Sparsity based Efficient Cross-Correlation Techniques in Sensor Networks

2 MB·English

by  Prasant Misra

1
Sparsity based Efficient Cross-Correlation
Techniques in Sensor Networks
Prasant Misra, Senior Member, IEEE, Wen Hu, Senior Member, IEEE, Mingrui Yang, Member, IEEE,
Marco Duarte, Senior Member, IEEE, Sanjay Jha, Senior Member, IEEE
Abstract—Cross-correlationisapopularsignalprocessingtechniqueusedinnumerouslocationtrackingsystemsforobtainingreliable
rangeinformation.However,itsefficientdesignandpracticalimplementationhasnotyetbeenachievedonmoteplatformsthatare
typicalinwirelesssensornetworkduetoresourceconstrains.Inthispaper,weproposeSparseS-XCorr:cross-correlationvia
structuredsparserepresentation,anewcomputingframeworkforrangingbasedon(cid:96)1-minimization[1]andstructuredsparsity.The
keyideaistocompresstherangingsignalsamplesonthemotebyefficientrandomprojectionsandtransferthemtoacentraldevice;
6
1 whereaconvexoptimizationprocessestimatestherangebyexploitingthesparsesignalstructureintheproposedcorrelation
0 dictionary.Throughtheoreticalvalidation,extensiveempiricalstudiesandexperimentsonanend-to-endacousticrangingsystem
2 implementedonresourcelimitedoff-the-shelfsensornodes,weshowthattheproposedframeworkcanachieveuptotwoordersof
magnitudebetterperformancecomparedtootherapproachessuchasworkingonDCTdomainanddownsampling.Comparedtothe
n
standardcross-correlation,itisabletoobtainrangeestimateswithabiasof2-6cmwith30%andapproximately100cmwith5%
u
compressedmeasurements.Itsstructuredsparsitymodelisabletoimprovetherangingaccuracyby40%underchallengingrecovery
J
conditions(suchashighcompressionfactorandlowsignal-to-noiseratio)byovercominglimitationsduetodictionarycoherence.

1
1 IndexTerms—Ranging,LocationSensing,Positioning,Cross-Correlation,SparseApproximation,CompressedSensing,
(cid:96)1-Minimization,StructuredSparsity
]
H (cid:70)
O
.
s
c
[ 1 INTRODUCTION measuring the travel time1 of the ranging signal. However,

3 Location sensing is a vital enabling technology for numer- the resources required for signal detection are a deciding
v ous applications in the field of binaural science, acoustic factorforthecost,sizeandweightofthesensingplatform;
3 and this essentially strikes a trade-off between (localization
sourcedetection,targetmotionanalysis,sensornetworking,
7 accuracy, coverage range) and energy efficiency. Low-cost
mobilerobotnavigation,mobilecomputing,etc.WhileGPS
4
remains to be the de facto solution for outdoor positioning, and low-power systems estimate the arrival time of the
6
pulsebyutilizingsimpledetectionschemes(suchasempiri-
0 its limitation to service GPS denied environments (such as
calthresholdingoftheleadingpulseedge[2]).Nevertheless,
. indoor and obstructed outdoor) makes location estimation
1
- still - a fundamental problem. Localization is a two step they turn out to be less reliable due to their limited com-
0
5 process.Thefirststepistomeasuretheseparationdistance putational capability to counter environmental noise and
multipath reflections [3]. An established methodology to
1 (orrange)oftheunknownentity(thatneedstobelocalized)
: fromatleastthreepositionedentities(orknownlocations). overcome these limitations is to broaden the range of signal
v
These measurements are subsequently utilized in the sec- frequencies and distribute the energy between the various
i
X ond step that multilaterates the position estimate using a multiplepaths;andsubsequentlyapplyamatchedfilteratthe
r spatially constrained optimization framework. This strong receiverendtocounttheelapsedtimesamplesbyresolving
a those multiple propagation paths. Its benefits are two fold
dependencyofthereliabilityofpositioningaccuracyonthe
distance measurement makes ranging a crucial prerequisite as broadband signals reduce the chance of the entire signal
fading at any particular time, while matched filters allow
forlocalization.
for their processing and form a strong pulse at the line-
Challenges. Acoustic and radio ranging technologies have of-sight(LoS)pathbyincreasingtheoverallsignal-to-noise
maturedsignificantlyinthelastfewdecades.Itisnowwell ratio(SNR)withoutusingexcesstransmissionpower.
understood that highly accurate results can be achieved by Therearenumerousin-airandunderwaterrangingsys-
tems [4]–[8] that have widely used these techniques to
• P. Misra is with the TCS Innovation Labs, Bangalore, India. E-mail: deliver remarkable (accuracy vs. range) performance, but
[email protected]
atthe expenseofspecialized computing platforms(suchas
• M. Yang is at the Case Western Reserve University, USA. E-mail: DSP processors) that are both power intensive and costly.
[email protected] Such stringent needs pose a major challenge to the field
of wireless sensor networks (WSN) that aim to achieve
• M.DuarteisattheUniversityofMassachusetts,Amherst,USA.E-mail:
[email protected]
1.Traveltimeisinterchangeablyreferredtoas:time-of-flight(ToF),
• W. Hu and S. Jha are at the University of New South Wales, Sydney,
time-of-arrival(ToA),propagationdelay,ortimedelayintherangefind-
Australia.E-mail:{wen.hu,sanjay}@cse.unsw.edu.au
ingliterature.
2
similar functional capability on constrained devices with ranging system and its evaluation in Section 3. Finally, we
high restrictions on data sensing rates, link bandwidth, surveyrelatedworkinSection4,andsummarizethepaper
computationalspeed,batterylifeandmemorycapacity(less withconcludingremarksinSection5.
than 50kB of code memory and 10kB RAM) [9], [10]. This
has been the primal factor that has greatly limited the
2 THE DESIGN OF STRUCTS-XCORR
realisationofsophisticatedalgorithms(suchasthematched
filter).Thisproblemcanbesimplifiedbydesigningalight- To ground our discussion, in this section, we first present
weightsignaldetectionandpost-processingmechanismthat thedetailsoftime-basedrangingusingcross-correlationand
notonlyservesthepurposeofsamplecounting,butisalso structuredsparseapproximation.Webuildontheselearning
suitable for running on constrained embedded platforms to cast the ranging problem into the new computation
typically used in WSNs. Motivated by the need to design framework of StructS-XCorr, and then follow it up with its
suchamechanism,weproposeStruct-Sparse-XCorr. empiricalanalysis.
