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Physical Communication 2 (2009) 265–273

Contents lists available at ScienceDirect

Physical Communication

journal homepage: www.elsevier.com/locate/phycom

Full length article

Hybrid coherent and frequency-shifted-reference ultrawideband radioI

H. Liu a,∗, A.F. Molisch b, D. Goeckel c, P. Orlik da School of Electrical Engineering and Computer Science, Oregon State University, 97331 Corvallis, OR, United Statesb Department of Electrical Engineering, University of Southern California, Los Angeles, CA, United Statesc Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA, United StatesdMitsubishi Electric Research Lab, Cambridge, MA, United States

a r t i c l e i n f o

Keywords:Pulsed ultrawidebandFrequency-shifted referenceCoherent rakeHybrid modulation

a b s t r a c t

Ultrawideband communications often occur in heterogeneous networks where differentreceivers have different complexity and energy consumption requirements. In this caseit is desirable to have a modulation scheme that works well with coherent receiversas well as simpler receivers, namely transmitted-reference (TR) receivers. In particular,we consider a TR scheme that employs slightly frequency-shifted-reference (FSR) signals[D.L. Goeckel, Q. Zhang, Slightly frequency-shifted-reference ultrawideband (UWB) radio,IEEE Trans. Commun. (2007)] and thus avoids one of themaindrawbacks of conventional TRschemes, namely the need to implement a delay line.Wepropose and analyze amodulationscheme thatworkswellwith both FSR receivers (where it has at least the sameperformanceas conventional TR modulation), and coherent receivers. Coherent receivers receivingconventional TR modulation suffer a 3 dB penalty, because they cannot make use of theenergy invested into the reference pulse. Our proposed scheme avoids this drawback byincluding adata preprocessor that canbe viewedas a nonsystematic rate-1/2 convolutionalcode. These codes give 1.5 dB gain over our previously proposed constraint-length-twosystematic codes at a BER of 1 × 10−4 in 802.15.3a CM4 multipath fading channels. Wealso develop a sliding-window based scheme to derive the template waveform that isneeded for coherent rake receivers. This scheme exploits the data preprocessor structureand flexibly uses the received signal over a certain window. The distortion to the self-derived template waveform is a decreasing function of the window length; in the extremebut unrealistic case of a very long window length in a slowly fading channel, the self-derived template waveform is noiseless (i.e., the ideal template without distortion).

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

One of the key challenges for pulsed ultrawideband(UWB) systems [1–4] is the construction of low-cost re-ceivers that work well in multipath environments. In im-pulse radio, each symbol is represented by a series of

I This paper was presented in part at the IEEE Globecom’07,Washington DC, USA, Nov. 2007. The work by D. Goeckel is supported inpart by the National Science Foundation under ECS-0725616.∗ Corresponding author. Tel.: +1 541 737 2973.E-mail addresses: hliu@eecs.oregonstate.edu (H. Liu),

molisch@usc.edu (A.F. Molisch), goeckel@ecs.umass.edu (D. Goeckel),porlik@merl.com (P. Orlik).

pulses. In multipath environments, pulses suffer delaydispersion, and their energy has to be collected by rakereceivers or equivalent structures [5]. Coherent rake re-ceivers that combine all resolvable multipath components(MPCs) [6] are optimum; however, tracking, estimating,and combining a large number of MPCs (e.g., tens oreven hundreds in indoor environments [7,8]), or even asubset of all the resolvable paths [9], result in complexreceivers. Consequently, simpler alternatives, in particulartransmitted-reference (TR) receivers, have drawn signifi-cant attention in recent years [10–14].In conventional TR systems, the symbol is represented

by a series of pulse doublets, where each doublet consists

1874-4907/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.phycom.2009.06.003

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266 H. Liu et al. / Physical Communication 2 (2009) 265–273

of two pulses separated by a fixed delay: the first,unmodulated, pulse serves as a reference pulse; the secondpulse is data modulated and is referred to as the datapulse. The received signal can simply be demodulatedby correlating it with a delayed version of itself, andintegrating the resulting signal. The main problem forimplementing TR receivers is the delay unit, which musthandle wideband analog signals and must be precise,making it difficult to build in low-power integratedfashion [15,16]. In order to avoid the delay unit, analternative, called slightly frequency-shifted reference(FSR), was recently proposed by one of us [15]. In thisscheme, the pulse doublet does not consist of pulsesthat are offset in the delay domain, but rather in thefrequency domain. A receiver thus only has to implementa mixer, and not a delay line, and is thus much easier tobuild; furthermore such a receiver can actually performslightly better than the conventional TR receiver overboth additivewhite Gaussian noise (AWGN) andmultipathfading channels [15].Devices in a UWB network often have different cost/

complexity/performance requirements. For example, someUWB devices may be battery-operated and thus require asimple, energy-saving receiver, while other devices maybe connected to a power supply and thus can use ahigh-complexity, power hungry receiver that offers bettercommunication range [17]. The resulting heterogeneousnetwork requires a ‘‘universal’’ modulation method com-patible with different types of receivers such as coherentrake and FSR or TR receivers. Technically, it is possible todemodulate FSR signals with a coherent receiver, by sim-ply ‘‘throwing away’’ the reference pulses. However, thisimplies a 3 dB signal energy penalty compared to a sys-tem that is designed to use coherent receivers only. Onthe other hand, signals designed for coherent receivers(i.e., those without reference pulses) obviously cannot bedemodulated by a FSR receiver. In [18], we proposed a hy-bridmodulation scheme that enables efficient reception byboth coherent and TR receivers. The key idea is to makethe ‘‘reference pulse’’ information bearing, without modi-fying the phase relationship between the reference pulseand data pulse. This makes sure that the energy in thereference pulse is not ‘‘wasted’’ for the coherent receiver,and recovers the 3 dB loss by ‘‘normal’’ TR signaling. Fur-thermore, the information in the reference pulse is madedependent on the previous information symbol, which in-troducesmemory into themodulation, and leads to furtherperformance gain for the coherent receiver.In the current paper, we show how our hybrid modu-

lation scheme can be modified to enable coherent rake re-ceivers and FSR receivers in the samewireless network, andpresent severalmodifications that further improve theper-formance. In particular, our contributions are:

