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Generalized Spatial Modulation in Large-Scale Multiuser MIMO Systems

T. Lakshmi Narasimhan, P. Raviteja, A. Chockalingam

arXiv:1503.03997v1cs.ITmath.IT

TL;DR

Large-scale uplink multiuser GSM had not been studied for performance and detection, while optimal detection in large dimensions is prohibitively complex. The paper derives an ML-detection ABEP bound and proposes low-complexity message-passing detection and channel-estimation algorithms. At equal spectral efficiency, GSM-MIMO outperforms SM-MIMO and conventional multiuser MIMO by about 2 to 9 dB at 10^-3 BER.

  • Problem

    Large-scale uplink multiuser GSM had not been studied for performance and detection, while optimal detection in large dimensions is prohibitively complex.

    Source: §IV · p. 6

  • Method

    The paper derives an ABEP upper bound for ML detection and proposes low-complexity message-passing algorithms for GSM-MIMO detection and channel estimation.

    Source: §I · p. 2

  • Results

    At equal spectral efficiency, GSM-MIMO outperforms SM-MIMO and conventional multiuser MIMO by about 2 to 9 dB at 10^-3 BER; its ML ABEP bound is tight at moderate-to-high SNRs.

    Source: §I · p. 2

  • Takeaways & Limitations

    More spatial-index bits let GSM use lower-order, more power-efficient QAM at equal spectral efficiency, explaining its reported SNR gains.

    Source: §I · p. 2

Abstract

from arXiv · show

Generalized spatial modulation (GSM) uses $n_t$ transmit antenna elements but fewer transmit radio frequency (RF) chains, $n_{rf}$. Spatial modulation (SM) and spatial multiplexing are special cases of GSM with $n_{rf}=1$ and $n_{rf}=n_t$, respectively. In GSM, in addition to conveying information bits through $n_{rf}$ conventional modulation symbols (for example, QAM), the indices of the $n_{rf}$ active transmit antennas also convey information bits. In this paper, we investigate {\em GSM for large-scale multiuser MIMO communications on the uplink}. Our contributions in this paper include: ($i$) an average bit error probability (ABEP) analysis for maximum-likelihood detection in multiuser GSM-MIMO on the uplink, where we derive an upper bound on the ABEP, and ($ii$) low-complexity algorithms for GSM-MIMO signal detection and channel estimation at the base station receiver based on message passing. The analytical upper bounds on the ABEP are found to be tight at moderate to high signal-to-noise ratios (SNR). The proposed receiver algorithms are found to scale very well in complexity while achieving near-optimal performance in large dimensions. Simulation results show that, for the same spectral efficiency, multiuser GSM-MIMO can outperform multiuser SM-MIMO as well as conventional multiuser MIMO, by about 2 to 9 dB at a bit error rate of $10^{-3}$. Such SNR gains in GSM-MIMO compared to SM-MIMO and conventional MIMO can be attributed to the fact that, because of a larger number of spatial index bits, GSM-MIMO can use a lower-order QAM alphabet which is more power efficient.

I. INTRODUCTION

The introduction frames large-scale multiuser uplink GSM as a previously unreported setting and presents ABEP analysis alongside message-passing receiver algorithms. It motivates GSM through the information carried by both active-antenna indices and modulation symbols.

I. INTRODUCTION

  • Earlier work studied GSM rates, but performance analysis and signal detection for large-scale multiuser GSM uplink had not been reported.Source: §I · p. 2
  • GSM activates n_rf of n_t antennas and conveys information through their indices and transmitted symbols; SM and spatial multiplexing are special cases.Source: §I · p. 2
  • For maximum-likelihood detection, the paper derives an ABEP upper bound that is tight at moderate-to-high SNRs.Source: §I · p. 2
  • At equal spectral efficiency, GSM-MIMO outperforms multiuser SM-MIMO and conventional multiuser MIMO by about 2 to 9 dB at 10^-3 BER.Source: §I · p. 2
  • More spatial-index bits let GSM use lower-order, more power-efficient QAM, the stated explanation for its SNR gains.Source: §I · p. 2
  • Message-passing algorithms for GSM-MIMO detection and channel estimation at the base station scale well in complexity and achieve near-ML performance in large dimensions.Source: §I · p. 2
  • The proposed detection algorithms have lower complexity than MMSE detection while achieving significantly better performance than MMSE detection.Source: §I · p. 2

II. MULTIUSER GSM-MIMO SYSTEM MODEL

The section models uplink GSM-MIMO, where users select active antennas to carry index and modulation bits, and specifies the received signal and detection problem at a large-antenna base station.

