PaperlayerView exact revision on arXiv

Source-linked AI summary

Exact Degrees of Freedom of Spatially Sparse MIMO Channels Without Prior CSI

Yifeng Xiong, Weijiang Zhao, Fan Liu, Shi Jin, Jianhua Zhang

arXiv:2609.07926v2cs.ITeess.SPmath.IT

TL;DR

This paper addresses the question of how to characterize capacity pre-log when both geometric state and transmitted data are unknown. It characterizes the DoF of blockwise memoryless channels without prior CSI, finding 1−1/T for K=1 and K(1−3/(2T)) for K≥2 under specified antenna and block-length conditions.

  • Problem

    The central remaining question is how to characterize capacity pre-log when both geometric state and transmitted data are unknown.

    Source: A. Background and Related Works · p. 3

  • Method

    A quotient-rank argument accounts for the net loss of 3K real dimensions for K≥2.

    Source: §VI · p. 23

  • Results

    For K=1, DoF is 1−1/T; for K≥2, it is K(1−3/(2T)) when N_r≥K+1, N_t≥max{K,2}, and T≥K.

    Source: A. Main Result and Interpretation · p. 7, §VI · p. 23

  • Takeaways & Limitations

    The characterized pre-log applies to arbitrary fixed distinct array positions, including ULAs.

    Source: §VI · p. 23

    For more general array structures, achievability is established, but the converse remains open.

    Source: §V · p. 19

Abstract

from arXiv · show

We characterize the degree of freedom (DoF) of a point-to-point blockwise memoryless channel without prior channel state information (CSI), with a fixed number $K$ of propagation paths, where the transmitter (Tx) and the receiver (Rx) are equipped with nonuniform linear arrays (NULAs) of $N_t$ and $N_r$ antennas, respectively. The positions of array elements are fixed, known, pairwise distinct, and need not be equally spaced. The uniform linear array (ULA) is a special case. In each block of length $T$, the continuous angles of arrival (AoAs), angles of departure (AoDs), and independent complex Gaussian path gains are redrawn. Both Tx and Rx know the state distributions but are not given the current realizations before transmission. The receiver may estimate the channel from reference signals or decode without explicit channel estimation, with reference symbols counted in $T$ and their energy counted against the power constraint. Under the aforementioned model, we show that the DoF is $1-\frac{1}{T}$ for $K=1$, and $K(1-\frac{3}{2T})$ for $K \geq 2$, when $N_r\ge K+1$, $N_t\ge\max\{K,2\}$, and $T\ge K$. The analytical results are further demonstrated by their applications to the DoF tradeoff analysis in integrated sensing and communication (ISAC). For more general array structures, an achievability result is established, while the converse remains open in general.

I. INTRODUCTION · A. Background and Related Works · B. Contributions of this Paper

The paper studies noncoherent communication over continuous geometric channels, characterizing how channel-state uncertainty and transmitted data shape received signals and capacity pre-logs in the NULA-SV model. It also examines antenna and Fourier-mode selection on rectangular continuous apertures and extends achievability to more general finite-ray channels.

A. Background and Related Works

  • The central challenge is determining capacity pre-log when both geometric state and transmitted data are unknown, since some state variations remain distinguishable while others are absorbed into the data or effective gain.Input design affects the visible channel uncertainty, so counting channel parameters alone does not determine the achievable pre-log.Source: A. Background and Related Works · p. 3

