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Can Reciprocal RIS Perform as Non-Reciprocal RIS?

Matteo Nerini, Juan Vidal Alegría, Bruno Clerckx

arXiv:2610.30896v1cs.ITeess.SPmath.IT

TL;DR

The paper asks when reciprocal RIS can match non-reciprocal RIS, whose additional hardware complexity makes the performance value of that extra freedom important. It derives an equivalence condition based on symmetry of an optimal design’s round-trip channel, refined to the transmitted streams under precoding. Rank-one channels and optimal single-stream transmission are supported cases where reciprocal RIS matches non-reciprocal RIS.

  • Problem

    Non-reciprocal RIS requires additional hardware complexity, so it matters when that added capability brings a performance gain.

    Source: §I · p. 1

  • Method

    The paper derives necessary and sufficient equivalence conditions based on symmetry of the round-trip channel, refining the condition to the transmitted streams when precoding is considered.

    Source: §I · p. 1, §IV · p. 4

  • Results

    Reciprocal RIS matches non-reciprocal RIS when an optimal non-reciprocal design makes the round-trip channel symmetric, including when the transmitter-to-RIS channel has rank one or single-stream transmission is optimal.

    Source: §III · p. 4, §V · p. 5

  • Takeaways & Limitations

    In rank-one or single-stream-optimal systems, non-reciprocal RIS brings no performance gain, while it can be beneficial when optimal performance requires breaking round-trip reciprocity.

    Source: §V · p. 5

Abstract

from arXiv · show

Yes, if and only if some optimal non-reciprocal RIS (i.e., maximizing a given performance metric) makes the round-trip channel transmitter-RIS-transmitter symmetric, which holds if the channel transmitter-RIS has rank one or if single-stream transmission is optimal.

I. INTRODUCTION

The paper asks when reciprocal RIS can match non-reciprocal RIS despite the latter’s additional hardware complexity. It gives conditions for equivalence, including a refinement for linear precoding.

I. INTRODUCTION

  • Non-reciprocal RIS can use arbitrary unitary scattering matrices, while reciprocal RIS additionally requires a symmetric scattering matrix.The non-reciprocal design is therefore at least as capable, but requires non-reciprocal components such as circulators, isolators, or gyrators.Source: §I · p. 1
  • The paper derives a necessary and sufficient condition for reciprocal RIS to match non-reciprocal RIS: an optimal non-reciprocal RIS must make the transmitter-to-RIS-to-transmitter round-trip channel symmetric.The condition concerns an optimal design for the considered performance metric.Source: §I · p. 1
  • When the transmitter-to-RIS or RIS-to-receiver channel has rank one, reciprocal RIS matches non-reciprocal RIS, including in single-user SISO, SIMO, and MISO systems.The paper also refines the condition when linear precoding is considered.Source: §I · p. 1
  • With linear precoding, symmetry is required only for the round-trip channel seen by the transmitted streams; it is automatic when single-stream transmission is optimal.For one stream, this round-trip channel is a scalar.Source: §I · pp. 1–2

II. RECIPROCAL VERSUS NON-RECIPROCAL RIS

The section compares the feasible scattering matrices of reciprocal and non-reciprocal RIS for metrics that depend on the RIS through its channel effect. It reduces equivalence to whether a reciprocal RIS can reproduce the output of an optimal non-reciprocal RIS.

II. RECIPROCAL VERSUS NON-RECIPROCAL RIS

  • The paper models non-reciprocal RIS as unitary scattering matrices and reciprocal RIS as the symmetric matrices within that unitary set.It considers metrics of the form f(ΘHT), with other design variables already optimized; results also apply to metrics of the stated HRΘ or full-channel forms.Source: §II · p. 2
  • The comparison assumes the considered metric attains its maximum over the unitary matrices, and reciprocal RIS is sufficient when it can attain that same optimum.The reciprocal feasible set is a subset of the non-reciprocal set.Source: §II · p. 2
  • Equivalence reduces to whether a reciprocal RIS can reproduce ΘHT produced by an optimal non-reciprocal RIS, since the metric depends on Θ through ΘHT.The section frames this as comparing which matrices each architecture can produce.Source: §II · p. 2
  • Lemma 1 characterizes when a unitary matrix can map X to Y; its necessity follows from preserving the Gram matrix, YᴴY = XᴴX.The proof is included for completeness, and its structure is mirrored in the symmetric-unitary lemma.Source: §II · p. 2

1 XHXV1Σ−1

A unitary RIS can map vectors while preserving their norms and inner products, but requiring its scattering matrix to be symmetric adds constraints for multiple vectors. For one vector, symmetry adds no restriction.

