Time-Slotted Round-Trip Carrier Synchronization in Large-Scale Wireless Networks

Transcription

1 Time-Slotted Round-Trip Carrier Synchronization in Large-Scale Wireless etworks Qian Wang Electrical and Computer Engineering Illinois Institute of Technology Chicago, IL Kui Ren Electrical and Computer Engineering Illinois Institute of Technology Chicago, IL Abstract Carrier synchronization has been considered to be a big challenge to achieve distributed beamforming in wireless networks. In this paper, we investigate the feasibility of synchronizing the carriers of distributed transmitters with a distant Base Station receiver. Under the duplexing constraints of the wireless channel, we present a time-slotted round-trip carrier synchronization technique required for large-scale wireless networks in a master-slave architecture. The performance analysis presented in terms of average received power at BS demonstrates the robustness of the beamforming gains against synchronization errors. Our simulation results validate the analysis and indicate that the potential beamforming gains outweigh the synchronization overhead in implementation of the protocol. I. ITRODUCTIO In wireless networks, distributed beamforming provides a potential way to increase the energy efficiency. A group of cooperative transmitters can emulate a centralized antenna array by transmitting a common transmission to a intended Base Station [1. By adjusting the carrier phase of each transmitter, the individual transmissions will combine coherently at the receiver. The energy efficiency gains from beamforming are well-known, however, the fundamental limitation in realizing these gains are the duplexing constraints of the wireless channels and the requirement of precise phase and frequency synchronization. The research on synchronization in distributed beamforming algorithms also has its application in cooperative communication systems, where coherent combining of and relay transmissions are required at the destination. The focus of the growing literature on cooperative transmission has been on obtaining diversity gains [4-5. In contrast to diversity schemes, beamforming can not only reduce the probability of outage but also offer SR gains. However, most prior work on cooperative communication, including studies of the distributed coding techniques for space-time coding gains, assumes that synchronization is perfect [6-7. The performance of distributed beamforming has been previously well studied in the literature. In [8, the authors proposed a synchronization scheme in which a feedback from BS is utilized to minimize coordination among the transmitters while achieving coherent combining at the BS. However, this method requires significant coordination with the distant BS. Further, a protocol with minimized coordination with the BS in a master-slave architechture was proposed in [1, where a beacon is used to estimate the phase delay between transmitting nodes and master node in order to perform phase calibration. The use of channel reciprocity allows the transmitting nodes to carry out channel estimation for phase precompensation. The concept of round-trip carrier synchronization was first described in [, where continuously transmitted beacons were used at different frequencies. Unfortunately, this system has a degraded performance in multipath channels. In recent work, the authors in [3 propose a time-slotted round-trip carrier synchronization protocol for two transmitters. The performance is investigated in terms of the phase offset at the destination. In this paper, we present a new time-slotted round-trip carrier synchronization protocol required for large-scale cooperative communication. Fixed transmitters placement are considered in the analysis and the synchronization process is described in detail with single-path channels. The cumulative phase errors which limit the achievable average beamforming gains are identified in the time-duplexed operation of the synchronization process and the performance of the protocol in terms of average received power is compared with the fundamental Cramer-Rao estimation bounds. Our results show that even with imperfect synchronization due to cumulative phase errors, a large fraction of beamforming gains can be realised. II. SYSTEM MODEL In a wireless sensor network, we consider a cluster of sensors (transmitters), communicating a common message m(t) to one destination (BS receiver). The system model is shown in Figure 1. In the network, each sensor has a seperate local clock and a maximum likelihood (ML) estimator for frequency and phase estimation is employed. The channel from node i to node j is modeled as an LTI system with impulse response h ij (t) and stays roughly constant for several timeslots. We also assume channel reciprocity such that h ij (t) = h ji (t) and the underlying noise is additive white Gaussian noise (AWG). The idea behind the time-slotted round-trip carrier synchronization technique was first described in [3. The authors consider a two- one-destination model and each channel is single-path with delay τ ij. Because the total propagation /08/$ IEEE 5087

