3736 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011
Training-Based Synchronization and DemodulationWith Low Complexity for UWB Signals
Tiejun Lv, Member, IEEE, Yongwei Qiao, and Zicheng Wang
Abstract—In this paper, we propose a low-complexitydata-aided (DA) synchronization and efficient demodulationtechnique for an ultrawideband (UWB) impulse radio system.Depending on the autocorrelation property of a judiciously chosentraining sequence, a redundance-included demodulation template(RDT) can be extracted from the received signal by separating,shifting, and realigning two connected portions in the observationwindow. After constructing the RDT, two receiver designs areavailable. One approach is to demodulate transmitted symbolsby correlating the RDT in a straightforward manner, which doesnot require the explicit timing acquisition and, thus, considerablyreduces the complexity of the receiver. An alternative approachis accomplished with the assistance of a non-RDT (NRDT). TheNRDT-based receiver is able to remove the redundant noisycomponent of the RDT by acquiring timing offset via a simplesynchronization scheme, therefore achieving a better bit errorrate (BER) performance. Both the schemes can counteract theeffects of interframe interference (IFI) and unknown multipathchannels. Furthermore, analytical performance evaluations of theRDT- and NRDT-based receivers are provided. Simulations verifythe realistic performance of the proposed receivers in the presenceof multiuser interference (MUI) and timing errors.
U LTRAWIDEBAND (UWB) impulse radio has attractedgrowing interest in short-range wireless communications.
Several features are motivated by UWB transceivers, includingthe potential for very high data rates with a commensurateincrease of user capacity, promisingly small size and process-ing power, enhanced capability to penetrate through obstacles,and ultra-high-precision ranging at the centimeter level [1]–[3]. However, the realization of this vision calls for advancedtechniques to accomplish tasks such as synchronization, inter-ference cancellation, and low-complexity demodulation. Con-sideration for hardware implementation has also been taken.In [4], both high- and low-data-rate systems are proposed forUWB impulse radio with energy-efficient architectures andcircuits. For the digital UWB receiver, the high sampling rates
Manuscript received November 19, 2010; revised April 18, 2011 andAugust 12, 2011; accepted August 21, 2011. Date of publication September 1,2011; date of current version October 20, 2011. This work was supported bythe National Natural Science Foundation of China under Grant 60972075. Thispaper was presented in part at the IEEE International Symposium on Personal,Indoor, and Mobile Radio Communications, Tokyo, Japan, September 2009.The review of this paper was coordinated by Prof. A. M. Tonello.
The authors are with the School of Information and Communication Engi-neering, Beijing University of Posts and Telecommunications, Beijing 100876,China (e-mail: [email protected]).
Digital Object Identifier 10.1109/TVT.2011.2166815
of analog-to-digital converters can be remarkably reduced bycompressive sampling [5].
Accurate synchronization and effective demodulation arecrucial for ultra-short-duty-cycle UWB transmission, partic-ularly when the interframe interference (IFI) and unknownmultipath channels are present [6], [7]. The traditional synchro-nization scheme, where a locally generated template correlatesthe received signal with different delays, is proposed for time-hopping (TH) UWB systems under the assumption that multi-path is absent [8]. Maximum-likelihood [9] and least squaressynchronization approaches [10] for UWB systems are avail-able, while they are computationally complicated consideringthe high sampling rates. Tian and Giannakis [11] propose ablind timing acquisition scheme that exploits the cyclostation-arity nature of UWB transmissions. However, this scheme is notrobust against IFI. A synchronization scheme that exploits themean normalized energy profile is developed in [12]; TH codesare nevertheless absent.
The rich UWB multipath diversity gives rise to an impellentneed for cost-effective receiver design with the reasonablepower consumption and low complexity. However, Rake re-ceiver requires complicated channel estimation and a largenumber of correlation operations to capture a significant partof the signal energy [13], [14]. To circumvent these drawbacks,a considerable part of the research interest has been shifted tosuboptimum, noncoherent schemes [15]. The autocorrelationreceivers (AcRs) achieve efficient energy capture by correlatingtwo consecutively transmitted pulses, one of which serves as anoisy demodulation template for the other [16], [17]. An alter-native approach is to implement the energy detector (ED) withlower complexity [18]. To mitigate the noise effect, a completereceiver design for the weighted ED technique with channelestimation has been proposed recently [19]. However, bothAcRs and ED require additional synchronization schemes priorto demodulation. Analogous to the schemes that implement thetiming-with-dirty-templates (TDT) technique1 [20], [21], theproposed UWB algorithm integrates timing acquisition with de-modulation. In this framework, a judiciously designed trainingsequence is first transmitted for synchronization and templateextraction. After that, the proceeded data symbols can be simplycorrelated with the extracted template for demodulation.
The aim of this paper is to develop a low complexity data-aided (DA) synchronization and efficient demodulation tech-nique. The proposed scheme can avoid the use of a Rake
1The TDT-based timing algorithm mentioned in this paper refers to refers to[20, Proposition 4], which has been proven to obtain the best performance inall propositions. Further extension to TDT-based receiver is studied in [21].
LV et al.: TRAINING-BASED SYNCHRONIZATION AND DEMODULATION FOR UWB SIGNALS 3737
receiver and remains operational in general UWB settings withTH codes, unknown dense multipath propagation, and evenwhen IFI is present. The main contributions of this paper arelisted as follows.
1) Depending on a judiciously designed training patternwith a particular autocorrelation property, two connectedportions in a symbol-long segment can be separated,shifted, and realigned to rebuild an integral demodulationtemplate. Therefore, the redundance-included demodula-tion template (RDT), with a redundant part of noise, isextracted from the received signal without explicit timingacquisition. The RDT is able to capture the completeinformation of a symbol.
2) When the RDT is available, two receiver designs can beselected for demodulation. One approach is to estimatethe transmitted symbols by correlating the RDT in astraightforward manner, which captures the full multipathenergy with low complexity.
3) An alternative approach is carried out in two steps. Tim-ing acquisition is first implemented by invoking simpleenergy detection, which is employed to shear the RDTand obtain a non-RDT (NRDT). After that, demodulationis executed in the newly extracted template. Comparedwith the RDT-based receiver, the NRDT-based receiverobtains a better bit error rate (BER) performance withhigher complexity.
