Multi Detector Fusion of Dynamic TOA Estimation using Kalman Filter

0.25 MB·English

by  Vijaya Yajnanarayana

1
Multi Detector Fusion of Dynamic TOA Estimation
using Kalman Filter.
Vijaya Yajnanarayana, Satyam Dwivedi, Peter Ha¨ndel
Abstract—In this paper, we propose fusion of dynamic which matched filter is suitable including [12]–[14]. Digital
TOA (time of arrival) from multiple non-coherent detectors receiver structure offers several benefits such as flexibility in
like energy detectors operating at sub-Nyquist rate through
design, reconfigurability and scalability [15]. However, UWB
5 Kalman filtering. We also show that by using multiple of these
signal occupy extremly large bandwidth and thus requires
1 energy detectors, we can achieve the performance of a digital
0 matched filter implementation in the AWGN (additive white high-speedanalog-to-digitalconverters(ADCs). These speeds
2 Gaussian noise) setting. We derive analytical expression for demand the use of interleaved flash ADC [14] or a bank of
number of energy detectors needed to achieve the matched polyphase ADCs [12]. In addition, the ADC must support a
n filter performance. We demonstrate in simulation the validity
large dynamic range to resolve the signal from the strong
a of our analytical approach. Results indicate that number
J narrowband interferes. These aspects makes digital UWB
of energy detectors needed will be high at low SNRs and

2 converge to a constant number as the SNR increases. We also architecture which operates at Nyquist-rate to be costly and
study the performance of the strategy proposed using IEEE power hungry [14], [16], [17].

] 802.15.4a CM1 channel model and show in simulation that two
T
sub-Nyquist detectors are sufficient to match the performance
Duetotheabovediscussedpracticalconcerns,energydetec-
I of digital matched filter.
. torsarean interestingalternativeastheyarelesscomplexand
s
c Index terms: Time of arrival (TOA), ultra wideband (UWB), can operate at sub-Nyquistrate [18]–[20]. However,these de-
[ UWB ranging. tectorssufferfromthe noise dueto the square-lawdeviceand

1 are sub-optimal in AWGN channels. Performance analysis of
v standaloneenergydetectorforTOAestimationinsub-Nyquist
6 I. INTRODUCTION rate has been studied in [19], [21], [22]. However, how to
0
An ultra wideband (UWB) system is based on spreading achievethe performanceofa digitalmatchedfilter usingmul-
4
a low power signal in to large bandwidth. There are sev- tipleenergydetectorsfora dynamicTOAmodelistothebest
0
0 eral UWB techniques, impulse radio based UWB (IR-UWB) of the authors knowledge unavailable. In this paper, we first
. schemes are most popular as they provide better performance showthatTOAestimationusingdigitalmatchedfilterachieves
1
and complexity trade-offs compared to other UWB schemes Crame´r-Rao lower bound (CRLB) asymptotically with SNR.
0
5 [1]–[3]. Narrow impulse signals that are used in IR-UWB We show analytically that we can achieve the performanceof
1 schemes yield very fine time resolution and thus, can be used a Nyquist rate matched filter, by using multiple sub-Nyquist
v: for accurate measurement of time of arrival (TOA) [4], [5]. rateenergydetectorestimates.Weshowthatnumberofenergy
i Accurate TOA is essential in several applications includ- detectors needed to meet the matched filters performance is
X
ing localization and communication. In localization, when high at low SNRs and reduces as SNR increases, and finally
r the nodes are synchronized, the TOA of the signal can be converges to 4 as SNR increases asymptotically. We propose
a
used directly to obtain the range estimate. If nodes are not joint fusion and tracking of dynamic TOA using multiple
synchronized,TOAestimateisstillneededforseveralranging energy detector estimates through a suitable Kalman filter
protocolstoestimatetherange.Localizationinformationabout design. We assess the performance of fused TOA estimate in
the node can be derived from its range to anchor nodes [5], simulation for a dynamic TOA model from multiple energy
[6]. In many IR-UWB communication systems, the message detectors and compare it with the digital matched filter and
informationisembeddedinthelocationoftheIR-UWBpulse. demonstrate the validity of the claims.
