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Finite-SNR Closed Form for the Square Mar\v{c}enko-Pastur MIMO Capacity

Mohamed Akrout, Robert W. Heath

arXiv:2609.03356v1cs.ITmath.IT

TL;DR

At the square antenna ratio, the finite-SNR full-CSIT capacity had no closed form because the waterfilling cutoff stays inside the spectrum. The paper parametrizes both SNR and capacity by one angle, yielding a curve that can be traced by sweeping that angle.

Abstract

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For the canonical independent and identically distributed (IID) single-user multiple-input multiple-output (MIMO) channel with full channel state information at the transmitter (CSIT), the asymptotic capacity per receive antenna is a waterfilling integral over the Mar\v{c}enko-Pastur law, whose shape depends only on the ratio of transmit to receive antennas. Closed forms are known for every ratio other than the square one, and there only above a finite signal-to-noise ratio (SNR). The square ratio behaves differently because the Mar\v{c}enko-Pastur support reaches the origin. The waterfilling cutoff then stays strictly inside the support at every finite SNR, and the water level has been available only numerically. This letter closes that gap with a classical trigonometric parametrization of the square law, which removes the edge singularity and replaces the moving cutoff by a single angle. Both the SNR and the capacity become explicit functions of that angle, so the capacity curve is traced by sweeping the angle rather than by solving a scalar constraint at each operating point. The same parametrization delivers the low-SNR behavior in closed form, where the slope of capacity in SNR equals the upper edge of the limiting spectrum.

I. INTRODUCTION

At the square antenna ratio, the finite-SNR full-CSIT capacity lacked a closed form because the waterfilling cutoff remains inside the spectrum. The letter derives a single-angle parametric solution and places it in the context of the channel model and capacity framework.

A. Motivation and prior work

  • The full-CSIT capacity of the canonical IID channel is a waterfilling functional over the Marčenko–Pastur law, whose shape depends on the antenna ratio.Prior finite-SNR square-case work characterized capacity using an integral and a numerically found water level.Source: A. Motivation and prior work · p. 1

B. Contributions

  • At the square ratio, the cutoff remains interior at every finite SNR, preventing the capacity integral from becoming a full-support integral.For non-square ratios, raising SNR eventually floods the full support.Source: B. Contributions · p. 1
  • A classical trigonometric parametrization gives explicit SNR and capacity functions of one angle for the square ratio.The parametrization turns the interior cutoff into an explicit parameter rather than a numerically solved water level.Source: B. Contributions · p. 1

C. Outline

  • The paper verifies the closed form numerically over the full SNR range and against the exact high-SNR asymptote.The outline places the derivation before numerical verification and conclusion.Source: B. Contributions · p. 1, C. Outline · p. 1

II. BACKGROUND ON FULL-CSI MIMO CAPACITY

  • The background section sets up the signal model, Shannon transform, and waterfilling solution used throughout the letter.The system model considers a single-user link with transmit and receive antennas in a large-antenna regime at fixed aspect ratio.Source: §II · p. 1, A. System model · pp. 1–2

B. Channel-state information and capacity functional

The section defines capacity through the Shannon transform of the effective channel Gram matrix and specializes the input covariance to different levels of transmitter channel knowledge. With full CSIT, capacity is the maximum transform over admissible inputs.

B. Channel-state information and capacity functional

  • Full CSIT lets the transmitter adapt the input covariance to the instantaneous channel realization, while statistical CSIT uses the channel distribution and no CSIT uses isotropic input.The square-case waterfilling input with full CSIT is the open case addressed here; only the infinite-SNR limit had been known in closed form.Source: B. Channel-state information and… · p. 2
  • The Shannon transform averages log(1 + SNR x) over a limiting eigenvalue law and represents asymptotic per-dimension mutual information.The matrix form is log det(I + SNR A).Source: B. Channel-state information and… · p. 2
  • Per-receive-antenna capacity is the largest Shannon transform of the effective Gram matrix HΦH∗ over admissible inputs.The input covariance is optimized subject to the power constraint tr{Φ} = Nt.Source: B. Channel-state information and… · p. 2

C. Full-CSIT waterfilling solution

For full CSIT, the optimizer allocates power by waterfilling over channel Gram-matrix eigenvalues. In the large-antenna limit, those eigenvalues converge to the Marčenko–Pastur law, yielding the capacity integral whose behavior near zero governs the square case.

C. Full-CSIT waterfilling solution

  • Full-CSIT optimization diagonalizes H∗H and allocates power across its eigenvalues by waterfilling.The scalar water level is chosen to satisfy the total-power constraint, and only directions above the power threshold receive allocation.Source: C. Full-CSIT waterfilling solution · p. 2
  • The empirical spectral distribution of H∗H converges almost surely to the Marčenko–Pastur law as the antenna counts grow at a fixed ratio.The finite-dimensional eigenvalue average is thereby replaced by an average over the limiting law.Source: C. Full-CSIT waterfilling solution · pp. 2–3
  • The full-CSIT capacity integral is the starting point for the square-case analysis, whose difficulty lies in the density near zero.The capacity expression follows by taking the large-system limit of the spectral average.Source: C. Full-CSIT waterfilling solution · p. 3

III. MARˇCENKO–PASTUR LAW WITH THREE ASPECT-RATIO

The limiting Marčenko–Pastur spectrum determines whether the waterfilling integral can be reduced to a full-support form. The square case is distinguished by its support reaching the origin and the resulting interior cutoff at finite SNR.

