveform xxx determines both target and clutter response, and
the detection performance [5]–[7].
thus has a key role in the performance of the radar system.
AF: The joint optimization of the waveform and the
• The baseband signal received at the sensor n , which
amplifying gains of the receive sensors is studied for AF, by ∈ N
is backscattered by a stationary target, can be expressed as
concentratingonnon-orthogonalmultiple-accessbackhaul.We
r (t)=h s(t τ )+c (t)+w (t), (2)
adopt the performancecriterion and main assumptions of CF. n n − n n n
Furthermore, we consider both instantaneous and stochastic where h is the random complex amplitude of the target
n
CSI on the receive sensors-to-fusion center channels. return,whichincludestheeffectsofthechannelandfollowsa
Throughout,we assume tractableandwellacceptedmodels SwerlingItarget-typemodelhavinga Rayleighenvelope,i.e.,
in order to gain insight into the problem at hand. With this h (0,σ2 ); c (t) represents the clutter component;
n ∼ CN t,n n
insight gained, subsequent work may explore more detailed w (t) is a Gaussian random process representing the signal-
n
configurations. independent interference, which aggregates the contributions
3
of thermal noise, interference and jamming and is assumed to the fusion center for all n and to change sufficiently
∈ N
to be correlated over time, as detailed below; and τ is the slowly so as to enable the adaptation of the waveform and of
n
propagation delay for the path from the transmit element to the transmission strategy of the sensors to the values of the
the target and thereafter to the sensor n, which is assumed capacities C for all n .
n
∈N
to satisfy the condition τ KT in order for the target Non-orthogonal Multiple-access Backhaul: For the non-
n c
≥
to be detectable. The clutter component c (t) consists of orthogonal multiple-access backhaul, the signal received at
n
signal echoes generated by stationary point scatterers, whose the fusion center is the superposition of the signals sent by
echoes have independent return amplitudes and arrival times. all receive sensors, where channels are subject to mutual
Accordingly, the clutter component c (t) is expressed as interference. Accordingly, the received signal at the fusion
n
Nc centerrr˜r=[r˜1 r˜K]T is given by
c (t)= g s(t τ ), (3) ···
n n,v n,v
− N
v=1
X rr˜r = f ttt +zzz, (7)
whereN isthenumberofpointscatters;g istheamplitude n n
c n,v
n=1
ofthereturnfromscattererv;andτ isthepropagationdelay X
n,v
for the path from the transmit element to the scatterer v and wheretttn =[tn,1 tn,K]T is the signal sent by the receive
···
to the sensor n, which satisfies the condition τn,v KTc. sensor n on the backhaul to the fusion center; fn is the
≤
After matched filtering of the received signal (2) with the complex-valuedchannelgainbetweenthereceivesensornand
impulse response φ∗( t), and after range-gatingby sampling the fusion center; and zzz = [z1 zK]T is the noise vector
− ···
the output of the matched filter at the chip rate, the discrete- having a zero-mean Gaussian distribution with correlation
time signal at receive sensor n for n can be written matrix ΩΩΩz, i.e., zzz (000,ΩΩΩz). Based on prior information
∈ N ∼ CN
as or measurements, the second-order statistics of the channel
r =h x +g˜ x +w , (4) gains between the target and the receive sensors, and of the
n,k n k n k n,k
noise terms, namely σ2 , σ2 ,ΩΩΩ andΩΩΩ , are assumed to
where r is the output of the matched filter at the receive t,n c,n w,n z
n,k be known to the fusion center for all n . The channel
sensor n sampled at time t=(k 1)T +τ ; the term g˜ = ∈ N
vN=c1gn,vΨ(τn −τn,v) with Ψ(−t) , c−∞∞φn(τ −t)φ∗(τn)dτ abreetwaleseonarsescuemiveedsteonbsoerksnaonwdnfuatsitohnecfuesnitoenr ffcfen=te[rf,1via··t·rafinNin]Tg
beingtheauto-correlationfunctionofφ(t),representsthecon-
P R and channel estimation.
tributionofclutterscatterers,whichcanbemodeled,invoking
the central limit theorem, as a zero mean Gaussian random
variable with a given variance σ2 (see [7, Appendix A]); III. CF BACKHAUL TRANSMISSION
c,n
and wn,k is the kth sample of wn(t) after matchedfiltering at Inthissection,weconsiderorthogonal-accessbackhauland
the sensor n. CF transmission. With CF, each receive sensor quantizes the
In vector notation, we can write (4) as receivedvectorrrr in (6), andsendsa quantizedversionofrrr
n n
rrrn =sssn+cccn+wwwn, (5) to the fusion center. Note that, since the receive sensor does
not know whether the target is present or not, the quantizer
where we defined rrrn , [rn,1 ··· rn,K]T, sssn , hnxxx and cannotdependonthecorrecthypothesis or .Inorderto
cccn , g˜nxxx; and the noise vector wwwn , [wn,1 ··· wn,K]T facilitate analysis and design, we followHth0e staHn1dard random
follows a zero-mean Gaussian distribution with temporal cor-
coding approach of rate-distortion theory of modeling the
relation ΩΩΩ , i.e., www (000,ΩΩΩ ). The variables h , g˜
w,n n ∼CN w,n n n effect of quantization by means of an additive quantization
and www for all n , are assumed to be independent for
n ∈ N noise (see, e.g., [17], [18]) as in
different values of n under the assumption that the receive
sensors are sufficiently separated [6]. Moreover,their second- rr˜rn =rrrn+qqqn, (8)
orderstatisticsσ2 ,σ2 andΩΩΩ areassumedtobeknownto
t,n c,n w,n where rr˜r = [r˜ r˜ ]T is the quantized signal vector
thefusioncenter,foralln ,e.g.,frompriormeasurements n n,1 ··· n,K
or prior information [15],∈[1N6]. ofrrrn; andqqqn ∼CN(000,ΩΩΩq,n) is the quantization error vector,
which is characterized by a covariance matrix ΩΩΩ . Based
Tosummarize,thesignalreceivedatsensorncanbewritten q,n
on random coding arguments, while (8) holds on an average
as
over randomly generated quantization codebooks, the results
:rrr =ccc +www , (6a)
H0 n n n derived in this paper can be obtained by means of some (de-
1 :rrrn =sssn+cccn+wwwn, n , (6b) terministic)high-dimensionalvectorquantizer(see,e.g.,[19]).