Contributions.Struct-Sparse-XCorr(orStructS-XCorr):cross-
2.1 AnOverviewofTime-basedRanging
correlation via structured sparse representation is a
new computing framework for ranging based on (cid:96)1- Previousstudieshaveshownthemostsuccessfultechniques
minimization [1] and structured sparsity. It is based on a forestimatingtheprecisedistancebetweentwodevicesare
mechanism to compress and transmit the condensed rang- basedonmeasuringthetraveltimeofthesignalpropagation
ing data to a more resourceful offloading device (or base- between them [14]. The reliability of this measurement
station), wherein the time delay of the ranging signal can dependsonmanyfactors,ofwhichrobustnessofexamining
be efficiently recovered to determine the range. Cross- and estimating the energy of the received signal is one of
correlation is the conventional method of obtaining this pa- them. In this regard, matched filter is the state-of-the-art in
rameter;but,givenitssparseinformationcontentandstruc- detectiontechnology.
ture, we make use of the theoretical results in structured A matched filter is implemented by cross-correlating the
sparseapproximationtoachieveasimilarperformance.The receivedsignalx(t)withthetransmittedsignalreplicap(t).
underlyinginformationtheorysuggeststhatasignalcanbe Cross-correlation(X-Corr)ofp(t)andx(t)isasequences(τ)
recoveredby(cid:96)1-minimization[1],whenitsrepresentationis definedas:
(cid:90) t=+∞
sufficientlysparsewithrespecttoanover-completedictionary s(τ)= p(t+τ)x(t) (1)
of base elements. The recovery model (or the optimization t=−∞
frameworkthatbearresemblancetoLassoinstatistics[11], wheretheindexofτ ∈Risthetimeshift(orlag)parameter.
[12]), instead of penalizing the number of nonzero coeffi- This operation s(τ) results in correlation peaks where the
cientsdirectly(e.g.,(cid:96)0-norm)[13],penalizesthe(cid:96)1-normof
positionofthepeaksprovidesameasureofthearrivaltime
thesparsecoefficientsinthelinearcombination.
of the different multipaths. The index of the first tallest
We propose a new dictionary that combines the infor-
correlation peak is the estimate of the pulse arrival time
mation sparsity along the time-delay search dimension,
of the LOS path, which is a direct measure of the range.
and achieves up to two order of magnitude better sparse Generally,x(t)isacquiredfora(finite)minimumtimet=ta
representation and performance compared to standard ap-
givenby:
proaches such as working on DCT domain and down- (cid:18)d (cid:19)
sampling. StructS-XCorr overcomes ranging inaccuracies ta ≥ vc +tp+tr (2)
s
induced by dictionary coherence by approximately 40%
for signals subjected to high compression factor and/or where, dc is the channel length between the transmitter
receivedwithlowSNRlevels. and the receiver, vs is the speed of the ranging signal in
Weempiricallyvalidateourhypothesisinreal-worldin- the medium, tp is the time-period of the transmitted signal
door and outdoor setups. With respect to cross-correlation, p(t), and tr is the approximate reverberation time within
we show that StructS-XCorr obtains range estimates with a which the echoes from the transmitted pulse should have
relative error of less than 2cm by using 30% compressed fallen below an acceptable level before the next pulse is
measurements,andapproximately60cmrelativeerrorwith emitted.Thecorrespondingdiscrete-timesignalofp(t)and
5% measurements only. We also address the problems of x(t)obtainedatasamplingrate(hertz/samplespersecond)
slower compression speed and incorrect peak identification of Fs is given as: p[np] = p[tpFs] and x[na] = x[taFs]
(important for estimating range) by devising a divide-and- 0≤np,na ≤∞.Therefore,p(t)andx(t)canberepresented
conquermethod. asvectorsp∈Rnp andx∈Rna.Thetimedelayisobtained
We present the design and implementation of an end- byfinding:
to-end acoustic ranging system consisting of Tmote Invent τˆ=argmax|s(τ)|2. (3)
τ
(receiver) nodes and a custom built audio (transmitter)
Road-map. The computing operation of τˆ (Eq. 3) is ex-
node. The results show a relative ranging and 2D position
pensive, and demands high memory, computation and en-
error of less than 4cm over cross-correlation using 30-
ergy resources. Considering the constraints of typical WSN
40%compressedmeasurements,butwithsignificantenergy
platforms, it is desirable to scale down its complexity by
savingsofanorderofmagnitudetwo.
a simpler process while still being capable of precisely
To support our contributions; we present the design of
estimatingτˆ.Thismotivatesthescopeforanewframework.
SparseS-Xcorr and its emprirical studies in the next section,
Fig. 1(a) shows a received signal trace recorded for a
which is then followed by the description of the acoustic
durationof0.1ssampledat48kHz,anditscross-correlation
3
andΨsatisfiestheRestrictedIsometryProperty(RIP),then
1.0 Time based Representation 1 Cross−correlation based Representation the (cid:96)0-minimization problem (Eq. (5)) has the same sparse
Normalized Magnitude−00..055 Normalized Magnitude0000....2468 L1osSt M2Pnd MP scmoaenluthtbiooedn(ss(cid:96)o.a1lsv)e:tdheˆsi1nf=oplolaolrywgnimonmgini(cid:96)a(cid:107)1ls-(cid:107)ti1mmisenuibbmyjeiczlitantteiooa:nrxpp=rroogΨbrlsaemmmtih(n6ag)t
−1.0 0 However, due to noise (white Gaussian) v ∈ Rn present
0 0.02 0.04 0.06 0.08 0.1 0.02 0.03 0.04 0.05 0.06 0.07
Time (second) Time (second) in real data, x may not be exactly expressed as a sparse
(a) (b) superpositionofs,andso,Eq.(4)needstobemodifiedto:
Fig. 1: Range estimation by cross-correlation. The infor- x=Ψs+v (7)
mationcontentissparseasthetimedelayvaluecorresponding
to the correlation peak is only useful. It also depicts the same where v is bounded by (cid:107)v(cid:107)2 < (cid:15). The sparse s can still be
receivedwaveformintwodifferent(timeandcross-correlation) recoveredaccuratelybysolvingthefollowingstable(cid:96)1-min-
representations.Notethatinthefigure:LOSstandsforline-of-
imizationproblemviathesecond-orderconeprogramming.
sightandMPisexpandedasmultipath.
((cid:96)1s): ˆs1 =argmin(cid:107)s(cid:107)1 subjectto:(cid:107)Ψs−x(cid:107)2 ≤(cid:15) (8)
withthereferencecopy(alinearchirpof1-20kHz/0.01s)is It is important to note that RIP is only a sufficient but
depicted in Fig. 1(b). Ideally, only a single dominant peak not a necessary condition. Therefore, (cid:96)1-minimization may
should be observed at the correct time shift; however, due still be able to recover the sparse s accurately, even if the
tosignalandnoiseinterference,peaksofsmallermagnitude sensing matrix Ψ does not satisfy RIP. In fact, the use of
may also coexist. Fig. 1(b) exactly reiterates this principle, (cid:96)1-minimization to find sparse solutions has a rich history.
where the correlation peak is the only useful information, It was first proposed by Logan [18], and later developed
and is representative of the signal’s time delay. Therefore, in [1], [19]–[24]. Here, we use (cid:96)1-minimization to solve the
our idea is to exploit the underlying information sparsity cross-correlationproblemviasparserepresentation.
in the signal model to design a simpler acquisition scheme
Dimensionality reduction by random linear projections.