B. Contributions of this Paper

  • For fixed regular inputs, the achievable pre-log is the joint-minus-conditional real information-dimension difference divided by twice the block length, with Jacobian ranks enabling evaluation under explicit conditions.The analysis then establishes an upper bound uniform over admissible inputs, including SNR-dependent distributions approaching degenerate inputs.Source: B. Contributions of this Paper · p. 3
  • The exact capacity pre-log is 1 − 1/T for K = 1 and K(1 − 3/(2T)) for K ≥ 2, attained by Gaussian signaling through one antenna or a fixed K-dimensional subarray, respectively.The model retains continuous arrival and departure directions, independent complex Gaussian path gains, and known, pairwise-distinct array positions that need not be equally spaced.Source: B. Contributions of this Paper · p. 3
  • For multistream signaling, the AoA contribution cancels between joint and fixed-data output dimensions, leaving the complex gains and visible departure geometry to determine the noncoherent loss.Source: B. Contributions of this Paper · p. 4
  • A matching converse covers SNR-dependent inputs, including mixtures approaching rank-1 matrices at multiple scales, and bounds reference-signal-assisted schemes to complete the DoF characterization.For NULA-SV, the second singular value controls AoD resolvability and proximity to the rank-1 set; entropy bounds yield the converse.Source: B. Contributions of this Paper · p. 4
  • The achievability analysis extends to general separable finite-ray channels with fixed precoders, deriving a Gaussian-signaling pre-log formula and sufficient conditions from path-response matrices and array-response derivatives.The results are illustrated for uniform planar arrays.Source: B. Contributions of this Paper · p. 4
  • The paper studies UPA antenna and Fourier-mode selection on rectangular continuous apertures, showing that the transmit subspace controls distinguishable directional uncertainty.Source: B. Contributions of this Paper · p. 4

II. CHANNEL MODEL AND THE MAIN RESULT … A. Proof Architecture

The paper establishes exact no-CSI degrees of freedom for spatially sparse channels under fixed-array and blockwise state assumptions, and outlines an information-dimension proof for achievability. The result is limited when angular subspaces persist across gain blocks.

II. CHANNEL MODEL AND THE MAIN RESULT

  • The model permits fixed, known, pairwise-distinct array element positions, with uniform spacing as a special case.The element positions are fixed as SNR changes; the array origin can be normalized without restricting geometry.Source: §II · p. 5
  • The transmitter and receiver know the state distributions but not the current block’s realization, and channel states are independently redrawn across blocks.If angles are shared across gain blocks, the transition kernel no longer factors and the theorem does not apply.Source: §II · p. 6

A. Main Result and Interpretation

  • Theorem 1 addresses the degrees of freedom of the NULA-SV channel without CSI under Assumption 1, for Nr ≥ K + 1, Nt ≥ max{K, 2}, and T ≥ K.Source: §II · p. 6, A. Main Result and Interpretation · p. 7
  • For K ≥ 2, the K receive-direction dimensions cancel between the joint and fixed-data output images; the remaining loss reflects 2K real gain dimensions and K AoD dimensions.The cancellation reflects locally distinguishable AoA variations in both output maps on probability-1 regular sets.Source: A. Main Result and Interpretation · p. 7
  • Single-antenna input distributions attain 1 − 1/T for every K, and at K = T = 2 this matches the capacity pre-log 1/2 attained by a two-stream Gaussian input.Thus, pre-log optimality does not generally require a unique input rank.Source: A. Main Result and Interpretation · p. 7

A. Proof Architecture

  • For K ≥ 2, the proof uses a fixed Gaussian input and compares real Jacobian ranks of 2KT + K jointly varying dimensions and 4K dimensions conditioned on fixed input.It verifies absolute continuity, constant rank on probability-1 open sets, and finite output second moment so the ranks equal the corresponding information dimensions.Source: A. Proof Architecture · pp. 7–8

B. R´enyi Information Dimension and Its Conditional Version … E. The Jacobian Rank of the Gaussian Input

The section develops a framework connecting Jacobian ranks, information dimensions, and high-SNR pre-logs, and applies it to regularity and rank analysis for Gaussian inputs in spatially sparse channels. It presents achievable pre-logs and strategies, while clarifying that Gaussian-input achievability alone is not a capacity statement.

B. R´enyi Information Dimension and Its Conditional Version

  • R´enyi information dimension measures effective real dimension through fine-quantization entropy growth; it is zero for discrete laws and r for absolutely continuous laws on R^r,.The section adopts Geiger and Koch’s conditional information dimension [28].Source: B. R´enyi Information Dimension and… · p. 8

C. From Constant Rank to Information Dimension

  • Under the stated regularity conditions, information dimensions of the joint and conditional images equal their Jacobian ranks, a transfer result needed because parameter non-identifiability prevents reusing an existing theorem [14].The conditions include an almost-everywhere absolutely continuous parameter distribution and a C1 map on an open, full-probability domain.Source: C. From Constant Rank to Information… · pp. 8–9