II. MULTIUSER GSM-MIMO SYSTEM MODEL

  • Each uplink user selects nrf of nt transmit antennas for modulation symbols; the remaining antennas are silent, and the base station has tens to hundreds of receive antennas.The model includes K users and an nrf × nt switch connecting each user's RF chains to its antennas.Source: §II · p. 2
  • Users map information bits to selected activation patterns, choosing from the possible patterns with a shared, channel-independent mapping.For nt=4 and nrf=2, six activation patterns are possible, with a subset selected for signaling.Source: §II · pp. 2–3
  • GSM conveys bits through both active-antenna indices and modulation symbols, with each transmit vector containing exactly nrf nonzero entries.The signal set is constrained by the chosen activation patterns and modulation alphabet.Source: §II · p. 3
  • The received signal combines users' channel-weighted transmissions with noise; the channel gains are modeled as independent Gaussian variables.The model also accounts for received-power imbalance and assumes perfect synchronization.Source: §II · p. 3
  • ML detection minimizes the received-signal residual, while exact ML and MAP computation requires complexity exponential in K.The section introduces the low-complexity message-passing algorithms as approximate solutions to MAP detection.Source: §II · p. 3
  • At equal spectral efficiency, the example compares GSM with BPSK and two active antennas against SM and conventional MIMO configurations using 4-QAM.Each configuration conveys 4 bpcu per user.Source: §II · p. 3

III. AVERAGE BEP ANALYSIS

The analysis derives an upper bound on the average bit error probability of ML detection by averaging pairwise error probabilities over the channel and applying a union bound.

III. AVERAGE BEP ANALYSIS

  • For equally likely transmit vectors, the pairwise error probability is averaged over the channel using a central chi-square distribution with 2N degrees of freedom.The analysis evaluates the expectation of a Q-function with respect to the channel matrix.Source: §III · p. 4
  • A union bound on average bit error probability weights pairwise errors by the number of differing bits d(x, x~).The resulting expression sums over pairs of possible transmitted vectors.Source: §III · p. 4
  • Direct computation requires 2^η(2^η−1) pairwise error probabilities, so the bound's complexity grows exponentially in K, nt, and nrf.The following subsection introduces simplifications to reduce this computational cost.Source: §III · p. 4

A. Reduction of computation complexity in (12)

The section reduces the cost of computing the ABEP bound by grouping repeated pairwise-error calculations first by antenna-pattern overlap and then by symbol-distance sums.

A. Reduction of computation complexity in (12)

B. Numerical results

The numerical results compare multiuser GSM-MIMO with SM-MIMO and conventional MIMO at matched spectral efficiency, and assess an analytical ABEP upper bound against simulation. GSM-MIMO performs best in the reported comparison, while the upper bound is tight at moderate-to-high SNR.

B. Numerical results

  • The analytical ABEP upper bound is tight at moderate-to-high SNR for the evaluated multiuser GSM-MIMO configuration.Analysis and simulation also show that ABEP improves as the number of base-station antennas increases.Source: B. Numerical results · p. 5
  • At 6 bpcu per user, SM-MIMO achieves about 2–7 dB better performance than conventional multiuser MIMO at 10^-3 BER; GSM-MIMO performs better than SM-MIMO and two-RF-chain System 2.GSM-MIMO uses 4-QAM on two streams, conveying four bits through modulation symbols and two through antenna indexing.Source: Figure 4 · p. 6, B. Numerical results · p. 6
  • At 10^-3 BER, SM-MIMO improves on the two conventional MIMO systems by about 2 to 7 dB at the same 6 bpcu per user.The reported explanations are use of a smaller QAM alphabet than one conventional system and fewer interfering spatial streams than the other.Source: B. Numerical results · p. 6

IV. LOW-COMPLEXITY RECEIVER ALGORITHMS

Optimal detection for large-scale multiuser GSM-MIMO is prohibitively complex, motivating low-complexity receiver methods. The paper proposes two message-passing detection algorithms and a channel-estimation approach for use with one of them.