that supports efficient compression, and later recovery. In
As shown in [25] by the Johnson-Lindenstrauss Lemma,
other words, our problem statement is to: obtain the cross-
the (cid:96)2 distance is preserved in the projection domain with
correlationresult(unknown)susingsignificantlyfewer(known)
high probability by random projections. In other words,
observationsofxbasedonthesparsitysturectureoftheproblem.
all the useful information is preserved in the projection
Inthenextsection,wediscussthetheoryofsparseapproxi-
domain.Hence,(cid:96)1-minimizationcanstillbeusedtorecover
mationandstructuredsparsitythatcanexploitthisfeature.
the sparse s from the projected measurements with an
overwhelming probability, even though its dimension is
2.2 SparseApproximationandStructuredSparsity significantly reduced. More precisely, this projection from
high to low dimensional space can be obtained by using a
Motivation insight. One can accurately and efficiently re-
randomsensingmatrixΦ∈Rm×n as:
cover the information of a high dimensional signal (as x)
from only a small number of compressed measurements, y=Φx=Φ(Ψs) (9)
whenthesignal-of-interestissufficientlysparseinacertain
transformdomain(e.g.[15]). where m (cid:28) n and y ∈ Rm is the measurement vector.
The rationale of (cid:96)1-minimization. Using the sparsifying In practice, if s has k (cid:28) d nonzero coefficients, then the
domain, referred to as a dictionary Ψ ∈ Rn×d (with full numberofmeasurementsisusuallychosentobe[26]:
rank), any discrete time signal x ∈ Rn can be represented m≥2klog(d/m) (10)
asalinearcombinationofcolumnsofΨas:
The sparsity level of s can be verified if the reordered
d
(cid:88) entries of its coefficients decay like the power law; i.e., if
x=Ψs= siψi (4) s is arranged in the decreasing order of magnitude, then
i=1 the dth largest entry obeys |s|(d) ≤ Const·d−r for r ≥ 1.
where s ∈ Rd is a coefficient vector of x in the Ψ domain, Forsparses,the(cid:96)2-normerrorbetweenitssparsestandap-
and ψi is a column of Ψ. If s is sparse enough, then the proximated solution also obeys a power law, which means
solution to an underdetermined system of the form x = that a more accurate approximation can be obtained with
Ψs (where the number of unknowns d is greater than the the sparsest s. However, for efficient recovery, the columns
numberofobservationsn)canbesolvedusingthefollowing of ΦΨ should be as independent as possible so that the
(cid:96)0-minimization problem, where the (cid:96)0-“norm” counts the information regarding each coefficient of s is contributed
numberofnonzeroentriesinavector. byadifferentdirection;andthisisachievableifΦandΨare
more incoherent. Ensembles of random matrices sampled
((cid:96)0): ˆs0 =argmin(cid:107)s(cid:107)0 subjectto: x=Ψs (5) independentlyandidentically(i.i.d.)fromGaussianand±1
However, this problem of finding the sparsest solution Bernoullidistributionsarelargelyincoherentwithanyfixed
((cid:96)0-minimization) of an underdetermined system of linear dictionary, and hence, permit computationally tractable re-
equations is NP-hard [13]. As an alternative, Candes et al. coveryofs[1],[16].
in [16] and Donoho in [17] show that if s is sparse enough, Sparseapproximationwithstructuredsparsity.Thetheory
4
of sparse approximation is applicable to a sensing problem
Signal Representation in Different Dictionaries
iftheunderlyingsignalcanbesparselyrepresentedinsome
1
dictionary.Ausefulfeatureisthatthedimensionalityreduc- FFT
tionoperationiscompletelyindependentofitsrecoveryvia DCT
e
(cid:96)1-minimization.Asparsesignalcanbecapturedefficiently d Correlation
u
using a limited number of random measurements that is nit
proportional to its information level. The (cid:96)1-minimization g
a
process does its best to correctly recover this information M
d 0.5
with the knowledge of only the dictionary that sparsely e
z
describesthesignalofinterest,whenthenoisepower(cid:107)v(cid:107)2is ali
smallenoughandthedictionaryΨissufficientlyincoherent. m
r
The mutual coherence of Ψ ∈ Rna×(2na−1), denoted as No Threshold
µ(Ψ),isgivenas: 0.1
|ΨTΨ | 0
µ(Ψ)= max i j (11) 0 2000 4000 6000 8000 10000
1≤i<j≤(2na−1)(cid:107)Ψi(cid:107)(cid:107)Ψj(cid:107) Samples
Fig. 2: Signal representation in different dictionaries.
The worst-case coherence µ(Ψ) corresponds to the largest The signal has a more sparse representation in the correlation
absolute value of the inner product between two distinct dictionary than its FFT and DCT counterparts by an order of
dictionary elements, and is bounded as: 0 ≤ µ(Ψ) ≤ 1. magnitudeexceeding2.
WhileithasbeenproventhatanydesignedΨislargelyin-
coherentwithΦ,itstillmaynotbegoodenoughforparameter
estimation - especially under high noise conditions. There- if we design a representation dictionary having column
fore, it is important to prevent coherent pairs of dictionary element that enumerate over all possible time delay com-
elementsfromappearingintheapproximationprocess. binations.
Candesetal.[15,Theorem1.2],infact,haveshownthat Design execution. For realizing this design goal, we adopt
for conventional sparse approximation with coherent and a positive and negative time-shifted Hankel matrix design
redundant dictionaries, the reconstruction error is upper ofthetransmittedsignalvectorpasΨ.Notethatreversing
bounded by both the noise level and the best k-term ap- the time-shifting order results in a Hankel matrix. We refer
proximationerror.Inanotherrecentwork,DuarteandBara- tothisnewlydesignedΨasthecorrelationdictionary.
niuk[27,Theorem1]haveexaminedthatwithastructured Depending on the lengths of x and p, the following two
sparsitymodel(andusingagreedyrecoveryapproach),the categoriescanbeidentified.
upper bound of the reconstruction error decays exponen- • Case-1 (ta = tp) : Vectors p and x are of equal dimensions
tially to the noise level with an increase in the number of with na samples. The elements of Ψ ∈ Rna×(2na−1) are
iterations. Thus motivated by the significant benefits of the givenas:
structuredsparsitymodel,weproposeStructS-XCorrthatis
detailedinthenextsubsection.
[zeros(n −i)p(1:i)]T 1≤i≤n
 a a
2.3 DetailsofStructS-XCorr Ψ(:,i)= [p(i+1−na :na)zeros(i−na)]T
Inthissection,wepresentthedetailsofthenewdictionary  (n +1)≤i≤(2n −1)
a a
designfollowedbythecomputingmodelofStructS-XCorr.
