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50 Years of Permutation, Spatial and Index Modulation: From Classic RF to Visible Light Communications and Data Storage

Naoki Ishikawa, Shinya Sugiura, Lajos Hanzo

arXiv:1803.03939v1eess.SP

TL;DR

This treatise addresses overlooked historical and conceptual links between permutation modulation and spatial/index modulation. It surveys these links across communication and storage settings and reports that PM-aided schemes can retain reduced complexity while improving coding gain over conventional SMX in simulations.

  • Problem

    The survey addresses overlooked contributions on permutation and permutation-coded modulation in spatial and index modulation literature, including work dating to 1965.

  • Method

    The treatise synthesizes PM, SM, and related schemes using a three-dimensional spatial-temporal-frequency representation and interdisciplinary coverage of communications and data storage.

  • Results

    PM-aided coherent MIMO logarithmically increases transmission rate with antenna count, maintains reduced complexity, and achieves higher coding gain than conventional SMX across the transmission-rate region.

  • Takeaways & Limitations

    The survey positions PM as the historical and conceptual origin of current SM, PC, and SIM schemes.

  • Takeaways & Limitations

    High-frequency antenna switching remains a challenging task for single-RF spatial modulation transmitters.

Abstract

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In this treatise, we provide an interdisciplinary survey on spatial modulation (SM), where multiple-input multiple-output microwave and visible light, as well as single and multicarrier communications are considered. Specifically, we first review the permutation modulation (PM) concept, which was originally proposed by Slepian in 1965. The PM concept has been applied to a wide range of applications, including wired and wireless communications and data storage. By introducing a three-dimensional signal representation, which consists of spatial, temporal and frequency axes, the hybrid PM concept is shown to be equivalent to the recently proposed SM family. In contrast to other survey papers, this treatise aims for celebrating the hitherto overlooked studies, including papers and patents that date back to the 1960s, before the invention of SM. We also provide simulation results that demonstrate the pros and cons of PM-aided low-complexity schemes over conventional multiplexing schemes.

I. INTRODUCTION · II. PM PHILOSOPHY AND ITS RELATED FAMILY · A. Invention of PM in 1965

The treatise presents permutation modulation (PM) as the common philosophy underlying PM, parallel combinatory, spatial modulation, and subcarrier index modulation, tracing this family from Slepian’s 1965 proposal to modern communications. It surveys the family across microwave, visible-light, single-carrier, and multicarrier settings while relating PM codeword permutations to spatial, temporal, and frequency-domain signaling.

  • I. INTRODUCTION: Spatial modulation reduces transmitter and receiver complexity by selecting one antenna to convey additional information bits and limiting the number of transmitted data streams.Reduced-complexity detectors have been proposed for SM.
  • I. INTRODUCTION: SM performance advantages have been reported in STBCs, differential MIMO, millimeter-wave communications, visible light communications, and classic multicarrier communications.These studies compare SM with conventional MIMO schemes in specific scenarios.
  • II. PM PHILOSOPHY AND ITS RELATED FAMILY: The SM concept was first proposed in 2001, while OFDM later extended the concept as subcarrier index modulation.Theoretical analyses by Mesleh et al. helped drive a paradigm shift in coherent and non-coherent MIMO literature.
  • II. PM PHILOSOPHY AND ITS RELATED FAMILY: PM, parallel combinatory, SM, and SIM schemes share a permutation philosophy, with SM and SIM termed space-domain and frequency-domain index modulation, respectively.The treatise uses PM because it regards PM as the origin of current SM, PC, and SIM schemes.
  • I. INTRODUCTION: The survey celebrates overlooked contributions since Slepian coined permutation modulation in 1965 and adopts an interdisciplinary scope spanning microwave, visible-light, single-carrier, and multicarrier communications.It also details interactions among modulation constellations, spatial antenna domains, and frequency index domains.
  • A. Invention of PM in 1965: In 1965, Slepian defined PM codewords by permuting the order of a set of numbers, with repeated values allowed in the initial codeword.The number of possible codewords increases with factorial order, and PPM and PCM are subsumed by PM.
  • A. Invention of PM in 1965: The original PM framework links temporal and spatial signaling: PPM selects a single time index, whereas SSK selects a single antenna and acts as its spatial-domain counterpart.The cited codewords are identical for the illustrated PPM and SSK example.
  • A. Invention of PM in 1965: Permutation matrices generated from PM vectors provide the basis for SM-aided STBC and its differential counterpart.This establishes a direct connection between the PM construction and later spatial-modulation space-time schemes.