D. From Information Dimensions to an Achievable Pre-Log

  • The achievable high-SNR pre-log is the joint image rank minus the nuisance-only rank, measuring the output dimension added by the data-bearing variable.The result assumes the stated transfer conditions and independent Gaussian noise with deterministic positive-definite covariance that does not depend on SNR.Source: D. From Information Dimensions to an… · p. 10

E. The Jacobian Rank of the Gaussian Input

  • For almost every set of directions, distinct array-manifold frequencies yield rank K when N ≥ K and rank K + 1 for each augmented matrix when N ≥ K + 1.Direction distinctness alone does not guarantee these ranks deterministically for an arbitrary NULA; the exceptional events have probability zero under (U1).Source: E. The Jacobian Rank of the Gaussian… · p. 10
  • When S is unknown, the observations leave a 3K-dimensional local ambiguity corresponding to the AoDs and complex gains; fixing full-row-rank S removes it on the regular set.The candidate input uses the semi-unitary antenna-selection matrix B_K and S ∈ C^(K×T).Source: E. The Jacobian Rank of the Gaussian… · pp. 10–11
  • With an absolutely continuous full-row-rank Gaussian input S, the joint output map has constant real Jacobian rank 2KT + K, while its conditional channel-state map has rank 4K.These maps are C1 on the probability-one regular set.Source: E. The Jacobian Rank of the Gaussian… · p. 11
  • For K≥2 under Assumption 1, the Gaussian-input map satisfies regularity conditions (A1)–(A3) of Definition 2.The map is (ur, ut, α, S) ↦ H(ur, ut, α)BKS.Source: E. The Jacobian Rank of the Gaussian… · p. 11
  • For K≥2 under Assumption 1, Proposition 2 establishes an achievable pre-log for the Gaussian input.The supplied passage does not state the pre-log value.Source: E. The Jacobian Rank of the Gaussian… · p. 11
  • Proposition 2 is not a capacity statement because an optimizing input may lie on a lower-rank subset or have support that varies with SNR.Source: E. The Jacobian Rank of the Gaussian… · p. 12
  • For every K≥1, the fixed input X = e1sT with s ∼CN(0, IT ) independent of channel and noise achieves a per-channel-use pre-log of at least 1−1/T.Source: E. The Jacobian Rank of the Gaussian… · p. 12
  • The single-stream strategy removes AoA and AoD from the scalar observation, leaving only one complex fading coefficient.Selecting the first transmit and receive antennas gives fixed-position antenna phases.Source: E. The Jacobian Rank of the Gaussian… · p. 12

IV. CONVERSE VIA GENIE-AIDED BOUNDS · C. The Second Singular Value as a Common Scale for Rank-1 Proximity and AoD Resolvability

Genie-aided observations yield converse bounds that cover SNR-varying input distributions, treating K = 1 separately from K ≥ 2. For K ≥ 2, the second singular value links proximity to rank-1 inputs with AoD resolvability, supporting complementary output-entropy bounds.

IV. CONVERSE VIA GENIE-AIDED BOUNDS

  • The pathwise genie upper-bounds achievable mutual information for every admissible input distribution through a Markov-chain reduction.Its observation is physically unobservable, but the original channel output can be stochastically generated from it given the input.Source: §IV · p. 12
  • The converse establishes the exact K = 1 DoF by combining the single-path geometric-genie bound with the achievability result.The genie reveals the complete geometry; projection onto the receive steering vector then reduces the channel to scalar noncoherent block fading.Source: §IV · pp. 12–13
  • For K ≥ 2, the proof uses the pathwise observation because the geometric genie is too strong to recover the desired multiple-path pre-log.It first analyzes a fixed-amplitude shell with σ1(X) ≍ 1 and a common scale for σ2(X).Source: §IV · p. 13
  • Arbitrary bounded-peak inputs are handled by jointly binning log(ρσ2), and the argument is then extended to the average-power constraint.This addresses the obstacle that the optimal input distribution may depend on ρ.Source: §IV · p. 13
  • For K ≥ 2, the fixed-scale bound is D(a) = (1 − b)(T − 1) + bD⋆, with b = (1 − a)+ and D⋆ ≥ T − 1.Thus the bound is at most D⋆ at every second-singular-value scale.Source: §IV · p. 13

