Layered Processing for MIMO OFDM

1. Layered Processing for MIMO OFDM Yang-Seok Choi, yschoi@vivato.net Siavash M. Alamouti, siavash@vivato.net Yang-Seok Choi et al., ViVATO

2. Assumptions • Block Fading Channel • Channel is invariant over a frame • Channel is independent from frame to frame • CSI is available to Rx only • Perfect CSI at RX • No feedback channel • Gaussian codebook Yang-Seok Choi et al., ViVATO

3. Motivations … • To fully exploit Space- and Frequency-diversity in MIMO OFDM • Each information bit should undergo all possible space- and frequency-selectivity • Subcarriers should be considered as antennas (Space and frequency should be treated equally) • Apply Space-Time code (STC) over all antennas and subcarriers • STC • STC encoder generates multiple streams • Large dimension STC decoding is prohibitively complex in MIMO OFDM • Not only decoding, but also “designing good code” is complex STC Yang-Seok Choi et al., ViVATO

4. Motivations (cont’d)… • Serial coding : Use Single stream code and apply Turbo-code style detection/decoding • Serial code generates single stream (convolutional code, LDPC, Turbo-code,..) • MAP, ML or simplified ML with iterative decoding is complicated in MIMO OFDM (calculating LLR, large interleaver size,…) • Is there any efficient way of maximizing both Space- and Frequency-diversity while achieving the capacity? • Use existing code (No need of finding new large dimension STC) • Reduce decoding complexity of ML or MAP (linearly increase in the number of subcarriers and antennas) Serial Coding Yang-Seok Choi et al., ViVATO

5. Parallel Coding • Parallel coding : Multiple Encoders • Encoder generates single stream • Each layer carries independent information bit stream • In order to reduce decoding complexity, equalizer can be adopted Parallel Coding Yang-Seok Choi et al., ViVATO

6. System Model where Yang-Seok Choi et al., ViVATO

7. Linear Equalizers (LE) • MF : • LS (or ZF) : • MMSE : Yang-Seok Choi et al., ViVATO

8. Layered Processing (LP) • LP • Loop • Choose a layer whose SINR (post MMSE) is highest among undecoded layers • Apply MMSE equalizer • Decode the layer • Re-encode and subtract its contribution from received vector • Go to Loop until all layers are processed Yang-Seok Choi et al., ViVATO

9. “Instantaneous” Capacity • Capacity under given realization of channel matrix with perfect knowledge of channel at Rx from this point on for convenience the conditioning on H will be omitted • If transmitted frames have spectral efficiency less than above capacity, with arbitrarily large codeword, FER will be arbitrarily small • If transmitted frames have spectral efficiency greater than above capacity, with arbitrarily large codeword, FER will approach 100%. Yang-Seok Choi et al., ViVATO

10. Mutual Information in LE • Theorem1 (LE) For any linear equalizer • Equality (A) holds where A is a non-singular matrix • Equality (B) holds iff and are diagonal Proof : See [1] Yang-Seok Choi et al., ViVATO

11. Mutual Information in LE (cont’d) • In general equality (A) can be met in most practical systems. • In general the equality (B) is hard to be met. • In most cases, the sum of mutual information in LE is strictly less than the capacity • There is a loss of information when is used as the decision statistics for • This means that only is not sufficient for detecting since the information about is smeared to as a form of interference. • Hence, we need joint detection/decoding such as MLSE across not only time but all layers as well. • However, MLSE can be applied prior to equalization  No need for an equalizer Yang-Seok Choi et al., ViVATO

12. Mutual Information in LP • Theorem2 (LP) In LP (use MMSE at each layer) where is the SINR (post MMSE) at k-th layer Proof : See [1] LP is an optimum equalizer !!! Yang-Seok Choi et al., ViVATO

13. Mutual Information in LP (cont’d) • Chain rule says : • Note where is the modified received vector at k-th stage in LP • Decoder complexity can be reduced in LP • In LP, according to Theorem 2, MMSE equalizer output scalar is enough for decoding while the chain rule shows that vector is required Yang-Seok Choi et al., ViVATO

14. Mutual Information in LP (cont’d) • There is no loss of information in LP  Perfect Equalizer • is a perfect decision statistic for • The received vector y is ideally equalized through LP • Hence, through “parallel ideal code”, k-th layer can transfer without error • In LP it is natural that the coding should be done not across layers but across time (parallel coding) • Don’t need to design large dimension Space-Time code Yang-Seok Choi et al., ViVATO

15. Practical Constraints • Error propagation problem • No ideal code yet • Layer capacity is not constant • Even if the sum of layer capacity is equal to the channel capacity, individual layer capacity is variant over layers • Unless CSI is available to Tx and adaptive modulation is employed, we cannot achieve the capacity • Optimum decoding order • SINR calculations: determinant calculations • One of bottlenecks in LP Yang-Seok Choi et al., ViVATO