• we propose a hybrid modulation scheme that workswith coherent and FSR receivers.• we proposed a modified encoding structure. The datapreprocessor in the basic hybrid scheme proposedin [18] can be viewed as a rate-1/2 systematicconvolutional code with a constraint length two. Weshow in this paper that the systematic code can beextended into a nonsystematic convolutional codewith

longer constraint lengths that gives a higher codinggain, while the desired properties between the datapulse and reference pulse are still maintained for FSRreceivers.• we develop a channel estimation scheme that effi-ciently exploits the proposed signaling structure to ob-tain nearly noise-free channel coefficient estimates forcoherent reception in slowly fading channels.• we derive analytical bounds for the coding gain of thenew preprocessing structure.• we show that a frequency shift proposed in [15] can bereduced by a factor of two, improving performance, andallowing for the transmission of higher-data rates.• we analyze the resulting receiver structure and perfor-mance for the resulting UWB transmission scheme. Wederive closed-form equations for the BER in multipathchannels.

The remainder of the paper is organized in the followingway. Section 2 explains the improved hybrid FSR-coherentUWB scheme, and coherent and FSR receivers for the pro-posed hybridmodulation scheme are analyzed in Section 3.Section 4 presents simulation results of the proposedscheme with a coherent receiver. Comparison is madebetween the proposed improved hybrid FSR-coherentscheme and the basic hybrid TR-coherent scheme de-scribed in [18]. Concluding remarks are given in Section 5.

2. Hybrid FSR-coherent system

2.1. Transmitter

2.1.1. Transmitted signalWe consider a time-hopping impulse radio system,

where each bit interval Tb is partitioned into Nf frames,each having a duration Tf (Tb = Nf Tf ). Let us define a basictemplate waveform for the transmission of a bit

u(t) =Nf−1∑j=0

djp(t − jTf − cjTc) (1)

where p(t) is the basic UWB pulse shape normalizedso that

∫∞

−∞p2(t)dt = 1/Nf and

∫∞

−∞u2(t)dt = 1.

The elements of the pseudorandom sequence cj, whichdetermines the position of the pulse within each frame,are uniformly distributed integers between 0 and Tf /Tc ,where Tc is the chip duration. The elements of thepseudorandom sequence dj are taken from −1,+1, andeffect a randomization of the polarities, which is usefulfor smoothing the transmit spectrum and enhancing themultiuser separation. For ease of notation, we henceforthomit the dj, since they do not influence the operation ofthe TR receiver (where their impacts on reference pulseand data pulse cancel out) or the coherent receiver (wherethe receiver cancels its effect by multiplying the receivedsignal with the sequence dj).We furthermore define waveforms for the reference

and data signals (assuming equal energies)

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H. Liu et al. / Physical Communication 2 (2009) 265–273 267

Fig. 1. Block diagram of the slightly frequency-shifted-reference hybrid system transmitter.

xr(t) =∞∑

i=−∞

√Eb/2sr [i]u(t − iTb) (2a)

xd(t) =∞∑

i=−∞

√Eb/2sd[i]u(t − iTb). (2b)

The data streams sr [i] and sd[i] (sr [i], sd[i],∈ −1,+1)modulating the reference and data signals will be dis-cussed in Section 2.1.2.In an FSR scheme, the total transmitted waveform x(t)

is

x(t) = xr(t)+ xd(t)√2 cos(2π f0t)

=

∞∑i=−∞

√Eb2

sr [i] +

√2sd[i] cos(2π f0t)

× u(t − iTb) (3)

where f0 is the shift of the center frequency of the referencepulses relative to that of the data pulses.In order to allow simple demodulation, it is essential

that f0 be chosen such that the first and second term ofthe r.h.s. of Eq. (3) are orthogonal or at least approximatelyorthogonal. At the same time, f0 should be chosen to beas small as possible; this ensures that the reference signaland the (frequency-shifted) data signal ‘‘see’’ the samepropagation channel even in frequency-selective channels.In the following, we derive the minimum frequencyspacing. Without loss of generality, let us focus on the firstbit interval, i.e., i = 0:∫ Tb

0xr(t)xd(t) cos(2π f0t)dt

=Eb2sr [0]sd[0]

∫ Tb

0

Nf−1∑j=0

p2(t − jTf − cjTf )

× cos(2π f0t)dt

≈Eb2sr [0]sd[0]

Nf−1∑j=0

cos(2π f0jTf )∫ Tb

0p2(t − jTf )dt

=Eb2Nfsr [0]sd[0]

Nf−1∑j=0

cos(2π f0jTf ) = 0 (4)

where the approximation is obtained by considering thatp(t − jTf ) is a very narrow pulse, compared with a bitinterval Tb, located at t = jTf . When Nf is large and theNf pulses are equally spaced over Tb, choosing f0 = 1

2Tbwill imply that the summands cos(2π f0jTf ) in the last line

are the uniform samples of the sinusoidal signal cos(2π f0t)over the first half of its fundamental period from 0 to Tb,which is approximately equal to zero.This choice of the frequency-shift reduces the differ-

ence in the center frequencies of the data pulse and ref-erence pulse by a half compared with the suggested valueof f0 = 1

Tbin [15].