H ∈N
For instance, as discussed in [20], a Gaussian quantization
where and represent the hypotheses under which the
0 1
H H noiseqqq with any covarianceΩΩΩ can be realized in practice
target is absent or present, respectively. n q,n
viaalineartransform,obtainedfromtheeigenvectorsofΩΩΩ ,
In the rest of this section, we detail the assumed model q,n
followedbyamulti-dimensionalditheredlatticequantizersuch
forbothorthogonal-accessandnon-orthogonalmultiple-access
as Trellis Coded Quantization (TCQ) [21].
backhaul.
Based on (8), the signal received at the fusion center from
Orthogonal-access Backhaul: For the orthogonal-access
receive sensor n is given as
backhaulcase,eachreceivesensornisconnectedtothefusion
center via an orthogonal link of limited capacity Cn bits per H0 : rr˜rn =cccn+wwwn+qqqn, (9)
received sample. The capacity C is assumed to be known : rr˜r =sss +ccc +www +qqq .
n 1 n n n n n
H
4
As further elaborated in the following, the covariance matrix used to model the effect of the quantizers at the receive
ΩΩΩ determines the bit rate required for backhaul communi- sensors.
q,n
cationbetweenthereceivesensornandthefusioncenter[17], Bhattacharyya Distance: For two zero-mean Gaussian
[18] and is subject to design. distributions with covariance matrix of ΣΣΣ and ΣΣΣ , the Bhat-
1 2
To set the model (9) in a more convenient form, the signal tacharyya distance is given by [5]
B
received at the fusion center is whitened with respect to the
0.5(Σ +Σ )
overall additive noise cccn+wwwn+qqqn, and the returns from all =log | 1 2 | . (12)
sensors are collected, leading to the model B Σ1 Σ2 !
| || |
: yyy (000,III),
H0 ∼CN (10) Therefore,forthesignalmodepl(9),theBhattacharyyadistance
: yyy (000,DDDSSSDDD+III),
H1 ∼CN between the distributions under the two hypotheses can be
whereyyy =[yyyT yyyT]T,yyy =DDD rr˜r ,DDD is the whitening calculated as
1 ··· N n n n n
matrix associated with the receive sensor n and is given by III+0.5DDDSSSDDD
DDD =(σ2 xxxxxxH+ΩΩΩ +ΩΩΩ )−1/2,DDD istheblockdiagonal (xxx,ΩΩΩq)=log | |
n c,n w,n q,n B III +DDDSSSDDD !
matrix DDD = diag DDD ,...,DDD , and SSS is the block diago- | |
1 N
nal matrix SSS = d{iag σ2 xxxxxxH},...,σ2 xxxxxxH . The detection N p
problemformulatedin{(1t,01) has the stta,Nndard}Neyman-Pearson = Bn(xxx,ΩΩΩq,n)
n=1
solution given by the test X
N
1+0.5λ
1 = log n , (13)
yyyHTTTyyyHRν, (11) n=1 (cid:18) √1+λn (cid:19)
X
0
H where we have made explicit the dependence onxxx andΩΩΩ ;
q,n
where we have defined TTT = DDDSSSDDD(DDDSSSDDD +III)−1, and the ΩΩΩ collects all the covariance matrices of quantization noise
q
thresholdν issetbasedonthetoleratedfalsealarmprobability and is given as ΩΩΩ = ΩΩΩ ; and we have defined
q q,n n∈N
[22]. { }
In the rest of this section, we aim to find the optimum λn =σt2,nxxxH σc2,nxxxxxxH +ΩΩΩw,n+ΩΩΩq,n −1xxx. (14)
codevectorxxxandquantizationerrorcovariancematricesΩΩΩ
q,n (cid:0) (cid:1)
We observe that (13) is valid under the assumption that the
in (9), for given backhaul capacity constraints C , for all
n
effect of the quantizers can be well approximated by additive
n . Before we proceed, for reference, we first discuss
∈ N Gaussian noise as per (9). This is discussed next.
thestandarddistributeddetectionapproachthatcombineshard
local decisions at the receive sensors and a majority-rule Quantization: From rate-distortion theory, a vector quan-
detection at the fusion center (see, e.g., [23], [24]). tizerexiststhatisabletorealizetheadditivequantizationnoise
model (8), when operating over a sufficiently large number
of measurement vectors (6), as long as the capacity C is
n
A. Distributed Detection
no smaller than the mutual information I(rrr ;rr˜r )/K [19].
n n
Here, we describe the standard distributed detection ap- For example, a dithered lattice vector quantizer achieves this
proach applied to multistatic radar system (see, e.g., [23], result [20]. These considerationsmotivate the selection of the
[24]).Withthisapproach,eachreceivesensornmakesitsown mutual information I(rrr ;rr˜r ) as a measure of the backhaul
n n
H1 rate required for the transmission to the fusion center.
decisionbasedonthelikelihoodtestgivenbyyyyHnTTTnyyyn R γn, While the mutual information I(rrr ;rr˜r ) depends on the
H0 n n
where γ is the threshold for receive sensor n, which is actual hypothesis or , it is easy to see that I(rrr ;rr˜r )
n 0 1 n n
H H
calculated based on the tolerated false alarm probability [22], is larger under hypothesis . Based on this, the mutual
1
H
and we have defined TTT = DDD SSS DDD (DDD SSS DDD + III)−1 information I(rrr ;rr˜r ) evaluated under is adopted here as
n n n n n n n n n 1
H
with yyy = DDD rrr , DDD = (σ2 xxxxxxH + ΩΩΩ )−1/2 and the measure of the bit rate required between receive sensor
n n n n c,n w,n
SSS = σ2 xxxxxxH, for all n . The receive sensors transmit n and the fusion center. This can be easily calculated as
n t,n ∈ N
the obtained one-bit hard decision to the fusion center. Note I(rrr ;rr˜r )= (xxx,ΩΩΩ )byusingtheexpressionofthemutual
n n n q,n
I
that this scheme is feasible as long as the backhaul capacity information for multivariate Gaussian distribution (see, e.g.,
available for each receive sensor-to-fusion center channel is [18]) with
larger than or equal to 1/K bits/sample, i.e., C 1/K, for
n ≥ n(xxx,ΩΩΩq,n)=log III +(ΩΩΩq,n)−1ΩΩΩw,n
n . The fusion center decides on the target’s presence I
∈ N +log 1+(σ2 +σ2 )xxxH(ΩΩΩ +ΩΩΩ )−1xxx , (15)
based on the majority rule: if the number of receive sensors t,n (cid:12) c,n w,n (cid:12)q,n
(cid:12) (cid:12)
k that decide for satisfies k N/2, the fusion center
0 (cid:0) (cid:1)
H ≥ where again we have made explicitthe dependenceof mutual
chooses , and vice versa if k N/2.
