Sequential joint signal detection and signal-to-noise ratio estimation

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by  M. Fauß

SEQUENTIALJOINTSIGNALDETECTIONANDSIGNAL-TO-NOISERATIOESTIMATION
M.Fauß∗,K.G.Nagananda†,A.M.Zoubir∗,andH.V.Poor‡
∗ SignalProcessingGroup,DarmstadtUniversityofTechnology,D-64283Darmstadt,Germany
† Dept. ofElectronicsandCommunicationsEngineering,PESUniversity,Bangalore560085,India
‡ Dept. ofElectricalEngineering,PrincetonUniversity,Princeton,NJ08544,USA
ABSTRACT signal-to-noiseratio(SNR),specificallysignalandnoisepowers,un-
7 derthe“signalpresent”hypothesis.Forexample,inspeechprocess-
1 Thesequentialanalysisoftheproblemofjointsignaldetectionand ing,itwasshownin[17],[18]thattheperformanceofvoicedetec-
0 signal-to-noiseratio(SNR)estimationforalinearGaussianobser- tion systems can be drastically improved by jointly estimating the
2 vationmodelisconsidered.Theproblemisposedasanoptimization noisepowerandaprioriSNR.In[19],itwasshownthataschedul-
n setupwherethegoalistominimizethenumberofsamplesrequired ing scheme performed detection in an energy-efficient manner by
a toachievethedesired(i)typeIandtypeIIerrorprobabilitiesand(ii) jointly estimating the SNR. However, [20] reported that the tech-
J meansquarederrorperformance. Thisoptimizationproblemisre- niquesdevelopedinsomeofthepapersmentionedabovewerenot
9 ducedtoamoretractableformulationbytransformingtheobserved readily applicable to the problem of joint detection and signal and
1 signalandnoisesequencestoasinglesequenceofBernoullirandom noisepowerestimation. ForaBayesianformulation,itwasshown
] vBaerrinaobulelsli;sjeoqinutendceet.ecTtihoinstarnandsfeosrtimmaattiioonnreisndtheersnthpeerpforormbleedmoenastihlye [o2f0t,heSedcis.trIiIbIu]titohnatokfntohweluendkgneoowfnthpearpamrioertserPsrw(Has0n)oatnadmPern(aHbl1e)foorr
T solvable, and results in a computationally simpler sufficient statis- problemsaddressedin[17]-[19]. Instead,anoptimalsolutionfor
I tic compared to the one based on the (untransformed) observation Gaussianobservationmodelswaspresentedusingconjugatepriors
.
s sequences. Experimentalresultsdemonstratetheadvantagesofthe onthesignalandnoisepowers[20,Sec.IV].
c proposed method, making it feasible for applications having strict
In this paper, we extend the problem of joint signal detection
[ constraintsondatastorageandcomputation.
and SNR estimation, without a priori knowledge of the signal or

1 Index Terms— Sequential analysis, Bernoulli transformation, noise powers, to a sequential setting and propose a novel method
v jointdetectionandestimation. toaddressthisproblem. Tothebestofourknowledge,thesequen-
7 tialanalysisofthisproblemhasnotbeenreportedintheliterature.
9 1. INTRODUCTION Theproblemofdistinguishingbetweentwohypotheses(signalab-
2 sentandsignalpresent)andatthesametimeestimatingtheSNRina
5 The joint problem of distinguishing between different hypotheses
Gaussianobservationmodelisposedasanoptimizationsetup,where
0 and estimating the unknown parameters based on the outcome of
weseektominimizethenumberofsamplesrequiredtoachievethe
. thehypothesestesthasreceivedconsiderableattentionintheliter-
1 desired(i)typeIandtypeIIerrorprobabilitiesand(ii)meansquared
ature [1] - [6]. Such a problem arises in a wide range of applica-
0 error (MSE) performance. Our approach comprises transforming
tions,including(i)radiographicinspectionfordetectinganomalies
7 the observed signal and noise sequences to a single sequence of
inmanufacturedobjectsand estimating theirpositionandsize[7],
1 Bernoullirandomvariables,andthenperformingthedetection-and-
(ii)retrospectivechangepointhypothesestestingtodetectchangein
: estimationtaskontheresultingBernoullisequence.