(12)
where Ψ(:,i) denotes the ith column, [·] denotes a vector
2.3.1 DesignofRepresentationDictionary
of length na, zeros(i) denotes a zero vector of length i,
Design guidelines. The general criteria for designing a ·T denotes the transpose of a vector (matrix), and p(i : j)
reliablerepresentationdictionaryΨrequiresittosufficiently denotesavectorofelementswithindicesfromitoj ofthe
sparsify the signal x. This one dimension search over the inputsamplesetp.
time delay space introduces an important design criteria; • Case-2 (ta > tp) : The size of x is greater than p, and
where, Ψ should be able to preserve the propagation chan- so,thesystemisbalancedbyrightzero-padding(na−np)
nel profile information while adhering to the basic design entriestop.
guidelinesoutlinedbytheunderlyingtheory.Wealsodefine Other popular dictionaries such as the FFT and DCT,
an additional criteria where Ψ should facilitate a faster in contrast, do not provide as good a sparse depiction as
recovery mechanism that implicitly derives the time delay theproposedcorrelationdictionary,andalso,donotsatisfy
result without reconstructing the original signal. Therefore, the remaining two requirements (important for ranging).
thedesigncomplexityistoidentifyandconstructabefitting Fig. 2 compares their sparsity levels (for an indoor high
representation dictionary that satisfies all of the aforesaid multipath channel) by sorting the samples by their magni-
requirements. tudes. The fastest decay characteristic (or the smallest k) is
Design intuition. To this end, we were guided by Eq. 1 observedinthecorrelationdomain,andso,offersthemost
where the locally generated reference copy ensembles val- sparse representation. This implies that the most accurate
ues from a sweep over all possible (positive and negative) approximations (or range estimates) can be obtained with
time delay values. This suggests that the received signal x the correlation dictionary using the smallest number of
could be sparsely represented by a single dimension space measurementsm(Eq.(10)).
5
XCorr: 1440 Samples with 480 Samples S−XCorr: 432 Samples with 480 Samples StructS−XCorr: 432 Samples with 480 Samples
1 1 1
Lag: −ve 1 Lag: +ve Lag: −ve 1 Lag: +ve Lag: −ve 1 Lag: +ve
de de de
u u u
nit nit nit
g g g
ormalized Ma0.5 43 2 ormalized Ma0.5 43 2 ormalized Ma0.5
N N N
0 480 960 1440 1920 2400 2880 0 480 960 1440 1920 24002800 0 480 960 1440 1920 24002800
Samples Samples Samples
(a) Standardcross-correlation (b) Recoveryviasparseapprox. (c) Recoveryviastructuredsparseapprox.
Fig.3:Validation: StructS-XCorr vs. {S-XCorr, XCorr}. The detection accuracy of StructS-XCorr is on a par with S-XCorr and
XCorr,butwithbetterrobustnessagainstmultipathsandlow-noisepeaks.
2.3.2 CompressionandRecovery s can be recovered by solving Eq. (8). However, as the
Compression. The dimensions of x ∈ Rna are significantly dimensions of x are reduced significantly via Eq. (9)2, the
reduced at the receiver by multiplying it with a random reduced (cid:96)1- minimization problem for a given tolerance (cid:15) is
sensing matrix Φ ∈ Rm×na resulting in the measurement givenas:
vbyectthoer cyom∈pRremss(iomn(cid:28)factnoar)αags:ivyen=asΦ:mx. m=αisnraelawtehderetoαn∈a ((cid:96)1r): ˆs1 =argmin(cid:107)s(cid:107)(cid:96)1 s.t:||ΦΨs−y||2 ≤(cid:15) (14)
[0,1]. For example, α = 0.10 means that the information
in x has been compressed by 90%. Φ is a binary sensing ((cid:96)1)isknownasLasso3 instatisticalliterature,andregular-
r
matrixwithitsentriesidenticallyandindependently(i.i.d.)
izes highly undetermined linear systems when the desired
sampled from a balanced symmetric Bernoulli distribution solution is sparse. The correlation domain coefficients ˆs1 are
of±1.
related to the various propagation (direct and reflected) paths,
Φ= √1mΦ¯ whereΦ¯i i.i.d. Pr(Φ¯i,j =±1)=0.5 (13) wthheeersetitmheatienodfexthoefptuhlesefiarrsrtivtaallletsimt ecoorfretlhaetidoinreccotepffiatchie,natnpdetahkuiss,
providestherange.
Binary ensembles have a shorter memory representation
than Gaussian ensembles, and also, alleviate operational Recovery via Structured Sparsity and Sparse Approxima-
complexity;hence,theyareeconomicalforsensorplatforms. tion(StructS-XCorr).Inourcase,Ψisnotstrictlyincoherent
AbalancedΦconsistsof±1atequalprobability,whereeach duetotherepetitivenatureoftheelementsalongeachrow
rowcontainsequalnumberof1’sand-1’s.Therefore,ineach of the matrix (which is an artifact of Hankel matrices). To
rowofΦ,thesumoftheelementsisalwayszero.Abalanced
Φ provides a higher probability of detection (at recovery) Procedure1StructuredSparseXcorrss =Sk(s,µ0)
if the noise in x is Gaussian [28]. The receiver transfers m Input: Coefficientvectors,targetcoherenceµ0
samplesofytothebase-station(BS)forpost-processing. Output: Structuredsparsecoefficientvectorss
RecoveryviaSparseApproximation(S-XCorr)[29].TheBS 1: Initializess =0,i=1
uploadsthecompressedmeasurementstoaserviceapplica- 2: whilei<kands(cid:54)=0do
tiononthecontrolserver.Itrequiresthea-prioriknowledge 3: l(cid:63) =argmax1≤l≤2n−1|s(l)|
of the seed that generates Φ, and the dictionary Ψ. Since x 4: ss(l(cid:63))=s(l(cid:63))
c(tahnebreecreeipveredsesnigtendal)spisarksneloywans,sthine dtheesirdeidctisopnaarrsyeΨsolauntdionx 56:: Λs|Λ=={λ0: (cid:107)ψ|ψλλT(cid:107)(cid:107)ψψl∗l∗|(cid:107) ≥µ0}
7: i=i+1
8: endwhile
OUTPUT
R
CHANNEL overcometheshortcomingsduetodictionarycoherence,we
L apply the principles of structured sparsity to the S-Xcorr
E
AuSdtieor eCoa rd L Pulselectronic PulseAcoustic cteinaoffiimnPceipredouncvtteiindhagueluirmnereios1adtritechpl.[ar2Aot7glw]trhaaoomfrtukemgsrhieanxstghe,fceowuoltlepionatwigdmsoE.apqIltn.sa1ol4licun.oteImito3pinsuacptnaardnteisobleinennaotelebldy4-
INPUT
R
CHANNEL 2.Directcross-correlationintheprojectiondomain(usingy)didnotproduce
desirablerangingresultsbecauseyconsistsofrandomprojections.
3.Theminimizerof(cid:107)x−Ψs(cid:107)22+λ(cid:107)x(cid:107)1 isdefinedastheLassosolution;
Fig.4:Systemarchitectureofacustomdesignedacousticrang- whereλcanbereferredastheinverseoftheLagrangemultiplierassociatedwith
aconstraint(cid:107)x−Ψs(cid:107)2≤(cid:15).Foreveryλ,thereisan(cid:15)suchthatthetwoproblems
ingsystemforempiricalcharacterizationofStructS-XCorr. 2
havethesamesolution.
