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4656 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008
Robust Ultra-Wideband Signal Acquisition
¸
Ersen Ekrem, Student Member, IEEE, Mutlu Koca, Member, IEEE, and Hakan Delic, Senior Member, IEEE
Abstract—Ultra-wideband (UWB) communication is envisaged to be more challenging than that of the traditional spread
to be deployed in indoor environments, where the noise distribu- spectrum systems are the low duty cycle with very short
tion is decidedly non-Gaussian. A critical challenge for impulse duration pulses and the low-power constraint. The former is
radio UWB is the synchronization of nanosecond-long pulses. In
this paper, we propose two acquisition schemes which are robust necessary to spread the UWB signals and to allow for more
to uncertainties in the noise distribution and which operate efficient multiple-accessing. The latter is due to the rules
directly on samples taken at symbol-rate. The only channel and regulations imposed by the U.S. Federal Communications
state information that the algorithms need can be obtained by Commission [5], which aim at preventing UWB signals from
estimation of an aggregate channel gain, avoiding channel-related interfering with legacy systems.
complications. Following Huber’s M -estimates for the Gaussian
mixture noise model, the proposed robust acquisition systems The low duty cycle with very short pulses makes detection-
outperform their traditional counterparts designed according based UWB synchronization algorithms difficult to implement.
to the Gaussian noise assumption. Necessary modifications for To allow multiple access and to remove the spectral lines
operation under multiple access interference are also introduced. resulting from the pulse repetition pattern present in the
Theoretical and simulation-based performance evaluations that
reflect the asymptotic variance, normalized mean-square error transmitted UWB signal [4], long spreading codes are used
and the bit error rates demonstrate the gains offered by the causing a large search space. Unlike narrowband systems,
robust procedures. synchronization via linear search techniques with phase-locked
Index Terms—Ultra-wideband, impulse radio, timing acquisi- loops or early-late gate synchronizers is not viable. In [6]
tion, non-Gaussian noise, robust detection. and [7], alternative techniques such as bit reversal search
and look-and-jump by K bins are employed, but their mean
acquisition times are high.
I. I NTRODUCTION
The low-power constraint makes sufficient energy capture
I MPULSE radio ultra-wideband (IR-UWB) communication
can be characterized by the huge transmission bandwidth
that is obtained through the use of pulses on the order of
in the samples harder for an estimation-based approach. To
that end, template selection for optimal detection plays a key
role at the receiver [8]. The optimal template is the chan-
a few nanoseconds. The UWB technology brings out new nel output waveform without noise, which can be generated
potentials for low-power, bandwidth-demanding wireless ap- with the knowledge of the channel impulse response at the
plications such as the design of low-cost and low-complexity receiver. In addition to the unavailability of the channel state
baseband receivers with high user capacity. Despite these information (CSI) during synchronization, it seems almost
advantages, UWB also introduces unique signal processing impossible to acquire the impulse response of a UWB channel
challenges, especially for timing synchronization and channel because the number of paths for 85% energy capture varies
estimation [1]. Although there exist blind approaches aiming typically between approximately 20 to 120 [9]. Moreover, the
to bypass explicit channel estimation and synchronization channel statistics may not be reliable enough to obtain useful
tasks for simpler receiver structures, inherent performance loss information which disallows making prior assumptions.
in such schemes is unavoidable [2]. In particular, because even
The impact of low duty cycle and low power constraints
minor misalignments may result in the lack of sufficient energy
on synchronization performance in UWB systems has been
capture rendering symbol detection nearly impossible [3], low-
investigated in [6], [8], [10]-[13], with the assumption that the
complexity timing synchronization mechanisms are needed.
ambient noise is additive white Gaussian (AWGN). However,
The general solution to the synchronization of UWB signals
as reported in [14], the indoor environments, where the UWB
is to divide the task into two parts: timing acquisition and
devices are envisioned to be deployed, are subject to interfer-
fine synchronization. The first stage aims at reducing the
ence produced by photocopiers, printers, elevators, etc. Such
timing uncertainty to the order of a pulse duration while the
interference exhibits bursts of high-amplitude emissions whose
second makes a fine tuning within pulse duration accuracy
rate of occurrence depends on the device usage frequency.
[4]. Two major issues causing the UWB synchronization
Despite the processing gain offered by UWB systems, it is
Manuscript received July 4, 2007; revised November 17, 2007 and March shown in [15] that even a tone interference with 16 MHz
7, 2008; accepted April 25, 2008. The associate editor coordinating the review bandwidth can have detrimental effects on the performance
of this paper and approving it for publication was N. Arumugam. This work of transmitted-reference (TR) systems. A proper probabilistic
was supported by TUB˙¨ ITAK under contracts 105E034 and 105E077. Parts of
this paper were presented in IEEE WCNC, Hong Kong, March 2007 and in model that accounts for man-made interference, as well as
IEEE VTC, Dublin, Ireland, April 2007. ambient and thermal noise simultaneously has to be of non-
g ¸
E. Ekrem was with Boˇ azic i University, Bebek 34342 Istanbul, Turkey. Gaussian form, characterized by heavy tails that represent
He is now with University of Maryland, College Park, MD 20742, USA (e-
mail: ersen@umd.edu). the impulsive outliers, and it will hereafter be referred to as
M. Koca and H. Delic are with Wireless Communications Laboratory,
¸ impulsive noise.
Boˇ azic i University, Bebek 34342 Istanbul, Turkey (e-mail: {mutlu.koca,
g ¸
delic}@boun.edu.tr). Conventional synchronization methods suffer from a per-
Digital Object Identifier 10.1109/T-WC.2008.070730. formance drop in case of non-Gaussian disturbance just like
1536-1276/08$25.00 c 2008 IEEE
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EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4657
the detectors designed solely to counteract Gaussian noise an asymptotic performance analysis. Section VI extends the
as demonstrated in [16]. In this paper, we focus on the robust acquisition procedure to multiuser environments. Sec-
synchronization of the signal within frame-level accuracy, tion VII presents the performance evaluation via simulations,
referred to as timing acquisition, for which estimation-based and Section VIII makes some concluding remarks.
approaches seem to be the strongest candidates. Taking into
account the impulsive noise phenomenon, a robust UWB II. S IGNAL AND N OISE M ODEL
acquisition system is proposed to combat occasional devia-
In an IR-UWB communication system, every symbol is
tions in the noise probability density function (PDF) from
transmitted over a duration of Ts in which Nf frames, each
the nominal Gaussian assumption. The main approach is to
with a duration of Tf , are sent, i.e., Ts = Nf Tf . In each frame,
combine minimax design for non-Gaussian noise with the
a pulse p(t) with a duration of Tp is transmitted. Typically, the
TR methodology. In TR systems, each symbol consists of a
pulse duration is much shorter than the frame duration, i.e.,
modulated part (data) and an unmodulated part that serves as
Tp Tf , which leads to the low-duty cycle characteristic of
a template for the demodulation of the former. Because half
UWB. For multiple access and channel separation, the pulse is
of the signal is reserved for the unmodulated non-data part
hopped by an amount of cu (n)Tc where cu (n) is the TH code
and also because noisy signals are used as the template, the
corresponding to uth user’s transmitted pulse in the nth frame
associated bit error rate (BER) performance is poor [17]. On
of the symbol and Tc is the duration of the bins to which
the other hand, such a paradigm suits the timing task well as
the pulses are allowed to hop. The allowable range for TH
stated in [10] because of the fact that it greatly simplifies the
codes is [0, Nc ) where Nc = Tf /Tc . We adopt binary pulse
channel estimation task. Here, as in [10], instead of allocating
amplitude modulation (PAM) for the acquisition procedure,
half of the transmitted symbols to the templates (unmodulated
and the transmitted signal for uth user is given by
signal), we use short training sequences which force the
∞
received samples to be related with the delay parameters.
