Gaussian Identification in MIMO Channels

Mohammad Javad Salariseddigh; Christian Deppe

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Gaussian Identification in MIMO Channels

Mohammad Javad Salariseddigh, Christian Deppe

arXiv:2610.22931v1Oct 16, 2026cs.ITmath.IT

TL;DR

Deterministic identification capacity for peak-constrained Gaussian MIMO channels raises the question of how spatial channel structure affects achievable rates. The paper derives rank-dependent bounds for the static white-noise setting, showing linear scaling with rank alongside super-exponential codebook growth.

Problem
Deterministic identification capacity for Gaussian MIMO channels under peak-amplitude constraints had not been characterized, leaving the role of spatial rank in achievable rates open.
Source: §I · p. 3
Method
The paper derives achievability and converse bounds for deterministic identification using rank-space coding and geometric separation arguments.
Source: §I · p. 4, C. Achievable Coding Scheme in the… · p. 18
Results
Achievable identification rates scale linearly with channel rank for codebooks growing as M_n(R) = 2^(n log n R).
Source: §V · p. 32
Takeaways & Limitations
The bounds characterize a rank-dependent gain in achievable identification performance without requiring an exact multiplicative relationship between codebook sizes.
Source: §V · p. 32
The analysis considers static MIMO Gaussian channels with white noise, full CSI at both ends, and deterministic peak-constrained encoders.
Source: §I · p. 3

Abstract

from arXiv · show

We study the identification capacity of discrete-time Gaussian multiple-input multiple-output (MIMO) channels under per-antenna, per-symbol peak-amplitude constraints. We establish upper and lower bounds on the identification capacity that scale linearly with the channel rank $r.$ The number of reliably identifiable messages can grow super-exponentially with blocklength $n,$ following $M_n(R)=2^{(n\log n)R},$ where $R$ is an achievable rate. The channel rank provides a spatial-multiplexing advantage, enabling higher achievable rates and, consequently, stronger super-exponential codebook growth. These results characterize the rank-dependent gain in achievable identification performance.

I. INTRODUCTION

Deterministic identification over MIMO Gaussian channels is studied under peak-amplitude constraints, with the channel rank governing achievable-rate scaling. The paper places this problem against prior identification growth laws and specifies a static, full-CSI channel model.

I. INTRODUCTION

Identification asks whether a particular message was transmitted, unlike classical channel coding, and its codebook-growth law depends on the channel model.Randomized identification can grow doubly exponentially, whereas continuous-alphabet deterministic identification follows super-exponential growth.Source: §I · pp. 1–2
Channel rank plays an analogous structural role in Shannon transmission and identification, multiplying the leading-order growth relative to a rank-one channel.The cited comparison gives n log(SNR) growth for Shannon transmission and n log n for identification.Source: §I · p. 3
Deterministic identification over MIMO Gaussian channels is presented as an open problem, addressed here for static channels with white noise, full CSI at both ends, and deterministic peak-constrained encoders.The introduction contrasts this gap with prior randomized-identification work on Gaussian MIMO channels and deterministic-identification results for SISO channels.Source: §I · p. 3
Achievable identification rates scale linearly with channel rank, with super-exponential codebook growth M_n(R) = 2^(n log n R); explicit lower and upper bounds characterize the rank-dependent gain.The lower bound uses directional gains and high-dimensional sphere packing, while the converse uses pairwise separation and output-space volume arguments.Source: §I · p. 4

A. Related Works in Shannon Problem

Under an average-power constraint Gaussian signaling is optimal, whereas a hard peak constraint requires optimization over bounded-support input distributions.Prior work cited in the passage finds finite discrete structure in amplitude or radius for some peak-constrained Gaussian channels.Source: A. Related Works in Shannon Problem · p. 5

A. MIMO Gaussian Communication Model

The model uses a deterministic channel matrix fixed throughout each block, with rank r nonzero spatial modes that may have unequal gains and a nontrivial null space.The same matrix is known at transmitter and receiver; the block-constant assumption corresponds to coherence time at least as long as the transmission interval.Source: A. MIMO Gaussian Communication Model · pp. 7–8

B. Channel Inputs

The system model imposes per-symbol peak-amplitude limits on transmitted antenna symbols and uses additive Gaussian noise. Rank reduction isolates the signal-bearing spatial dimensions, while the coding preliminaries define the code and its error requirements.