Hybrid PM Signaling:

Hybrid permutation modulation divides input bits between frequency selection and APSK symbol generation, while a generalized view groups several index- or permutation-based schemes into PM families. The factorial growth of PM codewords has also supported applications in physical data storage.

  • Hybrid PM Signaling:: Hybrid PM splits input bits into frequency-set selection and conventional APSK symbol generation on the selected frequencies.This differs from the original PM scheme, which outputs a permuted sequence.
  • Hybrid PM Signaling:: Generalized PM families convey additional bits by selecting spreading sequences, subcarriers, transmit antennas, LEDs, dispersion or permutation matrices, and transmission subarrays.These schemes are interpreted as sharing the basic permutation structure of the original and hybrid PM concepts.
  • Hybrid PM Signaling:: The combination matrix represents on-off states for arbitrary elements selected from M candidates, including transmit antennas and subcarrier indices.Its rows can correspond to SSK or PM codewords in example cases such as (M, P) = (4, 1) and (4, 2).
  • Hybrid PM Signaling:: PM codeword counts increase factorially, motivating applications in steganography, volume holographic storage, flash memory, and solid-state storage.The passage links this growth to increased physical data-storage capacity.

B. Invention of PC in 1991 … A. Coherent MIMO

The paper traces parallel combinatory modulation from its 1991 spread-spectrum origin through code-index and OFDM applications, then presents spatial modulation as a lower-complexity alternative to conventional MIMO. It also specifies the channel model and situates coherent MIMO through diversity, space-division access, capacity, and beamforming developments.

  • B. Invention of PC in 1991: In 1991, parallel combinatory modulation conveyed additional bits by selecting P out of M spread sequences, subsuming conventional M-ary spread spectrum when P = 1.The concept later supported PC-aided OFDM and related modulation schemes.
  • B. Invention of PC in 1991: PC-based research extended to phase-rotation PC-OFDM, secret communications, and code-index modulation, whose generalized form had analyzed BER and complexity.The conventional PC-aided OFDM principle is described as the same as that of generalized spatial modulation.
  • C. Invention of SM in 2001: Spatial modulation was proposed to reduce transmitter and receiver complexity without decreasing the spectrum efficiency of conventional systems.The motivation is the high hardware and energy cost of multiple RF chains, especially in massive MIMO.
  • C. Invention of SM in 2001: SM reduces transmitter-side RF-chain hardware and cost, while requiring the same number of receive RF chains as classic MIMO.A transmit RF chain typically includes converters, filters, synchronizers, and an amplifier.
  • C. Invention of SM in 2001: Full-RF-aided SM offers higher minimum Euclidean distance and lower computational complexity than spatial multiplexing, supporting open-loop large-scale MIMO.Similar advantages were observed in microwave and visible-light channels with strong line-of-sight components.
  • III. SYSTEM MODEL: The system model uses narrowband Rayleigh, Rician, or Jakes channels with M transmit antennas, N receive antennas, and 2^B legitimate space-time codewords.Transmission is represented by space-time codewords S(i) ∈ C^M×T over T symbol intervals, with additive noise and channel-dependent received SNR.
  • A. Coherent MIMO: Before modern MIMO, patented work addressed diversity reception in 1942 and space-division multiple access in 1973, followed by ergodic MIMO capacity analysis in 1987.The capacity analysis was inspired by a dually polarized SISO channel equivalent to a 2 × 2 MIMO channel.
  • A. Coherent MIMO: Beamforming improves received SNR, spectrum efficiency, and inter-user interference through a beamforming gain, including via conjugate beamforming.With many transmit antennas, the channel Gram matrix H^H H converges to a diagonal form on average.

Channel Model: … C. MIMO-Aided Millimeter-Wave Communications

The paper models wireless channels through multipath, fading, Doppler effects, and narrowband Rayleigh/Rician assumptions, then reviews detection and differential MIMO before addressing millimeter-wave capacity, propagation loss, beamforming, and low-rank channels. It highlights the tradeoffs between coherent detection’s accuracy and complexity, differential MIMO’s channel-estimation avoidance and 3 dB loss, and hybrid beamforming’s mitigation of millimeter-wave hardware and propagation challenges.