Then, these delay parameters are determined through the M - su (t) = bu (n)pu,s (t − nTs ) (1)
n=0
estimation procedure, which is a minimax design when the
uncertainty about the system stems from the PDFs. In M - where bu (n) = ±1 denotes the PAM symbols and pu,s (t) is
estimation, the worst case performance of the system at the the transmitted symbol waveform of the uth user given by
neighborhood of the PDF is found, and at that point the overall Nf −1
performance of the system is optimized [18]. √
pu,s (t) = E u,s p t − nTf − cu (n)Tc (2)
In this paper, two data-aided robust acquisition algorithms n=0
using the robust estimation techniques mentioned above are
where Eu,s is the energy of the transmitted symbol of the uth
proposed. The first method uses the received symbol waveform
user.
as the template while the second employs an averaged version
The quasi-static channel with Lu resolvable paths observed
of the received symbols so that the level of noise enhancement
by user u is modeled as
which mainly occurs due to cross noise terms is decreased.
The robust signal acquisition procedures require an analog Lu
implementation of delay line and operate with samples taken hu (t) = hu,l δ(t − τu,l )
at symbol rate with/without time-hopping (TH) codes or l=1
direct-sequence (DS) spreading. Note that all the CSI needed where hu,l and τu,l denote the path gain and the time-
by the proposed receivers is the estimation of a tap gain delay of lth multipath component of the uth user’s channel
resulting from the correlation of the received signals. Respec- response, respectively. The frame duration is set to satisfy
tive performances of both methods are investigated through |τu,Lu −τu,1 |+Tp < Tf and cu (Nf −1) = 0 to avoid intersym-
asymptotic analyses and computer simulations. It is shown bol interference (ISI). Note that interframe interference (IFI)
that the averaged-template approach performs better than the is allowed by these restrictions. Although this assumption on
one employing successive sampling at the expense of added the frame length is suitable for low rate UWB systems, it can
complexity due to averaging in analog domain. As depicted be relaxed for high rate systems as long as guard frames are
by the simulation results, both robust algorithms have superior inserted between the symbols to prevent ISI [19].
performance compared to the Gaussian-optimal maximum In this paper, the estimation of the first path’s delay, τu,1 , is
likelihood solution, which results in a linear structure. Finally, considered while it is restricted to lie in one symbol duration,
the averaged-template approach is also extended to the mul- Ts . The time delay of the first path can be expressed as
tiuser environments where the multiuser interference (MUI) τu,1 = nu,f Tf + θu where nu,f = τu,1 /Tf and θu =
is first approximated with a Gaussian-distribution and then τu,1 − nu,f Tf represent the time delay at frame-level and
modeled as impulsive noise. It is illustrated via simulations pulse-level, respectively. The delay of the other paths can be
that the performances of both robust multiuser detectors are written as τu,l = τu,l,1 + τu,1 for l = 1, . . . , Lu where τu,l,1
similar, or in other words, that taking the impulsiveness of is the relative time delay of the lth path with respect to the
MUI into account does not provide any additional robustness. first one. The goal is to estimate nu,f from the received signal
This paper is organized as follows: Section II outlines which can be written as
the signal and noise model. Section III presents some pre- Nu ∞
liminaries about robust estimation. Section IV describes the r(t) = bu (n)pu,r (t − nTs − τu,1 ) + w(t) (3)
proposed robust acquisition procedures. Section V carries out u=1 n=0
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4658 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008
Lu
where pu,r (t) = l=1 hu,l pu,s (t − τu,l,1 ) is the received ML estimate with respect to this least favorable PDF such
symbol waveform and Nu denotes number of users. In (3), that it can be obtained by minimizing a cost function of the
w(t) is the impulsive noise modeled as a two-term Gaus- form, ρ(x) = − log fLF (x), which turns out to be
sian mixture, which is more appropriate for UWB systems ⎧ x2
whose main application will be in indoor environments as ⎨ 2σ2 for |x| ≤ kσ 2 ,
demonstrated in [14]. The two-term Gaussian mixture is ρ(x) = (8)
⎩ k2 σ 2 2
an approximation to the more general Middleton’s Class A 2 − k|x| for |x| > kσ
noise model, which consists of an infinite sum of Gaussian where k is found by solving (7). Moreover, it can be
PDFs with identical means and increasing variances [20], and shown that the M -estimates are asymptotically consistent
it adequately describes the Class A model when impulsive and normal-distributed. The asymptotic variance of the robust
interference constitutes about 0.1-1% of the total disturbance mean estimator using the PDF in (6) is given by [18]
(noise plus interference) [21]. The two-term Gaussian mixture
E ψ 2 (x) ψ 2 (x)fLF (x)dx
noise can be expressed as ν2 = = (9)
2 2
2 2 (E [ψ (x)]) ψ (x)fLF (x)dx
f (x) = (1 − )g(x; 0, σ ) + g(x; 0, κσ ) (4)
where ν 2 denotes the asymptotic variance and ψ (x) is the
where g(x; μ, σ 2 ) is the Gaussian PDF with mean μ and
first derivative of ψ(x), which itself is the derivative of the
variance σ 2 at point x. In the mixture, the first term accounts
cost function ρ(x).
for the nominal noise with higher prior probability, while the
second term represents the impulsive interference components
IV. ROBUST T IMING ACQUISITION FOR UWB
with its heavier tails. Here, κ is the impulsive part’s relative
variance with respect to the nominal noise variance, and is Templates play a critical role because for a typical UWB
the relative frequency of the outliers. system, the multipath effect combined with low-power trans-
Notational Convention: The subscript u, which denotes the mission make it difficult to capture an estimate of the transmit-
number of user is dropped for notational simplicity (e.g. b(m) ted symbol with sufficiently high energy. The possible lack of
denotes the information bit of any user in a single user link) prior information about the channel impulse response calls for
until needed in Section V, where a multiuser environment is alternative approaches for template selection. In [8] and [10],
considered. the received symbols and an averaged form of the received
symbols are proposed as templates for transmission without
III. P RELIMINARIES and with TH, respectively. Both approaches originate from
the TR methodology [17], where for each data symbol, an
In this section, we briefly outline the robust estimation
unmodulated reference symbol is sent for demodulation.
techniques, namely M -estimates, which are used throughout
the paper. Robust estimation procedures are generally designed
in a minimax sense such as the M -estimates by Huber [18], A. Successive Sampling
which was originally developed for location parameter estima- Here, instead of allocating half of the symbol duration to a
tion. In Huber’s solution, the maximal asymptotic variance of reference signal, successive symbols are used as the template
the estimator is minimized by finding the least favorable PDF with the aid of a training sequence as in [8], [10]. When
within a certain family of the distributions, which turns out to ISI is absent, each symbol-length waveform at the receiver
possess the minimum Fisher information. This minimax pro- consists of two successively transmitted symbols [10]. By
cedure is completed by determining the maximum-likelihood using each received symbol-length waveform as a template
(ML) estimator of the least favorable PDF. for the next one, the mean of the samples is directly governed
For the so-called -contaminated Gaussian mixture model by the frame-level delay, and thus, the estimate of the mean
in (4), the PDF family can be described by the set of the training sequence can be employed as a tool for timing
acquisition.