2.1.2. Data preprocessorIn the conventional FSR scheme, the reference bits are

fixed, i.e., sr [i] = 1. However, coherent receivers canachieve better performance if the reference signal alsocarries information; of course, it is essential to retain asignaling structure that allows demodulation by an FSRreceiver.The proposed hybrid scheme is shown in Fig. 1, where

the ‘reference’ bit br [i] and ‘data’ bit bd[i] can be consideredas the outputs of a rate-1/2 convolutional encoder. In therelated scheme of [18], we proposed a convolutional en-coder with constraint length two, resulting in a minimumfree distance of

√6. We now investigate a new encoding

scheme with a constraint length three and code genera-tor polynomial [1 ⊕ D ⊕ D2, 1 ⊕ D2], where ⊕ denotesmodulo-two addition. Note that the ‘reference’ bit is not re-ally a reference bit as in a conventional TR system; rather,it is one of the two output bits of the data preprocessor,i.e., the rate-1/2 convolutional encoder. Obviously, a co-herent receiver can detect the transmitted bits by demod-ulating both the reference bit and the data bit, and thenapplying a Viterbi decoder to recover the original informa-tion bits.This encoder structure must satisfy certain conditions

so that FSR receivers can detect the transmitted signalas well. Table 1 shows the polarities of the modulatedreference and data waveforms, and their phase/polaritydifference for all eight combinations of three consecutiveinput bits. As seen from the table, the phase difference ofthe pair of pulses – the reference and data pulses – solelydepends on input information bit b[i]; therefore, an FSRreceiver can also demodulate the received signals.The ith data bit and the ith reference bit are expressed

as

bd[i] = b[i+ 1] ⊕ b[i− 1] (5a)br [i] = b[i+ 1] ⊕ b[i] ⊕ b[i− 1]. (5b)

For bipolar signaling, the symbol mapper simply performsthe following mapping: br [i] = 0/1 → sr [i] = −1/1;bd[i] = 0/1→ sd[i] = −1/1.

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268 H. Liu et al. / Physical Communication 2 (2009) 265–273

Table 1Input–output combinations of data preprocessor designed as a rate-1/2convolutional code with constraint length three.

b[i+ 1] b[i] b[i− 1] sr [i] sd[i] Phase difference betweensr [i] and sd[i] ()

0 0 0 −1 −1 00 0 1 +1 +1 00 1 0 +1 −1 1800 1 1 −1 +1 1801 0 0 +1 +1 01 0 1 −1 −1 01 1 0 −1 +1 1801 1 1 +1 −1 180

For coherent rake receivers, the data preprocessoris effectively a rate-1/2 convolutional encoder with aconstraint length three. The minimum Euclidean distanceis√10ε, resulting in a 4 dB coding gain, which is 2.22 dB

higher than the basic hybrid TR-coherent scheme in [18].The proposed data preprocessor can be further ex-

tended to a rate-1/2 convolutional encoder with a longerconstraint length. For example, the convolutional codewith generator polynomial [1⊕D⊕D2⊕D3, 1⊕D2⊕D3]is a good choice with a constraint length four. The relation-ship among the input bits, the reference bit, and the databit for this preprocessor is shown in Table 2. TheminimumEuclidean distance with this preprocessor becomes

√12ε,

which results in an additional 0.8 dB coding gain over thepreprocessor shown in Fig. 1, or 3 dB coding gain over thescheme in [18].For even higher constraint lengths, it becomes difficult

to find good codes with the maximum, or near maximum,free distance while the phase relationship between thereference pulse and data pulse required by FSR receiversis still maintained. A longer constraint length wouldgenerally result in a higher coding gain at the expense ofa higher decoding complexity. Thus, system complexity-performance tradeoff can be made flexible by choosing anappropriate preprocessor structure.

2.2. Receiver

2.2.1. Received signalThe standardized 802.15.3a channel model [7] de-

scribes the channel impulse response as

h(t) =L−1∑l=0

hlδ(t − τl) (6)

where L is the total number of multipath components,hl is the channel fading coefficient for the lth path, τl isthe arrival time of the lth path relative to the first path(τ0 = 0 assumed), and δ(t) is the Dirac delta function. Thechannel gain hl is modeled as hl = λlβl, where λl takeson the values of−1 or 1 with equal probability and βl hasa lognormal distribution [7]. Since multipath componentstend to arrive in clusters [7], τl in (6) is expressed as τl =µc + νm,c , where µc is the delay of the cth cluster thatthe lth path falls in, νm,c is the delay (relative to µc) of themth multipath component in the cth cluster. The relativepower of the lth path to the first path can be expressedas E|hl|2 = E|h0|2e−µc/Γ e−νm,c/γ , where E· denotes

Table 2Input–output combinations of data preprocessor designed as a rate-1/2convolutional code with constraint length four.

b[i+ 1] b[i] b[i− 1] b[i− 2] sr [i] sd[i] Phase differencebetween sr [i] & sd[i] ()

