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On the Fundamental Tradeoff of Integrated Sensing and Communications Under Gaussian Channels

Yifeng Xiong, Fan Liu, Yuanhao Cui, Weijie Yuan, Tony Xiao Han, Giuseppe Caire

arXiv:2204.06938v8cs.IT

TL;DR

This paper studies the fundamental tradeoff between communication capacity and sensing CRB in point-to-point ISAC. It introduces a CRB-rate region, characterizes its corner points and bounds, and reveals subspace and deterministic-random tradeoffs shaping ISAC signaling.

  • Problem

    The paper addresses how to characterize the fundamental tradeoff between communication capacity and sensing CRB in simultaneous point-to-point ISAC systems.

  • Method

    The paper introduces the CRB-rate region, analyzes its sensing- and communication-optimal corner points, and constructs inner bounds using Gaussian, Stiefel-manifold, and covariance-shaping strategies.

  • Results

    The analysis yields a pentagon inner bound and shows that ISAC tradeoffs depend on waveform subspace alignment and deterministic-random signaling.

  • Takeaways & Limitations

    The results provide analytical guidance for designing practical Pareto-optimal ISAC signaling strategies under resource allocation and modulation choices.

Abstract

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ISAC is recognized as a promising technology for the next-generation wireless networks, which provides significant performance gains over individual S&C systems via the shared use of wireless resources. The characterization of the S&C performance tradeoff is at the core of the theoretical foundation of ISAC. In this paper, we consider a point-to-point ISAC model under vector Gaussian channels, and propose to use the CRB-rate region as a basic tool for depicting the fundamental S&C tradeoff. In particular, we consider the scenario where a unified ISAC waveform is emitted from a dual-functional ISAC Tx, which simultaneously performs S&C tasks with a communication Rx and a sensing Rx. In order to perform both S&C tasks, the ISAC waveform is required to be random to convey communication information, with realizations being perfectly known at both the ISAC Tx and the sensing Rx as a reference sensing signal as in typical radar systems. As the main contribution of this paper, we characterize the S&C performance at the two corner points of the CRB-rate region, namely, $P_{SC}$ indicating the max. achievable rate constrained by the min. CRB, and $P_{CS}$ indicating the min. achievable CRB constrained by the max. rate. In particular, we derive the high-SNR capacity at $P_{SC}$, and provide lower and upper bounds for the sensing CRB at $P_{CS}$. We show that these two points can be achieved by the conventional Gaussian signaling and a novel strategy relying on the uniform distribution over the Stiefel manifold, respectively. Based on the above-mentioned analysis, we provide an outer bound and various inner bounds for the achievable CRB-rate regions. Our main results reveal a two-fold tradeoff in ISAC systems, consisting of the subspace tradeoff (ST) and the deterministic-random tradeoff (DRT) that depend on the resource allocation and data modulation schemes employed for S&C, respectively.

I. INTRODUCTION … III. MAIN RESULTS

The paper formulates ISAC’s fundamental tradeoff through a CRB-rate region for a point-to-point vector Gaussian system with a shared, sensing-known random waveform. It characterizes the region’s corner points and identifies subspace and deterministic-random tradeoffs governing resource allocation and signal distributions.

  • A. Background and Related Works: ISAC shares one hardware platform and radio waveform across communications and sensing, improving energy, spectral, and hardware efficiencies.
  • B. Contribution of This Paper: The study analyzes communication capacity and sensing CRB jointly in a point-to-point system where one random waveform serves both receivers.
  • B. Contribution of This Paper: The sensing-optimal CRB requires deterministic sample-covariance trace and support restricted to the optimal deterministic CRB-minimization solution set.
  • III. MAIN RESULTS: The paper derives high-SNR capacity at P_SC, bounds sensing CRB at P_CS, and reveals subspace and deterministic-random tradeoffs with outer and inner CRB-rate bounds.Uniform sampling over the Stiefel manifold asymptotically achieves the sensing-optimal communication capacity.
  • A. System Model: The model uses communication and sensing channel matrices with a transmitted dual-functional waveform known to the transmitter and sensing receiver.
  • B. S&C Performance Metrics: Communication is measured by ergodic achievable rate, while sensing accuracy is measured by a Miller-Chang-type Bayesian CRB relevant when the waveform is known at sensing reception.
  • B. S&C Performance Metrics: The CRB-rate region contains feasible ordered pairs of sensing CRB and communication rate, with its boundary representing Pareto-optimal performance tradeoffs.
  • III. MAIN RESULTS: The main results establish a pentagon inner bound achievable by time-sharing between the two CRB-rate corner-point strategies.The corner points are the sensing- and communication-oriented operating points, with degenerate cases allowing R_SC = 0 or ε_CS = ∞.