0
H ≤ information onxxx and ΩΩΩ .
q,n
In the following, we formulate and solve the problem of
B. Performance Metrics and Constraints
jointlyoptimizingtheBhattacharyyadistancecriterionoverthe
To start the analysis of the cloud radar system, we discuss waveform xxx at the transmit element and over the covariance
thecriterionthatisadoptedtoaccountforthedetectionperfor- matricesΩΩΩ ofthequantizersatthe receivesensorsinSection
q
mance, namely the Bhattacharyya distance and the approach III-C and in Section III-D, respectively.
5
C. Problem Formulation outer iteration, MM requires to solve the problems (A.3)
and (A.6), whose sizes of the optimization domains are K
The problem of maximizing the Bhattacharyya distance in
and NK2, and numbers of constraints are N + 1 and 2N,
(13) over the waveform xxx and the covariance matrices ΩΩΩ
q
respectively [25], [29].
under the backhaul capacity constraints is stated as
N
minimize ¯(xxx,ΩΩΩ )= ¯ (xxx,ΩΩΩ ) (16a) Algorithm1Jointoptimizationofwaveformandquantization
xxx,ΩΩΩq B q n=1Bn qn noise covariances (16)
X
s.t. (xxx,ΩΩΩ ) KC =C¯ , n , (16b) Initialization(outer loop):Initializexxx(0) CK×1,ΩΩΩ(0)
n q,n n n q
I ≤ ∈N 0 and set i=0. ∈ (cid:23)
xxxHxxx P , (16c)
≤ T Repeat (BCD method)
ΩΩΩ 0, n , (16d) i i+1
q,n (cid:23) ∈N In←itialization (inner loop): Initializexxx(i,0) =
where we have formulated the problem as the minimization xxx(i−1) and set j =0.
of the negative distance ¯(xxx,ΩΩΩ )= N ¯ (xxx,ΩΩΩ ), with Repeat (MM method for xxx(i))
¯(xxx,ΩΩΩ ) = (xxx,ΩΩΩ )Band ¯q (xxx,ΩΩΩn=)1 =Bn (qxxx,n,ΩΩΩ ), j j+1
B q −B q Bn Pq,n −Bn q,n Fi←nd xxx(i,j) by solving the problem (A.3) with
following the standard convention in [25]. The power of the
ΩΩΩ =ΩΩΩ(i−1).
waveform xxx is constrained not to exceed a prescribed value q q
Until a convergencecriterion is satisfied.
of transmit power P . We observe that the constraint (16b)
T Updatexxx(i) xxx(i,j)
ensures that the transmission rate with K chips between each Initialization←(inner loop): InitializeΩΩΩ(i,0) =
receive sensor and the fusion center is smaller than C¯ , q
n ΩΩΩ(i−1) and set j =0.
according to the adopted information-theoretic metric. Note q
Repeat (MM method for ΩΩΩ(i))
also that the problem (16) is not a convex program, since q
j j+1
the objective function (16a) and the constraints (16b) are not Fi←nd ΩΩΩ(i,j) by solving the problem (A.6) with
convex. q
xxx=xxx(i).
Until a convergencecriterion is satisfied.
D. Proposed Algorithm UpdateΩΩΩq(i) ΩΩΩq(i,j)
Until a convergen←cecriterion is satisfied.
noSn-incocenvbeoxthinfuxxxncatinodnsΩΩΩB¯n(,xxxt,hΩΩΩeqnop)taimndizIanti(oxxxn,ΩΩΩprqonb)leinm(1(166))aries Solution:xxx←xxx(i) andΩΩΩq ←ΩΩΩq(i)
q,n
notconvex,andhenceitisdifficulttosolve.Toobtainalocally
optimalsolution, we approachthe joint optimization ofxxx and
ΩΩΩ in(16)viasuccessiveconvexapproximations.Specifically, IV. AFBACKHAUL TRANSMISSION
q
in an outer loop, Block Coordinate Descent (BCD) is applied In this section, we consider AF transmission on a non-
to update xxx and ΩΩΩq one at a time, while an inner loop orthogonalmultiple-access backhaul. With AF, sensor n
∈N
implemented via Majorization-Minimization(MM) solves the amplifies the received signal rrr in (6) and then forwards the
n
optimization of xxx and ΩΩΩq separately. This approach was first amplified signaltttn =αnrrrn to the fusion center, where αn is
introducedin [26] for a sum-capacitybackhaulconstraint. By is the amplification coefficient at the receive sensor n. From
thepropertiesofMM(see,e.g.,[27],[28]),thealgorithmpro- (7), the fusion center is faced with the following detection
videsasequenceoffeasiblesolutionswithnon-increasingcost hypothesis problem
function, which guarantees convergence of the cost function. N N
Note that, due to the non-convexityof the problem, no claim 0 : rr˜r= fntttn+zzz = fnαn(cccn+wwwn)+zzz,
H
of convergence to a local or global optimum is made here. n=1 n=1
X X
Attheithiterationoftheouterloop,theoptimumwaveform N N
xxx(i) is obtained by solving (16) for matrices ΩΩΩq = ΩΩΩ(qi−1) H1 : rr˜r= fntttn+zzz fnαn(sssn+cccn+wwwn)+zzz.