  • Channel Model:: Multipath creates delay spread, fading, and Doppler-induced time variation, making wireless channels unreliable; Rayleigh and Rician models are assumed for narrowband analysis.Delay spread becomes significantly distortive when it exceeds the reciprocal bandwidth, and Rayleigh fading models many uniformly distributed scatterers.
  • Detection:: ML detection achieves the lowest possible error rate but requires high complexity, while MAP detection reduces to likelihood maximization under equiprobable inputs.The hard detector searches over Nc = 2^B trials, so complexity grows exponentially with the input-bit segment length B.
  • B. Differential MIMO: Coherent MIMO estimates the channel matrix using pilot symbols, but noisy channel estimates reduce reliability and pilots lower the effective transmission rate.Pilot overhead increases linearly with the number of transmit antennas, and inaccurate estimates can produce an error floor in uncoded scenarios.
  • B. Differential MIMO: Differential space-time block coding avoids pilot insertion and explicit channel estimation by encoding successive unitary codewords through a data-carrying matrix.The receiver uses the previous received symbol as the coherent scheme’s pilot, although some schemes use non-unitary matrices to increase transmission rate.
  • B. Differential MIMO: 3 [dB] SNR loss results in differential MIMO because both consecutive received symbols contain additive noise, despite eliminating channel estimation.The differential BER curve is shifted by 3 [dB] relative to the coherent case.
  • C. MIMO-Aided Millimeter-Wave Communications: Millimeter-wave communications offer higher capacity through wider bandwidths at 30 to 300 [GHz], but their short wavelengths cause high propagation loss.Large antenna arrays and beamforming gain are needed to counteract path loss, while many RF circuits are complex, expensive, and power-hungry.
  • C. MIMO-Aided Millimeter-Wave Communications: Indoor millimeter-wave channels can be dominated by line-of-sight components, motivating specialized channel models and antenna-alignment techniques.The reported Rician K factor is between 8.34 and 12.04 [dB] in 60 [GHz] indoor communications scenarios.
  • C. MIMO-Aided Millimeter-Wave Communications: Dense antenna packing can produce low-rank indoor millimeter-wave channels, reducing MIMO gains; optimum antenna alignment increases the channel rank to rank(H) = min(M, N) in pure line-of-sight scenarios.An example uses 60 [GHz], λ = 0.5 [cm], d = λ/2 = 0.25 [cm], and 16-element transmitter and receiver arrays partitioned into four subarrays.

D. MIMO-Aided Visible Light Communications

MIMO-aided visible light communications exploit license-free, security-aware bandwidth on the order of several dozen MHz, despite practical LED and modulator limits. VLC requires positive real-valued signaling and amplitude-constrained transmission, with quasi-static positive real channel coefficients modeled using intensity modulation and direct detection.

  • Motivation: Practical VLC bandwidth is typically limited to the MHz range, although license-free and security-aware bandwidths of several dozen MHz remain attractive.Visible light spans 430–770 THz, but attainable bandwidth depends on modulator and LED specifications.
  • Signal Constraints: VLC symbols must be positive and real-valued, while OFDM requires frequency-domain Hermitian symmetry and remains vulnerable to LED nonlinearity and clipping distortion.The output signal must satisfy an amplitude constraint to avoid clipping distortions.
  • Channel Model: VLC channels have positive, real-valued, quasi-static coefficients in both outdoor and indoor scenarios, with white LEDs and photodetectors supporting channel analysis.CMOS imaging sensors have also been considered for VLC receivers.
  • Channel Model: The simulations use a simplified path-loss VLC channel with intensity modulation and direct detection, replacing the general MIMO channel matrix with a VLC-specific matrix.The received model incorporates photodetector response, area, source-to-detector distance, incidence angle, and receiver field-of-view semi-angle.

E. Multicarrier Communications · IV. APPLICATIONS OF THE PM CONCEPT · A. PM-Based Coherent MIMO

The paper places PM-based coherent MIMO within the broader PM application landscape, alongside established multicarrier communication, and develops GSTSK as a flexible framework unifying several MIMO schemes. It also describes activation-pattern design, extensions such as MBM and ASTSK, and key implementation limitations of OFDM.

