Differential Reflecting Modulation (DRM)

• DRM is a noncoherent RIS communication scheme that conveys information by jointly encoding the permutation order of reflecting-pattern activations and PSK symbols.
• It employs block-to-block differential encoding to achieve CSI-free detection, eliminating the need for explicit channel state estimation.
• Enhancements such as decision-feedback detection and DRM-DSTM coding improve robustness in time-varying channels while optimizing spectral efficiency and reducing complexity.

Differential Reflecting Modulation (DRM) is a noncoherent reconfigurable intelligent surface (RIS) communication scheme in which information is conveyed jointly by the permutation order of RIS reflecting-pattern activations across a block and by the PSK symbols transmitted during that block. Introduced for RIS-assisted links as a differential alternative to coherent RIS modulation, DRM encodes information in the relative evolution of successive transmit–reflect states, so detection can proceed without channel state information (CSI) at the transmitter, RIS, or receiver. Subsequent work has extended the baseline formulation to time-varying fading through decision-feedback differential detection and to coded variants through differential space-time modulation (Guo et al., 2020, Qiu et al., 30 Jun 2026, Qiu et al., 14 Aug 2025).

1. Foundational system model and conceptual role of the RIS

The original DRM formulation considers a RIS-assisted $(N_r,N,r)$ single-input multiple-output system with one transmit antenna, $N_r$ receive antennas, $N$ RIS reflecting elements, and total transmission rate $r$ bits per block. The propagation model contains a transmitter-to-RIS channel $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$, a RIS-to-receiver channel $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$, and a direct link $\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$, all under quasi-static Rayleigh fading. The RIS is configured from a finite set of $K$ candidate reflecting patterns,

$$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$

where each $\mathbf{\Phi}_k$ is diagonal and its $N_r$0-th diagonal entry has the form

$N_r$1

with $N_r$2 indicating ON/OFF and $N_r$3 the applied phase shift (Guo et al., 2020).

A central structural feature is that each information block contains $N_r$4 symbol slots, one slot for each activated reflecting pattern in a permutation order. This makes the RIS part of the signaling alphabet rather than a purely passive beamforming device. In DRM, the surface state is not chosen only to enhance an instantaneous channel realization; it is one of the information-bearing degrees of freedom. A plausible implication is that DRM should be viewed as a form of differential index modulation over reflecting states, with the RIS configuration entering the codebook itself rather than merely shaping an auxiliary propagation environment.

2. Joint permutation–symbol encoding

At the $N_r$5-th block, DRM conveys

$N_r$6

bits. These are partitioned into $N_r$7 bits that select a $N_r$8 permutation matrix $N_r$9, and $N$0 bits that select $N$1 $N$2-PSK symbols. If

$N$3

then $N$4 is the diagonal matrix whose diagonal entries are the chosen PSK symbols, and the information-carrying matrix is

$N$5

The resulting code object is therefore a permutation matrix multiplied by a diagonal PSK matrix (Guo et al., 2020).

Only $N$6 of the $N$7 legitimate permutation matrices need be used for bit mapping. For example, when $N$8 and $N$9, four of the six valid $r$0 permutation matrices are selected. This restriction is a direct consequence of the $r$1 bit allocation and is typical of index-modulated systems whose native combinatorial cardinality is not a power of two. The practical effect is that DRM jointly maps bits onto two coupled domains: the order in which the RIS patterns appear across the block and the PSK symbols transmitted over those same slots.

This joint structure is the defining difference between DRM and ordinary differential PSK. The RIS activation sequence is not a side parameter; it is part of the transmitted message. Later work preserved this same algebraic decomposition, writing $r$2 for uncoded DRM and replacing $r$3 by a unitary group-code matrix in coded variants (Qiu et al., 14 Aug 2025).

3. Differential recursion and CSI-free reception

DRM applies block-to-block differential encoding according to

$r$4

Repeated recursion yields a product of successive permutation and PSK factors. The original analysis notes that $r$5 remains a product of a permutation matrix and a diagonal matrix whose diagonal entries belong to the $r$6-PSK alphabet. Consequently, if $r$7 denotes the $r$8-th column of $r$9, then each column has exactly one nonzero entry and can be written as

$\mathbf{h}_1\in\mathbb{C}^{N\times 1}$0

where $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$1 is a basis vector and $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$2 is an $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$3-PSK symbol. In slot $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$4 of block $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$5, the transmitter therefore sends $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$6 while the RIS activates the corresponding reflecting pattern $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$7 (Guo et al., 2020).

The received signal in that slot is

$\mathbf{h}_1\in\mathbb{C}^{N\times 1}$8

with $\mathbf{h}_1\in\mathbb{C}^{N\times 1}$9. Stacking the $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$0 slots gives the block relation

$\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$1

Using the differential recursion, the receiver obtains

$\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$2

Under the high-SNR and slowly varying channel assumption, the last two terms are treated as effective noise, so $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$3 can be detected from $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$4 and $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$5 without explicit knowledge of $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$6, $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$7, $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$8, or $\mathbf{H}_2\in\mathbb{C}^{N_r\times N}$9 (Guo et al., 2020).