0 0 0 0 −1 −1 00 0 0 1 +1 +1 00 0 1 0 +1 +1 00 0 1 1 −1 −1 00 1 0 0 +1 −1 1800 1 0 1 −1 +1 1800 1 1 0 −1 +1 1800 1 1 1 +1 −1 1801 0 0 0 +1 +1 01 0 0 1 −1 −1 01 0 1 0 −1 −1 01 0 1 1 +1 +1 01 1 0 0 −1 +1 1801 1 0 1 +1 −1 1801 1 1 0 +1 −1 1801 1 1 1 −1 +1 180

expectation, Γ is the cluster decay factor, and γ is the raydecay factor. Note that, different from common basebandmodels of narrow-band systems, hl is real-valued.Given the transmitted signal x(t) in (3) and the channel

impulse response h(t) given by (6), the received signal isexpressed as

r(t) =L−1∑l=0

hlx(t − τl)

=

∞∑i=−∞

√Eb2

L−1∑l=0

hlu(t − iTb − τl)sr [i]

+√2sd[i] cos(2π f0(t − τl)) + n0(t) (7)

where n0(t) is the additive white Gaussian noise with thetwo-sided power spectral density N0/2.

2.2.2. FSR receiverThe FSR receiver is similar to the one described in [15]

and is shown in Fig. 2 for convenience. The receivedsignal is bandpass-filtered,1 squared, multiplied with√2 cos(2π f0t), and then integrated over each bit interval.Using an analysis similar to [15], we can show that for

the ith bit, the integrator output ri in the absence of noiseand multipath fading can be expressed as

ri =∫ (i+1)Tb

iTb

(√Eb2sr [i] +

√2sd[i] cos(2π f0t)

× u(t − iTb)

)2√2 cos(2π f0t)dt

= Ebsr [i]sd[i] (8)

1 The bandpass filter (BPF) has bandwidth B, which is chosen to beequal to or larger than the signal’s 10 dB bandwidth, to avoid severe,filter-induced distortion to the signal. For simplicity of analysis, it is alsoassumed that the BPF is ideal and its bandwidth B is an integermultiple of1/(2Tb). Let the noise at the output of the BPF be n(t). The filtered noiseis no longer white, but its autocorrelation can be very narrow in time dueto the ultrawide filter bandwidth B. The BPF does not change the receivedsignal component. This assumption applies to the coherent receiver to bediscussed in the next section.

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H. Liu et al. / Physical Communication 2 (2009) 265–273 269

Fig. 2. FSR receiver for the proposed hybrid FSR-coherent scheme.

where the second equation can be derived using therelationship given in (4). The analysis for multipathchannels follows the approach in [15], which also leads tothe product term sr [i]sd[i] in the decision variable givenby (8). From Tables 1 and 2, we observe that the productterm sr [i]sd[i] maps exactly to the symbol-mapped inputbits. Therefore, if the relationship between the referenceand data bits as described in Section 2.1.2 is maintained,FSR receivers can demodulate the transmitted signal.

2.2.3. Coherent receiver and channel estimationThe coherent receiver for the hybrid FSR scheme is

shown in Fig. 3. The received signal for the ith bit intervalis first passed through a BPF (see Section 2.2.2) and thencorrelated with wi(t) =

∑L−1l=0 hlu(t − iTb − τl). This

correlator requires the knowledge of hl and τl; thus,it is equivalently a coherent rake receiver. The correlatoroutput is processed further to generate the estimate ofboth the reference bit and the data bit. For the data bit,the correlator output is integrated over each bit interval;for the reference bit, the the correlator output is firstmultiplied by

√2 cos(2π f0t) and then integrated over each

bit interval. Finally, both estimates for the data bits sd[i]and reference bits sr [i] are applied for Viterbi decoding,yielding the estimated input bits b[i]. This is illustratedin Fig. 3.Note that in practice it is usually impossible to combine

all received paths considering receiver complexity; a par-tial rake can be realized simply by letting the coefficientshl that the receiver does not combine equal to zero.For low- to medium-data rates, it is reasonable to

assume that there is no inter-symbol interference (ISI). Thederivation of the decision variables for sr [i] and sd[i], rr [i]and rd[i], is similar; thus, let us focus on deriving rd[i]:

rd[i] =∫ (i+1)Tb

iTb

√Ebsr [i]

L−1∑l=0

h2l u2(t − iTb − τl)

× cos(2π f0t)dt

+

∫ (i+1)Tb

iTb

√Eb2sd[i]

L−1∑l=0

h2l u2(t − iTb − τl)

× 2 cos(2π f0t − 2π f0τl) cos(2π f0t)dt

+

∫ (i+1)Tb

iTbn(t)

L−1∑l=0

hlu(t − iTb − τl)√2

× cos(2π f0t)dt. (9)

For low-data-rate applications for which maxτl Tb,the first term of the r.h.s. of (9) is approximately equal tozero for the same reason as explained below Eq. (4). Thus

Fig. 3. Coherent receiver for the proposed hybrid FSR-coherent scheme.

we have

rd[i] ≈

√Eb2sd[i]

L−1∑l=0

h2l

∫ (i+1)Tb

iTbu2(t − iTb − τl)

× cos(2π f0τl)dt

+

√Eb2sd[i]

L−1∑l=0

h2l

∫ (i+1)Tb

iTbu2(t − iTb − τl)

× cos(4π f0t − 2π f0τl)dt + ξ [i]

≈

√Eb2sd[i]