v thestatisticsandsimultaneouslyestimatethetimeofchange[8],[9],
i (iii)jointlydetectingthepresenceofmultipleobjectsandestimating Oneofthemainadvantagesofthistransformationisthatitsig-
X
nificantlyreducesthecomplexityoftheoptimizationproblemsothat
theirstatesusingimageobservations[10],and(iv)distinguishingbe-
r tweentwohypothesesandatthesametimeestimatingtheunknown it can be solved more efficiently. Secondly, we obtain a computa-
a
tionallysimplersufficientstatisticcomparedtotheonethatemerges
parameters in the accepted hypothesis in a distributed framework
whensolvingtheproblemdirectly.Moreover,weshowthatthepro-
[11]. Somepopulartechniquestoaddressthisproblemincludere-
posedmethodallowsformoredegreesoffreedomthananequiva-
formulating the composite detection problem as a pure estimation
lent Bayesian solution. Experimental results show that (i) the ex-
problem[12],whilethemaximumaposterioriestimatewasshown
pectednumberofmeasurementsrequiredtoachievethedesiredper-
toprovideasolutiontothejointdetectionandestimationproblemin
formancealmostremainsconstantforincreasingvaluesofSNR,and
aBayesiancontext[13]. Theproblemhasalsobeenaddressedina
(ii)manyoftheconstraintsinthetransformedoptimizationproblem
sequentialsetting,wheretheobjectiveistominimizethenumberof
areinactivewhichrenderstheproblemeasilysolvable.Assuch,the
samplessubjecttoaconstraintonthecombineddetectionandesti-
methoddevelopedinthispaperisfeasibleespeciallyforapplications
mationcost[14],[15]. Thegeneralizedsequentialprobabilityratio
withstrictconstraintsondatastorageandcomputation.
testwaspresentedin[16],whereadecisionwasobtainedusingthe
maximumlikelihoodestimateoftheunknownparameter. InSection2,wepresenttheproblemstatement.InSection3,we
Thereisanotherclassofproblemswhereitisdesirabletodis- detailthetransformationoftheobservationstoaBernoullisequence,
tinguish between the hypotheses and simultaneously estimate the andshowhowtheoriginaloptimizationproblemcanbereformulated
into a setup which can be solved efficiently. Results of computer
ThisresearchwassupportedinpartbytheU.S.NationalScienceFoun- simulationsarepresentedinSection4.Concludingremarksarepro-
dationunderGrantsCNS-1456793andECCS-1343210. videdinSection5.
2. PROBLEMFORMULATION of the samples from both sequences and take an additional sam-
ThefollowinglinearGaussiansignalmodelisconsidered: ple from the one whose sum is smaller. Whenever the additional
samplechangestheorderofthetwosums,theprocedureoutputsa
x[n]=s[n]+w[n], n=1,2,..., (1) 0 or 1 depending on which sequence the sample was drawn from.
wszeh(cid:44)roer{emsxe[1a]n(cid:44),sG[{2ax]u,[s1.s].i,a.xn}[2ara]n,nd.d.wo.m}(cid:44)dve{anrwioa[tb1el]se,swthc[e2os]r,er.et.sop.fo}ondbdesinneorgvteatotsieotthnseso,sfwiig.hin.idalel. wEmt˜ossiswn[˜ekns{],txi(cid:44)aosl[rlky(cid:80)v],,icgwkn˜ei=vsv[e1knew]r˜}sx[a.n,Ia]n2on,ctdhcaeuwn˜rds,eayqwtuteeh(cid:44)nedceke{fiiytyhn[e,1sla]xe,mtystp[[hk2lee]],.i.(cid:44)th.T.thr}(cid:80)ea,nnknwst=ihhti1eeorxnBe[fneyrr]o[n2kmo]auxnl=ldsi
uanndknnoowisne,,arnedsptehcetivSeNlyR. iTshgeivveanribanycθes=σs2σ≥s2.0Tahnedpσrow2bl>em0isatroe sequenceisb(cid:44){b[1],b[2],...},where
distinguishbetweenthetwohypotheses σw2 b[m]=(cid:40)0, ifw˜s[km]<xs[km], (7)
1, otherwise.