4) This paper extends the development of [22] to providean analytical insight into the proposed schemes. Errorprobability evaluation reveals the dominant factor leadingto the performance degradation of the RDT-based re-ceiver. Comparisons between the proposed schemes andthe TDT-based receiver are made in terms of training,timing, demodulation, and error probability. The analysisof the computational complexity is carried out as well.Simulations further validate the effectiveness of the pro-posed technique.
5) In addition to pronounced reduction of the complexity,the NRDT-based receiver achieves improved BER per-formance and is more robust to mistiming in comparisonwith the TDT-based receiver. Moreover, the RDT-basedreceiver obtains a good tradeoff between the generalaccuracy and rapid demodulation.
The rest of this paper is organized as follows. The sys-tem model of the impulse radio UWB system is describedin Section II. The original RDT-based receiver and the im-proved NRDT-based receiver are presented in Section III. InSection IV, we investigate the error probability of the RDT-based receivers as well as the NRDT-based receivers; mean-while, comparisons with the TDT-based receiver are made.Corroborating simulations and numerical results are providedin Section V, followed by concluding remarks in Section VI.
II. SYSTEM MODEL
Consider a point-to-point UWB system, where packet trans-mission is employed. The packet is composed of the followingtwo distinct parts: 1) a preamble for template recovery andsynchronization and 2) a data field for symbol estimation.
For UWB communication, each symbol is transmitted by em-ploying a stream of ultrashort pulses. The transmitted symbolwaveform can be modeled as
ws(t) =Ns−1∑j=0
w(t − jTf − cjTc) (1)
where w(t) is the ultrashort pulse (referred to as monocycle)with the duration Tp and
∫ Tp
0 w2(t)dt = 1. The symbol witha duration of Ts = NsTf consists of Ns frames, each of du-ration Tf . The user-specific TH code sequence {cj} satisfies0 ≤ cj ≤ Nh − 1. Tc is the duration of an addressable timebin with Tf ≥ (Nh − 1)Tc + Tp. For the proposed schemes,we consider binary antipodal pulse amplitude modulation.The UWB-modulated signal at the transmitter can be ex-pressed as
us(t) =√
ε
+∞∑i=0
s(i)ws(t − iTs) (2)
where ε is the energy for each monocycle, and {s(i) ∈ {±1}}is the transmitted symbol sequence. At the stage of templaterecovery, training symbols are carried by {s(i)} to extract noisytemplate from the received signal, while for the demodulationstage, data symbols are assigned to {s(i)} for informationtransmission.
The impulse radio UWB system typically employs a mul-tipath channel whose impulse response can be described as aquasi-static tapped delay line, i.e.,
h(t) =L−1∑l=0
αlδ(t − τl) (3)
where L is the number of resolvable multipath taps, and αl
and τl are the gain and delay of the lth path, respectively. Toseparate the multipath spreading effects from the propagationdelay τ0, all relative path delays can be uniquely cast intoτl,0 := τl − τ0 with {τl,0}L−1
l=0 satisfying τ0,0 = 0 and τl,0 <τl+1,0. To avoid intersymbol interference, cNs−1 and Tf areselected to satisfy Tf ≥ cNs−1Tc + Tp + τL−1,0. The receivedsignal is then given by
r(t) =√
ε
+∞∑i=0
s(i)L−1∑l=0
αlws(t − iTs − τ0 − τl,0) + n(t) (4)
where n(t) is the additive noise, which is modeled as a zero-mean white Gaussian process with two-sided power spectraldensity N0/2 and bandwidth W indicated by the low-passfront-end filter’s cutoff frequency. For simplicity, we introducethe following received symbol waveform:
wR(t) =L−1∑l=0
αlws(t − τl,0). (5)
Therefore, the received signal can be further expressed as
r(t) =√
ε
+∞∑i=0
s(i)wR(t − iTs − τ0) + n(t). (6)
3738 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011
Fig. 1. Process of template recovery for RDT and NRDT in a training-basedUWB system.
III. TIMING ACQUISITION AND DEMODULATION
In this section, a low-complexity DA synchronization anddemodulation scheme is developed. The overall algorithm de-pends on a judiciously designed training sequence with a par-ticular autocorrelation property. Based on that, the RDT-basedreceiver is constructed by extracting a noisy template from thearriving signal. To further enhance the detection performance,the NRDT-based receiver is proposed with explicit timingoperation.
As shown in Fig. 1 (the noise is ignored for the sake ofclarity), we assume that the receiver initiates timing acquisitionat t1 (t1 > τ0). t1 can be expressed as a sum of an integermultiple Nϕ of symbol duration and a residue ϕ, i.e.,
t1 = NϕTs + ϕ, ϕ ∈ [0, Ts). (7)
The propagation delay can be served as a reference and set asτ0 = 0 since the receiver is unaware of the starting time of thetransmission. The received signal initiating timing acquisitioncan be formulated as
x(t) := r(t + t1)
=√
ε+∞∑i=0
s(i)wR(t − iTs + t1) + n(t + t1) (8)
for t ∈ [0,+∞). Note that in Fig. 1 the transmitted symbolswith perfect synchronization are separated by the dashed verti-cal lines. The observation signal taken from the received signaloriginates at time t1 and persists for a duration of MTs. Eachsegment in the observation window lasts for a duration of Ts
and can be distinguished by the solid vertical lines. The mthsegment waveform of the observation signal, starting at timet1 + (m − 1)Ts, m ∈ [1,M ], is denoted by
xm(t) = x (t + (m − 1)Ts) , t ∈ [0, Ts). (9)
For simplicity, we assume that noise is negligible. Then, usingt1 = NϕTs + ϕ, xm(t) can be derived as follows:
xm(t) =√
ε
+∞∑i=0
s(i)wR (t − (i − Nϕ − m + 1)Ts + ϕ) .