For example,pulse position modulation(PPM) and its variant
modulation schemes. To demodulate the IR-UWB symbols in This paper is organized as follows: In Section II, we
these modulationschemes, TOA estimation techniquescan be discuss the signal and system model employed. In Section
used [7]–[9]. III, we will discuss different types of detectors. Here, we
IR-UWB system with matched filter can provide optimal will study the energy collection strategy for matched filter
estimate of TOA for additive white Gaussian noise (AWGN) and energy detectors. In Section IV, performance analysis
limited channels. The TOA is estimated by finding the peak of these two detectors are studied for a static TOA case.
location at which the filter attains the maximum value. Section VI, discussesthe Kalman filter design forjointfusing
The impulse response of the matched filter should closely and tracking of the energy detector estimates for a dynamic
approximate the transmit pulse shape. When implemented TOA model. Section VII, demonstrates the performance in
digitally, matched filter requires Nyquist-rate sampling [10], simulation. Finally, Section VIII details the conclusions from
[11]. There are several digital UWB receiver architecturesfor the design and results demonstrated.
2
Receiver-1 τˆt1oa Filter)
n
τˆ2 ma
Receiver-2 toa Kal τˆtoa
. (
.. .. nter
. Ce
n
Receiver-N τˆtNoa Fusio
Fig.1. Jointfusion andtracking structure usingN distributed receivers usinglow costenergydetectors operating atsub-Nyquist rate, tracking adynamic
TOAofamovingtarget.
II. MODEL of analysis, we assume cj = 0 and dj = 1. TOA estimation
A. Signal Model problemis the estimate the first arraivalpath,τ1 =τtoa, in the
received signal (3).
The signal model comprises of N frames each having a
f
unit energy pulse s(t), given by
B. System Model
Nf We define a system model as shown in Fig 1, where
ωtr(t)= djs(t jTf cjTc), (1) N receivers each having an energy detector is employed to
− −
j=0 estimate TOA. When the target is moving then the estimated
X
where each frame is of duration T and the frame index is TOAis dynamicinnature.In these situationsjointfusingand
f
represented by j. The chip duration is represented by T and tracking using multiple sub-optimal estimates from the sub-
c
c 0...N indicatesthetime-hoppingcode.d 1 is Nyquist receivers can yield better performance. We employ
j c j
the∈po{larityco}de,whichcanbe usedalongwith tim∈e-h{o±pp}ing Kalman structure as shown in Fig. 1 to jointly fuse and track
tosmooththesignalspectrum.Weconsidertheframeduration the TOA. The multiple receivers can be distributed in nature
longer than the delay spread of the channel. and makes independentestimates of the TOA, which are then
A widerangeofpulseshapeshavebeenexploredforUWB fused using a Kalman filter at the fusion center. All receivers
applicationsfrom rectangularto Gaussian. Gaussian and their should return the TOA for a reference location. Without loss
derivatives, usually called monopulses, are effective due to of generality, we assume that the all receivers are equidistant
the ease of construction and good resolution in both time from the target1. The proposed tracking and fusion strategy
and frequency.In manycosteffectivehardwaredesigns, these can also be employed in a single receiver architecture having
shapes are generated without any dedicated circuits [8], [9], a bank of energy detectors.
[23],[24].Foranalyticalandsimulationanalysis,wehaveused
C. UWB Hardware
the 2nd order Gaussian pulse [21],
There are several low-complexity sub-Nyquist receivers
4πt2 2πt2
based on energy detector available in literature [25]–[28].
s(t)=A 1 exp − , (2)
− ζ2 ζ2 Figure2showsagraphicaldepictionofanin-housedeveloped
(cid:18) (cid:19) (cid:18) (cid:19)
The amplitude is adjusted through parameter, A, and pulse IR-UWB platform for ranging and communication. It uses a
width is adjusted through parameter, ζ. low cost pulse generator to generate sub-nano second pulses
The received signal is the distorted version of the transmit using step recovery diode (SRD), as described in [29]. The
pulsewithmultipaths.TheTOAisdefinedasthetimeelapsed characterization and modeling of the UWB platform for a
for the first arrival path to reach the receiver from the trans- distance measurement system can be found in [30] [31]. A
mitter. The received signal can be represented by detailed architectural description and experimental ranging
results from a prototype of the platform have been published
Nf
in [28]. The power and range of the transceiver can be easily
ω (t)= d r(t jT c T )+n(t). (3)
rx j − f − j c tradedbycontrollingtheamplitude,dutycycleandnumberof
j=0
X pulsesperbitoftransmission.Eventhoughproposedmethods
L in this paper are not tied to a particular hardware, using
where r(t)= Eb α r (t τ ), where E , is the captured
Nf l l − l b the above discussed UWB hardware can yield significant
l=1
l=Lq P performance benefits in terms of cost and power since they
energyand α2 =1.ThegainandreceivedUWBpulsefor
l are implemented using low-complexity analog circuits.
