This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings. Convergence Property of Transmit Time Pre-Synchronization for Concurrent Cooperative Communication Yong Jun Chang and Mary Ann Ingram School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250 Email: {yongjun.chang, mai}@gatech.edu Abstract—This paper proposes a preamble detection-based cooperative transmit time pre-synchronization technique specially for Concurrent Cooperative Transmission (CCT). CCT enables a collection of power-constrained radios to transmit as a group and achieve a transmit range that is much greater than the range of a single device. In the absence of a GPS system or a globally synchronized clock, each cooperating node can derive its transmit time autonomously by detecting the preamble from a received signal. The paper first compares two methods for estimating the start of packet (SOP) received in the presence of a carrier frequency offset (CFO): the peak method, which is optimal in a single diversity channel without CFO, and a mean method, which outperforms peak with large CFO. Next, the SOP is estimated by combining all diversity channels using an approximated Best Linear Unbiased Estimator (BLUE). Using theory and measurement, we show that the time error statistics are convergent as the number of hop increases. I. I NTRODUCTION Cooperative transmission (CT) is a physical layer wireless communication scheme in which spatially separated sensor nodes collaborate to transmit the same source message so that the transmission power required for each node can be reduced or the range can be extended [1][2]. Collaborating nodes can be chosen in a centralized or distributed manner and grouped into a cluster. Therefore, we can consider cooperative communication to be cluster-to-cluster transmission (MIMO) or cluster-to-single-node transmission (MISO). Concurrent CT (CCT) is a special form of CT which allows cooperating nodes transmit the diversity versions of the message at the approximately same time. Any relative synchronization errors in the transmit times of the cooperators roughly add to the delay spread of the multipath channel, so our aim is for the root mean square (RMS) transmit time spread to be at most comparable to the channel delay spread. This paper considers how synchronization time errors evolve over consecutive CCT hops without single source redistribution within a cluster. We show by both measurement in a typical indoor office environment as well as theoretical analysis that the error statistics are convergent with the number of hops. The measurement data shows that the error spread is comparable to delay spreads of the indoor environment. The impairment caused by time synchronization errors on the performance of cooperative MIMO was theoretically treated in [3]. The authors also showed that jitters as large as 10% of the bit duration do not have much effect on the BER performance of the system. Using a globally synchronized clock by network time protocol or a GPS has been proposed for CCT [4]. These approaches require additional devices or a convergence period for accurate time synchronization. In this paper, we show experimental results for CCT, using binary frequency shift keying (BFSK), with non-coherent demodulation. Orthogonality is achieved in the frequency domain, by having different cooperating radios transmit on different orthogonal carriers. Because CCT is not supported by any off-the-shelf radios, we use software-defined radios (SDRs) to implement our approach. II. S YSTEM M ODEL AND P REAMBLE D ETECTION Without using a globally synchronized clock, cooperating nodes can decide their cooperative transmission time by adding fixed amount of period to the reception time of the message to be relayed. It can be achieved by locating the preamble or sync word embedded in the packet. Suppose that s(t) is the modulated baseband signal of a known preamble which has energy |s(t)|2 dt = 1. The received signal r(t) has a