Table Of ContentIEEESIGNALPROCESSINGLETTERS,VOL.*,NO.*,MONTHYYYY 1
Non-colocated Time-Reversal MUSIC:
High-SNR Distribution of Null Spectrum
D. Ciuonzo, Senior Member, IEEE and P. Salvo Rossi, Senior Member, IEEE
Abstract—We derive the asymptotic distribution of the null scatterer. On the other hand, TR-MUSIC imaging is based
spectrum of the well-known MultipleSignal Classification (MU- on a complementary point of view and relies on the noise
SIC) in its computational Time-Reversal (TR) form. The result subspace (viz. orthogonal-subspace1), leading to satisfactory
pertainsto a single-frequencynon-colocated multistatic scenario
performance as long as the data space dimension exceeds
and several TR-MUSIC variants are here investigated. The
7 analysis builds upon the 1st-order perturbation of the singular thesignalsubspacedimensionandsufficientlyhighSignal-to-
1 value decomposition and allows a simple characterization of Noise-Ratio(SNR)ispresent.TR-MUSICwasfirstintroduced
0 null-spectrum moments (up to the 2nd order). This enables a for a Born Approximated (BA) linear scattering model [7]
2 comparison in terms of spectrums stability. Finally, a numerical
and, later, successfully applied to the Foldy-Lax (FL) non-
analysis is provided to confirm the theoretical findings.
n linear model [8]. Also, it became popular mainly due to: (a)
a Index Terms—Time-Reversal (TR), Radar imaging, Null- algorithmic efficiency; (b) no need for approximate scattering
J spectrum, Resolution, TR-MUSIC.
models; and (c) finer resolution than the diffraction limits
5
(especially in scenarios with few scatterers). Recently, TR-
2
I. INTRODUCTION MUSIC has been expanded to extended scatterers in [9].
] TIME-REVERSAL(TR)referstoallthosemethodswhich ThoughavastliteratureonperformanceanalysisofMUSIC
T
[10] for Direction-Of-Arrival (DOA) estimation exists (see
exploit the invariance of the wave equation (in lossless
I
. and stationary media) by re-transmitting a time-reversed ver- [11], [12] for resolution studies and [13]–[16] for asymptotic
s
Mean Squared Error (MSE) derivation, with more advanced
c sionofthescattered(orradiated)fieldmeasuredbyanarrayto
[ studiespresentedin[17]–[19]),suchresultscannotbedirectly
focus on a scattering object (or radiating source), by physical
applied to TR-MUSIC. Indeed, in TR framework scatter-
1 [1] or synthetic [2] means. In the latter case (computational
ers/sources are generally assumed deterministic and more
v TR), it consists in numerically back-propagating the field
6 data by using a known Green’s function, representative of importantlyasinglesnapshotisused,whereasMUSICresults
1 for DOA refer to a differentasymptotic condition (i.e. a large
the propagation medium. Since the employed Green function
5 numberofsnapshots).Also,toourknowledge,nocorrespond-
dependsontheobject(orsource)position,animageisformed
7
ingtheoreticalresultshavebeenproposedintheliteraturefor
0 by varying the probed location (this procedure is referred
TR-MUSIC, except for [20], [21], providing the asymptotic
. to as “imaging”). Computational TR has been successfully
1
(high-SNR) localization MSE for point-like scatterers. Yet,
0 applied in different contexts such as subsurface prospecting
a few works have tackled achievable theoretical performance
7 [3], through-the-wall[4] and microwave imaging [5].
1 ThekeyentityinTR-imagingistheMultistaticDataMatrix both for BA and FL models via the Cramér-Rao lower-bound
: [22].