l=0
the l-th tap Pis given by αl and rl(t). The n(t) is the AWGN 1Ifthisconditionisnotmet,theneachreceiverneedstoappropriatelyscale
process with zero mean and double-sided power spectral
theTOAbasedonthegeometryofthereferencelocationandactuallocation
densityofN0/2.Withoutlossofgenerality,andforsimplicity ofthereceiver.
3
RF LNA BPF ADC m(n) iN=p0m(i+n)s(i) xn argnmax nˆtoa
P
Ts s(i)
(a)MatchedFilter
(a)Radiomodule (b)Printedcircuit board(PCB) RF LNA BPF (.)2 xn f(x) nˆtoa
Fig.2. Iconicmodelofthein-housedevelopedimpulseradioUWB-platform R Tb
of size 6cmx4cm for ranging and communication working in the 6GHz (b)EnergyDetector
regimewithseparateRXandTXantennasfromGreenwavescientific. (details
available at[32]) Fig. 3. Receiver structure with different detectors. The s(i) and m(n)
denotes thesampledversionoftransmitpulseandreceived signal.
1.5
III. DETECTORS
The two commonly employed receiver structures are as 1
shown in the Fig. 3. The received signal passes through the n
x
low noise amplifier (LNA) and band pass filter (BPF) of 0.5
bandwidth B. The output signal of BPF is converted in to
energysamples, x= x ,x ,...,x , from which TOA is 0
1 2 Nobs 0 50 100 150 200
estimated. Here, N , defines the number of energy samples n
obs
(cid:0) (cid:1)
usedintheestimation.Basedontheenergycollectionstrategy
different types of detectors exists. In this paper, we consider Fig.4. Variation ofenergysamples,xn,versesblockindex,n.Parameters
Tf =200ns,Tb=1ns,SNR=0dBandNobs=200areconsidered.
the matched filer which operates at Nyquist rate (1/T ) and
s
energy detectors operating at sub-Nyquist rate (1/T ).
b
where, f(), is the estimator function, which estimates the
·
A. Matched filter block-index/sample-indexof the first arriving path. The func-
tion,f(),ischosenbasedonthechannelmodel.Equation(8),
In the matched filter, energy is collected by correlating the ·
representsτˆ asthemid-pointofthecorrespondingestimated
received samples with the transmit pulse shape as shown in toa
block, assuming that the true TOA is uniformly distributed
Fig. 3a [33]. Matched filter can be mathematically expressed
with in this block.
as
Withoutlossofgenerality,weconsiderthesub-Nyquistrate,
Np T =T ,indetectorstructureshowninFig.3.Therefore,each
b c
xn = m(n+i)s(i), (4) frame consists of, N = T /T , blocks, each with energy,
obs f c
Xi=0 xn, n [1,...,Nobs], given by
nˆ = argmax(x), (5) ∈
toa
τˆtoa = nˆtoanTs, (6) xn = Nf (j−1)Tf+nTb r(t)2dt. (9)
| |
where, s(i) = s(iTs), represent the digitized transmit pulse Xi=0Z(j−1)Tf+(n−1)Tb
and m(n)=ωrx(nTs), representthe digitized receivedsignal, Typicalvariationof energysamples, xn, versesn is as shown
sampled at interval T . In matched filter, the energy samples, in the Fig. 4. In energy detector, the energy samples are at
s
xn, are at Nyquist rate, 1/Ts. Energy detectors, operating at sub-Nyquist rate, 1/Tb.
the sub-Nyquist rates are an interesting alternative to digital
matched filter. IV. AWGN CHANNEL ANALYSIS
Thebestperformanceintermsofmean-square-error(MSE)
B. Energy Detector for an unbiased estimator is given by the CRLB and for a
TOA estimation problem this is given by [4], [34]:
The structure for energy detector is as shown in Fig. 3b.