(10)
Since the received symbol waveform wR(t) has finite support[0, Ts), xm(t) is zero, except for a finite number of i values. Forany given m and t1, it is not difficult to find that
i = Nϕ + m − 1 + q (11)
where q takes a value of 0 or 1. Substituting (11) into (10) yields
xm(t) =√
ε
1∑q=0
s(Nϕ + m − 1 + q)wR(t − qTs + ϕ). (12)
Note that wR(t − qTs + ϕ) only depends on ϕ for any fixed q.Moreover, the summation in (12) suggests that each segmentof the observation signal consists of the following two specialparts: 1) the truncated tail of the received symbol waveformfrom the previous symbol and 2) the head of the receivedsymbol waveform originating in the current symbol.
The aforementioned analysis is inspiring. If the tail andthe head on the segments of the observation window can beseparated, we would be able to construct a complete receivedsymbol waveform wR(t) as the demodulation template. Wemaintain that a possible answer is to design a special trainingsequence to achieve the separation. The analysis (which will bepresented later) suggests that the proposed training sequence isable to accomplish the objective.
At the stage of synchronization and template extraction, thetraining sequence {s(i)} consists of a repeated pattern (1, 1, 1,−1) [or any of its circularly shifted versions, e.g., (1, 1, −1, 1)].The periodic autocorrelation function (PACF) of the trainingsequence is defined as
Rs(j) =14
l+4∑i=l+1
s(i)s(i + j) ∀l. (13)
This training pattern is particularly attractive, because its PACFhas the following property:
Rs(j) ={
1, j = 4n0, j �= 4n
, n = 0,±1,±2, . . . . (14)
Note that only when the variable j is a multiple of four, thevalue of the PACF is one; in any other case, it is zero. Actually,the training sequence {s(i)} is the only known binary sequencewith an out-of-phase PACF value of zero [23]. It should bementioned that the proposed training sequence serves to extractthe demodulation template from the received signal, althoughit has the potential application of synchronization and rangingsystems.
LV et al.: TRAINING-BASED SYNCHRONIZATION AND DEMODULATION FOR UWB SIGNALS 3739
A. RDT-Based Receiver
The attractive autocorrelation property of the training se-quence provides a solution to separate the truncated tail andthe head of the received symbol waveform for an integraldemodulation template. To begin with, we set the observationwindow as M = 4 symbol-long and the observation interval as[0, 4Ts) by using the periodicity of the training symbols. Twosignals are constructed to acquire the truncated tail and the headof the received symbol waveform wR(t). The signal vT
k (t) forextracting the truncated tail is defined as
vTk (t) =
14√
ε
4∑m=1
xm(t)s(m − 1 + k) (15)
for k ∈ [0, 3] and t ∈ [0, Ts). As derived in Appendix A, vTk (t)
can be further expressed as
vTk (t) = Rs(Nϕ − k)wR(t + ϕ)
+ Rs(Nϕ − k + 1)wR(t − Ts + ϕ). (16)
The PACF of the training sequence shows that if Nϕ − k =4n, vT
k (t) = wR(t + ϕ). Therefore, the truncated tail of thereceived symbol waveform wR(t) is obtained. Similarly, thehead of the waveform wR(t) can be extracted by formulatingvH
k (t) as
vHk (t) =
14√
ε
4∑m=1
xm(t)s(m + k)
=Rs(Nϕ − k − 1)wR(t + ϕ)
+ Rs(Nϕ − k)wR(t − Ts + ϕ) (17)
for k ∈ [0, 3] and t ∈ [0, Ts).If Nϕ − k = 4n, vH
k (t) = wR(t − Ts + ϕ) represents thehead of the received symbol waveform wR(t). However, thetraining sequence mismatches if Nϕ − k �= 4n. For the caseof Nϕ − k − 1 = 4n, vH
k (t) represents the tail, and vTk (t) be-
comes an empty template.Note that the signal vH
k (t) and vTk (t) have been constructed
as follows: the observation signal is multiplied by the desig-nated training sequence at first, and then, the summation is per-formed to formulate a symbol-long signal. The only differencebetween the two signals is the designated portion of the trainingsequence. It can be seen in (15) and (17) that the correspondingtraining sequence of vH
k (t) is one chip backward from thatof vT
k (t). Since the tail and the head of the segment xm(t)originate from the previous symbol and the current symbol,respectively, they are assigned by different chips of the trainingsequence. Therefore, the separation of the tail and the headis possible by applying the uniquely designed signal vH
k (t)and vT
k (t). In (16) and (17), we find that vHk (t) and vT
k (t)contain the same PACF Rs(Nϕ − k). If the determined valuekϕ among k satisfies Nϕ − kϕ = 4n, the tail and the head canbe separated.
To construct the demodulation template, two signals vTk (t)
and vHk (t) are required to be reassembled. First, the truncated
tail vTk (t) is shifted Ts time forward to follow the head vH
k (t).
Then, vHk (t) and vT
k (t) are concatenated to rebuild an integraltemplate waveform vk(t) with duration 2Ts. Corresponding todifferent k values, vk(t) is expressed as
vk(t) = vHk (t) + vT
k (t − Ts)
= Rs(Nϕ − k − 1)wR(t + ϕ)
+ Rs(Nϕ − k)wR(t − Ts + ϕ)
+ Rs(Nϕ − k + 1)wR(t − 2Ts + ϕ) (18)
for k ∈ [0, 3] and t ∈ [0, 2Ts). By applying the unique auto-correlation property of the training sequence, the demodula-tion template is extracted from the observation signal. Notethat when the value of k changes, vk(t) can represent thefollowing four possible segments: 1) the truncated tail of thereceived symbol waveform; 2) the complete received symbolwaveform; 3) the head of the received symbol waveform; and4) nothing, i.e.,
vk(t) =
wR(t + ϕ), Nϕ − k − 1 = 4n
wR(t − Ts + ϕ), Nϕ − k = 4n
wR(t − 2Ts + ϕ), Nϕ − k + 1 = 4n
0, Nϕ − k + 2 = 4n
(19)
for n = 0,±1,±2, . . . and t ∈ [0, 2Ts).According to the analysis above, the complete received
symbol waveform wR(t − Ts + ϕ) has the maximum energyamong the four possible segments. Therefore, it can be pickedup by means of simple energy detection. The value of kϕ isobtained by
kϕ = arg maxk∈[0,3]
J(k) (20)
where J(k) =∫ 2Ts
0 v2k(t)dt. Note that the searching process for
kϕ can be regarded as sequence acquisition with the assumptionthat the initial chip of the training sequence at t1 is unknown.At the receiving end, four possible circularly shifted versionsof the training pattern (with four possible values of k) arematched to obtain the exact sequence for template extraction inthe observation window. However, no explicit timing operationis performed since the timing offset ϕ is not acquired in thisprocedure.
