Timing Synchronization for Fading Channels with Different Characterizations Using Near ML Techniques Md

404 JOURNAL OF COMMUNICATIONS, VOL. 4, NO. 6, JULY 2009

Timing Synchronization for Fading Channels with Different Characterizations using Near ML Techniques Md. Tofazzal Hossain #, Sithamparanathan Kandeepan ∗, and David Smith & #&National ICT Australia, Canberra Research Laboratory, ACT, Australia ∗CREATE-NET International Research Centre, Trento, Italy # ∗ &Research School of Information Sciences and Engineering, Australian National University Email: #[email protected], ∗[email protected], [email protected]

Abstract— In this paper, we analyze the performance of the incoming signal. Non-synchronous sampling is used a non-data aided near maximum likelihood (NDA-NML) in this paper to estimate the symbol timing on a received estimator for symbol timing recovery in wireless commu- baseband signal. The optimum symbol strobing point is nications. The performance of the estimator is evaluated for an additive noise only channel. Performance analysis estimated using the samples with minimum intersymbol is extended to fading channels characterized by Rayleigh interference. Obtained samples from nonsynchronously fading, Weibull fading and log-normal fading, appropriate sampled signal may not include the sample which is at to a variety of transmission scenarios. The probability the optimum sampling point and therefore interpolation distribution of the maxima and probability distribution of is used to reduce the error produced by nonsynchronous timing estimates are derived, presented and compared with simulation results. The performance of the estimator is sampling process. While performing interpolation and presented in terms of the bit error rate (BER) and the error upsampling with the available high processor speeds of variance of the estimates. The BER is computed when the the latest digital signal processor, no significant limita- estimator is operating under additive white Gaussian noise tions are noted. In this paper, interpolations are assumed (AWGN) channel and fading channels. The variance of the available for convenience. A good review on interpolation estimates is computed for the noise only case and compared with the Cramer Rao bound (CRB) and modified Cramer techniques for symbol timing estimation is given by Gard- Rao bound (MCRB). ner [2, 3] where some useful interpolation filter functions and the use of them for timing recovery to improve Index Terms— BER, CRB, Rayleigh fading, Weibull fading, Log-normal fading, Near Maximum Likelihood Estimation, performance of nonsynchronously sampled systems are MCRB, Symbol Timing Error, Timing Synchronization. discussed. Many feedback and/or feedforward symbol timing estimation techniques for synchronous and/or nonsyn- I.INTRODUCTION chronous sampled systems for either continuous time or Timing in a receiver must be synchronized to the burst mode transmissions are available in the literature [4]. symbols of the incoming transmitted signal. In the analog Techniques for synchronizing receivers can be divided implementation of digital modems a free running clock at into data aided and non-data aided. The former is the the symbol rate is phase adjusted in a feedback manner case where synchronization relies on knowledge of the to find the optimum sampling point of the symbol. In information symbols. The latter is the case where train- feedforward arrangement, timing wave is generated from ing sequences are impractical or inconvenient, and the incoming signal. Mueller and Muller [1] presented a novel decision process is not sufficiently reliable for decision technique for symbol timing recovery in the analog era. feedback [5]. With the advancement of the digital and microproces- The performance of the estimator is evaluated in terms sor systems, came digital signal processors and FPGAs, of BER performance and measuring the variance of the which allows us to implement complicated open loop estimator and comparing with CRB and MCRB. The systems. BER performance under the assumption of perfect timing In this paper, we present a non-data aided symbol estimates is well documented for various modulation timing estimation technique and evaluate its performance. formats [6]. In practice timing estimates exhibit small Symbol timing can be performed with synchronous sam- random fluctuations (jitter) about their optimum values pling where exploiting some sort of error signal, the signal that give rise to a BER degradation when compared sampling clock is adjusted in a feedback manner, and non- to perfect synchronization [7], which degrades further synchronous sampling where the sampling is not locked to when the signal undergoes fading in wireless communications [8]. A joint symbol timing estimation and data Manuscript received January 30, 2009; revised March 27, 2009; detection on flat fading channel based on particle filtering accepted May 11, 2009. This work was presented in part at IEEE 11th International Conference on Computer and Information Technology is demonstrated in [9]. (ICCIT 2008). We evaluate the performance of the symbol timing