A. The Structure of Sensing-optimal Signals

The section characterizes sensing-optimal transmit covariances through the BFIM structure and a fixed-trace reformulation. It shows that minimum CRB requires a deterministic covariance trace, with uniqueness characterized by rank conditions and accompanied by a rank upper bound.

  • BFIM characterization: Proposition 2 establishes the BFIM structure for the sensing parameter η conditioned on the transmitted signal X.The BFIM includes signal-dependent terms and a prior-distribution contribution.
  • Fixed-trace reformulation: When tr{R_X} = γ is fixed, Corollary 1 provides an alternative representation of the sensing objective Φ(·).The reformulation uses matrices defined in the corollary and has an auxiliary rank bounded by r_3 ⩽ KM.
  • Sensing-optimal covariance: The minimum achievable CRB ε_min requires tr{R_X} to be deterministic, with the support of p(R_X) restricted to solutions of a deterministic convex optimization problem.If that optimization has a unique solution, the sensing-optimal covariance R_X is itself deterministic.
  • Uniqueness conditions: Uniqueness holds if and only if Ξ(U*opt ⊗ U_opt) has full column rank for a maximum-rank optimal solution R_opt.Additional sufficient conditions cover generic cases, K ⩾ M, and K = 1.
  • Rank characterization: When the sensing-optimal covariance is unique, Corollary 2 upper-bounds its rank.The supplied passage states the existence of the bound but does not include its explicit expression.

B. Point PSC–Achieving Strategy … IV. DISCUSSIONS

The paper characterizes the two CRB-rate corner strategies: PSC uses sensing-optimal covariance and Stiefel-manifold signaling, while PCS uses communication-optimal Gaussian signaling and incurs quantifiable sensing-DoF loss. The discussion emphasizes how covariance structure, channel knowledge, and modulation determine these tradeoffs.

  • B. Point PSC–Achieving Strategy: The PSC optimization is solved case by case, with the optimal sample covariance matrices denoted RSC; its detailed structure is deferred to future work.This leaves the general optimal covariance characterization open beyond the notation introduced for RSC.
  • X |Hc) + I(Yc; X|Hc, RSC: When the sensing-optimal covariance is unique, Theorem 1 characterizes the PSC rate in the high-SNR regime.The theorem applies when the sensing-optimal sample covariance matrix is unique and PT/σ2 is in the high-SNR regime.
  • X |Hc) + I(Yc; X|Hc, RSC: At high SNR, PSC is asymptotically achieved by a waveform whose data matrix Q is uniformly sampled from a rescaled Stiefel manifold.Q contains the modulated data and is semi-unitary, satisfying QQH = IMSC.
  • X |Hc) + I(Yc; X|Hc, RSC: A PSC-achieving waveform need not incorporate Hc: Corollary 4 constructs XSC from the eigendecomposition of the sensing-optimal covariance and uniform sampling of Q.The construction uses Us and Λs from the eigendecomposition of the sensing-optimal covariance, while Q is sampled from V.
  • C. Point PCS–Achieving Strategy: The PCS strategy uses the communication-only capacity-achieving transmission, with water filling over each Hc realization and Gaussian columns distributed as CN(0, eRCS).This produces the communication-limited minimum CRB after maximizing the communication rate.
  • C. Point PCS–Achieving Strategy: Theorem 2 bounds the high-SNR sensing CRB for independent, identically distributed circularly symmetric complex Gaussian columns under its stated invertibility conditions.The conditions require existence of [Φ(RX) − eJP]−1 almost surely and invertibility of eJP.
  • C. Point PCS–Achieving Strategy: Communication-optimal signaling can lose up to min{K, rank(eRX)} sensing DoF, while a random covariance with sensing DoF νs has the CRB of deterministic covariance observations with T = νs.When RX = eRX, the sensing DoF is T, interpreted as the number of independent observations.
  • IV. DISCUSSIONS: The discussion section interprets the preceding PSC and PCS results as implications of the proposed CRB-rate tradeoff.It explicitly frames the section as discussing the intuitions and implications of the results from Section III.