Now that kϕ is available, the demodulation template can bewritten as
vkϕ(t) = wR(t − Ts + ϕ), t ∈ [0, 2Ts). (21)
It should also be mentioned that the demodulation template isextracted by consuming only four symbol-long segments. If alarger number of segments are available, further performanceimprovement can be achieved by averaging out the additivenoise effect. In practice, the received signal x(t) is averagedto obtain the estimation for xm(t), i.e.,
xm(t) =1N
N−1∑n=0
x (t + (m − 1)Ts + 4nTs) (22)
3740 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011
Fig. 2. Block diagram of the RDT-based receiver.
for t ∈ [0, Ts) and m ∈ [1, 4], where N = �M/4 is the numberof segments to be averaged for each xm(t), and �· denotes thefloor function.
In the end, the proposed algorithm for template extraction issummarized as follows.
Step 1) Make use of xm(t) to establish vHk (t) and vT
k (t) by (15)and (17).
Step 2) Construct the template waveform vk(t) in terms of (18).Step 3) Find an estimation of kϕ by energy detection, i.e., kϕ =
arg maxk∈[0,3]
∫ 2Ts
0 v2k(t)dt.
Step 4) Substitute kϕ into (18) to obtain the demodulationtemplate vkϕ
(t).
For the template signal vkϕ(t) of duration 2Ts, the introduc-
tion of the additive noise is inevitable despite the fact that xm(t)has been averaged. As shown in Fig. 1, a redundant part ofduration Ts is contained in the demodulation template, whichis nothing but noise. Thus, vkϕ
(t) is called the RDT.The block diagram of the RDT-based receiver is shown in
Fig. 2. After obtaining the RDT, the transmitted symbols are di-rectly demodulated with the RDT-based correlator. Specifically,the following decision statistic is given:
d(k1) =
2Ts∫0
vkϕ(t)r (t + t1 + (k1 − 1)Ts) dt. (23)
Finally, the estimated symbols are acquired with a sign detector:s(k1) = sign[d(k1)]. The RDT-based receiver is counteractiveto the effect of IFI by reason that a symbol level template isextracted from the received signal. The RDT contains completeinformation of the received symbol despite of the interferenceon the frame level.
B. NRDT-Based Receiver
The RDT-based receiver has considerably low complexityand is able to directly demodulate the transmitted symbols with-out explicit timing acquisition. It is noted that its BER perfor-mance depends primarily on the noise level of the constructedRDT. In practice, considering the efficiency of the system, it isimpossible to transmit a large number of training symbols toaverage out the noise, which calls for further improvement toenhance the detection performance.
Our objective here is to find a solution that can improvethe performance of the receiver while maintaining the limitednumber of training symbols. We have to point out that theadditive noise involved in the redundant part of the RDT isthe major cause for the deterioration of BER performancein the RDT-based receiver. To eliminate the redundant noisepart, the NRDT is constructed with the assistance of explicit
Fig. 3. Block diagram of the NRDT-based receiver.
timing operation. If the RDT vkϕ(t) is available, timing offset
is acquired by means of the following estimate:
τϕ = arg maxτ∈[0,Ts)
Ts∫0
v2kϕ
(t + τ)dt. (24)
Therefore, by eliminating the redundant noise, the NRDTvτϕ
(t) of duration Ts is obtained by
vτϕ(t)= vkϕ
(t + τϕ)= wR(t − Ts + ϕ + τϕ), t ∈ [0, Ts).(25)
Note that when no timing error exists in (24), τϕ = Ts − ϕholds. Thus, the NRDT vτϕ
(t) can be simplified as
vτϕ(t) = wR(t), t ∈ [0, Ts). (26)
For the extraction of the NRDT, the procedure of searchingover a continuous range [0, Ts) is added to obtain the timingoffset τϕ. In practice, the search is implemented over a grid offinite values τ = nTsam, where Tsam denotes the sampling pe-riod for timing operation, and n ∈ [0, �Ts/Tsam) is an integer.Thus, (24) becomes
nϕ = arg maxn∈[0,�Ts/Tsam)
Ts∫0
v2kϕ
(t + nTsam)dt. (27)
Eventually, the desired timing estimation τϕ = nϕTsam approx-imates τϕ within the ambiguity of Tsam.
In summary, the algorithm for constructing the NRDT iscarried out as follows.Step 1)–4) Apply the same four procedures as in RDT.Step 5) Find an estimate of nϕ in terms of nϕ =
arg maxn∈[0,�Ts/Tsam)∫ Ts
0 v2kϕ
(t + nTsam)dt.
Step 6) Acquire the timing estimation τϕ = nϕTsam and obtainthe NRDT vτϕ
(t), which can be expressed as
vτϕ(t) = vkϕ
(t + τϕ), t ∈ [0, Ts). (28)
Moreover, the block diagram for the NRDT-based receiveris illustrated in Fig. 3. Similar to the RDT-based receiver, thefollowing decision statistic is given:
d(k1) =
Ts∫0
vτϕ(t)r (t + t1 + τϕ + (k1 − 1)Ts) dt. (29)
Here, the transmitted symbol is demodulated with a sign detec-tor: s(k1) = sign[d(k1)].
LV et al.: TRAINING-BASED SYNCHRONIZATION AND DEMODULATION FOR UWB SIGNALS 3741
Finally, we conclude that the NRDT-based scheme outper-forms the RDT-based approach at the cost of higher complexitywith the introduction of explicit timing acquisition. The size ofthe sampling period Tsam directly determines the complexityand the performance of the NRDT-based algorithm, thus, Tsam
should be set accordingly. Moreover, simulations will show thatthe NRDT-based algorithm is robust to timing errors.