obtained at the previous iteration; subsequently, the matrices nX=1 nX=1
(17)
ΩΩΩ(i) are calculated by solving (16) with xxx=xxx(i). These two
q
separate optimizations are carried out by the MM method, The variables h , g˜ , www , f and zzz, for all n , are
n n n n
∈ N
which, as described in Appendix A, requires the solution of assumed to be mutually independent. Since only the second-
a quadratically constrained quadratic programs (QCQP). The order statistics of the channel gains h , n , are known
n
∈ N
proposed algorithm coupling BCD and MM to solve problem to the receive sensors and the fusion center, no coherent
(16),issummarizedinTableAlgorithm1.InAlgorithm1,we gains may be achieved by optimizing the amplifying gains,
usethesuperscriptitoidentifytheiterationsoftheouterloop, and hence one can focus, without loss of optimality, only
and the superscriptj as the index of the inner iteration of the on the receive sensors’ power gains ppp = [p p ]T, with
1 N
···
MM method (e.g.,xxx(i,j) indicates the waveform optimized at p = α 2, for n .
n n
| | ∈N
the jth iteration of the inner loop of the MM method and the As in the CF backhaul transmission in Section III, we can
ithiterationoftheouterloop).InAppendixA,wepresentthe write the hypotheses (17) in a standard form by whitening
MM steps and the overall proposed algorithm in detail. the signal received at the fusion center, and consequently
The complexity of Algorithm 1 by using standard convex the detection problem can be expressed as (10), where we
optimization tools is polynomial in K and N since, at each have redefined yyy = DDDrr˜r; DDD = ( N (f 2p σ2 xxxxxxH +
n=1 | n| n c,n
P
6
f 2p ΩΩΩ )+ΩΩΩ )−1/2 is the whitening filter with respect We proposean algorithmto solvetheoptimizationproblem
n n w,n z
t|o t|he overall additive noise N f α (ccc +www )+zzz; and (19). As in Section III-D, due to the difficulty of obtaining
n=1 n n n n
SSS = N f 2p σ2 xxxxxxH is the correlation matrix of the a global optimal solution, we develop a descent algorithm,
n=1| n| n t,n P
desired signal part. Accordingly, the detection problem has and adopt the BCD method coupled with MM. The proposed
the stPandard estimator-correlator solution given by the test algorithm is summarized in Table Algorithm 2 and further
in (11). In the rest of this section, we seek to optimize the detailed in Appendix B. The complexity of the Algorithm 2
detection performance with respect to the waveform xxx and by using standard convex optimization tool is polynomial in
the power gains ppp, under power constraints on the transmit K and N since, at each outer iteration, MM requires to solve
element and receive sensors. As done above, we adopt the the problems(B.2) and (B.4), whose sizes of the optimization
Bhattacharyya distance as the performance metric. As per domains are K and N, and numbers of constraints are 1 and
(12),theBhattacharyyadistancebetweenthedistributions(17) N +1, respectively [25], [29].
of the signals received at the fusion center under the two
hypotheses and can be calculated as Algorithm 2 Short-term adaptive design of waveform and
0 1
H H amplifier gain (19)
III+0.5DDDSSSDDD
(xxx,ppp;fff) = log | | Initialization (outer loop): Initializexxx(0) CK×1, ppp(0)
B III +DDDSSSDDD ! 0 and set i=0. ∈ (cid:23)
| |
Repeat (BCD method)
1+0.5λ
= log p , (18) i i+1
√1+λ In←itialization (inner loop): Initializexxx(i,0) =
(cid:18) (cid:19)
xxx(i−1) and set j =0.
where λ = fffHPPPΣΣΣtfffxxxH(fffHPPPΣΣΣcfffxxxxxxH +(fff IIIK)H (PPP Repeat (MM method for xxx(i))
⊗ ⊗
IIIanKd)ΩΩΩΣΣΣw(fff=⊗dIIIiaKg)σ+2 ΩΩΩ,.z.).−,1σxxx2; ΣΣΣtar=e tdhieagd{iσatg2,o1,n.a.l.m,aσtrt2i,cNe}s Fji←ndjxxx+(i,j1) by solving the problem (B.2) with
whoseccomponen{tsca,1re the sec,cNon}d-order statistics of channel ppp=ppp(i−1).
Until a convergencecriterion is satisfied.
amplitudes of target return and clutter, respectively; ΩΩΩ =
w Updatexxx(i) xxx(i,j)
diag{ΩΩΩw,1,...,ΩΩΩw,N} ∈ RNK×NK is a block diagonal ma- Initialization←(inner loop): Initializeppp(i,0) =
trix containingall the noise covariancematricesat the receive
ppp(i−1) and set j =0.
sensors;andPPP =diag{ppp}∈RN×N isthediagonalmatrixthat Repeat (MM method for ppp(i))
contains the receive sensors’ power gains. Note that we have j j+1
made explicit the dependence of the Bhattacharyya distance Fi←nd ppp(i,j) by solving the problem (B.4) with
(xxx,ppp;fff) on the channels fff at the fusion center, as well as xxx=xxx(i).
Bon the waveformxxx and the receive sensors’ power gainsppp. Until a convergencecriterion is satisfied.
Updateppp(i) ppp(i,j)
Until a convergen←cecriterion is satisfied.
A. Short-Term Adaptive Design Solution:xxx xxx(i) andppp ppp(i)
← ←
We first consider the case in which design of the wave-
form xxx and of the receive sensors’ gains ppp depends on
the instantaneous gain of the CSI of the receive sensors-to- B. Long-Term Adaptive Design
fusion center channels fff. Note that this design requires to Here, in order to avoid the possibly excessive feedback
modify the solution vector (xxx,ppp) at the time scale at which overhead between fusion center and the transmit element
the channel vector fff varies, hence entailing a potentially and receive sensors of the short-term adaptive solution, we
large feedback overheadfrom the fusion center to the receive adoptthe average Bhattacharyyadistance, as the performance
sensorsandthetransmitelement.Theproblemofmaximizing criterion, where the average is taken with respect to the
the Bhattacharyya distance (18) over the waveformxxx and the distribution of the receive sensors-to-fusion center channels
powergainspppunderthepowerconstraintsfortransmitelement fff. In this way, the waveform xxx and receive sensors’ gains
and receive sensors, is stated as ppp have to be updated only at the time scale at which the
minimize ¯(xxx,ppp;fff) (19a) statistics of channelsand noise terms vary.Then,the problem
xxx,ppp B for the long-termadaptivedesign is formulatedfrom problem
s.t. xxxHxxx PT, (19b) (19) by substituting the objective function ¯(xxx,ppp;fff) with
111Tppp≤≤PR, (19c) Efff[B¯(xxx,ppp;fff)]m,iyniiemldiizneg E ¯(xxx,ppp;fff) B (20a)
p 0, n , (19d) fff
n ≥ ∈N xxx,ppp B
where we have defined ¯(xxx,ppp;fff)= (xxx,ppp;fff) to formulate s.t. (19(cid:2)b)−(19d).(cid:3) (20b)
B −B
theproblemastheminimizationofthenegativeBhattacharyya Note that the problem (20) is a stochastic program with a
distance ¯(xxx,ppp;fff). We observethat the problem(19) may be non-convexobjective function (20a).
