Multi-Dimensional OFDM

• Multi-Dimensional OFDM is a versatile transmission scheme that distributes signaling across multiple dimensions (frequency, space, time, and Doppler) to enhance performance.
• It integrates techniques such as index modulation, eigen-domain signaling, and Doppler coding to optimize the rate–reliability–complexity trade-off in various MIMO systems.
• The design advances offer improved error performance and flexibility, addressing challenges like CSI requirements and detection complexity in non-stationary channels.

Multi-Dimensional OFDM (MD-OFDM) denotes a class of OFDM-derived transmission frameworks in which signaling, multiplexing, or orthogonality is intentionally distributed across more than one domain rather than being confined to the classical per-subcarrier symbol domain. In the cited literature, the term covers frequency–space index modulation, constellation-mode indexing, joint space-frequency indexing, Doppler-domain multiplexing, and channel-adaptive eigen-domain modulation; it is also used as the explicit name of a per-subcarrier transmit-antenna-selection MIMO-OFDM variant. The common thread is the use of additional bit-bearing or interference-separating dimensions—such as active-subcarrier indices, antenna indices, constellation modes, slow-time Doppler shifts, or multidimensional eigenwaves—to alter the rate–reliability–complexity trade-off relative to conventional OFDM and MIMO-OFDM (Basar, 2015, Yarkin et al., 2020, Dogukan et al., 2020, Velez et al., 2021, Zou et al., 2022, Lang et al., 2023, Gulia, 18 Jul 2025).

1. Terminological scope and canonical dimensions

The term MD-OFDM is not attached to a single canonical waveform. In the relevant papers, it functions both as a broad design perspective and as the proper name of a particular antenna-selection scheme. As a broad perspective, MD-OFDM refers to OFDM systems that exploit multiple axes—frequency, space, time, mode, Doppler, delay-Doppler, or channel eigenstructure—as information-bearing or orthogonality-bearing resources. As a specific waveform name, it refers to a MIMO-OFDM architecture in which only one transmit antenna is active per subcarrier (Gulia, 18 Jul 2025).

This heterogeneity is visible across representative formulations. MIMO-OFDM-IM embeds information in active subcarrier indices on each transmit antenna and uses spatial multiplexing across antennas, yielding a frequency–space multi-dimensional OFDM architecture (Basar, 2015). Q-MM-OFDM-IM and SuM-OFDM-IM treat the constellation mode as an additional dimension, jointly with conventional symbol modulation and, in SuM-OFDM-IM, with subcarrier activation patterns (Yarkin et al., 2020, Dogukan et al., 2020). PT-GSFIM overlays space- and frequency-domain indexing on OFDM for downlink MU-MIMO and supplements it with precoding and signal space diversity (Velez et al., 2021). MEM treats MD-OFDM as modulation in a channel-dependent eigen-domain, where the subcarriers are jointly orthogonal eigenwaves extracted from a multidimensional kernel (Zou et al., 2022). DDM uses the Doppler or slow-time axis as an additional orthogonal resource for MIMO OFDM joint sensing and communications (Lang et al., 2023).

Representative scheme	Exploited dimensions	Defining mechanism
MIMO-OFDM-IM	Frequency and space	Active-subcarrier indices per antenna with spatial multiplexing
Q-MM-OFDM-IM / SuM-OFDM-IM	Mode and frequency	Constellation-mode indexing, MAP/SAP selection, repeated symbols
PT-GSFIM	Space and frequency	Active antennas and active subcarriers with MU precoding
MEM	Space, time-frequency, delay-Doppler	Channel eigenwaves from HOGMT
DDM	Doppler, time, frequency, space	Antenna-specific slow-time phase ramps
MD-OFDM (TAS)	Space and frequency	One active transmit antenna per subcarrier

A common misconception is that MD-OFDM is synonymous with index modulation alone. The literature does not support that restriction: MEM and DDM are explicitly positioned as MD-OFDM without relying on classical active-index signaling (Zou et al., 2022, Lang et al., 2023).