C. The Second Singular Value as a Common Scale for Rank-1 Proximity and AoD Resolvability

  • The second singular value controls distance to the rank-1 input set and, in turn, higher singular values and approaches to that set.This supplies a common scale for rank-1 proximity and AoD resolvability on a fixed-amplitude shell.Source: C. The Second Singular Value as a… · pp. 14–15
  • A uniform AoD-sensitivity result lower-bounds resolvability using σ2(X), with its constant depending only on fixed transmit positions, Nt, and T.The Wronskian construction establishes this result for the fixed NULA.Source: C. The Second Singular Value as a… · p. 14
  • Together, the two meanings of σ2 support an output-entropy upper bound and a conditional output-entropy lower bound, respectively.Greater resolvable AoD variation raises the conditional output lower bound, while smaller σ2 tightens the rank-1-tube output upper bound.Source: §IV · p. 13, C. The Second Singular Value as a… · p. 15

D. Conditional-Output Entropy from AoD Resolvability · E. Output Entropy from Rank-1 Tube Concentration

Section D turns AoD resolvability into uniform single-path and shell-wise conditional-output entropy lower bounds. Section E uses rank-1 tube concentration to upper-bound output entropy and obtain single-scale and multiscale DoF bounds.

D. Conditional-Output Entropy from AoD Resolvability

  • The projective preimage-volume bound yields an ε² gain cross section and an admissible AoD-length factor min{1, ε/(|α|δ)}, producing a conditional-entropy lower-bound route.The bound holds on a good set whose excluded measure is O(τ^κ).Source: D. Conditional-Output Entropy from AoD… · pp. 15–16
  • Proposition 7 gives a uniform noisy single-path fixed-input output-entropy lower bound under bounded, nondegenerate AoD density and independent Gaussian gain and noise assumptions.For sufficiently large ρ, its o(log ρ) remainder can be replaced by a deterministic function that is o(log ρ) uniformly over the shell.Source: D. Conditional-Output Entropy from AoD… · p. 16
  • The pathwise-genie argument lifts the single-path bound to a shell-wise conditional-output entropy bound valid for arbitrary distributions of σ2(X), with a separate single-scale specialization.It sums independent path contributions and averages over X; the expectation bound does not require a single σ2(X) scale.Source: D. Conditional-Output Entropy from AoD… · p. 16

E. Output Entropy from Rank-1 Tube Concentration

  • When σ2(X) is small, the noiseless output lies near the rank-1 output set, and the pathwise channel map takes a rank-1 input tube to a corresponding output tube.A constraint on σ2 and a Frobenius-norm rank-1 tube differ only by a dimension-dependent radius change.Source: E. Output Entropy from Rank-1 Tube… · p. 17
  • h(Zρ) ≤ (K + T − 1) log ρ + (KT − K − T + 1) log(1 + ρδ^2) + o(log ρ) uniformly over the stated rank-1 tube distributions.For ρ ≥ e^2, the remainder is bounded by C(1 + log log ρ); C1 and δmax are fixed independently of ρ, while δ may vary with ρ.Source: E. Output Entropy from Rank-1 Tube… · p. 18
  • Under the fixed-amplitude shell and single-scale condition, the input tube radius is O(ρ^−a/2); combining the tube entropy upper bound with the conditional lower bound yields a single-scale DoF upper bound.The bound is expressed using D(a), defined in (23).Source: E. Output Entropy from Rank-1 Tube… · p. 18
  • The multiscale extension applies the two component bounds within every cell of the two-dimensional effective-SNR grid.This carries the shell-wise expectation bound into the generic multiscale case.Source: D. Conditional-Output Entropy from AoD… · p. 16, E. Output Entropy from Rank-1 Tube… · p. 18

F. Full Multiscale Converses · V. DISCUSSIONS · A. Generic Achievability

The section develops multiscale converse arguments for peak and average power, then discusses the scope of the capacity result and gives rank-based achievability results for more general array structures.