(cid:40)
H :θ=0 (signalabsent),
0 (2)
H1 :θ θmin (signalpresent), In [21], it was proved that irrespective of the actual values of σs2
≥ andσ2,theoutputbisasequenceofi.i.d. Bernoullirandomvari-
w
and at the same time estimate the SNR under the hypothesis H ableswithsuccessprobabilityρ= 1 ,therebyestablishingaone-
1 θ+2
usingasfewsamplesaspossible, whilesatisfyingpredefinedcon- to-onecorrespondencebetweenθ andρ. Therefore, (2)canbere-
straints on the type I and type II error probabilities. In (2), θ hypothesizedintermsofρ,i.e.,
min
denotestheminimumSNRforwhichreliabledetectionistobeguar- (cid:40)
anteed. Weattempttojointlysolvetheproblemofsignaldetection H0 :ρ=0.5 (signalabsent), (8)
andSNRestimationinasequentialsetting. Whilethelatterenables H :ρ ρ (signalpresent),
1 max
≤
onetoadaptthenumberofsamplestothequalityofrealizations,the
formerensuresthatthedualobjectiveofdetectionandestimationis whereρ = 1/(θ +2). Theoptimizationsetup(3)-(6)can
max min
achievedwithadesiredperformance. Essentially,theproblemcan bereformulatedas
beformulatedasthefollowingoptimizationsetup:
min Eρ∗[M] (9)
min Eθ∗[N] (3) ψ,δ,θ˜
ψ,δ,θˆ subjectto
subjectto
P0(δM =1) α, (10)
≤
P0(δN =1)≤α, (4) Pρ(δM =0) β(ρ), ρ ρmax, (11)
≤ ∀ ≤
PEθθ((cid:104)δ(NθˆN=0)θ)≤2(cid:105)β(θγ),(θ)∀,θ≥θθminθ,min, ((56)) Eρ(cid:104)(θ˜M − ρ1)2(cid:105)≤γ(ρ), ∀ρ≤ρmax, (12)
− ≤ ∀ ≥
where θ˜ = θˆ 2 is an estimator for θ that is biased in order to
whereN denotesthesamplenumberatwhichthesequentialtestis simplifytheex−pressionundertheexpectedvalue. P0()andPρ()
terminated, ψn,δn ∈ {0,1}denotethestoppinganddecisionrule denote the probabilities of an event under hypothesis H· 0 and H1·,
after the nth sample has been observed, and θˆn is an estimator for respectively, corresponding to (8). M denotes the sample number
θ.Theconstantθ∗isanominalSNRvalueunderwhichtheaverage of b at which the procedure is terminated. Since it takes multiple
samplenumber(ASN)istobeminimum.P0()andPθ()denotethe observationsofxandw˜ togenerateoneobservationofb,N isin
· ·
probabilitiesofaneventunderhypothesisH0andH1,respectively. generallargerthanM. Moreover,N includesobservationsofw˜ as
ThetypeIandtypeIIerrorprobabilitiesareboundedbyαandβ, wellasx. However,byformulatingtheproblemintermsofband
respectively,withβ beingallowedtodependonthetrueSNR.The ρ, weaimatminimizingtherequiredsamplesofb, irrespectiveof
meansquareerror(MSE)oftheestimatorθˆisboundedbyafunction thenumberofobservationsofxandw˜ areusedtogeneratethese
γ(). samples. Duetothisdifferenceintheobjective,bothproblemfor-
·
Weassumeknowledgeofasequenceofnoise-onlyrealizations mulationsarenotstrictlyequivalentsothattheproposedprocedure
w˜ (cid:44) [w˜ ,w˜ ,...], that can either be recorded before performing cannotbeguaranteedtobestrictlyoptimal. Amoredetailedanal-
1 2
thetest,orcanbegeneratedonthefly,forexample,viaanidentical ysisofthelossincurredbyapplyingBirnbaum’stransformationis
sensor that is shielded from the external signal, but otherwise ex- asubjectoffutureresearch. Alsonotethat, dependingonthecost
posedtothesameenvironmentalconditions.Withoutw˜,thetesting involvedinsamplingfromxandw˜,onecanmodifythetransforma-
problemcannotbesolvedforthesetupconsideredinthispaper. tiontorequiremoreorfewersamplesfromacertainsequence.This
additionalpotentialforoptimizationisnottakenintoaccountinthis
3. SOLUTIONMETHODOLOGY work.Formoredetailsonthetransformationanditsnear-optimality
Our approach comprises the following two steps: (i) the two se- propertiessee[21].
quencesxandw˜ aretransformedtoasinglesequenceofBernoulli
3.2. Jointdetectionandestimation
random variables, whose success probability is determined by the
Inordertosolve(9),wefirstcalculateitsLagrangiandual.Forfixed
trueSNR,and(ii)asequentialjointdetectionandestimationproce-
Lagrangemultipliers,itresultsinanunconstrainedoptimalstopping
dureisappliedtothisBernoullisequence.
