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Generalized Space and Frequency Index Modulation

T. Datta, H. S. Eshwaraiah, A. Chockalingam

arXiv:1506.08291v1cs.ITmath.IT

TL;DR

Multi-antenna systems face RF-chain hardware complexity, size, and cost, while conventional modulation conveys information through symbols alone. This paper introduces GSIM and GSFIM, which also encode information through active-antenna and subcarrier indices. The schemes show rate and BER advantages over spatial multiplexing and MIMO-OFDM in the reported comparisons.

  • Problem

    Multi-antenna systems face RF-chain hardware complexity, size, and cost, while conventional modulation conveys information only through modulation symbols.

    Source: §I · p. 1

  • Method

    The paper introduces GSIM, which indexes active antennas, and GSFIM, which indexes both active antennas and subcarriers, alongside conventional modulation.

    Source: §I · pp. 1–2

  • Results

    GSIM can exceed spatial multiplexing in achievable rate with fewer RF chains and better BER, while GSFIM can offer higher rates and potential BER gains over MIMO-OFDM.

    Source: §I · p. 2, §IV · pp. 13–14

  • Takeaways & Limitations

    Indexing active transmit entities provides an additional way to convey information beyond conventional modulation symbols.

    Source: §IV · p. 12

    Low-complexity detection methods for GSFIM remain future work.

    Source: §IV · p. 14

Abstract

from arXiv · show

Unlike in conventional modulation where information bits are conveyed only through symbols from modulation alphabets defined in the complex plane (e.g., quadrature amplitude modulation (QAM), phase shift keying (PSK)), in index modulation (IM), additional information bits are conveyed through indices of certain transmit entities that get involved in the transmission. Transmit antennas in multi-antenna systems and subcarriers in multi-carrier systems are examples of such transmit entities that can be used to convey additional information bits through indexing. In this paper, we introduce {\em generalized space and frequency index modulation}, where the indices of active transmit antennas and subcarriers convey information bits. We first introduce index modulation in the spatial domain, referred to as generalized spatial index modulation (GSIM). For GSIM, where bits are indexed only in the spatial domain, we derive the expression for achievable rate as well as easy-to-compute upper and lower bounds on this rate. We show that the achievable rate in GSIM can be more than that in spatial multiplexing, and analytically establish the condition under which this can happen. It is noted that GSIM achieves this higher rate using fewer transmit radio frequency (RF) chains compared to spatial multiplexing. We also propose a Gibbs sampling based detection algorithm for GSIM and show that GSIM can achieve better bit error rate (BER) performance than spatial multiplexing. For generalized space-frequency index modulation (GSFIM), where bits are encoded through indexing in both active antennas as well as subcarriers, we derive the achievable rate expression. Numerical results show that GSFIM can achieve higher rates compared to conventional MIMO-OFDM. Also, BER results show the potential for GSFIM performing better than MIMO-OFDM.

I. INTRODUCTION

The paper treats indexing transmit entities as an additional way to convey information, and introduces GSIM and GSFIM to encode bits through active antennas and, for GSFIM, subcarriers. It analyzes achievable rates and detection performance for these schemes.

I. INTRODUCTION

  • Index modulation conveys additional bits through indices of participating transmit entities, addressing the RF-chain complexity, size, and cost associated with multi-antenna systems.The paper applies this idea to antennas in GSIM and to antennas and subcarriers in GSFIM.Source: §I · pp. 1–2

II. GENERALIZED SPATIAL INDEX MODULATION

  • GSIM can exceed spatial multiplexing in achievable rate under an analytically established condition, while using fewer transmit RF chains and potentially achieving better BER.The paper also derives easy-to-compute rate bounds and proposes Gibbs sampling based detection.Source: §I · p. 2

I. INTRODUCTION

  • GSFIM combines spatial and frequency indexing and can offer higher rates with fewer RF chains than conventional MIMO-OFDM, with potential BER gains.Its frequency-domain bits are encoded across multiple sub-blocks within a space-frequency block.Source: §I · p. 2

A. System model

  • A GSIM transmitter activates nrf of nt antennas, sends M-ary symbols on them, and leaves the other antennas silent.An nrf × nt switch connects the RF chains to the transmit antennas.Source: A. System model · p. 2

B. Achievable rates in GSIM

GSIM combines antenna-index bits with modulation-symbol bits, and its maximum achievable rate can exceed spatial multiplexing under a stated antenna-to-alphabet condition. Its rate-maximizing RF-chain count can be below the antenna count.