F = {(1 − )g(x; 0, σ 2 ) + v(x); v(x) is a symmetric PDF},
The block diagram of the successive sampling-based ac-
(5)
quisition unit is given in Fig. 1. As a first step, the received
for which the least favorable PDF is the one that obeys
symbol-length waveform is correlated with the previous one,
the Gaussian distribution at the center and then decays with
yielding
exponential tails such that
⎧ Ts
⎪ √
⎪ 1− exp − 2σ2 x2
for |x| ≤ kσ 2 , rSUC (m) = r (t + mTs ) r t + (m + 1)Ts dt,
⎨ 2πσ 0
fLF (x) =
⎪ 1−
⎪ √ which results in
⎩ 2 2
exp k 2 − k|x|
σ
for |x| > kσ 2 ,
2πσ
(6) ˜
rSUC (m) = b(m)Cg b(m − 1)˜ f + b(m + 1)(Nf − nf )
n
where k, σ and are related through + ωSUC (m) (10)
φ(kσ) T 2
L
− Q(kσ) = (7) where Cg = t=0 f
l=1 hu,l p(t − τu,l )
u
dt is the energy of
kσ 2(1 − )
√ ˜
the received frame and nf = nf + χ with χ being the random
and√ φ(u) = (1/ 2π) exp(−u2 /2), Q(u) = uncertainty induced by the TH codes and/or the pulse-level
∞
(1/ 2π) u exp(−t2 /2)dt [18]. The M -estimate is then the ˜
uncertainty [8]. Here, χ ∈ [0, 1) so that the integer floor of nf
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EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4659
r(t) (k+1)Ts Robust Channel ˆ
Cg Robust Frame ˆ
nf
dt
kTs
Gain Estimator Offset Estimator
Delay Ts
Fig. 1. Block diagram of the successive sampling based robust acquisition system.
is equal to nf , the frame-offset time delay. In (10), ωSUC (m) (6) and (8), respectively, with k now satisfying
is the noise sample which consists of three parts, one noise-
1 − (1 − )3 φ kσSUC
noise term and two signal-noise terms. The overall noise PDF = − Q kσSUC .
is given by 2(1 − )3 kσSUC
The estimates for the means of the two parts of the training
fSUC (x) = (1 − )3 g x; 0, σ 2 (2Er + 4Ts W σ 2 ) sequence can be found as
+ 2 (1 − )2 g x; 0, σ 2 ((1 + κ)Er + 4Ts W σ 2 ) ˆ
Cg,SUC =
+ (1 − )2 g x; 0, σ 2 (2Er + 4κ2 Ts W σ 2 ) Nt /2−1
2 2 2 2 arg min ρSUC rSUC (m) − b(m)b(m + 1)Nf Cg,SUC ,
+ 2 (1 − )g x; 0, σ ((1 + κ)Er + 4κ Ts W σ ) Cg,SUC
2 m=0
+ (1 − )g x; 0, σ 2 (2κEr + 4Ts W σ 2 ) (12)
3 2 2 2 ˆ
μSUC =
+ g x; 0, σ (2κEr + 4κ Ts W σ ) (11)
Nt
where Er = Nf Cg is the energy of received symbol and W is arg min ˆ
ρSUC rSUC (m) − b(m)b(m + 1)Cg,SUC μSUC
µSUC
the bandwidth of the front-end-filter at the receiver, which is m=Nt /2
typically given by W ≈ 1/Tp . The derivation of (11) is given (13)
in Appendix A. ˆ
where μSUC and μSUC are defined by μSUC (Nf −
˜
The primary goal is to estimate nf whose integer floor is 2˜ f ) and μSUC
n ˆ (Nf − 2˜ f ). The closed form solutions
n
ˆ
the estimate of the frame-offset, nf,SUC . Note that to these equations do not exist, but they can be computed
1) if b(m − 1) = b(m + 1), (10) reduces to rSUC (m) = iteratively via the modified residuals method [18]. Assuming
b(m)b(m + 1)Nf Cg + ωSUC (m), from which Cg can be ˆ ˆ ˆ ˆ
that Cg,SUC and μSUC are the estimates of Cg and μSUC at
estimated; and the th step, respectively, recursive solutions of (12) and (13)
2) if b(m − 1) = −b(m + 1), (10) reduces to rSUC (m) = are given by
b(m)b(m + 1)Cg (Nf − 2˜ f ) + ωSUC (m), from which
n Nt /2−1
˜
nf can be estimated. ˆ +1 ˆ
Cg,SUC = Cg,SUC + α b(m)b(m + 1)Nf
Therefore, a training sequence of length Nt whose first m=0
Nt /2 elements are identically +1 and the remaining half is ˆ
× ψSUC rSUC (m) − b(m)b(m + 1)Nf Cg,SUC (14)
{+1, +1, −1, −1, . . . , +1, +1, −1, −1}, makes it possible to
obtain the necessary equations for the estimation of Cg and Nt
ˆ +1
μSUC = μSUC + α
ˆ ˆ
b(m)b(m + 1)Cg,SUC
˜
nf . The second half of this preamble structure is also used
in [10], where timing-estimation is basically search-oriented m=Nt /2
relying on peak-picking, and it is designed for Gaussian noise. ˆ ˆ
× ψSUC rSUC (m) − b(m)b(m + 1)Cg,ss μSUC
Huber’s robust solution corresponds to a situation where (15)
the nominal PDF is Gaussian and in the neighborhood of the
nominal PDF, there exist symmetric contaminations, as in (5). where α is the step-size, taken about 0.1, and ψSUC (x) =
If we consider the noise PDF in (11), there is a dominating ρSUC (x) is the first derivative of ρSUC (x). All estimates are
Gaussian term with a priori probability of (1 − )3 and five set to zero initially, and convergence is typically established
more Gaussians with lower prior probabilities. Because the in less than 10 steps. Convergence is ensured so long as
contaminations are symmetric and the dominating PDF is |ψSUC (x)| < 1/α [22]. Once the final channel gain estimate
Gaussian, Huber’s solution to -contaminated Gaussian models ˆ
Cg,SUC is obtained through the recursions in (14), it is
can be adopted here, as well. To that end, one needs to employed in (15) for the estimation of μSUC whose final value
define the nominal situation and the corresponding PDF. We ˆ
at the end of the recursions is denoted by μSUC . Then, nf ˜
define the nominal case as the one in which both of the is determined, whose integer floor gives the estimate of the
two successively received symbols face just thermal noise, frame-offset as
and the contamination consists of all terms where either one ˆ
Nf − μSUC
or both of the received successive symbols contain outliers. ˆ ˆ
˜
nf,SUC = nf,SUC =
2
Therefore, the least favorable PDF and the cost function ρSUC
for successive sampling can be determined by replacing σ 2 ˆ
˜ ˜ ˆ
where nf,SUC denotes the value of nf calculated using μSUC
2
with σSUC σ 2 (2Er + 4Ts W σ 2 ) and with 1 − (1 − )3 in ˆ
and nf,SUC represent the estimate of the frame-offset obtained
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4660 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008
r(t) (k+1)Ts Robust Channel ˆ
Cg Robust Frame ˆ
nf
dt
kTs
Gain Estimator Offset Estimator
Template Estimator ¯
r(t)
(Averager)
Fig. 2. Block diagram of the averaged template based robust acquisition system.
via successive sampling. In the next subsection, an alternative which can be derived in a similar way as (11), whose
technique decreasing the degree of noise enhancement that derivation is in Appendix A.
occurs due to using noisy templates is presented. The samples in (18) differ from those in (10) and a
˜
new training sequence should be designed for estimating nf .
B. Averaged Template Hence, we observe that
In TR systems, or in autocorrelation-based receivers, one
significant issue is the double-noise (noise-noise) terms which 1) if b(m) = b(m − 1), (18) reduces to rAVG (m) =
result in noise emphasis as in our case. It is possible to lower b(m)Cg Nf + ωAVG (m) from which Cg can be esti-
the noise variance of the samples by employing a template mated;
that is obtained by averaging over the training sequence. The 2) if b(m) = −b(m − 1), (18) reduces to rAVG (m) =
cost of performance improvement is a complexity increase ˜
Cg b(m)(Nf − 2˜ f ) + ωAVG (m) from which nf can be
n
that results from averaging in the analog domain. The block estimated.
diagram of the corresponding acquisition system is given
in Fig. 2. Suppose that during the first half of the training Therefore, a training sequence whose first half consists of
sequence, the received symbols are averaged as only +1’s and the remaining half having a structure of
Nt /2−1
{+1, −1, +1, −1, . . . , −1, +1} enables us to estimate Cg and
2 ˜
nf separately. In addition, the template can be obtained by
¯
r (t) = ¯
r(t+iTs ) = pr (t+Ts −τ1 )+pr (t−τ1 )+w(t)
Nt averaging the received symbols of the first half of the training
i=0
(16) sequence. It should be noted that this preamble structure is
¯
where w(t) is the noise averaged over Nt /2 symbol-length same as the one proposed in [8] for timing-estimation under
waveforms whose PDF is given by Gaussian noise.