F. Full Multiscale Converses

  • For K ≥2 and T ≥K, Theorem 2 establishes a multiscale converse under peak power by partitioning logarithmic coordinates into rectangular cells.Inputs in the low-SNR cell contribute only O(∆log ρ) mutual information.Source: F. Full Multiscale Converses · pp. 18–19
  • The average-power converse follows by splitting at a growing input-power threshold, rescaling the typical event, and controlling the high-power tail with Markov’s inequality.Lemma 12 extends the bound (40) under Theorem 2’s conditions.Source: F. Full Multiscale Converses · p. 19

V. DISCUSSIONS

  • The capacity result is specific to the blockwise memoryless NULA-SV channel, while its achievability argument extends to more general array structures for fixed input distributions.The discussion also addresses implications for integrated sensing and communication (ISAC).Source: §V · p. 19

A. Generic Achievability

  • For a fixed full-column-rank precoder and independent Gaussian input, Proposition 8 expresses the achieved pre-log through real ranks of the joint and conditional noiseless output maps.Its assumptions include an absolutely continuous state distribution and constant ranks on a probability-1 open set.Source: A. Generic Achievability · p. 20
  • Proposition 9 replaces the geometric condition with sufficient rank conditions and computes explicit ranks for the resulting Gaussian-input achievability.The conditions require receive-side parameter variations to change the receive subspace and count transmit variations after removing gain-absorbable complex scaling.Source: A. Generic Achievability · pp. 20–21
  • For the NULA subarray precoder, Lemma 3 verifies the required receive ranks and gives q_BK = 1 for K ≥2.Source: A. Generic Achievability · p. 21

B. Examples of Other Array Structures … VI. CONCLUSION

The paper applies quotient-rank analysis to other array structures and an ISAC tradeoff, then concludes with the NULA-SV capacity pre-log result and the limits of extending its converse.

B. Examples of Other Array Structures

  • The two other-array examples evaluate qB under an absolutely continuous joint state distribution, with uniformly bounded responses and additional conditions verified to support the analysis.The text says those conditions ensure (G2) through the computed qB and Proposition 9.Source: B. Examples of Other Array Structures · p. 21

1) Uniform planar array with antenna selection:

  • For a UPA with antenna selection, selecting at least three noncollinear elements gives qB = 2, while selecting merely collinear elements with L ≥2 gives qB = 1.The example (0, 0), (1, 0), (0, 1) retains both dimensions.Source: 1) Uniform planar array with antenna… · p. 21

2) Rectangular aperture with Fourier-mode selection:

  • For a rectangular aperture with Fourier-mode selection, a single mode gives qB = 0, while at least two modes sharing a second index give qB = 1 almost everywhere.Selecting (0, 0), (1, 0), and (0, 1) gives qB = 2 generically.Source: 2) Rectangular aperture with… · p. 22
  • The quotient-rank method provides useful achievable DoF lower estimates for generic array structures, but useful converse bounds and optimality proofs remain difficult in general.The UPA and rectangular-aperture examples illustrate achievability; matching converses for additional degeneracy sets remain future work.Source: 2) Rectangular aperture with… · p. 22, §VI · p. 24

C. Application to ISAC: DoF Tradeoff

  • In NULA-SV, unknown-CSI DoF loss is at most 3K/2T, below the K^2/2T sensing-constraint loss for K ≥ 4 when the sensing-optimal sample covariance is unique; thus, a sensing-communication DoF tradeoff may persist without communication-receiver CSI.Source: C. Application to ISAC: DoF Tradeoff · pp. 22–23
  • The ISAC example considers estimating the full target response matrix with linear estimators and MSE, for which the LMMSE estimator is optimal.The discussion limits these findings to the specific example, especially the use of linear estimators.Source: C. Application to ISAC: DoF Tradeoff · p. 23

VI. CONCLUSION

  • For K ≥2, the quotient-rank argument yields a net loss of 3K real dimensions attained by fixed K-antenna Gaussian input; for K = 1, single-antenna transmission reduces the channel to scalar block fading.The uniform converse rules out higher DoF from input distributions that depend on SNR.Source: §VI · p. 23

APPENDIX A INFORMATION-DIMENSION RESULTS … C. Proof of Lemma 5

The appendices establish information-dimension and Jacobian-rank results using local chart and analytic-rank arguments. They also derive the stated mutual-information bound and verify local differential identifiability under the required array conditions.