problemthatcanbesolvedbymeansofdynamicprogramming.We
3.1. TransformationtoaBernoullisequence thenchoosetheLagrangemultiplierssuchthattheproceduresatis-
ThetwoGaussiansequencesxandw˜ aretransformedintoasingle fiestheconstraintsontheerrorprobabilitiesandthedesiredestima-
BernoullisequencebusingBirnbaum’ssequentialprocedure[21]as tionaccuracy.Foranalyticaltractability,werelaxtheconstraintsun-
follows: At every time step, we calculate the sum of the squares derH1toholdforallρ P,whereP(cid:44) ρ1,...,ρK isadiscrete
∈ { }
subsetof[0,ρmax].Thatis,weboundPρk(d=0)andEρk[(θ˜−ρ1)2] mintermin(20)signifiesthedetectioncostifthedecisionrule(17)
onlyatafinitenumberofgridpoints;β(θ)andγ(θ)will,therefore, isemployed,whilethelasttwotermscorrespondtotheestimation
beapproximatedforpointsin-between. Fortheproblemconsidered cost, i.e., the deviation of the estimator θ˜∗ from the true SNR. At
inthiswork,theseapproximationsareshowntobereasonablyaccu- firstglance,theoptimaldecisionruleaswellastheoptimalestima-
rate(seeexamplesinSection4). torseemequivalenttoBayesiansolutionsbecauseofthefollowing
TheLagrangiandualproblemof(9),with[0,ρmin]replacedby reason: ThetermEλm0,m1 canbeinterpretedastheposteriorprob-
PandLagrangemultipliersλ RK+1andµ RK,isgivenby abilityofH1 giventheobservations b[1],...,b[m] andtheprior
∈ ∈ λ(ρ)thathasbeenscaledbyacostc{oefficient. Sim}ilarly, i 0,
(cid:40) (cid:88)K (cid:41) thetermsEµm,i0,m1 canbeinterpretedastheconditionalmom∀en≥tsof
max L(λ,µ) λ0α (λkβ(ρk)+µkγ(ρk)) , (13) theposteriordistributionofρ(or, θ)withpriorµ, andtheoptimal
λ,µ≥0 − −k=1 estimatorθ˜∗astheposteriorexpectedvalueofθ 2.
−
However,theproposedschemeisnotequivalenttotheBayesian
(cid:40)
procedure.Notethat,λandµbothbehaveas“priors”forρ,butcan
L(λ,µ)= min Eρ∗[M]+λ0P0(δM =1) bechosenindependently.FortheproposedapproachtobeBayesian,
ψ,δ,θ˜
onewouldrequireλ = µ. Inthecaseofasingleconstraintunder
(cid:88)K (cid:16) (cid:104) (cid:105)(cid:17)(cid:41) eitherhypothesis,thecorrespondingLagrangemultipliercanalways
+ λkPρk(δM =0)+µkEρk (θ˜M − ρ1k)2 , (14) beinterpretedasapriordensityscaledbyacostcoefficient,sothat
k=1 theoptimalmethodnecessarilyhasaBayesianequivalent;compare
where ρk ∈ P. Following the techniques developed in [22], [23], tthweocLlaasgsriacnlgikeemlihuoltoipdl-ireartsioλteasntd[2µ4]c.orIrnesopuornadpptorotawcoh,dhifofwereevnetr,cothne-
(14)canbestraightforwardlysolvedasfollows,whereweomitthe
straintsunderthesamedistribution,sothatthereisnoequivalence
detailsintheinterestofspace: Letm andm denotethenumber
0 1 totheBayesiansetup.