frequency offset f which is introduced by carrier frequency difference between the transmitter and the receiver. Assuming a quasi-static fading channel with a channel coefficient α by r(t) = h · s(t − τ ) · ej2πf t + w(t) (1) where τ is a propagation delay and w(t) is zero-mean complex additive white Gaussian noise (AWGN) with a power spectral density of N0 . If the preamble is known a priori at the receiver, the beginning of the preamble can be found by cross-correlating the received signal r(t) with a preamble signal s(t) and finding a threshold or peak. Mathematically, the cross-correlation of the preamble-bearing signals s(t) and the received signals r(t) which has encountered a frequency offset f can be described as d(t, τ, f ) = h · q(t, τ, f ) + z(t) (2) ∞ where q(t, τ, f ) = −∞ s(x − τ )s∗ (t + x)ej2πf (x−τ ) dx ∞ and z(t) = −∞ w(x)s∗ (t + x)dx. z(t) is zero-mean complex AWGN with a variance N0 . Under the assumption of high signal-to-noise ratio (SNR) and f = 0 [5] the estimation of 978-1-4244-5637-6/10/$26.00 ©2010 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings. Fig. 2. Illustration of the multiple Concurrent Cooperative Transmission (CCT) hops Fig. 1. Decision function Λ(t) of a single preamble in the presence of carrier frequency offset (CFO) and the accuracy of peak and mean estimator in the presence of CFO SOP time can be achieved by choosing t to maximize Λ(t) = P −1 2 i=0 |d(t − iT )| where P is the length of preamble. The accuracy of correlation rule is proportional to the SNR and sidelobe suppression of the correlation output [6]. With a carrier frequency offset (CFO), however, peak searching suffers performance degradation because CFO makes the filter mismatched. Example filter output shown in in-set figure in Figure 1, which is the decision function Λ(t) over (−T, T ) where h = 1, P = 1, τ = 0, and noise-free z(t) = 0. It is shown that as CFO increases, signal strength diminishes. A. Mean Estimation Instead of searching a peak of correlation output to determine a SOP time, we propose a mean (center-of-mass) estimation. Without loss of generality, let us assume τ = 0 which implies that the truth time of SOP is 0. We also assume that the soft-valued signal Λ(t) is sampled with an oversampling rate 1/Ts . The finding a mean of Λ(t) can be realized as follows: K −K k · Λ[k] . (3) ω = Ts K −K Λ[k] In Appendix A, we show that the estimation error ω has an approximately zero mean Gaussian distribution in high SNR and its variance σω2 can be approximated by σω2 = where K= 2 i j N0 KT s2 |h|2 (4) ij|q[i, f ]||q[j]|(s s)2 [i − j] . ( i q 2 [i])2 (5) Figure 1 shows a MATLAB simulation of the root-meansquare error (RMSE) of peak and mean estimator in the presence of CFO = {0, 5, 10, 15}ppm. For this simulation, 1MHz sampling rate Ts = 10−6 and a BFSK waveform with a rectangular shaping pulse are used. It also shows the theoretical RMSE treated in Equation 4 by using a dotted line. While peak estimation outperforms mean estimation in the absence of or relatively small CFO, mean estimation gives better performance in the presence of relatively large CFO. Intriguingly, it is shown that RMSE of peak estimator at 15ppm CFO rises rapidly because the output of correlator Λ(t) becomes blunt and finally it becomes bimodal as shown in the in-set figure in Figure 1. III. C OOPERATIVE T RANSMIT T IME S YNCHRONIZATION In this section, we analyze the Start of Packet (SOP) time estimation error at each relay in a sequence of relay clusters, such that each cluster has N relays and does concurrent cooperative transmission (CCT). We assume that the packet is originally transmitted by a single-antenna source (S) at time T , as shown in Figure 2 for N = 2. To simplify the analysis, we assume that the N nodes in each cluster are co-located (not shown this way in the figure) and that the clusters are arranged on a line with equal spacing. This implies that the propagation distances between nodes in different clusters