2σ 2 2κσ 2 For the implementation of the robust estimation procedure,
fw (x) = (1− )g x; 0,
¯ +(1− )g x; 0, . (17) the nominal situation is assumed to be acting when neither the
Nt Nt
averaged template nor the sampled symbol contains outliers.
Through averaging, the noise variance of the template is
Thus, the term with the a priori probability of (1 − )3 is the
reduced by a factor of Nt /2 according to (17), which is crucial
nominal PDF, and the rest represent the contamination with
for suppressing the double-noise terms. Using the template in
impulsive components. In this case, the least favorable PDF
(16), the samples are obtained as
and the cost function ρAVG can be found by replacing σ 2 with
2
σAVG σ 2 (Er + 4Ts W σ 2 /Nt ) and with 1 − (1 − )3 in (6)
Ts
rAVG (m) = r(t + mTs )¯(t)dt, m = 0, 1, . . . , Nt − 1,
r
0 and (8), respectively, with k being the solution to
which results in [8]
rAVG (m) = Cg b(m − 1)˜ f + b(m)(Nf − nf ) + ωAVG (m)
n ˜ 1 − (1 − )3 φ (kσAVG )
= − Q (kσAVG ) .
(18) 2(1 − )3 kσAVG
where ωAVG (m) is the sequence of noise samples having a
PDF of
σ2 The estimated values for the means of the two parts of the
fAVG (x) = (1 − )3 g x; 0, σ 2 Er + 4Ts W
Nt training sequence are given by
σ2
+ (1 − )2 g x; 0, σ 2 Er + 4κ2 Ts W
Nt
κ σ2 Nt /2−1
+ (1 − )2 g x; 0, σ 2 2Er + 4Ts W ˆ
Cg,AVG = arg min ρAVG r(m) − Nf Cg ,
Nt Nt Cg
σ2
m=0
2 2 κ 2
+ (1 − )g x; 0, σ 2Er + 4κ Ts W Nt −1
Nt Nt ˆ
μAVG = arg min ˆ
ρAVG r(m) − b(m)Cg,AVGμAVG
2 σ2 µAVG
m=Nt /2
+ (1 − )g x; 0, σ κEr + 4Ts W
Nt
2 σ2
+ g x; 0, σ 2 κEr + 4κ2 Ts W (19)
Nt ˆ
where μAVG = (Nf − 2˜ f ) and μAVG
n (Nf − 2˜ f ). The
n
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EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4661
2 1
10 10
1
10 0
10
0
10
−1
10
)
log (ν2 )
SUC
AVG
−1
log (ν2
10
−2
10
−2
10
Linear−SUC (ε = 0.1) −3 Linear−AVG (ε = 0.1)
−3 10
10 Linear−SUC (ε = 0.01) Linear−AVG (ε = 0.01)
Robust−SUC (ε = 0.1) Robust−AVG (ε = 0.1)
Robust−SUC (ε = 0.01) Robust−AVG (ε = 0.01)
−4 −4
10 10
2 3 2 3
10 10 10 10
log (κ) log (κ)
Fig. 3. Asymptotic variance of successive sampling-based acquisition when Fig. 4. Asymptotic variance of averaged template-based acquisition when
is fixed and ((1 − ) + κ)σ2 = (0.1)2 . is fixed and ((1 − ) + κ)σ2 = (0.1)2 .
modified residuals method in [18] yields the iterations where N (0, σ 2 ) represents a Gaussian random variable with
Nt /2−1 zero mean and variance σ 2 . Moreover, because M -estimates
ˆ +1 ˆ
Cg,AVG = Cg,AVG + α r(m)Nf can be viewed as a generalization of the ML-estimates, the
m=0
asymptotic variance of the linear estimator designed to be
ˆ optimal under AWGN but operating in impulsive noise can
× ψAVG r(m) − b(m)Nf Cg,AVG , (20)
be calculated via (9) by setting ψ(x) = x. In this case,
Nt −1
ˆ the corresponding linear receivers use the quadratic penalty
ˆ +1 ˆ
μAVG = μAVG + α b(m)Cg,AVG
function ρ(x) = x2 . Because the noise PDFs in (11) and (19)
m=Nt /2
are complicated, the asymptotic variances of the estimators,
ˆ
× ψAVG r(m) − b(m)Cg,AVG μAVG
ˆ (21) 2 2
νSUC and νAVG , are calculated numerically and plotted in
ˆ +1 Figures 3 and 4. It is seen in these two figures that when and
ˆ +1
where ψAVG (x) is the derivative of ρAVG and Cg,AVG , μAVG
the total noise variance ((1− )+ κ)σ 2 are fixed, the variances
denote the estimates of Cg,AVG and μAVG at the th step,
of the robust estimators are monotonically decreasing in κ
respectively. As with successive sampling, α is the step size
whereas those of the linear estimators, i.e., ML-optimal ones
around 0.1, and all initial estimates are set to zero. The
under AWGN, are almost invariant to changes in κ. This was
recursive use of (20) produces the final channel gain estimate,
ˆ also observed and stated in [22] for robust multiuser detectors
Cg,AVG , which is then employed in (21) for determining
in non-Gaussian channels.
μAVG , whose final value at the end of the recursions will be 2 2
Assuming that νSUC and νAVG are available, the probability
ˆ ˜ ˆ
denoted by μAVG . Finally, nf,AVG is calculated using μAVG ,
mass functions (PMFs) of the robust estimators are derived
and its integer floor gives the estimate of the frame-offset as
next to calculate normalized mean square error (NMSE). First
ˆ ˆ
Nf − μAVG the conditional probability of frame offset’s estimate obtained
nf,AVG = nf,AVG =
ˆ ˜
2 via successive sampling is considered, which is given by
ˆ ˆ
Nf − μSUC
˜ ˜
where nf,AVG denotes the value of nf calculated using ˜
Pr{ˆ f,SUC = x | nf } = Pr x ≤
n < x + 1 nf
˜
2
ˆ ˆ
μAVG and nf,AVG represent the estimate of the frame-offset ⎛ ⎞ ⎛ ⎞
obtained via averaged template. In the next section, asymptotic Nf − 2x − 2 − μSUC ⎠ Nf − 2x − μSUC ⎠
= Q⎝ − Q⎝ .