estimator operating under fading channels with different distributed with the same statistics as w(n). Now if z(t) characterizations, suitable for a variety of applications. is sampled at times kTs, k = 0, ±1, ±2... where Ts is the We consider the timing estimator similar to that of de- sampling interval, we have scribed in [4] and analyze its performance by finding X∞ the statistical distribution of the estimates, the distribu- z(kTs) = cmgo(kTs − mT − τ) + n(kTs). (4) tion can then be used to characterize the average BER m=−∞ theoretically. The analysis is further extended to fading channels using Monte-Carlo simulations. A tighter CRB and MCRB are also derived to study the performance of the estimator for the noise only case. Although the considered timing estimator is not optimal, we consider it here because of its design simplicity, such that it can be easily used in software defined radios at the baseband. The rest of the paper is organized as follows. In the next section we present the system model. In section III, the timing estimator is presented. Section IV presents the statistical analysis of the timing estimator. Performance Figure 1. Block diagram of signal detection using square root raised of the estimator is presented in section V. The timing cosine filter. estimator operating under fading channels for different characterizations are then presented in section VI before When the symbols are transmitted over fading channel the paper is finally concluded in section VII. h(t), the faded discrete signal at the baseband can be written as, II.SYSTEM MODEL X∞ f The system considered here is depicted in Figure 1. z (kTs) = h(kTs)cmg0(kTs − mT − τ) + n(kTs) In our model we consider both the additive noise only m=−∞ channel as well as the fading channel. A conventional (5) baseband pulse amplitude modulation (PAM) signal can where h(kTs) denotes the multiplicative fading pro- be represented as cess introduced by the fading channel sampled at times kTs, k = 0, ±1, ±2.... As is shown in Figure 1, square ∞ X root raised cosine (SRC) filter is used as the pulse shaping s(t) = c g(t − mT ) (1) m filter and also as the receiving filter, the overall response m=−∞ is therefore similar to a raised cosine filter. The impulse where cm’s are the transmitted symbols, g(·) is the pulse response of SRC filter is given by: shape used to control the spectral characteristics of the 1 gsrc(t) = transmitted signal and T is the signaling interval ( T is the symbol rate). The cm’s are assumed to be independently  π(1−α)+4α and identically distributed, and take on the values ±1 with  √ t = 0  π T equal probability. The channel introduces a delay of τ, and  ³ (1+α)πt sin((1−α)πT ) ´  cos( T )+ 4αt 4√α T T additive white Gaussian noise is considered at the receiver 4αt 2 t 6= 0, ± 4α (6) π T 1−( T ) N0  α π with a double sided power spectral density of 2 W/Hz.  √ ((π − 2) cos( )+  π 2T 4α Then the signal becomes  π T (π + 2) sin( 4α )) t = ± 4α y(t) = s(t − τ) + w(t) where α is called the roll-off factor, which takes values ∞ X in the range 0 ≤ α ≤ 1 and T is the symbol period. The = cmg(t − mT − τ) + w(t) (2) theoretical bit error probability Pb for bipolar signaling is m=−∞ Ãr ! where the noise process w(t) is stationary and Gaussian 2Eb Pb = Q (7) with a mean of zero. The frequency response of the pulse N0 shaping filter in the transmitter is G(f). The received signal is passed through a receiving filter. In general, the where Q(x) is called the complementary error function or co-error function and is defined as optimum filter at the receiver is matched to the received Z signal pulse g(t). Hence the frequency response of this 1 ∞ ³ u2 ´ ∗ Q(x) = √ exp − du. (8) filter is G (f). Then the received signal is 2π x 2 X∞ z(t) = cmgo(t − mT − τ) + n(t) (3) III.TIMING ESTIMATOR m=−∞ Here a near maximum likelihood (NML) timing esti- where go is the signal pulse response of the receiving mation technique is presented that estimates the symbol ∗ 2 filter-that is Go(f) = G(f)G (f) = |G(f)| -and n(t) is timing error. This NML timing estimation technique is a bandlimited Gaussian noise independent and identically similar to the estimator described in [4]. We assume that

the sampled received baseband waveform for the additive where noise only channel as given by (4) is: XN L−1 f 0 0 0 0 X z (k ) = h(k Ts)cmg0(k Ts − mT − τ) + n(k Ts). z(kT ) = c g (kT − iT − τ) + n(kT ). (9) i s i 0 s s k0=1 i=0 (17)

The noise process n(kTs) is a zero mean bandlimited Gaussian process with a variance of σ2 and L is the number of symbols used to estimate timing offset. In this IV. STATISTICAL ANALYSIS OF THE TIMING STIMATOR case we ignore the channel h(kTs) to derive the estimator. E The probability density function of n(kTs) is given by: " # The time estimates relate to the maximum energy of 2 1 {n(kTs)} the signal. The instant of time at which signal energy fn(n) = √ exp − . (10) 2πσ2 2σ2 is maximum, is taken as the time estimate. Hence, the distribution of the time estimate corresponds to the distri- If we let R to be a vector of the sample set z(kTs) then bution of the maximum of the signal. We ﬁnd expressions using (9) and (10) for the distribution of maxima and the probability mass ( P ) L−1 2 function of time estimates. The analytical expressions are 1 i=0 [E − F ] fR(R|τ) = L exp − 2 (11) then compared with the simulated distribution. For both (2πσ2) 2 2σ the cases, nice matches are found. where

