Receive Quadrature Spatial Modulation (RQSM)

• Receive Quadrature Spatial Modulation (RQSM) is a scheme that encodes information by mapping the in-phase and quadrature components of a complex symbol to separate receive antenna indices.
• It employs RIS-assisted phase control and a one-tap zero-forcing pre-equalizer to optimize signal separation and boost effective channel gains.
• RQSM enhances conventional spatial modulation by using index and quadrature mapping to improve spectral efficiency and reliability in SIMO systems.

Searching arXiv for recent and foundational papers on Receive Quadrature Spatial Modulation and closely related schemes. Receive quadrature spatial modulation (RQSM) is a receive-side spatial-modulation scheme in which the receive antenna indices and the quadrature components of a complex symbol jointly convey information. In its canonical form, one receive antenna is associated with the in-phase component and another with the quadrature component, so the tuple of receive indices and the underlying symbol together forms the transmitted message. In that sense, RQSM is the receive-side analogue of quadrature spatial modulation (QSM), which on the transmit side attains $q + 2\log_2(n_T)$ bits/s/Hz by mapping a complex symbol and two antenna indices into one channel use (Mohaisen et al., 2017). Most explicit RQSM constructions to date are RIS-assisted, where a reconfigurable intelligent surface is used to shape the effective channel so that the real and imaginary components appear at selected receive antennas with large magnitude (Dinan et al., 2023).

1. Conceptual position within spatial and quadrature index modulation

RQSM belongs to the broader family of spatial-modulation schemes in which indices, rather than only constellation points, carry bits. Conventional spatial modulation maps bits to a modulation symbol and an active transmit-antenna index, while receive spatial modulation focuses energy on one receive antenna so that the receive index itself carries information. QSM extends transmit-side SM by exploiting the in-phase and quadrature dimensions independently, using one antenna index for the real branch and another for the imaginary branch. RQSM transfers that quadrature-indexing principle to the receive side, so that the receiver-side spatial entity for the real component need not coincide with the entity for the imaginary component (Dinan et al., 2021).

Two adjacent schemes are especially important for situating RQSM. The first is RIS-assisted receive quadrature space-shift keying (RIS-RQSSK), which is essentially an SSK-style receive-quadrature scheme: it uses receive indices and polarity bits, but no conventional QAM/PSK symbol, and therefore represents the “index-only” end of the receive-quadrature spectrum (Dinan et al., 2021). The second is generalized receive quadrature spatial modulation (GRQSM), in which multiple receive antennas are activated independently for the real and imaginary parts, bringing generalized spatial modulation into the receive-quadrature setting (Dinan et al., 2023). Together, RQSSK, RQSM, and GRQSM span the main receive-side quadrature architectures currently discussed in the arXiv literature.

A recurring misconception is that RQSM is simply QSM with transmit and receive labels exchanged. The available formulations indicate otherwise. Receive-side quadrature indexing generally requires some mechanism that shapes the propagation channel so that chosen receive antennas carry the intended real and imaginary components; in current work, this role is typically played by RIS phase control or transmit-side precoding rather than by a purely passive reinterpretation of the channel (Dinan et al., 2023).

2. Canonical signal model and bit mapping

A canonical RIS-assisted RQSM model uses a SIMO downlink with one single-antenna transmitter, an $N_r$-antenna receiver, and an RIS with $N$ passive reflecting elements. The RIS-reflected link is the only link considered, the RIS-to-receiver channel is $\mathbf{H} \in \mathbb{C}^{N_r \times N}$ with i.i.d. entries $\sim \mathcal{CN}(0,1)$, and the receive noise is $n_l \sim \mathcal{CN}(0,N_0)$ on antenna $l$ (Dinan et al., 2023). In this formulation, each symbol interval carries

$\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$

bits. The $\log_2 M$ bits choose a complex symbol $x=x^{\mathcal R}+j x^{\mathcal I}$, while the remaining $N_r$0 bits independently choose the receive indices $N_r$1 assigned to the real and imaginary components (Dinan et al., 2023).

A notable structural feature is that the RF source has one RF chain and transmits a positive real PAM symbol

$N_r$2

The RIS phases and a scalar one-tap ZF pre-equalizer $N_r$3 then synthesize the desired IQ symbol at the receiver. For the selected receive antennas $N_r$4 and $N_r$5, the per-component input-output relations are

$N_r$6

$N_r$7

The RIS phase vector is chosen so that the effective real and imaginary gains match the sign and ratio of the target QAM symbol (Dinan et al., 2023).