  • E. Multicarrier Communications: OFDM multiplexes symbols over orthogonal subcarriers and has supported WLAN, cellular, and digital television standards.It exploits limited bandwidth through simultaneous frequency-domain transmission.
  • E. Multicarrier Communications: OFDM faces out-of-band radiation and high PAPR, motivating spectrum-edge null symbols and single-carrier frequency-domain equalization.High PAPR requires a high dynamic range amplifier per transmit antenna.
  • E. Multicarrier Communications: Wideband multipath causes intersymbol interference, which OFDM mitigates by adding a guard interval that converts linear convolution into circular convolution.The assumed guard interval is longer than the maximum channel delay.
  • A. PM-Based Coherent MIMO: PM-based coherent MIMO originated with SSK in 2001, and its codewords use zero and non-zero symbols following an on-off structure.The surveyed schemes are illustrated for M = 4 transmit antennas.
  • A. PM-Based Coherent MIMO: GSTSK represents SM, SSK, GSM, and BLAST through flexible dispersion matrices designed offline for comprehensive space-time block-code analysis.The framework is denoted GSTSK(M, N, T, Q, P).
  • A. PM-Based Coherent MIMO: GSTSK partitions B = B1 + B2 input bits between DM-pattern selection and P APSK symbols, with Na = 2⌊log2(Q choose P)⌋ activation patterns.The selected DM indices are represented by sorted activation vectors, mapped using NBC or LUT methods.
  • A. PM-Based Coherent MIMO: LUT activation patterns affect achievable performance, with coding gain maximized when each index is selected with equal probability.The framework includes SM, SSK, GSM, GSSK, and ASTSK as conventional or extended schemes.
  • A. PM-Based Coherent MIMO: MBM conveys data by changing radio propagation and offers capacity growth with scattering paths, whereas ASTSK increases block length to T ≥ 2 and achieves at most diversity order T.SSK transmission rate is limited by the number of transmit antennas, while MBM uses RF mirrors.

B. PM-Based Differential MIMO · 1) Differential Spatial Modulation:

PM-based differential MIMO replaces coherent channel estimation with differential encoding and noncoherent detection. Within this family, DSM uses unitary, sparse space-time codewords to flexibly trade transmission rate against diversity order while encompassing related differential schemes.

  • B. PM-Based Differential MIMO: Differential PM schemes dispense with the channel-estimation overhead required by coherent MIMO counterparts.The section reviews differentially encoded and noncoherently detected versions of coherent PM schemes.
  • 1) Differential Spatial Modulation:: DSM maps B input bits to a transmission-indexed space-time matrix using Q prepared dispersion matrices, each with one unit-magnitude nonzero element per row and column.The nonzero element is denoted aq,m, with |aq,m| = 1 enforcing the unitary constraint.
  • 1) Differential Spatial Modulation:: The first B1 = log2(Q) bits select a dispersion matrix, while the remaining B2 = log2(L1 · L2 · · · · · L ¯ M) bits select PSK symbols.The selected symbols are packed into an M×1 vector, with repetition determined by the number of embedded symbols.
  • 1) Differential Spatial Modulation:: DSM varies the number of embedded PSK symbols ¯ M to trade transmission rate against diversity order.Embedding ¯ M = M symbols maximizes transmission rate but gives maximum diversity order M/ ¯ M = 1.
  • 1) Differential Spatial Modulation:: D = 2 is achieved for M = 4 and ¯ M = 2 when two BPSK symbols are each spread over two successive transmissions.This configuration embeds the symbol pair into a space-time codeword.
  • 1) Differential Spatial Modulation:: Increasing transmission rate R reduces the maximum diversity order D in DSM configurations with M = 2, 4, 8, and 16 transmit antennas.The evaluated number of embedded symbols ranges from ¯ M = 2^0 to 2^log2(M).
  • 1) Differential Spatial Modulation:: DSM subsumes DSTSK and BDSM, is equivalent to DSTSK when ¯ M = 1, and generalizes BDSM by allowing complex-valued nonzero elements.Unlike BDSM, DSM can support Q > 2⌊log2(M!)⌋ and avoid the fixed diversity order D = 1 limitation.