B. Achievable rates in GSIM

  • The achievable rate combines antenna-activation-pattern bits with nrf log2 M modulation bits.The antenna-index term depends on the number of available activation patterns.Source: B. Achievable rates in GSIM · p. 3
  • For a fixed nt, the rate-maximizing nrf need not equal nt, the spatial-multiplexing setting.The optimum can occur at nrf<nt.Source: B. Achievable rates in GSIM · p. 3
  • GSIM's maximum rate exceeds spatial multiplexing iff nt≥2M; when nt<2M, it does not exceed it.The proof establishes the greater-rate case using nrf=nt−1.Source: B. Achievable rates in GSIM · pp. 3–4
  • At nt=32 with 4-QAM, nrf=24 yields 71 versus 64 bpcu, an 11% rate gain with 25% fewer RF chains.The cited comparison identifies the 64 bpcu spatial-multiplexing rate.Source: B. Achievable rates in GSIM · p. 4
  • For nt=32 and 4-QAM, GSIM can match spatial multiplexing's 64 bpcu using 18 RF chains, a 43% saving.The comparison is against spatial multiplexing with 32 transmit RF chains.Source: B. Achievable rates in GSIM · p. 4

C. Bounds on achievable rates in GSIM

The section derives upper and lower GSIM rate bounds designed for easier computation than the exact rate expression. The bounds also approximate the maximum rate closely in the reported examples.

C. Bounds on achievable rates in GSIM

D. GSIM signal detection

GSIM detection is formulated for a rich-scattering channel and seeks the maximum-likelihood transmit vector. Because exhaustive ML search becomes prohibitive for medium and large systems, the paper proposes a lower-complexity Gibbs-sampling approach.

D. GSIM signal detection

  • The channel matrix H is modeled with independent circularly symmetric complex Gaussian entries of zero mean and unit variance.The received vector includes additive white Gaussian noise.Source: D. GSIM signal detection · p. 5
  • ML detection selects the feasible transmit vector minimizing the squared residual between the received signal and Hx.The activation-pattern set and its mapping to antenna-selection bits are known at both ends.Source: D. GSIM signal detection · p. 5
  • Exhaustive ML search becomes computationally prohibitive for medium and large nt and nrf, motivating a low-complexity Gibbs sampler.The sampler forms a Markov chain whose states are possible transmitted vectors.Source: D. GSIM signal detection · pp. 5–6

1) Proposed modified Gibbs sampler:

The modified Gibbs sampler preserves the fixed number of active antennas by jointly considering coordinate pairs. It compares no-swap and swap candidates, then updates the best vector until a stopping condition or iteration limit is reached.

1) Proposed modified Gibbs sampler:

2) Stopping and restart criterion:

The detector uses a stagnation metric to stop each restart and a repeat-count metric to decide when the best valid output is reliable enough to terminate the full search.

2) Stopping and restart criterion:

  • A restart stops when the current best vector has remained unchanged for Θs(z) iterations.The best vector among restart outputs is checked for validity before it contributes to the overall output.Source: 2) Stopping and restart criterion · p. 6
  • The algorithm terminates when the best valid vector has been returned Θr(s) times or the maximum restart count is reached.Θr(s) is computed from the best vector’s ML cost.Source: 2) Stopping and restart criterion · p. 6
  • The detector’s complexity is analyzed in three parts: starting-vector generation, computing yMF and R, and sampling and updating.The first restart uses the MMSE output as its starting vector; subsequent restarts use random starting vectors.Source: 3) Complexity · p. 6, 2) Stopping and restart criterion · p. 6

3) Complexity:

The proposed Gibbs-sampling detector has cubic complexity in the number of transmit antennas and performs near ML detection in the reported GSIM tests. At matched rates, GSIM improves BER over spatial multiplexing in medium-to-high SNR regimes, with low-SNR performance favoring spatial multiplexing.

3) Complexity:

  • The detector’s sampling and updating work is O(nrf) per iteration, and simulations find O(nrf(nt − nrf)) iterations, yielding total cubic complexity in nt.The analysis separates this work from starting-vector generation and computation of yMF and R.Source: 3) Complexity · p. 6

E. BER performance results

  • At 6 bpcu, (4,2)-GSIM performs about 1 dB better than (2,2)-SM at 0.01 uncoded BER; (4,1)-GSIM is within 0.5 dB of SM at medium-to-high SNR.The comparison uses brute-force ML detection and nr = 2; GSIM’s gain uses extra antennas rather than extra transmit RF chains.Source: Figure 4 · p. 7, E. BER performance results · p. 7
  • The proposed detector nearly matches ML performance in the tested (4,3)-GSIM and (8,7)-GSIM systems, while MMSE performs very poorly.The proposed detector has cubic complexity in nt, versus exponential complexity for ML detection.Source: Figure 5 · p. 7, E. BER performance results · p. 8
  • At low SNR, the SM schemes have better BER than corresponding GSIM schemes; GSIM’s antenna-pattern errors can corrupt both index and modulation bits.At medium-to-high SNR, modulation-symbol errors dominate instead, and GSIM’s larger constellation minimum distance supports its advantage.Source: E. BER performance results · p. 8
  • At 24 bpcu, (12,8)-GSIM beats (8,8)-SM by about 2 dB at high SNR and nearly matches (12,12)-SM, while using fewer RF chains than the latter.The GSIM detector is much lower complexity than the generalized sphere decoder used for (12,12)-SM.Source: Figure 6 · p. 8, E. BER performance results · p. 8
  • At 48 bpcu, (22,16)-GSIM outperforms (16,16)-SM by about 2 dB at medium-to-high SNR using the same RF resources.GSIM uses six extra transmit antennas, and its proposed detector has much lower complexity than sphere decoding.Source: Figure 7 · p. 8, E. BER performance results · p. 8

III. GENERALIZED SPACE-FREQUENCY INDEX MODULATION

GSFIM extends GSIM by encoding information through indexing in both spatial and frequency domains. Its transmitter maps antenna indices, subcarrier indices, and modulation symbols before OFDM transmission.