The structureis amenablefora low-complexityanalogimple- 1
σ2 , (10)
mentation at sub-Nyquist rates [18]–[20]. In energy detector, τ ≥ 8π2SNRβ2
the outputsignalfromBPF isconvertedin toenergysamples,
where β, is the effective signal bandwidth defined by
x = x ,x ...,x . TOA is estimated from the energy
1 2 Nobs
sampl(cid:0)es using the equa(cid:1)tion below β2 = −∞∞f2|S(f)|2df , (11)
nˆtoa =f(x), (7) "R ∞ |S(f)|2.df #
−∞
1 where S(f), is the FouriRer transform of the transmit pulse,
τˆ = nˆ T , (8)
toa toa− 2 b s(t).
(cid:18) (cid:19)
4
InAWGNchannelmodel,weconsiderasingle-pathmodel,
104
with L = 1, α1 = 1 and τ1 = τtoa in (3). The probability Energydetector
density of the x , for matched filter and energy detector Matchedfilter
n 102 CRLB
depends on whether the particular energy sample, x , is
n
signal+noise sample or noise-only sample. For matched filter
ihtycpaonthbeseiss,hHow1n∼=thxantitshesigpnroalb+anboiliisteysdaemnpsiltey, afunndcHtio0n∼=s uxnndeisr 2SE[ns]100
noise-only sample, is given by M10−2
p(x ) N(√Eb,σ2) under H1, (12) 10−4
n ∼( (0,σ2) under H0,
N
10−6
where σ2 = N /2, is the variance of the noise samples and 8 10 12 14 16 18 20 22 24 26
0 Eb/N0
denotes the Gaussian distribution.
N
To derive the probabilitydistribution function (PDF) of the Fig.5. Variation ofMSEwithSNRforenergydetector andmatched filter.
energy samples in energy detectors case, the function f() in MSE is evaluated by averaging the estimated TOA for 1000 random τtoa
(7) need to be defined. Under the AWGN assumption,· the drawn from U[0,Tf = 200 ns]. Matched filter asymptotically reaches the
CRLBbound.
receivedsignalwillhaveasinglepath,whosedelayrepresents
the TOA. Thus, optimal TOA estimation strategy here would
be to pick the energy sample having maximum energy as the noise-only energy sample is higher than the signal+noise
nˆ , this selection criteria is called maximal energy selection energy sample. That is
toa
(MES) and is given by [21]
x >x , (21)
nno nsn
where x and x are energies of signal+noise and noise
nsn nno
nˆtoa = max(x), (13) only sample. For AWGN channel, there will be only one
1 signal+noise energy sample and all other samples are noise-
τˆ = nˆ T , (14)
toa toa− 2 b only.Thecorrectselectionwillhappenwhenxn <xsn,forall
(cid:18) (cid:19)
theenergysamplesexceptforthesignal+noiseenergysample.
The complementary cumulative distribution function (CCDF)
Thus, the probabilty of correct selection, P , and probalility
s
of energy samples, xn for energy detector, is given by [21], of error selection, P , is given by [36]
e
[34]–[36]
QM′/2 Eσb,√ση under H1 Ps = Z∞ (cid:18)1−Q(cid:18)xnσ−noµno(cid:19)(cid:19)Nobs−1p(xn)dxn,
P(xn >η)= (cid:16) M′(cid:17)/2 1 i xn=0
exp −ηNNof − i1! ηNN0f under H0 Pe = 1−Ps, (22)
i=0
where QZ(a,b)denot(cid:16)es the M(cid:17)arcPum-Q-f(cid:16)uncti(cid:17)on with par(a1m5-) fwohretrheepd(extenc)toirsstuhnedderenhsyitpyotfhuenscistioHn1.ofQthdeeneonteersgtyhesaCmCpDleFs
eter Z, P denotes the probability and M′ ≈ Nf(2BTb+1), function of xn. Here µno, denotes the mean values of xn
denotes the degrees of freedom (DOF). At high SNRs the for matched filter and energy detectors and is given by 0
above equation can be approximated to [21] and NMσ2 respectively. σ2 , denotes the variance of x for
f no n
matched filter and energy detectors and is given by σ2 and
p(x ) N(µH1,σH21) under H1, (16) 2NfMσ4 respectively.
