IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 2, FEBRUARY 2013 353

Optimal BER-Balanced Combining forWeighted Energy Detection of UWB OOK Signals

Xiantao Cheng, Yong Liang Guan, and Shaoqian Li, Senior Member, IEEE

Abstract—For the demodulation of ultra-wideband (UWB) on-off keying (OOK) signals, some weighted energy detection (WED)schemes have been proposed. However, the schemes availablesuffer from unbalanced bit error rates (BERs) for bits ‘0’ and‘1’. Therefore, in this letter we will address the optimal BER-balanced WED. Specifically, we incorporate the BER balancerequirement into the WED formulation and obtain the optimalweighting vector by solving a nonlinear convex optimizationproblem. To bypass the computational complexity required bythe convex optimization, we derive a simple but near-optimalweighting vector. Simulations illustrate that the proposed BER-balanced WED schemes can achieve almost the same BERperformance as the maximum likelihood (ML) demodulator,which is optimal in terms of BER performance but producesunbalanced BERs for different bits.

Index Terms—Convex optimization, energy detection (ED),impulse radio (IR), on-off keying (OOK), weighted combining,ultra-wideband (UWB).

I. INTRODUCTION

IN contrast to coherent rake reception of formidably high-complexity, non-coherent ultra-wideband (UWB) receivers

can be implemented at an affordable complexity and thusare preferred in practice [1]. Among them, energy detection(ED), in combination with on-off keying (OOK) signallingor pulse position modulation (PPM), has motivated muchresearch efforts due to its low-complexity implementation [2],[3]. In ED, the received signal passes though a square-lawdevice followed by an integrator, and the demodulation isachieved by comparing the integrator’s output with a pre-determined threshold.

Despite its simple struture, conventional ED (CED) suffersfrom non-trivial performance degradation, compared with rakereceiver. This can be attributed to the enhanced noise effect inED. The noise power captured in ED is linearly proportionalto the product of bandwidth and integration interval length,which is typically large for UWB systems. To mitigate theaggregated noise, weighted energy detection (WED) is oftensuggested, wherein the integration interval is chopped intoseveral sub-intervals and the outputs of sub-intervals are com-bined using appropriate weights to form the decision statistics

Manuscript received November 14, 2012. The associate editor coordinatingthe review of this letter and approving if for publication was W. Zhang.

This work was supported in part by the Chinese Important National Scienceand Technology Project under Grant 2011ZX03004-002-02, and in part by theAgency for Science, Technology and Research (A*STAR), Singapore underSERC Grant 0521210087.

X. Cheng and S. Li are with the National Key Laboratory of Sci-ence and Technology on Communications, University of Electronic Scienceand Technology of China (UESTC), Chengdu, China 611731 (e-mail: xi-antaocheng@163.com, lsq@uestc.edu.cn).

Y. L. Guan is with the School of EEE, Nanyang Technological University(NTU), Singapore (e-mail: eylguan@ntu.edu.sg).

Digital Object Identifier 10.1109/LCOMM.2013.010313.122540

[4]–[6], [10]. In particular, for UWB OOK signals, a WEDdemodulator is derived based on minimum mean square error(MMSE) criterion [6]. However, it is found that the MMSEWED is inferior to the CED in terms of bit error rate (BER)performance and moreover produces unbalanced BERs for bits‘0’ and ‘1’. Besides, another WED scheme is proposed in [10]using maximum likelihood (ML) criterion, which can achievethe minimal average BER but suffers from unbalanced BERsfor different bits.

In view of the above, this letter focuses on the BER-balanced WED scheme. The contribution is : 1) A BER-balanced WED scheme is proposed, wherein the BER balanceconstraint is satisfied by setting an appropriate decision thresh-old. 2) To minimize the BER, the optimal weighting vectoris determined by solving a nonlinear convex optimizationproblem. Furthermore, low-complexity near-optimal weightingvectors are derived. The proposed BER-balanced WED usingthe optimal or near-optimal weighting vector can producealmost the same average BER performance as the ML WEDin [10].

