Now we discuss how the end-to-end distributed relay forwarding could be modeled as conventional OFDM transmission. We are interested in amplify-forwarding distributed relay network with OFDM modulation throughout end-to-end transmission. Assume relay network is composed of L single-antenna relays. Both source and destination terminal are equipped with single-antenna. Note the discussion in this chapter could be intuitively extended to the cases of multi-antenna source or destination. Again we apply two-phase relay transmission as assumed in Chap. 2. In the first slot, source transmits one OFDM symbol. Each relay performs respective time synchronization to determine correct symbol boundary, then remove cyclic prefix (CP) and acquire a complete OFDM symbol with N subcarriers. To realize amplify-forwarding, the ith (1 ≤ i ≤ L) relay amplify received OFDM symbol by ri, then add a new CP and transmit. Note that the procedure of removing CP then add another seems redundant. Actually, he received CP samples contain IBI (inter-block interference) signals. If these CP samples remain untouched and are forwarded to destination, the end-to-end transmission would supper two stages of IBI corruption. Thus removing the received CP samples would equivalently remove the IBI from source and therefore would reduce the overall IBI.
We assume a global synchronization for relays transmission is available so that relays could transmit at the same time. Upon receiving the forwarded OFDM symbol, destina-tion works as a convendestina-tional OFDM receiver to process received OFDM symbols. The relays reception and forwarding process is depicted in Fig. 5.2.
To model the end-to-end OFDM transmission, we start by focusing on single relay.
Fig. 5.2: Relay operations for OFDM transmission For the ith relay, its frequency-domain reception could be stated as
zi = XW efi+ ni = Xfi+ ni
= Fix + ni (5.14)
where zi ∈ ZN, X denotes a diagonal matrix whose diagonal terms are equal to x and composed of signals from source, niis the additive noise, W the Fourier transform matrix, fei and fi represent the impulse response and frequency response of channel between source and ith relay, respectively. Fi is defined to be a diagonal matrix whose diagonal terms are equal to fi. Note that though relays do not perform DFT or IDFT throughout the signal reception and forwarding, we use zi as frequency-domain representation of received signal for sake of concise and convenient system modeling.
At destination, amplified zi travels through another multipath channel and is summed up with OFDM signals from other relays. The frequency-domain reception of destination is described as
where nR is the overall forwarding noise, gei and gi represent the impulse response and frequency response of channel between destination and ith relay, respectively. Gi is defined to be a diagonal matrix whose diagonal terms are equal to gi, and diagonal matrix H ,P
∀i , GiFi means frequency response of the equivalent end-to-end channel. Let h be the vector composed of diagonal terms of H. Observing the subcarrier-wise product
in the definition of H, we may conclude that
where denotes element-wise product, ⊗ stands for circular convolution. Observing (5.16) we understand that the equivalent channel impulse response of end-to-end channel could be modeled as follows: for each relay, do circular convolution for the two-stage channel efi andgei, then sum up the L copies of convolution results. Based on this model, it is clear that the equivalent channel is of widely dispersive channel impulse response.
5.3 Numerical Results
Let the DFT size in OFDM be 256, with 12 subcarriers assigned for pilots. The pilot locations are randomly determined. Let QPSK be employed for each data subcarrier.
Consider transmission over a 4-path channel. The path coefficients vary randomly from one OFDM symbol to another, each following a complex Gaussian distribution. Besides the first path delay τ0 = 0, other path delays are uniformly distributed in the range [0, τmax), but stay constant during the OFDM symbol group used in GMP channel esti-mation. We let τmax = 25 and Lg = 10.
Two MP-based approaches are simulated: OMP and GMP. As mentioned previously, to the best of our knowledge there does not exist prior techniques suitable for subspace-based OFDM channel estimation under arbitrary pilot assignments that may vary from symbol to symbol. Thus we cannot compare with eigen-decomposition based schemes such as that in [46] or [32]. However, we simulate channel estimation methods based on linear interpolation and spline interpolation, for a comparison.
Fig. 5.3 shows the mean-square channel estimation errors of different approaches, and Fig. 5.4 the average symbol error rates for each simulated scheme. In the figures, the labels
“GMP+MS” and “OMP+SS” mean “GMP approach for multi-symbol estimation” and
“OMP for single-symbol estimation,” respectively. While interpolation-based methods suffer from scarcity of pilots and are not able to estimate the shape of channel frequency responses accurately, MP-based methods can use the limited resource (pilots) efficiently and result in clearly superior estimation. The proposed GMP algorithm enjoys the great-est “diversity gain” from multi-symbol processing and thus has the better performance among all.