2) Non-Coherent Generalized Spatial Modulation:

NCGSM is the differential counterpart of GSTSK, retaining its encoding principle with real-valued PAM symbols and Hermitian dispersion matrices. Its Cayley-transform construction yields a unitary space-time code and subsumes several differential space-time coding schemes, including differential LDC when P = Q.

  • Construction: NCGSM is the differential counterpart of the GSTSK scheme, using essentially the same encoding principle with two specified exceptions.The exceptions concern the embedded symbols and dispersion matrices.
  • Construction: The embedded symbols s1, · · · , sP are real-valued, as in pulse amplitude modulation.This imposes PAM-style real-valued signaling on the embedded symbols.
  • Construction: The dispersion matrices A1, · · · , AQ ∈CM×M are Hermitian, satisfying Aq = AHq.Hermitian structure ensures the relevant summed codeword matrix is Hermitian when combined with real-valued PAM symbols.
  • Cayley transform: The NCGSM codeword construction produces a Hermitian matrix, which is mapped to a unitary matrix through the Cayley transform.The transform maps the skewed-Hermitian counterpart j ˜X(i) to the unitary matrix X(i).
  • Relations to existing schemes: With P = Q, NCGSM is equivalent to differential LDC and achieves a high transmission rate using multiplexed PAM symbols.The architecture also subsumes conventional differential space-time block-coding schemes relying on the Cayley transform.

C. PM-Based MIMO-MWC · D. PM-Based MIMO-VLC

The PM-based MIMO-MWC section traces spatial-index modulation from RF switching and SSK/GSM toward lower-complexity transmission, while the MIMO-VLC section applies PM to low-rank, highly correlated optical channels. In VLC, OSM and power-imbalanced OSM address channel limitations, with benefits shown in uncoded BER and low-SNR coded mutual information.

  • C. PM-Based MIMO-MWC: Hybrid beamforming reduces transmitter RF-chain requirements in MIMO-MWCs but can remain complex as transmission rate and independent-stream count increase.The section motivates PM-based schemes as a response to practical resource constraints.
  • C. PM-Based MIMO-MWC: RF-switching modulation generates conventional I/Q symbols by changing electromagnetic boundary conditions, inspiring PM-like switching of appropriate MIMO-MWC subarrays.The RF-switching concept was extended from earlier work to MIMO-MWCs.
  • C. PM-Based MIMO-MWC: SSK was first directly applied to MIMO-MWCs by Liu and Springer, then extended to GSM and combined with analog phase shifters.The section identifies as seminal research in SSK-based MWC schemes.
  • C. PM-Based MIMO-MWC: GSM-based MIMO-MWC conveys additional information bits by selecting among beam patterns determined by the positions of non-zero codeword elements.The cited examples include x = [s1, s2, 0, 0]T and [s1, 0, s2, 0]T.
  • D. PM-Based MIMO-VLC: In PD-aided MIMO-VLC, strong LoS elements typically produce low-rank channel matrices that erode both MIMO diversity and spatial-multiplexing gains.PM was introduced to combat this limitation, beginning with Mesleh et al.’s optical spatial modulation scheme.
  • D. PM-Based MIMO-VLC: OSM was difficult to improve in highly correlated realistic LoS channels, motivating power-imbalanced OSM with real-valued precoding and careful power allocation.The power-allocation method mitigated channel correlations associated with the light sources.
  • D. PM-Based MIMO-VLC: Most OSM studies demonstrated performance advantages through uncoded BER, while coded analyses found higher constrained mutual information in the low SNR region.These results span the PM-based MIMO-VLC family’s BER and information-theoretic evaluations.
  • D. PM-Based MIMO-VLC: The conventional PAM-RC VLC transmitter emits the same PAM symbol from every light source, and PAM-RC was shown capable of outperforming OSTBC in free-space optical wireless communications.The section introduces PAM-RC before presenting representative OSM schemes.

1) Equal-Power OSM Scheme: · 2) Power-Imbalanced OSM Scheme: · E. PM-Based Multicarrier Communications

The merged sections develop OSM from equal-power light-source selection with PAM, through power-imbalanced transmission tailored to VLC channels, to PM-based multicarrier schemes that activate selected frequencies or subcarriers. Across these schemes, index selection conveys additional bits while maintaining structured, sparse signal representations and low-complexity processing.