E = zi(kTs) and A. Distribution of Maxima LX−1 F = cig0(kTs − iT − τ). Let N = 2N 0 + 1. Considering the signal components i=0 Z−N , ..., Z0, ...ZN as random variables, the maximum of We use the Maximum Likelihood (ML) principles to the signal Q can be found derive the symbol timing estimator assuming no a-priori knowledge of the received symbol sequence ci. Let a 0 Q = Max(Z−N 0 , ..., Z0,Z1, ..., ZN ). (18) matrix of samples be given as     Z1 z1(1) z1(2) ·· z1(N) Then the cumulative distribution function of the maxima      Z2   z2(1) z2(2) ·· z2(N)  can be written as      ·   ·····      Z =  Zi  =  zi(1) zi(2) ·· zi(N)  (12) FQ(q) = P (Q ≤ q)  ·   ·····  0     = P (Z−N 0 ... ≤ Z0 ≤ Z1...ZN ≤ q)  ·   ·····  = F 0 (q, q...q). (19) Z−N0 ...Z ZL zL(1) zL(2) ·· zL(N) N 0 where, If Z−N 0 , ..., Z0, ...ZN are independent XL 0 Z = z (k ) (13) F (q) = F (q)...F 0 (q) (20) i i Q Z−N0 ZN i=1 and, where FZi (q) is the cumulative distribution function of Z . To ﬁnd the probability density function of the maxima XN i 0 0 0 fQ(q), differentiating (20) with respect to q gives, zi(k ) = cmg0(k Ts − iT − τ) + n(k Ts). (14) k0=1 d 0 fQ(q) = FZ 0 (q) (FZ 0 (q)...FZ (q)) Here, k0 = mod(k, N), L is the number of symbols used −N δq −N +1 N +(F (q)...F (q))f (q) to estimate the timing offset and N is the number of Z−N0+1 Z−N0 Z−N0 samples per symbol. The estimator is then given by: PN 0 QN 0 = i=−N 0 fZi (q) j=−N 0,j6=i FZj (q) (21) XL 0 0 τˆ = arg max{Γ1(k , τ) = |zi(k , τ)|}. (15) where fZi (q) is the probability density function of Zi. i=1 Figure 2 shows the analytical and simulated probabil- The validity of the technique demands the constraint on ity density function of maxima for signal-to-noise-ratio number of samples to be integer multiple of symbols. (SNR) of 15 dB, roll-off factor of the SRC ﬁlter α = 0.6, When the estimator operates under fading channel, it number of samples per symbol N = 100 with a contri- becomes, bution of L = 300 symbols to the estimator. Figure 2 demonstrates that the simulated probability density func- XL 0 f 0 tion of maxima closely match the analytical probability τˆ = arg max{Γ2(k , τ) = |zi (k , τ)|} (16) i=1 density function.

analytical results closely match with the simulated pdf. 1 Simulated pdf of maxima 0.9 Analytical pdf of maxima 0.8 0.14 0.7 Simulated pdf of τ Analytical pmf of τ (q))

Q 0.6 0.12

0.5 0.1 τ 0.4 E /N = 15 dB b 0 pdf of maxima (f 0.3 0.08

0.2 0.06 0.1

0 Probability density of 0.04 220 221 222 223 224 225 226 227 228 229 Maxima (q) 0.02 Figure 2. Probability Density Function of Maxima 0 0 20 40 60 80 100 τ/T s B. Probability Distribution of the Timing Estimator Figure 3. Analytical and simulated probability distribution function for In order to determine the probability of bit error in the timing estimates. receiver, the distribution of the timing estimate errors must be taken into account. When using feedback techniques, the probability density function (pdf) of the timing error may be modelled using a Tichanov pdf [6]. Alternatively V. PERFORMANCEOFTHE ESTIMATOR if the signal to noise ratio (SNR) is greater than the The statistics of the estimates are the key elements inverse of twice the variance of the timing offset, a in determining the performances of the symbol timing Gaussian pdf may be used [10]. However, in our case recovery method. If the estimates are biased, the resulting the timing estimates relate to the maximum energy of sampling location selected by the symbol timing estimator the signal. The instant of time at maximum signal energy will be biased from the true optimum and performance is the estimate. Considering this, we find an expression will suffer. Similarly, if the estimates have a high variance, for the probability mass function of timing estimates. the variance of the selected sampling location will also be The analytical expression is then compared with the high. Clearly it is desirable to keep the variance of the simulated distribution. Let N = 2N 0 + 1, and con- estimates as low as possible. The performance bounds for sider the two vectors, τ = [τ−N 0 , .., τ0, .., τk0 .., τN 0 ] and the symbol timing estimator are the CRB and MCRB. We 0 0 0 0 Γ1(k , τ) = [Γ1(−N ), .., Γ1(0), .., Γ1(k ), .., Γ1(N )] evaluate the performance of the estimator finding bit error 0 with q = max(Γ1(k , τ)). Then, when the particular time probability with matched filter detection and measuring instance τk0 is chosen as the timing estimate τˆ, that is variance of the estimator. Bit error probability is com- τˆ = τk0 , the probability can be written as pared to the theoretical bit error probability for AWGN 0 and variance is compared with the lower bounds CRB P (ˆτ = τk0 ) = P (q = Γ1(k , τ)) and MCRB. Figure 4 shows the bit error performance for the NDA-NML estimator in an AWGN channel when 0 0 0 = P (q > Γ1(−N ), .., q = Γ1(k ).., q > Γ1(N )) roll-off factor α = 0.6 in both transmitter and receiver, YN 0 N = 20 samples per symbol, and L = 100 symbols 0 = fΓ1(k )(q) FΓ1(j)(q) (22) used for timing estimates; with the further assumption that j=−N 0,j6=k0 there is negligible inter-symbol interference. NDA-NML