B
easily modified to include individual power constraints at the Since the stochastic program (20) has a non-convex ob-
receivesensors,butthisisnotfurtherexploredhere.Moreover, jective function, we apply the stochastic successive upper-
the problem (19) is not a convexprogram,since the objective boundminimizationmethod(SSUM)[30],whichminimizesat
function (19a) is not convex. each step an approximate ensemble average of a locally tight
7
1 0.8
0.9
0.7
0.8
0.6
0.7
Bhattacharyya distance000...456 Pd000...345 U(Dpipsetrri bbuotuendd) Hwaitrhd iddeecails iboanc k+h maualjority rule
0.3 (Cloud) No opt.
(Cloud) Waveform opt.
Upper bound with ideal backhaul 0.2 (Cloud) Quantization opt.
0.2 No opt. (Cloud) Joint opt.
Waveform opt.
0.1
0.1 Quantization opt.
Joint opt.
0 0
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
C¯ C¯
Fig.2. BhattacharyyadistanceversusthebackhaulcapacityC¯n=C¯,n∈N Fig. 3. Probability ofdetection Pd versus the backhaul capacity C¯n =C¯
fσoc2r,1CF=b0a.c1k2h5a,ulσtc2r,a2ns=mi0s.s2io5n,,σwc2i,t3h=PT0.=5a1n0dd[BΩΩΩ,wK,n=]i,j13=,N(1=−30,.1σ2t2n,n)|=i−1j|, fσot2r,nCF=ba1c,kσhac2u,1lt=ran0sm.1i2ss5i,onσ,c2n,2∈=N0,.2w5i,thσPc2,T3 ==100.5d,B[,ΩΩΩKw,n=]i1,j3,=N(=1−3,
forn∈N. 0.12n)|i−j| forn∈N andPfa=0.01.
1
upper bound of the cost function. Specifically, we develop a Upper bound with ideal backhaul
(Distributed) Hard decision + majority rule
BCD scheme similar to the one detailed in Table Algorithm 0.9 (Cloud) No opt.
(Cloud) Waveform opt.
2 that uses SSUM in lieu of the MM scheme. Details are 0.8 (Cloud) Quantization opt.
(Cloud) Joint opt.
provided in Appendix C. The final algorithm for long-term
0.7
adaptivedesigncanbesummarizedasinTableAlgorithm2by
0.6
substituting(B.2)and(B.4)with(C.1)and(C.2),respectively.
Convergence of the SSUM algorithm is proved in [30] and Pd0.5
the algorithm guarantees feasible iterates. The complexity of 0.4
theproposedalgorithmbyusingstandardconvexoptimization 0.3
tool is polynomial in K and N since, at each outer iteration,
0.2
SSUM requires to solve the problems (C.1) and (C.2), whose
0.1
sizes of the optimizationdomainsare K and N, and numbers
of constraints are 1 and N +1, respectively [25], [29]. 0
10(cid:237)6 10(cid:237)5 10(cid:237)4 10(cid:237)3 10(cid:237)2 10(cid:237)1 100
P
fa
V. NUMERICAL RESULTS Fig. 4. ROC curves for CF backhaul transmission with PT = 10 dB,
In the following, the performance of the proposed algo- K = 13, N = 3, σt2,n = 1, σc2,1 = 0.125, σc2,2 = 0.25, σc2,3 = 0.5,
rithms that perform joint optimization of the waveformxxx and [ΩΩΩw,n]i,j =(1−0.12n)|i−j| andC¯n=C¯=5forn∈N.
capacitybackhaullinks,distributeddetectionusingtheBarker
of the quantization noise covariance matrices ΩΩΩ for the CF,
q
waveform (see, Section III-A), and the following strategies:
and of the waveform xxx and of the power gains ppp for AF,
(i) No optimization (No opt.): Set xxx = P /Kbbb and
are investigated via numerical results in Section V-A and in T 13
ΩΩΩ = ǫIII, for n , where ǫ is a constant that is found
Section V-B, respectively. Throughout, we set the length of q,n ∈ N p
bysatisfying the constraint(16b) with equality;(ii) Waveform
the waveform to K = 13 and the variances of the target
amplitudes as σ2 = 1 for n . For reference, we optimization (Waveform opt.) : Optimize the waveform xxx by
t,n ∈ N using the algorithm in [7], which is given in Algorithm 1
consider a baseline waveform with Barker code of length 13,
i.e., bbb = [1 1 1 1 1 1 1 1 1 1 1 1 1]T. by setting ΩΩΩq,n = 0 for n , and set ΩΩΩq,n = ǫIII, for
13 ∈ N
− − − − n , as explainedabove;(iii) Quantizationnoise optimiza-
Moreover, unless stated otherwise, we model the noise with
∈N
covariancematrices[ΩΩΩ ] =(1 0.12n)|i−j|and[ΩΩΩ ] = tion (Quantizationopt.):Optimizethe covariancematricesΩΩΩq
w,n i,j z i,j
(1 0.6)|i−j| as in [7], hence −accounting for temporally as per Algorithm 1 with xxx = PT/Kbbb13. In the following,
− we set the number of receive sensors, the transmit power and
correlated interference. The channel coefficients fn have unit p
variance, i.e., σ2 =1. the varianceof the clutter amplitudesasN =3,PT =10dB,
fn σ2 = 0.125, σ2 =0.25 and σ2 = 0.5, respectively. Also,
c,1 c,2 c,3
thebackhaulrateconstraintsC¯ are assumedto beequal,i.e.,
n
A. CF Backhaul Transmission C¯ =C¯ for all n .
n
∈N
In this section, the performance of the proposed joint In Fig. 2 the Bhattacharyya distance is plotted versus the
optimization of the waveformxxx and of the quantization noise available backhaul capacity C¯. For intermediate and large
covariancematricesΩΩΩ inSectionIIIisverifiedvianumerical values of C¯, the proposed joint optimization of waveform
q
results. Note that some limited results for a sum-backhaul and quantization noise is seen to be significantly beneficial
constraint were presented in [26]. For reference, we consider over all separate optimization strategies. In order to study the
the performance of the upper bound obtained with infinite actual detection performance and validate the results in Fig.