v (MDM), whose entries are the scattered field due to each
In this letter we provide a null-spectrum2 analysis of TR-
i Transmit-Receive (Tx-Rx) pair. Two popular methods for
X MUSIC for point-like scatterers, via a 1st-order perturbation
TR-imaging are the decomposition of TR operator (DORT)
r of Singular Value Decomposition (SVD) [24], thus having
a [6] and the TR Multiple Signal Classification (TR-MUSIC)
asymptotic validity (i.e. meaning a high SNR regime). The
[7]. DORT imaging exploits the MDM spectrum by back-
present results are based on a homogeneous background
propagatingeacheigenvectoroftheso-calledsignalsubspace,
assumption and neglecting mutual coupling, as well as po-
thus allowing to selectively focus on each (well-resolved)
larization or antenna pattern effects. Here we build upon
Manuscript received 2ndDecember2016;accepted 25thJanuary2017. [25] (tackling the simpler colocated case) and consider a
D.CiuonzoiswithDIETI,UniversityofNaples“FedericoII”,Naples,Italy. general non-colocated multistatic setup with BA/FL models
P. Salvo Rossi is with the Dept. of Electronics and Telecommunications,
where severalTR-MUSIC variants, proposed in the literature,
NTNU,Trondheim,Norway.
E-mail:{domenico.ciuonzo, salvorossi}@ieee.org. are here investigated. The obtained results complement those
0Notation-Lower-case(resp.Upper-case)boldlettersdenotecolumnvec- found in DOA literature [23] and allow to obtain both the
tors(resp.matrices),withan(resp.an,m)beingthenth(resp.the(n,m)th) mean and the variance of each null-spectrum, as well as to
element of a (resp. A); E{·}, var{·}, (·)T, (·)†, (·)∗, Tr[·], vec(·),
(·)−, ℜ(·), δ(·), k·k and k·k denote expectation, variance, transpose, draw-outitspdf.Also,theyhighlightperformancedependence
F
Hermitian, conjugate, matrix trace, vectorization, pseudo-inverse, real part, of null-spectrum on the scatterers/arrays configurations and
Kronecker delta, Frobenius and ℓ2 norm operators, respectively; j denotes compare TR-MUSIC variants in terms of spectrum stability.
the imaginary unit; 0N×M (resp. IN) denotes the N × M null (resp.
identity)matrix;0N (resp.1N)denotesthenull(resp.ones)columnvectorof
length N;diag(a)denotes thediagonal matrixobtained fromthevector a; 1Suchtermunderlines thatitisorthogonal tothesignalsubspace.
x1:M , (cid:2)xT1 ··· xTM(cid:3)T denotes the vector concatenation; NC(µ,Σ) 2WeunderlinethattheMUSICimagingfunctioniscommonlyreferredtoas
denotes aproper complex Gaussian pdfwith meanvector µ andcovariance “pseudo-spectrum”inDOAliterature.Thoughlessused,inthispaperwewill
Σ;Cχ2 denotesacomplexchi-squaredistributionwithN(complex)Degrees insteadadopttotheterm“null-spectrum”employedin[23],asthelatterwork
N
ofFreedom(DOFs);finallythesymbol∼means“distributed as”. represents theclosestcounterpart inDOAestimation tothepresent study.
2 IEEESIGNALPROCESSINGLETTERS,VOL.*,NO.*,MONTHYYYY
We recall that stability property is important for TR-MUSIC, where U CNR×NRdof is the matrix of left singular vectors
n
and has been investigated by numerical means [26], [27] of K span∈ning the noise subspace, g¯ (x),g (x)/ g (x)
n r r r
or using compressed-sensing based approaches [28]. Finally, is the eunit-norm Rx Green vector function and Pk ,k
r,n
a few numerical examples, for a 2-D geometry with scalar (U U†) (i.e. the “noisy” projector into the left noise sub-
n n
scattering, are presented to confirm our findings. space). A dual approach, denoted as Tx mode TR-MeUSIC,
The letter is organized as follows: Sec. II describes the coensteructs the null spectrum (assuming M <NT):
systemmodelandreviewsclassicresultsonSVDperturbation
2
analysis. Sec. III presents the theoretical characterization of (x;V ),g¯ (x)T P g¯ (x)∗ = V†g¯∗(x) , (4)
Pt n t t,n t n t
TR-MUSIC null-spectrum, whereas its validation is shown in
(cid:13) (cid:13)
Sec. IV via simulations. Finally, conclusions are in Sec. V. whereVn e CNT×NTdof iesthematrixof(cid:13)(cid:13)rieghtsingu(cid:13)(cid:13)larvectors
of K spa∈nning the noise subspace, g¯ (x),g (x)/ g (x)
n t t t
II. SYSTEMMODEL istheuneit-normTxGreenvectorfunctionandPt,n ,(kVnVn†k)
We consider localization of M point-like scatterers3 at (i.e. the “noisy” projector into the right noise subspace).
