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arxiv: 2604.26618 · v1 · submitted 2026-04-29 · 📡 eess.SP

SEP Analysis of Quantized SIMO Systems with M-PSK over Correlated Fading Channels

Amila Ravinath , Bikshapathi Gouda , Italo Atzeni , Antti T\"olli This is my paper

Pith reviewed 2026-05-07 11:14 UTC · model grok-4.3

classification 📡 eess.SP
keywords symbol error probabilityphase quantizationcorrelated Rayleigh fadingSIMO systemsM-PSK modulationdiversity gaincoding gainmaximum ratio combining
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The pith

In phase-quantized SIMO systems using M-PSK over correlated Rayleigh fading, receive correlation degrades coding gain but preserves diversity order when the covariance matrix has full rank.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the average symbol error probability for a single-input multiple-output wireless receiver that quantizes the phase of incoming signals before combining them with maximum ratio combining. It focuses on M-ary phase-shift keying modulation over spatially correlated Rayleigh fading channels plus noise. By introducing an approximation to the combiner that holds at high signal-to-noise ratios, the authors obtain closed-form expressions for the error probability that separate the effects of correlation, quantization resolution, and modulation order. These expressions show that correlation mainly increases the required signal strength for a target error rate while the number of independent fading paths, and thus the diversity order, stays intact if the antennas still provide a full-rank covariance. The formulas are checked against Monte Carlo runs that match closely over a broad range of signal strengths.

Core claim

The paper derives closed-form high-SNR expressions for the symbol error probability of phase-quantized SIMO systems with M-PSK modulation under correlated Rayleigh fading. These expressions characterize the diversity and coding gains as explicit functions of the receive correlation structure, phase-quantization resolution, and modulation order, up to a bounded scaling factor between 1 and 2. The analysis establishes that channel correlation primarily reduces the coding gain and therefore imposes an SNR penalty, while the diversity gain remains equal to the rank of the channel covariance matrix and is therefore preserved whenever that matrix is full rank.

What carries the argument

An asymptotically equivalent MRC combiner that replaces the exact maximum-ratio combiner at high SNR to permit closed-form SEP expressions under spatial correlation.

Load-bearing premise

The analysis relies on replacing the exact maximum-ratio combiner with an asymptotically equivalent version whose accuracy holds for the correlation strengths and quantization levels considered.

What would settle it

A Monte Carlo simulation at extremely high SNR that measures a diversity order below the number of receive antennas for a full-rank yet strongly correlated covariance matrix would falsify the claim that diversity is preserved.

Figures

Figures reproduced from arXiv: 2604.26618 by Amila Ravinath, Antti T\"olli, Bikshapathi Gouda, Italo Atzeni.

Figure 2
Figure 2. Figure 2: corroborates Proposition 4, with Nr = 4, M ∈ {4, 8}, and the same correlation model used in view at source ↗
read the original abstract

The average symbol error probability (SEP) of a phase-quantized single-input multiple-output system with M-ary phase-shift keying modulation and maximum ratio combining (MRC) is analyzed under correlated Rayleigh fading and additive white Gaussian noise. Building on our prior framework for the independent and identically distributed case, we extend the analysis to spatially correlated channels by introducing an asymptotically equivalent MRC combiner that enables tractable SEP characterization. Using this approach, we derive closed-form expressions at high signal-to-noise ratio (SNR) that explicitly characterize the diversity and coding gains as functions of the receive correlation structure, phase-quantization resolution, and modulation order, up to a scaling factor bounded between 1 and 2. The results show that channel correlation primarily degrades the coding gain, leading to an SNR penalty, while the diversity gain is preserved when the channel covariance matrix is full-rank. The analytical findings are validated through Monte Carlo simulations, demonstrating a tight match across a wide SNR range.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript analyzes the average symbol error probability (SEP) of phase-quantized SIMO systems with M-PSK modulation and MRC under correlated Rayleigh fading channels. Building on the authors' prior i.i.d. framework, it introduces an asymptotically equivalent MRC combiner to derive closed-form high-SNR SEP expressions that characterize diversity order and coding gain explicitly as functions of the receive correlation matrix, quantization resolution, and modulation order, up to a scaling factor bounded in [1,2]. The results indicate that full-rank correlation preserves diversity while degrading coding gain (inducing an SNR penalty), with Monte Carlo simulations confirming a tight match over a wide SNR range.