are equal and deterministic. We assume that each relay in a cluster transmits a preamble that is orthogonal to the other N − 1 preambles being transmitted in the cluster. These orthogonal preambles define the N diversity channels for synchronization, and are transmitted simultaneously. We assume that each receiver has a bank of N correlators, with one correlator for each diversity channel. We assume independent Rayleigh fading in all diversity channels. For all terms used in this paper, a superscript and subscript represent the cluster and node index, respectively. For exam(j) ple, Tk is a time of transmission of Relay k in Cluster j. (j) For example of two subscripts case, h21 is a channel between (j) Relay 1 in Cluster j − 1 and Relay 2 in Cluster j. T̂kn is (j−1) an estimate of Tn based only on correlation in channel (j) (j) hkn at the Node k in the Cluster j. ωkn is the “correlator (j) estimation error” of T̂kn transmitted from the Node n of (j) previous cluster. T̂k is an approximated Best Linear Unbiased Estimate (BLUE) of T without clock error, which is a linear (j) (j) combination of T̂kn ’s. ξk is a clock compensation error of the Node k in the Cluster j. Each relay autonomously estimates the time of the SOP. (j) (j) Let Rk be the kth relay in the jth cluster, and let T̂k be its 978-1-4244-5637-6/10/$26.00 ©2010 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings. estimate of the SOP. The details on how that estimate is formed are explained below. Next, ideally, each relay determines its transmit time by adding an universally known constant, Tproc , (j) to T̂k , to allow enough time for the signal processing on all relays in a cluster to be completed [7]. However, because each relay has a slightly different oscillator frequency from the other relays, some clocks run fast and some run slow; this is “clock skew.” Each relay will attempt to compensate its clock skew, but the compensation is not perfect, so some relays will still have Tproc expire faster than others. The deviation in Tproc expiration on Node k of Cluster j from an imaginary universal (j) (j) reference clock is designated as the “clock error,” ξk . ξk is assumed to be zero mean and Gaussian random variable with a variance σξ2 . Therefore, the transmit time of the kth relay in (j) (j) (j) the jth cluster is Tk = T̂k + ξk . (j) To compute T̂k , the estimate of the SOP at the kth node in (j) the jth cluster, the node first creates N estimates, T̂kn , n = 1, 2, . . . , N , of the SOP in each of its N diversity channels, respectively, by defining each estimate to be a centroid of the correlator output as discussed in Section II. Since only one of the previous cluster nodes transmits in a given diversity (j) channel, T̂kn is actually an estimate of the SOP of the packet (j−1) (j) (j−1) (j) received from Rn , i.e. T̂kn = Tn + ωkn . A. Approximated Linear Unbiased Estimation The performance of cooperative transmit time synchroniza(j) tion can be assessed by a covariance of transmission time T̂k . In this subsection, we design an approximated linear unbiased (j) estimator to minimize the variance of transmission time T̂k (j) (j) where T̂k is a linear combination of T̂kn for 1 ≤ n ≤ N . (j) Let akn denote a coefficient of combiner. By best linear unbiased estimator (BLUE) and under the assumption that the average transmission time errors of previous cluster nodes are (j) (j) (j) (j) same, the coefficient vector ak = [ak1 , ak1 , . . . , akN ]T = (V−1 1)/(1T V−1 1) where V is a covariance matrix of (j) (j) (j) (j) T̂ k = [T̂k1 , T̂k2 , . . . , T̂kN ]T and 1 is the column vector of (j) N ones. Since single correlation errors ωkn are uncorrelated over index n, the covariance matrix becomes a diagonal matrix V = diag[σω2 1 , σω2 2 , . . . , σω2 N]. Therefore, the estimator N (j) (j) 2 (j) 2 . coefficient becomes akn = |hkn | / i=1 |hki | To simplify analysis, we assume that all deterministic times are equal to zero; this includes the propagation time and