L−1∑l=0

h2l + ξ [i] (10)

where we have applied the approximation∫ (i+1)TbiTb

u2(t −iTb − τl) cos(2π f0τl)dt ≈ 1 because maxτl Tb orf0τl ≈ 0, and the second term on the r.h.s. is approximatelyequal to zero for the same reason given in deriving Eq. (4).In the analysis so far, we have assumed that wi(t) =∑L−1l=0 hlu(t − iTb − τl) is available for detection of the

ith bit. This requires the receiver to estimate hl andτl and then construct the receiver template waveformwi(t) based on the transmitted template waveform givenin (1). We analyze a scheme that efficiently exploitsthe proposed signaling structure to provide an estimateof wi(t) for the coherent receiver. In a slowly fadingchannel, hl and τl changes very little over a blockof many bit intervals. When a coherent rake receiveris applied to detect the conventional TR signals, thereference bits could be exploited for channel estimation. Adisadvantage of making the reference bits data dependentin the hybrid scheme is that the reference bits cannotbe exploited directly to estimate hl. Without an explicitreference signal, a decision-directed approach could beemployed [14]. However, the decision-directed approachcould potentially suffer from error propagation [14].We investigate a sliding-window approach which

exploits the FSR receiver structure to estimate thecorrelation-template wi(t) for detecting the ith symbol.This approach does not suffer from the error propagationeffects. Note that a coherent receiver has all the maincomponents that an FSR requires such as a BPF, correlator,integrator, oscillator, etc., except a squaring device. For

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270 H. Liu et al. / Physical Communication 2 (2009) 265–273

clarity of description, let us write the received signal at theBPF output in the ith bit interval as

ri(t) =

√Eb2sr [i]

L−1∑l=0

hlu(t − iTb − τl)

+

√EbL−1∑l=0

hlu(t − iTb − τl)sd[i]

× cos(2π f0(t − τl))+ ni(t) (11)where the first term of the r.h.s. is exactly the corre-lation template subject to data-dependent scaling factor√Eb2 sr [i].

Let b[i] be the bit decisions by the FSR receiver andb[i − j], i = 1, . . . ,N , be the temporary FSR bit decisionsover a block of N bit intervals preceding the ith bit. Thetemporary estimates of the reference and data symbolsover the same block of N bit intervals, sr [i − j] and sd[i −j], j = 1, . . . ,N , can be obtained by passing b[i] throughthe data preprocessor. With the FSR bit decisions, we can‘‘data-demodulate’’ the received signal in each bit intervalgiven in (11), forming the estimate ofwi(t) as

wi(t) =

√2Eb

1N

N∑j=1

sr [i− j]ri−j(t). (12)

The estimated correlation template is distorted byone multiplicative distortion and two forms of additivedistortions: (a) due to the slightly frequency-shifted datacomponent and (b) due to noise. The multiplicativedistortion is due to the FSR bit decision errors, whichcauses a constant scaling factor to the template. Thisdistortion term is written as 1N

∑Nj=1 sr [i − j]sr [i − j]. In

the limiting case when N → ∞, it becomes (1 − Pb,sr ),where Pb,sr is the error rate of the re-encoded reference bitsfrom b[i]. This distortion does not degrade the performanceof the coherent rake. The two additive distortions areexpressed as

Isd(t) =

(1N

N∑j=1

sr [i− j]sd[i− j]

)

×

L−1∑l=0

hlu(t − iTb − τl)√2 cos(2π f0(t − τl)) (13a)

In(t) =2

N√Eb

N∑j=1

sr [i− j]ni−j(t) (13b)

where in (13a) we have applied the fact that over the Ndifferent bit intervals, all quantities except sr [i] and sd[i]remain the same. When N is large (e.g., 20), the term∑Nj=1 sr [i− j]sd[i− j]/N in (13a) approaches Gaussian with

a zero mean and variance 1/N . In(t) is also zero-meanGaussian with a power spectral density 2

N1

Eb/N0, which is

inversely proportional to Eb/N0 and block size N . In thelimiting but unrealistic case when N →∞, the estimatedtemplate is expressed as

limN→∞

wi(t) = (1− Pb,sr )L−1∑l=0

hlu(t − iTb − τl) (14)

which is precisely the ideal template waveform, sincethe constant scaling factor (1 − Pb,sr ) does not affectthe detection performance. For the realistic case when Nis finite (e.g., 10), the self-derived template for coherentreceivers is a slightly distorted version of the idealtemplate.

3. Performance analysis

3.1. FSR receiver in multipath

As described in Section 2.2.1, the received signal at theBPF output is expressed as r(t) = x(t) ∗ h(t) + n(t),where h(t) is given in (6), x(t) is given in (3), ∗ denotesconvolution, andn(t) is the zero-meanGaussian noisewitha one-sided spectral density N0. Again using an analysissimilar to [15], the BER over multipath channels can bewritten as

PFSR = Eh

QEbL−1∑l=0h2l cos(2π f0τl)√

52EbN0

L−1∑l=0h2l + TbN

20B

(15)

where Eh[·] denotes the expectation over the set of channelcoefficients hl. While a numerical evaluation of theexpectation operator has been used in the past, we derivein the following a closed-form expression.In the following, we again assume that f0τl 1,

which is fulfilled for signaling at low- to medium-datarates. Let γ =

∑L−1l=0 h

2l =

∑L−1l=0 β

2l . Note that βl

are independent of one another. Since βl is a lognormalrandom variable (RV), β2l is also a lognormal RV. Thus, γ isa sum of independent lognormal RVs. Let βl = evl , wherevl is a normal RV, i.e., vl∼N(µvl , σ

2vl). The kth moment of

βl is given as

Eβkl = ekµvl+k

2σ 2vl/2. (16)

Although an exact closed-form expression of the pdfof a sum of independent lognormal RVs does not exist,such a sum can be approximated by another lognormalRV [20,21]. The parametersµ andσ of this randomvariableγ can be obtained from the following set of nonlinearequations [19]:NG∑i=1

wi√πexp

[−sm exp

(√2σai + µξ

)]

