Presenter: Chun-Hsien Peng ( 彭俊賢 ) Advisor: Prof. Chong-Yung Chi ( 祁忠勇 教授 )

1. Blind Beamforming for Multiuser OFDM Systemsby Kurtosis Maximization Based on Subcarrier Averaging Presenter: Chun-Hsien Peng (彭俊賢) Advisor: Prof. Chong-Yung Chi (祁忠勇 教授) Institute of Communications Engineering & Department of Electrical Engineering National Tsing Hua University Hsinchu, Taiwan 30013, R.O.C. E-mail: d905610@oz.nthu.edu.tw

2. OUTLINE 1. Introduction 2. MIMO Models for Beamforming of Multiuser OFDM Systems 3. Post-FFT Fourier Beamformer by Subcarrier Averaging 4. Blind Post-FFT KMBFA by Subcarrier Averaging 5. Simulation Results 6. Conclusions and Future Researches KMBFA:Kurtosis Maximization Beamforming Algorithm

3. 1. Introduction • Wireless communication problems: ISI, MAI, and CCI suppression in cellular wireless communication systems (Noise) f5 f6 f4 ISI f1 MAI (Multipath channel) f7 f3 f2 f1 ISI:Intersymbol Interference (due to Multipath) CCI MAI:Multiple Access Interference (Caused by Multiple Users) in a Cell CCI:Co-channel Interference A multiuser OFDM system with antenna arrays such as the pre-FFT and post-FFT beamfoming receivers have been considered for combating CCI, MAI and ISIin the receiver design.

4. approximately Gaussian (by Central Limit Theorem) 2. MIMO Models for Beamforming of Multiuser OFDM Systems • Transmitter for “Quasi-synchronous” Multiuser OFDM Systems : length of guard interval (GI) : data sequence of user : number of subcarriers

5. Baseband discrete-time received signal: :totalnumber of paths (or DOAs) associated with user where ( steering vector) : ( ) total number of paths (or DOAs) of all the users (Received signals) (Transmitted signals) 1 2 time delay DOA path gain : number of receive antennas (Noise vector) DOA:Direction of Arrival

6. total number of paths ofall the users , andL is known. (A2) ,for all ; number of receive antennas Quasi-synchronous OFDM Systems (A3) (A4) is zero-mean white Gaussian with and statistically independent of 's. Non-Gaussianprocess • Some general assumptions : (A1)are i.i.d. QPSK symbol sequences (i.e., for each k is a random variable with uniform probability mass function over the sample space ), and is statistically independent of for . i.i.d.:Independent Identically Distributed QPSK:Quadriphase-shift Keying :identity matrix

7. Pre-FFT Beamforming Structure (Pre-FFT BFS) [18,19] [18] M. Okada and S. Komaki, “Pre-DFT combining space diversity assisted COFDM,” IEEE Trans. Vehicular Technology,vol. 50, pp. 487-496, Mar. 2001. [19] Z. Lei and F.P.S. Chin, “Post and pre-FFT beamforming in an OFDM system,” IEEE 59th Vehicular Technology Conference, vol. 1, Milan, Italy, May 17-19, 2004, pp. 39-43.

8. (A2) ,for all ; full column rank with by Assumption (A2) ( DOA matrix) approximately Gaussian (by Central Limit Theorem) • MIMO Model for Pre-FFT BFS

9. (A1)are i.i.d. QPSK symbol sequences (i.e., for each k is a random variable with uniform probability mass function over the sample space ), and is statistically independent of for . (A3) • SOS based blind beamforming algorithms can be applied, butHOS based blind beamforming algorithms are not applicable because is approximately a Gaussian vector process. • Each column of the mixing matrix A only comprise the energy from a single path. Though beamforming algorithmsusing SOS can be applied to extract each source , their performance is limited due to lack of path diversity. • MIMO Model: Remarks: • By Assumptions (A1) and (A3), one can observe that for each fixed n is a zero-mean L ×1 random vector with all the L random components being mutually statistically independentwith .