n ∼(N(µH0,σH20) under H0, Itisnotstraightforwardtoarriveattheclosedformexpres-
sion for energydetector’sP and P due to complexdistribu-
s e
Themeanandvarianceofxnunderbothhypothesesisgiven tion functions (15). There is however a loss of performance
by in energy detectors due to squaring and integration operation
at sub-Nyquist rates compared to matched filter, operating at
µ = NMσ2+E , (17)
H1 f b Nyquist rate. Figure 5, shows the mean square error (MSE),
σH21 = 2NfMσ4+4σ2Eb2, (18) performance for matched filter and energy detector. MSE is
µ = NMσ2, (19) evaluated by averaging the estimated TOA for 1000 random
H0 f
σ2 = 2NMσ4, (20) TOAs(τtoa)drawnfrom [0,Tf =200ns].Thesamplingrate
H0 f (1/T )usedbymatchedUfilteris1000GHz2 andsamplingrate
s
whereM =2BTb+1,is the degreesof freedomof noiseand (1/Tb) used by energy detector is 1/1000-th that of matched
σ2 =N /2. filter and is equal to 1 GHz.
0
If we choose the MES criteria for TOA estimation for both
2SinceTOAiscontinuousintime,veryhighsamplingrateisemployedto
matched filters and energy detectors as given in (4) - (6) and
demonstrate that matched filter will indeed reach the CRLB bound without
(13) - (14) respectively,then an error can occur, if any one of muchambiguityduetodiscretization oftimeduetosampling.
5
Algorithm1:KalmanfilterfortrackingandfusingofTOA The probability distribution function of e is given by3
from multiple sub-Nyquist energy detector estimates
σ2
Input: Prior state, sinit, prior state’s covariance, Minit, and e∼N µH1, KH1 . (25)
covariance of plant noise, Q. (cid:18) (cid:19)
Output:Tracked and fused TOAestimates, τˆtoa and variance of Let σ2 = σ2 and σ2 = σ2 denote the variance of the x
estimate, σˆt2oa underMHF for matcheEdD filterHa1nd energy detector. From (12n)
1 s[1]←sinit ⊲ Initial state 1
2 M[1]←Minit ⊲ Initial states Covariance and(25)toachievetheperformanceofthematchedfilterusing
3 τˆtoa[1]←sinit[1] ⊲ Initial TOA estimate multiple energy detectors, we need
4 σˆt2oa[1]←Minit[1](1,1) ⊲ Initial MSE estimate 1
65 for isp←rd1←toAns[id]o ⊲ Pridicted state σM2F = NEDσE2D, (26)
7 Mprd ←AM[i]A′+Q ⊲ Predicted MSE where, NED, is the number of energy detectors needed to
⊲ Compute Kalman gain
have the same performance as that of matched filter. For
8 K=(MprdHT)(C+HMprdHT)−1 normalized energy per bit (E = 1), and using the variance
⊲ Update state from observations (y[n]) b
for signal+noise sample from (12), (16) and (18) we get
9 s[i+1]=sprd+K(y[i]−Hsprd)
10 τˆtoa[i+1]=s[i+1](1) ⊲ Save fused TOA (τˆtoa) σ2
1112 Mσˆt2o[ai[i++11]]==(IM2−[i+K1H]()1M,1p)rd ⊲ U⊲pSdaavtee MMSSEE NED = SNlRim→∞σME2DF (27)
13 return τˆtoa,σˆt2oa NED = σl2i→m0(cid:0)2Mσσ42+4σ2(cid:1), (28)
= 4. (29)
At high SNRs, to achieve the performance of the matched Thus, from (29), asymptotically, with the increase of SNR
filter using multiple energy detectors, we need the covariance (Eb/N0), the number of energy detectors needed to achieve
of the signal+noise energy sample (under hypothesis H ) to the same performance as digital matched filter is equal to 4.