IV. LOW-COMPLEXITY RECEIVER ALGORITHMS

  • Optimal detection of multiuser GSM-MIMO signals in large dimensions is prohibitively complex, and generalized sphere decoding can have exponential complexity at low-to-medium SNRs.The paper therefore presents low-complexity, near-optimal detection and channel-estimation algorithms for large-scale systems.Source: §IV · p. 6
  • MP-GSM forms messages from the basic multiuser GSM-MIMO system model, while CHEMP-GSM forms them from its matched-filtered version.Both are approximate message-passing detection algorithms.Source: §IV · p. 6
  • The channel-estimation approach directly estimates HHH for use in CHEMP-GSM.The proposed receiver algorithms are described as scaling well in complexity and achieving near-ML performance in large dimensions.Source: §IV · p. 6

A. MP-GSM detection algorithm

MP-GSM approximates multiuser GSM-MIMO detection through message passing on a factor graph, modeling interference as Gaussian. Simulations compare its BER and complexity with other detectors and MIMO configurations, including under different antenna counts and stream settings.

A. MP-GSM detection algorithm

  • MP-GSM models the system as a fully connected factor graph with K user-variable nodes and N observation nodes, approximating interference as Gaussian to obtain an approximate MAP solution.The Gaussian approximation is stated to be accurate for large K, for example systems with tens of users.Source: A. MP-GSM detection algorithm · p. 6
  • Messages from observation nodes to user-variable nodes contain scalar mean and variance values, while reverse messages are probability vectors over candidate symbols.The algorithm iterates message updates until convergence or a maximum iteration count.Source: A. MP-GSM detection algorithm · p. 7
  • MP-GSM complexity is O(NK|Snrf_nt,A|), less than MMSE detection, whose complexity is O(N^2Knt).The double summation in the MP-GSM message calculations can be simplified using FFT as a convolution.Source: A. MP-GSM detection algorithm · p. 7
  • The reported Fig. 6 performance evaluation assumes perfect receiver CSI for GSM-MIMO with K = 16, N = 64 or 128, and 6 bpcu per user.The paper states that it relaxes the perfect-CSI assumption later.Source: A. MP-GSM detection algorithm · p. 7
  • At 10^-3 BER, GSM-MIMO with MP-GSM gains about 4 dB over SM-MIMO and about 7 dB over conventional MIMO at 6 bpcu per user.The comparison uses K = 16, N = 64 or 128, and assumes perfect channel state information at the receiver.Source: A. MP-GSM detection algorithm · pp. 7–8, Figure 6 · p. 8
  • At 10^-3 BER, MP-GSM outperforms MMSE, MMSE-SIC, and MB-MMSE-DF detection, with lower complexity than all three.For MMSE detection, the reported MP-GSM advantage is 9 dB at N = 64 and 3 dB at N = 128.Source: Figure 7 · p. 8, A. MP-GSM detection algorithm · p. 8
  • At 4 bpcu per user, GSM-MIMO performs best among the four evaluated systems; M-MIMO loses performance with higher-order QAM or more spatial-stream interference.The comparison uses K = 16 and N = 128.Source: A. MP-GSM detection algorithm · pp. 8–9, Figure 8 · p. 9
  • When Knt ≤ N, GSM-MIMO requires about 9 dB less SNR than M-MIMO and 4 to 6 dB less than SM-MIMO at 10^-3 BER.For smaller N where Knt > N, the systems become underdetermined and required SNRs rise sharply.Source: A. MP-GSM detection algorithm · p. 9

B. CHEMP-GSM detection algorithm

CHEMP-GSM combines message passing with a channel-hardening approximation to detect GSM signals at lower complexity than MMSE detection. The section compares its performance with MMSE and MP-GSM across system loadings.

B. CHEMP-GSM detection algorithm

  • CHEMP-GSM uses a matched-filtered system model, where each user’s observation is represented as a concatenation of its own signal term and other users’ contributions.The algorithm exchanges approximate a posteriori probability messages among user nodes and damps updates to improve convergence.Source: B. CHEMP-GSM detection algorithm · pp. 9–10
  • At equal spectral efficiency, GSM-MIMO outperforms SM-MIMO and conventional MIMO in the reported comparison, which the paper attributes to GSM’s use of smaller QAM.The comparison uses 6 bpcu per user and perfect CSI.Source: Figure 9 · p. 9, B. CHEMP-GSM detection algorithm · p. 11
  • The detector approximates each user’s interference-plus-noise as multivariate Gaussian, using channel hardening’s strong diagonal terms in J relative to off-diagonal terms.CHEMP-GSM is a message-passing detector; its overall complexity is less than MMSE detection complexity.Source: B. CHEMP-GSM detection algorithm · pp. 9–10
  • 9 dB for N = 64 and 3 dB for N = 128: CHEMP-GSM outperforms MMSE at BER 10^-3.The comparison uses K = 16, n_t = 4, n_rf = 2, 4-QAM, and 6 bpcu per user.Source: Figure 12 · p. 11, B. CHEMP-GSM detection algorithm · p. 11
  • At K/N = 0.25, MP-GSM and CHEMP-GSM outperform MMSE by about 9 dB at BER 10^-3.Their advantage is about 2–3 dB at lower loading factors; MP-GSM and CHEMP-GSM perform almost the same, while MP-GSM has lower complexity.Source: Figure 13 · p. 11, B. CHEMP-GSM detection algorithm · p. 11