IV. PERFORMANCE ANALYSIS AND COMPARISONS
In this section, the error probabilities of the RDT- and theNRDT-based receivers are studied. The analytical results revealthe major factor for the performance degradation of the RDT-based receiver. After that, comparisons between the proposedschemes and the TDT-based algorithm are made. For the sake ofclarity, we will focus on a point-to-point link with the presenceof IFI.
A. Performance of the RDT-Based Receiver
As described in Section III, the RDT-based receiver does notrequire explicit timing operation, thus, timing estimation τϕ isnot involved in this analysis. The estimated information symbolcan be rewritten as
s(k1) = sign [d(k1)]
= sign[∫ 2Ts
0
vkϕ(t)r (t + t1 + (k1 − 1)Ts) dt
]. (30)
We assume that the RDT is correctly extracted. With theaddition of 2Ts-long Gaussian noise, the RDT vkϕ
(t) can beexpressed as
vkϕ(t)=wR(t−Ts+ϕ)+nH(t)+nT (t−Ts), t ∈ [0, 2Ts)
(31)
where nH(t) and nT (t) are noise terms of duration Ts and aredefined as
nH(t) =1
4N√
ε
4∑m=1
{N−1∑n=0
n(t + t1 + (m − 1)Ts + 4nTs)
}
× s(m + kϕ), t ∈ [0, Ts) (32)
nT (t) =1
4N√
ε
4∑m=1
{N−1∑n=0
n (t + t1 + (m − 1)Ts + 4nTs)
}
× s(m − 1 + kϕ), t ∈ [0, Ts). (33)
The received signal r(t + t1 + (k1 − 1)Ts) with a finiteinterval of [0, 2Ts) is expressed as
r (t + t1 + (k1 − 1)Ts)
=√
εs(Nϕ + k1)wR(t − Ts + ϕ)
+√
ε1∑
l=0
s (Nϕ + k1 + (2l − 1)) wR(t + ϕ − 2lTs)
+ n (t + t1 + (k1 − 1)Ts) . (34)
It is composed of the designated data symbol, two adjacentsymbols, and the noise term. Substituting (31) and (34) into(30), the decision variable d(k1) is computed as
d(k1) =√
εs(Nϕ + k1) {ER(0) + ςH1 + ςT1} + ςkϕ
+√
εs(Nϕ + k1 − 1)ςH2 +√
εs(Nϕ + k1 + 1)ςT2 (35)
where ER(u) :=∫ Ts
0 w2R(t − u)dt, ςH1 :=
∫ Ts
0 nH(t)wR(t −Ts + ϕ)dt, ςT1 :=
∫ 2Ts
TsnT (t − Ts)wR(t − Ts + ϕ)dt, ςkϕ
:=∫ 2Ts
0 vkϕ(t)n(t+ t1+ (k1−1)Ts)dt, ςH2 :=
∫ Ts
0 nH(t)wR(t +
ϕ)dt, and ςT2 :=∫ 2Ts
TsnT (t − Ts)wR(t − 2Ts + ϕ)dt.
As shown in Appendix B, the average signal-to-noise ratio(SNR) of d(k1) can be derived as
SNRRDT = εE2R(0)
N0
2ER(0) +
N0
4NER(0) +
N20
4NεWTs.
(36)
As N increases, the linear noise term (N0/4N)ER(0) as-ymptotically converges to zero. However, the time–bandwidthproduct WTs is large enough to counteract the increase of N .Therefore, the noise-by-noise term (N2
0 /4Nε)WTs cannot beignored. If the number of segments to be averaged is largeenough, SNRRDT can be simplified to
SNRRDT =εE2
R(0)N0
2 ER(0) + N20
4NεWTs
. (37)
This leads to the following approximate expression for the errorprobability of the RDT-based receiver:
PRDTe = Q(
√SNRRDT) = Q
(√4N(Eb/N0)2
2N(Eb/N0) + WNsTf
)(38)
where Eb := εER(0) is the energy for each symbol at thereceiver, and Q(·) is the Gaussian Q-function. From (38), wefurther observe that if a clean template is available, i.e., N →+∞, then the lower bound of PRDT
e follows, i.e.,
PRDTl = Q
(√2(Eb/N0)
). (39)
Based on the above analysis, it can be concluded that for apoint-to-point link, the RDT-based receiver can capture all themultipath energy without explicit timing acquisition. However,the expense for low complexity is the redundant symbol-longnoise included in the RDT. In Section IV-B, the error probabil-ity of the NRDT-based receiver is evaluated to verify the effectof noise cancellation.
B. Performance of the NRDT-Based Receiver
For the NRDT-based receiver, frame-level synchronization isperformed. We assume that the timing offset τϕ is estimated and
3742 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011
that the NRDT vτϕ(t) is available. The estimated data symbol
can be rewritten as
s(k1) = sign [d(k1)]
= sign[∫ Ts
0
vτϕ(t)r (t + t1 + τϕ + (k1 − 1)Ts) dt
].