performance analysis of these estimators is carried out. νSUC Cg Nt /2 νSUC Cg Nt /2
(22)
V. A SYMPTOTIC P ERFORMANCE A NALYSIS
Using the fact that μSUC = Nf − 2˜ f , (22) can be expressed
n
Since the M -estimates are asymptotically consistent and as
normal-distributed, denoting the asymptotic variances for the
2 2
successive sampling and averaged template as νSUC and νAVG , Pr{ˆ f,SUC = x | nf } =
n ˜
respectively, the estimation errors obey a normal PDF such ⎛ ⎞ ⎛ ⎞
that 2˜ f − 2x − 2
n 2˜ f − 2x
n
Q⎝ ⎠−Q⎝ ⎠,
Nt νSUC Cg Nt /2 νSUC Cg Nt /2
2
ˆ
Cg (μSUC − μSUC ) ∼ N (0, νSUC ),
2 (23)
Nt
ˆ 2
Cg (μAVG − μAVG ) ∼ N (0, νAVG ) ˜ ˜
where nf can be decomposed as nf = nf + ϑ with ϑ being
2 the random time-uncertainty induced by the TH codes and/or
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4662 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008
pulse-level uncertainty, θ, which is assumed to be uniformly sequence is adopted for the multiuser environment because
distributed in [0, 1). To alleviate the dependency of (23) on ϑ, of its superior performance as attested by the asymptotic
expectation of (23) should be taken over ϑ resulting in variance comparison in Section IV. In the presence of MUI,
1 the averaged template becomes
Pr{ˆ f,SUC = x | nf } =
n Pr(ˆ f = x | nf , ϑ)dϑ
n Nt /2−1 Nu Nt /2−1
⎛ 0 ⎞ 2 2
¯
r (t) = r(t + iTs ) = ¯
ru (t) + w(t)
i
2nf − 2x Nt Nt
= 2(nf − x)Q ⎝ ⎠ i=0 u=1 i=0
νSUC Cg Nt /2 ¯
where w(t) is the averaged noise whose PDF is given in (17)
⎛ ⎞ and ru (t) is given by
i
2nf − 2x − 2
− (nf − x − 1)Q ⎝ ⎠ ru (t) = bu (i − 1)pu,r (t + Ts − τ1 ) + bu (i)pu,r (t − τ1 ). (26)
i u u
νSUC Cg Nt /2
⎛ ⎞ The received samples are formed via
2nf − 2x + 2
− (nf − x + 1)Q ⎝ ⎠ Ts
r(m) = r(t + mTs )¯(t)dt
r
νSUC Cg Nt /2 0
2 which is explicitly given by
νSUC nf − x − 1
+ √ exp −Nt Cg
2Cg πNt νSUC r(m) = Cd,g bd (m − 1)˜ d,f + bd (m)(Nf − nd,f )
n ˜
2
nf − x + 1 + ωMUI (m) + ωMA (m)
+ exp −Nt Cg
νSUC where the subscript d denotes the desired user, ωMUI (m) and
nf − x
2 ωMA (m) denote the MUI and noise term, respectively. The
− 2 exp −Nt Cg (24) PDF of ωMA (m) is given by
νSUC
σ2
which is the PMF of the frame-offset estimate conditioned on fMA (z) = (1 − )3 g z; 0, σ 2 Ed,r + 4Ts W
ˆ
the actual value of nf . Therefore, the NMSE of nf,SUC can Nt
be calculated as σ2
+ (1 − )2 g 0, σ 2 Ed,r + 4κ2 Ts W
2 Nt
ˆ
nf,SUC − nf
NMSE(ˆ f,SUC )
n E ET,r σ2
Nf + (1 − )2 g z; 0, σ 2 2κ + 4Ts W
Nt Nt
Nf −1
1 σ2
= 2 Pr{nf = x}E (ˆ f,SUC − nf ) | nf = x
n
2 + 2 g z; 0, σ 2 κEd,r + 4κ2 Ts W
Nf Nt
x=0
(25) σ2
+ (1 − )g z; 0, σ 2 κEd,r + 4Ts W
Nt
where E (ˆ f,SUC − nf )2 | nf = x is the conditional mean
n ET,r σ2
square error. Assuming nf to be uniform over [0, Nf − 1], + 2 (1 − )g z; 0, σ 2 2κ + 4κ2 Ts W
Nt Nt
(25) can be expressed as (27)
NMSE(ˆ f,SUC ) =
n N
where ET,r = u=1 Eu,r is the sum of the received energies
u
Ts
1
Nf −1 ∞
2
of all users and Eu,r = t=0 p2 (t)dt. The derivation of (27)
u,r
3 Pr{ˆ f,SUC = y | nf = x} (y − x)
n is given in Appendix B. Now, we turn our attention to the
Nf x=0 y=−∞ MUI term whose PDF is
where Pr{ˆ f,SUC = y | nf = x} is given in (24). Because
n (Nu −1)Nt /2
(Nu − 1)Nt
the NMSE of the frame-offset estimate obtained via averaged fMUI (z) = 2−(Nu −1)Nt /2 C ,l
2
template can be calculated with the same procedure described, l=0
its derivation is omitted. 2((Nu − 1)Nt − 2l)(Nf Cg )2
× g z; 0, , (28)
3Nt2
VI. M ULTIUSER E NVIRONMENT as shown in Appendix C, where C(n, l) denotes the number
So far, a point-to-point link has been considered with the of l-groupings of n. As noted in [23], [24], the MUI in UWB
assumption that user separation is accomplished via channel- systems is of non-Gaussian nature. Moreover, in [24], it is
ization [10]. In this section, the performances of the proposed shown that the MUI term for TH pulse position modulation
algorithms are explored when asynchronous users which are (PPM) UWB systems follows Middleton’s Class A noise
assumed to transmit independent and identically distributed model. Equation (28) indicates that the MUI in the signal
(i.i.d.) data symbols interfere with the desired user. The goal acquisition paradigm is also non-Gaussian. Note that in [23],
is to achieve timing acquisition with only one user in the [24], the origin of MUI is the collision of independent user’s
presence of both impulsive noise and multiuser interference. pulses whereas in acquisition, the correlation of same user’s
The averaged template method with the associated training symbol-length waveforms contribute to the MUI term, as well.
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EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4663
Therefore, as long as the collision probability of two different B. Impulsive MUI
user’s pulses is low, i.e., the frame length is long enough, the We propose a robust synchronizer which takes the impulsive
latter dominates the MUI. In the next two subsections, we character of MUI into design consideration. To that end, first
design two robust synchronizers where one uses a Gaussian reexpress the noise term in (27) as
approximation for the MUI term while the other takes the
impulsiveness into consideration. σ2
fMA (z) = (1 − )3 g z; 0, σ 2 Ed,r + 4Ts W + ζ1 (z)
Nt
A. Gaussian Approximation and the MUI term in (28) as
In this subsection, we design a robust synchronizer by (Nu −1)Nt (Nu −1)Nt
approximating the MUI term as a Gaussian random variable C 2 , 4
fMUI (z) =
with zero mean and variance 2(Nu −1)Nt /2
2 4(Nu − 1) 2 (Nu − 1)Nt (Nf Cg )2
σMUI = (Nf Cg ) . × g z; 0, + ζ2 (z)
3Nt 3Nt2
Although the accuracy of the Gaussian assumption for small where the distributions ζ1 (z) and ζ2 (z) denote the
number of users is poor, here we incorporate this approach interference-contaminated parts of noise and MUI terms, re-
to construct a comparison platform for the next robust syn- spectively. For the MUI term, the most probable Gaussian
chronizer which is built using (28) instead of a Gaussian PDF in (28) is selected as the nominal part, and all others are
assumption. Note that MUI is asymptotically Gaussian in the declared as interference-contaminated. Thus, the PDF of the
product Nf Nc which represents the processing gain, provided overall disturbance, noise plus MUI, can be expressed as
that Nf and Nc grow while the ratio Nf /Nc remains constant 2
[25]. The PDF of overall disturbances, noise plus MUI, can fDIST (z) = (1 − )g z; 0, σDIST + ζ3 (z) (30)
2
be found as fDIST,G (z) = fMA (z) ∗ g z; 0, σMUI resulting where (1 − ), the prior probability of the nominal part of
in (30), is given by
fDIST,G (z) = (Nu −1)Nt (Nu −1)Nt
C ,
3 2 σ2 2
1− = (1 − ) 3 2 4
(1 − ) g z; 0, σ Ed,r + 4Ts W + σMUI ,
Nt 2(Nu −1)Nt /2
σ2 2
and σDIST is given by
+ (1 − )2 g 0, σ 2 Ed,r + 4κ2 Ts W 2
+ σMUI
Nt
σ2 (Nu − 1)Nt (Nf Cg )2
ET,r σ2 2
σDIST = σ 2 Ed,r + 4Ts W + .