6
oftheprocedure,weselecttheentrys(l∗)ofthecoefficient to obtain the multipath profile5, but it is not accurate as
vectorswithlargestmagnitudethatisthenpushedintothe it does not follow the same height-to-position relationship
outputofthestructuredsparsevectors.Topreventcoherent (observethepositionofpeak-2&3inFig.3(b)assuggested
elementsfromappearingsimultaneouslyinΨss,wedefine by the corresponding XCorr result shown in Fig. 3(a)).
(line5)thesetΛofallindicesλforwhichtheinnerproduct Although, these parameters are not important for distance
of ψλ and ψl∗ is larger than some predefined threshold µ0. estimation, they are - nevertheless - legitimate sources of
This set indices all dictionary elements that are coherent erroneousdetection.StructS-XCorr,ontheotherhand,does
with the newly selected one, and their future selection is not recover multipath/low-noise peaks apart from the LoS
prevented in line 6 by setting the corresponding entries of path(Fig.3(c));andtherefore,alleviatessuchanomalies.
thevectorstozero. Analysis: space and time complexity. The functionality
algorithm(XCorrvs.compression)onthereceiveristhevital
2.4 AnalysisofStructS-XCorr point of difference. The running time of XCorr is O(n2) in
thetimedomain(TD-XCorr)andO(nlogn)inthefrequency
Inthissection,weanalyzetheperformanceofStructS-Xcorr
domain(FD-XCorr)onconventionalreceiversystems.How-
andidentifychallengesindetectionreliability.
ever,forWSNnodes,additionalsignalprocessingplatforms
Experimentalsystem.Weconductedthisstudyusingacus-
havetobeaddedtocompensateforthelackofhardwaredi-
tomdesignedacousticrangingsystem(Fig.4)withdifferent
videorfloatingpointsupportunits.Therefore,wepropose
assembled units. The front-end of the transmitter consisted
analternatedatacompressionfunctionalitythathasasimi-
ofaCOTSribbon(speaker)transducer,butdrivenbyancus- lartimecomplexity(mn≈O(nlogn)),butamuchsmaller
tomassembled(external)widebandpoweramplifierwitha
spacecomplexity(competentwiththemoteconstraints).
tunable (5-20 times) gain controller. The receiver front-end
We compared their performance on the experimental
comprisedofacustomdesignedpreamplifierunitinterfaced
system, for which we performed the same ranging process
with a COTS Knowles microphone (SPM0404UD5). The but configured the receiver to record for 0.1s (i.e., 4800
controller for this system was setup on a laptop, where
acquired samples). Table 1 shows the individual running
synchronization and ranging signals were generated, cap-
time of the TD-XCorr, FD-XCorr and compression for dif-
tured and analyzed for range estimation. The operational ferent compression factors α. We note that FD-XCorr is
sequencecommencedwiththegenerationofthelinearchirp ≈ 30 times faster than TD-XCorr as expected from their
[01-20]kHz/0.01sthatwasthendirectedintotwoseparate
asymptoticresults.However,thecompressiontime(shown
streams:first,leftinputchanneloftheADCoftheaudiocard as‘Compression1-Buf’)variesfordifferentα,andisslower
(i.e., an electronic chirp) and second, wideband amplifier thanFD-XCorrforallexceptα=0.05.
(i.e., an acoustic chirp). The electronic chirp is equivalent
We overcome this drawback by using the simple idea
to an RF pulse and marks the transmission time of the
of buffer-by-buffer compression rather than one-step com-
acoustic chirp, which is thereafter detected by the receiver
pression. This method divides the acquired signal vector
unit and directed into the right input channel of the ADC. x of length na across b buffers of equal sizes, compresses
The received acoustic signal is considered from the time
the information in each buffer, and finally, assembles the
marker provided by the electronic chirp so as to discard
measurements in their correct order. The signal in each
the delays incurred during the transmission stage4. At the buffer ˜x is of length n˜, where n˜ = na/b. The random
processing station (that functionally replicates the receiver sensing matrix Φ for compressing the data in each buffer
post-processing stage and BS), the acquired samples are is of size [m˜ ×n˜], where m˜ = αn˜ = α(na/b) = m/b.
firstcompressedandsubsequentlyrecoveredtoestimatethe The resultant measurement vector y˜ (for each buffer) is of
range. length of m˜. The number of iterations required to process
each buffer is (m˜n˜). Therefore, the compression time for b
2.4.1 RangingChallengesandMitigation bufferstake(bm˜n˜)=(mna/b)iterations.Thisimprovement
Analysis: basic ranging performance. In this experimental can be identified in Table 1 (shown as ‘Compression 10-
setup, the transmitter and the receiver were placed 1.5m Buf’), where we divide the 4800 samples across 10 buffers
apart.Therangingprocesswasperformedwiththereceiver and record their individual compression time for different
configuredtorecordfor0.03s-justlongenoughtocapture α. The results show a worst-case to best-case improvement
therangingsignalalongwithitsmultipaths.Theaudiocard of6×to60×overFD-XCorr.AsresourceconstrainedWSN
wasconfiguredtosampleat48kHz;hence,thetransmitted motesdonotsupportfloatingpointoperation,ourproposed
signalpandtheacquiredtracexconsistedof480and1440 method is expected to yield better performance (shown in
samples respectively. Using α = 0.30, x was compressed to Section3)onsuchplatformsthanonastandardPC.
obtain the measurement vector y of 432 samples followed Analysis: signal detection and post-processing. The pro-
byitsrecoverytoobtains(Section2.3.2)usingS-XCorrand cess of detection is not without errors as the reconstructed
StructS-XCorr, and its accuracy is then validated against coefficientssmayhavebeenwronglyapproximateddueto
XCorr (Eq. 3). Fig. 3(a), Fig. 3(b) and Fig. 3(c) show the measurementnoisethatcontributestohighercoefficientval-
respective results, where we observe that all the methods ues at incorrect locations. To overcome these inaccuracies,
obtainexactlythesameestimateforthepositionofthefirst
tallestpeakatanegativelagof220samples.S-XCorrisable
5.Thegenerationofthedictionarycoefficientsandcross-correlation
peaksareinthenegativelagpartsincewehavereversedtheorderof
4.Theexperimentalsetupmimicstheconceptofvelocity-difference operation,whereinthereferencesignalwasoperatedwiththeacquired
TDOA(V-TDOA)[14] signal.