unknown positions x M in Rp with unknown scattering Finally,a combinedversionoftwo modes,nameedgeneeraliezed
potentials τ M {inkC}k.=T1he Tx (resp. Rx) array consists TR-MUSIC, is built as (assuming M <min NT,NR ) [8]:
{ k}k=1 { }
of NT (resp. NR) isotropic point elements (resp. receivers) (x;U ,V ), (x;V )+ (x;U ). (5)
located at r˜ NT in Rp (resp. r¯ NR in Rp). The il- Ptr n n Pt n Pr n
{ i}i=1 { j}j=1
luminators first send signals to the probed scenario (in a Usually, the M largest local maxima of (x;U )−1,
e e e ePr n
known homogeneous background with wavenumber κ) and (x;V )−1 and (x;U ,V )−1 are chosen as the esti-
t n tr n n
P P
the transducerarray recordsthe received signals. The (single- mates xˆ M . Indeed, it can be shown that Eq. (3)e(resp.
{ k}k=1
frequency) measurement model is then [30]: Eq. (4)e) equals zero whenex eequals one among {xk}Mk=1 in
the noise-free case, since when U = U (resp. V = V )
K = K(x ,τ)+W (1) n n n n
n 1:M this reduces to the eigenvector matrix spanning the left (resp.
= G (x )M(x ,τ)G (x )T +W (2)
r 1:M 1:M t 1:M right) noise subspace of K(x1:M,eτ) [7]. Similar coenclusions
swuhreedre(Kresnp∈. nCoNisRe-×fNreTe)(rMesDp.MK. (Dxif1f:Mere,nτt)ly) dWenotesCthNeRm×NeaT- hold for Ptr(x;Un,Vn) in a noise-free condition.
Nis ,a NnoisNe m. aAtrdidxitiso.tn.alvlye,c(wWe h)av∼e dNenCo(t0edN:,(σi)w2t∈IhNe v),ecwtohreoref B. Review of Reesultseon SVD Perturbation
scatterinTg cRoefficients as τ , τ1 ··· τM T ∈ CM×1; δ <Wme icnonRsi,dTer a, wrahnoksedeSfiVcDienAt m=atUrixΣAV∈† iCsRre×wTriwtteitnharsa:nk
(ii) (b) the Tx (resp. Rx) array matrix as Gt(x1:M) { }
CNT×M (resp. G (x ) C(cid:2)NR×M), whose(cid:3)(i,j)th entr∈y Σ 0 V†
equals G(r˜i,xj) (rresp1.:MG(r¯∈i,xj)), where G(·,·) denotes the A= Us Un (cid:18) 0δ¯×sδ 0δδ¯××δδˇˇ (cid:19)(cid:18) Vsn† (cid:19) , (6)
(scalar) background Green function [7]. Also, jth column (cid:0) (cid:1)
g (x ) (resp. g (x )) of G (x ) (resp. G (x )) denotes where δ¯ , (R δ) and δˇ , (T δ), respectively. Also,
t j r j t 1:M r 1:M U CR×δ a−nd V CT×δ (r−esp. U CR×δ¯ and
the Tx (resp. Rx) Green’s function vector evaluated at x . s s n
In Eq. (2) the scattering matrix M(x ,τ) CM×Mj Vn ∈CT×δˇ)denotethele∈ftandrightsingularve∈ctorsofsignal
equals M(x ,τ) , diag(τ) for BA1:Mmodel∈[7], while (res∈p. orthogonal) subspaces in Eq. (6), while Σs Rδ×δ
1:M ∈
M(x ,τ) , diag−1(τ) S(x ) −1 in the case of collects the (> 0) singular values of the signal subspace.
1:M − 1:M Then,considerA=(A+N), whereN is a perturbingterm.