8
70 70
Sensor 1: w1 Sensor 1: w1
Sensor 2: w2 Sensor 2: w2
60 Sensor 3: w3 60 Sensor 3: w3
50 50
wer spectral density3400 wer spectral density3400
Po Po
20 20
10 10
0 0
(cid:237)0.5 (cid:237)0.4 (cid:237)0.3 (cid:237)0.2 (cid:237)0.1 0 0.1 0.2 0.3 0.4 0.5 (cid:237)0.5 (cid:237)0.4 (cid:237)0.3 (cid:237)0.2 (cid:237)0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency (Hz) Frequency (Hz)
(a) Low-frequency interference (b) High-frequency interference
45 180
Barker waveform
Optimal waveform Barker waveform
40 Sensor 1: q1 160 OSepntismora 1l :w qaveform
Sensor 2: q2 Sensor 2: q1
35 Sensor 3: q3 140 Sensor 3: q2
3
Energy/power spectral density12235050 Energy/power spectral density1102680000
10 40
5 20
(cid:237)00.5 (cid:237)0.4 (cid:237)0.3 (cid:237)0.2 (cid:237)0.1 0 0.1 0.2 0.3 0.4 0.5 (cid:237)00.5 (cid:237)0.4 (cid:237)0.3 (cid:237)0.2 (cid:237)0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency (Hz) Frequency (Hz)
(c) Optimal waveform and quantization noise with low-frequency (d) Optimal waveform and quantization noise with high-frequency
interference interference
Fig. 5. Comparison ofthe energy/power spectral densities ofthe waveforms obtained with a Barker code (Barker waveform) and with anoptimal codexxx
(Optimalwaveform), andofoptimal quantization noise{qqqn}Nn=1 obtained byAlgorithm 1whenPT =10dB,K=13,N =3,σt2,n=1,σc2,1=0.125,
σc2,2 =0.25, σc2,3 =0.5, and C¯n =C¯ =5 forn∈N:(a) and (c) consider receive sensors with low-frequency interference having temporal correlation
[ΩΩΩw,n]i,j = (1−0.12n)|i−j|, (b) and (d) consider receive sensors with high-frequency interference having temporal correlation [ΩΩΩw,n]i,j = (−1+
0.12n)|i−j|.
2, Fig. 3 shows the detection probability P as a function approach. For instance, for P = 0.01, joint optimization
d fa
of the available backhaul capacity C¯ when the false alarm yieldsP =0.7251,while waveformoptimizationonly yields
d
probabilityisP =0.01.ThecurvewasevaluatedviaMonte P =0.4556.
fa d
Carlo simulations by implementing the optimum test detector
(11). We also implemented the distributed detection scheme Fig. 5 shows the energy/power spectral density functions
described in Section III-A by setting the threshold γ to be of the waveform with Barker code (Barker waveform) and
n
equalforn forsimplicity.Itcanbenotedthattherelative with optimal code xxx (Optimal waveform), and of optimal
gainspredic∈tedNbytheBhattacharyyadistancecriterioninFig. quantization noise {qqqn}Nn=1 obtained by Algorithm 1 when
2 are consistent with the performanceshown in Fig. 3. More- a square root Nyquist chip waveform φ(t) with duration Tc
over, for small values of C¯, distributed detection outperforms is adopted, and C¯n = C¯ = 5 for n . We consider two
∈ N
cloud detection due to the performance degradation caused types of interference at the receive sensors, namely (a) low-
by the large quantization noise on the cloud-based schemes. frequency interference with temporal correlation [ΩΩΩw,n]i,j =
However, as the available backhaul capacity C¯ increases, the (1 0.12n)|i−j| which has a single spectral peak at zero
−
clouddetectionapproachconsiderablyoutperformsdistributed frequency; and (b) high-frequencyinterference with temporal
detection. correlation[ΩΩΩw,n]i,j =( 1+0.12n)|i−j|, having a minimum
−
at zero frequency. It is observed in Fig. 5(c) and Fig. 5(d)
Fig. 4 plots the Receiving Operating Characteristic (ROC), that the spectrum of the optimal waveform concentrates the
i.e.,thedetectionprobabilityP versusfalsealarmprobability transmitted energy at frequencies for which the interference
d
P , for C¯ = 5. It is confirmed that the proposed joint opti- powerisless pronounced,while thespectrumofthequantiza-
fa
mization method provides remarkable gains over all separate tion noise concentrates at frequencies and sensors for which
optimizationschemesas well as overthe distributeddetection the interference power is more pronounced.
9
0.35 0.2
No opt. No opt.
(Long(cid:237)term) Joint opt. (Long(cid:237)term) Joint opt.
0.3 ((LLoonngg(cid:237)(cid:237)tteerrmm)) GWaaivne ofoprt.m opt. 0.18 ((LLoonngg(cid:237)(cid:237)tteerrmm)) GWaaivne ofoprt.m opt.
(Short(cid:237)term) Joint opt. (Short(cid:237)term) Joint opt.
(Short(cid:237)term) Gain opt. 0.16 (Short(cid:237)term) Gain opt.
0.25 (Short(cid:237)term) Waveform opt. (Short(cid:237)term) Waveform opt.