+ (1 − )2 g z; 0, σ 2 2κ + 4Ts W 2
+ σMUI Nt 3Nt2
Nt Nt
σ2 In this case, the least favorable PDF can be found by setting
+ 2 g z; 0, σ 2 κEd,r + 4κ2 Ts W 2
+ σMUI 2
σ 2 to σDIST and to . Moreover, k can be found through
Nt
σ2 φ (kσDIST )
+ (1 − )g z; 0, σ 2 κEd,r + 4Ts W 2
+ σMUI − Q (kσDIST ) = .
Nt kσDIST 2(1 − )
ET,r σ2
+ 2 (1 − )g z; 0, σ 2 2κ + 4κ2 Ts W 2
+ σMUI . Again, the first and second halves of the training sequence
Nt Nt
(29) ˜
are used to estimate Cd,g and nd,f according to (14)-(15) and
(20)-(21) for successive sampling and averaged template-based
Note that MUI introduces an additional term to the variance acquisition, respectively.
of the noise PDF given in (19), thereby resulting in (29), which
is independent of signal-to-noise ratio (SNR) defined as the
VII. S IMULATION R ESULTS
ratio of symbol energy to the variance of the background noise.
The MUI induces a noise floor which cannot be lowered by In this section, we test the performance of the robust
increasing the transmitted power. estimators. The pulse is selected as the second derivative of the
The nominal part of (29) is the first term with a pri- Gaussian function with duration Tp = 1 ns. The CM1, CM2,
ori probability (1 − )3 and all other terms are treated as CM4 channel models are implemented [9]. The frame duration
interference-contaminated. In this case, the least favorable is selected to be 40 ns. Each symbol consists of Nf = 30
PDF and the cost function can be found by setting σ 2 frames resulting in Ts = 1200 ns. The chip duration is chosen
2
to σDIST,G 2
σ 2 Ed,r + 4Ts W σ 2 /Nt + σMUI , and to as Tc = 1 ns, and the TH codes are randomly generated
3
1 − (1 − ) in (6) and (8), respectively, and with k found from a uniformly distributed set of {0, 1, . . . , Nf − 1}. The
through tracking level uncertainty in time delay, i.e. θ, is randomly
generated from a uniform distribution on [0, Tf ). The linear
φ (kσDIST,G ) 1 − (1 − )3 acquisition system that is considered as the competitor here is
− Q (kσDIST,G ) = .
kσDIST,G 2(1 − )3 the Gaussian-optimal ML solution as developed in [8]. Com-
˜
The estimation of Cd,g and nd,f can be carried out as indi- parison with the acquisition method in [10] is also presented.
cated in (14)-(15) and (20)-(21) using the first and second In all simulations, SNR Es / (1 − )σ 2 + κσ 2 , where
2 2
halves of the corresponding training sequence for successive (1 − )σ + κσ is the total noise variance at the input of the
sampling and averaged template-based robust timing acquisi- receiver and it is fixed. The NMSE is calculated through sam-
tion, respectively. ple averaging. Although nf ∈ {0, . . . , Nf − 1}, the estimated
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4664 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008
3 0
10 10
Robust−SUC (no TH)
Linear−SUC (no TH)
2
10 Robust−SUC (TH)
Linear−SUC (TH)
1 −1
10 Dirty Templates 10
Robust−SUC (Theoretical)
0 Linear−SUC (Theoretical)
10
NMSE
NMSE
−1 −2
10 10
−2
10
Linear−SUC
−3 −3
10 10 Robust−SUC
Linear−AVG
−4
10 Robust−AVG
−5 −4
10 10
0 5 10 15 20 25 30 0 5 10 15 20 25 30
SNR (dB) SNR (dB)
Fig. 5. Simulation results for the NMSE of the successive sampling-based Fig. 7. NMSE performances of all receivers in the presence Gaussian noise,
robust acquisition and the corresponding linear acquisition for = 0.1, κ = when = 0.
100.
3
10
2
10 Robust−SUC (N =30)
Robust−AVG (no TH) t
2 Linear−SUC (N =30)
Linear−AVG (no TH) 10 t
1
10 Robust−AVG (TH) Robust−SUC (Nt=60)
Linear AVG (TH) 1
10 Linear−SUC (N =60)
Dirty Templates t
0
10 Robust−AVG (Theoretical) Robust−SUC (N =120)
t
Linear−AVG (Theoretical) 0
10 Linear−SUC (N =120)
t
NMSE
−1
10
NMSE
−1
10
−2
10
−2
10
−3
10
−3
10
−4
10
−4
10
−5 0 5 10 15 20 25 30
10
0 5 10 15 20 25 30 SNR (dB)
SNR (dB)
Fig. 8. NMSE of successive sampling-based robust acquisition and corre-
Fig. 6. Simulation results for the NMSE of the averaged template-based sponding linear acquisition under = 0.1, κ = 100 for various values of
robust acquisition and the corresponding linear acquisition for = 0.1, κ = Nt .
100.
and the ‘dirty templates’approach in [10] significantly. For
ˆ
frame-offsets nf often go out of that range in simulations the SNR range of 0-20 dB, the performance of Gaussian-
with linear acquisition. In such cases, instead of discarding optimal acquisition is unacceptable, justifying the design of
the out-of-range frame-offset estimates, we incorporated them robust procedures. Specifically, to attain NMSE = 1 × 10−3
to reflect the system’s performance more realistically, with when = 0.1 and κ = 100, linear acquisition requires
NMSE values exceeding unity by a large margin at low 12 dB higher SNR compared to the robust scheme with
SNR. Throughout the simulations we assume that the exact averaged template. The same margin grows to 17 dB for
values of the noise model parameters are known; these can be = 0.01 and κ = 1000, when the interference intensifies.
estimated efficiently via the algorithm in [26]. A simplification (The corresponding graph is available in [27], [28].) The
is possible by setting k = 1.5/σ where the estimation of σ is gap is higher for dirty template although at low SNR, the
rather straightforward and no serious performance loss occurs performance is better than the Gaussian-optimal ML solution.
[22]. This better performance originates from the fact that ‘dirty
First, we evaluate the performance of the robust receivers templates’ in [10] is a search-based technique ensuring that
under a CM4 channel for which noise parameters are selected the estimated frame-offset lies in {0, . . . , Nf − 1} contrary
as = 0.1, κ = 100 and the number of training symbols to the estimation-based ones for which the estimated frame-
is set to 30. The corresponding results are given in Figs. offset goes out of range due to impulses. We are concerned
5 and 6, which demonstrate that robust acquisition methods with the estimation of only frame-level time-offset and the
outperform the conventional Gaussian-optimal linear system random pulse-level uncertainty results in error floors in all
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EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4665
2 2
10 10
Robust−AVG (CM1)
Robust−AVG (Nt=30)
Linear−AVG (CM1)
1 Linear−AVG (N =30) 1
Robust−AVG (CM2)
10 t 10
Robust−AVG (N =60) Linear−AVG (CM2)
t
Linear−AVG (Nt=60) Robust−AVG (CM4)
0 0 Linear−AVG (CM4)
10 10
Robust−AVG (Nt=120)
Linear−AVG (N =120)
t
NMSE
NMSE
−1 −1
10 10
−2 −2
10 10
−3 −3
10 10
−4 −4
10 10
0 5 10 15 20 25 30 0 5 10 15 20 25 30
SNR (dB) SNR (dB)
Fig. 9. NMSE of averaged template-based robust acquisition and corre- Fig. 11. NMSE of averaged template-based robust acquisition and corre-
sponding linear acquisition under = 0.1, κ = 100 for various values of sponding linear acquisition under = 0.1, κ = 100 for various channel
Nt . models.