A. Proof of Lemma 1

  • On each regular chart, the output has information dimension r by absolute continuity and bi-Lipschitz invariance [10],.The proof constructs a countable cover of the probability-one regular set using the constant-rank theorem and second countability [39, Thm. 4.12].Source: A. Proof of Lemma 1 · p. 24
  • The countable-chart argument, with tail contributions controlled using finite second moments, yields dI(z) = r.The proof also invokes finite entropy of the integer parts to establish the dimension identity [28, Lemma 2].Source: A. Proof of Lemma 1 · pp. 24–25

B. Proof of Proposition 1

  • Noise independence gives a Markov chain from (θ, u) through z to yρ; whitening, a Gaussian entropy bound, and dominated convergence yield (13).The proof combines Lemmas 1 and 2 before substituting the resulting bound into the mutual-information difference.Source: B. Proof of Proposition 1 · p. 25

A. Proof of Lemma 3

  • Real-analytic minors are nontrivial, so A(u) has full column rank almost everywhere and the augmented matrix [A(u), ȧ(uk)] has rank K + 1 almost everywhere.The proof uses distinct frequencies and a Hermite evaluation matrix to show the relevant minors are nonzero.Source: A. Proof of Lemma 3 · p. 26

B. Proof of Lemma 4

  • Under the stated full-rank and distinct-position conditions, a zero first-order channel variation forces the AoA, AoD, and gain variations to vanish.The proof uses a projector to isolate direction variations and linear independence of each steering vector and its derivative, with Nt ≥ 2.Source: B. Proof of Lemma 4 · pp. 26–27

C. Proof of Lemma 5

  • The transformed map (ur, C) ↦ ArC has real Jacobian rank K + 2KT, and the original joint map has the same local image and rank.The proof establishes that the parameter transformation is a local submersion and uses full column rank of Ar.Source: C. Proof of Lemma 5 · pp. 27–28

D. Proof of Lemma 6 … A. Proof of Proposition 4

The proofs establish the regularity and rank conditions needed for the achievability argument, derive the proposition bounds, and construct a randomized genie channel for Proposition 4. Together, they support the paper’s DoF analysis under the stated channel assumptions.

D. Proof of Lemma 6

  • The conditional Jacobian has rank 4K because its real kernel is trivial on a 4K-dimensional parameter domain.The proof uses distinct selected positions and an invertible matrix to establish the needed linear independence.Source: D. Proof of Lemma 6 · p. 28

E. Proof of Corollary 1

  • The joint and conditional maps satisfy (A2): their Jacobians have ranks 2KT + K and 4K, respectively, on probability-one regular sets.Analyticity and the zero-measure exceptional sets establish the regularity claim.Source: E. Proof of Corollary 1 · p. 28

F. Proof of Lemma 7

  • The Gaussian input and Corollary 1 verify conditions (A1)–(A3), completing the proof of Lemma 7.The argument also invokes the unit norms of the steering vectors and semi-unitarity.Source: F. Proof of Lemma 7 · p. 29

A. Proof of Proposition 2 (APPENDIX C ACHIEVABILITY)

  • Lemma 6 gives ru = 4K for every fixed full-row-rank S; the passage invokes Proposition 1 and invariance of mutual, then introduces the per-channel-use pre-log.Source: A. Proof of Proposition 2 · p. 29

B. Proof of Proposition 3 (APPENDIX C ACHIEVABILITY)

  • A scalar Rayleigh block-fading reduction gives the stated bound for every K ≥ 1, including K = 1 and T = 1.The proof uses only the first receive antenna and an input transmitted through the first transmit antenna.Source: B. Proof of Proposition 3 · p. 30

A. Proof of Proposition 4 (APPENDIX D GENIE BOUNDS)

  • Randomizing the receive steering vectors and adding conditioned Gaussian noise constructs a kernel yielding the Markov chain X → Zρ → eY.The randomization depends on X only through Zρ, and the resulting indexed variables have the specified channel distribution.Source: A. Proof of Proposition 4 · pp. 30–31

B. Proof of Proposition 5 … B. Proof of Lemma 9

These proofs combine a genie-aided scalar-channel bound with uniform geometric and exponential-polynomial estimates. Together, the arguments control constants, zero sets, and projector level sets under fixed array and frequency conditions.