B. Channel Inputs

Each antenna’s transmitted symbol is subject to a per-symbol peak-amplitude constraint.The constraint applies to every antenna and channel use, with maximum allowable amplitude √Pmax.Source: B. Channel Inputs · p. 8

C. Channel and Noise Model

Across the n channel uses, the noise matrix Z is independent across receive antennas and channel uses, with distribution N(0, σ² I_NRn).Source: C. Channel and Noise Model · p. 9

D. Rank-Reduced Channel Representation

Only the input projection onto H’s r-dimensional right-singular subspace affects the received signal; the remaining dimensions lie in channel null spaces.The reduced representation is an r-dimensional channel with matrix Σr.Source: D. Rank-Reduced Channel Representation · p. 10

III. CODING SCHEME AND PRELIMINARIES

An identification code pairs a codebook with decoding regions, uses rank-space codewords precoded into the physical transmit space, and bounds type-I and type-II errors.The peak constraint is imposed on the physically transmitted codeword entries.Source: §III · p. 11
Identification capacity is the supremum of rates achievable with both prescribed error tolerances for all sufficiently large blocklengths.The definition applies to codes with the stated message count and peak-amplitude constraint.Source: §III · p. 11

A. Operational Interpretation of Identification Rates

The section distinguishes the dominant n log n growth of identification codebooks from a system-dependent order-n contribution. It presents deterministic matrix-valued encoding and decoding, states main capacity results, and notes that the available bounds do not yet yield a finite second-order characterization under one normalization.

A. Operational Interpretation of Identification Rates

The first-order rate captures dominant codebook growth on the n log n scale, while the second-order term captures system-dependent contributions at the next asymptotic order.The latter may distinguish schemes with the same leading growth if the corresponding asymptotic expansion is established.Source: A. Operational Interpretation of… · p. 12

B. Encoding

Each message is mapped deterministically to a matrix-valued codeword c_i in R^(r×n).The transmitted codeword has one representation as channel-use vectors and another as antenna-wise sequences.Source: B. Encoding · p. 13

C. Decoding

To test whether message j was sent, the decoder checks whether the received output lies in that message's decoding set.The decoding threshold is defined using fixed positive constants a and b.Source: C. Decoding · p. 13

A. Main Results

Theorem 1 treats identification capacity for a rank-r MIMO Gaussian channel under a peak-amplitude constraint and codebook size M_n(R) = 2^(n log n R).The standard identity-channel and SISO cases are also identified as corollaries or special cases.Source: A. Main Results · pp. 14–15
Theorem 2 gives an achievable order-n contribution relative to the leading coefficient L1 = (1−b)r/4.The available converse does not match the achievability coefficient, so this is not a finite second-order characterization under a shared normalization.Source: A. Main Results · p. 14
The achievable and converse bounds have different coefficients on the leading n log n term, preventing one normalization from yielding finite order-n remainders for both.The stated bounds therefore do not establish a finite second-order characterization under a common leading coefficient.Source: A. Main Results · p. 14

B. Converse Proof via Fidelity of Output Distributions

The converse bounds codebook size by separating codewords and packing non-overlapping spheres, then derives bounds on achievable identification rates and capacity.

B. Converse Proof via Fidelity of Output Distributions

A minimum Frobenius distance between distinct codewords permits non-overlapping spheres to be placed around them inside a bounded hypercube.The resulting volume argument bounds the codebook size.Source: B. Converse Proof via Fidelity of… · pp. 15–16
For fixed rank r, the dominant codebook-size term scales as n log n, yielding M_n(R) = 2^(n log n R).The passage gives this scaling when setting the codebook size from the dominant term.Source: B. Converse Proof via Fidelity of… · p. 17
The converse translates finite-blocklength bounds into an asymptotically achievable-rate bound and then a capacity bound.Capacity is defined as the supremum of asymptotically achievable rates.Source: B. Converse Proof via Fidelity of… · p. 17

C. Achievable Coding Scheme in the Rank-Space

The coding scheme designs codewords in the channel’s effective rank space, maps them into the physical transmit space, and enforces peak amplitude after precoding.