Fig. 5.5 shows the average symbol error rates when the four path delays are fixed at [0, 3, 6, 9]. The simulation demonstrates even better performance for GMP than that in Fig. 5.4. This is because subspace-based algorithms for OFDM channel estimation has a resolution limitation depending on the pilot ratio [32]. When some paths are close together, as occasionally happened in the simulation resulting in Fig. 5.4, MP-based
0 5 10 15 20 25 0
0.2 0.4 0.6 0.8 1 1.2 1.4
random path delay and gain, QPSK, pilot=12, 256 FFT
SNR (dB)
Normalized Mean Square Channel Estimation Error
linear spline OMP+SS GMP+MS
Fig. 5.3: Normalized mean-square channel estimation errors of different channel estima-tion methods
schemes may have difficulty telling them apart. But this is certainly not the case with the simulation resulting in Fig. 5.5, for the paths are well separated.
In Fig. 5.6 we simulate and examine channel estimations for OFDM distributed relay networks. We set the number of relay terminals as 10. It is assumed there are Stmultipath delay taps randomly spaced between [0, τmax]. In OFDM transmission with 256-point FFT/IFFT we assign Sp pilot subcarriers and estimate Se (the column size of W in (5.5)) tap gains. In Fig. 5.6 we simulate two sets of configurations. For solid lines we examine the performance with fewer pilots and smaller delay range, thus we set St = 3, τmax= 4, Se = 8 and Sp = 10. For dotted lines we try wider delay range with more pilots by setting St= 4, τmax = 8, Se = 18 and Sp = 20. To compare the performance of various approach we realize channel estimation based on linear interpolation, time-domain least-square method (discussed in Sec. 5.1.2) and OMP-based algorithm. It is clear that OMP-based outperforms the others in both configurations.
0 5 10 15 20 25 10−2
10−1 100
random path delay and gain, QPSK, pilot=12, 256 FFT
SNR (dB)
Symbol Error Rate
linear spline OMP+SS GMP+MS
Fig. 5.4: Average symbol error rates (SERs) at data subcarriers with different channel estimation methods.
0 5 10 15 20 25
10−3 10−2 10−1 100
fixed sparse delay=[0, 3, 6, 9], QPSK, pilot=12, 256 FFT
SNR (dB)
Symbol Error Rate
linear spline OMP+SS GMP+MS
Fig. 5.5: Average SERs at data subcarriers with different channel estimation methods when multipath delay are spaced apart.
−20 −15 −10 −5 0 5 10 15 20 10−2
10−1 100
(σx/σD)2 (dB)
SER
linear LS OMP linear LS OMP
dotted: Sp = 20 solid: Sp = 10
Fig. 5.6: Average SERs of OFDM distributed relay network at data subcarriers with different channel estimation methods when multipath delay are randomly spaced.
Chapter 6 Conclusion
In this thesis we studied the design of distributed AF MIMO relay networks. Specifically, we focused on the capacity improvement and present noise-dominant models to simplify design problem. As to channel estimation we applied proposed subspace-based algorithm to handle equivalent dispersive channel.
In Chap. 2 and Chap. 3, We considered the design of distributed amplify-and-forward relay networks for two-hop MIMO transmission. More specifically, we considered the determination of relay gains for maximization of system capacity. As no closed-form ana-lytical solution could be found for the problem, we considered two alternative approaches.
One approach was algorithmic, for which we derived an efficient iterative algorithm. Since the algorithmic solution gave little insight into the analytical properties of the solution, we also took an analytical approach, assuming some asymptotic noise conditions. The analytical approach resulted in several relay selection-type of solutions and facilitated an analysis of the diversity behavior of the solutions. It turned out that their capacity di-versity performance behaved similarly to some single-hop point-to-point MIMO antenna selection systems previously analyzed by other researchers. Some simulation results were presented. The results showed that, not surprisingly, the iterative algorithm did yield better designs than the relay selection methods, but at the cost of a substantially higher computational complexity. More significantly, they also confirmed our outage diversity analysis and verified that increasing the number of relays could enhance the outage di-versity performance.
In Chap.4 we considered the design of distributed amplify-and-forward relay networks for two-hop MIMO transmission. With the help of noise-dominating models we simplify the originally intractable capacity optimization problem. Further simplification is realized by designing modified criterion of maximal ratio of norms, and shrinking the solution space with zero-forcing. Since no relay selection is required, we could control the computation cost even with large size of relay network. The performance of proposed algorithms is verified by simulations and is proven to be comparable or better than selection-based algorithm.
Time-domain channel estimation techniques can obtain relatively accurate channel estimates for OFDM transmission with relatively few pilot subcarriers. But it requires knowledge of the multipath delays. In Chap. 5 We proposed a group matching pursuit technique for multipath delay estimation. Unlike previous techniques, the proposed tech-nique allows arbitrary pilot structures that may vary from one OFDM symbol to the next. Simulation results showed that the proposed algorithm has superior estimation performance. We also presented the channel modeling of distributed AF relay network with OFDM transmission. It turns out the relay network of interest actually suffer equiv-alent highly dispersive channel. Thus we examined applying subspace-based algorithm to handle the channel estimation.