The main receive-quadrature variants can be organized succinctly as follows.

Scheme	Rate	Core mapping
RIS-RQSSK	$N_r$8	two receive indices + two polarity bits
RIS-RQSM	$N_r$9	two receive indices + QAM symbol
RIS-GRQSM	$N$0	$N$1 receive indices for I, $N$2 for Q, plus polarity bits

The table captures a progression from pure index modulation to mixed symbol/index modulation and then to generalized receive-side activation. In all three cases, the real and imaginary branches are treated as separate index-bearing dimensions (Dinan et al., 2021, Dinan et al., 2023, Dinan et al., 2023).

3. RIS-assisted realizations and receive-side generalization

RIS-assisted receive-quadrature schemes are unified by a max-min phase-design objective. In RIS-RQSM, the RIS phases are selected to maximize the useful signal power on the two selected antennas while enforcing the correct sign pattern and real-to-imaginary ratio of the desired symbol. With

$N$3

the phase-design problem is

$N$4

subject to

$N$5

where $N$6 and $N$7 are the effective real and imaginary gains at the selected antennas. The optimal RIS coefficients admit a closed-form structure parameterized by a scalar $N$8, and for large $N$9 the constructed symbol has mean on a circle of radius

$\mathbf{H} \in \mathbb{C}^{N_r \times N}$0

This underlies the one-tap pre-equalization strategy and the low-CSI receiver design in RIS-RQSM (Dinan et al., 2023).

RIS-RQSSK uses the same receive-quadrature idea but without a conventional QAM symbol. Bits select one receive antenna index $\mathbf{H} \in \mathbb{C}^{N_r \times N}$1 for the real part, another receive antenna index $\mathbf{H} \in \mathbb{C}^{N_r \times N}$2 for the imaginary part, and optionally polarity bits $\mathbf{H} \in \mathbb{C}^{N_r \times N}$3. The RIS then maximizes the corresponding real and imaginary SNR components jointly. A key point in that formulation is that all RIS elements are optimized for both targets simultaneously, unlike RIS-RQRM, where the RIS is split into two groups and each dimension gets only $\mathbf{H} \in \mathbb{C}^{N_r \times N}$4 elements. Numerical comparisons show sizeable gains: for $\mathbf{H} \in \mathbb{C}^{N_r \times N}$5 and $\mathbf{H} \in \mathbb{C}^{N_r \times N}$6 bpcu, RIS-RQSSK gains about $\mathbf{H} \in \mathbb{C}^{N_r \times N}$7 dB over RIS-RQRM for $\mathbf{H} \in \mathbb{C}^{N_r \times N}$8 and about $\mathbf{H} \in \mathbb{C}^{N_r \times N}$9–$\sim \mathcal{CN}(0,1)$0 dB over RIS-SM as $\sim \mathcal{CN}(0,1)$1 increases from $\sim \mathcal{CN}(0,1)$2 to $\sim \mathcal{CN}(0,1)$3; for $\sim \mathcal{CN}(0,1)$4 and $\sim \mathcal{CN}(0,1)$5 bpcu, the gains over RIS-SM grow to about $\sim \mathcal{CN}(0,1)$6–$\sim \mathcal{CN}(0,1)$7 dB (Dinan et al., 2021).

RIS-GRQSM generalizes receive-side quadrature indexing by activating sets

$\sim \mathcal{CN}(0,1)$8

of receive antennas for the real and imaginary branches. The phase design becomes a max-min problem over $\sim \mathcal{CN}(0,1)$9 effective branch gains. Using Lagrange duality, the non-convex optimization over all RIS phases reduces to a convex dual problem in $n_l \sim \mathcal{CN}(0,N_0)$0 real variables $n_l \sim \mathcal{CN}(0,N_0)$1, after which the optimal RIS phases follow in closed form. In the large-$n_l \sim \mathcal{CN}(0,N_0)$2 regime, the sub-optimal but nearly optimal choice $n_l \sim \mathcal{CN}(0,N_0)$3 becomes accurate. At $n_l \sim \mathcal{CN}(0,N_0)$4, $n_l \sim \mathcal{CN}(0,N_0)$5, $n_l \sim \mathcal{CN}(0,N_0)$6, and $n_l \sim \mathcal{CN}(0,N_0)$7 bpcu, RIS-GRQSM outperforms RIS-RQSM by about $n_l \sim \mathcal{CN}(0,N_0)$8 dB and generalized RIS-RQSSK by about $n_l \sim \mathcal{CN}(0,N_0)$9 dB; for $l$0, the advantage increases further because the benchmark RIS-RQSM must resort to very high-order QAM to match throughput (Dinan et al., 2023).