1
be same. We can approximate the probability distribution We will show in simulation that this phenomenon is indeed
of energy samples for energy detectors in to a Gaussian true in the later section.
distribution at high SNR as shown in (16). If there are K
receivers having energy detectors to independently estimate V. MULTIPATHCHANNEL ANALYSIS
TOA, then signal+noise block in each detector will have a Many UWB ranging applications have channel response
probability distribution given by with several multipath components, i.e. the received pulse in
(3) has (α ,α ...α ;τ ,τ ...,τ ), where L is the number
1 2 L 1 2 L
ei ∼N µH1,σH21 ,i∈[1,··· ,K]. (23) of multipaths. The TOA estimation problem is to identify
the leading edge (first arriving path, τ ). In multipath UWB
Since e s are indep(cid:0)endent and(cid:1) identically distributed, then 1
i
channels, the matched filtering performance is not optimal
the best estimate for the signal+noise block using K energy
since the shape of the pulse is lost in the channel due to
detectors is given by [37] ,
the frequency selective fading. Also, the magnitude of the
K
1
e= ek. (24) 3See [38] to obtain the probability distribution function of a function of
N
ED Gaussianrandomvariable.
k=1
X
200 200
100 180
0 160
0 20 40 60 80 100
200
140
100
[ns] 0 [ns]120
A 0 20 40 60 80 100 A100
O200 O
T T 80
100
60
00 20 40 60 80 100 True TOA
200 40 Observed TOA
Fused TOA after Kalman Formulation
100 20
0 0
0 20 40 60 80 100 0 20 40 60 80 100
τ[s] τ[s]
(a)ObservedTOAbytheenergydetectors (b)ORfusion
Fig.6. FusionofmultipleDetectors usingKalmanFormulation.
6
104 1
Energydetector(MES)
Matchedfilter 0.9
Energydetector(WMESS)
y0.8
g
er0.7
n
e
el0.6
2[ns]103 ann0.5
E h
S c
M e0.4
g
a
er0.3
v
A
0.2
0.1
102 0
10 15 20 25 30 0 10 20 30 40 50
Eb/N0 Delay[ns]
Fig.7. VariationofMSEwithSNRforenergydetector(withselectioncriteria Fig.8. AveragedChannelenergyprofilefor802.15.4aCM1channelmodel.
MES and WMESS) and matched filter. MSE is evaluated by averaging the 100different channel realizations areaveraged.
estimated TOAfor1000randomτtoa drawnfromU[0,100ns].
from x starting from k and E is the N 1 vector denot-
e
energy sample containing first arriving path may be smaller ing the a-priori channel energy information×which is similar
than the peak energy sample, therefore, using MES criteria to correlating received energy samples with PDP. Averaged
for TOA estimation, as represented in (13) and (5) does not channelenergyprofile,E,forIEEE802.15.4aresidentialLOS
always yeild true TOA [39]. Figure 7, shows the performance channel model is as shown in Fig. 8. The performance of
in terms of MSE for matched filter and energy detector for energydetectorwithWMESSalgorithmisasshowninFig.7.
802.15.4a, residential LOS channel (CM1 model) [39]. MSE Even though, the WMESS estimation algorithm require more
isevaluatedbyaveragingtheestimatedTOAfor1000random computation, its performance is better than matched filter at
τtoa drawn from [0,100 ns]. The sampling rate (1/Ts) used lowSNRsforIEEE802.15.4aCM1multipathchannelmodel.
U
by matched filter is 8 GHz and sampling rate (1/Tb) used by However,athighSNRthereisalossofperformancecompared
energydetectoris 1/8-ththatof matchedfilter andis equalto to matched filter.
1 GHz. The performance degradation at high SNR with WMESS
The performance of the energy detectors can be improved
criteria for energy detector can become a problem when the
by considering the a-priori information such as power delay
TOA of the estimated target is dynamic in nature. In the next
profile (PDP). To accomplish this, we can repose the TOA
section, we will discuss a mechanism to fuse the estimates
estimation problem as a multiple hypothesis testing problem.
from multiple receivers to achieve improved performance for
The energy samples vector, x, is a 1 Nobs vector, out a dynamic TOA system.
×
of which, N blocks are having multipath signals, N T ,
e e b
represents the excess delay of the channel. If we define H ,
k VI. MULTIDETECTORFUSION USING KALMAN FILTER
as the hypothesis that the k-th sample denotes the first path
arrival, then H =x, with, WhentheestimatedTOAofthetargetisdynamicinnature,
k
then joint fusing and tracking using multiple independent
nTb TOA estimations from the energy detectors can yield better
n2(t).dt,
performance.We employKalmanstructureasshowninFig.1
 Z(n−1)Tb to accomplish this.