2. Index-modulated MD-OFDM in frequency, space, and mode

A major MD-OFDM lineage extends OFDM through index modulation. In OFDM-IM, only $k$ out of $n$ subcarriers in a subblock are activated according to a bit-driven index selection, and the remaining $n-k$ subcarriers are set to zero. Each subblock therefore carries index bits, which select the active-subcarrier pattern, and data bits, which select the constellation symbols on the active positions. MIMO-OFDM-IM extends this idea to the spatial dimension by combining OFDM-IM with V-BLAST-style MIMO. Each transmit antenna independently generates and transmits its own OFDM-IM frame; there is no joint index selection across antennas. With $N_F=Gn$, the $N_F$ subcarriers are partitioned into $G$ subblocks of length $n$, each subblock contains exactly $k$ active entries, and the per-frame spectral efficiency is

$R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$

with a useful per-subcarrier expression

$$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$

The paper’s matched-rate comparisons choose $n$0 such that $n$1 equals the classical OFDM rate $n$2, thereby keeping spectral efficiency constant while improving BER (Basar, 2015).

Q-MM-OFDM-IM generalizes the signal space by introducing $n$3 disjoint $n$4-ary constellations $n$5 and coding the mode indices across an $n$6-subcarrier block with a $n$7-ary MDS code satisfying

$n$8

The resulting code has parameters $n$9, the number of index codewords is $n-k$0, and the spectral efficiency is

$n-k$1

This construction enlarges the local alphabet to the disjoint union $n-k$2 and the global codebook to $n-k$3 codewords, while retaining low-complexity subcarrier-wise detection (Yarkin et al., 2020).

SuM-OFDM-IM adopts a different multidimensional structure. Rather than leaving some subcarriers null, it activates all $n-k$4 subcarriers in a subblock by jointly selecting a mode activation pattern (MAP) and a subcarrier activation pattern (SAP): $n-k$5 subcarriers use one mode and the complement uses a second mode. Conventional symbols are repetition coded over subcarrier pairs, which yields at least second-order diversity. Its per-subblock payload is

$n-k$6

The paper also distinguishes a separate-selection variant, but joint MAP–SAP selection provides higher spectral efficiency (Dogukan et al., 2020).

PT-GSFIM extends multidimensional indexing to MU-MIMO downlink. Bits select active antennas in an $n-k$7-length GSM vector and active subcarriers in an $n-k$8-subcarrier block, while the active antenna–subcarrier resource elements carry $n-k$9-ary APM symbols. With LUT-based independent mapping, the total bits per PT-GSFIM symbol for user $N_F=Gn$0 are

$N_F=Gn$1

This makes PT-GSFIM an explicitly multidimensional index modulation over space, frequency, and symbol domains (Velez et al., 2021).

3. Doppler-domain and eigen-domain formulations

A second MD-OFDM lineage departs from fixed index sets and instead redesigns the basis itself. MEM addresses non-stationary channels in which OFDM and OTFS lose orthogonality because fixed Fourier or SFT bases no longer diagonalize the channel. The paper models the channel as a multidomain linear operator with kernel $N_F=Gn$2 and invokes Higher Order Mercer’s Theorem to obtain a decomposition

$N_F=Gn$3

where $N_F=Gn$4 and $N_F=Gn$5 are jointly orthonormal eigenfunctions over the receive and transmit domains. Transmit symbols are mapped onto transmit eigenwaves,

$N_F=Gn$6

and, after projection onto receive eigenwaves, the received coefficients satisfy

$N_F=Gn$7

The cross terms vanish by orthonormality, so symbol-wise independence is achieved in the eigenwave domain. In this sense, MEM is “OFDM in the eigen-domain” and reduces to conventional OFDM for stationary separable channels and to OTFS for WSSUS channels with static delay-Doppler statistics (Zou et al., 2022).

DDM exploits a different multidimensional resource: Doppler or slow time. In MIMO OFDM joint sensing and communications, each transmit antenna radiates the same subcarrier symbols but with an antenna-specific phase ramp across the OFDM symbol index,

$N_F=Gn$8

Choosing $N_F=Gn$9 with distinct integers $N_F$0 shifts each antenna’s return by exactly $N_F$1 Doppler bins in the range–Doppler map. DDM therefore uses Doppler as a controllable separation dimension while keeping all subcarriers active for all antennas. Its principal communications consequence is a heavily time-varying effective channel frequency response,

$N_F$2

which must be explicitly estimated and tracked (Lang et al., 2023).