  • 1) Equal-Power OSM Scheme:: Equal-power OSM partitions B=B1+B2 input bits into B1=log2(M) light-source-selection bits and B2=log2(L) L-PAM symbol bits.A single light source q is selected from M lights, while the remaining bits map to an L-PAM symbol s.
  • 1) Equal-Power OSM Scheme:: Equal-power OSM generates a time-domain vector with one non-zero element, encoding information through the selected transmit light source and PAM symbol.The original OSM formulation constrains the PAM constellation size to one, s=1.
  • 2) Power-Imbalanced OSM Scheme:: Power-imbalanced OSM communicates over VLC-specific LoS channels by multiplying the OSM symbols by per-light power-allocation factors a1,…,aM.Its encoding principle is basically the same as the GSM transmitter, with L-PAM symbols subsequently weighted by the power-allocation factors.
  • 2) Power-Imbalanced OSM Scheme:: The power-allocation factors satisfy Σm=1^M am=M and can be designed to maximize constrained MI or maximum achievable rate.For the reported (M,N)=(4,4) setup, β was set to 1, 3, and 4 dB.
  • E. PM-Based Multicarrier Communications: PM-based multicarrier communications began with simultaneous activation of multiple frequencies, followed by FSK–PSK combinations and PC-based spreading sequences carrying additional bits.The cited chronology places these developments in 1967, 1986, and 1989, respectively.
  • E. PM-Based Multicarrier Communications: PM was exported to OFDM in 1999 by activating only a fraction of subcarriers, using an encoding principle later associated with GSM and termed subcarrier index modulation.The SIM encoding principle matches GSM, although SIM predates GSM by eleven years.
  • E. PM-Based Multicarrier Communications: SIM(M,P) activates P subcarriers out of M, creating sparse frequency-domain symbols whose activation patterns encode information.The frequency-domain symbols are concatenated, transformed by IFFT, and protected with a cyclic prefix against inter-channel interference.

V. PERFORMANCE METRICS … Constrained AMI:

The paper evaluates general MIMO communication systems using average mutual information, minimum Euclidean distance, and receiver decoding complexity, distinguishing unconstrained and constrained AMI. Constrained AMI assumes finitely many codewords, equal codeword probabilities, and is bounded by the code rate while estimating the turbo-cliff SNR.

  • V. PERFORMANCE METRICS: The performance-metric framework for general MIMO systems comprises AMI, MED, and receiver decoding complexity.These metrics are presented in Sections V-A, V-B, and V-C, respectively.
  • A. Average Mutual Information (AMI): Mutual information represents the maximum number of information bits conveyed per channel use, motivating unconstrained and constrained AMI metrics.The two metrics differ in their input-symbol assumptions.
  • Unconstrained AMI:: Unconstrained AMI assumes continuous Gaussian-distributed input symbols.Its general MIMO expression is introduced through the system model of Eq. (8).
  • Unconstrained AMI:: Unconstrained AMI depends only on the channel matrix H and received SNR ρ after averaging over random channel matrices.The metric uses eigenvalues of the Hermitian matrix Q and assumes complex-valued Gaussian signals sampled at discrete intervals.
  • Constrained AMI:: Constrained AMI is an effective upper bound based on a finite set of Nc = 2^B space-time codewords associated with B input bits.The codewords S(1), · · · , S(Nc) belong to C^M×T.
  • Constrained AMI:: The constrained-AMI maximization is attained when all codewords are selected with equal probability 1/Nc.This assumption simplifies the constrained-AMI expression before evaluating the likelihood terms under Y = HS(f) + V.
  • Constrained AMI:: The constrained AMI satisfies ID ≤ B/T = R [bits/symbol] and applies directly to SISO and spatial-modulation codewords.It estimates the turbo-cliff SNR at which powerful channel coding drives BER to an infinitesimal value.

B. Reliability · C. Complexity

The reliability analysis uses pairwise-error probability and BER bounds to characterize uncoded performance, with rank and determinant criteria governing diversity and coding gains. Complexity is measured by receiver real-valued multiplications, under which reduced-stream PM schemes are less complex than conventional multiplexing and are compared with multiple coherent and non-coherent schemes.