FL model [22], where the (m,n)th entry of S(x ) equals
(cid:2) (cid:3) 1:M Similarly to (6), the SVD A=UΣV† is rewritten as
(x ,x ) when m = n and zero otherwise. We recall that
m n
G 6 e
our null-spectrum analysis of TR-MUSIC is general and can Σ 0 V†
be applied to both scattering models. A= Us Un e0 see eΣδ×δˇ Vs† , (7)
NanTdFd,ionffaol,rly,n(woNteaTtdioefinanMle)tchoaensSvetNhnReiedn,icmek,eKnNs(iRxodn1o:sMf o,,fτt)h(kNe2FRl/ef(−tσw2aMnNd)TriNagnhRdt) sAh,owhiingeghlitghh(cid:16)etienefgfecthteeofchN(cid:17)an goenoeδ¯tf×heδthsepeelcentfrtala!nred prreieegsnehntt!aptriionnc4ipoafl
orthogonal subs−paces, whereas Ndof ,(NRdof +NTdof). directions. We are here concerned with the perturbations
peertaining to Un and Vn, stressed as Un = Un+∆Un and
A. TR-MUSIC Spatial Spectrum V =V +∆V ,where∆()termsaregenerallycomplicated
n n n
·
Several TR-MUSIC variants have been proposed in the functions of Ne. Howeever, when N haes a “small magnitude”
literature for the non co-located setup [8]. A first approach ceompared to A (see [31]), a 1st-order perturbation (i.e. ∆()
·
are approximated as linear with N), will be accurate [24].
consists in using the so-called Rx mode TR-MUSIC, which
The key result is that perturbed orthogonal left subspace U
evaluatesthe null(or spatial) spectrum (assuming M <N ): n
R
(resp. right subspace V ) is spanned by U + U B (resp.
n n s
Pr(x;Un),g¯r(x)†Pr,ng¯r(x)= Un†g¯r(x) 2 , (3) ℓVno+r VsB¯)n,owrmh)eroefnBor(emres(apn.yB¯s)uibs-omfutlhteipslaicmateivoerdoenreo,fsuthcahteoasf
(cid:13) (cid:13) 2 k·kF
3Thenumbereofscatterers M iseassumedtobek(cid:13)(cid:13)noewn,asusu(cid:13)(cid:13)allydonein
array-processing literature [29]. 4Indeed, asopposedtoEq.(6),Amaybefull-rank ingeneral.
e
CIUONZOETSALVOROSSI:NON-COLOCATEDTR-MUSIC:HIGH-SNRDISTRIBUTIONOFNULLSPECTRUM 3
N. Intuitively, a small perturbation is observed at high-SNR. E ξT ξT T = 0 , exploiting E W = 0 .
The∆exUpnre=ssion(As f−o)r†∆NU†nUann;d ∆∆VVnn, =at 1s(tA-o−rd)eNr, aVren5;[32(]8:) SEe{{c(cid:2)ξoknr},dk=ly,0tNth,kdeo(cid:3)f)c}oisvagriivaenNncdeionf mclaotsreixd-fΞorkm a,s{: E}{ξkξk†}NR(×siNncTe
− − σ2 t 2 I 0
where we have exploited A− =VsΣ−s1Us† [33]. Ξk = w0k r,kk NRdof σ2 NtRdof×2NITdof . (12)
(cid:20) NTdof×NRdof w k t,kk NTdof(cid:21)
The above result is based on circularity of the entries of W,
III. NULL-SPECTRUMANALYSIS
alongwiththeirmutualindependence.Thirdly,aimingatcom-
First, we observe that the null spectrums at scatterer po-
pletingthestatisticalcharacterization,weevaluatethepseudo-
sU{i1ti,o.=n.s.,UPMr(}+x,ki∆n;UeUEnq)s,.an(P3dt)(,Vx(k4;)=VeannVd) (a+5n)d∆caPVntrb(xeanksd;imUeepxnlp,ifiVleoenidt)i,,nugksitnh∈ge cwohvoasreiacnlcoesemd-aftorrixmΨiskΨ,k E={ξ0kNξdkTof}×N(sdionfc.eTEhe{ξlkat}te=r re0sNudltofi)s,
n n n n n n basedoncircularityofthe entriesofW, alongwith theirmu-
properties6 Un†g¯r(xk)=0NRdof and Vn†g¯t∗(xk)=0NTdof, as tualindependenceandexploitingthe resultsV†t =0
e r(xk;Un)= ξr,ke 2 , t(xk;Vn)= ξt,k 2, (9) and Un†tt,k = 0NRdof, arising from subspacens orr,kthogonNaTlditoyf
P k k P k k V†V =0 and U†U =0 .