10. Post-FFT BFS [19,20] KMBFA:Kurtosis Maximization Beamforming Algorithm [19] Z. Lei and F.P.S. Chin, “Post and pre-FFT beamforming in an OFDM system,” IEEE 59th Vehicular Technology Conference, vol. 1, Milan, Italy, May 17-19, 2004, pp. 39-43. [20] D. Bartolome and A. I. Perez-Neira, “MMSE techniques for space diversity receivers in OFDM-based wireless LANs,” IEEE J. Sel. Areas Commun.,vol. 21, pp. 151-160, Feb. 2003.

11. full column rank and ( matrix) • MIMO Model for Post-FFT BFS • After the processes of the removal of GI, S/P conversion, N-point FFT operation, and P/S conversionat each receive antenna, the MIMO model for each subcarrier k of the post-FFT BFS can be established as follows: ( vector)

12. All the components 's (QPSK signals) of the P × 1 random input vector are zero-mean non-Gaussian mutually statistically independent with . • Each column of the mixing matrix comprises multipath energy implying apath diversity gain in the estimation of each sourcecan be foreseen. • However, a set of N estimatorsis needed each for one subcarrierk because of for all for all kj. This leads to high computational complexity. ( vector) • MIMO Model: Remarks:

13. Blind algorithms associated with post-FFT BFS using HOS (such as FKMA) are applicable to the estimation of , but in general, they also require many OFDM data blocks with the assumption that the channel is static over these OFDM data blocks, and, again, a set of N estimators is needed. [19] Z. Lei and F.P.S. Chin, “Post and pre-FFT beamforming in an OFDM system,” IEEE 59th Vehicular Technology Conference, vol. 1, Milan, Italy, May 17-19, 2004, pp. 39-43. [20] D. Bartolome and A. I. Perez-Neira, “MMSE techniques for space diversity receivers in OFDM-based wireless LANs,” IEEE J. Sel. Areas Commun.,vol. 21, pp. 151-160, Feb. 2003. [21] M. Budsabathon, Y. Hara, and S. Hara, “Optimum beamforming for pre-FFT OFDM adaptive antenna array,” IEEE Trans. Vehicular Technology, vol. 53, pp. 945-955, Jul. 2004. Remarks: • Theoretically, the nonblind MMSE beamformer associated with post-FFT BFS is optimum and performs much better than that associated with pre-FFT BFSowing to lack of path diversity for the latter. However, the latter only needs one OFDM pilot block for the estimation of channel matrix, but the former may needmany pilots to accurately estimate the channel matrix (and thus needs many OFDM blocks provided over which the channel is static)[19,20,21].

14. To design a block-by-block blind beamforming algorithm which is exactly the same for all the subcarriers, and attains “maximum multipath diversity gain” in the meantime. GOAL

15. 3. Post-FFT Fourier Beamformer by Subcarrier Averaging • Notations: Time Delays L× 1 vector DOAs L× 1 vector Path gains L× 1 vector where Lp× 1 vector Lp× 1 vector Lp× 1 vector

16. where (correlated sources) zero-mean wide-sense non-Gaussian stationary process by treatingk as a time index • Alternative form for :

17. NOTE • Subcarrier average of Lemma 1. Under the assumptions (A1)and (A3), it can be shown that, where denotes “convergence in probability” as . In spite of , , which implies that all the components of become “uncorrelated”by subcarrier averaging.

18. Post-FFT <Fourier> beamformer: where for sufficiently large where . • Let be a beamformer (a spatial filter) with the input being . Its output is then where

19. and • Under the noise-free assumption, and the assumptions (A1)through (A3),the rth column of A can be estimated by input- output cross-correlation (IOCC) as follows: Channel of interest Bias • For finite Q,the output of spatial filter is then where

20. NOTE is an estimate of and is an estimate of implying that the better the estimation accuracy ofboth and , the larger the value of . • A “blind performance index” for post-FFT <Fourier> beamformer: (magnitude of the normalized cross correlation between and and ).