=

K∏i=1

ΨX (sm;µi, σi) (17)

wherem = 1, 2 and

M(−s) =∫∞

0

[1+

sΩm

]−m 10/ ln(10)√2πσ 2Ω

× e−(10 log10 Ω−µ)

2

2σ2 dΩ (18)

and the weights, wi, and abscissas, ai, of the Gaussianquadrature for different orders, NG, are tabulated in

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H. Liu et al. / Physical Communication 2 (2009) 265–273 271

standard mathematical references. The parameters s1 ands2 are the arguments of the moment-generating functionat which the exact distribution of the sum of lognormalvariables should match the distribution of the equivalentlognormal variable.The approximated pdf of γ is given as

f (γ ) =1

γ√2πσ 2

exp[−(ln(γ )− µ)2

2σ 2

]. (19)

The average BER can be calculated by averaging the

conditional BER P FSR(γ ) = Q

(Ebγ√

5EbN0γ /2+TbN20 B

)over f (γ )

as

PFSR =∫∞

0PFSR(γ )f (γ )dγ . (20)

3.2. Coherent receiver

Because of the convolutional decoding involved in thecoherent receiver for the hybrid scheme, the exact BERexpression is difficult to obtain. However, a union boundcan be derived in the following manner. First, let T (D, I) =∑i,j n(i, j)I

iDj be the convolutional code transfer function,where n(i, j) is the number of paths with i input 1’s andj output 1’s that diverge from the all zero’s path andre-merge with it later. Assume that h(t) does not varyappreciably between such a divergence and re-merge forthe shorter paths that dominate the error performance(i.e., no temporal diversity is achieved). Then, note that:(1) the proposed codes are linear, and (2) the noise in onebranch of the receiver is independent of the noise in theother branch of the receiver (since they are uncorrelatedand jointly Gaussian). Then, applying a union bound:

Pb ≤∑i,j

i n(i, j)P(0j → 1j) (21)

where 0j and 1j are vectors of j 0’s and j 1’s, respectively,and P(0j → 1j) is the probability of deciding the latterwhen the formerwas sent. Continuing, Pb is upper boundedas:≤

∑i,j

i n(i, j)Eh[P(0j → 1j|h)]

=

∑i,j

i n(i, j)Eh

Q√√√√EbjN0

L−1∑l−0

h2l

=

∑i,j

i n(i, j)∫∞

0Q

(√EbjN0γ

)f (γ )dγ

=

∫∞

0

[∑i,j

i n(i, j)1π

∫ π2

0e−

Eb2N0

jγsin2 θ dθ

]f (γ )dγ

=1π

∫∞

0

∫ π2

0

∑i,j

i n(i, j)[e−

Eb2N0

γ

sin2 θ

]jf (γ )dθdγ

=1π

∫∞

0

∫ π2

0

∂

∂ IT (D, I)

∣∣∣∣∣I=1,D=exp

(−Eb2N0

γ

sin2 θ

)× f (γ )dθdγ .

0 1 2 3 4 5 6 710–5

10–4

10–3

10–2

10–1

Eb/N

0 (dB)

BE

R

Hybrid TR–coherent scheme in [18]Proposed scheme, preprocessor constraint length = 3Proposed scheme, preprocessor constraint length = 4

Fig. 4. BER versus Eb/N0 curves of the proposed hybrid system with acoherent rake receiver operating over an AWGN channel. Performance ofthe hybrid TR scheme proposed in [18] is provided for comparison.

For small numbers of states, T (D, I) can be found by hand,but this becomes unwieldy for larger numbers of states. Insuch cases, Appendix 6A1 of [22] gives an efficient methodthat can be extended to the case here.

4. Numerical results

Since the performance of the FSR receiver for theproposed hybrid scheme is the same as the FSR schemein [15], we only focus on simulating the performance withthe coherent rake receiver. In all simulations, a carrier-modulated, truncated root-raised-cosine pulse with a roll-off factor 0.25 is used as the UWB pulse shape p(t). The10-dB signal bandwidth is 1 GHz. We adopt channelmodels from the IEEE 802.15.3a [23] with a largedelay spread (CM4) and consider a low-data-rate systemoperating at 1 Mbps and the number of pulses to representa bit is Nf = 10. We assume perfect symbol timingand simulate the performances with ideal and estimatedtemplate waveforms. For each bit, the receiver combinesten strongest paths using maximal ratio combining (MRC).This is achieved by setting the delays τl of the correlatorto those of the ten strongest paths. The integrator outputsare fed into a Viterbi decoder, which approximates themaximum likelihood sequence detector. The simulationalso assumes a quasi-static fading model, i.e., the channelcoefficients and delays do not change over each packetduration but change independently over different packets.The total number of packets is 2000, each with 256 bits;thus, the BER values are effectively the average over 2000realizations of the channel coefficients.Fig. 4 shows the performance of the proposed hybrid

FSR-coherent system with coherent detection and softinput Viterbi decoding over AWGN channels. BER perfor-mances of the proposed hybrid scheme with the data pre-processor of constraint length three and constraint lengthfour over multipath fading channels modeled by IEEE802.15.3a CM4are shown in Fig. 5. For comparison, the per-formance of the hybrid TR-coherent system with a coher-ent receiver in [18] in the same environment is included in

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272 H. Liu et al. / Physical Communication 2 (2009) 265–273

Eb/N

0 (dB)