B. Proof of Proposition 5

  • For K = 1, revealing geometry makes the effective output a sufficient statistic for a scalar constant-block noncoherent channel, whose capacity upper bound is uniform over average-power input distributions.The proof combines this bound with preceding bounds to complete the argument.Source: B. Proof of Proposition 5 · p. 31

APPENDIX E AOD RESOLVABILITY AND THE CONDITIONAL-OUTPUT ENTROPY BOUND

  • Converse constants may depend on fixed model parameters, but not on SNR, the input distribution, tube radius, quantization cell, or shell index; Lemma 9 constants depend only on (Λ, I, T), with multiscale constants also depending on (A, ∆).Source: Appendix E · p. 31

A. Proof of Proposition 6

  • For rank X ≤ 1, wX ≡ 0; for rank X ≥ 2, distinct-frequency Vandermonde independence makes κ positive on the compact Stiefel manifold, giving it a positive minimum.Source: A. Proof of Proposition 6 · p. 32

B. Proof of Lemma 9

  • For fixed distinct frequencies Λ on compact interval I, every nonzero exponential polynomial has uniformly bounded zeros and sublevel sets, with constants depending only on (Λ, I), using Turán–Nazarov [41, Theorem I].Source: B. Proof of Lemma 9 · pp. 32–33
  • Applying the sublemma to v and w with frequency sets Λ and Λ + Λ yields a bad-set bound |I\Gτ| ≤ Cτ^κ.Source: B. Proof of Lemma 9 · p. 33
  • The proof controls projector-coordinate level sets using exponential polynomials with frequencies in Λ − Λ, a one-dimensional area formula, and Fubini’s theorem to obtain its final bound.Source: B. Proof of Lemma 9 · p. 34

C. Gaussian quantization lower bound … A. Proof of Lemma 10

The appendix gives a Gaussian quantization lower-bound result and proves a uniform remainder bound for Proposition 7. It also derives the rank-1 manifold dimension and a covering bound for its tube.

C. Gaussian quantization lower bound

  • Lemma 13 gives a Gaussian quantization lower bound when x is independent of standard Gaussian noise and H(floor x) is finite, with a finite constant depending only on n for s ≥ 1.The statement also holds for fixed positive-definite Gaussian noise covariance, with the constant allowed to depend on that covariance.Source: C. Gaussian quantization lower bound · p. 35

D. Proof of Proposition 7

  • The Proposition 7 proof combines shell bounds with angular and Gaussian small-ball estimates to bound the output-entropy contribution across the shell.The argument applies a directional estimate with δ = c*σ2 and τρ = e^−√log ρ, then uses the angular estimate and Gaussian small-ball bound.Source: D. Proof of Proposition 7 · p. 36
  • R_sp(ρ)/log ρ → 0, and this deterministic remainder is independent of the entire matrix sequence.This is the concluding uniform remainder property in the proof of Proposition 7.Source: D. Proof of Proposition 7 · p. 37

APPENDIX F RANK-1-TUBE OUTPUT ENTROPY BOUNDS

  • The rank-1-tube output-entropy appendix sets m = 2KT and d1 = 2(K + T − 1).Source: Appendix F · p. 37

A. Proof of Lemma 10

  • The unit-norm rank-1 matrix manifold has real dimension (2K − 1) + (2T − 1) − 1 = d1 − 1, with the rank loss arising from common phase ambiguity.Source: A. Proof of Lemma 10 · p. 37
  • The tube covering bound is O(R/ϵ)^d1(1 + r/ϵ)^(m−d1), with constants depending only on (K, T).The proof covers the tube using coordinates for scale, the rank-1 manifold, and the orthogonal residual.Source: A. Proof of Lemma 10 · p. 38

B. Proof of Lemma 11 · APPENDIX G MULTISCALE CONVERSE AND FINAL ASSEMBLY · A. Proof of Theorem 2

The proofs establish a uniform tube-entropy remainder, derive a distribution-independent converse bound through multiscale cells, and assemble the converse with achievability.