C. Achievable Coding Scheme in the Rank-Space

Codewords are generated directly in the r-dimensional effective signal space rather than the full transmit space.The rank-space codewords are matrices in R^(r×n).Source: C. Achievable Coding Scheme in the… · p. 18
The operator V_r embeds rank-space codewords into the physical transmit subspace while preserving Frobenius distances.The embedding is isometric because V_r has orthonormal columns.Source: C. Achievable Coding Scheme in the… · p. 18
Peak-amplitude constraints are enforced on antenna-domain codewords after precoding, with a rank-space amplitude bound ensuring compliance.The rank-space peak-amplitude bound is stated to ensure the physically precoded codeword satisfies the constraint.Source: C. Achievable Coding Scheme in the… · pp. 18–19

D. Rank-Space Signaling

Rank-space signaling uses the channel’s r-dimensional signal subspace: dimensions outside it are invisible at the receiver, while the smallest nonzero singular value governs a distance lower bound.

D. Rank-Space Signaling

Transmit components outside the rank-r signal space lie in the channel null space and are invisible at the receiver.The received signal depends only on projection onto the right-singular subspace.Source: D. Rank-Space Signaling · p. 19
Using all r dimensions captures the full spatial-multiplexing gain, whereas using fewer dimensions does not exploit the full signal space.The passage allows signaling over fewer than r dimensions but says they leave some supported spatial dimensions unused.Source: D. Rank-Space Signaling · p. 20
Channel-domain distance is lower-bounded by rank-space distance scaled by the smallest nonzero singular value, σ_r(H).The bound applies to physical codewords obtained from rank-space codewords through V_r.Source: D. Rank-Space Signaling · p. 20

E. Rate Analysis

This section establishes an achievability lower bound by turning a saturated sphere-packing construction into a covering, counting its centers by volume, and bounding the relevant error events. For fixed rank and sufficiently large blocklengths, these arguments yield identification codes whose log-cardinality has a dominant $n\log n$ term.

Assigning codewords to sphere centers keeps their 1-norm and arithmetic mean bounded by Ar, as required by the construction.Source: E. Rate Analysis · p. 21
Saturated radius-τn sphere packings yield a 2τn-covering of the hypercube, so volume comparison lower-bounds the number of packing centers and codewords.This covering argument bypasses the difficulty of obtaining sufficiently sharp packing-density bounds in bounded geometries.Source: E. Rate Analysis · pp. 20–22
Vectorization preserves Frobenius distances and volumes, allowing the packing count to use Euclidean hypersphere volumes in rn dimensions.Source: E. Rate Analysis · pp. 22, 24
For fixed rank r, the achievable codebook log-cardinality has a dominant n log n term, yielding super-exponential message growth and a lower bound on identification capacity.The authors use this growth to define achievable rates and establish the capacity lower bound.Source: E. Rate Analysis · p. 26

E. Rate Analysis

For every e1, e2 > 0 and sufficiently large n, an (n, Mn(R), e1, e2)-DI code exists, completing Theorem 1’s achievability proof.The proof shows Pe,2(i, j) ≤ e2 for sufficiently large n and arbitrarily small e2 > 0.Source: E. Rate Analysis · p. 32

V. CONCLUSIONS AND PERSPECTIVES

The paper characterizes MIMO Gaussian identification rates with bounds scaling linearly in channel rank, and establishes super-exponential codebook growth. It identifies open directions including non-asymptotic performance, imperfect channel knowledge, multi-user settings, and diversity–multiplexing tradeoffs.

V. CONCLUSIONS AND PERSPECTIVES

The achievable identification rate grows at least linearly with channel rank r, supporting stronger super-exponential codebook growth.The codebook size scales as Mn(R) = 2(n log n)R, without requiring an exact multiplicative relationship between codebook sizes.Source: §V · p. 32
A unified general theory, including non-asymptotic performance and imperfect channel knowledge, remains an open research direction.The paper also highlights MIMO multiple-access and broadcast identification rates under input constraints such as average-power and peak-amplitude limits.Source: §V · p. 32
Extending the diversity–multiplexing framework of Zheng and Tse to identification could clarify tradeoffs among identification performance, diversity, and multiplexing gains.The paper also points to multi-user extensions in uplink MIMO multiple-access and downlink MIMO broadcast channels.Source: §V · p. 32

A. Open Problems and Conjectures

The paper conjectures that super-exponential identification-codebook scaling persists for MIMO Gaussian channels with ISI and spatially and temporally correlated noise, subject to a spectral condition. It specifies a Kronecker covariance case and a corresponding decoding metric.