A. S&C Tradeoff as a Two-fold Tradeoff: ST and DRT … 2) Statistical covariance shaping:

The paper characterizes ISAC’s two-fold tradeoff through subspace alignment and signal randomness, quantifies their DoF effects, and constructs inner bounds connecting the corner points via time sharing and covariance shaping.

  • A. S&C Tradeoff as a Two-fold Tradeoff: ST and DRT: The S&C tradeoff comprises a subspace tradeoff and a deterministic-random tradeoff.The former depends on covariance alignment, while the latter depends on the signal’s randomness.
  • 1) Subspace tradeoff (ST): Covariance alignment with the sensing subspace improves sensing performance at the cost of communication performance, and vice versa.Us characterizes the sensing subspace, whereas Uc characterizes the communication subspace.
  • 2) Deterministic-random tradeoff (DRT): Higher communication rates favor increasingly random waveforms, whereas sensing favors deterministic signals for stable performance.At the sensing-optimal point, semi-unitary signal rows sacrifice communication DoFs; communication-optimal signals sacrifice sensing DoFs.
  • 3) ST and DRT in terms of DoF: The communication DoF efficiency at PSC is reduced by the deterministic-random term (1 − MSC/2T), which is below 1 for finite T.The communication subspace overlap coefficient α(e_RX) lies in [αSC, 1] and indicates the subspace tradeoff.
  • 3) ST and DRT in terms of DoF: At PCS, the sensing DoF is lower bounded by T − min{K, rank(e_RX)}, with sensing DoF loss determined by min{K, rank(e_RX)}.When K ≤ rank(e_RX), the maximum sensing DoF loss at PCS is MCS; when K > rank(e_RX), it becomes K.
  • B. Achievable Inner Bounds Connecting PSC and PCS: The genuine CRB-rate boundary is generally intractable for vector Gaussian channels because mutual information and CRB terms lack explicit tractable forms.The paper therefore develops separate or combined strategies to obtain inner bounds.
  • 1) Time sharing: Time sharing convexifies achievable inner bounds by selecting one signaling scheme with probability p and another with probability 1 − p.This generalizes the pentagon construction to arbitrary pairs of achievable CRB-rate points.
  • 2) Statistical covariance shaping: Statistical covariance shaping uses α ∈ [0, 1] to balance sensing and communication, yielding Gaussian, semi-unitary, and time-shared semi-unitary–Gaussian inner bounds.When the optimal objective in (49) is not identical for all α ∈ [0, 1], the semi-unitary–Gaussian inner bound is tighter than the pentagon inner bound.

C. The Connection Between Existing ISAC Schemes and PSC– & PCS–Achieving Strategies … V. CASE STUDY

The paper relates existing sensing- and communication-centric ISAC designs to CRB-rate boundary strategies, showing how waveform structure and power allocation affect sensing and communication performance. It also distinguishes the PSC-achieving semi-unitary strategy from USTM through sensing alignment, optimality rationale, and communication degrees of freedom.