16. Solutions • Error propagation problem • Iterative Interference cancellation • Ordered Serial Iterative Interference Cancellation/Decoding (OSI-ICD) • Minimize error propagation and the number of iterations • Layer capacity is not constant • Spreading at Tx : Spread each layer’s data over all layers  Regulate Received Signal power • Ordered detection/decoding at Rx : Serial Detection/Decoding  No loss of information rate • Grouping Increase Layer size • Layer Interleaver • Minimize variance of SINR over layers  Maximize Diversity Gain • Decoding Order • Layer Interleaver and Spreading : Less sensitive to decoding order Yang-Seok Choi et al., ViVATO

17. Spreading • Without Spreading • Received Signal power for : • With Spreading where T is a unitary matrix • is carried by which is a linear combination of • Received Signal power for : Yang-Seok Choi et al., ViVATO

18. Spreading for Orthogonal channel • Assume that channel vectors are orthogonal each other • Example : Single antenna OFDM under time-invariant multipath -- The channel matrix is diagonal (OFDM w/ Spreading called MC-CDMA[2]) • Assume • Then, the received signal power is constant • SINR after MMSE is constant as well Yang-Seok Choi et al., ViVATO

19. Spreading for Orthogonal channel (cont’d) • : SINR of after MMSE equalizer with Spreading matrix • Constant SINR over k regardless of choice of T • Constant Received Signal Power, SINR and Layer Capacity Maximum diversity gain • Note is a harmonic mean of • Hence, Yang-Seok Choi et al., ViVATO

20. Spreading for Orthogonal channel (cont’d) • Although constant layer capacity is achieved, layer capacity is less than the mean layer capacity from Jensen’s inequality or Theorem 1 • Spreading destroys orthogonality of the channel matrix  Inter-layer interference Yang-Seok Choi et al., ViVATO

21. Spreading for iid MIMO channel • There is no benefit when spreading is applied to iid MIMO channel • Since the spreading matrix is a unitary matrix, the channel matrix elements after the spreading are iid Gaussian • Spreading may provide some gain in Correlated MIMO channel (when the layer size is smaller than number of Tx antennas) Yang-Seok Choi et al., ViVATO

22. Spreading for Block Diagonal Channel • MIMO OFDM : Block Diagonal channel matrix • Spreading Matrix • : Spreading over Space • : Spreading over Frequency Yang-Seok Choi et al., ViVATO

23. Spreading for Block Diagonal Channel (cont’d) • New channel matrix where • Assume Then SINR at k-th subcarrier and n-th antenna where is the SINR when (No spreading over frequency) • Again, Yang-Seok Choi et al., ViVATO

24. Spreading for Block Diagonal Channel (cont’d) • Spreading regulates received signal power and SINR at the output of the MMSE equalizer, and hence maximizes diversity • Inverse matrix size for MMSE is nT instead of nTK because the channel matrix is a block diagonal matrix and the spreading matrix is unitary • Spreading increases interference power since it destroys orthogonality Yang-Seok Choi et al., ViVATO

25. Ordered Decoding at RX • Corollary1 In LP, different ordering does not change the sum of layer capacity which is equal to channel capacity. Proof : Clear from the proof of Theorem 2 • Thus, even random ordering does not reduce the information rate. • However, different ordering changes individual layer capacity and yields different variance. • Hence, optimum ordering is required to maximize minimum layer capacity Yang-Seok Choi et al., ViVATO

26. Ordered Decoding at RX (cont’d) • Assume that channel vectors are orthogonal • Without Spreading the layer capacity is where the decoding order is assumed to be k • With Spreading (see [1] for proof) Yang-Seok Choi et al., ViVATO

27. Grouping • A simple way of reducing layer capacity variance is to reduce the number of layers by grouping (i.e. increasing layer dimension) • Namely, coding over several antennas or subcarriers • N element data vector d is decomposed to subgroups (or layers) • In general, each layer may have a different size Yang-Seok Choi et al., ViVATO

28. Grouping (cont’d) • Is there an equalizer which reduces decoder complexity without losing information rate? • Generalized Layered Processing (GLP) • Assuming a decoding order to be k, at the k-th layer, the received vector can be written as where • MMSE Equalizer (L is the layer size) • Let MMSE equalizer output Yang-Seok Choi et al., ViVATO

29. Grouping (cont’d) • Theorem3 (GLP) GLP does not lose information rate when is full rank and MMSE equalizer is applied Proof : See [1] • At each layer, MMSE equalized vector is used instead of for the decoding • Under certain conditions [1] Yang-Seok Choi et al., ViVATO

30. Layer Interleaving (LI) • Layer Interleaving provide Layer diversity • Doesn’t require memory and doesn’t introduce any delay • Doesn’t require synchronization • Diversity gain is less significant than spreading L=1 case Yang-Seok Choi et al., ViVATO