∆whVenr†eg¯t∗ξ(rx,kk),e∈C∆NTUdon†f×g¯1r,(xreks)pec∈tiveClyN. RSedimof×il1arlya,nd ξt,k , prnoTpheserrecfoomrNep,Tldeoixnf×GMsuamusmsiaanrynveξckstor∼[3N4RN]d.oCfS×(i0mMNildaorfl,yΞ, ikt),isi.ree.ada-
wherPetrξ(xk,;Unξ,VTn)=ξTkξtT,kk2+CkNξdro,kf×k12.=Tkhξuks,k2to, cha(r1a0c-) iia.lneyd. tihnξefte,ykrraerd∼e tihnNadteCp(ξe0rnN,dkTedno∼tf,pσrow2NpekCrt(t0,GkNkaR2udsoIfs,iNaσTndw2ovfk)e,tcrt,orkerkss2.peCIcNlteiRvaderollfyy),,
k e re,k t,k ∈
fitecreizsetoPrs(txukd;yU(cid:2)tnh)e, rPatn(dxokm(cid:3);Vvne)ctaonrdξP.trI(nxdke;eUd,n,thVen)m,airtgsinuaf-l scionvcaeriaξnrc,ke, athned cξotr,rkesphoanvdeinzgervoarmiaenacne-naonrdmaslcizaeledd-eindeerngtiietys
k
pdfs of ξr,k eand ξt,k aere easily drawnefroem that of kξr,kk2/(σw2 ktr,kk2)∼Cχ2NRdof andkξt,kk2/(σw2 ktt,kk2)∼
ξ . As a byproduct, ξ definition also allows an ele- χ2 , respectively (i.e. they are chi-square distributed).
k k C NTdof
gant and simpler MSE analysis with respect to [21], as InterestinglytheseDOFscoincidewiththoseavailableforTR-
it can be shown that the position-error of the estimates MUSIC localization through Rx and Tx modes, respectively.
with Tx mode (∆x ), Rx mode (∆x ) and general- Based on these considerations, the means of the null-
T,k R,k
a−izneΓddT−,∆1(k∆ℜxx{JTTRT,,kk)VnTξRt-,MkΓ}U−, S1∆ICxRca,[kn(Jb≈e† e−xUΓpr−Re),s1ksℜe(dJ{JTaRs†,k∆VUx)nTξ],kξr,k≈}, sσsppw2eeccktttirvur,emklky,2wfNohrRerdeToafxsafnoadrngdEen{ekRrξxatl,ikzkme2do}dneu=sll-σsapw2reeckttrtEu,mk{kkE2ξrN,kTξkkd2o}f2, r==e-
respectiveTlRy,,kwh≈ere−J TR,,kJℜ{ , ΓR,k, Γn anTd,kΓn kar}e E ξr,k 2 + E ξt,k 2 (by linearity). By simi{lakr rekas}on-
T,k R,k T,k R,k TR,k {k k } {k k }
ing, the variances for Tx and Rx modes are given by
suitably defined known matrices (see [21]). Clearly, finding
cthoemHeopxwliacecavtteeprdd,f∆fuoUnfcnξtikoannissdho∆afrVtdh,neaaussns∆kunmUoewnana(nptdrear∆ctutarVbbnlien)agrcemlogasteerndixefroWarlmly. nσvauw4rll{k-kstpξt,erk,ckktkr4u2Nm}Tvd=aorf{,kσrξew4kskpk2et}crt,=ikvkev4lyaN,r{wRkdhξoerf,rkekaa2sn}df+orvvaatrhr{{ekkξgξtet,,nkkekkr22a}}liz(be=yd
with a 1st-order approximation (see Eq. (8)). This approxi- independence of ξr,k and ξt,k).