0
3 10
10
Robust−SUC (CM1)
2
Linear−SUC (CM1)
10 Robust−SUC (CM2) −1
10
Linear−SUC (CM2)
1 Robust−SUC (CM4)
10
Linear−SUC (CM4)
−2
10
0
10
NMSE
BER
−1
10 −3
10
−2 Perfect Timing
10
Robust−SUC (no TH)
−4
10 Robust−SUC (with TH)
−3 Linear−SUC (no TH)
10
Linear−SUC (with TH)
No Timing
−4 −5
10 10
0 5 10 15 20 25
0 5 10 15 20 25 30 SNR (dB)
SNR (dB)
Fig. 12. BER of successive sampling-based robust acquisition and the
Fig. 10. NMSE of successive sampling-based robust acquisition and corresponding linear acquisition under = 0.1, κ = 100.
corresponding linear acquisition under = 0.1, κ = 100 for various channel
models.
CM4) implying the robustness of the estimators with respect
cases. The error floor is higher when TH codes are employed to the channel statistics. This stems from the use of received
because they induce IFI that results in an additional timing symbols as templates in TR systems which results in the same
ˆ
uncertainty in nf as stated in [8]. From the figures, it is clear amount of energy capture in the samples regardless of the
that average-template based robust acquisition outperforms delay spread and multipath.
successive sampling alone due to better noise suppression at To judge the impact of timing mismatch in an impulsive en-
the expense of a complexity increase due to averaging in vironment, BER graphs are plotted assuming that the receiver
analog domain. The theoretical NMSEs are also plotted for has perfect knowledge of the channel and that impulsive noise
both robust and linear methods without TH in Figures 5 and occurs only during transmission of the training sequence. This
6, showing good agreement with the simulation results. In Fig. simulation set-up is selected so as to observe the impact of
7, receiver performances are presented for the case of AWGN only the timing acquisition on the BER performance [10].
only. Robust methods do not incur any noticeable performance Otherwise, both channel estimation errors (or the effect of
loss in the nominal case. Then, the performances of the robust the unknown channel in blind detection) and the effect of
estimators are evaluated for different lengths of the training the impulsive noise on symbol detection combine with the
sequence and it can be shown in Fig. 8 and Fig. 9 that as timing acquisition error. In the BER simulations, data are
the length of the training sequence increases, the performance sent through bursts which consist of 30 training symbols and
improves. Fig. 10 and Fig. 11 show the performance of the 1000 information symbols. At each SNR, these bursts are
estimators for different channel models (CM1, CM2, and transmitted until at least 100 symbol errors are recorded. The
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4666 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008
0
10
PDF is similar to Gaussian. Therefore, it can be argued that
as long as the ambient noise and disturbances contain outliers,
−1
10
they dominate MUI in UWB systems regardless of the number
of interfering users. On the other hand, robust techniques are
largely ineffective (except at low SNRs where they offer a
−2
10
less than 2 dBs of gain over linear acquisition) when the only
source of impulsiveness is MUI and noise is white Gaussian.
BER
−3
10 VIII. D ISCUSSION AND C ONCLUSION
In this paper, we have presented UWB timing acquisition
Perfect Timing
−4
Robust−AVG (no TH) schemes that combine TR methodology with robust estima-
10 Robust−AVG (with TH)
Linear−AVG (no TH)
tion techniques for operation in the presence of interference
Linear−AVG (with TH) modeled as impulsive noise. For a single user link, successive
−5
No Timing
sampling- and averaged template-based robust acquisition are
10
0 5
SNR (dB)
10 15 proposed. The former suffers from noise enhancement because
of noisy signal correlations, while the latter provides improve-
Fig. 13. BER of averaged template-based robust acquisition and the ment through averaging. Both robust methods offer substantial
corresponding linear acquisition under = 0.1, κ = 100. performance gains over their linear counterparts, i.e., ML
2
estimators designed under the Gaussian noise assumption.
10 Robust−MUI−Gauss (N =5)
u Two robust acquisition techniques are proposed for mul-
Robust−MUI−Impulsive (Nu=5) tiuser environments, one of them being a simple extension
10
1 Robust−MUI−Gauss (N =20)
u of averaged template-based acquisition by assuming MUI as
Robust−MUI−Impulsive (N =20)
u Gaussian-distributed. The other is a new design that takes
0
Linear−MUI−Gauss (N =5)
u the inherent impulsiveness of MUI in UWB systems into
10
Linear−MUI−Gauss (N =20)
u consideration. Although the performance of the multiuser
receiver is satisfactory for moderate number of users, the
NMSE
−1
10 specific correlation structure of MUI is not exploited here for
further suppression. Synchronization methods employing MUI
10
−2 suppression in the ISI channels may be explored as future
work.
−3
10
A PPENDIX A
−4
In this Appendix, the noise PDFs of the successive sam-
10
0 5 10 15 20 25 30 pling, (11), is derived. With following the same line of
SNR (dB) arguments, (19) can be also derived, hence it is omitted. The
noise term in successive sampling, ωSUC (m), consists of three
Fig. 14. NMSE of the robust acquisition methods designed for the multiuser
environment and of the linear counterpart when = 0.1, κ = 100. parts:
ωSUC (m) ωSUC mTs
performance gains obtained by the robust methods become = ωSUC,1 mTs + ωSUC,2 mTs + ωSUC,3 mTs
clearer with Figures 12 and 13. The Gaussian-optimal ML where ωSUC,1 mTs , ωSUC,2 mTs and ωSUC,3 mTs are
estimators cannot track the UWB signal in impulsive noise, given by
whereas the robust methods can within an acceptable range of
Ts
SNR. Moreover, when the averaged template is employed, its ωSUC,1 mTs = rs (t + mTs )w t + (m + 1)Ts dt,
performance moves within 1 dB of perfect timing when no TH 0
is employed. In these figures, the effect of IFI induced by the Ts
TH codes can again be observed considering the performance ωSUC,2 mTs = rs t + (m + 1)Ts w(t + mTs )dt,
0
gaps between no-TH and TH cases. Ts
Lastly, we examine the performance of the MUI receivers ωSUC,3 mTs = w(t + mTs )w t + (m + 1)Ts dt,
where 5 and 20 users transmit at 5 dB below the desired 0
user. Fig. 14 depicts the rise in the NMSE error floor due respectively, where rs (t + mTs ) = b(m − 1)pr (t + Ts −
to MUI. Interestingly, robust estimators designed for the τ1 ) + b(m)pr (t − τ1 ) and τ1 is the time delay of the first
Gaussian approximation-based and the impulsive model-based path. The autocorrelation function and the PDF of ωSUC (m)
MUI have about the same performance. This is due to the fact are derived in [17] for Gaussian w(t) and it is shown that
that in TR systems, the main performance limiting factor is the PDF of wSUC (m) can be approximated as Gaussian [17],
the double-noise term irrespective of whether the system is in [29]. This approximation is valid as long as both the time-
a single-user link or not. The MUI becomes comparable to bandwidth product of the system and the number of pulses per
the double-noise term for large number of users, but then its symbol are large enough. Moreover, under these conditions,
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EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4667
noise components can be assumed to be mutually uncorrelated. represented by Gaussian mixture model with zero mean and
Here, following the same theory as in [17], [29], the PDF of variances
ωSUC (m) is derived for the case where w(t) is modeled as σ2 σ2
2
w(t) = wB (t) + wI (t) with wB (t), wI (t) denoting thermal E ωMA,1 (m) = ET,r (1 − )2 +2 κ ,
Nt Nt
Gaussian and shot noise, respectively, which are zero-mean,
mutually independent random processes. One can conclude
2
E ωMA,2 (m) = Ed,r (1 − )σ 2 + κσ 2 ,
that wSUC,1 (mTs ) and wSUC,2 (mTs ) are also mixtures whose 2 σ4 σ2
E ωMA,3 (m) = 4Ts W (1 − ) + κ2
total variance is Cg Nf (1 − )σ 2 + κσ 2 and PDF is Nt Nt
fSUC,1 (x) = (1 − )g x; 0, Cg Nf σ 2 + g x; 0, Cg Nf κσ 2 .