Bibliography
[1] J. Adler, B. D. Rao, and K. Kreutz-Delgado, “Comparison of basis selection meth-ods,” in Asilomar Conf. Signals Systems Computers, 1996, pp. 252–257.
[2] J. Andersen, “Antenna arrays in mobile communications: Gain, diversity, and chan-nel capacity,” IEEE Antennas Propagat. Mag., vol. 42, pp. 12–16, 2000.
[3] Y. Bae and J. Lee, “Power allocation for MIMO systems with multiple non-regenerative single-antenna relays,” in Proc. IEEE Veh. Technol. Conf. (VTC-Spring), 2010.
[4] J. Bezdek and R. Hathaway, “Some notes on alternating optimization,” in Advances Soft Computing, 2002, pp. 187–195.
[5] H. B¨olcskei, R. U. Nabar, O. Oyman, and A. J. Paulraj, “Capacity scaling laws in MIMO relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1433–
1444, 2006.
[6] C. B. Chae, T. Tang, R. W. Heath, Jr., and S. Cho, “MIMO relaying with linear processing for multiuser transmission in fixed relay networks,” IEEE Trans. Signal Process., vol. 56, no. 2, pp. 727–738, Feb. 2008.
[7] M. X. Chang and Y. T. Su, “2D regression channel estimation for equalizing OFDM signals,” in Proc. IEEE Veh. Technol. Conf., 2000.
[8] H. Chen, A. B. Gershman, and S. Shahbazpanahi, “Distributed peer-to-peer beam-forming for multiuser relay networks,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), 2009.
[9] ——, “Filter-and-forward distributed beamforming for relay networks in frequency selective fading channels,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process-ing (ICASSP), 2009.
[10] E. Chong and S. ˙Zak, An introduction to optimization. Wiley-interscience, 2008.
[11] L. Dong, Z. Han, A. Petropulu, and H. Poor, “Improving wireless physical layer security via cooperating relays,” IEEE Trans. Signal Process., vol. 58, no. 3, pp.
1875–1888, 2010.
[12] M. Elkashlan, P. L. Yeoh, R. H. Y. Louie, and I. B. Collings, “On the exact and asymptotic SER of receive diversity with multiple amplify-and-forward relays,” IEEE Trans. Veh. Technol., vol. 59, no. 9, pp. 4602–4608, 2010.
[13] P. Fertl, A. Hottinen, and G. Matz, “A multiplicative weight perturbation scheme for distributed beamforming in wireless relay networks with 1-bit feedback,” in Proc.
IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), 2009.
[14] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading en-vironment when using multiple antennas,” Wireless personal communications, vol. 6, no. 3, pp. 311–335, 1998.
[15] Y. Fu, L. Yang, and W. P. Zhu, “A nearly optimal amplify-and-forward relaying scheme for two-hop MIMO multi-relay networks,” IEEE Commun. Lett., vol. 14, no. 3, pp. 229–231, 2010.
[16] A. Gorokhov, D. Gore, and A. Paulraj, “Performance bounds for antenna selection in MIMO systems,” in Conf. Rec., IEEE Int. Conf. Commun., vol. 5, 2003.
[17] ——, “Receive antenna selection for MIMO flat-fading channels: theory and algo-rithms,” IEEE Inf. Theory, vol. 49, no. 10, pp. 2687–2696, Oct. 2003.
[18] R. Hunger, “Floating point operations in matrix-vector calculus,” Munich University of Technology, Tech. Rep., 2007.
[19] IEEE Standard for Local and Metropolitan Area Networks - Part 16: Air Interface for Fixed Broadband Wireless Access systems, IEEE Std. 802.16-2004, Oct. 2004.
[20] S. Jin, M. R. McKay, C. Zhong, and K. Wong, “Ergodic capacity analysis of amplify-and-forward MIMO dual-hop systems,” IEEE Inf. Theory, vol. 56, no. 5, pp. 2204–
2224, 2010.
[21] Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,”
IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524–3536, 2006.
[22] Y. Jing and H. Jafarkhani, “Network beamforming using relays with perfect channel information,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), vol. 3, 2007.
[23] S. G. Kang, Y. M. Ha, and E. K. Joo, “A comparative investi-gation on channel estimation algorithms for OFDM in mobile communications,” IEEE Trans. Broad-casting, vol. 49, no. 2, pp. 142–149, 2008.