  • C. The Connection Between Existing ISAC Schemes and PSC– & PCS–Achieving Strategies: Existing ISAC designs are interpreted through their relationships with CRB-rate boundary-approaching strategies.
  • 1) Sensing-Centric Designs:: Sensing-centric schemes incorporate communication into radar infrastructures while preserving sensing performance.
  • 1) Sensing-Centric Designs:: Index modulation uses deterministic semi-unitary codewords and permutation matrices, producing deterministic sample covariance.
  • 1) Sensing-Centric Designs:: log2 M! bits per transmission is the index-modulation throughput, typically below the sensing-optimal capacity asymptotically achieved by uniform Stiefel-manifold signaling.
  • 2) Communication-Centric Designs:: Communication-centric OFDM designs use protocol-compatible waveforms whose symbols are independently selected from fixed constellations such as PSK and QAM.
  • 2) Communication-Centric Designs:: Equal power allocation across subcarriers is required for optimal CIR estimation, although statistical and sample covariance matrices need not coincide.
  • D. Semi-unitary Signalling: The PSC–Achieving Strategy vs. Non-coherent Communication: The PSC-achieving semi-unitary strategy resembles USTM but aligns precoding with the sensing subspace and obtains communication optimality through uniform Stiefel-manifold sampling.
  • D. Semi-unitary Signalling: The PSC–Achieving Strategy vs. Non-coherent Communication: M^2/T is the communication DoF loss for the PSC-achieving strategy, half the non-coherent loss when MSC = M.

A. Target Angle Estimation · 1) Sensing DoF: · 2) Communication DoF:

The angle-estimation analysis models a single-target MIMO radar with nuisance echo amplitude and characterizes sensing and communication DoF at the CRB-rate corner points. It shows that sensing-optimal signaling can impose a communication DoF loss, while covariance shaping and signaling choices yield multiple CRB-rate inner and outer bounds.

  • A. Target Angle Estimation: The sensing model estimates target angle θ from a co-located MIMO radar response, with transmitting and receiving steering-vector mappings and a single target assumed.The target response uses θ, complex echo amplitude α, and conjugate-symmetric transmit and receive arrays.
  • A. Target Angle Estimation: The angle’s equivalent Bayesian Fisher information treats the complex amplitude α as a nuisance parameter under independent priors and circular symmetry.Circular symmetry gives E{α} = E{α∗} = 0 and simplifies the cross-information term.
  • 1) Sensing DoF:: At P_CS, the single-target sensing DoF is lower-bounded by (78), and the bound is achieved when the communication channel H_c is rank-1.For rank-1 H_c, the communication-optimal covariance is also rank-1, enabling the stated equality condition.
  • 2) Communication DoF:: When the largest eigenvalue of M has multiplicity 1, the sensing-optimal sample covariance R_X^SC is unique and rank-1.The sensing-optimal covariance concentrates its columns in the eigenspace associated with M’s largest eigenvalue.
  • 2) Communication DoF:: (2T −1)/2T is the communication DoF, with a DRT-induced communication DoF loss of 1/2T.The high-SNR sensing-limited capacity is nonzero only when the sensing covariance column space is not orthogonal to H_c.
  • 2) Communication DoF:: For a single-antenna communication receiver, communication and sensing subspaces are spanned by h_c and the sensing-optimal steering vector u_s, respectively.Their overlap is used to depict the tradeoff under deterministic communication-channel assumptions.
  • 2) Communication DoF:: Statistical covariance shaping produces refined CRB-rate inner bounds and an outer bound, while Gaussian, semi-unitary, and time-shared signaling provide alternative inner bounds.The shaping parameter λ controls sensing-versus-communication preference, and time-sharing between Gaussian and semi-unitary regions yields a tighter convex-envelope inner bound.

B. Target Response Matrix Estimation · 1) Sensing DoF: ( · 2) Communication DoF:

The target-response matrix estimation task characterizes sensing and communication performance at P_SC and P_CS through sensing DoF, CRB behavior, communication DoF, and signaling bounds. The results identify sensing DoF loss, high-SNR communication behavior, and null-space completion by semi-unitary signals.