Bhattacharyya distance0.01.52 Bhattacharyya distance00..011.241
0.1
0.08
0.05
0.06
(cid:237)015 (cid:237)10 (cid:237)5 0 5PT (dB)10 15 20 25 30 0.041 2 3 4 N 5 6 7 8
Fig.6. Bhattacharyya distance versusthetransmitelement’s powerPT for Fig.7. BhattacharyyadistanceversusthenumberreceivesensorsN forAF
1A,Fσbt2,anckh=aul1,trσanc2s,1mi=ssio0n.2w5,ithσc2P,2R==01.50,dσBc2,,3K==1,13[ΩΩΩ, wN,n=]i,j3,=σf2(n1−= σbat2c,nkh=aul1t,raσnc2s,m1i=ssio1n, σwc2i,t2h=PT0.=9,5σc2d,B3,=PR0.7=5,1σ0c2d,4B,=K0.=5,1σ3c2,,5σf2=n0=.351,,
0.12n)|i−j| and[ΩΩΩz]i,j =(1−0.6)|i−j| forn∈N. σc2,6=0.25andσc2,7=0.125,σc2,8=0.05,[ΩΩΩw,n]i,j =(1−0.12n)|i−j|
and[ΩΩΩz]i,j =(1−0.6)|i−j| forn∈N.
B. AF Backhaul Transmission 1
No opt.
Inthissection,weevaluatetheperformanceoftheproposed 0.9 ((LLoonngg(cid:237)(cid:237)tteerrmm)) JGoaiinnt oopptt..
(Long(cid:237)term) Waveform opt.
algorithmsthatperformthejointoptimizationofthewaveform 0.8 (Short(cid:237)term) Joint opt.
(Short(cid:237)term) Gain opt.
xxx and of the amplifying power gains ppp for the short-term 0.7 (Short(cid:237)term) Waveform opt.
(SectionIV-A)andlong-term(SectionIV-B)adaptivedesigns.
0.6
Forreference,weconsiderthefollowingschemes;(i)Noopt.:
Set xxx = P /Kbbb and ppp = P /N111 ; (ii) Waveform opt.: Pd0.5
T 13 R N
Optimize the waveform xxx as per Algorithm 2 (with (C.1) in 0.4
p
lieu of (B.2) for the long-term adaptive design) with ppp =
0.3
P /N111 ; and (iii) Gain optimization (Gain opt.): Optimize
R N
0.2
the gainsppp as per Algorithm2 (with (C.2) in lieu of (B.4) for
the long-term adaptive design) with xxx= P /Kbbb . We set 0.1
T 13
the totalreceivesensors’powerasPR =p10dB. Note thatthe 0 10(cid:237)2 10(cid:237)1 100
upper bound with ideal backhaul is far from the performance Pfa
achieved with AF over a non-orthogonal backhaul even for
glaarpgebestewnesoenrst’hpeoAwFerpPerRfo,ramnadnicteisanhdenthceeunpoptesrhboowunndhiesred.uTehtoe σ1F0ic2g,3.d8B=., 1KR,O[=ΩΩΩCw1c,3nu,r]viN,ejs==for(31A,−Fσf2b0na.c1=k2hna1)u|,li−σtrt2ja,|nnsamn=dis1s[ΩΩΩi,ozσn]c2iw,,1jith==P(0T1.2−=5,05σ.6c2d),B2|i,−=PjR|0f.o=5r,
thefactthat,inordertoobtainanidealbackhaul,oneneedsto n∈N.
code acrosslong blocklengthswhereasAF operateson block
length of size equal to the waveform K (here K =13). optimizationseekstodesignthetransmittedsignalxxxsuchthat
Fig. 6 shows the Bhattacharyya distance as a function of itreducesthepowertransmittedatthefrequenciesinwhichthe
the transmit element’s power P , with N = 3, σ2 = 0.25, receive sensors observe the largest interference, while, at the
T c,1
σ2 = 0.5 and σ2 = 1. For small values of P , optimizing same time, allocating morepowerto receivesensorssuffering
c,2 c,3 T
the waveform is more advantageous than optimizing the am- fromlessinterferenceand,withtheshort-termadaptivedesign,
plifying gains, due to the fact that performance is limited by having better channels to the fusion center.
thetransmitelement-to-receivesensorsconnection.Incontrast, In Fig. 7, the Bhattacharyya distance is plotted versus the
for intermediate and large values of P , the optimization of number receive sensors N with P = 5 dB, σ2 = 1,
T T c,1
the receive sensors’ gains is to be preferred, since the perfor- σ2 =0.9, σ2 =0.75, σ2 =0.5, σ2 =0.35, σ2 =0.25,
c,2 c,3 c,4 c,5 c,6
mance becomes limited by the channels between the receive σ2 =0.125andσ2 =0.05.Optimizingthereceivesensors’
c,7 c,8
sensors and the fusion center. Joint optimization significantly powergainsisseentobeespeciallybeneficialatlargeN,due
outperforms all other schemes, except in the very low- and to the ability to allocate more power to the receive sensors
large-powerregimes, in which, as discussed, the performance in better condition in terms of interference and channels
is limited by either the transmit element-to-receive sensors to the fusion center. For instance, even with the long-term
or the receive sensors-to-fusion center channels. In addition, adaptive design, optimizing the receive sensors’ power gains
we observe that the long-term adaptive scheme loses about outperforms waveform optimization with short-term adaptive
30% in terms of the Bhattacharyya distance with respect to design for sufficiently large N.