(40)
Without loss of generality, we suppose that τϕ ≤ τϕ, and vτϕ(t)
is expressed as
vτϕ(t) = vkϕ
(t+τϕ)
=wR(t−τe)+nH(t+τϕ)+nT (t−Ts+τϕ) (41)
for t ∈ [0, Ts), where τe := τϕ − τϕ denotes the timing residualerror. The received signal r(t + t1 + τϕ + (k1 − 1)Ts) with afinite interval [0, Ts) is expressed as
r (t + t1 + τϕ + (k1 − 1)Ts)
=√
εs(Nϕ + k1)wR(t − τe)
+√
εs(Nϕ + k1 − 1)wR(t + Ts − τe)
+ n (t + t1 + τϕ + (k1 − 1)Ts) . (42)
Compared with the RDT-based receiver, it is composed of thedesignated symbol, the noise term, as well as a portion of theprevious symbol due to the timing error. Substituting (41) and(42) into (40), the decision variable can be calculated as
d(k1) =√
εs(Nϕ + k1) (ER(τe) + ςH3 + ςT3)
+ςτϕ+√
εs(Nϕ + k1 − 1)ςH4 (43)
where ςτϕ:=∫ Ts
0 vτϕ(t)n(t+t1+τϕ+(k1−1)Ts)dt, ςH3 :=
intTs0 nH(t+τϕ)wR(t−τe)dt, ςT3 :=
∫ Ts
0 nT(t − Ts + τϕ) ×wR(t − τe)dt, and ςH4 :=
∫ Ts
0 nH(t + τϕ)wR(t + Ts − τe)dt.As derived in Appendix B, the average SNR is expressed as
SNRNRDT =εE2
R(τe)N02 ER(τe) + N0
8N ER(0) + N20
8NεWTs
. (44)
Similar to the case of RDT, as N increases, the SNRNRDT issimplified to
SNRNRDT =εE2
R(τe)N02 ER(τe) + N2
08NεWTs
. (45)
Due to the existence of the timing residual error τe, the NRDT-based receiver cannot capture all the multipath energy. How-ever, if proper range of τe is given, such as |τe| < Tf , whichcan be achieved by the NRDT-based synchronizer, ER(τe) isapproximately equal to ER(0). Accordingly, the error probabi-lity of the NRDT-based receiver can be approximately given by
PNRDTe = Q
(√SNRNRDT
)
= Q
(√4N(Eb/N0)2
2N(Eb/N0) + (1/2)WNsTf
). (46)
Compared with the error probability of the RDT-based receiverin (38), it is observed that due to the elimination of the re-dundant noise, half of the time–bandwidth product WNsTf isreduced, which gives rise to the performance improvement ofthe NRDT-based receiver.
C. Comparisons
In this section, the analytical comparisons between the pro-posed algorithms and the TDT-based algorithm are presented[20], [21]. It is obvious that there is a common signal structurefor all the schemes, which is originated by a preamble withtraining sequence for timing acquisition and template extrac-tion, followed by a data field for detection. However, thesealgorithms also have some noticeable differences with respectto the following aspects.
1) Training Pattern: The proposed algorithms employ a re-peated sequence (1, 1, 1, −1) with a particular autocorrelationfunction to separate the truncated tail and the head of thereceived symbol waveform, which motivates the constructionof the RDT. The TDT-based scheme, however, depends on arepeated pattern (1, 1, −1, −1) to maximize the samples of thecross-correlation at the correct timing.
2) Timing Offset Estimation: The major difference betweenthe two synchronization schemes lies in the fact that the NRDT-based algorithm implements an ED to search the maximumenergy over a continuous range [0, Ts), while the motiva-tion for TDT is to locate the maximum of the square ofthe cross-correlation between pairs of successive symbol-longsegments [21]. For the sake of comparison, the complexity ofthe NRDT-based acquisition algorithm and TDT is discussed.Computational complexity of the NRDT-based synchronizer isdetermined mainly by the definite integral presented in (27).Taking a closer look at (27) and focusing on frame-level timingacquisition (i.e., Tsam = Tf ), we rewrite (27) as
nϕ = arg maxn∈[0,Ns)
n+Ns−1∑m=n
α(m)
where α(m) :=∫ (m+1)Tf
mTfv2
kϕ(t)dt with m ∈ [0, 2Ns) is an
integral over the interval Tf . The implementation of the max-imum energy search only needs 2Ns Tf -long integral oper-ations. In addition, the extraction of the RDT is considered,because it is performed prior to timing offset estimation. Asshown in (20), 8Ns Tf -long integral operations is executed inthis procedure. Note that the data for obtaining the RDT canbe reused for frame-level timing acquisition. Therefore, theNRDT-based synchronizer consumes a total of 8Ns Tf -longintegral operations. Similarly, the computation load of TDT isalso determined mainly by the definite integral (for details, see[20, eq. (25)]), which requires Ns Ts-long integral operations toachieve the frame-level timing acquisition. Since Ts = Ns Tf ,the computation load of a Ts-long integral operation is equiv-alent to that of Ns Tf -long integral operations. The computa-tional requirement of TDT is thus equal to N2
s Tf -long integraloperations. The inequality N2
s > 8Ns holds when Ns > 8. Forframe-level synchronization, the assumption of Ns > 8 is valid
LV et al.: TRAINING-BASED SYNCHRONIZATION AND DEMODULATION FOR UWB SIGNALS 3743
TABLE ICOMPLEXITY COMPARISONS BETWEEN THE NRDT- AND THE
TDT-BASED RECEIVERS IN TERMS OF Tf -LONG INTEGRAL OPERATION
to acquire explicit timing offset. If synchronization at the chipresolution is required, the advantage of the proposed algorithmbecomes more prominent, since the complexity of the integraloperations increases linearly with Ns, compared with that ofTDT. We remind that the above complexity analysis does notindicate either analog or digital operation. If digital operation isemployed, the complexity of the RDT extraction can be furtherreduced because the selection among four possible templatesdoes not require the same precision as that of explicit timingacquisition. Therefore, sampling rate can be considerably re-duced for the acquisition of RDT. In conclusion, the NRDT-based synchronization algorithm enjoys lower complexity incomparison with TDT.
3) Demodulation: For the proposed receivers, the templateextracted from the training sequence can be directly correlatedwith the data symbol for detection. However, the template fromTDT has a sign ambiguity and could not be implemented fordemodulation. After the training sequence, additional K posi-tive training symbols are transmitted to obtain the ambiguity-free template [21]. Therefore, a total of KNs Tf -long integraloperations is implemented. The proposed receivers clearly havelower complexity compared with the TDT-based receiver. Thecomplexity differences between the NRDT-based receiver andthe TDT-based receiver are summarized in Table I.
4) Error Probability: The error probabilities for the pro-posed receivers are derived in the previous section. For the sakeof comparison, the error probability of the TDT-based receiveris derived similarly and expressed as
PTDTe =
12Q
√√√√ ε2E2
R(0)N02 εER(0) + N2
04N WTs
+12Q
√√√√ε2 (ER(τe) − ER(Ts − τe))
2
N02 εER(0) + N2
04N WTs
. (47)
Considering that Q(·) is a decreasing function and
ε2 (ER(τe) − ER(Ts − τe))2
N02 εER(0) + N2
04N WTs
≤ ε2E2R(0)
N02 εER(0) + N2
04N WTs
we have
PTDTe ≥ Q
(√4N(Eb/N0)2
2N(Eb/N0) + WNsTf
)= PRDT
e . (48)
According to (38), (46), and (48), the error probabilities ofthe RDT-based receiver, the NRDT-based receiver, and the
TDT-based receiver are compared as follows:
PNRDTe < PRDT
e ≤ PTDTe . (49)
The analytical results show that the NRDT-based receiverachieves the best performance with the reduced noise effect.The RDT approach, which does not employ explicit timingoperation, is slightly better than the TDT-based receiver. Theseresults are further verified by the simulations in Section V.