  • B. Target Response Matrix Estimation: The sensing objective is to estimate the entire target response matrix H_s in a statistical MIMO radar setting.The model assumes an a priori distribution for each entry of H_s and uses an affine map involving a unitary matrix.
  • 1) Sensing DoF: (: At P_SC, the sensing-optimal sample covariance matrix is deterministic, yielding the minimum CRB and sensing DoF ν_s,max = T.The minimum CRB can also be written to expose the contribution of a priori knowledge as an additional effective SNR.
  • 1) Sensing DoF: (: At P_CS, the minimum achievable CRB is determined by a complex Wishart sample covariance matrix with degree of freedom T.For CWM(I,T), the eigenvalue distribution converges to the Marchenko–Pastur distribution as M →∞ with β = M/T fixed.
  • 1) Sensing DoF: (: The Marchenko–Pastur limit closely approximates the P_CS CRB even for M = 2, while the general covariance analysis requires eigenvalues bounded by positive constants.The sensing DoF analysis additionally requires the sample covariance matrix to have full rank; otherwise, sensing DoF is zero.
  • 1) Sensing DoF: (: The sensing DoF loss for target response matrix estimation achieves the upper bound indicated by Theorem 2.This conclusion follows from the full-rank sensing DoF analysis.
  • 2) Communication DoF:: At P_CS, the maximum achievable rate equals the unconstrained channel capacity, and the communication DoF is M_SC.The covariance matrix determining this capacity is obtained by water-filling.
  • 2) Communication DoF:: At P_SC, the high-SNR communication DoF loss is mainly caused by row-orthogonality in the sensing-optimal coding strategy, since α_SC reaches its maximum 1.The sensing-induced loss is therefore not attributed to communication subspace overlap.
  • 2) Communication DoF:: The communication tradeoff admits Gaussian and semi-unitary inner bounds, while semi-unitary signals can complete the null space without reducing communication rate.When M_CS < M, the Gaussian inner-bound scheme is not purely Gaussian; at P_CS, rank deficiency leaves the target-response null space dependent entirely on a priori knowledge.

VI. NUMERICAL RESULTS … VII. CONCLUSIONS

Numerical results demonstrate how subspace overlap, coherent sensing duration, and antenna dimensions shape the CRB-rate tradeoff across angle and target-response estimation. The conclusions identify subspace alignment and deterministic-random signaling as the two governing tradeoffs, with covariance shaping improving beyond the basic pentagon bound.

  • A. Target Angle Estimation: For target angle estimation, the semi-unitary–Gaussian inner bound is tighter than the PSC–PCS inner bound because it combines time sharing with power allocation.The PSC–PCS bound uses time sharing alone, while the semi-unitary–Gaussian strategy also adjusts power allocation.
  • A. Target Angle Estimation: As the correlation coefficient ρ increases, the semi-unitary–Gaussian inner bound and outer bound approach the rectangular boundary, reflecting stronger sensing–communication subspace overlap.The two inner-bound corner points correspond to sensing-optimal and communication-optimal signaling strategies.
  • A. Target Angle Estimation: As T increases, the gap between the semi-unitary–Gaussian inner bound and outer bound vanishes, making the deterministic-random tradeoff irrelevant as T →∞.This numerical behavior indicates that finite coherent sensing duration drives the deterministic-random tradeoff.
  • A. Target Angle Estimation: As M and Ns increase, ρ decreases from ρ ≈0.82 to ρ ≈0.61, and the room for improvement from subspace-tradeoff adjustment expands.The reported parameter change is from M = Ns = 7 to M = Ns = 10.
  • B. Target Response Matrix Estimation: For target response matrix estimation with spatially uncorrelated Rayleigh fading, the outer bound is close to a rectangle, resembling the ρ = 1 angle-estimation scenario.The example uses coherent sensing period T = 4M = 16.
  • B. Target Response Matrix Estimation: For rank-1 LoS communication, the Gaussian inner bound is achieved through null-space completion, with semi-unitary signals transmitted in an M −1 dimensional null space.Here MCS = 1 < M, so the communication subspace is one-dimensional and its overlap with the sensing subspace is significantly smaller.
  • VII. CONCLUSIONS: The CRB-rate region has a pentagon inner bound connecting the communication-optimal point PCS and sensing-optimal point PSC, with PSC achieved by deterministic-trace sample covariance.The conclusions also state that covariance-shaping strategies combined with time sharing can achieve more favorable tradeoffs than this pentagon.
  • VII. CONCLUSIONS: The S&C tradeoff is two-fold: subspace alignment governs the waveform generally, while deterministic-random signaling governs performance in the finite coherent sensing period regime.These findings motivate practical Pareto-optimal ISAC signaling strategies.