Tproc . This allows us to focus only on the behavior of time estimation errors. Since all the errors are zero mean, we have that all estimators are trying to estimate the same thing, T , which is the original transmit time of the source (S). B. Markov Process Model (j) (j) (j) Let the vector T (j) = [T1 , T2 , . . . , TN ]T represent the transmit times of each relay in Cluster j (or “hop j”). From the arguments above, each element of T(j) is an unbiased estimate of the original packet transmit time, or E{T(j) } = T · 1. Let the estimation error vector of T (j) be denoted as e(j) = (j) (j) (j) [e1 , e2 , . . . , eN ]T . Our objective is to get a recursion for T the covariance of transmit time error C(j) = E e(j) e(j) , from which we derive the sample variance. Because our estimator coefficients at Cluster j do not depend on the multi-path channels of previous hops, we can see that the statistics of the errors at Cluster j are independent of the errors of past hops, assuming T (j−1) is given. In other words, the conditional joint probability density function (PDF) of T (j) , given the entire past transmit times {T (j−1) , T (j−2) , . . . , T (1) }, is the same as the conditional joint probability density function (PDF) of T (j) , given only the latest vector of transmit times T (j−1) . This makes T (j) , j = 1, 2, . . . , N , a vector-valued, continuous-state, discrete-time (j) (j) (j) Markov Process. ek with substitution of T̂k and T̂kn can be written as (j) (j) (j) ek = T̂k + ξk − T = N (j) (j) (j) akn T̂kn − T + ξk n=1 = N (j) akn e(j−1) + n n=1 N N (j) (j) (j) akn ωkn + ξk n=1 (j) (j) noting that and n=1 akn = 1 for all j and k. Let A (j) be a estimator coefficient and correlator error matrix ω (j) (j) whose elements of kth column and nth row are akn and ωkn (j) (j) (j) T (j) respectively, and ξ = [ξ1 , ξ2 , . . . , ξN ] . The covariance matrix C(j) has an iterative form as T T C(j) = E{e(j) e(j) } = E A(j) C(j−1) A(j) (j) (j) (j) (j) T (j) (j) [A ◦ ω ]1 + ξ (6) + E [A ◦ ω ]1 + ξ where ◦ denotes an element-wise product operator. In the following sections, we show that the expected value of the (j) sample variance of estimation error ek is convergent even though the absolute covariance matrix is not. C. Convergence Property The performance of transmit time pre-synchronization can be assessed by relative transmit time error of relay nodes within a cluster. The relative transmit time error of jth cluster’s relay nodes is basically equivalent to the sample variance of estimation error e(j) . To simplify analysis of a convergence property of the covariance matrix C(j) , we assume that the statistics of all channels, clocks and noises do not vary with relay index within a cluster. These assumptions yield that the variance and the covariance of the estimation error are not functions of a node index. Therefore, the diagonal terms of (j) the matrix C(j) , denoted as Cd , are identical. Also, the off(j) diagonal terms, denoted as Co , are identical. In Appendix B, (j) (j) we show that Cd and Co can be written as 2 N − 1 (j−1) (j−1) Cd C + + H + N σξ2 N +1 N +1 o 1 (j−1) N − 1 (j−1) Co Co(j) = Cd + N N N (j) (j) where H = n=1 E{agn ωgn }. (j) Cd = 978-1-4244-5637-6/10/$26.00 ©2010 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings. (a) Transmit Power = 0dBm Fig. 3. A floor map and photo of the experiment which had conducted on the fifth floor of the Centergy Building at the Georgia Institute of Technology In [8], the expected value of sample variance in which the samples are not mutually independent and pairwise covariances are constant γ, is given by E{σs2 } = σ 2 − γ where s2 is the sample variance and σ 2 is the variance. From this relationship, we can get the linear difference equation form of the expected value of the sample variance at jth cluster as E{σs2 (j) }= N −1 (j−1) E{σs2 } + H + N σξ2 . N (N + 1) The expected value of the sample variance is convergent to E{σs2 } = N (N + 1) (H + N σξ2 ) as j → ∞. N2 + 1 