Hence, once we have obtained the mean and the variance
mation holds tightly at high-SNR, as W will be statistically
“small”comparedtonoise-freeMDMK(x1:M,τ). Hence,at ofPr(xk;Un),Pt(xk;Vn)andPtr(xk;Un,Vn),respectively,
we can consider the Normalized Standard Deviation (NSD),
high-SNR,ξ is(approximately)expressedintermsofW as:
k
genericallyedefined as e e e
ξ U†W t
ξk =(cid:20)ξtr,,kk(cid:21)≈(cid:20)−−Vn†nW†trt,,kk(cid:21) , (11) NSDk , var{P(xk; ·)}/E{P(xk; ·)}. (13)
where t , K−(x ,τ)g¯ (x ) CNT×1 and t , Clearly, the lowerpthe NSD, the higher the null-spectrum
K−(x r,k,τ)†g¯*(x )1:M CNRr×1k ar∈e deterministic. tS,kince stability at xk [23]. For Rx and Tx modes it follows that
the vec1t:Mor ξ istlinkear7∈with the noise matrix W, it will NSDr,k =1/√NRdof andNSDt,k =1/√NTdof,respectively.
k It is apparent that in both cases the NSD does not depend (at
be Gaussian distributed; thus we only need to evaluate its
highSNR)onthescatterersandmeasurementsetup,aswellas
moments up to the 2nd order to characterize it completely.
σ2,butonlyonthe(complex)DOFs,beingequaltoN and
Hereinafterweonlysketchthemainstepsandprovidethede- w Rdof
N , respectively.Thus, the NSD becomes(asymptotically)
tailed proofas supplementarymaterial. First, the mean vector Tdof
small only when the number of scatterers is few compared to
5We notice that in obtaining Eq. (8), “in-space” perturbations (e.g. the the Tx (resp. Rx) elements of the array. Those results are
contribution to ∆Un depending on Un) are not considered, though they analogous to the case of MUSIC null-spectrum for DOA,
havebeenshowntobelinearwithN (andthusnotnegligible atfirst-order) whose NSD depends on the DOFs, namely the difference
[32]. The reason is that these terms do not affect performance analysis of
TR-MUSIC null-spectrum when evaluated at scatterers positions {xk}Mk=1, between the (Rx) array size and the number of sources [23].
duetothenullspectrum orthogonality property. Differently, the NSD for generalized null spectrum equals
6Such conditions directly follow from orthogonality between left (resp.
right)signalandorthogonal subspaces Us andUn (resp.Vs andVn). tr,k 4NRdof + tt,k 4NTdof
7Inthefollowingoftheletterwewillimplicitlymeanthattheresultshold NSD = k k k k . (14)
“approximately” inthehigh-SNRregime. k qt 2N + t 2N
r,k Rdof t,k Tdof
k k k k
4 IEEESIGNALPROCESSINGLETTERS,VOL.*,NO.*,MONTHYYYY
0 0.34
−1 0.32 Generalized − Scatt. 1 − BA
−2 Generalized − Scatt. 2 − BA
0.3 Generalized − Scatt. 1 − FL
−3 Generalized − Scatt. 2 − FL
λY / −4 SD0.28 RTxx mmooddee
−5 N0.26
Targets
−6
Tx array 0.24
−7 Rx array
−8 0.22
−8 −6 −4 −2 0 2 4 6 8
X / λ
0.2
−10 −8 −6 −4 −2 0 2 4 6 8 10
d / λ
Figure1. Geometryfortheconsidered imagingproblem in2-Dspace.
Figure3. TheoreticalNSDvs.scatterersrigidshiftd;twotargetslocatedat
0.32 (x1/λ)=(cid:2) (−1−d) −6 (cid:3)T and(x2/λ)=(cid:2) (1−d) −6 (cid:3)T.
Scatt. 1 − FL
Scatt. 2 − FL
0.3
Scatt. 1 − BA
Scatt. 2 − BA
0.28 T
consider M = 2 targets located at (x /λ) = 1 6
1
NSD0.26 and (x2/λ) = +1 −6 T and having sca(cid:2)tte−ring−coef(cid:3)fi-
T
0.24 cients τ = 3 4 ; thus η =(0.7445).