0 estimator exhibits good bit error performance compared where, fΓ1(k )(q) and FΓ1(j)(q) are the probability density 0 with theoretical BER of AWGN. function and cumulative distribution function of Γ1(k ) respectively. The expression for the probability mass function (pmf) for the timing estimates is given by, A. Cramer Rao Bound (CRB) ½ P (ˆτ = τ); τ ∈ [τ−N 0 , .., τk0 .., τN 0 ] We consider the additive noise only case to derive fτˆ(τ) = (23) 0; τ 6∈ [τ−N 0 , .., τk0 .., τN 0 ]. the CRB here, assuming that the received continuous waveform as given in (3) has complex envelope given Figure 3 shows the analytical probability mass function by and the simulated probability density function for the z(t) = s(t) + n(t) (24) timing estimates when SNR is 10 dB, roll-off factor α = 0.6, samples per symbol N = 100, and L = 300 which is observed over an interval T . s(t) is the infor- symbols used for timing estimates. It is seen that the mation bearing signal and n(t) represents complex-valued

0 with respect to time t evaluated at t = 0. For a raised 10 cosine channel when the signal energy normalized to 00 −1 unity, g (0) is given by [13] 10 0 ³ 2 ´ 00 π 2 2 −2 g0 (0) = − + α (π − 8) (28) 10 3 where α is the roll-off factor of the raised cosine ﬁlter. −3 10 α = 0.6 BER L = 100 B. Modiﬁed Cramer Rao Bound (MCRB) −4 10 The MCRB by [14] to the variance of λˆ(z) − λ is the

−5 Simulation with RCF and NML following 10 AWGN Theory 1 MCRB(λ) = . (29) −6 nh i2o 10 ∂ ln p(z|u,λ) −2 0 2 4 6 8 Ez,u E /N (dB) ∂λ b 0 This bound is found by [14] observing that Figure 4. Bit error performance: NDA-NML estimator in an AWGN n o n o ˆ 2 ˆ 2 channel. Ez,u [λ(z) − λ] = Eu Ez|u[(λ(z) − λ) ] ( ) 1 ≥ Eu h³ ´2i additive white Gaussian noise with two-sided power spec- ∂ ln p(z|u,λ) Ez|u ∂λ tral density. Some of the parameters are unknown, such 1 as the carrier phase θ, the symbol epoch τ, the carrier ≥ ( ) h³ ´2i frequency error ν, etc. ∂ ln p(z|u,λ) Eu Ez|u In order to estimate of a single element of {θ, τ, ν}, ∂λ denoted by λ, which is assumed to be deterministic (non- 1 random), all the other parameters, including the data, are = nh i2o (30) ∂ ln p(z|u,λ) collected in a random vector u having a known probability Ez,u ∂λ density function p(u) which does not depend on λ. If ˆ where the first inequality derives from application of the λ(z) is any unbiased estimator of λ, a lower bound to the ˆ ˆ CRB to the estimator λ(r) for a fixed u, while the second variance of the error λ(z)−λ is given by the Cramer-Rao is true in view of Jensen’s inequality and the convexity formula [11] 1 of the function x for x > 0. 1 In spite of having same structure of CRB and MCRB, CRB(λ) = nh i o (25) 2 MCRB is much easier to use. For Gaussian Channel, the E ∂ ln p(z|λ) z ∂λ probability density for (29) is a well-known exponential where Ez denotes statistical expectation with respect to function whose argument is a quadratic form in the dif- the subscripted variable, and p(z|λ) is the probability ference between z and the signal s. Thus the logarithm of density function of z for a given λ. p(z|u, λ) equals this quadratic form and the expectation To compute CRB(λ) we need p(z|λ) which, in prin- in (29) is readily derived. ciple, can be obtained from the integral The MCRB of symbol epoch (τ) by [5] is given by Z ∞ 1 T 2 MCRB(τ) = (31) p(z|λ) = p(z|u, λ)p(u)du (26) 8π2Lξ E /N −∞ s 0 where p(z|u, λ), the conditional probability density func- where L is the number of symbol intervals on which tion of r given u and λ is easily available, at least for receiver bases its timing estimate on the observation on additive Gaussian channels. Unfortunately, in most cases the received signal, T is the symbol interval, Es is the of practical interest, the computation of (25) is impossible energy of one symbol waveform and ξ is an adimensional because either the integration in (26) cannot be carried out coefficient depending on the shape of g0(t): R analytically or the expectation in (25) poses insuperable ∞ 2 2 2 −∞ T f |G(f)| df obstacles. Using the general form of CRB given by [12], ξ = R ∞ . (32) |G(f)|2df the CRB for timing estimates for AWGN channel can be −∞ written as: MCRB for a normalized τ is given by [15] where τ is " # ³ ´ between 0 and 1. The MRCB is τ 2Eb 00 2 CRB = − g0 (0)LT (27) 1 T N0 MCRB(τ) ≥ (33) 2LΓ(Es/N0) where L represents the number of symbols used to make where 2 Z T the estimate of the timing offset, T is the symbol period, T 2 00 Γ = gd(t)dt (34) g (0) is the second derivative of the pulse shaping filter Es 0