The maximum-likelihood detector is

$\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$0

equivalently

$\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$1

The method assumes quasi-static channels across at least two adjacent blocks, symbol duration longer than the multipath delay spread so that delayed paths do not cause intersymbol interference, synchronization between transmitter and RIS, and an initial reference block $\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$2 that carries no information (Guo et al., 2020).

A common misconception is that “CSI-free” implies complete insensitivity to channel dynamics. DRM does eliminate explicit channel estimation, but its conventional differential detector still depends on block-to-block channel coherence. Later work on time-varying fading makes this limitation explicit (Qiu et al., 30 Jun 2026).

4. Spectral efficiency, reflecting-pattern design, and algorithmic scaling

Because the first block is a non-informative reference, the effective transmission rate over $\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$3 blocks is

$\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$4

bits per channel use. For large $\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$5, Stirling’s approximation gives

$\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$6

This makes the rate contribution of the permutation domain explicit: beyond the PSK term, there is an additional component due to the ordering of the $\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$7 reflecting patterns (Guo et al., 2020).

The reflecting-pattern set itself is not arbitrary. Since DRM is designed to avoid CSI at the RIS as well as at the endpoints, the pattern subset is selected to maximize a minimum Euclidean separation criterion. In the original formulation, the design objective is

$\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$8

and the authors use the stepwise depletion algorithm from earlier reflecting modulation work to select a small subset of patterns. This matters because only $\mathbf{h}_d\in\mathbb{C}^{N_r\times 1}$9 patterns are retained from a larger candidate pool; the selected subset governs pairwise separability and hence BER (Guo et al., 2020).

The major algorithmic cost is in exhaustive ML detection over the valid information matrices. For uncoded DRM, the ML complexity is

$K$0

multiplications. Increasing $K$1 therefore improves spectral efficiency but also expands both the differential alphabet and the search burden. This tradeoff remains central in later DRM literature: larger $K$2 is attractive from a rate standpoint, yet it generally worsens BER and receiver complexity unless compensated by coding or improved differential detection (Guo et al., 2020, Qiu et al., 14 Aug 2025).

5. Baseline performance and comparison with coherent reflecting modulation

The original performance study compares DRM against non-differential reflecting modulation (NDRM), which uses

$K$3

and assumes perfect CSI for coherent detection. In the reported setup, the comparison is rate-matched and the studied detectors have similar complexity. The simulation model uses a RIS-assisted SIMO system with $K$4 RIS elements, 1-bit encoded elements so that $K$5, a total of $K$6 possible reflecting patterns, and a subset with $K$7 patterns selected by the stepwise depletion algorithm; DRM is also tested with $K$8. The SNR is defined as

$K$9

(Guo et al., 2020).

Under perfect CSI, DRM performs worse than coherent NDRM by about $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$0–$$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$1 dB, which the paper characterizes as an acceptable SNR penalty given that DRM removes the need for CSI and the associated channel-estimation overhead. When $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$2 increases from $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$3 to $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$4, BER worsens, consistent with the larger differential alphabet and more difficult detection. Pattern selection is also consequential: the stepwise depletion algorithm yields better BER than random pattern selection (Guo et al., 2020).

The comparison changes once CSI is imperfect. The reported results show that NDRM degrades substantially with channel-estimation error. At estimation-error factor $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$5, DRM and NDRM become close; at $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$6 and $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$7, DRM can outperform coherently detected NDRM in the plotted SNR region. The significance of this result is methodological rather than merely numerical: DRM should not be compared only against an ideal coherent reference with perfect CSI, because the entire motivation of RIS differential signaling is that CSI acquisition is intricate and resource-consuming in cascaded RIS channels (Guo et al., 2020).

6. Time-varying fading and decision-feedback differential detection

Later work studied DRM over time-varying Rayleigh fading generated by Jakes’ model, with symbol-rate autocorrelation

$$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$8

where $$\Psi=\{\mathbf{\Phi}_1,\mathbf{\Phi}_2,\cdots,\mathbf{\Phi}_K\},$$9 is the normalized Doppler frequency. In that setting, conventional differential demodulation relies on the approximation $\mathbf{\Phi}_k$0, but this approximation degrades as Doppler grows, producing severe performance loss and high-SNR error floors (Qiu et al., 30 Jun 2026).