(cid:2) (cid:3)
Then, we compare the asymptotic NSD (Eq. (14), solid
0.22 (cid:2) (cid:3)
lines) with the true ones obtained via Monte Carlo (MC)
0.2 0 5 10 15 20 simulation (dashed lines, 105 runs), focusing only on the
SNR [dB]
generalized null-spectrum for brevity. To this end, Fig. 2
Figure2. NSD(generalized nullspectrum)vs.SNR;theoretical (Eq.(14), depicts the null-spectrum NSD vs. SNR for the two targets
solidlines)vs.MC-based(dashedlines)performance. being considered, both for FL and BA models. It is apparent
that, as the SNR increases, the theoretical results tightly
approximatetheMC-basedones,withapproximationsdeemed
Eq.(14)underlines(i)acleardependenceofgeneralizednull-
accurate above SNR 10dB. Differently, in Fig. 3, we
spectrum NSD on scatterers and measurement setup and (ii) ≈
plot the asymptotic NSD of the three TR-MUSIC variants
independencefromthenoiselevelσw2.Also,itisapparentthat vs. d, where (x /λ) = ( 1 d) 6 T and (x /λ) =
when ktr,kk ≈ 0 (resp. ktt,kk ≈ 0) the expression reduces (1 d) 6 1T (i.e. a r−igid−shift o−f the two sca2tterers),
to NSDk 1/√NTdof (resp. NSDk 1/√NRdof), i.e. the − − (cid:2) (cid:3)
≈ ≈ in order to investigate the potentially improved asymptotic
NSDisdominatedbyTx(resp.Rx)modestability.Finally,the
(cid:2) (cid:3)
stability(viz.NSD)ofthegeneralizedspectrumincomparison
same equation is exploited to obtain the conditions ensuring
to Tx and Rx modes. It is apparentthat the gain is significant
that generalized spectrum is “more stable” than Tx and Rx
when d ( 5,5), while outside this interval the NSD
modes(NSD NSD andNSD NSD ,respectively),
k t,k k r,k ∈ −
≤ ≤ expression is either dominated by Tx or Rx mode, which
expressed as the pair of inequalities
for the present case NSD = 1/√11 2 0.33 and
t,k
12[1−NRdof/NTdof]≤(ktt,kk/ktr,kk)2 (Tx) (15) NSDr,k = 1/√17−2 ≈ 0.26, with the−gene≈ralized NSD
(21[1−NTdof/NRdof]≤(ktr,kk/ktt,kk)2 (Rx) never above that of NSDt,k (as dictated from Eq. (15)).
Clearly, when N > N (resp. N > N ) the inequality
R T T R
regardingtheTx(resp.Rx)modeisalwaysverifiedastheleft-
V. CONCLUSIONS
hand side is always negative. Also, in the special case N =
T
NR the left-hand side is always zero for both inequalities. We provided an asymptotic (high-SNR) analysis of TR-
MUSIC null-spectrum in a non-colocated multistatic setup,
IV. NUMERICAL RESULTS by taking advantage of the 1st-order perturbation of the SVD
of the MDM. Three different variants of TR-MUSIC were
Inthissectionweconfirmourfindingsthroughsimulations,
focusing on 2-D localization, with Green function8 being analyzed (i.e. Tx mode, Rx mode and generalized), based
(x′,x) = H(1)(κ x′ x ). Here H(1)() and κ = 2π/λ on the characterization of a certain complex-valued Gaussian
G 0 k − k n · vector.ThisallowedtoobtaintheasymptoticNSD(ameasure
denote the nth order Hankel function of the 1st kind and
ofnull-spectrumstability)forallthethreeimagingprocedures.
the wavenumber (λ is the wavelength), respectively. First, we
While similar results as the DOA setup were obtained for Tx
considerasetupwithλ/2-spacedTx/Rxarrays(N =11and
T
andRxmodes,itwasshownacleardependenceofgeneralized
N = 17, respectively, see Fig. 1). Secondly, to quantify the
R
levelofmultiplescattering(asin[8])wedefinetheindexη , null-spectrum NSD on the scatterer and measurement setup.
Finally, its potential stability advantage was investigated in
K (x ,τ) K (x ,τ) / K (x ,τ) , where
k f 1:M − b 1:M kF k b 1:M kF comparison to Tx and Rx modes. Future works will analyze
K (x ,τ) and K (x ,τ) denote the MDMs generated
b 1:M f 1:M
mutualcoupling,antennapatternandpolarizationeffects[35],
viaBAandFLmodels,respectively.Finally,forsimplicitywe
[36],andpropagationininhomogeneous(random)media[37].
8Wediscardtheirrelevant constant termj/4.
CIUONZOETSALVOROSSI:NON-COLOCATEDTR-MUSIC:HIGH-SNRDISTRIBUTIONOFNULLSPECTRUM 5
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