These two formulations show that MD-OFDM need not be limited to additional bit-bearing indices. It can also denote multidomain orthogonalization: adaptive eigenwaves in MEM, or deterministic slow-time phase coding in DDM.

4. Detection, equalization, and computational structure

The receiver architecture in MD-OFDM is determined by which dimensions are activated. In MIMO-OFDM-IM, joint ML detection over all active-index and symbol hypotheses has complexity that grows roughly as $N_F$3 per subblock, so the paper proposes a low-complexity detector consisting of per-tone MMSE filtering, LLR-based activity detection, and ML symbol decisions on the detected active tones. The per-subcarrier arithmetic is $N_F$4 complex multiplications, compared with $N_F$5 for classical MIMO-OFDM with MMSE detection (Basar, 2015).

Q-MM-OFDM-IM also avoids exhaustive ML. Its optimal blockwise ML detector has complexity $N_F$6, whereas the proposed LC-ML receiver detects independently on the $N_F$7 strongest subcarriers over the union constellation, recovers the final mode index by the MDS parity constraint, and attains complexity order $N_F$8 (Yarkin et al., 2020). SuM-OFDM-IM similarly replaces exhaustive search over $N_F$9 subblock realizations with an LLR-based reduced-complexity ML detector. The paper gives the reduced complexity as

$G$0

and reports that this detector achieves the same error performance as the ML detector in the reported scenarios (Dogukan et al., 2020).

PT-GSFIM combines index detection with MU precoding. Block diagonalization is applied per subcarrier so that the downlink reduces to equivalent single-user channels, after which three detectors are proposed: OB-MMSE, sMMP, and ADMM. The ADMM formulation solves a constrained ML problem through variable splitting and alternating projections onto the APM, GSM-support, and active-subcarrier constraint sets, while sMMP uses a greedy parallel pursuit and OB-MMSE ranks joint space-frequency candidates by reliability (Velez et al., 2021).

MEM produces the most diagonal receiver structure once the channel decomposition is known. After HOGMT-based extraction of transmit and receive eigenwaves, the receiver performs matched filtering and per-eigenwave equalization, with matched filtering and equalization scaling as $G$1 per block for $G$2 eigenwaves. The main computational burden is moved to kernel or tensor formation and HOSVD/Tucker-type factorization (Zou et al., 2022). DDM lies between these extremes: transmit processing is lightweight, but the receiver requires preamble-based synchronization, ECIR/ECFR estimation via BLUE, pilot-aided common phase error tracking, and bundle-wise LMMSE combining across four repeated OFDM symbols (Lang et al., 2023). In the 2025 MD-OFDM architecture, by contrast, detection is scalar because only one transmit antenna is active per subcarrier; the essential transmitter-side operation is per-subcarrier antenna selection,

$G$3

followed by scalar equalization at the receiver (Gulia, 18 Jul 2025).

5. Performance characteristics and operating trade-offs

The reported gains of MD-OFDM are scenario-dependent and usually arise from a specific trade-off rather than from a universally dominant design. For MIMO-OFDM-IM, simulations with $G$4, $G$5, $G$6, and MMSE detection show substantially better BER than classical V-BLAST MIMO-OFDM at matched spectral efficiency. In the BPSK case with $G$7 and $G$8, the $G$9 MIMO-OFDM-IM system achieves approximately $n$0 dB SNR gain over classical $n$1 MIMO-OFDM at $n$2, and gains persist for $n$3 and $n$4 configurations (Basar, 2015).

Q-MM-OFDM-IM emphasizes spectral efficiency and low-complexity detection. Its index-only mode ($n$5) achieves diversity order two, because the index codewords have minimum Hamming distance two. For LC-ML detection, the reported SNR penalties relative to ML at $n$6 are about $n$7 dB for Q-MM $n$8, $n$9 dB for Q-MM $k$0, and $k$1 dB for Q-MM $k$2. In uncoded QAM or PSK comparisons, Q-MM $k$3 surpasses OFDM-OFSPM and MM-OFDM-IM by about $k$4 dB and OFDM or OFDM-IM by about $k$5 dB at $k$6 (Yarkin et al., 2020).