APPENDIX A PROOF OF PROPOSITION 2 · APPENDIX B PROOF OF COROLLARY 1

Appendix A derives Proposition 2 by extending the parameter vector, incorporating the prior into the BFIM, and analyzing linear super-operators through Choi representations. Appendix B proves Corollary 1 by decomposing the relevant operators, constructing their Choi representations, and replacing eigendecomposition with singular value decomposition when necessary.

  • APPENDIX A PROOF OF PROPOSITION 2: The Proposition 2 proof first ignores the prior distribution pη(η) and defines an extended parameter vector for further BFIM derivation.
  • APPENDIX A PROOF OF PROPOSITION 2: The extended parameter vector’s BFIM is expressed using the resulting formulation, with component matrices Fi having dimensionality K × M.
  • APPENDIX A PROOF OF PROPOSITION 2: After restoring the prior contribution, the proof defines linear super-operators Φ1(·) and Φ2(·) to complete the BFIM analysis.
  • APPENDIX A PROOF OF PROPOSITION 2: The proof uses the Choi representation of Φ1 and the one-to-one correspondence between operator realizations and Choi representations to obtain the expected operator representation.
  • APPENDIX B PROOF OF COROLLARY 1: Appendix B begins by decomposing the relevant expression and deriving its Choi representation from the component terms.
  • APPENDIX B PROOF OF COROLLARY 1: The proof separately identifies the Choi representation of ΦP(γ, A) and combines it with eΨ2 to form the Choi representation of Φγ(A).
  • APPENDIX B PROOF OF COROLLARY 1: Because eΨγ is not Hermitian, the eigendecomposition used earlier is unavailable, so the proof applies the singular value decomposition of eΨγ.
  • APPENDIX B PROOF OF COROLLARY 1: The resulting representation has r3 ⩽ KM because eΨ belongs to C^KM×KM.

APPENDIX C PROOF OF PROPOSITION 3 … 1) (Generic):

The appendices establish optimal-distribution structure, characterize uniqueness through maximum-rank and convexity arguments, and derive sufficient conditions using KKT and duality analyses across several dimensional cases.

  • APPENDIX C PROOF OF PROPOSITION 3: The sensing-optimal RX distribution is formulated through a deterministic optimization problem, with equality achieved when γ is deterministic.The resulting distribution is supported on optimal solutions satisfying tr{R_X}=P_TM.
  • APPENDIX C PROOF OF PROPOSITION 3: Lemma 1 establishes that f(γ) is convex, completing the argument for the optimization result.The proof combines optimal solutions at γ_1 and γ_2 with convexity of the relevant function.
  • APPENDIX D PROOF OF PROPOSITION 4: Any optimal solution can be represented relative to a maximum-rank optimal solution R_opt using a Hermitian perturbation matrix D.Otherwise, averaging with another optimal solution would produce an optimal solution of larger rank, contradicting maximality.
  • 1) Sufficiency:: The sufficiency proof uses strict convexity and injectivity of the linear map from D to the objective representation to establish uniqueness.A zero-trace perturbation satisfying the relevant condition would otherwise generate another optimal solution.
  • 2) Necessity:: The necessity proof constructs a positive-semidefinite alternative solution unless the associated subspace is orthogonal to the null space of Ξ.The argument uses dim(V)=r^2−1 and concludes that Ξ(U_opt⊗U_opt) has full column rank r^2.
  • 1) (Generic):: The generic uniqueness condition is derived by rewriting problem (14), applying KKT conditions, and forming its Lagrange dual with strong duality.The dual analysis relates optimal solutions to the maximum eigenvalue of Φ_a^PTM(A−2).
  • 1) (Generic):: For K ≥ M, full column rank of Ξ implies full column rank of the associated Kronecker product, so problem (14) has a unique optimal solution.The rank relations use r^2 ≤ M^2 ≤ K^2.
  • 1) (Generic):: When K=1, Ξ becomes a row vector and the uniqueness condition is analyzed through the scalar dual variable and the maximum-eigenvalue eigenspace.If that eigenspace has dimensionality 1, condition (155) must hold to avoid an unbounded optimum.