II. SYSTEM MODEL

We consider UWB OOK signalling in a single-user sce-nario. Each symbol interval Ts consists of Nf consecutiveframes of duration Tf , leading to Ts = NfTf . The binaryinformation symbol s is conveyed by transmitting Nf ultra-short pulses or not, depending on s = 1 or s = 0. Specifically,the transmitted signal corresponding to s can be described by

s(t) =

√Es

Nf

Nf−1∑k=0

sp(t− kNf ) (1)

where Es is the symbol energy and is related to the averagebit energy Eb as Es = 2Eb, p(t) represents the ultra-shortpulse with unit energy, i.e.,

∫∞−∞ p2(t)dt = 1, and the width

Tp � Tf .After passing through the multipath channel, denoted as

h(t), the received signal is given by

r(t) =

√Es

Nf

Nf−1∑k=0

sg(t− kNf ) + n(t) (2)

where g(t) � p(t)⊗h(t) represents the normalized compositechannel, n(t) denotes the filtered Gaussian noise with two-sided power spectrum N0/2 and bandwidth B. In (1) and (2),we assume that the frame duration Tf is chosen such thatthere’s no inter-frame interference and inter-symbol interfer-ence. Therefore, we can consider each symbol’s transmissionand reception separately.

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A WED receiver divides Tf into M equi-length sub-intervals, each of duration TM = Tf/M . Squaring r(t) andintegrating it over each sub-interval, the receiver obtains M -tuple outputs as

ym =

Nf−1∑k=0

∫ kTf+mTM

kTf+(m−1)TM

r2(t)dt, m = 1, 2, ...,M (3)

By invoking the sampling theorem [7] and Gaussian ap-proximation, ym’s can be shown to be statistically independentand ym can be modeled as a Gaussian variable with mean andvariance as shown below [4]–[6], [10]

μm = NfN0BTM + sEm

σ2m = NfN

20BTM + 2sN0Em (4)

where Em = Es

∫mTM

(m−1)TMg2(t)dt. For compactness, we

collect ym’s into a vector y = [y1, ..., yM ]T , whose meanvector and diagonal covariance matrix, respectively, are

m = [μ1, μ2, ..., μM ]T

C = diag{σ21 , σ

22 , ..., σ

2M} (5)

III. OPTIMAL BER-BALANCED WED

The key in WED is the determination of weighting vectorw ∈ R

M and threshold ρ. Once w, ρ are given, the transmittedsymbol s can be recovered as follows

z = wTy, s =

{0, z ≤ ρ1, z > ρ

(6)

Based on (4) and (5), the decision statistic is Gaus-sian distributed as z ∼ N(wTm0,w

TC0w) or z ∼N(wTm1,w

TC1w), depending on s = 0 or s = 1, wherem0,C0 (m1,C1) are, respectively, the mean vector and thecovariance matrix in (5) corresponding to s = 0 (s = 1).

Accordingly, the BER of the WED using (6) is derived as:

pb =1

2pb,0 +

1

2pb,1

=1

2Q( ρ−mT

0 w√wTC0w

)+

1

2Q( mT

1 w− ρ√wTC1w

) (7)

where pb,0, pb,1 are the BER expressions for data bits ‘0’ and‘1’, respectively, Q(·) is the complementary error function. Toachieve the balanced BER, we set pb,0=pb,1 and thus have

ρ =mT

1 w√wTC0w+mT

0 w√

wTC1w√wTC0w+

√wTC1w

(8)

pb = Q( hTw√

wTC0w+√wTC1w︸ ︷︷ ︸

�x

)(9)

where h � m1 − m0 = [E1, E2, ..., EM ]T , such channel-related information can generally be obtained through trainingsymbols.

Now we are in the position to determine the optimalweighting vector (denoted as wo) which achieves the min-imal pb, denoted as pb,o. Since Q function is monotonically

decreasing, minimizing pb in (9) is equivalent to the followingminimization problem:

minw

f(w) =

√wTC0w+

√wTC1w

hTw(P1) (10)

Note that w and its scaling γw with γ being an arbitrarypositive scalar, lead to the same f(w). Therefore, P1 can betransformed into

minw

g(w) =√wTC0w+

√wTC1w (P2)

subject to hTw = 1(11)

Since square root function is increasing in the argument, andits argument is convex with respect to w, so the function g(w)is convex. As a result, (P2) is a convex minimization problemand can be easily solved for wo [8].

To bypass the computation burden required by (11), can wefind a simple but near-optimal weighting vector? To answerthis question, we want to derive lower and upper bounds onpb,o, from which we will gain some hints.

Lemma 1: The minimal BER can be bounded as:Q(12‖C−1/2

0 h‖2) ≤ pb,o ≤ Q(12‖C−1/21 h‖2) .