where gd(t) denote the time derivative of g0(t), Es is the well known that the envelope of the sum of two quadrature 4π2 Gaussian noise signals obeys a Rayleigh distribution [16– symbol energy. For a raised cosine pulse Γ = 3 . Figure 5 shows variance of the NDA-NML estimator 18]. The Rayleigh distribution has a probability density with respect to CRB and MCRB when roll-off factor α = function given by ( ¡ ¢ 0.6 and N = 40 samples per symbol. It is demonstrated r r2 σ2 exp − 2σ2 , 0 ≤ r ≤ ∞ in [14] that MCRB is generally lower than (at most equal) f(r) = h h (35) to the true CRB. The simulation result exhibits that the 0, r < 0 CRB is always greater than MCRB and the variance of where σh is the root mean square (rms) value of the the estimator is greater than CRB and MCRB as well. received signal before envelope detection. In our system The variance of the estimator decreases as the number of model the sampled received signal experiencing Rayleigh symbols contributed for the timing estimation increases. fading can be expressed by (5) where the fading process With the increasing number of symbols used for estima- h(kTs) is Rayleigh distributed. For fading amplitude we tion, the performance of the estimator improves in terms use h in place of h(kTs) which can be expressed as: of variance. If the number of symbols contributed for the ¡q ¢ estimation is signiﬁcantly small, then the performance of 2 2 h = hI + hQ (36) the estimator will be degraded. It is easy to implement the NDA-NML estimator and the complexity is low as where hI and hQ are uncorrelated and independent of well. each other and follow correlated Gaussian distribution. The correlations among the samples are achieved

−1 passing the Gaussian distributed samples through a 10 spectral shaping ﬁlter. In our case we use a digital

−2 one dimensional filter to correlate the samples. The 10 coefficients of the filter are given by: h ³ í −3 γ 10 A = 1 − exp − , and B = γ (37) F s −4 NML L=100 10 CRB L=100 where γ defines the amount of correlation in h(kTs) and MCRB L=100 F s is sampling frequency. The filter is a “Direct form −5 NML L=200 10 CRB L=200 II Transposed” implementation of the standard difference MCRB L=200 equation:

Variance of timing estimates −6 10 NML L=50 CRB L=50 A(1)y(n) = B(1)x(n) − A(2)y(n − 1) (38) MCRB L=50 −7 10 −5 0 5 10 15 20 25 30 where X is the input to the filter and Y is the filtered E /N b 0 output. The channel energy is maintained to be unity so that the received SNR doesn’t change. Figure 5. NDA-NML estimator error variance performance. Figure 6 shows the bit error performance for timing estimator operating under a Rayleigh fading channel when roll-off factor α = 0.6, N = 15 samples per symbol, and L = 100 symbols used for timing estimates. The VI.TIMING ESTIMATOR OPERATING UNDER FADING amount of correlation has a significant effect on BER CHANNELS performance. Figure 6 exhibits that the BER performance One of the many impairments inherently present in any for the timing estimator improves as the γ increases under wireless communication system, that must be recognized fading conditions. and mostly mitigated for a system to function well, is fading. In this paper, we consider Rayleigh fading, B. Timing Estimator Operating Under Weibull Fading Weibull fading and log-normal fading channels and assess Conditions the performance of the timing estimator computing BER under each fading condition assuming flat fading. Last of The Weibull distribution is useful for modeling mul- all, the three fading scenarios are compared. tipath fading signal amplitude. Weibull fading channel model exhibits an excellent fit both for indoor [19–21] and outdoor environments [22]. It provides flexibility A. Timing Estimator Operating Under Rayleigh Fading in describing the fading severity of the channel and Conditions subsumes special cases such as the well-known Rayleigh The statistical characteristics of a fading channel can fading. The appropriateness of the Weibull distribution to be modelled using various probability density functions. model fading channels was also reported in [23], where In mobile radio channels, the Rayleigh distribution is a path-loss model for the Digital Enhanced Cordless commonly used to describe the statistical time varying Telecommunications system at 1.89 GHz, was studied. nature of the received envelope of a flat fading signal , or A physical justification for modeling wireless fading the envelope of an individual multipath component. It is channels with Weibull fading distribution has been given

0 0 10 10

γ = 0.1 γ = 0.5 γ = 0.9 −1 10

−1 10

−2 10 BER BER

−2 λ = 1.0, k = 1.0 10 W W

−3 λ = 1.0, k = 1.5 10 W W λ = 1.0, k = 2.0 W W λ = 1.0, k = 3.0 W W

−3 −4 10 10 −2 0 2 4 6 8 10 −2 0 2 4 6 8 E /N (dB) E /N (dB) b 0 b 0

Figure 6. Bit error performance of NDA-NML estimator under Rayleigh Figure 7. Bit error performance of NDA-NML estimator under Weibull fading. fading.