7
TABLE1:TimeComplexityAnalysis
XCorr in Buffers: [Buf−1] XCorr in Buffers [Buf−2]
480 samples with 480 samples 480 Samples with 480 Samples
α TD-XCorr FD-XCorr Comp. Comp. 1 1
000...013500 (000s...111)999333222 (000s...000)000666222 1000-...000B002u471f278fer(s) 10000...000-000B000u136ffers(s) malized Magnitude0.5Lag: −ve21 Lag: +ve malized Magnitude0.5Lag: −ve 42 31Lag: +ve
0.50 0.1932 0.0062 0.0361 0.0010 Nor Nor
0 200 4S00amples600 800 960 0 200 4S00amples600 800 960
weusethesameprincipleofbuffer-by-bufferreconstruction (a) (b)
attheBSaswell,whichnotonlyprovidesanadditionalclue
StructS−XCorr in Buffers [Buf−1] StructS−XCorr in Buffers [Buf−2]
forcorrectdetection,butalso,servesasaguidelinetochoose 144 Samples with 480 Samples 144 Samples with 480 Samples
1 1
toohffestTbhahmueefpfrenelreufescmroienbunenceretaobcsf.higbbnuuafffleferpsr,biis.iest.h,cehn˜ossa=emnetspuaFscsht.htFehoasrtatmehxepalmneupcmloeub,neirft malized Magnitude0.5 Lag: −ve1 Lag: +ve malized Magnitude0.5Lag: −ve 1 Lag: +ve
p contains 100 samples and x consists of 1000 samples, Nor Nor
then b is 10. There are two benefits in making this choice. 0 200 4S0a0mples600 800 0 200 4S00ampes600 800
First, it restricts the direct path signal (in the total acquired
trace) to be spread across a maximum of 2 buffers, and (c) (d)
so, guarantees that the magnitude of the corresponding Fig.5:Buffer-by-BufferProcessing.Thedetectionaccuracy
recovered coefficient would always remain at least 50% ofStructS-XCorr,inregardstothepositionoftheLoSpeakand
above its original estimate. Increasing b beyond 2 buffers thetallestpeakineachbuffer,isatparwithXCorr.
decreasestheindividualpeakheightstosmallermagnitudes
that poses a difficult detection task to differentiate them
2.4.2 CharacterizationStudiesandBenchmarks
from the noise-floor. Second, it provides easy processing at
theBS,wheretheoperationofrightzero-paddingptomake Rangingerrorvs.{compressionfactor,SNR}.Theoptimal
its dimensions equal to x is substituted by fragmenting x choice of the compression factor α that achieves the best
intobbufferstomatchthesizeofp(Section2.3.2). accuracy with the least measurements (or projections) m
Thereconstructionprocessisperformedonallbbuffers, is a key design decision as a smaller m leads to lower
whichisfollowedbythesignaldetectionandrangeestima- storage and transmission cost. α depends on the sparsity
tionalgorithm. k (Eq. 10) of the received signal in the correlation domain,
• Phase-1: It identifies the various correlation domain coef- whichinturndependsonthereceivedSNRthatvarieswith
ficient peaks and selects the first tallest peak in each of the transmissionpowerandrangingdistance.Inthissubsection,
b buffers that is at least 6 standard deviations above the we empirically study the relationship between SNR and α.
mean. The detection is considered to have failed for those Thestudywasconductedinthefollowingenvironments.
bufferswherenopointqualifiesasapeak.Thisreducesthe • Case-A {outdoor, very low multipath}: A less frequently
validationspaceforphase-2to˜b(≤b)buffers.
used urban walkway, and the weather being sunny with
• Phase-2: If there are valid peaks in more than one buffer occasionalmildbreeze.
(i.e.,˜b>1),thenthetallestpeak(acrossall˜bbuffers)among • Case-B {indoor, low multipath}: A quiet lecture theatre
themisselectedastherangingpeak.Thedetectioniscorrect, ([25×15×10]m) with a spacious podium at one end of
ifthispeakinbufferbi hasalagthatis: thelargeroom.
• Positive: ⇒ The peak in the previous buffer bi−1 • Case-C {indoor, high multipath}: A quiet meeting room
musthaveanegativelag. ([7×6×6]m) with a big wooden table in the center and
• Negative: ⇒ The peak in the next buffer bi+1 must otherofficefurnitures.
haveapositivelag. The transmitter and the receiver were fixed at a constant
separation distance of 5m. The transmit power was varied
This relationship is a result of the manner in which the
suchthatthereceivedSNRwererecordedwithinthelimits:
signal gets aligned in different buffers and its equivalent
[0-5)dB, [5-10)dB, [10-20)dB, [20-30)dB. For reasons that
representation in the correlation domain/cross-correlation
will be explained in the next subsection, we slightly modi-
(Fig.5).
fiedthepeakselectioncriteriaofthedetectionalgorithmto
If˜b=1(i.e.,onlyasinglebufferhasavalidpeak),thenthe choosethetallestpeakiftherewasnovalidpeak(6standard
peakidentifiedinphase-1qualifiesastherangingpeak.The
deviationabovethemean).100observationswerecollected
estimatedrangerisobtainedas:
for every experiment. We show the relative mean error and its
r =((n˜bi−1+ˆl)/Fs)×vs (15) deviationwithrespecttothe(best-case)XCorrinalltheresultsin
thissegment.
where bi−1 is the buffer count before the detection buffer,ˆl Fig.6(a),Fig.6(b)andFig.6(c)showsthedependenceof
is the lag (in samples) of the ranging peak in the detection α-compressionanditsrecoveryaccuracyontheSNRofthe
buffer, and vs is the temperature compensated speed of rangingsignalusingS-XCorr.Acrossallfigures,weobserve
soundinair. thatapplyingahigherαonalowerSNRsignalresultsinan
8
Correlation Domain Correlation Domain Correlation Domain
Outdoor Walkway Indoor: Lecture Theatre Indoor: Meeting Room
140 140 140
[20-30) dB [20-30) dB [20-30) dB
120 [10-20) dB 120 [10-20) dB 120 [10-20) dB
[05-10) dB [05-10) dB [05-10) dB
m)100 [00-05) dB m)100 [00-05) dB m)100 [00-05) dB
c c c
or ( 80 or ( 80 or ( 80
Err 60 Err 60 Err 60
n n n
a 40 a 40 a 40
e e e
M M M
20 20 20
0 0 0
0 0.05 0.10 0.15 0.20 0.25 0.30 0 0.05 0.10 0.15 0.20 0.25 0.30 0 0.05 0.10 0.15 0.20 0.25 0.30
Compression Factor Compression Factor Compression Factor
(a)Case-A: (b)Case-B: (c)Case-C:
Multipath:very-low Multipath:low Multipath:high
Fig.6:S-XCorr.CharacterizationofcompressionfactorαwithSNR.
Correlation Domain Correlation Domain Correlation Domain
Outdoor Walkway Indoor: Lecture Theatre Indoor: Meeting Room
140 140 140
[20-30) dB [20-30) dB
[20-30) dB
120 [10-20) dB 120 [10-20) dB 120 [10-20) dB
[05-10) dB [05-10) dB
m)100 [00-05) dB m)100 [00-05) dB m)100 [[0050--1005)) ddBB
c c c
or ( 80 or ( 80 or ( 80
Err 60 Err 60 Err 60
n n n
ea 40 ea 40 ea 40
M M M
20 20 20
0 0 0
0 0.05 0.10 0.15 0.20 0.25 0.30 0 0.05 0.10 0.15 0.20 0.25 0.30 0 0.05 0.10 0.15 0.20 0.25 0.30
Compression Factor Compression Factor Compression Factor
(a)Case-A: (b)Case-B: (c)Case-C:
Multipath:very-low Multipath:low Multipath:high
Fig.7:StructS-XCorr.CharacterizationofcompressionfactorαwithSNR.