10–6

10–5

10–4

10–3

10–2

10–1

100

BE

R

Hybrid TR–coherent scheme in [18]Proposed scheme (perfect template), preprocessor CL = 3Proposed scheme (perfect template), preprocessor CL = 4Estimated template (window length = 10), preprocessor CL = 4Estimated template (window length = 20), preprocessor CL = 4

0 1 2 3 4 5 6 7 8 9 10

Fig. 5. BER versus EB/N0 curves of the proposed hybrid system withdifferent preprocessor constraint lengths (CL) and a coherent rakereceiver operating over an IEEE 802.15.3a multipath fading channelmodeled by CM4. Performance of the hybrid TR scheme proposed in [18]is provided for comparison.

the two figures. Compared with the case of a perfect tem-plate waveform for coherent rake, with an estimated tem-plate derived using a window length of N = 20, the BERperformance is degraded mainly in the low-Eb/N0 region.It is also observed that with the data preprocessor of con-straint length four, the proposed scheme achieves a gain ofabout 1.5 dB at a BER of 1× 10−4 over the scheme in [18];with the data preprocessor of constraint length three, theproposed scheme achieves a gain of about 1.2 dB over thescheme in [18].

5. Conclusion

We have proposed a hybrid UWB modulation schemethat allows reception by both coherent and FSR receivers,and derived closed-form equations and bounds for itsperformance. For coherent receivers, we introduce a datapreprocessing scheme where both the reference signaland data signal in the proposed hybrid FSR-coherentsystem are generated from the input bits by the datapreprocessor. This allows the coherent receiver to exploitnot only the energy invested in the reference pulses,but also the memory introduced by the preprocessor forimproved performance. The proposed scheme extendsthe preprocessor in [18] from an effective convolutionalencoder with a constraint length two to three or more,improving the performance by more than 2 dB. Comparedwith coherent detection of the conventional uncoded TRscheme, the proposed scheme not only recovers the 3 dBloss due to the energy wasted in the reference signals,but also achieves an additional 4 dB coding gain. Wealso proposed a channel estimation scheme that exploitsthe FSR receiver structure for the coherent rake receiverand derived analytical bounds for the performance of thedecoder in multipath environments.

References

[1] M.G. diBenedetto, T. Kaiser, A.F. Molisch, I. Oppermann, C. Politano,D. Porcino (Eds.), UWB Communications Systems: A ComprehensiveOverview, Hindawi, 2006.

[2] L. Yang, G.B. Giannakis, Ultra-wideband communications — An ideawhose time has come, IEEE Signal Process. Mag. 21 (2004) 26–54.

[3] R.C. Qiu, H. Liu, X. Shen, Ultra-wideband for multiple accesscommunications, IEEE Comm. Mag. 43 (2005) 80–87.

[4] M.Z. Win, R.A. Scholtz, Impulse radio: How it works, IEEE Commun.Lett. 2 (1998) 36–38.

[5] A.F. Molisch, Wireless Communications, IEEE-Press, Wiley, 2005.[6] M.Z. Win, R.A. Scholtz, On the energy capture of ultrawidebandwidth signals in densemultipath environments, IEEE Commun.Lett. 2 (9) (1998) 245–247.

[7] A.F. Molisch, J.R. Foerster, M. Pendergrass, Channel models forultrawideband personal area networks, IEEE Wireless Commun. 10(2003) 14–21.

[8] A.F. Molisch, et al., A comprehensive model for ultrawidebandpropagation channels, IEEE Trans. Antennas Prop. 54 (2006)3151–3166.

[9] D. Cassioli, M.Z. Win, A.F. Molisch, F. Vatelaro, Performance ofselective Rake reception in a realistic UWB channel, in: Proc. ICC2002, 2002, pp. 763–767.

[10] R.T. Hoctor, H.W. Tomlinson, An overview of delay-hopped,transmitted-reference RF communications, Technical InformationSeries, GE Research and Development Center, 2002, pp. 1–29.

[11] J.D. Choi, W.E. Stark, Performance of ultra-wideband communica-tions with suboptimal receiver in multipath channels, IEEE J. SelectAreas Commun. 20 (9) (2002) 1754–1766.

[12] K. Witrisal, G. Leus, M. Pausini, C. Krall, Equivalent system modeland equalization of differential impulse radio UWB systems, IEEE J.Selected Areas Commun. 23 (2005) 1851–1862.

[13] T.Q.S. Quek, M.Z. Win, Analysis of UWB transmitted-referencecommunication systems in dense multipath channels, IEEE J.Selected Areas Commun. 23 (2005) 1863–1874.

[14] S. Zhao, H. Liu, Z. Tian, Decision directed autocorrelation receiversfor pulsed ultra-wideband systems, IEEE Trans. Wireless Commun.5 (2006) 2175–2184.

[15] D.L. Goeckel, Q. Zhang, Slightly frequency-shifted reference ultra-wideband (UWB) radio, IEEE Trans. Commun. (2007).

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[17] IEEE 802.15 Working Group, Alternative PHY layer amendment forpersonal area networks, IEEE 802.15.4a standard, 2007.

[18] S. Zhao, P. Orlik, A.F. Molisch, H. Liu, J. Zhang, Hybrid ultrawidebandmodulations compatible for both coherent and transmit-referencereceivers, IEEE Trans. Wireless Commun. 6 (7) (2007) 2551–2559.

[19] N.B. Mehta, A.F. Molisch, J. Wu, J. Zhang, Approximating the sum ofcorrelated lognormal or lognormal-Rice random variables, in: Proc.IEEE ICC’06, 2006, pp. 1605–1610.