B. Proof of Lemma 11

  • Lemma 11 uses a finite ρ^-1/2-cover on the good event, where the scaled residual has norm at most 1, to bound conditional output entropy uniformly over cover indices.The cover-size bound combines a rank-one direction with the remaining dimensions.Source: B. Proof of Lemma 11 · p. 39
  • The tube-entropy bound has remainder R_tube(ρ) = C(1 + log log ρ) for ρ ≥ e², uniformly over input distributions and 0 ≤ δ ≤ δ_max.The proof first fixes M and then chooses c sufficiently large to control gain tails.Source: B. Proof of Lemma 11 · pp. 39–40

APPENDIX G MULTISCALE CONVERSE AND FINAL ASSEMBLY

  • Appendix G proceeds from multiscale cell bounds uniform over peak-power-constrained inputs, to lifting away the peak-power constraint, and then to assembling the converse with achievability.The multiscale theorem applies the two component bounds inside finitely many two-dimensional cells.Source: Appendix G · p. 40

A. Proof of Theorem 2

  • Theorem 2 partitions the scale intervals into cells; for fixed A and ∆, their label count is O_A,∆(1), so label entropy is o(log ρ).The low-SNR cell contribution can be neglected as ∆ tends to zero.Source: A. Proof of Theorem 2 · pp. 40–41
  • On each positive-probability grid cell, h(Zρ | κ∆ = C) ≤ (K + T − 1)s + (KT − K − T + 1)q + o_A,∆(log ρ).This is the conditional output-entropy bound used in the multiscale converse.Source: A. Proof of Theorem 2 · p. 41
  • For the conditional-output lower bound, normalizing by σ1(X) makes the lower-SNR observation a stochastic degradation of the actual one, and Proposition 7 is applied across K conditionally independent rows.Data processing makes the output entropy nondecreasing in SNR.Source: A. Proof of Theorem 2 · pp. 41–42
  • p_k(ρ; A) ≤ D⋆log ρ + C_A,K,T(1 + ∆log ρ) + C_A,∆ + R_A,∆(ρ), independently of the input distribution.The bound is obtained after informing the receiver of the cell label.Source: A. Proof of Theorem 2 · p. 42

B. Proof of Lemma 12 · C. Proof of Theorem 1

The proofs derive an input-independent finite-SNR mutual-information upper bound and use upper and lower bounds to establish the capacity pre-log limits in Theorem 1.

B. Proof of Lemma 12

  • Revealing whether Eϵ occurs gives a decomposition of the mutual-information bound according to an event determined by the input.This event label is revealed because Eϵ is determined by X.Source: B. Proof of Lemma 12 · p. 42
  • On the high-power tail, revealing Gpath and using Gaussian maximum-entropy and Jensen bounds leads to a uniform bound on the weighted tail term.The argument also uses that conditioning on the input-determined event preserves independence between X and Gpath.Source: B. Proof of Lemma 12 · p. 43
  • On Eϵ, the proof rescales X to a unit-Frobenius-ball input and relates the conditional channel to the definition of pk(γ; 1).It chooses eX = ρ^-ϵ/2 X and γ = ρ^(1+ϵ), then invokes the asymptotic bound in (40).Source: B. Proof of Lemma 12 · p. 43
  • Combining the bounds yields an input-independent finite-SNR mutual-information upper bound with displayed leading term (D⋆+η)(1+ϵ) log ρ.The displayed inequality also includes log 2 and a term involving ρ^-ϵ; the supplied passage truncates the remainder.Source: B. Proof of Lemma 12 · p. 43

C. Proof of Theorem 1

  • For K = 1, the proof combines Proposition 3’s fixed-input lower bound with a capacity upper bound to obtain the asserted value.The proof states that these bounds establish the limit and its asserted value.Source: C. Proof of Theorem 1 · p. 44
  • For K ≥ 2, Proposition 4 supplies an upper bound for every admissible input, while Proposition 2 provides an admissible Gaussian input for the lower bound.The proof takes the supremum under the average-power constraint and applies Lemma 12 to the upper bound.Source: C. Proof of Theorem 1 · p. 44
  • The liminf lower bound and limsup upper bound coincide, so the capacity pre-log limit exists also for K ≥ 2, proving (8) in both cases.Source: C. Proof of Theorem 1 · p. 44

APPENDIX H PROOFS FOR OTHER ARRAY STRUCTURES … 2) Rectangular aperture with Fourier-mode selection:

The appendix proves rank-based conditions for the array propositions and verifies them for UPA antenna selection and rectangular-aperture Fourier-mode selection.

A. Proof of Proposition 8

B. Proof of Proposition 9

C. Verification of the Array Examples

1) UPA with antenna selection:

2) Rectangular aperture with Fourier-mode selection:

Related papers

Loading 2609.07926v2…