Proof: For a diagonal matrix, the diagonal elements areits eigenvalues. Exploiting this fact and (4), (5), we have

λmin(C1) = minm

(NfN0BTM + Em)

≥ λmax(C0) = NfN0BTM

(12)

where λmax(·), λmin(·) represent the maximum and minimumeigenvalues of a matrix, respectively. Therefore, we can readilyobtain wTC1w ≥ wTC0w, which leads to

hTw

2√wTC1w

≤ x ≤ hTw

2√wTC0w

(13)

Let us first consider the left-hand inequality

maxw

x ≥ maxw

hTw

2√wTC1w

=1

2λ1/2max(C

−T/21 hhTC

−1/21︸ ︷︷ ︸

�A

)

=1

2‖C−1/2

1 h‖2

(14)

where the inequality in the first step is obvious, in thesecond step we use the Rayleigh Ritz inequality, and the

equality is achieved when w′ = C121 w is the principal

eigenvector corresponding to λmax of A and hTw′ > 0.Considering that A is rank-one, its maximal eigenvalue is‖C−T/2

1 h‖22 = ‖C−1/21 h‖22, which leads to the third step,

and its principal eigenvector is αC−T/21 h = αC

−1/21 h with

α denoting an arbitrary non-zero scalar. To satisfy hTw′ > 0,α > 0 is required. Therefore, the weighting vector enablingthe second and third steps is wl � C

−1/21 w′ = αC−1

1 h withα being an arbitrary positive scalar. Note that wl with differentpositive α’s lead to the same result, which can be verified bysubstituting wl into (14) or (10). So different α’s make nodifference for data demodulation.

For the right-hand inequality, in a similar way we can arriveat

maxw

x ≤ 1

2‖C−1/2

0 h‖2 (15)

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CHENG et al.: OPTIMAL BER-BALANCED COMBINING FOR WEIGHTED ENERGY DETECTION OF UWB OOK SIGNALS 355

where the weighting vector used is wr � βC−10 h = βC−1

0 hwith β being an arbitrary positive scalar (the explanation isthe same as α). Combining (14) and (15) leads to Lemma 1.

Remark 1: If C0 ≈ C1, the bounds on maxw

x will be tight.In this case, wl,wr will be close to the optimal weightingvector wo and thus are good candidates for being a near-optimal weighting vector. Fortunately, it is verified throughsimulations that at the signal-to-noise ratios (SNRs) of interest,C0,C1 are close. Thus we can choose wl, wr as near-optimalalternatives.1 In particular, wr is preferred since it bears asimpler form wr = h.2 Note that wr is also presented in [6],where, however, it is derived by maximizing the deflectioncoefficient and is used for timing synchronization.

For data demodulation of UWB OOK signals, one counter-part scheme is used in [6], which works as shown below

z = aTy, s =z −NfN0BTM (aT1M )

aThs = sign(s− 0.5)

(16)

where a represents the weighting vector, 1M represents acolumn vector of M 1’s. In particular, the following MMSE-based weighting vector is recommended in [6]:

ammse =M

1TM (C0 +C1)−11M

(C0 +C1)−11M (17)

where aT1M = M .Remark 2: Using s(i) to represent s in (16) corresponding

to s = i, we have s(0) ∼ N(μ0 = 0, σ20 = aTC0a

(aTh)2 ) and

s(1) ∼ N(μ1 = 1, σ21 = aTC1a

(aTh)2 ). Since s(0) and s(1) havedifferent variances, the threshold ‘0.5’ will lead to differentBERs for bits ‘0’ and ‘1’. To minimize the BER of s = 0,we should minimize σ2

0 , which can be achieved by using theweighting vector a = C−1

0 h.3 Similarly for s = 1, the BER-minimizing weighting vector is a = C−1

1 h. It is easily seenthat both a and a impose larger (smaller) weight on ym withlarger (smaller) signal energy Em. However, ammse in (17)behaves in the opposite way. In other words, ammse weightsym of larger (smaller) Em with smaller (larger) weightingcoefficient. Therefore, ammse is an improper weighting vector.Simulations in next section show that ammse leads to poorBER performance.

IV. SIMULATION AND DISCUSSION

The second derivative Gaussian pulse with width Tp = 1nsis used. Multipath channels are generated according to theCM1 model in [9] with 1ns resolution and maximum spreadtruncated to be 100ns. The other parameters are: B = 5GHz,Tf = 100ns and Nf = 10. Besides, we assume h is per-fectly known for all WED schemes. The information symbols ∈ {0, 1} is randomly generated with equal probability. Forclarity, in the figures, we use the following notations:

1In fact, these two vectors’ performance curves merge together.2Since stretching wr makes no difference for the demodulation, we omit

the scalar:α/(NfN20BTM ).

3This can be derived by using the Rayleigh Ritz inequality, as in the proofof Lemma 1.

12 13 14 15 16 17 18 19 20 21 220.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

Eb/N

0 (dB)

BE

R(0

/BE

R(1

)

MMSEMLBER−balanced (optimal)BER−balanced (near−optimal)

Fig. 1. BER ratios of different WED schemes with BER(i) representing theBER conditioned on s = i, M = 10, TM = 10ns.