A. Canonical channel and its limiting spectrum

  • For the canonical IID complex Gaussian channel, the empirical spectrum converges almost surely to the Marčenko–Pastur law with support [a, b].The limiting law depends on the transmit-to-receive antenna ratio.Source: A. Canonical channel and its limiting… · p. 3
  • The lower support edge determines the distance from the origin, while the density’s 1/λ factor becomes decisive as that edge approaches zero.When the support stays away from zero, the factor is bounded on [a, b].Source: A. Canonical channel and its limiting… · p. 3

B. Full-CSIT capacity and waterfilling cutoff

C. Solvability of the non-square cases

For non-square ratios, the positive lower support edge lets waterfilling eventually cover the whole spectrum, while reciprocity reduces the calculation to β < 1.

C. Solvability of the non-square cases

  • When β ≠ 1, the positive lower edge creates a gap from zero, so sufficiently high SNR lets waterfilling reach the full support.With 1/λ bounded on the support, the power constraint becomes an algebraic equation solvable for ν in closed form.Source: C. Solvability of the non-square cases · p. 3
  • Exchanging transmit and receive roles preserves the nonzero Gram-matrix eigenvalues, maps β to 1/β, and rescales per-receive-antenna normalization by β.Thus the β > 1 branch follows from evaluating the full-support integrals for β < 1.Source: C. Solvability of the non-square cases · p. 3
  • The full-support integrals for β < 1, extended to β > 1 by reciprocity, yield the known closed form,,.The passage attributes the closed form to,, and.Source: C. Solvability of the non-square cases · p. 3

D. The obstacle at the square ratio

At the square ratio, the spectrum reaches zero, leaving an interior waterfilling cutoff at every finite SNR and a singular density near that cutoff.

D. The obstacle at the square ratio

  • At β = 1, the support reaches the origin and the gap between zero and the smallest eigenvalue vanishes.The hard edge contrasts with the positive lower edge at non-square ratios.Source: D. The obstacle at the square ratio · p. 3
  • The 1/λ factor makes the density unbounded near zero, where the cutoff approaches as SNR increases.The partial-support integral therefore encounters a density that blows up at the edge toward which its cutoff slides.Source: D. The obstacle at the square ratio · pp. 3–4
  • At finite SNR, the square-case full-CSIT water level was available only numerically, with a closed form previously known only as SNR tends to infinity.The cited finite-antenna result adapts power across time under a long-term constraint; this work instead waterfills per realization under an instantaneous constraint, and the two coincide in the large-antenna limit.Source: D. The obstacle at the square ratio · p. 4

IV. CLOSED-FORM FULL-CSI CAPACITY AT β = 1

A trigonometric change of variable maps the square Marčenko–Pastur law to a smooth angular form and encodes its moving cutoff by one angle.

IV. CLOSED-FORM FULL-CSI CAPACITY AT β = 1

  • The substitution λ = 4 sin^2 θ maps the support (0, 4] bijectively to θ ∈ (0, π/2], taking the spectral edges to the angular endpoints.Since 4 − λ = 4 cos^2 θ, the square-root edge factor becomes cot θ.Source: §IV · p. 4
  • The Jacobian cancels the 1/λ blow-up at zero, leaving a bounded, smooth trigonometric-polynomial weight.The transformed law is described as a bounded, everywhere-smooth raised-cosine weight.Source: §IV · p. 4
  • The interior cutoff is represented by a reference angle θ0, encoding the water level and reducing the power-constraint integrals to trigonometric expressions.This parametrization removes the edge singularity and makes SNR explicit in θ0.Source: §IV · p. 4

A. SNR as a function of the reference angle θ0

The square-case operating SNR is parametrized by a reference angle whose mapping is decreasing and covers all positive SNR values. This lets the capacity curve be traced by sweeping the angle instead of numerically solving for the waterline at each SNR.

A. SNR as a function of the reference angle θ0

B. Full-CSI capacity in closed form

For the square channel, the full-CSIT capacity is given parametrically by a reference angle that also determines the SNR. Sweeping that angle traces the capacity curve, including its low- and high-SNR limits.

B. Full-CSI capacity in closed form

V. NUMERICAL VALIDATION

The closed form agrees with numerical waterfilling across the full SNR range, including finite-size simulations at 64 transmit and receive antennas. The high-SNR asymptote is informative above about 20 dB, while the waterfilling gain is largest at low SNR.

V. NUMERICAL VALIDATION

  • Six decimal places of agreement with brute-force waterfilling hold over the full SNR range, and the Nt = Nr = 64 finite-size points lie on the curve.Source: §V · p. 5, Figure 1 · p. 5
  • Above about 20 dB, capacity approaches the log SNR − 1 asymptote, which is not meaningful at low SNR.Source: Figure 1 · p. 5, §V · p. 6
  • The waterfilling gain over isotropic input without CSIT is widest at low SNR and closes as SNR tends to infinity.Source: Figure 1 · p. 5, §V · p. 6

VI. CONCLUDING REMARKS

The letter gives a closed-form full-CSIT capacity for the square canonical MIMO channel, replacing a previously numerical finite-SNR quantity with a one-angle parametrization. Its derivation is specific to the IID square setting, though the change of variable may apply to other laws with a hard edge at the origin.

VI. CONCLUDING REMARKS

  • The closed form evaluates both SNR and capacity at β = 1, turning the interior cutoff into an explicit parameter and allowing the curve to be traced by sweeping one angle.Source: §VI · p. 6
  • The trigonometric parametrization is classical,; the new contribution is its closed-form evaluation for both SNR and capacity in the square case.Source: §VI · p. 6
  • The derivation covers the canonical IID full-CSIT channel at β = 1; correlated, Rician, and non-square models require their own limiting spectra.Source: §VI · p. 6
  • The same change of variable may apply to other limiting laws with a hard edge at the origin, although their resulting integrals may differ.Source: §VI · p. 6
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