APPENDIX F PROOF OF COROLLARY 2 · APPENDIX G PROOF OF THEOREM 1

Appendix F reduces Corollary 2 to the case K < M and analyzes the rank structure of the optimal eR_X. Appendix G proves Theorem 1 by reducing the communication observation through singular-value decomposition and a sufficient statistic, then bounding entropy using complex Stiefel-manifold geometry and asymptotic estimates.

  • APPENDIX F PROOF OF COROLLARY 2: When K ≥ M, rank(eR_X) ≤ M, so the proof only needs to consider K < M.
  • APPENDIX F PROOF OF COROLLARY 2: For the optimal eR_X, the proof proceeds through an eigendecomposition, while uniqueness of (14) determines the column rank of Ξ(U∗).
  • APPENDIX G PROOF OF THEOREM 1: Theorem 1’s proof begins by expressing R_SC under the stated assumptions and taking the singular value decomposition of H_cX.
  • APPENDIX G PROOF OF THEOREM 1: The first M_SC left-singular-vector columns are retained, and a receiver-side linear combiner eU^H yields a reduced observation model.
  • APPENDIX G PROOF OF THEOREM 1: In the reduced model, H_Zc has i.i.d. circularly symmetric complex Gaussian entries, eΣ contains the nonzero singular values, and eV^H H_X contains the first M_SC columns of V^H H_X.
  • APPENDIX G PROOF OF THEOREM 1: Because eU^H Y_c is sufficient for estimating X, I(Y_c|H_c; X) can be expressed using the reduced observation.
  • APPENDIX G PROOF OF THEOREM 1: The entropy bound uses the logarithmic volume of a rescaled complex Stiefel manifold, whose unscaled volume is denoted V_T,M_SC.
  • APPENDIX G PROOF OF THEOREM 1: The proof approximates the volume of a small ε-tube around the manifold using Vol(S)(1 + O(ε^2)), then evaluates entropy through a chi-squared variable with DoF M_SC,6 and completes the asymptotic argument.The orthogonal component eZ_c^⊥ is measured relative to the tangent space of S, and the remaining term is shown to be o(T) as T →∞.

APPENDIX H PROOF OF LEMMA 2

The proof derives Lemma 2 by comparing Riemannian volume forms under an alternative Stiefel-manifold parametrization and evaluating the resulting Jacobian through tangent-space coordinates. Unitary invariance reduces the calculation to a canonical base point.

  • Riemannian volume and parametrization: The manifold volume is expressed through the Riemannian metric tensor and its induced volume form, with the transformed representation viewed as an alternative parametrization of the original Stiefel manifold.The volume form combines |G_eQ| with the exterior product over tangent-vector components.
  • Tangent-space structure: At the canonical point Q0, tangent perturbations decompose into a skew-Hermitian component Δ∥ and an arbitrary component Δ⊥.The proof uses Q0 = [I, 0] and represents arbitrary points as Q = Q0U for a unitary matrix U.
  • Unitary invariance: Right multiplication by a unitary matrix preserves matrix inner products and exterior products, so the Riemannian metric and volume form at an arbitrary point equal those at the canonical point.This invariance allows the result established at Q0 to apply throughout the manifold.
  • Jacobian evaluation: The Jacobian determinant is obtained by constructing real vector representations of the canonical tangent differentials and using orthonormal bases for skew-Hermitian matrices.The proof introduces a matrix B whose columns form such an orthonormal basis before deriving the determinant.

2 MSC , (205) … APPENDIX J PROOF OF THEOREM 2

The appendices prove the signaling characterizations and matrix inequalities underlying the stated corollaries and Theorem 2. The arguments use unitary equivalence, Haar invariance on the complex Stiefel manifold, and positive-map techniques.