21. 4. Blind Post-FFT KMBFA by Subcarrier Averaging • Proposed Blind Post-FFT KMBFA at the th stage Source Extraction Using Hybrid-<Fourier> -<FKMA> TDEC Update throughBMRC No Yes Deflation Classification Updateby ? KMBFA:Kurtosis Maximization Beamforming Algorithm TDEC:Time Delay Estimation and Compensation

22. =1 =1 0 • By Lemma 1, it can be easily shown that (may depend on ifor other modulations such as BPSK signals) Lemma 1. • Kurtosis Maximization Based on Subcarrier Averaging • Let us define the kurtosis of by subcarrier averaging as follows:

23. Maximization ? • The objective function to be maximized for the design of the beamformer : “magnitude of normalized kurtosis” of

24. Assumptions: (A5) if , where are distinct integers. (A6) if , where are distinct integers. (A7) and if and Theorem 1:Under the assumptions (A1) through (A3), (A5) through(A7), and the noise-free assumption, attains maximum, and where is an unknown constant, and is anunknown integer.

25. (SEA) Algorithm nential -expo Super • Post-FFT <FKMA> by Subcarrier Averaging at the th iteration Yes Compute To the thiteration ? No Update through a gradient type optimization algorithm such that ( matrix)

26. The proposed post-FFT <FKMA> may fail to extract the sources when any of the assumptions (A5),(A6) and (A7) are not satisfied, while the probability of the event that violation of any of the three assumptions occurs depends on values of (lengthof GI) and (number of paths of each user). Remarks: • An initial condition is needed to initialize the proposed post-FFT <FKMA>. For finite N and finite SNR, • Empirically, we find that the proposed iterative post-FFT <FKMA> also shares the fast convergence and guaranteed convergence advantages of the conventional FKMA.

27. (A8) Three ExtraAssumptions and if (A9) if and (A10) (for the QPSK case) The proposed post-FFT <FKMA> is also applicable to thecase of BPSK symbol sequence. × • Therefore, the proposed post-FFT <FKMA> may fail to extract a source with higher probability for the BPSK case than for the QPSK case due to the above three extra assumptions required.

28. Multistage Source Extraction Cancellation (or Deflation Processing) Assume that and , are the source estimate and the associated channel estimateobtained at stage . Source Extraction Using Hybrid-<Fourier> -<FKMA> ... At th stage: Deflation which basically corresponds to the MIMO signal withall the contributions from removed.

29. Source Extraction Using Hybrid-<Fourier> -<FKMA> ... Deflation Initial Condition: Post-FFT <Fourier> beamformer where (Output) (Channel estimate)

30. Post-FFT <FKMA>:initialized by Source Extraction Using Hybrid-<Fourier> -<FKMA> (Output of <FKMA>) (Channel estimate) ... Deflation Hybrid-<Fourier>-<FKMA>: <FKMA> <Fourier>

31. unknown time delay where is an unknown constant, , Remark: • The proposed blind Hybrid-<Fourier>-<FKMA> performs well only with Assumptions (A1) through (A4) required, and it is applicable to both the BPSK case and the QPSK case.

32. Time Delay Estimation and Compensation (TDEC) unknown time delay at the th stage Source Extraction Using Hybrid-<Fourier> -<FKMA> TDEC Update throughBMRC No Yes Deflation Classification Updateby ?