2 (node ) (node ) (node 1) (node 5) (node 4) Fig. 1. (node 3) system model. Destination (node 0) times for the circuit D S 1 S D and the circuit D S S 1 D are identical, the carriers at each will arrive with identical phase at the destination. Our protocol designed for large-scale cooperative communication is based on the same idea, i.e. all the signals finally received at the destination will have the same propagation time. The following section describes a round-trip carrier synchronization protocol based on a master-slave architechture, i.e. one of the s will serve as a master sensor to coordinate the transmissions among the sensors. III. PROTOCOL DESCRIPTIO In this section, we present a protocol for achieving carrier phase synchronization based on a master-slave architechture. Assume there are s and one destination in the wireless sensor network, we now list activities in each timeslot: TS 1 : The destination S 0 broadcasts a sinusoidal primary beacon to all s. All s generate phase and frequency estimates from their local observations. From TS to TS :S,...,S transmits a sinusoidal secondary beacon to S 1 in TS,...,TS, respectively. The phase and frequency of each beacon is estimated from their observations in TS 1.S 1 generates phase and frequency estimates from the local observations in the corresponding timeslot. S 1 will then have phase/frequency estimates-one from the beacon transmitted by the destination and one from each of the other -1 nodes. From TS +1 to TS -1 : S 1 transmits a sinusoidal secondary beacon to S,...,S in TS +1,...,TS -1, respectively. Each beacon uses an optimal frequency estimate and a phase generated by summing up the -1 phase estimates. For example, the phase of beacon transmitted to S is the sum of - phase estimates from S 3,...,S and one from destination. S j for j {,..., } generates phase and frequency estimates from its local observation in the corresponding timeslot. TS : All s transmit simultanesously to the destination as a distributed beamformer using the phase and frequency estimates in the earlier timeslots. A. Protocol Implementation in Single-Path Channels To simplify the exposition, we first describe the implementation of the protocol with single-path channels, i.e. the channel response from sensor i to j is h ij (t) =a ij δ(t τ ij ). Further let η ij denote the additive white Gaussian noise (AWG) in the channel. In the first timslot, the destination S 0 broadcasts a sinusoidal beacon of duration T 1 : x(t) =cos(ω c t + φ 0 ) t [0, T 1 ) (1) The signals received at S j can be written as y 0j (t) =a 0j cos(ω c t + φ 0j )+η 0j (t) t [t 0j,t 0j + T 1 ) where φ 0j = φ 0 ω c τ 0j and t 0j = τ 0j for j {1,..., }. Then S j for j {1,..., } computes estimates of the received frequency and phase, which are denoted as ˆω 0j and ˆφ 0j, respectively. From TS to TS,S,..., S generates a sinusoidal secondary beacon using their previous frequency and phase estimates and then transmits to S 1. The transmitted signal can be written as x j1 (t) = cos(ˆω 0j t + ˆφ 0j ) t [t j,t j + T j ) where t j = t 0j + j 1 T i for j {,..., }. ote that we have assumed the signals received at S 1 do not interfere with each other. To achieve this, each could wait for a guard time before transmitting. The signal received at S 1 can be written as y j1 (t) =a j1 cos(ˆω 0j t + φ j1 )+η j1 (t) t [t j1,t j1 + T j ) where φ j1 = ˆφ 0j ˆω 0j τ j1 and t j1 = t j +τ j1 for j {,..., }. S 1 generates estimates of the received frequency and phase which are denoted as ˆω j1 and ˆφ j1, respectively. From TS +1 to TS -1,S 1 transmits a sinusoidal beacon to S,...,S in TS +1,...,TS -1, respectively. The transmitted signal can be written as x 1j (t) = cos(ˆω 1 t + ˆφ j ) t [t 1j,t 1j + T +j 1 ) where t 1j = t 1 + T + +j i=+1 T i (when j =, t 1j = t 1 + T ), ˆφ j = ˆφ i= i1 ˆφ j1 + ˆφ 01 and ˆω 1 is an optimal frequency estimate. The signal received at S j for j {,..., } can be written as y 1j (t) =a 1j cos(ˆω 1 t + φ 1j )+η 1j (t) t [t j,t j + T +j 1 ) where φ 1j = ˆφ j ω 1 τ 1j and t j = t 1j + τ 1j for j {,..., }. S j for j {,..., } generates estimates of the received frequency and phase which are denoted as ˆω 1j and ˆφ 1j, respectively. In the final timeslot, all s transmit to the destination S 0 as a distributed beamformer for a duration of T. These carriers are generated from the phase and frequency estimates obtained in the previous timeslots. The unmodulated carrier transmitted by S j for j {,..., } can be written as x j (t) = cos(ˆω j t + ˆφ 1j ) () ote that S 1 transmits x 1 (t) with frequency ˆω 1 and phase ˆφ 1 = i= ˆφ i1. The signal received at the destination during 5088