  • 2 MSC , (205): A square matrix with orthonormal columns is unitary, completing the derivation from (199).The proof explicitly uses B ∈ CM2 and the fact that the matrix is square.
  • A. Proof of Corollary 3: The high-SNR P_SC-achieving signal has the same distribution as the optimal construction from Appendix G.The proof identifies eΣQ with the optimal eΣeV and concludes that the P_SC-achieving X satisfies the stated condition.
  • A. Proof of Corollary 3: The signaling construction separates information-bearing X_inf from orthogonal non-informative padding X_⊥.X_⊥ lies in a subspace of Col(e R_SC X) orthogonal to Col(H_c X), subject to the stated validity conditions.
  • A. Proof of Corollary 3: Uniform sampling in Corollary 3 generates Haar-uniform samples over the complex Stiefel manifold.This establishes the sampling procedure used by the construction.
  • B. Proof of Corollary 4: Any two square roots of the same positive semidefinite Hermitian matrix are related by right multiplication with a unitary matrix.Lemma 3 is proved through singular-value decompositions, shared eigenstructure, semi-unitary factors, and augmented matrices.
  • B. Proof of Corollary 4: The unitary-equivalence lemma yields the required relation A = BU and validates the candidate X_SC,1.The proof also uses invariance of Haar measure under unitary multiplication and selects X_⊥ accordingly.
  • APPENDIX J PROOF OF THEOREM 2: Theorem 2 follows by representing the sample covariance through a complex Wishart matrix and applying positive unital-map inequalities.The proof constructs a Cholesky-based map, applies Choi’s inequality, and derives the required bounds through spectral decompositions and monotonicity of matrix-valued covariance terms.
  • APPENDIX J PROOF OF THEOREM 2: The final theorem inequality is obtained by combining the intermediate bounds and the sign of covariance between increasing and decreasing commuting matrix functions.The proof concludes after establishing the relevant spectral and covariance relations.

APPENDIX K PROOF OF COROLLARY 5 · APPENDIX L PROOF OF PROPOSITION 6

Appendix K proves Corollary 5 by identifying conditions under which equality in (36) holds, including unitary transformations and invertibility of e RX. Appendix L proves Proposition 6 by showing that suitable inner-bound points and endpoint slope behavior tighten the pentagon inner bound.

  • APPENDIX K PROOF OF COROLLARY 5: When K ⩽ rank(e RX), equality in (36) is achieved under the stated condition from (231).The proof then uses rank(e RX) = M when e RX is invertible.
  • APPENDIX K PROOF OF COROLLARY 5: When r1 + r2 = 1, F is unitary because Φunital is unital with respect to V (W), so (244) holds.The proof also considers the limit σs → 0, where e JP can be neglected.
  • APPENDIX K PROOF OF COROLLARY 5: If F st is unitary, the proof concludes the required relation leading to Corollary 5.The final equality uses the distribution T RX ∼CWM(e RX, T ).
  • APPENDIX L PROOF OF PROPOSITION 6: Proposition 6 suffices to prove that at least one point (ǫ, R) on the Gaussian or semi-unitary inner bound lies above the segment connecting PSC and PCS.Time-sharing among this point, PSC, and PCS then yields a tighter inner bound than the pentagon inner bound.
  • APPENDIX L PROOF OF PROPOSITION 6: Under mild assumptions, the proof compares the semi-unitary–Gaussian inner-bound slope at PSC and PCS with the slope of the segment connecting them.It requires that the optimal objective value is not identical for every α ∈ [0, 1].
  • APPENDIX L PROOF OF PROPOSITION 6: As α → 0, the relevant slope expression for the semi-unitary inner bound tends to infinity.This follows after expressing the slope using the Lagrange-dual variable λα associated with constraint (49c).
  • APPENDIX L PROOF OF PROPOSITION 6: The semi-unitary–Gaussian inner-bound slope is positive infinity at PSC and zero at PCS, so the connecting segment’s slope lies between these endpoint slopes.The argument uses equality with the semi-unitary inner bound around PSC and analogous reasoning at PCS.
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