[24] R. Krishna, Z. Xiong, and S. Lambotharan, “A cooperative MMSE relay strategy for wireless sensor networks,” IEEE Signal Process. Lett., vol. 15, pp. 549–552, 2008.
[25] C. Li, L. Yang, and W. P. Zhu, “Joint power allocation based on link reliability for MIMO systems assisted by relay,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), 2009.
[26] C. Li, L. Yang, and W. Zhu, “Two-way MIMO relay precoder design with channel state information,” IEEE Trans. Commun., vol. 58, no. 12, pp. 3358–3363, 2010.
[27] S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,”
IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397–3415, Dec. 1993.
[28] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000.
[29] H. Minn and V. K. Bhargava, “An investigation into time-domain approach for OFDM channel estimation,” IEEE Trans. Broadcast., vol. 46, no. 4, pp. 240–248, Dec. 2000.
[30] O. Mu˜noz, J. Vidal, and A. Agust´ın, “Non-regenerative MIMO relaying with chan-nel state information,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), vol. 3, 2005, pp. 361–364.
[31] R. Nabar, H. B¨olcskei, and F. Kneub¨uhler, “Fading relay channels: performance limits and space–time signal design,” IEEE J. Select. Areas Commun., vol. 22, no. 6, p. 1099, 2004.
[32] M. Oziewicz, “On application of MUSIC algorithm to time delay estimation in OFDM channels,” IEEE Trans. Broadcast., vol. 51, no. 2, pp. 249–255, Jun. 2005.
[33] R. Pabst, B. Walke, D. Schultz, P. Herhold, H. Yanikomeroglu, S. Mukherjee, H. Viswanathan, M. Lott, W. Zirwas, M. Dohler, H. Aghvami, D. D. Falconer, and G. P. Fettweis, “Relay-based deployment concepts for wireless and mobile broadband radio,” IEEE Commun. Mag., vol. 42, no. 0, pp. 80–89, Sep. 2004.
[34] B. D. Rao, K. Engan, and S. Cotter, “Diversity measure minimization based method for comupting sparse solutions to linear inverse problems with multiple measurement vectors,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP),, 2004.
[35] M. Sadek, A. Tarighat, and A. H. Sayed, “A leakage-based precoding scheme for downlink multi-user MIMO channels,” IEEE Trans. Wireless Commun., vol. 6, no. 5, p. 1711–1721, 2007.
[36] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” in Proc.
RADC Spectral Estimation Workshop, 1979, pp. 234–258.
[37] G. A. F. Seber, A matrix handbook for statisticians. Wiley-Interscience, 2007.
[38] T. J. Shan, M. Wax, and T. Kailath, “On spatial smoothing for direction-of-arrival estimation fo coherent signals,” IEEE Trans. Acoust., Speech, Signal Process., vol. 33, pp. 806–811, 1985.
[39] O. Simeone, Y. Bar-Ness, and U. Spagnolini, “Pilot-based channel estimation for OFDM systems by tracking the delay-subspace,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 315–324, Jan. 2004.
[40] S. Sun and Y. Jing, “Channel training and estimation in distributed space-time coded relay networks with multiple transmit/receive antennas,” in Proc. IEEE Wireless Commun. Networking Conf., 2010.
[41] I. E. Telatar, “Capacity of multi-antenna gaussian channels,” Europ. Trans. Telecom-mun., vol. 10, pp. 585–595, Nov. 1999.
[42] L. Trefethen and D. Bau, Numerical Linear Algebra. Philadelphia, PA: SIAM, 1997, no. 50.
[43] J. A. Tropp, A. C. Gilbert, and M. J. Strauss, “Simultaneous sparse approximation via greedy pursuit,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP),, May 2005.
[44] E. Van Der Meulen, “Three-terminal communication channels,” Adv. Appl. Prob., vol. 3, no. 1, pp. 120–154, 1971.
[45] C. J. Wu and D. W. Lin, “Sparse channel estimation for OFDM transmission based on representative subspace fitting,” in Proc. IEEE Veh. Technol. Conf., May 2005.
[46] B. Yang, K. B. Letaief, R. S. Cheng, and Z. Cao, “Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling,”
IEEE Trans. Commun., vol. 49, no. 3, pp. 467–478, Mar. 2001.
[47] E. Yilmaz and M. O. Sunay, “Effects of Transmit Beamforming on the Capacity of Multi-Hop MIMO Relay Channels,” in Proc. IEEE Personal Indoor Mobile Radio Commun., Sep. 2007, pp. 1–5.
[48] E. Zeng, S. Zhu, X. Liao, Z. Zhong, and Z. Feng, “On the performance of amplify-and-forward relay systems with limited feedback beamforming,” IEICE Trans. Commun., vol. 91, no. 6, pp. 2053–2057, 2008.