x(n)= Z(nn−Tffb1oo)rrTbnn|ω==rxk1(t,,)..+....,,nkk(t−+)|21N.det,−1 (30) bgyivTethhneebssyttaatteeoefqtuhaetissoy[nnst]ien=mtAhaensdK[niats−lmd1ya]nn+afimlutiec[nrn]fa,oturmreuilsartieopnr,esaenn(d3te2ids)
Here, x(n), denotesZ(tnnh−Tefb1on)rT-btnhn=2e(letk)m.+detnN,teo,f..x.,aNndobsH. ntoa, is the where s[n], A, and u[sn[]ni]s=gi(cid:20)vve1tnoτa[[bnny1]](cid:21), (33)
true hypothesis [21]. If the PDP for the channel is available, A[n]= , (34)
0 ν
thenwecanmodifythehypothesistestfromMEStoweighted (cid:20) (cid:21)
maximum energy sum selection (WMESS), and is given by 0
u[n]= , (35)
u [n]
nˆtoa =arg max x(k :k+Ne)E , (31) (cid:20) v (cid:21)
k∈[1,...,Nobs]{ } where uv[n] is the noise with variance σp2 and ν < 1 is a
where,x(k :k+N )isa1 N vectorhavingtheN elements constant. The τ[n] is the TOA at n-th time intervaland v is
e e e toa
×
7
anAR(1)processeswithconstantν, representingthedynamic

nature of TOA. The plant noise u[n] (0, ), where is 10−1
∼N Q Q
given by
0 0
= . (36)
Q 0 σ2
(cid:20) p(cid:21) E10−2
The measurementmatrix, y=[τˆ1 ,τˆ2 ,...,τˆN], is the in- MS
toa toa toa
dependentTOAestimatesfromN receivers.Themeasurement
equation is given by
NED=1
NED=4
y[n]=H[n]s[n]+w[n], (37) 10−3 NED=10
Matchedfilter(NMF=1)
where
0 5 10 15 20 25
1 0 SNR
1 0
H[n]=.. .., (38) Fig. 10. Steady state MSE verses time, for multiple energy detectors and
. . digital matchedfilter.
 
1 0
 
 
w[n] (0,diag(σ2,...σ2 )). N denotesthe numberof de- 9
∼N 1 N
teelcetmoresntasndredpiraegse(·n)teinddiincsaitdeestthheedbiraagcokneatsl.matrixwithdiagonal )ED8
N
( 7
The Kalman filter based fusion of dynamic TOA from rs
multiple energy detector estimates, y[n], is as shown in cto 6
e
Algorithm 1. At each iteration, the Kalman filter yields fused det 5
y
TOA estimate, τˆtoa[n], and its variance, σˆt2oa[n], from multiple erg 4
energydetectorestimates, y[n]. In ouranalysis, we have used en
same kind of detectors and also the reliability or importance of 3
of each of the detectors are assumed same. By appropriately mber 2
selecting the measurement matrix H and covariance of the Nu 1
measurements,wecanextendtheKalmanfilterformulationto 0
2 4 6 8 10 12 14 16 18 20 22 24
accommodate different types of detectors with varying char- SNR
acteristics.Inthenextsection,wewillassess theperformance
of the proposed fusion and tracking method. Fig. 11. Number of sub-Nyquist energy detectors need to match the
performance ofmatchedfilteroperating atNyquistrate.
VII. SIMULATION STUDY
A. AWGN Channel Figure 11, shows the number of energy detectors operating
in sub-Nyquist (F /8) rate needed to match the performance
AWGN channel is simulated with F = 8 GHz with s
s of matched filter operating at Nyquist rate (F ). At lower
T = 200 ns, N = 1 and B = 4 GHz. We set T = 1 ns, s
f s b SNRs, more number of energy detectors, (N ), are required
thus resulting sampling rate of the energy detector is 1/8-th ED
to meet the matched filter performance. The number of en-
the Nyquist rate. Consider a linear variation of TOA (τ )
toa ergy detectors, N , reduces with the increase of SNR and
observed by a set of four independent energy detectors (refer ED
asymptoticallyapproachesto4,confirmingwiththeanalytical
Fig. 1). The observedTOA of the fourdetectorsare asshown
derivation of the previous section.
in Fig. 6a at SNR = 12 dB. The tracked TOA using the
KalmanformulationisshownintheFig.6b.Weconsiderprior
state, covariance of prior state and plant noise variance equal B. Multipath Channel
to [20,1], 0.01I and σ2 =0.0001 respectively. IEEE 801.15.4a CM1 channel model is simulated with
2 p
The mean square error (MSE) variation with time for the F = 8 GHz with T = 200 ns, N = 1 and B = 4 GHz.