To address this, decision feedback differential detection (DFDD) forms a predicted reference from multiple previously received blocks: $\mathbf{\Phi}_k$1 The coefficients $\mathbf{\Phi}_k$2 are chosen by minimizing the mean-square error between the current effective channel term and the predicted reference, which reduces to the linear prediction system

$\mathbf{\Phi}_k$3

with $\mathbf{\Phi}_k$4 and $\mathbf{\Phi}_k$5. The resulting DFDD decision rule is

$\mathbf{\Phi}_k$6

(Qiu et al., 30 Jun 2026).

The reported trend is that $\mathbf{\Phi}_k$7 is the best practical choice, $\mathbf{\Phi}_k$8 is inferior but often usable, and $\mathbf{\Phi}_k$9 frequently degrades high-SNR performance because feedback error propagation becomes more severe. For BPSK with $N_r$00 and $N_r$01, the $N_r$02 detector exhibits an error floor of about $N_r$03 around $N_r$04 dB, whereas $N_r$05 reaches $N_r$06 at $N_r$07 dB. For $N_r$08 and $N_r$09, the floor is about $N_r$10 for $N_r$11, $N_r$12 for $N_r$13, and $N_r$14 for $N_r$15. QPSK shows the same qualitative behavior and is slightly more sensitive to Doppler. The chief benefit identified for DFDD is lower error floors over time-varying fading, at the expense of a small increase in complexity; the full demodulation complexity scales as $N_r$16, compared with $N_r$17 for uncoded DRM with conventional differential detection (Qiu et al., 30 Jun 2026).

These results sharpen the operational scope of DRM. The baseline scheme is well matched to quasi-static or slowly varying channels, but on time-varying channels the quality of the differential reference becomes the dominant bottleneck. DFDD does not alter the DRM code structure; it alters only the way the receiver synthesizes the reference against which the current block is tested.

7. Coded variants and relation to adjacent modulation paradigms

A major extension of DRM replaces the diagonal PSK matrix by a unitary group-coded space-time matrix, yielding DRM-DSTM. In this construction,

$N_r$18

with initialization $N_r$19 such that $N_r$20. The permutation matrix $N_r$21 still carries the RIS-pattern-order information, but the simple PSK-symbol matrix is replaced by a unitary group-code matrix $N_r$22. The detector retains the same differential ML form,

$N_r$23

so CSI-free operation is preserved while coding gain is introduced (Qiu et al., 14 Aug 2025).

The coded framework supports cyclic group codes for general $N_r$24 and dicyclic group codes for even $N_r$25. The paper provides code tables for $N_r$26. A notable consequence is that DRM-DSTM can reduce search complexity relative to uncoded DRM: for cyclic codes the reported detection complexity is

$N_r$27

and for dicyclic codes it is

$N_r$28

whereas uncoded DRM requires a search over $N_r$29 PSK combinations. In the reported quasi-static Rayleigh simulations with $N_r$30 and $N_r$31, DRM-DSTM significantly outperforms uncoded DRM: for $N_r$32, the gains at BER $N_r$33 are about $N_r$34 dB for BPSK and $N_r$35 dB for QPSK; for $N_r$36, the gains are roughly $N_r$37–$N_r$38 dB at BER $N_r$39, depending on $N_r$40; and for $N_r$41, both cyclic and dicyclic coded systems outperform uncoded DRM across the SNR range (Qiu et al., 14 Aug 2025).

Conceptually, DRM is closely related to differential generalized spatial-modulation ideas in which information is jointly embedded in an index pattern and in modulation symbols. In differential GSM, the dual carriers of information are active antenna combinations and transmitted symbols; in DRM, the analogous carriers are RIS reflecting-pattern order and PSK or group-coded matrix state. This suggests that DRM belongs to a broader class of differential indexed noncoherent schemes, with the RIS pattern space playing the role that antenna-index space plays in differential spatial modulation (Jose et al., 2021).

DRM should also be distinguished from other RIS-based reflection modulations. Hybrid Reflection Modulation (HRM), for example, conveys information through the number and configuration of active versus passive RIS sub-groups, relies on absolute amplitude levels, and assumes perfect CSI for phase alignment and detection. The HRM literature explicitly contrasts this with DRM, which is differential and noncoherent in the sense that information is embedded in relative changes of RIS states over time rather than in absolute CSI-dependent amplitudes (Yigit et al., 2021).

Taken together, these developments position DRM as a CSI-free RIS modulation architecture with a clear internal logic: joint indexing over reflecting-pattern order and signal-domain symbols, differential recursion across blocks, and noncoherent detection from consecutive observations. Its main limitations are equally clear from the literature: exhaustive-search complexity grows rapidly with $N_r$42, BER degrades as $N_r$43 increases, and conventional differential detection is fragile under Doppler. The extensions developed so far—pattern-set optimization, DFDD, and DRM-DSTM—can be read as successive attempts to preserve the CSI-free premise while improving separability, robustness, and high-SNR behavior (Guo et al., 2020, Qiu et al., 30 Jun 2026, Qiu et al., 14 Aug 2025).