[20] H. Liu, Error performance of a pulse amplitude and positionmodulated ultra-wideband system in lognormal fading channels,IEEE Commun. Lett. 7 (2003) 531–533.

[21] F. Rajwani, N.C. Beaulieu, Simplified bit error rate analysis of PAPM-UWB with MRC and EGC in lognormal fading channel, in: Proc. IEEEICC, 2005, pp. 2886–2889.

[22] S. Wilson, Digital Modulation and Coding, Prentice-Hall, UpperSaddle River, 1996.

[23] A.F. Molisch, K. Balakrishnan, C.-C. Chong, S. Emami, A. Fort, J.Karedal, J. Kunisch, H. Schantz, U. Schuster, K. Siwiak, IEEE 802.15.4achannel model — final report, Document IEEE 802.15-04-0662-02-004a, 2005.

H. Liu (S’95-M’97-SM’08) received his B.S. andM.S. degrees from Nanjing University of Postsand Telecommunications, Nanjing, P.R. China,in 1987 and 1990, respectively, and his Ph.D.degree from New Jersey Institute of Technology,Newark, NJ, in 1997, all in electrical engineering.From July 1997 to August 2001, he was withLucent Technologies, New Jersey. He joined theSchool of Electrical Engineering and ComputerScience at Oregon State University in September2001, and is currently an associate professor.

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H. Liu et al. / Physical Communication 2 (2009) 265–273 273

His research interests include ultra-wideband systems, MIMO OFDMschemes, channel coding, and modulation and detection techniques formultiuser time-varying environments. He is currently an associate editorfor the IEEE Transactions on Vehicular Technology and an associate editorof IEEE Communications Letters.

A.F. Molisch (S’89, M’95, SM’00, F’05) receivedthe Dipl. Ing., Dr. techn., and habilitation degreesfrom the Technical University Vienna (Austria)in 1990, 1994, and 1999, respectively. From1991 to 2000, he was with the TU Vienna.From 2000–2002, he was with the WirelessSystems Research Department at AT&T (Bell)Laboratories Research in Middletown, NJ. From,2002–2008, he was with Mitsubishi ElectricResearch Labs, Cambridge, MA, most recentlyas Distinguished Member of Technical Staff and

Chief Wireless Standards Architect. At the same time he was alsoprofessor and chairholder for radio systems at Lund University, Sweden.Since 2009 he is Professor of Electrical Engineering at the University ofSouthern California.Dr. Molisch has done research in the areas of SAW filters, radiative

transfer in atomic vapors, atomic line filters, smart antennas, andwideband systems. His current research interests are cooperativecommunications, UWB, MIMO systems, and measurement and modelingofmobile radio channels. Dr. Molisch has authored, co-authored or editedfour books (among them the textbook ‘‘Wireless Communications’’,Wiley-IEEE Press), eleven book chapters, some 120 journal papers, andnumerous conference contributions, aswell as 60 standards contributionsand 70 patents.Dr. Molisch is area editor for antennas and propagation of the IEEE

Trans.Wireless Comm., co-editor of special issues of various internationaljournals. He has been general chair, TPC chair, and symposium chair ofseveral international conferences. He has chaired several standardizationgroups, been chairman of Commission C (signals and systems) of URSI(International Union of Radio Scientists), and chairman of the IEEEComSoc Radio Communications Committee. Dr. Molisch is a Fellow of theIEEE, Fellow of the IET, an IEEE Distinguished Lecturer, and recipient ofseveral awards.

D. Goeckel (S’89-M’92-SM’04) split time be-tween PurdueUniversity and Sundstrand Corpo-ration from 1987–1992, receiving his BSEE fromPurdue in 1992. From 1992–1996, he was a Na-tional Science Foundation Graduate Fellow andthen Rackham Pre-Doctoral Fellow at the Uni-versity of Michigan, where he received his MSEEin 1993 and his Ph.D. in 1996, both in ElectricalEngineering with a specialty in CommunicationSystems.In September 1996, he joined the Electri-

cal and Computer Engineering Department at the University of Mas-sachusetts, where he is currently a Professor. His current researchinterests are in the areas of communication systems and wirelessnetwork theory.Dr. Goeckel was the recipient of a 1999 CAREER Award from the Na-

tional Science Foundation for ‘‘Coded Modulation for High-Speed Wire-less Communications’’. He was a Lilly Teaching Fellow at UMass-Amherstfor the 2000–2001 academic year and received the University of Mas-sachusetts Distinguished Teaching Award in 2007. He served as an Edi-tor for the IEEE Journal on Selected Areas in Communications: WirelessCommunication Series during its transition to the IEEE Transactions onWireless Communications from 1999–2002, and as a Technical ProgramCommittee Co-Chair for the Communication Theory Symposium at IEEEGlobecom 2004 and the Wireless Communications Symposium at IEEEGlobecom 2008.He is currently an Editor for the IEEE Transactions on

Communications.

P. Orlik (S’97, M’99) was born in New York, NYin 1972. He received the B.E. degree in 1994 andthe M.S. degree in 1997 both from the State Uni-versity of New York at Stony Brook. In 1999 heearned his Ph.D. in electrical engineering alsofrom SUNY Stony Brook.He is currently a principal technical staff

member at Mitsubishi Electric Research Labora-tories Inc. located in Cambridge,MA.His primaryresearch focus is on sensor networks, ad hoc net-working and UWB. Other research interests in-

clude mobile cellular and wireless communications, mobility modeling,performance analysis, queuing theory, and analytical modeling.