s f s
detection schemes with multiple energy detectors at various We set T = 1 ns, thus resulting sampling rate of the
b
SNRs (E /N ) are illustrated in the Fig. 9. Notice that the energy detector is 1/8-th the Nyquist rate. We use WMESS
b 0
steady state variance decreases as the number of energy algorithm discussed in the previous section to arrive at TOA
detector estimates increases. Also, fromFig. 9, at high SNRs, estimates from the energysamples. We employ similar fusion
onlyfewenergydetectors(operatingat1/8-thrateofmatched and tracking strategy using the Kalman filter described in the
filter) are sufficient to achieve same performance as digital previous section for the dynamic TOA model. The prior state
matched filter. information, covariance of prior state and other initialization
Figure 10, shows the variation of steady state MSE with parameters of the Kalman filter are kept same as in the
number of energy detectors (operating at F /8). Notice that previous section. Figure 12, shows the variation of MSE
s
steady state MSE of fusing 4 independent energy detector with time at SNR of 20 dB for different number of energy
estimates reaches that of digital matched filter estimate. detectors.As expectedthe steady state variancedecreasewith
8
0.45
NED=1 0.18
0.4 NED=4 NED=1
NED=7 0.16 NED=4
0.35 NED=10 NED=7
MachedFilteratNyquistRate 0.14 NED=10
MachedFilteratNyquistRate
0.3
0.12
2[ns]0.25 2ns] 0.1
E [
MS 0.2 SE0.08
M
0.15 0.06
0.1 0.04
0.05 0.02
0 0
0 20 40 60 80 100 0 20 40 60 80 100
t[s] t[s]
(a)MSEversestime(SNR=Eb/N0=2dB) (b)MSEversestime(SNR=Eb/N0=6dB)
0.012
0.08
00..0067 MNNNNaEEEEcDDDDhe====d1471F0ilteratNyquistRate 0.01 MNNNNaEEEEcDDDDhe====d1471F0ilteratNyquistRate
0.008
2E[ns]00..0045 2E[ns]0.006
MS MS
0.03
0.004
0.02
0.002
0.01
0 0
0 20 40 60 80 100 0 20 40 60 80 100
t[s] t[s]
(c)MSEversestime(SNR=Eb/N0=10dB) (d)MSEversestime(SNR=Eb/N0=20dB)
Fig.9. Variation ofMSEwithtimefordigital matched filterandfusionofmultiple sub-Nyquistenergydetectors usingKalmanfilteratdifferent SNRs
the increase in the number of detectors. Figure 13, shows the
5
variation of steady state MSE with number of sub-Nyquist
energy detectors, as discussed earlier a single energy detector 4.5 NED=1
NED=4
withWMESSselectioncriteriaoutperformsthematchedfilter 4 NED=7
NED=10
at low SNRs, however, at high SNRs more energy detectors 3.5 MachedFilteratNyquistRate
are need to match the matched filter performance. Figure 2s] 3
n
13, also illustrates that fusing two energy detector estimates E[2.5
S
(NED = 2) can match the matched filer performance across M 2
the SNR region. 1.5
1
VIII. CONCLUSION AND DISCUSSION 0.5
0
Estimating TOA with good accuracy is very important for 0 20 40 60 80 100
t[s]
localization and several other applications. UWB with its
large bandwidth can yield high precision ranging as evident
Fig.12. VariationofMSEforIEEE802.14.4aCM1channelmodel,withtime
from CRLB Equation (10). For AWGN channel, matched for digital matched filter and fusion of multiple energy detectors (WMESS
filter based estimation yields optimum performance at high criteria isemployed).
SNRs.However,matchedfilterrequiresNyquist-ratesampling
and is inefficient in terms of cost and power consumption.
Energydetectorsoperatingatsub-Nyquistrateisaninteresting channel, we can achieve the performance of a matched filter,
alternative as it can be designed using cost effective analog by using a multiple sub-Nyquist energy detector estimates.
circuitsandispowerefficient,however,itlackstheprecisionin We showed that number of energy detectors needed to meet
rangemeasurements.Inthispaper,weshowedthatforAWGN the matched filters performance is high at low SNRs and
9
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