Buffer-by-Buffer vs. Single Buffer Detection Method Buffer-by-Buffer vs. Single Buffer Detection Method Buffer-by-Buffer vs. Single Buffer Detection Method
Outdoor Walkway Indoor: Lecture Theatre Indoor: Meeting Room
104 104 104
Buf-by-Buf Buf-by-Buf Buf-by-Buf
Single Buf Single Buf Single Buf
Error)102 Error)102 Error)102
n n n
a a a
e e e
M M M
g (100 g (100 g (100
o o o
L L L
10-2[ 00-05) [05-10) [10-20) [20-30) 10-2[ 00-05) [05-10) [10-20) [20-30) 10-2[ 00-05) [05-10) [10-20) [20-30)
SNR (dB) SNR (dB) SNR (dB)
(a)Improvement: (b)Improvement: (c)Improvement:
Orderofmag.1 Orderofmag.1.5 Orderofmag.4
Fig.8:StructS-XCorr:Buffer-by-Buffervs.SingleBufferDetection.Foracompressionfactorof0.30,thebuffer-by-bufferdetection
showsanorderofmagnitude1-4improvementoversinglebufferdetectionmethod.
increaseinestimationerror.Fig.6(a)forCase-Apresentsthe in Case-C (due to high multipath), and so, the errors are
mostclearcharacterizationbynegatingtheeffectofchannel as large as 1m with α = 0.05, but attain stability after
multipaths (though introducing an increased background α = 0.25. The cumulative probability results suggest that
noise level), where observations with a high SNR of [20- there is a 95% probability of incurring an additional error
30)dB provide reliable range estimates by using only 15% of < 1.5cm in indoors and < 3cm in outdoors with
projections while those having low SNR of [0-5)dB show α=0.30withrespecttoitsXCorrestimate.Usingα>0.30
confident result only with α = 0.30 (i.e., using more pro- doesnotimprovetheaccuracysignificantlyconsideringthe
jections). Fig. 6(b) and Fig. 6(c) show the results for Case- additionaloverheads.Fig.6alsoshowsthatforapplications
B and Case-C. Due to a less dominant multipath profile that require lower accuracy (e.g., 100cm), α as less as 0.05
and background noise in Case-B, the accuracy levels show is sufficient. We also performed ranging experiments with
high confidence for α ≥ 0.20. The situation is challenging changes in distance over 1-10m. Although, smaller values
9
Domain and Algorithm Comparison Domain and Algorithm Comparison Domain and Algorithm Comparison
Outdoor Walkway Indoor: Lecture Theatre Indoor: Meeting Room
108 108 108
Corr Domain + CP + StructS−Xcorr Corr Domain + CP + StructS−Xcorr Corr Domain + CP + StructS−XCorr
Error)110046 DDCCSor T+r DDXooCmmoraariinn ++ CDPS ++ SSttrruuccttS−X−XCCoorrrr Error)110046 DDCCSor T+r DDXooCmmoraariinn ++ CDPS ++ SSttrruuccttSS−−XXccoorrrr Error)110046 DDCCSor T+r DDXooCmmoraariinn ++ CDPS ++ SSttrruuccttS−X−XCCoorrrr
n n n
a a a
Me102 Me102 Me102
g ( g ( g (
o o o
L100 L100 L100
10−2[ 00−05) [05−10) [10−20) [20−30) 10−2[ 00−05) [05−10) [10−20) [20−30) 10−2[ 00−05) [05−10) [10−20) [20−30)
SNR (dB) SNR (dB) SNR (dB)
(a)Improvement: (b)Improvement: (c)Improvement
Orderofmag.1-4 Orderofmag.2-4 Orderofmag.4
Fig.9:DomainandAlgorithmComparison.Foracompression(CP)/downsampling(DS)factorof0.30,StructS-XCorrmodelinthe
correlationdomainshowsanorderofmagnitude1-4higherdetectionaccuracycomparedto:(i)compressionwithStructS-Xcorrinthe
DCTdomain(ii)downsamplingwithXCorr(iii)downsamplingwithStructS-XCorr.
ofα(i.e.,lesserprojections)weregoodforhighSNRlevels, ministically choosing samples and discarding information
the results with α = 0.30 were optimal, even in the worst (i.e.,frequencycomponents)bydownsampling.Thisresult,
casetoobtainhigheraccuracy(<2cm). therefore, supports the theoretical underpinning that there
Fig.7showstheaboveanalysis,butusingStructS-XCorr. is an overwhelming probability of correct recovery via (cid:96)1-
Itrevelastwointerestingobservations.First,similartoFig.6, minimization for dimensionality reduction by random lin-
weobservethatapplyingahigherαonalowerSNRsignal ear projection (Section 2.2). Since the recovery techniques
results in an increase in estimation error. Second and more are based in l1-minimization, we direct the readers to [30]
imporantly,thereisa40%improvementinrangingaccuracy forasystematicbenchmarkoftheirperformance.
forcasesofhighcompressionfactorandlowSNRinCase-A Adaptiveestimation:compressionfactor.Thedesignofan
andCase-C.However,thereisn’tappreciableimprovement adaptive mechanism for α requires estimating the received
inCase-B(anenvironmentwithlownoise). SNR.Weproposetwodifferentapproaches:first,withaBS
Fig. 8 compares the detection accuracy between our feedbacktoreceiver,andsecond,onthereceiveritself.
proposedbuffer-by-buffermethodversusprocessingallthe For the BS-feedback mechanism, we utilize empirical
samplesinasinglebufferusingtheStructS-XCorrrecovery information from the peak detection algorithm. In Sec-
method.FromreasonsexplainedinSection2.4.1,theresults tion 2.4.1, we considered the scenarios where the valid
showatleast1orderofmagnitudeimprovement. buffer count˜b ≥ 1. If a valid peak (i.e., at least 6 standard
The sparse representation in the proposed correlation deviations above the mean) is not detected in any buffer
domain shows significantly better accuracy of an order of (i.e.,˜b = 0), then the detection is considered to have failed.
magnitude 2 (Fig. 9) compared to the DCT domain (for
Thisimpliesthattherecoveredcoefficientsarenoisydueto
α=0.30) due to the most sparse depiction of the ranging anon-optimalαfortherespectivemeasurements(character-
signal (Fig. 2). For DCT domain processing, the recovered
ized by its SNR). It was precisely the reason for modifying
coefficientsˆs1 weremultipliedwiththeDCTbasisΨ(Eq.4) thepeakselectioncriteriaintheprevioussubsection,where
to obtain an estimate of the received signal ˆx1, and then we observed large errors in peak positions for magnitudes
cross-correlated with the reference signal p. Here also, the
below the specified threshold. The BS-feedback algorithm
recoverymechanismisbasedonStructS-XCorr.
starts with the initial knowledge of whether a valid peak
Another simple (but deterministic) method of reducing was determined with α = 0.30. If the detection succeeds,
thesamplecountistodownsamplexbyafactorFdresulting thenαisdecrementedbyastepsizeof0.05andcompressed.
in yˆ. We verify its detection accuracy in the correlation
This process is iterated until the detection fails, in which
domain by using two different algorithms: (a) standard case, the previous α values is selected. On the other hand,
cross-correlation and (b) StructS-XCorr with the following
if no valid peaks were encountered for the starting case,
formulationofthesparseapproximationproblem: α is incremented in steps of 0.05 and the entire process is
repeateduntilthedetectionsucceeds.