IV. E XPERIMENTAL S ETUP AND R ESULTS The synchronization method we designed in Sections II and III was evaluated in a Software Defined Radio (SDR) testbed. Each wireless node in this experiment is composed of a RF-daughterboard (RFX-2400), an Universal Software Radio Peripheral (USRP1) board, a personal computer (PC), and the GNU radio software. The USRP1 board has an ADC/DAC and a FPGA to convert passband signal to baseband signal and vice versa. All baseband processing is done on the PC. Binary frequency shift keying (BFSK) with non-coherent envelope detection was used for the experiments. Orthogonal preambles were achieved by choosing orthogonal center frequencies. We used 64 kbps bit-rate with 1 Mhz sampling-rate. The total length of a packet is 24 bytes consisting of 4 bytes preamble, 6 bytes header, 10 bytes data and 2 bytes CRC. The source packet is scrambled at every transmission in which the index of scrambler is included in the header. All baseband processing for non-coherent BFSK modulator/demodulator and the transmit time synchronization schemes are programmed by using C++ and Python languages. (j) = The rms transmit time spread (RTTS), σs N i (b) Transmit Power = -5dBm Fig. 4. Measured RTTS of “ping-pong” experiment large, we designed the “ping-pong” experiment, where two groups of cooperating nodes transmit the source message back and forth up to 10 hops (or ”CT”s). Figure 3 shows a floor map and photo of the “ping-pong” experiment was conducted on the fifth floor of the Centergy Building at the Georgia Institute of Technology. The experiment was (j) repeated 500 times to get 500 trials of σs . USRP1 boards are programmed to generate a trigger pulse at the time of each transmission and the trigger pulse was captured by a customized FPGA board by wire. Figure 4 shows the empirical RTTS of the cooperative nodes with two different transmit powers. Each curve represents an empirical cumulative density function (CDF) of RTTS of each CT. We note that we measured sample variance of cooperative nodes so that the CDF does not follow the CDF of the error statistic we derived in Section III. As shown in the figure, the CDFs corresponding to −5dBm transmit power are shifted by approximately 20% compared to the CDFs corresponding to 0dBm. This was predicted in the simulation result in Figure 1. We also observe that RTTS tends to be converged after the third hop. This provides strong evidence that our analysis of convergence property is correct. Moreover 90% of the RTTS is less than 300ns in both transmit powers, which indicates that CCT can support up to 300 kbps data rate in narrow-band waveforms without significant ISI degradation [3]. V. C ONCLUSION (j) (Ti −T̄ (j) )2 N −1 of cooperative relay nodes in jth CT was used to evaluate the performance where T̄ (j) is the sample mean of transmission time of all relay nodes at Cluster j. To simulate multiple consecutive cluster hops in a In this paper, we proposed a preamble detection-based transmit time synchronization scheme for CCT. We designed a mean estimation method for non-coherent BFSK demodulation to estimate the start of packet (SOP) time. We showed that 978-1-4244-5637-6/10/$26.00 ©2010 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings. the mean estimation method outperforms the peak searching method in the presence of CFO by both analysis and simulation. Diversity channels are formed by having each cooperating node transmit on an orthogonal carrier frequency. The estimates of the SOP times in each diversity channel are combined using an approximate BLUE method, to produce an overall estimate of the SOP, which is the reference for the retransmit times. We show that the transmit time error statistics are convergent over consecutive CCT hops based on measurement in an SDR testbed in a typical indoor office environment, as well as by theoretical analysis. Our experiment and implementation show that CCT without using single-source redistribution within