3 the final timeslot can be written as the sum of all transmissions after their respective channel delays, i.e., y(t) = a j0 x j (t τ 0j )+η(t) (3) j=1 B. Optimum Frequency Estimation At the end of TS 1,S j for j {1,..., } computes the frequency estimate ˆω 0j. The estimation error ω 0j =ˆω 0j ω c is assumed to be Gaussian distributed with zero-mean and variance σω 0j. The errors ω 0j are independent due to the independe noise processes at each sensors. From TS to TS, S 1 computes the estimate ˆω j1. The estimation error ω j1 = ˆω j1 ω c, is a consequence of both the estimation error at S j in TS 1 and the estimation error at S 1 in the jth timeslot. Given ˆω j1 =ˆω 0j + ω j1, we can write ω j1 =ˆω 0j + ω j1 ω c = ω 0j + ω j1 The errors ω 0j and ω j1 are independent and Gaussian distributed with zero-mean and variance σω 0j and σω j1, respectively. It can be shown that these errors are also independent of each other for j {1,..., }. These results imply that the minimum variance frequency estimate at S 1 can be written as a linear combination of the frequency estimates in the first timeslots, i.e., ˆω 1 = µ 1 ˆω 01 + µ j ˆω j1 (4) j= where j=1 µ j = 1 and the optimum weighting can be derived as and where µ = k= µ 1 = (σ ω + σ 0k ω ) k1 µ µ j = σ ω 01 k=,k j (σ ω 0k + σ ω k1 ) µ (σω 0k + σω k1 )+σω 01 k= n= k=,k n (5) (6) (σ ω 0k + σ ω k1 ) (7) From TS +1 to TS -1, S j for j {,..., } computes the estimate ˆω 1j. The error in this estimate, denoted as ω 1j =ˆω 1j ˆω 1. Similarly, the minimum variance frequency estimate ˆω j at S j for j {,..., } can be written as a linear combination of the frequency estimates ˆω 1 and ˆω 0j. In the final timeslot, all s transmit simultanesously to the destination as a distributed beamformer using the optimum frequency estimates ˆω j. C. Discussion The carrier phase calibration is the most important step of the synchronization protocol. Authors in [1 proposed a closed-loop method to solve this problem. The basic idea is the master sensor broadcasts a reference signal and the slave sensors transmit back this signal to it. Then master sensor could measure the round-trip phase offset and perform the phase calibration, assuming a large SR and symmetry in the forward and reverse channels. Under this scheme the master sensor should synchronize each of the slave sensors individually. In order for the sensors to beamform towards the destination, a reciprocity based approach with channel estimation has been considered [1. In such scheme, the destination broadcasts a unmodulated carrier signal g(t). Each independently demodulates its received signal g i (t) to obtain an estimate ĥi of its own complex channel gain h i. In the final timeslot, all the s use the channel estimate ĥi to modulate the message signal to pre-compensate the channel phase responses. In our round-trip method, these problems can be avoided since the total propagation times (or cumulative phase offsets) for each circuit of the sensors are identical, thus the individual transmissions will combine coherently at the receiver. The implementation of the time-slotted round-trip carrier synchronization protocol with single-path channels is decribed in detail in the previous section. ow we consider the solution in multipath case. The authors in [3 suggest that with multipath channels the duration of each beacon must exceed the finite delay spread of each channel and the s should estimate the frequency and phase from the steady-state observations. This implies our protocol can also be effective with multipath channels if some modifications are made to the synchronization protocol. We can model the channel coefficients h ij as independent circularly symmetric complex normal random variables with zero mean and unit variance, as denoted by h ij C(0, 1). This allows us to evaluate the variation of beamforming gain in fading channels. In this paper, we consider only fixed sensors placement. It is also possilbe to implement our ideas in communication systems with potential or destination mobility. We conjecture that the time-slotted round-trip carrier synchronization protocol still applies in this case. These issues are not considered in this present work and a detailed performance study of such scenarios is a topic for future work. IV. PERFORMACE AALYSIS We consider a cluster of s, communicating a common (baseband) message signal m(t) to a distant destination, by modulating m(t) with a carrier signal at frequency f c. Using our time-slotted round-trip carrier synchronization algorithm the s can cooperatively transmit the message m(t) by beamforming, just like a centralized antenna array. The resulting received signal y(t) is the superposition of the channel-attenuated transmissions of all the s and 5089