N
where ET,r = u=1 Eu,r is the sum of the received energies
u
of all users. By defining the PDFs of ωMA,1 (m), ωMA,2 (m)
The shot noise wI (t) has a PDF of the form fI (x) = (1 − and ωMA,3 (m) as fMA,1 (x), fMA,2 (x) and fMA,3 (x), the
)δ(x) + g(x; μ, σ 2 ) [30], [31], which can be approximated overall PDF of ωMA (m) can be determined by fMA (z) =
as a Gaussian PDF. Therefore, following the analysis in [17], fMA,1 (z) ∗ fMA,2 (z) ∗ fMA,3 (z), which results in (27).
[29], the autocorrelation function of ωSUC,3 (mTs ) can be
found as
A PPENDIX C
E ωSUC,3 (mTs )ωSUC,3 (mTs − τ ) = The derivation of the MUI PDF for TH-PPM UWB systems
∞ is given in [23], [24] for an AWGN channel. Because these
2 2
2 Ts − | τ | RB (y) + RI (y) dy results are obtained for coherent receivers, they do not contain
−∞
the term originating from correlation of the same user’s re-
where RB (t) and RI (t) are the autocorrelation functions of ceived symbols, which is a consequence of TR-like reception.
wB (t) and wI (t), respectively. Similar to ωSUC,1 and ωSUC,2 , The MUI term is given by
ωSUC,3 also obeys a Gaussian mixture density because of that
its autocorrelation function is governed by two process, one ωMUI (m) = ωMUI,1 (m) + ωMUI,2 (m)
2
for impulsive part leading to RB (y) and one for nominal part where
2
leading to RI (y). Therefore, again following [17], [29], its Nt /2−1
Nu
variance can be found as 4Ts W σ 2 (1 − )σ 2 + κ2 σ 2 where 2 Ts
ωMUI,1 (m) = ru (t)ru (t)dt,
m i
W is the bandwidth of the front-end filter employed in the Nt t=0
u=1,u=d i=0
receiver and can be approximated as W ≈ 1/Tp . Moreover,
Nu Nu Nt /2−1
its PDF is of the form 2 Ts
ωMUI,2 (m) = ru (t)rl (t)dt
m i
4 2 4 Nt
fSUC,3 (x)= (1 − )g x; 0, 4Ts W σ + g x; 0, 4Ts W κ σ . u=1 l=1,l=u i=0 t=0
Finally, note that in the Gaussian-mixture model, a noise sam- where is given in (26). Note that ωMUI,2 (m) consists
ru (t)
m
ple is generated by the acting PDF in the mixture depending of cross-terms between different users and its value is gov-
on whether ambient interference is present or not. Therefore, erned primarily by the collision probability of different users’
2
as long as these three noise samples are mutually uncorrelated, pulses, which is given by 1/Nc [23], [24]. As long as Nc is
they are independent as well, implying that overall noise PDF sufficiently large, ωMUI,2 (m) becomes negligible with respect
of ωSUC is given by fSUC = fSUC,1 ∗ fSUC,2 ∗ fSUC,3 , where to ωMUI,1 (m). The reasoning behind this negligibility is that
∗ denotes linear convolution, which results in (11). the collision probability of the same user’s pulses is obviously
one. Therefore, we approximately get
A PPENDIX B 2
Nu Nt /2−1
ωMUI (m) ≈ ωMUI,1 (m) = βu,i
In this appendix, we furnish the derivation of (27). The Nt i=0
u=1,u=d
noise ωMA (m) consists of three parts as
where the PDF of βu,i can be found as
ωMA (m) = ωMA,1 (m) + ωMA,2 (m) + ωMA,3 (m)
fβu,i (x) = h1 (x) + h2 (x)
where
where h1 (x) and h2 (x) are given by
Nu Ts
ωMA,1 (m) = ¯
ru (t)w(t)dt,
m
h1 (x) = 0.25 δ x − Nf Cg,u + δ x + Nf Cg,u
u=1 0
0.25
2
Nu Nt /2−1 Ts h2 (x) = u x + Nf Cg,u − u x − Nf Cg,u
ωMA,2 (m) = ru (t)w(t)dt,
i Nf Cg,u
Nt 0
u=1 i=0 and u(x) is the unit-step function. We assume that all the
Ts
ωMA,3 (m) = ¯
w(t)w(t)dt interfering users have same received energy, hence Cg,u =
0 Cg,l = Cg , ∀l, u ∈ {1, . . . , Nu }. The PDF of ωMUI (m) can
be expressed as (Nu − 1)Nt /2-fold convolution; that is
where ru (t) and ru (t) are given in (26). Following the
i m
analysis carried out in the previous section, it can be shown 2 2
fMUI x = fβ (x) ∗ · · · ∗ fβ (x) (31)
that these noise samples are independent and their PDFs are Nt Nt
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EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4669
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Hakan Delic (SM’00) received the B.S. degree
2139, June 2007. (with honors) in electrical and electronics engineer-
[30] H. J. Larson and B. O. Shubert, Probabilistic Models in Eng. Sci.. ing from Boˇ azic i University, ˙
g ¸ Istanbul, Turkey, in
Malabar, FL: R.E. Krieger Pub. Co., 1989. 1988, and the M.S. and the Ph.D. degrees in elec-
[31] J. H. Miller and J. B. Thomas, “The detection of signals in impulsive trical engineering from the University of Virginia,
noise modeled as a mixture process,” IEEE Trans. Commun., vol. COM- Charlottesville, in 1990 and 1992, respectively. He
24, no. 5, pp. 559-563, May 1976. was a Research Associate with the University of
Virginia Health Sciences Center from 1992 to 1994.
Ersen Ekrem received the B. S. and M. S. de- In September 1994, he joined the University of
grees in electrical and electronics engineering from Louisiana at Lafayette, where he was on the Fac-
Boˇ azic i University, Istanbul, Turkey, in 2006 and
g ¸ ulty of the Department of Electrical and Computer
2007, respectively. Currently, he is working toward Engineering until February 1996. He was a Visiting Associate Professor
the Ph. D. degree in the Department of Electrical in the Department of Electrical and Computer Engineering, University of
and Computer Engineering, University of Maryland, Minnesota, Minneapolis, during the 2001-2002 academic year. Dr. Delic is ¸
College Park. His research interests include wireless g ¸
currently a professor of electrical and electronics engineering at Boˇ azic i
communications and multiuser information theory. University. His research interests lie in the areas of communications and signal
processing with current focus on wireless multiple access, ultra-wideband
communications, OFDM, robust systems, and sensor networks.
Mutlu Koca (M’01) received the B.S. degree from
g ¸
Boˇ azic i University, Istanbul, Turkey, in 1996 and
the M.S. and the Ph.D. degrees from the University
of California, Davis (UC-Davis), in 2000 and 2001,
respectively, all in electrical engineering. He was a
Postgraduate Researcher and a Lecturer at UC-Davis
between September and December 2001. He was a
Member of the Technical Staff at Flarion Technolo-
gies, Bedminster, NJ between December 2001 and
December 2002, and a Researcher at the Institut de
Recherche en Informatique et Systemes Aleatoires
(IRISA), Rennes, France between February 2003 and March 2004. In July
2003, he joined the Department of Electrical and Electronics Engineering,
g ¸
of Boˇ azic i University where he is currently an Assistant Professor. His
research interests are in communication theory, error correcting codes, signal
processing for communications, ultra-wideband and iterative (turbo) methods.
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