  • B. Reliability: Pairwise-error probability is the central uncoded reliability metric, evaluating the probability that transmitted symbol S(f) is decoded as S(g).Goldsmith’s analytical framework provides tight error-probability bounds, and the Q-function appears in the PEP formulation.
  • B. Reliability: Rank and determinant criteria maximize coding gain while preserving maximum diversity order in uncoded scenarios.Diversity order is the high-SNR error-probability slope, while higher coding gain corresponds to lower achievable BER.
  • B. Reliability: At high SNRs, diversity order is D = m′N, where m′ is the minimum rank of D, and coding gain is determined by the minimum Euclidean distance between codewords.The coding gain is expressed through Qm′, with µm denoting the mth eigenvalue of D.
  • C. Complexity: Receiver complexity is approximated by the number of real-valued multiplications, which is nearly the receiver’s multiplier count because complex additions are negligible.The treatise uses this measure to characterize PM schemes and explains that reduced data streams yield lower complexity than conventional multiplexing.
  • C. Complexity: For RF SM, ML detection tests 2^R candidates; each trial requires 4N real-valued multiplications for H · S and 2N for the Frobenius-norm calculation.Thus, the SM complexity is 2^R · 6N, with a lower bound of Ω(2^RN).
  • C. Complexity: Table IX summarizes ML complexities divided by codeword time slots T alongside diversity and transmission rates for coherent and non-coherent schemes.The compared schemes include single-stream APSK, SM, GSM, rectangular ASTSK, BLAST, square GSTSK, differential APSK, rectangular DSM, square DSM, and square NCGSM.

VI. PERFORMANCE COMPARISONS · A. PM-Based Coherent MIMO

The performance comparisons evaluate PM-based coherent MIMO schemes against conventional multiplexing using fixed total transmit power. Results cover constrained AMI, MED, BER relationships, and decoding complexity across antenna numbers and transmission rates.

  • VI. PERFORMANCE COMPARISONS: All comparisons fixed the total transmit power to unity across PM-based and conventional multiplexing schemes.The study considered coherent and differential MIMO, MIMO-MWC, MIMO-VLC, and multicarrier systems.
  • A. PM-Based Coherent MIMO: Constrained AMI comparisons examined SM, ASTSK, and single-stream QAM for transmit-antenna counts ranging from M = 2 to 1024.The number of receive antennas was N = 1, and unconstrained AMI curves were also plotted.
  • A. PM-Based Coherent MIMO: For M = 4, SM, ASTSK, and single-stream schemes exhibited similar AMI, while SM’s gain over single-stream increased with more transmit antennas.ASTSK could not be simulated for M = 64 and 1024 because of excessive complexity.
  • A. PM-Based Coherent MIMO: GSM achieved MED gains across the full transmission-rate range 2 ≤ R ≤ 32 [bits/symbol] against BLAST under constant transmission power.The comparison varied M from 2 to 32 and constellation size L from 2 to 1024; MED comparison is valid only at the same transmission rate.
  • A. PM-Based Coherent MIMO: At SNR = 15 [dB], constrained AMI and BER exhibited a relationship for randomly generated DMs in the GSTSK-based evaluation.A total of 7187 random DMs were generated, and the constrained AMI calculation related to MED through the high-SNR approximation.
  • A. PM-Based Coherent MIMO: ML decoding complexity was compared for BLAST, square STBC, SM, GSM, ASTSK, and single-stream APSK as M varied from 2 to 1024.The associated transmission rates ranged from R = 3 to 12 [bits/symbol], with N = 1 receive antenna.

B. PM-Based Differential MIMO … E. PM-Based Multicarrier Communications

The paper evaluates PM-based differential MIMO, MIMO millimeter-wave communications, MIMO visible-light communications, and multicarrier schemes using BER, AMI, MI, MED, and coding-related comparisons. Results identify Doppler-induced error floors, tilt sensitivity, constellation bias, and advantages of selected-subcarrier modulation in coded scenarios.

