1. Dimitris S. Papailiopoulos and George N. Karystinos Department of Electronic and Computer Engineering Technical University of Crete Kounoupidiana, Chania, 73100, Greece {papailiopoulos | karystinos}@telecom.tuc.gr N EAR ML D ETECTION OF N ONLINEARLY D ISTORTED OFDM S IGNALS 1 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

2. O VERVIEW OFDM signals. Nonlinear power amplifiers (PAs). Peak to average power ratio (PAPR) + PA nonlinear distortion. Iterative receiver. Near ML performance. 2 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

3. S YSTEM M ODEL ASSUMPTIONS Transmission of uncoded CP-OFDM sequence. Single-input single-output. Arbitrary constellation. Multipath Rayleigh fading channel. NOTATION N: sequence length. M: number of constellation points. G: size of cyclic prefix. L : length of channel impulse response. 3 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

4. S YSTEM M ODEL (cntd) Consider data vector. All elements selected from M-point constellation. IDFT of data vector where 4 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

5. S YSTEM M ODEL (cntd) Time-domain OFDM symbol, with and. How to avoid ISI ? Cyclic prefix. 5 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

6. S YSTEM M ODEL (cntd) exhibits Gaussian-like behavior high PAPR example M = 4. 6 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

7. S YSTEM M ODEL (cntd) Before transmission, the OFDM sequence is amplified by a nonlinear PA: with and. Families of PAs - Solid State Power Amplifiers (SSPA): WiFi, WiMAX. - Traveling Wave Tube (TWT): satellite transponders. 7 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

8. S YSTEM M ODEL (cntd) SSPA conversion characteristics 8

9. S YSTEM M ODEL (cntd) 9 N-point IFFTCP Transmitter model Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

10. D ETECTION Baseband equivalent received signal : zero-mean complex Gaussian channel vector. : additive white complex Gaussian (AWGN) vector. : convolution between two vectors. 10 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

11. D ETECTION (cntd) We remove the cyclic prefix and obtain. Fourier transform of. : N-point DFT of channel impulse response. : element-by-element multiplication. : zero-mean AWGN vector with covariance matrix. 11 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

12. D ETECTION (cntd) Channel coefficients known to the receiver Symbol-by-symbol one-shot detection. : Minimum Euclidean distance to the M-point constellation. ML only when PA is linear. Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 12

13. D ETECTION (cntd) Channel coefficients unknown to the receiver Transmit Training sequence. Best linear unbiased estimator (BLUE) of : with. : diagonal matrix whose diagonal is. : amplified training sequence. 13 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

14. D ETECTION (cntd) Channel coefficients unknown to the receiver (cntd) Symbol-by-symbol one-shot detection. : Minimum Euclidean distance to the M-point constellation. Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 14

15. D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 15 N-point FFT remove CP Reciever model Channel estimation One-shot detection

16. D ETECTION (cntd) However PA is not linear Detection is not ML Performance Loss! 16 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

17. ML D ETECTION We take into account the PA transfer function. ML detection rule: Complexity !!! Impractical even for small M and N. 17 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

18. I TERATIVE N EAR ML D ETECTION We propose to use the ML decision rule on a reduced candidate set. How to build such a set? 1) Perform conventional detection to obtain and use it as a “core” candidate. 2) Find the closest (in Hamming distance) vectors to and evaluate the ML metric for each one of them. 3) Keep the best neighboring vector, call it, and repeat steps 2-3 until convergence. 18 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

19. I TERATIVE N EAR ML D ETECTION (cntd) Conventionally detect. repeat Step 1: define consisting of closest vectors to Step 2: find Step 3: set Step 4: go to Step 1 until (max iterations OR convergence) denotes hamming distance of two vectors 19 Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos

20. I TERATIVE N EAR ML D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 20 N-point IFFT remove CP Iterative Detection model Channel estimation One-shot detection Hamming- distance-1 set ML metric

21. I TERATIVE N EAR ML D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 21 N = 12, L = 8, M = 2 (BPSK) Observe: proposed attains ML performance in 1 iteration!

22. I TERATIVE N EAR ML D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 22 N = 64, L = 17, M = 4 (QPSK), clip level = 0 dB Observe: Clipping DOES NOT work, don’t employ it!

23. I TERATIVE N EAR ML D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 23 N = 64, L = 17, M = 4 (QPSK), clip level = 0 dB PA operates in saturation, proposed outperforms all else!

24. I TERATIVE N EAR ML D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 24 N = 64, L = 17, M = 4 (QPSK), clip level = 0 dB PA operates in linear range, proposed outperforms all else!

25. I TERATIVE N EAR ML D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 25 N = 16, L = 17, M = 64 (64-QAM) Even for greater constellation orders the proposed excels!

26. I TERATIVE N EAR ML D ETECTION (cntd) Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 26 N = 64, L = 17, M = 4 (QPSK) Even with channel estimation proposed receiver works great!

27. C ONCLUSION Technical University of Crete Dimitris S. Papailiopoulos and George N. Karystinos 27 Near ML receiver for nonlinearly distorted OFDM signals. Efficient, bilinear complexity. Truly near ML, since it exhibits ML behavior! Much better than conventional. Works great with channel estimation.