1 XHXV1Σ−1

  • A unitary Θ maps columns of X to Y if and only if their norms and inner products are preserved.When X has orthogonal columns, this means Y must also have orthogonal columns with matching column norms.Source: §1 · p. 3
  • A symmetric unitary Θ maps X to Y only when the unitary conditions also satisfy the additional symmetry condition x_k^T y_k2 = x_k2^T y_k1.The cited lemma states this as a necessary and sufficient condition; the proof constructs a symmetric unitary solution when it holds.Source: §1 · p. 3
  • For K = 1, a symmetric unitary Θ maps any vector x to any vector y with the same norm, just as a unitary Θ can.For K ≥ 2, symmetric Θ additionally requires x_k^T y_k to be symmetric across vector pairs, imposing K(K −1)/2 constraints.Source: §1 · p. 3

III. A NECESSARY AND SUFFICIENT CONDITION

Reciprocal RIS matches non-reciprocal RIS exactly when some optimal non-reciprocal design produces a symmetric transmitter-side round-trip channel. Rank-one channel cases guarantee this condition.

III. A NECESSARY AND SUFFICIENT CONDITION

  • Reciprocal RIS performs as non-reciprocal RIS if and only if some optimal non-reciprocal RIS makes H_T^T Θ⋆ H_T symmetric.This is the round-trip channel from the transmitting antennas, via the RIS, back to the transmitting antennas.Source: §III · p. 4
  • For metrics of the form f(H_RΘ), the analogous condition is symmetry of the receiving-side round-trip channel H_RΘ⋆H_T^R.For metrics depending on H_RΘH_T, either the transmitter-side or receiving-side condition can be used.Source: §III · p. 4
  • If rank(H_T) = 1, reciprocal RIS performs as non-reciprocal RIS.This includes single-transmit-antenna systems and single-user SISO, SIMO, and MISO systems; for metrics involving H_RΘH_T, rank one in either H_T or H_R suffices.Source: §III · p. 4

IV. A NECESSARY AND SUFFICIENT CONDITION IN THE PRESENCE OF PRECODING

With precoding, reciprocal RIS need only reproduce the optimal non-reciprocal RIS output along the chosen precoder’s transmission directions. Thus, symmetry is required only for the round-trip channel seen by the streams.

IV. A NECESSARY AND SUFFICIENT CONDITION IN THE PRESENCE OF PRECODING

  • The metric is written as g(F, ΘH_TF), with F drawn from a feasible precoder set and K ≤ N_T data streams.The feasible set can, for example, enforce a power constraint; the analysis assumes the maximum over RIS and precoders is attained.Source: §IV · p. 4
  • Reciprocal RIS matches non-reciprocal RIS with precoding if and only if an optimal RIS-precoder pair has a symmetric stream-level round-trip channel F⋆T H_T^T Θ⋆ H_T F⋆.The reciprocal RIS need only reproduce Θ⋆H_TF⋆, rather than the full Θ⋆H_T matrix.Source: §IV · p. 4
  • The stream-level condition is weaker than requiring the full transmitter-side round-trip channel to be symmetric.A reciprocal RIS need preserve reciprocity only along the directions used for transmission; an analogous condition applies to a receiver combiner.Source: §IV · p. 5
  • If single-stream transmission (K = 1) is optimal with non-reciprocal RIS, reciprocal RIS performs as non-reciprocal RIS.The stream-level round-trip channel is scalar and therefore symmetric, even when H_T and H_R have arbitrary rank.Source: §IV · p. 5

V. CONCLUSION

Reciprocal RIS matches non-reciprocal RIS if and only if an optimal non-reciprocal design makes the transmitter-side round-trip channel symmetric. Rank-one transmitter-to-RIS channels or optimal single-stream transmission satisfy this condition; non-reciprocal RIS may help when optimal performance requires breaking that symmetry.

V. CONCLUSION

  • Reciprocal RIS matches non-reciprocal RIS if and only if an optimal non-reciprocal RIS makes the transmitter-side round-trip channel symmetric.With a precoder, symmetry need only hold for the round-trip channel seen by the transmitted streams.Source: §V · p. 5
  • Rank-one transmitter-to-RIS channels and systems where single-stream transmission is optimal are cases where reciprocal RIS performs as well as non-reciprocal RIS.In these systems, non-reciprocal RIS hardware brings no performance gain.Source: §V · p. 5
  • Non-reciprocal RIS is beneficial when optimal performance requires breaking round-trip channel reciprocity, as in full-duplex systems where a base station receives from an uplink user and transmits via the RIS to a downlink user in a different direction.A reciprocal RIS forces the channel from the uplink user to the base station to equal its reverse,,,.Source: §V · p. 5
  • Non-reciprocal RIS is expected to help when numerous streams pass through well-conditioned channels, including rich-scattering environments and systems serving transmitters and receivers in different directions.Examples include an RIS serving multiple transmitters and receivers in different directions.Source: §V · p. 5
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