clusters and without using a globally synchronized clock is practical for the indoor environment. A PPENDIX A Let us consider a single bit preamble P = 1. At any index k, Λ[k] can be considered as an absolute square of a non-zero mean complex Gaussian random variable whose normalized value has a non-central chi-square distribution with two degrees of freedom and non-centrality parameter λ = μ2 /σ 2 where μ and σ 2 are mean and variance of unnormalized complex Gaussian random variable. The mean and variance of normalized non-central chi-square distribution with two degrees of freedom are 2+λ and 4(1+λ) respectively. Therefore, the mean and variance of Λ[k] at any index k are μΛ[k] = N0 + |h · q[k]|2 2 σΛ[k] = N02 + 2N0 |h · q[k]|2 where q[k] is a sampled signal of q(t). Similarly, the mean and variance of k · Λ[k] are 2 = k 2 (N02 + 2N0 |h · q[k]|2 ) σk·Λ[k] By (CLT), a probability distribution central limit theorem of k · Λ[k] and Λ[k] can be approximately Gaussian distribution. Let X and Y be denoted as k·Λ[k] and Λ[k] respectively and define Z = X/Y . In Equation 2, z[k], a sampled version of z(t), is a filtered white Gaussian noise so called a colored Gaussian noise . Its auto-covariance is Cov(z[i], z[j]) = N0 · (s s)[i − j]. The mean and variance of X and Y are μΛ[i] μx = 0, μy = = i σy2 = i i σi·Λ[i] σi·Λ[j] (s s)2 [i − j] j (j) (j−1) Cd = Cd N 2 (j−1) E{(a(j) gn ) } + Co n=1 + N (j−1) Co(j) = Cd (j) E{a(j) gx agy } x=1 y=1 x=y (j) E{a(j) gn ωgn } + n=1 N N N E{ξn(j) } n=1 N 2 (j−1) (E{a(j) gn }) + Co n=1 N N (j) E{a(j) gx }E{agy } x=1 y=1 x=y (j) It can be shown that E{agn } = 1/N regardless any inN (j) (j) dexes [10]. Assuming that X = |hgn |2 and Y = k=1 |hgk |2 , (j) 2 E{(agn ) } can be written as E{X 2 }/E{Y 2 }. Since X and Y have exponential and gamma distribution respectively, (j) E{agn } can be simplified to N (N2+1) . Also we can get N N (j) E{a(j) gx agy } = E{( x=1 y=1 x=y N 2 a(j) gn ) } − E{ n=1 = 1 − E{ N N 2 (a(j) gn ) } n=1 2 (a(j) gn ) } = n=1 N −1 N +1 (j) It is noted that the third and fourth term of Cd do not depend on the index j because we assume that the statistics of all channels, clocks and noises do not vary. ACKNOWLEDGMENT The authors would like to acknowledge Zhen GAO in Tianjin University, China for his importance review and discussion. R EFERENCES μk·Λ[k] = k(N0 + |h · q[k]|2 ) σx2 From Equation 6, A PPENDIX B (j) and Co can be written as (j) Cd σΛ[i] σΛ[j] (s s)2 [i − j]. j By Geary-Hinkley transformation √ [9], a random variable ω can be transformed to ω = (Ts KN0 /|h|)u where u is a zero-mean unit variance Gaussian random variable. [1] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. on Inform. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [2] A. Scaglione and Y. Hong, “Opportunistic large arrays: cooperative transmission in wireless multihop ad hoc networks to reach far distances,” IEEE Trans. on Signal Processing, vol. 51, no. 8, pp. 2082– 2092, Aug. 2003. [3] S. Jagannathan, H. Aghajan, and A. Goldsmith, “The effect of time synchronization errors on the performance of cooperative miso systems,” IEEE GlobeCom Workshop, pp. 102 – 107, Nov. 2004. [4] A. Blair, T. Brown, K. M. Chugg, T. R. Halford, and M. Johnson, “Barrage relay networks for cooperative transport in tactical MANETs,” IEEE MILCOM, Nov. 2008. [5] G. Lui and H. Tan, “Frame synchronization for gaussian channels,” IEEE Trans. on Comm., vol. 35, no. 8, pp. 818 – 829, Aug 1987. [6] M. I. Skolnik, “Theoretical accuracy of radar measurements,” IRE Trans. on Aeronautical and Navigational Electronics, vol. ANE-7, no. 4, pp. 123 –129, Dec. 1960. [7] Y. J. Chang, M. A. Ingram, and S. R. Frazier, “Cluster transmission time synchronization for cooperative transmission using software defined radio,” ICC Workshop on CoCoNet3, June 2010. [8] R. Serfling. Expected value of the sample variance. [Online]. Available: http://www.utdallas.edu/ serfling/handouts/ [9] D. V. 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