of0’sand1’sobserved. Thelikelihood-ratiosofthecorresponding
Returningtothesolutionof(14),themainadvantageoftrans-
observationsunderP0andPρk,withrespecttoPρ∗ aregivenby formingGaussiansequencesintoBernoullisequencesisthatthepair
Z0m0,m1 = (cid:18)10.5ρ∗(cid:19)m0(cid:18)0ρ.∗5(cid:19)m1, (15) p(mari0s,omn,1d)ibreecctolmyessolavsinugffitchieenptrsotabtliesmticionft{hbe[1S],N.R..-d,ob[mma]i}n.rIenqcuoirmes-
− a more complicated sufficient statistic, which will include obser-
Zm0,m1 = (cid:18)1−ρk(cid:19)m0(cid:18)ρk(cid:19)m1. (16) vations from x and w˜ as well as their empirical variances. Given
k 1 ρ∗ ρ∗ a maximum sample number m¯, the matrices R,G Rm¯×m¯ can
− ∈
W(cid:80)eKk=de1fiρn−kei,µ∀kiZ≥km00,m,E1.λmT0h,me1op(cid:44)tim(cid:80)alKkd=e1ciλsikoZnkmru0l,emδ1∗ainsdgiEveµmn,i0b,my1 (cid:44) bRmeme¯rc,iamc¯lac.lulylSaitsnetcadebvlteihaefobsratcamktewodvaeardrraiatrebellcyeusrlsaairrogeneinwvtaietlghueesrssta,ortfthinmisg0rpeoacniundrtsimGonm1¯.i,sm¯Tnhu=e-
δm∗0,m1 =(cid:40)10,, λλ0ZZ0mm00,,mm11 ≤>EEλmm00,,mm11,, (17) finstinonapglpeoinlfegtmhreeuntletessot,f(gtih.ieve.e,rnmecb0uyr=s(i1om7n)i1sa=nRd00,(0)1,w9w)h,heirncehsthpieescotthpivetiecmloya,sltadarteecthuisesieobdnegaainnndd-
0 0 λ λ and µ are given, i.e., L(λ,µ) = R0,0(λ,µ). Once we are able
andtheoptimalestimatorθ˜∗by toevaluateL(λ,µ),theLagrangemultipliescanbedeterminedby
solving(13)forλandµ. SincebyconstructionL(λ,µ)isjointly
Em0,m1 concaveinλandµ,thisisaconvexoptimizationproblemthatcan
θ˜∗ = µ,1 . (18) besolvedusingstandardalgorithms,asshowninthenextsection.
m0,m1 Em0,m1
µ,0
4. EXPERIMENTALRESULTS
Theoptimalstoppingruleψ∗isobtainedas
In this section, we present results of two numerical experiments.
(cid:40) whichprovideinterestinginsightsintothestructureofthejointde-
ψm∗0,m1 = 01,, GGmm00,,mm11 =>RRmm00,,mm11,, (19) t[ec3tiodnB,a1n0ddeBst]iwmiatthiognridprpoobilnetms.atTinhteegSeNrRvaliusecsh,oi.see.,nθikn=the10rak1n−0g4e,
where k−= 1,...,14. The nominal SNR value θ∗ is set at 3dB. The
target error probabilities are identical for both experiments, with
Gm0,m1 =min[λ0Z0m0,m1, Eλm0,m1] αrela=tivβe((θo)rn=or0m.0al5i.zeAd)sMaSmEeaissuurseedfoarntdheboeustnidmeadtiboynaacccounrsatcayn,t.thIne
+Em0,m1 (Eµm,10,m1)2 (20) ordertomatchtheproblemformulationinSection2,theconstraint
µ,2 − Em0,m1 canbeexpressedintermsoftheabsoluteMSEandanSNRdepen-
µ,0
dentboundγ(θ) = cθ2 or,intermsofρ,asγ(ρ) = c(1/ρ 2)2.
andRm0,m1 isdefinedrecursivelyasfollows Forthefirstexperiment,c=0.1,whileforthesecondc=0.−25.
For the numerical solution of (13), we employed the Subplex
Rm0,m1 =min[Gm0,m1 , algorithm[25]asimplementedin[26]. ItappliestheNelder–Meat
1+ρ∗Rm0,m1+1+(1−ρ∗)Rm0+1,m1]. (21) sdiimmpelnesxioanlgaolrsiuthbmspa[2ce7s]ainndardeopeesanteodtrfeaqshuiiorentohnescuailtcaubllayticohnoosefngrlaodwi--
ThequantitiesGm0,m1 andRm0,m1 correspondtothecostin- ents,whichiscomputationallyintensivefortherecursively-defined
curredforstoppingimmediately,orstoppingattheoptimaltimein- cost function (13). By limiting the search to subspaces, the algo-
stant, giventhatm 0’sandm 1’shavebeenobserved. Thepro- rithm can exploit the sparsity in the optimal Lagrange multipliers.