V. SIMULATIONS
In this section, simulations are performed to evaluate the per-formance of the proposed synchronization and demodulationalgorithms. The basic assumptions and parameter settings areconcluded as follows.
1) The channel statistics are taken from the IEEE 802.15.3amodel CM1 [24], and the channel impulse responses aretruncated beyond 99 ns.
2) The monocycle pulse is taken as the second derivative ofa Gaussian function with unit energy, i.e.,
w(t) =[1 − 4π(t/σ)2
]exp
[−2π(t/σ)2]
(50)
where the pulsewidth σ is chosen based on the FederalCommunications Commission (FCC) requirements. Thepulse duration is set as Tp = 1 ns, and W ∼= 1 GHz is the3-dB bandwidth of the front-end low-pass filter.
3) Each symbol contains Ns = 32 frames with a duration ofTf = 100 ns.
4) The TH code is randomly picked up from the interval[0, Nh − 1], with Tc = 2 ns, and Nh = 50.
5) Without loss of generality, t1 is generated from a uniformdistribution over [0, Ts).
6) Considering the complexity and the validity, we focus onframe-level acquisition with Tsam = Tf .
A. Synchronization Performance
In this section, the comparisons of synchronization perfor-mance between the proposed NRDT-based scheme and theTDT-based scheme in [20] are investigated. Fig. 4 illustrates theacquisition probability with various length of the observationwindow, i.e., M = 16 and 64. It can be observed that theperformance gap between the two schemes is not distinct whenEb/N0 ranges from 0 to 18 dB. Therefore, we conclude thatthe timing estimator of the proposed NRDT-based scheme, withlower complexity, has almost the same accuracy as the TDTscheme. This conclusion can also be validated by employingnormalized mean square error (NMSE) metric in Fig. 5. Notethat the relationship between the NMSE and the acquisitionprobability is not linear. The NMSE is a more accurate metricthan the acquisition probability. The MSE, which is normalizedwith respect to T 2
s , is defined as
NMSE =Num∑k=1
(nϕ(k) · Tsam − τϕ
Ts
)2
/Num (51)
3744 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011
Fig. 4. Comparisons of acquisition probability between the NRDT- and TDT-based synchronizers.
Fig. 5. Comparisons of NMSE between the NRDT- and TDT-basedsynchronizers.
where nϕ(k) represents the kth estimated value of timingacquisition, and Num denotes the total number of Monte Carlotest.
In addition, the effect of signal averaging can be seen inFigs. 4 and 5. With increasing M , synchronization performancein terms of acquisition probability and NMSE is improved.The same trend can be observed by increasing Eb/N0. Thesimulations show that the noise can be mitigated effectively byaveraging the received signal.
In Fig. 6, the NMSE of the timing offset is demonstratedfor different frame numbers Ns and various sampling periodsTsam. Since Eb/N0 is dependent on Ns, we introduce Ep/N0,where Ep = Eb/Ns is the received average energy per pulse.Fig. 6 shows that for Ep/N0 = −6 dB and M = 64, with alarger number of frames Ns, the precision of the proposed syn-chronizer increases, which results in the decrease of the NMSE.
Fig. 6. NMSE of the NRDT-based scheme as a function of Ns for varioussampling periods.
Fig. 7. BER comparisons of the RDT, NRDT-, and TDT-based receivers alongwith their theoretical curves.
Moreover, the proposed synchronization technique works wellfor low complexity timing acquisition with flexible samplingperiod Tsam. We note that the curve for Tsam = Tf almostoverlaps with the one for Tsam = 2Tf , which demonstrates thepotential of complexity reduction with proper setting of Tsam.
B. BER in the Single-User Scenario
Fig. 7 illustrates the detection performance of the RDT-,NRDT-, and TDT-based receivers for M = 16 and 64. Asshown, the BER performance is improved as M increases. Ofall the three receivers, the NRDT-based receiver achieves thebest performance due to the elimination of the redundant noisypart of RDT. Moreover, the BER performance of the RDT-based receiver is slightly better than that of the TDT-basedreceiver when M = 16 and is almost the same when M = 64.
LV et al.: TRAINING-BASED SYNCHRONIZATION AND DEMODULATION FOR UWB SIGNALS 3745
Fig. 8. BER of the NRDT- and TDT-based receivers in the presence of timingerror.
With the advantage of considerably lower complexity, the RDT-based receiver is more attractive for implementation. The sim-ulations also support the conclusions of (49). For comparison,the theoretical BER curves of the three receivers are presentedin Fig. 7. The simulations are in good agreement with thetheoretical results of (38), (46), and (47), particularly for highervalues of M .
C. Robustness to Mistiming
Fig. 8 shows the influence of timing residual error τe on theBER performance of the NRDT- and TDT-based receivers withM = 64 for various Eb/N0. Note that the mistiming is mea-sured with τe/Tf since the search for timing residual is operatedon a time shift of the sampling period. With the variation of |τe|,the BER performance of the NRDT-based receiver significantlyoutperforms that of the TDT-based receiver for fixed Eb/N0.Therefore, the NRDT-based receiver achieves better robustnessto mistiming. In other words, the proposed algorithm enjoyswider range of potential timing error while maintaining thesame BER.
D. Robustness to Multiuser Interference (MUI)
Fig. 9 reveals the robustness of the RDT-, NRDT-, and TDT-based receivers in the presence of MUI with M = 64. All theactive users make asynchronous access to the channel. In accor-dance with the assumption in [20], during the stage of templateextraction and demodulation, the interfering users transmitzero-mean independent and identically distributed informationsymbols. The performance of the NRDT-based receiver showsmore robustness to MUI in comparison with the TDT- and RDT-based receivers for various numbers of interfering users Nu.Since no MUI suppression is invoked, MUI becomes the maincause for BER degradation at higher SNR.