in [24]. Based on fading channel data obtained from a C. Timing Estimator Operating Under Log-normal Fad- recent measurement program at 900 MHz, Tzeremes and ing Conditions Christodoulou also reported that the Weibull distribution The log-normal distribution has traditionally been used can be used to model outdoor multipath fading well to describe the noticeable change in mean signal level in some cases [25]. In the past few years, a renewed (slow fading) experienced by a mobile receiver moving interest has been expressed in studying the characteristics over long distances within a multipath environment. Rapid of Weibull fading channel and performances of different signal fluctuations about the local mean may also be wireless techniques operating on such channel. This is observed over much shorter distances, typically in the evident from the numerous publictions covering different order of tens of wavelengths [30]. Log-normal small- aspects of this fading model [26–29]. scale fading has been reported for some radio channels, The probability density function of the Weibull distri- particularly in indoor environments [31–33]. Recently, bution is extensive effort has been directed at characterization of b b−1 −( x )b the propagation channels that involve the human body. Re- fX (x) = ba x e a (39) search into on-body channels [32], where communications where b > 0 is the shape parameter which defines the is across the surface of human body and off-body [33], severity of fading and a > 0 is the scale parameter of the where a bodyworn radio device is communicating with distribution. Smaller value of shape parameter indicates remote terminal, have both reported small-scale fading more severity of fading. In the special case when b = 1, which has followed the log-normal distribution. A statis- the Weibull distribution becomes an exponential distribu- tical characterization of the narrowband dynamic human tion; when b = 2, the Weibull distribution specializes to on-body area channel was presented in [34] demonstrating a Rayleigh distribution. that log-normal distribution provides a good fitting model, For simulation purpose, for generating Weibull dis- particularly when the subject’s body is moving. tributed random variates, at first a random variate U drawn The log-normal probability density function p(r) for an from the uniform distribution in the interval (0, 1) is envelope R may be expressed as 1/b generated. Then the random variate hW = a(− ln(U)) ³ 2 ´ 1 −[log(r) − µL] is generated which has a Weibull distribution with pa- pR(r) = √ exp 2 (40) rσL 2π 2σL rameters b and a. hW is used as the fading amplitude for Weibull fading. While generating U, zero values are ex- for r > 0, where µL and σL are the log-mean and log- cluded to avoid natural log of zero. Sampled received sig- standard deviation and both of which are found from nal experiencing Weibull fading can be expressed by (5) log[p(r)]. where the fading process h(kTs) is the Weibull distributed For our simulation purpose, at first a random variate S fading amplitude hW . BER when the timing estimator is generated which is drawn from the normal distribution operates under Weibull fading channel is demonstrated in with 0 mean and 1 standard deviation and then the Figure 7 for various Weibull shape parameters for roll-off variate hL = exp(µ + σS) is generated which has a log- factor α = 0.6, N = 15 samples per symbol and L = 100 normal distribution with parameters log-mean µL and log- symbols used for the timing estimates. With increasing standard deviation σL. Sampled received signal experienc- value of shape parameter, the BER performance improves. ing log-normal fading can be expressed by (5) where the fading process h(kTs) is the log-normal distributed fading

amplitude hL. Figure 8 exhibits the BER performance for comparison of the Rayleigh fading, Weibull fading and the timing estimator operating under log-normal fading log-normal fading, in Figure 9. It is clear in Figure 9 when roll-off factor α = 0.6, N = 15 samples per that the estimator performs similarly considering these symbol, and L = 100 symbols used for timing estimates. fading conditions. Characterizations of these three types This performance is for the on-body wireless transmission of fading are reasonably different with respect to the scenarios where there is log-normal fading. Performance application of the NDA-NML estimator. decreases signiﬁcantly with the increasing of standard deviation. Small change of mean of the log-normal distri- 0 bution does not have signiﬁcant effect on the performance. 10 Rayleigh fading Weibull fading Log−normal fading

0 −1 10 10

−1

10 BER

−2 10 −2 10

µ = 0, σ = 0.75 L L BER µ σ −3 = 0, = 0.50 10 L L µ σ = 0, = 0.25 −3 L L 10 µ = 2, σ = 0.75 −4 −2 0 2 4 6 8 L L E /N (dB) −4 µ = 2, σ = 0.50 b 0 10 L L µ = 2, σ = 0.25 L L Figure 9. Bit error performance of NDA-NML estimator under various −5 10 fading. −2 0 2 4 6 8 E /N (dB) b 0

Figure 8. Bit error performance of NDA-NML estimator under log- VII.CONCLUSION normal fading. Statistical performance of a NDA-NML timing estimator was presented. Several interesting aspects of the timing Figure 9 shows the BER performance for three kinds of estimator were revealed examining the statistical distri- fading stated before. In every case, roll-off factor α = 0.6, bution of the estimator. The expression for maxima and N = 25 samples per symbol, and L = 300 symbols for expression for the probability mass function of the timing timing estimation are considered. In Figure 9, standard estimates were derived and compared with the simulated Rayleigh fading with fading amplitudes that are based on density function. The performance of the estimator was complex fading coefﬁcients with mean 0 and variance 1 assessed considering the BER, and the error variance was is considered. For the Weibull fading channel, the scale compared to the CRB and MCRB for AWGN channel. parameter a = 1 and shape parameter b = 1.5 are used. Finally, performance of the estimator was studied when For the log-normal fading channel, log-mean µL = 0 operating under different fading scenarios with different and log-standard deviation σL = 0.75 are used. The characterizations by calculating the BER and the estimator parameters for each of three different types of fading are was shown to be robust to change of fading conditions. appropriate to typical small-scale fading scenarios. From the Figure 9, it is demonstrated that BER performance ACKNOWLEDGMENT for the Rayleigh fading channel is better than that of the The authors would like to thank National Informa- Weibull fading and the log-normal fading channel and the tion and Communication Technology (ICT) Australia performance of the Weibull fading channel is better than (NICTA). NICTA is funded by the Australian Government that of the log-normal fading channel. The performance as represented by the Department of Broadband, Com- will vary depending on the value of fading parameters. munications and the Digital Economy and the Australian The estimator performs better for Rayleigh fading than Research Council through the ICT Centre of Excellence Weibull fading, because of the more fading severity of program. the Weibull fading. For the Weibull fading, 1.5 is used for shape parameter which is compared with a shape REFERENCES parameter of 2 corresponding to Rayleigh fading, which indicates more severity of the Weibull fading scenario. [1] K. H. Mueller and M. Muller, “Timing Recovery in Digital The estimator performs better for Rayleigh fading than Synchronous Data Receiver,” IEEE Trans. Communica- tions, vol. 24, no. 5, pp. 516–531, May 1976. that of log-normal fading because of the higher variance [2] F. M. Gardner, “Interpolation in Digital Modems-Part I: used in Figure 9 of the log-normal fading. The estimator Fundamentals,” IEEE Trans. Communications, vol. 41, is shown to be robust to a change of fading conditions in no. 3, pp. 501–507, Mar. 1993.