0 1
cedureisstoppedforthefirsttimewhenGm0,m1 = Rm0,m1. The Since there is no formal proof of convergence, the Subplex solu-
tionwassubsequentlyverifiedbyevaluatingitsfirst-orderoptimality ty 0.05 c=0.25
conditions. ThemaximumnumberofBernoullisampleswassetto bili c=0.1
m¯ =400,whichprovedtobesufficient. ba 0.04
o
r
P
0.03
r
c=0.25 o
150 c=0.1 Err 0.02
e I
u I
al e 0.01
V 100 yp
al T
m 0
Opti 50 −3 −2.5 −2 SNR−1in.5dB −1 −0.5 0
0 0.25
λ0 µ1 µ14
E 0.2
Lagrange Multipliers S
M
0.15
e
Fig. 1: Optimal Lagrange multipliers for α = β(θ) = 0.05 v
ti
and c 0.1,0.25 . The multipliers are plotted in the order a 0.1
∈ { } el
λ0,...,λ14,µ1,...,µ14. Thetwoclippedvaluesareµ1 =502.77 R
andµ3 =307.37. 0.05 c=0.25
c=0.1
0
InFig.1,theoptimalLagrangemultipliersaredepictedforboth 2 0 2 4 6 8 10
experiments. Itisinterestingtonotethatinbothcasesthesolution −
SNR in dB
is sparse, which implies that most of the performance constraints
areinactive. ConsideringtheverycoarseSNRgrid,thisoutcomeis
Fig.2: Detectionandestimationperformanceoftheproposedalgo-
ratherunexpectedandsuggeststhatinpracticeveryfewconstrains
rithmforα=β(θ)=0.05andc 0.1,0.25 .
canbesufficienttoboundtheperformanceoverlargeSNRintervals. ∈{ }
ThiscanalsobeseeninFig.2,wherethetypeIIerrorprobabilities
andtherelativeMSEareplottedovertherangeofSNRs.Theresults
wereobtainedbyaveragingover104 MonteCarlosimulationsand r From x, c=0.25
e
the SNR interval was sampled at intervals of size 0.1dB. Within mb 600 From w˜, c=0.25
the numerical accuracy, the performance requirements are met or Nu From x, c=0.1
exceededforallSNRvaluesinthefeasibleinterval. Especiallythe
e From w˜, c=0.1
typeIIerrorprobabilitiesarewellbelowtherequired5%forallSNR pl 400
m
values. For c = 0.1, the estimation constraint is so much tighter
a
than the detection constraint that the latter is virtually deactivated, S
e
i.e., the corresponding Lagrange multipliers are close to zero (see g 200
a
Fig.1). TheconstraintsontherelativeMSEarestrictersothatthe er
v
boundisreachedoverlargeregionsoftheSNRinterval.Considering A
thattheperformanceisrestrictedtointegervaluesoftheSNR,itis 0
2 0 2 4 6 8 10
remarkablethattherequirementsaresatisfiedovertheentireinterval. −
Theaveragenumberofsamplesdrawnfromtheobservationse- SNR in dB
quencexandthereferencesequencew˜ isshowninFig.3. Asex-
Fig.3: Theaveragenumberofsamplesdrawnfromtheobservation
pected,ASNsforbothcasesarehighforlowSNRsanddecreasefor
referencesequencesversusSNR.
higherSNRs. BothASNsreachaminimumataround3dB,which
correspondstothenominalSNRthatwastargetedintheminimiza-
tionprocedure.ForlargeSNRvalues,thenumberofsamplesdrawn
from the reference sequence increases again, while the number of
samples generated by the signal itself stays almost constant. This formationleadstoasimplerreformulationofthemainoptimization
isadesirableproperty,consideringthatgeneratingtrainingsamples problem, which can be efficiently solved. The expected minimum
isusuallyeasierthantakingphysicalmeasurements. Themodesat numberofsamplesrequiredtoachievethedesiredperformancere-
lowandhighSNRvaluesareduethedetectionandestimationcon- mainsalmostconstantforincreasingvaluesofSNR.Wealsoobtain
straints,respectively. asufficientstatisticforthetestwhichisveryeasytocompute. Ex-
perimentalresultsindicatethatmanyconstraintsontheoptimization
5. CONCLUDINGREMARKS setupareinactive,whichrenderstheproblemeasilysolvable.These
WehaveconsideredtheproblemofjointsignaldetectionandSNR results indicate the feasibility of the proposed method to practical
estimation for a linear Gaussian model in a sequential framework. applications.Understandingtheimplicationsofnon-stationarynoise
The central idea of our approach is to transform the observed se- processesontheperformanceoftheproposedapproachespeciallyin
quences to a sequence of Bernoulli random variables. This trans- thehigh-SNRregimeisoneofthemainavenuesforfutureresearch.
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