33. The unknown timedelay in the extracted source can be estimated also by subcarrier averaging where

34. <FKMA> • Classification and BMRC at the th stage Source Extraction Using Hybrid-<Fourier> -<FKMA> TDEC Update throughBMRC No Yes Deflation Classification Updateby ? BMRC:Blind Maximum Ratio Combining

35. MIMO signal models replaced by • Correlated Sources • Assume that is a cluster of correlated sources impinging on the receiver antenna array where : path gain of each correlated signal in the cluster : DOA of each correlated signal in the cluster

36. Remarks: • The proposed post-FFT KMBFA is able to accurately estimate the associated source signal as long as is sufficiently large, implying its robustness to correlated signals. • On the other hand, the Capon's MV beamformer is incapable of extracting the associated source regardless of the value of because is no longer a steering vector of a certain DOA required by the Capon's MV beamformer.

37. 5. Simulation Results • Parameters Used: Consider a four-user ( ) OFDM system with , and . • 's: zero-mean, i.i.d.BPSK (or QPSK) signalsused with • : i.i.d. zero-mean Gaussian with . • Input SNR: • performance index: average symbol error rate (SER)

38. Example 1 (Environment without Correlated Sources): MultipathChannelParameters • Fifty sets of time delay parameterswere generated randomly. For each set of time delay parameters, fifty sets of data were generated.

39. ( ) post-FFT KMBFA ifpost-FFT <Fourier> Beamformer is used (a) QPSK signals ( ) post-FFT KMBFA if Hybrid-<Fourier>-<FKMA> is used ( ) post-FFT KMBFA ifpost-FFT <FKMA> is used INPUT SNR (dB) (b) BPSK signals INPUT SNR (dB)

40. (a) QPSK signals INPUT SNR (dB) (b) BPSK signals INPUT SNR (dB)

41. Example 2 (Environment with Correlated Sources): total number of correlated sources of a cluster MultipathChannelParameters 's are all distinct DOAs 's ( ) were generated randomly

42. (a) QPSK signals INPUT SNR (dB) (b) BPSK signals INPUT SNR (dB)

43. 6. Conclusions and Future Researches • Conclusions • Under Assumptions (A1) through (A4), we have presented ablock-by-block processing algorithm based on subcarrier averaging, namely the post-FFT KMBFA, for the estimation of symbol sequences of all the active users of an OFDM system. It is also a multistage blind beamforming algorithm consisting of source extraction using the proposed blind Hybrid-<Fourier>-<FKMA>, TDEC processing,classification and BMRC processing at each stage. • The proposed blind Hybrid-<Fourier>-<FKMA>, which is the kernel of the proposed post-FFT KMBFA, is also a selection scheme by the performance of two blind beamformers, the post-FFT <FKMA> and thepost-FFT <Fourier> beamformer, using subcarrier averages of one OFDM block. Moreover, as the conventional FKMA, the post-FFT <FKMA> supported by Theorem 1 is also a computationallyfast source extraction algorithm.

44. Some simulation results were provided to support that the proposedpost-FFT KMBFA performs well no matter whether correlated sources are present or not, and its performance is close to the “optimal”nonblind MMSE beamformer associated with the post-FFT BFS, in addition to a performance comparison of some existing beamformers • The results of this chapter have been partly presented at • IEEE ISPACS'05 (Hong Kong, Dec. 13-16, 2005), co-authored by Chun-Hsien Peng, C.-C. Lin, Y.-H. Lin, and C.-Y. Chi, • and have been submitted to IEEE Trans. Signal Processing for publication, co-authored by Chun-Hsien Peng, K.-C. Huang, C.-Y. Chi, and W.-K. Ma. ISPACS: Intelligent Signal Processing and Communication Systems

45. The extension of the proposed blind beamforming algorithm to more general scenarios is a worthwhile research. The feasibility of other digital symbols such as 16-QAM isa future research in addition to the estimation of L. Subcarrieraveraging may open a door for efficient post-FFTblind beamforming algorithms of multiuser OFDM systemsusing one OFDM block. • Future Researches • We considered the uplink of a multiuser OFDM system under the scenario of multiple distinct DOAs for each user, and a flatfading channel for each DOA.

46. Thank you very much

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