• ‘CED’ represents the conventional energy detection in[3].

• ‘BER-balanced (optimal)’ represents the proposed BER-balanced WED using wo in (11) as the weighting vector.

• ‘BER-balanced (near-optimal)’ represents the proposedBER-balanced WED using wr = h as the weightingvector.

• ‘MMSE’ represents the MMSE WED in [6], which uses(16) and (17)) for data modulation

• ‘ML’ represents the non-linear ML WED [10], which isderived using the fact that ym’s are Gaussian distributed(see (4)). The ‘ML’ WED achieves the minimum averageBER and thus serves as the performance benchmark.

For fair comparison, all WED schemes use the same M in thesimulations.

Fig. 1 compares the ratios of BER(0) to BER(1) of differentWED schemes, where BER(i) represents the BER conditionedon that s = i is transmitted. The number of sub-intervals isM = 10 and thus TM = Tf/M = 10ns. It’s shown thatthe ratio curves corresponding to ‘BER-balanced (optimal)’and ‘BER-balanced (near-optimal)’ merge with the line ‘1’over varying SNRs. This means that the proposed schemescan provide balanced BERs for bits ‘0’ and ‘1’. In constrast,the curves of ‘ML’ and ‘MMSE’ schemes deviate from theconstant ’1’, which means that their BERs are unbalanced.Moreover, when increasing SNR, ‘ML’ and ‘MMSE’ incurmore unbalance. This is because for these two schemes, thedifference of the decision statistic’s variances for bits ‘0’ and‘1’ becomes larger with increasing SNR.

Fig. 2 depicts the BER curves of the CED and differentWED schemes , together with the lower and upper boundsderived in Lemma 1. The curves are obtained by averagingover the results of bits ‘0’ and ‘1’. Note that the three curvescorresponding to ‘ML’, ‘BER-balanced (optimal)’ and ‘BER-balanced (near-optimal)’ merge together. This means thatthe proposed BER-balanced WED schemes achieve the bestaverage BER performance. While for ‘MMSE’, it performsobviously worse than the CED and the other WED schemes,

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356 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 2, FEBRUARY 2013

14 16 18 20 22 24 2610

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

BE

R

MMSE BER−balanced (optima)lBER−balanced (near−optimal)Lower boundUpper boundMLCED

Fig. 2. BER performance of the CED and different WED schemes, M =10, TM = 10ns for WED schemes.

14 16 18 20 22 24 26

10−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

BE

R

CEDBER−balanced (near−optimal). M=5BER−balanced (near−optimal). M=10BER−balanced (near−optimal). M=20

Fig. 3. BER performance of the proposed near-optimal BER-balanced WEDusing different M ’s.

due to its improper weighting vector ammse. It’s worthy topoint out that due to different channel model and simulationparameters used, here ‘MMSE’ exhibits different performancefrom that in Fig. 11 of [6]. Moreover, we also conducted thesimulations of the BER-balanced WED using wl, the curve

of which merges with that of ‘BER-balanced (near-optimal)’and so is omitted for clarity.

Fig 3 illustrates the effect of M on the performanceof the proposed near-optimal BER-balanced scheme. Since‘BER-balanced (optimal)’ and ‘BER-balanced (near-optimal)’exhibit indistinguishable curves (as seen in Fig. 2), onlythe latter’s results are shown here for clarity. It can beseen that when enlarging M , the near-optimal scheme enjoysincreasingly more performance gain over the CED, but thegain increment is decreasing gradually. Note that larger Mmeans higher system complexity. So it is necessary to drawan proper trade-off between performance and complexity.

V. CONCLUSION

To counteract the enhanced noise in CED, weighted pro-cessing is often suggested. However, for the demodulationof UWB OOK signals, the existing WED schemes fail toprovide balanced BERs for different bits. In view of this, wepropose an optimal BER-balanced WED, which achieves theBER balance by setting a proper threshold and determines theoptimal weighting vector by solving a convex optimizationproblem. Furthermore, a simple near-optimal weighting vectoris derived to avoid the computation required by the optimaldetector. Simulations show that the proposed WEDs canachieve the best performance and provide desirable balancedBERs for different bits.

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[6] F. Wang, Z. Tian, and B. M. Sadler, “Weighted energy detection fornoncoherent ultra-wideband receiver design,” IEEE Trans. Wireless Com-mun., vol. 10, pp. 710–720, Feb. 2011.

[7] H. Urkowitz, “Energy detection of unknown deterministic signals,” Proc.IEEE, vol. 55, pp. 523–531, Apr. 1967.

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