SuM-OFDM-IM trades higher indexing complexity for both rate and diversity. For $k$7, $k$8, and varying $k$9, the paper reports approximately $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$0 spectral-efficiency improvement over comparable alternatives in several scenarios, and the SDR prototype shows that SuM-OFDM-IM outperforms OFDM and OFDM-IM in BER while maintaining that higher SE (Dogukan et al., 2020). PT-GSFIM’s main gain comes from combining space-frequency indexing with signal space diversity: for QPSK, CRM yields approximately $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$1 dB SNR gain at $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$2 over the no-CRM counterpart, and relative to conventional BD MU-MIMO the reported gains at $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$3 range from approximately $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$4 dB to approximately $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$5 dB depending on $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$6 (Velez et al., 2021).

MEM and DDM address non-stationarity and joint sensing/communications rather than classical BER alone. In MEM, Channel A and Channel B simulations show that MEM remains robust when OTFS suffers extensive IDI; MEM achieves the highest throughput because it uses no zero padding, while ZP-MEM lowers BER by pruning weak eigenwaves (Zou et al., 2022). In DDM, coded transmission with perfect channel knowledge and synchronization gives approximately $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$7 dB $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$8 gain over ESI, NeqDySI, and SISO at a given BER, but only by using $R=\frac{mT}{N_F+C_p}\ \text{bits/s/Hz},$9 repetition, which reduces raw throughput by a factor of $$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$0 (Lang et al., 2023). The antenna-selection MD-OFDM of 2025 reports absolute total powers of $$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$1 mW for MD-OFDM $$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$2 and $$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$3 mW for MMSE MIMO-OFDM $$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$4, alongside lower PAPR and better BER for MD-OFDM, but with the explicit caveat that MMSE can still achieve higher peak energy efficiency because its ideal spectral efficiency is $$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$5 times larger (Gulia, 18 Jul 2025).

6. Design tensions, limitations, and research directions

Across these formulations, MD-OFDM repeatedly exchanges one system resource for another. Index-modulated variants often trade nulls, sparse supports, or restricted codebooks for improved BER or lower detection complexity. The 2025 antenna-selection MD-OFDM trades spatial multiplexing for lower PAPR, lower absolute power, and scalar equalization. DDM trades raw throughput and unambiguous velocity for Doppler-domain separability and slow-time diversity. MEM trades fixed-basis simplicity for the need to estimate and decompose a multidimensional kernel (Basar, 2015, Zou et al., 2022, Lang et al., 2023, Gulia, 18 Jul 2025).

CSI requirements are a recurrent limitation. MIMO-OFDM-IM assumes perfect CSI at the receiver and standard MIMO-OFDM pilot design (Basar, 2015). PT-GSFIM requires accurate per-subcarrier CSIT for block diagonalization, and its hardware cost motivates hybrid precoding approximations (Velez et al., 2021). MEM explicitly requires CSI at both transmitter and receiver, with eigenwaves updated per block or per stationarity interval (Zou et al., 2022). The antenna-selection MD-OFDM assumes CSIT via feedback; a plausible implication is that per-subcarrier feedback overhead becomes substantial unless grouped subcarrier selection or differential updates are used, and the paper identifies grouped subcarriers and robust selection under CSI uncertainty as future extensions (Gulia, 18 Jul 2025).

Analytical completeness is also uneven. MIMO-OFDM-IM does not provide analytical error probability, and performance analysis with optimal $$R_{\text{MIMO-OFDM-IM}}=N_t\cdot \frac{p}{n}\ \text{bits/subcarrier}.$$6 selection is left for future work (Basar, 2015). By contrast, SuM-OFDM-IM derives a BER upper bound and proves at least second-order diversity through the minimum rank of the pairwise error matrix (Dogukan et al., 2020). Q-MM-OFDM-IM uses a union-bound framework and shows that its high-SNR diversity is dominated by the modulation symbols unless operated in index-only mode (Yarkin et al., 2020).

The broader significance of MD-OFDM lies in this plurality rather than in any single architecture. One branch enriches OFDM by adding index dimensions across subcarriers, antennas, or constellation modes; another replaces fixed carriers with channel-dependent eigenwaves; another makes Doppler a deliberate multiplexing axis; another simplifies MIMO-OFDM through per-subcarrier antenna selection. This suggests that MD-OFDM is best understood as a design regime in which OFDM is generalized from a one-domain carrier grid to a multi-domain signaling structure whose dimensions are chosen to match the target channel, hardware, and service constraints.