A. Open Problems and Conjectures

The codebook-size scaling Mn(R) = 2(n log n)R is conjectured to persist with ISI and correlated Gaussian noise when the channel spectrum is bounded away from zero.Analytical evidence and derivations reported in [36], [37] motivate the conjecture.Source: A. Open Problems and Conjectures · p. 33
For noise covariance Σz = Rt ⊗ Rs, with spatial and temporal covariance matrices Rs and Rt, the same scaling is conjectured under the corresponding generalized Mahalanobis decoding metric.This is a specified case of the broader conjecture for ISI and spatio-temporally correlated noise.Source: A. Open Problems and Conjectures · p. 33

APPENDIX A

The appendix derives a minimum-distance implication for reliable identification codes by linking error probabilities to output-distribution distance and Gaussian fidelity. It also shows that the fidelity of two Gaussian outputs depends on the squared distance between their channel-propagated codewords.

PROOF OF LEMMA 1; MINIMUM FROBENIUS DISTANCE

Reliable identification codes with type-I and type-II error probabilities summing to less than one induce output distributions separated in total variation, yielding a lower bound on codeword Frobenius distance.The proof uses the classical Fuchs–van de Graaf inequality and the Gaussian fidelity expression.Source: PROOF OF LEMMA 1; MINIMUM FROBENIUS… · pp. 33–34

APPENDIX B FIDELITY BETWEEN TWO GAUSSIAN DISTRIBUTIONS

The fidelity between Gaussian output distributions depends only on the squared Euclidean distance between the codewords after propagation through channel matrix H.The derivation obtains the fidelity by separating a codeword-dependent factor from a normalized Gaussian integral.Source: Appendix B · p. 37

APPENDIX C

The appendix proves a channel-domain distance relation using the rank-reduced channel representation and norm-preserving orthonormal factors. It also establishes vectorization as a linear isometry, supporting volume preservation.

PROOF OF LEMMA 2; CHANNEL-DOMAIN DISTANCE

The channel-domain distance argument uses the rank-reduced representation and singular values, with orthonormal factors preserving the Frobenius norm.The diagonal singular-value matrix has entries σ1 ≥ ··· ≥ σr > 0, with σk ≥ σr.Source: PROOF OF LEMMA 2; CHANNEL-DOMAIN… · p. 38

PROOF OF LEMMA 3; VOLUME PRESERVATION UNDER VECTORIZATION

Vectorization is a linear isometry from the matrix space to its vector representation, preserving inner products and enabling a change-of-variables argument for volume.The proof applies the change-of-variables formula to vec with the constant function g ≡ 1.Source: PROOF OF LEMMA 3; VOLUME PRESERVATION… · pp. 38–39

APPENDIX E

Appendix E uses Stirling’s expansion to obtain asymptotic expressions in the proof of Lemma 4, then establishes the distribution of V_i,j through mean, variance, and Gaussianity arguments in Lemma 5.

PROOF OF LEMMA 4; NEGATIVE LOG-VOLUME OF HYPERSPHERE

Stirling’s expansion and the vanishing-order terms yield asymptotic expressions (177) and (178) in the proof of Lemma 4.The proof notes that 1/2 log(πn) and O(n^-1) are o(n).Source: PROOF OF LEMMA 4; NEGATIVE LOG-VOLUME… · pp. 39–40

PROOF OF LEMMA 5; NOISE-CHANNEL-DISTANCE PRODUCT

The proof of Lemma 5 establishes the mean and variance of V_i,j using zero-mean noise and the additivity of variances for independent noise samples.Each noise sample has variance σ^2.Source: PROOF OF LEMMA 5;… · pp. 40–41
Because V_i,j is a deterministic linear combination of independent Gaussian noise variables, the proof concludes that it is Gaussian.This is the Gaussianity step in the three-part distribution argument.Source: PROOF OF LEMMA 5;… · p. 41