31. Numerical Experiments • General Tx Structure • Simulation Conditions • Without Interleaver • 2-by-2 MIMO OFDM, K=32 subcarriers N=64 • iid MIMO channel • Maximum delay spread is ¼ of symbol duration • rms delay spread is ¼ of Maximum delay spread • Exponential delay profile • Decoding order is based on maximum layer capacity • 32-by-32 Walsh-Hadamard code for frequency spreading • No spreading over space Yang-Seok Choi et al., ViVATO

32. Numerical Experiments (cont’d) • CDF of normalized layer capacity in MIMO OFDM, L=1 • Spreading yields steeper curve  Diversity • LP improves Outage Capacity • Recall by Theorem 1&2 Yang-Seok Choi et al., ViVATO

33. Numerical Experiments (cont’d) • CDF in MIMO OFDM, L=2(Grouped over antennas, ) • Grouping can significantly improve outage capacity • Unless Best grouping is employed, GLP has less outage capacity than LP • Spreading is still useful in reducing the variance of the layer capacity • Recall Yang-Seok Choi et al., ViVATO

34. Numerical Experiments (cont’d) • Effect of Layer size and Spreading in LP and GLP • w/o Spreading : distance of grouped subcarriers is maximized • w/ Spreading : neighboring subcarriers are grouped • SP is effective when layer size is small • Ideal “single stream code” is better thanIdeal “4-by-4 code” !!! • We don’t know optimum spreading matrix structure Yang-Seok Choi et al., ViVATO

35. Numerical Experiments (cont’d) • GLP performance with 2-by-2 STC • 16 state 2 bps/Hz QPSK STTC (1 bps/Hz/antenna) • L=2, 128 symbols per layer • Two iterations (hard decision) Serial STC w/o Spreading Parallel STC Yang-Seok Choi et al., ViVATO

36. Numerical Experiments (cont’d) • GLP of Parallel STC w/ SP has the best performance • Serial STC has less frequency diversity gain Ideal 2-by-2 STC w/ GLP & w/o SP 2.1 dB Gain Ideal N-by-N STC 3.5 dB Gain Ideal 2-by-2 STC w/ SP&GLP Loss due to non-ideal 2-by-2 STC Yang-Seok Choi et al., ViVATO

37. Comments on Serial code w/ SP • Spreading provides diversity gain (steeper curves) but increases interference • Unless ML or Turbo type decoding over antennas and subcarriers is applied, capacity cannot be achieved • Complexity grows exponentially with the number of subcarriers and antennas • Partial spreading • The spreading matrix T is unitary but some of elements are zero • Reduces interference • Reduces ML decoder complexity • Reduces diversity Yang-Seok Choi et al., ViVATO

38. More on Partial Spreading • Partial Spreading in MIMO OFDM • K : number of subcarriers • SF : Spreading factor, number of subcarriers spread over • SF> Max delay in samples  Negligible frequency diversity loss • Partial spreading over subcarriers • The partial spreading matrix is useful when K is not a multiple of 4 Yang-Seok Choi et al., ViVATO

39. Versatilities of Parallel coding • Allows LDMA (Layer Division Multiple Access) • Parallel coding can send multiple frames by nature • Different frames can be assigned to different users (Different spreading code are assigned to different users) • A convenient form of multiplexing for different users • Control or broadcasting channel can be established • Adaptive modulation • By changing not only modulation order but also the number of frames Yang-Seok Choi et al., ViVATO

40. MMSE or MF instead of LP • MMSE can be used instead of LP at first iteration in order to reduce latency or complexity • Then, it requires more iteration than LP because LP provides better SINR. • MF can also be used to reduce complexity. • But it will require more iterations and error propagation is more severe. • LP requires less number of iterations Yang-Seok Choi et al., ViVATO

41. Conclusions • Large dimension STC design/decoding is prohibitively complex • Serial code can have limited diversity gain or the complexity grows at least cubically with the number of subcarriers and antennas • Use parallel coding, apply SP at Tx and LP at Rx • Spreading increases diversity gain when layer size is small • LP does not lose the information rate while LE does • SP and Layer interleaver can reduce the sensitivity to decoding order in LP or GLP • Complexity of LP : Linearly increase in the number of subcarriers and antennas • LP needs less number of iterations • LP w/ SP is an efficient way of increasing diversity gain with reduced code design effort and decoding complexity Yang-Seok Choi et al., ViVATO

42. References • [1] Yang-Seok Choi, “Optimum Layered Processing”, Submitted to IEEE Transactions on Information Theory, 2003 • [2] Hara et al., “Overview of Multicarrier CDMA”, IEEE Transactions on Commun. Mag.,pp.126-133, Dec. 1997 Yang-Seok Choi et al., ViVATO

43. Thank you for your attention!! Questions? Yang-Seok Choi et al., ViVATO

44. Back-up • Different Spreading Matrix Yang-Seok Choi et al., ViVATO