III. GENERALIZED SPACE-FREQUENCY INDEX MODULATION

  • GSFIM generalizes GSIM by using index modulation in both spatial and frequency domains.The paper derives its achievable-rate equation and studies rate variation and gains over conventional MIMO-OFDM.Source: §III · p. 8

A. System model

  • The transmitter maps information bits to active antennas, frequency-index choices, and M-ary modulation symbols, then modulates the signal with OFDM.The system has nt transmit antennas, nrf RF chains, N subcarriers, and nr receive antennas; only nrf antennas are active at a time.Source: §III · p. 8, A. System model · p. 9
  • The transmitter forms an nrf × N matrix B from submatrices and feeds each row to an OFDM IFFT for transmission on one active antenna.Each submatrix contains a selected zero/nonzero pattern and nonzero entries from the modulation alphabet.Source: 1) Frequency indexing · p. 9, A. System model · p. 9

1) Frequency indexing:

GSFIM encodes frequency-index bits by selecting among permitted zero/nonzero activation patterns for each submatrix. The nonzero entries carry the modulation symbols.

1) Frequency indexing:

  • Each submatrix has nf = N/nb subcarriers, and its number k of nonzero entries is a design parameter.A frequency activation pattern specifies which entries are zero or nonzero.Source: 1) Frequency indexing · p. 9
  • The encoder selects one of 2^kf patterns from the frequency activation pattern set using kf bits per submatrix.Across nb submatrices, frequency indexing carries nbkf bits; the knb log2 M nonzero-entry bits carry modulation symbols.Source: 1) Frequency indexing · p. 9
  • Each row of B becomes an N-length OFDM symbol, and the nrf symbols are transmitted in parallel on the selected active antennas.The antenna activation pattern is selected from antenna-index bits.Source: 1) Frequency indexing · p. 9, 2) Antenna indexing · p. 9

2) Antenna indexing:

GSFIM carries information through antenna and subcarrier indices as well as modulation symbols, and its achievable rate can exceed MIMO-OFDM while using fewer transmit RF chains. The rate depends on the indexing parameters, with gains and RF-chain savings varying by modulation alphabet and system settings.

2) Antenna indexing:

  • GSIM selects an antenna activation-pattern set of 2^ka patterns, and an nrf × nt switch connects the RF chains to the selected antennas.The active antennas may change from one MIMO-OFDM symbol to the next.Source: 2) Antenna indexing · pp. 9–10

B. Achievable rate, rate variation, and rate gain

C. GSFIM signal detection and performance

The section formulates maximum-likelihood detection for GSFIM using antenna and frequency activation patterns. In the reported BER comparisons, GSFIM performs better at moderate to high SNRs, while MIMO-OFDM performs better at low SNRs.

C. GSFIM signal detection and performance

  • The received signal model uses the channel for the chosen antennas, a transmitted vector, and additive white Gaussian noise.The model is specified per subcarrier and then for each sub-matrix.Source: C. GSFIM signal detection and… · p. 11
  • The ML detector jointly chooses an antenna activation pattern and one transmit vector per sub-matrix to minimize the detection metric.Inverse mapping recovers antenna-index bits from the selected antenna pattern and frequency-index bits from the selected vectors.Source: C. GSFIM signal detection and… · p. 11
  • GSFIM performs better than MIMO-OFDM at moderate to high SNRs, while MIMO-OFDM performs better at low SNRs.The crossover is reported for both plotted comparisons under ML detection; the systems use different achieved rates in each comparison.Source: C. GSFIM signal detection and… · p. 12

IV. CONCLUSIONS

The paper concludes that indexing active antennas and subcarriers can carry information alongside conventional modulation symbols. It reports rate and BER advantages for GSIM and GSFIM in the stated comparisons, while identifying lower-complexity GSFIM detection as future work.

IV. CONCLUSIONS

  • GSIM can exceed spatial multiplexing in achievable rate under an analytically established condition, and the paper provides easy-to-compute upper and lower rate bounds..Source: §IV · pp. 12–13
  • GSIM is reported to achieve better BER than spatial multiplexing while using fewer transmit RF chains..Source: §IV · p. 13
  • GSFIM’s numerical results show higher rates than conventional MIMO-OFDM, with ML-detection BER results indicating potential gains at moderate to high SNRs.The authors identify low-complexity GSFIM detection as a direction for future work.Source: §IV · pp. 13–14
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