  • B. PM-Based Differential MIMO: BDSM and UDSM exhibit error floors for normalized Doppler frequencies FdTs ≥7.5 × 10−3, while UDSM retains performance advantages as Doppler frequency increases.The comparison uses BPSK-aided schemes with (M, Q) = (2, 2), eight scatterers, and transmission rate R = 1.5 [bits/symbol].
  • B. PM-Based Differential MIMO: Both UDSM and single-RF NCGSM gain diversity order as transmit antennas increase, while their performance gap also widens with increasing M.The comparison constrains the receiver to N = 1 and uses reduced-RF-chain transmitters.
  • C. PM-Based MIMO-MWC: Receiver tilt reduces constrained AMI for all schemes from θ = 0° to 30°, while BLAST and GSM exceed 4.0 [bits/symbol] for 0°≤θ ≤18°.The constrained AMI is evaluated at SNR = −5 [dB], with steered cases using accurate θAoD and θAoA adjustment.
  • D. PM-Based MIMO-VLC: Single-parameter PI-OSM constellations become biased at β = 3 and 4 [dB] because Eq. (39) exponentially increases power with β.The comparison considers equal-power OSM, PI-OSM with β = 1, 3, 4 [dB], and unconstrained PI-OSM for M = 4 and L = 2.
  • D. PM-Based MIMO-VLC: The unconstrained PI-OSM scheme is expected to outperform PAM-RC in channel-coded scenarios.The PM-based MIMO-VLC evaluation considers MI and BER, using MI because the channel coefficients are static.
  • E. PM-Based Multicarrier Communications: Reciprocal MED correlates with BER in QPSK-aided SIM, making MED useful for predicting uncoded BER across subcarrier selections.The study varies M = 4, 8, 16 and selected subcarriers P = 1 to M, with simulated BER at received SNR 30 [dB].
  • E. PM-Based Multicarrier Communications: Constrained SIM exceeds OFDM by 1.2 and 2.1 [dB] at coding rates 1/2 and 3/4, respectively, although unconstrained OFDM has higher AMI throughout the SNR region.The comparison uses BPSK-aided OFDM and 4−QAM-aided SIM(4,1) in Rayleigh fading channels with turbo coding.

VII. CONCLUSIONS … GLOSSARY

The treatise concludes that permutation modulation offers flexible, low-complexity designs across communications and storage, while exposing tradeoffs among rate, reliability, capacity, and sparsity. It also identifies cross-domain research opportunities, differential-MIMO scalability limits, and the glossary’s key terminology.

  • A. Summary and Design Guidelines: PM surveys coherent and differential MIMO, MIMO-MWC, MIMO-VLC, multicarrier RF communications, and data storage through a shared permutation-based structure.The concept originated with Slepian in 1965 and maps input bits by selecting permutations from arbitrary on/off-state sets.
  • A. Summary and Design Guidelines: Selecting P elements from a Q-sized set, with 1 ≤ P ≤ Q, provides design flexibility affecting transmission rate, reliability, AMI, and implementation complexity.The simulations and theoretical analyses are summarized as design guidelines for PM-aided systems.
  • A. Summary and Design Guidelines: PM-aided coherent MIMO increases transmission rate logarithmically with transmit-antenna count, maintains reduced complexity, and achieves higher coding gain than conventional SMX across the full rate region.Fewer data streams also enable low-complexity receiver detection.
  • 1) Cross-Pollination of SM and Data Storage Research:: SM and PM research can cross-pollinate by transferring low-complexity SM detectors to PM storage and low-latency PM storage algorithms to SM-aided MIMO.The proposed interaction targets low-latency reading and writing across both domains.
  • 2) Large-Scale High-Rate PM-Based Differential MIMO:: Sparse square matrices constrain PM-based differential MIMO because M = T limits space-time codeword design and performs poorly in large-scale, high-rate scenarios.A nonsquare-matrix scheme in is identified as the sole exception at the time of writing and achieves competitive performance for M = 1024 antennas.
  • 3) Striking the Tradeoff between Sparsity and Capacity:: PM-based OFDM transmits P independent streams over M subcarriers, so its unconstrained AMI is below classic OFDM, while constrained AMI can exceed OFDM only in specific scenarios.The achievable AMI gain may vanish at high rates, although sparse codewords reduce receiver computational complexity.
  • GLOSSARY: The glossary defines terminology spanning beamforming, modulation, coding, channel information, and visible-light communications, including VLC as Visible Light Communication.It also lists STBC as Space-Time Block Code and STSK as Space-Time Shift Keying.
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