Fig. 9. BER of the RDT-, NRDT-, and TDT-based receivers for variousnumbers of interfering users Nu.
VI. CONCLUSION
In this paper, a low-complexity synchronization and efficientdemodulation scheme has been proposed. Based on the autocor-relation property of the proposed training sequence, the RDTis constructed to demodulate the transmitted symbols withoutexplicit timing acquisition. The RDT-based receiver obtains agood tradeoff between general accuracy and rapid demodula-tion. Furthermore, when the RDT is available, the proposedalgorithm can acquire timing offset by energy detection anddemodulate the transmitted symbols with the assistance of theNRDT. Simulations and analysis demonstrate that the NRDT-based receiver achieves superior BER performance with thereduction of time–bandwidth product compared with the RDT-based receiver. Both the RDT- and NRDT-based receiversoutperform the TDT-based receiver in terms of BER andcomplexity. In addition, the NRDT-based receiver works wellfor low complexity timing acquisition with flexible samplingperiod and is robust to mistiming as well as MUI.
3746 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011
APPENDIX BPROOF OF (36) AND (44)
The SNR of the RDT-based receiver is derived first. ςkϕcan
be broken down into three terms as ςkϕ= ς1 + ς2 + ς3, where
the linear noise term is denoted by
ς1 =
2Ts∫0
wR(t − Ts + ϕ)n (t + t1 + (k1 − 1)Ts) dt. (53)
The noise-by-noise product terms are written as
ς2 =
Ts∫0
nH(t)n (t + t1 + (k1 − 1)Ts) dt (54)
ς3 =
2Ts∫Ts
nT (t)n (t + t1 + (k1 − 1)Ts) dt. (55)
The noise terms are similar to the ones found in transmitted ref-erence or TDT [20], [25]; thus, the derivations of the mean andvariance are not presented in detail. When the time–bandwidthproduct or Ns is large enough, the distribution of the noisecomponents ς1, ς2, and ς3 are conditionally Gaussian andmutually uncorrelated. The linear noise term is Gaussian withzero mean and variance of Var[ς1] = N0ER(0)/2, where Var[·]denotes variance operation. The noise-by-noise product termsς2 and ς3 can also be approximately Gaussian with zero meanand variance Var[ς2] = Var[ς3] = N2
0 WTs/8Nε.Similarly, ςkϕ
, ςH1, ςH2, ςT1, and ςT2 can be approximated asmutually uncorrelated Gaussian variables with zero mean andvariance of
Var[ςkϕ
]=
N0
2ER(0) +
N20 WTs
4Nε(56)
Var[ςH1] = Var[ςT2] =N0
8NεER(Ts − ϕ) (57)
Var[ςH2] = Var[ςT1] =N0
8NεER(−ϕ). (58)
In addition, according to the definition of ER(u) in Section IV,it is easily obtained that
ER(Ts − ϕ) + ER(−ϕ) = ER(0). (59)
Therefore, the SNR of the RDT-based receiver is expressed as
SNRRDT
=E[d2(k1)
]Var [d(k1)]
=εE2
R(0)
Var[ςkϕ
]+ε(Var[ςH1]+Var[ςH2]+Var[ςT1]+Var[ςT2])
=εE2
R(0)N02 ER(0)+ N0
4N ER(0)+ N20
4NεWTs
(60)
where E[·] denotes mean operation.
The SNR of the NRDT-based receiver is derived similarly.ςτϕ
, ςH3, ςT3, and ςH3 can be approximated as mutually uncor-related Gaussian variables with zero mean and variance of
Var[ςτϕ
]=
N0
2ER(τe) +
N20
8NεWTs (61)
Var[ςH3] =N0
8NεER(τϕ) (62)
Var[ςT3] =N0
8Nε(ER(τϕ − Ts) − ER(τe − Ts)) (63)
Var[ςH4] =N0
8NεER(τe − Ts). (64)
The SNR of the NRDT-based receiver is expressed as
SNRNRDT =εE2
R(0)Var[ςτϕ
]+ε (Var[ςH3]+Var[ςT3]+Var[ςH4])
=εE2
R(τe)N02 ER(τe)+ N0
8N ER(0)+ N20
8NεWTs
. (65)
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Tiejun Lv (M’08) received the M.S. and Ph.D. de-grees in electronic engineering from the Universityof Electronic Science and Technology of China,Chengdu, China, in 1997 and 2000, respectively.
From January 2001 to December 2002, he wasa Postdoctoral Fellow with Tsinghua University,Beijing, China. From September 2008 to March2009, he was a Visiting Professor with the Depart-ment of Electrical Engineering, Stanford University,Stanford, CA. He is currently a Professor with theSchool of Information and Communication Engi-
neering, Beijing University of Posts and Telecommunications. He is the authorof more than 100 published technical papers on the physical layer of wirelessmobile communications.
Dr. Lv is a Senior Member of the Chinese Electronics Association. He wasthe recipient of the “Program for New Century Excellent Talents in University”Award from the Ministry of Education, China, in 2006.
Yongwei Qiao received the B.S. and M.S. degreesin electronic engineering from Hefei University ofTechnology, Hefei, China, in 2003 and 2006, respec-tively, and the Ph.D. degree from the Beijing Uni-versity of Posts and Telecommunications, Beijing,China, in 2009.
He is currently with the Beijing University ofPosts and Telecommunications and an engineer ofthe 96669 Troops. His research interests include theareas of signal processing, communication theory,and virtual instruments. His current research focuses
on ultrawideband wireless communications.
Zicheng Wang received the B.S. degree in electricalengineering from Zhejiang University of Technol-ogy, Zhejiang, China, in 2009. He is currently work-ing toward the M.S. degree in electrical engineeringwith the School of Information and Communica-tion Engineering, Beijing University of Posts andTelecommunications, Beijing, China.
His research interests include communication, net-working, and signal processing. His current researchfocuses on ultrawideband communication and net-work coding.