4 additive noise η(t) ( y(t) =R m(t)e jπfct h i e jφi(t) ) + η(t) (8) where h i is the channel response from S i to destination and φ i (t) is the cumulative phase error from the synchronization process for S i. For power-limited sensor networks, the feasible communication range is limited by SR. Given that the total transmit power is P t = 1, the beamforming gain is defined as the normalized received power P r P r = 1 h i e jφi(t) (9) In this paper, we concentrate on the effect of phase errors on the average beamforming gain. For simplicity of analysis, assume that the duration of the beacons from TS 1 to TS -1 are equal. We use a maximum likelihood (ML) frequency and phase estimator instead of an PLL to improve the beamforming performance. When the the number of samples in the observation increases, the estimation errors converge to zero-mean Gaussian random variables with variances equal to the Cramer-Rao lower bounds (CRLB). The CRLB when estimating the unknown frequency and phase of a sampled sinusoid of amplitude a in white noise with PSD 0 / is given as [3[9 1fs 3 0 var[ ω a ( 1) 1 0 a D 3 (10) var[ φ f s 0 ( 1) a ( +1) 4 0 a D (11) where f s is the sampling rate, is the number of samples in the observation, and D is the duration of the observation in seconds. The approximations result from assuming that is large and the fact that /f s = D. The cumulative phase error at the destination during the final timeslot for S j can be written as φ 1 (t) = ω 1 t + ( φ 0i + φ i1 ) (1) and for j {,..., } φ j (t) = ω j t + i=,i j i= ( φ 0i + φ i1 )+ φ 01 + φ 1j (13) where all of the phase estimation errors are independent and Gaussian distributed according to the CRLB and we have assumed that any deviation in the phase response of the channels at ˆω j with respect to ω c is much smaller than the phase estimation errors, i.e. the channel responses at ˆω j are the same as the channel response at ω c. Under our assumption that the duration of the beacons T j for j {1,..., 1} are equal, the CRLB and the reciprocal channel assumption implies that var[ φ ij =var[ φ ji and var[ ω ij =var[ ω ji. Considering first the ω j term, we can use the results of optimum weighting established in Section III-B to solve for var[ ω j. ote that estimation errors in the frequency and phase estimates at each, result in unavoidable phase offset at the start of the final timeslot as well as linear phase drift at a rate ω j radians/second over the duration of the final timeslot. Considering now the remaining phase estimation error term, let φ Σj denote the sum of phase estimation errors for S j. Since ω j and φ Σj are both Gaussian distributed with zero mean, it is straightforward to show that the cumulative phase error at the destination during the final timeslot for S j is Gaussian distributed with zero mean and variance σφ j (t) :=var[φ j (t) = t σω j + σφ Σj +te[ω j φ Σj (14) The random variables ω j and φ Σj are functions of ML estimates that are affected by the same noise processes, hence they cannot be assumed to be independent. umerical simulations in [3 suggest that te[ω j φ Σj t σω j + σφ Σj. For s, the expected value of P r could be written as [( E[P r = 1 )( E ) h i e jφi(t) h i e jφi(t) = 1 E[ h i +E[ h i E cos(φ i (t) φ j (t)) i,j (15) Where 1 i, j,i j and φ i (t), φ j (t) are Gaussian distributed. When φ i (t) φ j (t) is very small, i.e. all s have comparable cumulative phase errors at the destination, the expected value of P r can be approximated as [ 1+ ( 1) 1 = (16) E[P r 1 where we have assumed h i are single-path channels with unit-gain. ote that E[P r for multipath channels can also be derived via (16). This result shows that when phase differences at the destination are small, the expected value of the received signal power increases linearly with, i.e. large gains can be realised using distributed beamforming. V. UMERICAL RESULTS We now present the simulation results to demonstrate the time-slotted round-trip carrier synchronization protocol presented in Section III. Consider the system model with unit-gain single-path channels and random propagation delays shown in Fig.1. We assume that the noise power spectral density (PSD) is 0 = and the primary beacon frequency is ω c =π radians/second. Two different methods are used here to estimate the variance of the phase offset: full ML estimation of the phase and frequency of each beacon to obtain an exact φ j (t) and approximate analytical predictions using CRLB for σφ j (t). Figure shows the average beamforming gain versus t for different number of s using variance results from 5090