[3] ——, “Interpolation in Digital Modems-Part II: Implemen- Proceedings of the International Symposium on Antennas tation and Performance,” IEEE Trans. Communications, and Propagation Society, vol. 1, 2002, pp. 232–235. vol. 41, no. 6, pp. 998–1007, June 1993. [26] N. C. Sagias, D. A. Zogas, G. K. Karagiannidis, and G. S. [4] S. Kandeepan and S. Reisenfeld, “DSP based symbol Tombras, “Channel Capacity and Second Order Statistics timing estimation techniques,” in Proceedings of the IEEE in Weibull Fading,” IEEE Communications Letters, vol. 8, ICICS-PCM’03, Dec. 2003, pp. 568–572. pp. 377–379, June 2004. [5] U. Mengali and A. N. D’Andrea, Synchronization Tech- [27] M. H. Ismail and M. M. Matalgah, “Performance of niques for Digital Receivers. New York and London: Dual Maximal Ratio Combining Diversity in Non-identical Plenum Press, 1997. Correlated Weibull Fading Channels using Pade Approxi- [6] W. C. Lindsley and M. K. Simon, Telecommunication mation,” IEEE Trans. Communications, vol. 54, no. 3, pp. Systems Engineering. New Jersey: Prentice Hall, 1973. 397–402, Mar. 2006. [7] K. Bucket and M. Moeneclaey, “Timing Jitter Sensitiv- [28] W. Xie and S. Liu, “Outage Probability Analysis of Dual- ity Comparison of Narrowband M-PSK and Bandlimited brach Transmit-selective System Over Correlated Modified DS/SS M-PSK Signals,” IEEE Trans. Communications, Weibull Fading Channels,” in Proceedings of the IEEE vol. 1, no. 3, pp. 453–457, May 1993. International Conference on Communications, Cicuits and [8] W. K. M. Ahmed and P. J. McLane, “On the Error Expo- Systems, vol. 2, June 2006, pp. 659–663. nent for Memoryless Flat Fading Channels with Channel- [29] M. H. Ismail and M. M. Matalgah, “BER Analysis of State-Information Feedback,” IEEE Communications Let- BPSK Modulation Over the Weibull Fading Channels with ters, vol. 3, no. 2, pp. 49–51, Feb. 1999. CCI,” in Proceedings of the IEEE Vehicular Technology [9] T. Ghirmai, “Data Detection and Fixed Symbol-Timing Conference, Sept. 2006, pp. 1–5. Estimation in Flat Fading Channels,” in Proceedings of [30] W. C. Jakes, Microwave Mobile Communications. New the IEEE CISS, Mar. 2007, pp. 49–51. York: Wiley, 1974. [10] J. J. Stiffler, Theory of Synchronous Communications. [31] R. Gonesh and K. Pahlavan, “Statistics of Short Time Prentice Hall, 1971. Variations of Indoor Radio Propagation,” in Proceedings [11] S. M. Kay, Fundamentals of Statistical Signal Processing: of the International Conference on Communications, June Estimation Theory. Prentice Hall PTR, 1993. 1991, pp. 1–5. [12] T. Jesupret, M. Moeneclaey, and G. Ascheid, “Dig- [32] A. Fort, C. Desset, P. D. Doncket, P. Wanbacq, and ital Demodulator Synchronisation-Performance Analy- L. V. Biesen, “An Ultra-wideband Body Area Propagation sis,” European Space Agency Rep. ESTEC Contact No. Channel Model-from Statistics to Implementation,” IEEE 8437/89/NL/RE, June 1991. Trans. Microw. Theory Tech., vol. 54, pp. 1820–1826, [13] C. N. Georghiades and M. Moenecleay, “Sequence Estima- 2006. tion and Synchronization from Nonsynchronised Samples,” [33] S. L. Cotton and W. G. Scanlon, “Indoor Channel Char- IEEE Trans. Inform. Theory, vol. 37, pp. 1649–1657, Nov. acterisation for a Wearable Antenna Array at 868 MHz,” 1991. in Proceedings of the IEEE Wireless Communication and [14] A. D’Andrea, U. Mengali, and R. Reggiannini, “The Mod- Networking Conference, Apr. 2006, pp. 1783–1788. ified Cramer Rao Bound and Its Application to Synchro- [34] D. Smith, L. Hanlen, D. Miniutti, and J. Zhang, “Statistical nization Problems,” IEEE Trans. Communications, vol. Characterization of the Dynamic Narrowband Body Area 2/3/4, pp. 1391–1399, Feb./Mar./Apr. 1994. Channel,” in Proceedings of the International Symposium [15] S. C. White and N. C. Beaulieu, “Improved Bounds for on Applied Sciences on Biomedical and Communication Timing Estimation Jitter,” in Proceedings of the IEEE Technologies, Oct. 2008, pp. 1–5. GLOBECOM’90, Dec. 1990, pp. 609–616. [16] J. P. E. Biglieri and S. Shamai, “Fading Channels: Informa- tion Theoretic and Communications Aspects,” IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998. Md. Tofazzal Hossain is currently a PhD candidate in In- [17] C. Loo and N. Secord, “Computer Models for Fading formation Engineering specifically in the areas of wireless Channels with Applications to Digital Transmission,” IEEE communications at Australian National University, Canberra, Trans. Vehicular Tech., vol. 40, no. 4, pp. 700–707, Nov. ACT, Australia. He was born in Mymensingh, Bangladesh in 1991. 1984. He received his Bachelor of Science in Computer Science [18] T. S. Rappaport, Wireless Communications, Principles and and Engineering from Khulna University of Engineering and Practice. Prentice Hall, 1996. Technology, Khulna, Bangladesh in 2005. He was awarded [19] H. Hashemi, “The Indoor Radio Propagation Channel,” in Prime Minister Gold Medal and University Gold Medal for his Proceedings of the IEEE, vol. 81, July 1993, pp. 943–968. academic Excellence. He received his Masters of Information [20] R. K. Mallik, “On Multivariate Rayleigh and Exponentials and Communication Technology from Australian National Uni- Distributions,” IEEE Trans. Inform. Theory, vol. 49, no. 6, versity in 2007. He was awarded National Information and Com- pp. 1499–1515, June 2003. munication Technology (ICT) Australia (NICTA) Scholarship [21] N. H. Shepherd, “Radio Wave Loss Deviation and Shadow for his Masters and PhD. Loss at 900 MHz,” IEEE Trans. Vehicular Tech., vol. 26, He was a lecturer of the department of Computer Science pp. 309–313, Nov. 1977. and Engineering at the Khulna University of Engineering and [22] N. S. Adawi, “Coverage Prediction for Mobile Radio Technology, Khulna, Bangladesh, 2005-2007. His research in- Systems Operating in the 800/900 MHz Frequency Range,” terests include body-area communications, cooperative com- IEEE Trans. Vehicular Tech., vol. 37, pp. 3–72, Feb. 1988. munications, low power communications and synchronization [23] A. Babich and G. Lombardi, “Statistical Analysis and techniques for modern digital receivers. Characterization of the Indoor Propagation Channel,” IEEE Trans. Communications, vol. 48, pp. 455–464, Mar. 2000. [24] J. Cheng, C. Tellumbura, and N. C. Beaulieu, “Perfor- mance Analysis of Digital Modulations on Weibull Fading Sithamparanathan Kandeepan was born in Colombo in 1975, Channels,” in Proceedings of the IEEE Vehucular Technol- he did his undergraduate level of studies in Colombo by com- ogy Conference, vol. 1, Oct. 2003, pp. 236–240. pleting the Engineering Council (UK) examinations majoring [25] G. Tzeremes and C. G. Christodoulou, “Use of Weibull in communications and control systems engineering, he then Distributions for Describing Outdoor Multipath Fading,” in went on to complete his Masters in Engineering (Telecommu-