A second misconception is that receive-quadrature schemes necessarily partition the RIS between I and Q. The literature shows two distinct strategies: some designs split the RIS into I- and Q-supporting subarrays, while others optimize all RIS elements jointly for both branches. The latter approach is central to RIS-RQSSK and RIS-GRQSM and is one reason those schemes avoid the SNR penalty associated with hard RIS partitioning (Dinan et al., 2021, Dinan et al., 2023).

4. Detection architectures and error analysis

The dominant low-complexity detector in RIS-assisted receive-quadrature work is an energy-based greedy detector. In RIS-RQSM, the real-branch index is detected as

$l$1

and the imaginary-branch index as

$l$2

Conditioned on the detected indices, the real and imaginary PAM components are then estimated independently: $l$3 The full ML benchmark jointly searches over all $l$4: $l$5 with complexity $l$6 per symbol plus the RIS-phase-design overhead (Dinan et al., 2023).

RIS-RQSSK uses the same greedy principle, but detection terminates at the index and polarity level: $l$7 followed by sign decisions for the polarity bits. The full ML detector would jointly search over $l$8, but simulations show that the greedy detector performs very close to ML, especially for large $l$9, while requiring no CSI at the receiver (Dinan et al., 2021).

RIS-GRQSM extends greedy detection to set-valued receive indices. The estimated real-branch set $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$0 is formed by the $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$1 indices with largest $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$2, and the estimated imaginary-branch set $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$3 by the $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$4 indices with largest $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$5. This avoids the $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$6 combinatorial burden of joint ML over all receive-index sets and polarity patterns (Dinan et al., 2023).

Error analysis in RIS-RQSM is built around an ABEP upper bound that separates index errors from symbol errors. The scalar PAM pairwise error probability conditioned on correct index detection is

$\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$7

and the total ABEP upper bound combines this term with the real-branch index error probability $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$8 and its symmetric imaginary-branch counterpart. In RIS-RQSSK, the analogous analysis is performed on the event $\log_2(MN_r^2)=\log_2 M + 2\log_2 N_r$9, leading to approximate closed forms and a characteristic $\log_2 M$0 decay of the PEP for large RIS sizes (Dinan et al., 2023, Dinan et al., 2021).

5. Capacity, constellation structure, and abstract design viewpoints

Although explicit capacity formulas for practical finite-alphabet RQSM are scarce, closely related generalized quadrature spatial modulation results provide an analytical template. For continuous-input GQSM, the average mutual information conditioned on the channel decomposes as

$\log_2 M$1

where $\log_2 M$2 is the symbol-domain contribution and $\log_2 M$3 is the activation-pattern contribution. A closed-form expression for $\log_2 M$4 removes the Monte Carlo integration error that otherwise grows exponentially with SNR, and the resulting analysis shows that equiprobable antenna selection slightly decreases AMI of symbols while significantly improving total AMI (Yukiyoshi et al., 2023). A dual receive-side interpretation has been proposed in which RQSM can be analyzed as a GQSM-like scheme on the transposed channel $\log_2 M$5, with transmit and receive roles exchanged. This suggests that receive-side quadrature index sets should also be evaluated through a split between symbol-domain and index-domain mutual information, and that equiprobable receive-index usage may improve total AMI even when it marginally reduces symbol-domain AMI.