5 both methods. The beacon durations were set at T j =1µs for j {1,..., 1}. The results obtained using the CRLB can approximate the results obtained through full ML estimation very accurately. These results also demonstrate that it is possible for the time required to transmit the beacons to be only a small percentage of the overall time that the beamformer is able to maintain good phase lock when is small. For example, in the case shown in Figure, for different values of, the beamformer can maintain the maximum amplitude (theoretical value) until t = 100µs. The total time spent synchronizing the carriers in this example was µs. This suggests that the gain obtained by beamforming could far outweigh the synchronization costs (timeslot overhead and energy loss) in many cases. When t>100µs, the beamforming gain will drop quickly especially for large number of users. After t reaches 10 3 s, the average gain will not deteriorate any more but remain at a constant value. Figure 3 shows the effect of the beacon durations T j on the average beamforming gain given t =10 3 s and using full ML estimation. It is interesting to note that when beamformer is not in a good phase lock, we can increase the beacon durations to get over it. For example, by setting the values of beacon duration larger than 10µs, the average beamforming gain could recover the losses. This is especially desirable when increases. This result can be explained, the cumulative phase error is mainly affected by the frequency estimation error. When the beacon durations increases, for a fixed sampling rate, the number of samples in the observation using for ML estimation increases, thus the variance of frequency estimation error decreases. This implies that longer beamforming times can be achieved with good performance if the durations of the beacon intervals are increased. Average beamforming gain E[P r = 6 MLE sim = 6 CRLB sim = 4 MLE sim = 4 CRLB sim = MLE sim = CRLB sim time (seconds) Fig.. Average beamforming gain E[P r versus t. VI. COCLUSIO We have investigated a time-slotted round-trip carrier synchronization protocol required for large-scale cooperative Average beamforming gain E[P r = 6 MLE sim = 4 MLE sim = MLE sim Beacon duration T j (seconds) Fig. 3. Average beamforming gain E[P r versus Beacon duration T j. communication. This method is based on a master-slave architecture. The master can coordinate the transmissions among all s in order to assure the individual transmissions combine coherently at the receiver. The cumulative phase errors are identified in the time-duplexed operation of the synchronization process. A detail performance analysis of the proposed synchronization scheme shows that even with imperfect synchronization, a large fraction of beamforming gains can be realised. Our simulation results with single-path channels also indicate that the gains obtained by distributed transmit beamforming outweigh the synchronization overhead. REFERECES [1 R. Mudumbai, G. Barriac, and U. Madhow, On the Feasibility o Distributed Beamforming in Wireless etworks, IEEE Transactions on Wireless Communications, vol. 6, no. 5, pp , May 007. [ D.R. Brown III, G. Prince, and J. Mceill, A method for carrier frequency and phase synchronization of two autonomous cooperative transmitters, in Proc.IEEE 6th Workshop on Signal Processing Advances in Wireless Communications, ew York, Y, June 5-8, 005, pp [3 Ozil and D.R. Brown III, Time-Slotted Round-Trip Carrier Synchronization, Proceedings of the 41st Asilomar Conference on Signals, Systems, and Computers. [4 A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversity -part I: System description, IEEE Trans. on Communications, vol. 51, no. 11, pp , ov [5 A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversity -part II: Implementation aspects and performance analysis, IEEE Trans. on Communications, vol. 51, no. 11, pp , ov [6 J.. Laneman and G.W. Wornell, Distributed spacectime-coded protocols for exploiting cooperative diversity in wireless networks, IEEE Trans. on Information Theory, vol. 49, no. 10, pp. 415C45, Oct [7 H. Ochiai, P. Mitran, H. Poor, and V. Tarokh, Collaborative beamforming for distributed wireless ad hoc sensor networks, IEEE Trans. Acoustics, Speech, Signal Processing, vol. 53, no , pp , 005. [8 G. Barriac, R. Mudumbai, and U. Madhow, Distributed beamforming for information transfer in sensor networks, in Information Processing in Sensor etworks (IPS), Third International Workshop, Berkeley, CA, April [9 D.C. Rife and R.R. Boorstyn, Single-tone parameter estimation from discrete-time observations, IEEE Trans. on Information Theory, vol. IT- 0, no. 5, pp , Sep