the short-term adaptive design in the high SNR regime. The Fig. 8 plots the ROC curves with P = 5 dB, N = 3,
T
results in Fig. 6 can be interpreted by noting that the joint σ2 =0.25,σ2 =0.5andσ2 =1.Thecurvewasevaluated
c,1 c,2 c,3
10
via Monte Carlo simulations by implementing the optimum Taylor approximation [27], which follows the same steps as
testdetector(11)asdiscussedinSectionIV.Itcanbeobserved in [7, eq. (34) and (50) in Section IV], and is given by
thatthegainsobservedinthepreviousfiguresdirectlytranslate
N
intoabetterROCperformanceofjointoptimization.Notealso B¯(xxx,ΩΩΩ xxx(i,j−1))= B¯(xxx,ΩΩΩ xxx(i,j−1))
that power gain optimization is seen to be advantageous due U q| Un q,n|
n=1
to sufficient value of P as predicted based on Fig. 6. X
T N
= φ(i,j−1)xxxH(ΩΩΩ +ΩΩΩ )−1xxx
n w,n q,n
VI. CONCLUDING REMARKS nX=1
H
We have studied a multistatic cloud radar system, where Re ddd(i,j−1) xxx , (A.1)
− n
the receive sensors and fusion center are connected via an where (cid:18)(cid:16) (cid:17) (cid:19)
orthogonal-access backhaul or a non-orthogonal multiple-
access backhaul channel. In the former case, each receive
β
sensor quantizes and forwards the signal sent by transmit φ(i,j−1) = n +β (1+0.5γ )
element to a fusion center following a compress-and-forward n 1+βnyn(i,j−1) n n
protocol, while amplify-and-forward of the received signal 0.5γn βn
+ ;
is carried out over the multiple-access backhaul. The fusion 1+λ(ni,j−1) 1+β y(i,j−1) 2
center collects the signals from all the receive sensors and n n
determines the target’s presence or absence. We have in- 2β(1+0.5γ(cid:16) ) (cid:17)
ddd(i,j−1) = n
vestigated the joint optimization of waveform and backhaul n (i,j−1)
transmission so as to maximize the detection performance. 1+βnyn (1+0.5γn)
As the performance metric, we adopted the Bhattacharyya +2βn(1+0.5γn))(ΩΩΩw,n+ΩΩΩq,n)−1xxx(i,j−1);
distanceandtheproposedalgorithmicsolutionswerebasedon y(i,j−1) = xxx(i,j−1) H(ΩΩΩ +ΩΩΩ )−1xxx(i,j−1);
successive convexapproximations.Overall, joint optimization n w,n q,n
was seen to have remarkable gains over the standard separate λ(i,j−1) =(cid:16)γ (cid:17) γn .
optimization of waveform and backhaul transmission. More- n n− 1+βnyn(i,j−1)
over, cloud processing is found to outperform the standard
with β =σ2 and γ =σ2 /β . A bound with the desired
distributeddetectionapproachaslongasthebackhaulcapacity n c,n n t,n n
property can also be easily derived for (xxx,ΩΩΩ ) by using
is large enough. In q,n
theinequalitylog(1+t) log(1+t(l))+1/(1+t(l))(t t(l)),
≤ −
for t=(σ2 +σ2 )xxxH(ΩΩΩ +ΩΩΩ )−1xxx, leading to
t,n c,n w,n q,n
APPENDIX A:DETAILSOF CFOPTIMIZATION I(xxx,ΩΩΩ xxx(i,j−1))=log III+(ΩΩΩ )−1ΩΩΩ
Un q,n| q,n w,n
A. Review of MM Method 1
+log(1+t(i,j−1))+ (cid:12) (σ2 +σ(cid:12)2 )
We start by reviewing the MM method. For a non-convex 1+(cid:12)t(i,j−1) c,n (cid:12)t,n
function f(ttt) of a generic variable ttt, which may appear xxxH(ΩΩΩ +ΩΩΩ )−1xxx (cid:0)t(i,j−1) . (A.2)
w,n q,n
−
either in the cost function or among the constraints, the MM
(cid:17)
methodsubstitutesatthelthiteration,aconvexapproximation Atthe jth iteration ofthe MM methodand the ithouter loop,
f(tttttt(l−1)) of f(ttt), such that the globalupperboundproperty we evaluate the new iterate xxx(i,j) by solving the following
|
f(tttttt(l−1)) f(ttt) is satisfied for all ttt in the domain, along QCQP problem
with| theloc≥altightnessconditionf(ttt(l−1)ttt(l−1))=f(ttt(l−1)). xxx(i,j) argmin B¯(xxx,ΩΩΩ xxx(i,j−1)) (A.3a)
| ← U q|
These properties guarantee the feasibility of all iterates and xxx
thedescentpropertythattheobjectfunctiondoesnotincrease s.t. UnI(xxx,ΩΩΩq,n|xxx(i,j−1))≤C¯n, n∈N, (A.3b)
along the iterations. xxxHxxx P . (A.3c)
T
≤
The MM methodobtainsthe solutionxxx(i) for the ith iteration
B. Details of the Proposed Algorithm 1
oftheouterloopbysolvingtheproblem(A.3)iterativelyover
In the following, we discuss the application of the MM j until a convergence criterion is satisfied.
method to performoptimizationsoverxxx andΩΩΩq in Algorithm Optimization over ΩΩΩq: In this part, we consider the
1, respectively. optimization of matrices ΩΩΩ(i) for a given xxx = xxx(i). Sim-
q
Optimizationoverxxx:Here,thegoalistoobtaintheoptimal ilar to the optimization over xxx(i), we use upper bounds of
valueofxxx(i) forproblem(16)givenΩΩΩ =ΩΩΩ(i−1).Tothisend, ¯(xxx,ΩΩΩ ) and (xxx,ΩΩΩ ) foroptimization.First, by rewriting
q q q n q,n
B I
we apply the MM method. Specifically, at the jth iteration of (xxx,ΩΩΩ ) as (xxx,ΩΩΩ )=log ΩΩΩ +(σ2 +σ2 )xxxxxxH+
In q,n In q,n | q,n t,n c,n
the MM method and the ith iteration of the outer loop, the ΩΩΩ log ΩΩΩ , we obtain difference-of-convex functions
w,n q,n
|− | |
MM method solves a QCQP and obtains a solution xxx(i,j) by with respect to ΩΩΩ . Then, by linearizing negative convex
q,n
substituting the non-convex objective function ¯(xxx,ΩΩΩ ) with component via its first-order Taylor approximation, upper
q
a tight upper bound B¯(xxx,ΩΩΩ xxx(i,j−1)) arounBd the current bounds I(xxx,ΩΩΩ ΩΩΩ(i,j−1))and B¯(xxx,ΩΩΩ ΩΩΩ(i,j−1))withthe
U q| Un q,n| q,n U q| q
iterate xxx(i,j−1). This bound is obtained by linearizing the desired properties of MM method are derived for functions
difference-of-convex functions in ¯(xxx,ΩΩΩ ) via the first-order (xxx,ΩΩΩ ) and ¯(xxx,ΩΩΩ ), respectively, as follows:
q n q,n q
B I B