A second abstract viewpoint comes from lattice-based spatial modulation. Spatial lattice modulation (SLM) treats the joint spatial, in-phase, and quadrature dimensions as coordinates of a lattice point in $\log_2 M$6, with cubic signal set

$\log_2 M$7

and dense-lattice generalizations based on $\log_2 M$8, $\log_2 M$9, and higher Barnes–Wall lattices. In that framework, conventional QSM is a constrained subset of the cubic lattice, and dense lattices such as $x=x^{\mathcal R}+j x^{\mathcal I}$0 and $x=x^{\mathcal R}+j x^{\mathcal I}$1 offer nominal coding gains of $x=x^{\mathcal R}+j x^{\mathcal I}$2 dB and $x=x^{\mathcal R}+j x^{\mathcal I}$3 dB, respectively (Choi et al., 2018). A useful interpretation is that RQSM can likewise be viewed as an extended lattice over receive-index and quadrature coordinates. This suggests that RQSM constellation design need not remain separable in the symbol and index domains; instead, dense-lattice or distance-optimized joint constellations may be used, with mutual information and ASVEP then governed mainly by pairwise Euclidean distances.

Transmit-side multidimensional quadrature index modulation pushes the same logic further by combining time indices, spatial indices, and RF-mirror indices. Its rate is

$x=x^{\mathcal R}+j x^{\mathcal I}$4

This is not an RQSM construction, but it shows how quadrature indexing naturally composes with other index domains. A plausible implication is that receive-side quadrature indexing could also be combined with time indexing or channel-state indexing to obtain multidimensional receive-side IM architectures (Elganimi et al., 2021).

6. Learning-based selection, adjacent architectures, and current directions

Recent RIS-RQSM work has begun to replace explicit combinatorial antenna selection with supervised learning. In a SIMO RIS-RQSM system with $x=x^{\mathcal R}+j x^{\mathcal I}$5 physically available receive antennas and a selected subset of $x=x^{\mathcal R}+j x^{\mathcal I}$6 antennas, a DNN is trained to emulate COAS, which ranks the channel columns $x=x^{\mathcal R}+j x^{\mathcal I}$7 by their norms and keeps the top $x=x^{\mathcal R}+j x^{\mathcal I}$8. The DNN input is the real-valued vector

$x=x^{\mathcal R}+j x^{\mathcal I}$9

of dimension $N_r$00, and the output is a softmax over the $N_r$01 possible receive-antenna subsets. The reported architecture uses four fully connected hidden layers with $N_r$02 neurons each, ReLU activations, Adam with learning rate $N_r$03, and $N_r$04 training samples (Ozden et al., 7 Jul 2025).

The learned selector improves BER over COAS-based RIS-RQSM. At BER $N_r$05, the reported SNR gains are $N_r$06 dB for $N_r$07, $N_r$08 dB for $N_r$09, and $N_r$10 dB for $N_r$11; when varying the RIS size with $N_r$12, the gains are $N_r$13 dB for $N_r$14 and $N_r$15 dB for $N_r$16 and $N_r$17 (Ozden et al., 7 Jul 2025). The cost is complexity: for a representative case with $N_r$18, $N_r$19, and $N_r$20, COAS requires $N_r$21 real multiplications whereas the DNN requires about $N_r$22 multiplications. The trade-off is therefore not between accuracy and feasibility in the abstract, but between explicit norm-based subset ranking and a much larger learned inference pipeline.

Two adjacent research directions are also relevant. First, fluid-antenna receive spatial modulation introduces correlation-aware port selection and low-complexity detectors such as MED and RTTD in a receive-spatial architecture with a fluid-antenna transmitter; it is not an RQSM scheme, but it suggests that correlation-aware port selection and hybrid detection can be transferred to receive-quadrature designs (Guo et al., 9 Jun 2025). Second, deep-learning-aided OFDM-based GOQSM shows that joint DNN detection can remove the error-propagation and noise-amplification effects of ML-MRC in quadrature-indexed systems; this suggests a natural detector class for OFDM-RQSM, where the receive-side I/Q index hypotheses would replace the transmit-side GOQSM hypotheses (Chen et al., 2021).

The current literature therefore presents RQSM less as a single fixed modulation format than as a design principle: separate the real and imaginary dimensions, attach them to receive-side spatial indices, and then shape or learn the effective channel so that those indices are both spectrally efficient and reliably detectable. RIS phase optimization, generalized receive activation, mutual-information-driven pattern design, lattice-based joint constellations, and learning-based antenna selection all fit within that principle, and the existing results indicate that the receive-quadrature perspective is especially advantageous when the system seeks high spectral efficiency without resorting to very high-order constellations (Dinan et al., 2023, Dinan et al., 2023).