C. Estimation of HHH

The proposed CHEMP-GSM receiver estimates J directly rather than estimating the channel matrix H. Its performance and complexity are compared with MMSE and MP-GSM receivers.

C. Estimation of HHH

  • Because the proposed detection operation uses H through J = H^H H, CHEMP-GSM estimates J directly instead of estimating H.The comparison uses the proposed J estimate for CHEMP-GSM and MMSE channel estimates for the MMSE and MP-GSM receivers.Source: C. Estimation of HHH · pp. 11–12
  • The channel is assumed to remain constant over a frame containing a pilot part followed by a data part.The pilot part uses K n_t channel uses.Source: C. Estimation of HHH · p. 11
  • The CHEMP-GSM receiver has the best performance among the three receivers; MP-GSM and CHEMP-GSM receivers have much lower complexity than MMSE.CHEMP-GSM receiver complexity is slightly higher than MP-GSM’s because it additionally computes z-hat.Source: Figure 14 · p. 12, C. Estimation of HHH · p. 12

V. MULTIUSER GSM-MIMO IN FREQUENCY SELECTIVE

The section extends multiuser GSM-MIMO to frequency-selective fading using cyclic-prefixed single-carrier transmission and frequency-domain detection. It reports comparisons against conventional multiuser MIMO with both perfect and estimated CSI.

V. MULTIUSER GSM-MIMO IN FREQUENCY SELECTIVE

  • The frequency-selective channel model has L multipath components, and the received signal is transformed to an equivalent frequency-domain system for detection.The transformed model is detected using the MP-GSM and CHEMP-GSM algorithms.Source: §V · pp. 12–13
  • CPSC frames contain a pilot block for channel estimation followed by data blocks, each with a cyclic prefix and Q information symbol vectors.The fade coefficients are assumed constant over one frame.Source: §V · p. 12, Figure 15 · p. 13
  • 12 dB: GSM-MIMO outperforms M-MIMO at BER 10^-4 with perfect CSI; MP-GSM and CHEMP-GSM also outperform MMSE by about 3 dB.The comparison uses CPSC transmission with L = 3, Q = 6, ξ = 3 dB, N = 128, and 6 bpcu per user.Source: §V · p. 13, Figure 16 · p. 14
  • 11 dB: GSM-MIMO CPSC outperforms M-MIMO CPSC at BER 10^-4 with estimated CSI.The comparison uses the same system and channel parameters as the perfect-CSI results.Source: Figure 17 · p. 14, §V · p. 14

VI. CONCLUSIONS

The paper studies uplink multiuser GSM-MIMO, deriving an ML-detection ABEP upper bound and proposing message-passing detection and channel-estimation algorithms. It concludes that GSM-MIMO can outperform conventional MIMO while using low-complexity reception.

VI. CONCLUSIONS

  • The analytical upper bound on ABEP for ML detection is tight at moderate-to-high SNRs.The conclusion reports that numerical results show multiuser GSM-MIMO can outperform conventional multiuser MIMO by several dBs at the same spectral efficiency.Source: §VI · p. 14
  • The proposed message-passing detection and channel-estimation algorithms are low-complexity, and results in large-scale systems with tens of users and hundreds of BS antennas show GSM-MIMO outperforming conventional MIMO.The paper attributes the advantage to lower-order QAM from spatial-index bits and to increased spatial interference in conventional MIMO when it uses more streams at the same spectral efficiency and QAM size.Source: §VI · p. 14
  • The paper presents large-scale multiuser GSM-MIMO as attractive for future wireless systems, citing its performance advantage and low RF hardware complexity.It also notes experimental validation of the SM concept in indoor and outdoor settings.Source: §VI · p. 14
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