nications) in 2000, and his PhD in Electrical Engineering in 2003, both at the University of Technology, Sydney. Kandeepan currently is a senior researcher with the Create-Net International Research Centre in Trento, Italy working on various Industrial and European Union funded projects on applied and fundamen- tal research in the areas of wireless, satellite and cognitive radio communications. In the past (2000 to 2004), he worked with the Cooperative Research Centre for Satellite Systems (CRCSS) on the FedSat micro satellite project in Sydney, whilst with the CRCSS he won the Earth Station Satellite Fellow award at the University of Technology, Sydney and from 2004 to 2008 he was with the National ICT Australia (NICTA) in Canberra as a Researcher working on wireless signal processing. Kandeepan a member of the IEEE since 2001.

David Smith was born in 1974. He received the B.E. degree in electrical engineering from the University of N.S.W. Australia in 1997, while studying toward this degree he was on a CO-OP scholarship. He obtained an M.E. (research) degree in 2001 and a Ph.D. in 2004 both from the University of Technology, Sydney (UTS) and both in telecommunication engineering, speciﬁcally in the area of wireless communications. He has also had a variety of industry experience in telecommunications planning; radio frequency, optoelectronic and electronic communications design and integration. He is currently a researcher at National ICT Australia (NICTA) and is an adjunct research fellow with the Australian National University, both since 2004. His research interests include: MIMO systems; coherent and non-coherent space-time coding; body-area communications; radio propagation and electromagnetic modelling; and antenna design.