((cid:96)1r): ˆsd1 =min(cid:107)s(cid:107)(cid:96)1 subjectto:||Ψ(cid:48)s−yˆ||2 ≤(cid:15) (16) A major drawback of the feedback approach is the ad-
The comparison results in Fig. 9 show that neither of ditional measurements (that translate to transmission over-
these two methods based on downsampling provide better head),anditsassociateddelayandpowerusageforderiving
estimates than the proposed method of (cid:96)1-minimization α.Therefore,weintroducethisfunctionalityonthereceiver
and structured sparsity in the correlation domain where by a simple power estimation algorithm. The ratio ρ of the
the improvement is of an order of magnitude 2 across peaksignalamplitudetotheaverageoftheabsolutevalues
all experimental environments. Information embedding in in the sampled signal is calculated, and a corresponding
random ensembles preserves the (cid:96)2-norm (or energy) of its α is selected according to the following empirically chosen
respective higher dimension representation, and therefore, criteria.α = {{0.05 : ρ > 30},{0.10 : 20 < ρ ≤ 30},{0.10 :
the recovery accuracy is significantly better than deter- 20 < ρ ≤ 30},{0.20 : 15 < ρ ≤ 20},{0.30 : 10 < ρ ≤
10
TABLE 2: Projections vs. Accuracy. A positive value indi-
responseof≈22dBabovethenoise-floorbetween1-10kHz.
cateshigherprojectionsorreconstructionerrorcomparedtothe
thresholdα=0.30 Receiver. TmoteInvent [32] was used as the listener node,
due to its low-cost and low-power (100 times more power
Scenario Projections(%) Accuracy(%) efficient than the DSP on the transmitter) features that are
expected from a WSN platform. The receiving front-end
BS-Feedback 101.16 -1.75
Receiver -17.55 5.26 consistedofanomni-directionalelectretmicrophone(Pana-
sonic WM-61B) attached to an Analog Devices SSM2167
15},{0.50:05<ρ≤10},{1.00:ρ≤05}}. preamplifier. It allows omni-directional acquisition in the
For our analysis, we randomly selected 1000 measure- range20Hz-10kHz,andhasanear-flatfrequencyresponse
ments pertaining to different SNR levels in the indoor between 3-7kHz that is 10dB above the noise floor. High-
lecture theatre (Case-B). The respective α was estimated rate audio data collection was achieved using the DMA
using the above two methods and their performance was controller packaged with the MSP430 MCU. However, the
compared against our empirically selected threshold value MSP430DMAcausestruncationofthe12bitsADCdatato
of α = 0.30. Table 2 reports their performance trade-off 8 bits rather than to two bytes, and so, results in a data
where the BS-feedback obtains high accuracy but requires resolutionlossof4bits.
2 times more measurements, while the receiver estimation Ranging/Detection methodology. The system uses the V-
approachtakesfewermeasurementsandobtainsonlya5% TDOAofRFandacousticsignalstomeasurethebeacon-to-
worseaccuracy. listenerdistance.Thebeaconinitiatestherangingprocessby
periodicallytransmittingaRFsignalfollowedbyaacoustic
pulse after a fixed time interval. The fast propagating RF
3 EVALUATION
pulse reaches the listener almost instantaneously and syn-
Wepresentthedesignandimplementationofanend-to-end chronizes the clocks on both the devices, following which,
acoustic ranging system using constrained WSN platforms theTDOAismeasuredafterthearrivaloftheacousticpulse.
in this section, followed by evaluation results. Fast data The ranging signal was a linear chirp of [3-7]kHz/0.01ms
acquisition and compression on the receiver node was the and was transmitted at an acoustic pressure level of 70dB.
underlying system rationale; hence, all design decisions The DAC on the audio codec of the beacon node was
were guided towards maximum RAM utilization rather programmed to sample at 48kHz, while the ADC on the
than external flash (that would introduce additional la- receiverTmotewasconfiguredtoacquireat15kHz.
tency). If the time taken for sound to travel a maximum range
dc at a speed vs is at most dvsc, and if the transmitted chirp
3.1 Systemdesignonconstrainedplatforms. length is tp, then the signal must reach the receiver within
ThesystemcomprisedoftheTmoteInvent(aslistener),our o[dvfscth+etspig].nFalormtupst=be0.c0o1mspalnedteddcby≈0.1003ms.,Wtheeinrcelcuodrdeinang
designed sensor mote (as beacon) and a network interface additional 0.01s to compensate for reverberation time (tc),
tothebase-station(Fig.10). and setup the recording time to 0.04s (Eq. 2). Following
Transmitter. The beacon node [31] comprised of our WSN the buffer-by-buffer compression method, the signal was
platform along with a custom designed audio daughter spreadacross5buffers.AmeasurementmatrixΦ¯ wasstored
board that included four TI TLV320AIC3254 audio codecs in the RAM that contained i.i.d. entries sampled from a
and the Bluetechnix CM-BF537E digital signal processor symmetric Bernoulli distribution (Eq. 13). We postponed
module.Thetransmittingfront-endofthebeaconmotecon- the multiplication operation on the matrix entities with the
√
sistedofapoweramplifierdrivingatweeter(speaker)trans- constant(1/ m)untiltherecoverystageattheBS.
ducer (VIFA 3/4” tweeter module MICRO). The tweeter Thelisteneracquirestheaudiosamples,compressesand
(size:[2×2×1]cm)hadafairlyuniformandhighfrequency stores these measurements in the RAM over a period of 5
iterations, and then, transfers them to the BS. These mea-
surements are again divided into their respective buffers
Transmitter Receiver and reconstructed to obtain the coefficients. The detection
RF process is the same as explained in Section 2.4.1, however
we made two minor modifications. First, due to a higher
Acoustic receiver noise floor, we set the criteria for selecting the
first tallest peak to 3 (instead of 6) standard deviations
RF above the mean. Second, as each sample corresponds to
2.2cm of distance (at a sampling rate of 15kHz), we used
Base Node a simple parabolic interpolation method [8] to obtain finer
resolution.Thisadditionalstepidentifiesthepositionofthe
SOFTWARE
Serial first neighboring peak on the left and right of the selected
Packet ranging peak, finds the parabola that passes through these
Reconstruction Simulator
Display/ & TCP IP/ points, and calculates the time coordinate of the maximum
Database Localization Directory Poll
Algorithm Packet ofthisparabolathatestimatestherange.
Listener Base Station
Fig. 10: System architecture. End-to-end acoustic ranging
systemusingconstrainedWSNplatforms.