Significance. If the asymptotic equivalence of the combiner rigorously preserves the leading high-SNR terms, the closed-form expressions would offer a practical tool for quantifying correlation-induced penalties in quantized reception systems, extending prior i.i.d. results with explicit dependence on the covariance structure. This could support design guidelines for antenna arrays in fading environments. The bounded scaling factor, however, restricts the precision of the coding-gain characterization, and the lack of an independent error analysis beyond simulation matching reduces the immediate utility for exact performance prediction.

major comments (3)
  1. [§III] §III (Asymptotically Equivalent MRC Combiner): The introduction of the asymptotically equivalent combiner is asserted to enable tractable SEP characterization with coding gain matching true MRC up to a factor in [1,2], but no derivation is provided showing that the approximation error vanishes faster than the leading SNR^{-d} term (where d is the diversity order) or that any residual SNR offset remains independent of the condition number of the correlation matrix. This is load-bearing for the central claim of explicit characterization of coding gain as a function of correlation structure.
  2. [High-SNR SEP expression] High-SNR SEP expression (around Eq. (high-SNR closed form)): The closed-form coding gain is expressed in terms of the correlation eigenvalues, quantization bits, and M, yet the derivation relies on the unproven equivalence; when the covariance is ill-conditioned or quantization is coarse (e.g., 1-2 bits), the bounded factor [1,2] may conceal a growing offset that alters the claimed explicit dependence, and no separate analysis quantifies this beyond the overall simulation match.
  3. [Validation section] Validation section (Monte Carlo results): While simulations are stated to match the analytical expressions across a wide SNR range, the paper does not report separate tests for the equivalence accuracy under varying correlation strengths (e.g., high condition numbers) or coarse quantization independently of the overall SEP; this leaves the weakest assumption unverified beyond the aggregate fit.
minor comments (2)
  1. Notation for the correlation matrix and its eigenvalues should be introduced earlier and used consistently in the high-SNR expressions to improve readability.
  2. The abstract and introduction could more explicitly state the precise limitation of the scaling factor (bounded rather than exact) to set reader expectations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment below with point-by-point responses and indicate the revisions we will make to strengthen the rigor of the asymptotic analysis and validation.

read point-by-point responses

  1. Referee: [§III] §III (Asymptotically Equivalent MRC Combiner): The introduction of the asymptotically equivalent combiner is asserted to enable tractable SEP characterization with coding gain matching true MRC up to a factor in [1,2], but no derivation is provided showing that the approximation error vanishes faster than the leading SNR^{-d} term (where d is the diversity order) or that any residual SNR offset remains independent of the condition number of the correlation matrix. This is load-bearing for the central claim of explicit characterization of coding gain as a function of correlation structure.

    Authors: We agree that an explicit derivation showing the approximation error is o(SNR^{-d}) and that any residual offset is independent of the condition number (under full-rank covariance) was not provided in Section III. In the revised manuscript we will add this derivation, either as an expanded subsection or appendix, based on the asymptotic expansion of the combiner output difference. This will rigorously justify the leading-term equivalence used for the coding-gain expression. revision: yes

  2. Referee: [High-SNR SEP expression] High-SNR SEP expression (around Eq. (high-SNR closed form)): The closed-form coding gain is expressed in terms of the correlation eigenvalues, quantization bits, and M, yet the derivation relies on the unproven equivalence; when the covariance is ill-conditioned or quantization is coarse (e.g., 1-2 bits), the bounded factor [1,2] may conceal a growing offset that alters the claimed explicit dependence, and no separate analysis quantifies this beyond the overall simulation match.

    Authors: The [1,2] bound follows directly from the norm properties of the equivalent combiner and preserves the explicit functional dependence on the eigenvalues. We acknowledge that the bound may be loose for ill-conditioned covariances or coarse quantization and that a tighter characterization would be desirable. We will add a remark and supporting bounds in the revised text discussing the behavior in these regimes while retaining the closed-form expression. revision: partial

  3. Referee: [Validation section] Validation section (Monte Carlo results): While simulations are stated to match the analytical expressions across a wide SNR range, the paper does not report separate tests for the equivalence accuracy under varying correlation strengths (e.g., high condition numbers) or coarse quantization independently of the overall SEP; this leaves the weakest assumption unverified beyond the aggregate fit.

    Authors: We will augment the validation section with dedicated Monte Carlo experiments that isolate the combiner equivalence. These will include separate curves or tables showing the relative difference between true MRC and the equivalent combiner (or the resulting SEP deviation) for correlation matrices with high condition numbers and for 1-2 bit quantization, independent of the aggregate SEP match. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior i.i.d. framework; new combiner and high-SNR analysis are independent

full rationale

The paper extends its authors' prior i.i.d. framework to correlated channels by introducing an asymptotically equivalent MRC combiner that enables closed-form high-SNR SEP expressions. These expressions characterize diversity order and coding gain explicitly in terms of the correlation matrix, quantization bits, and M, subject only to a bounded scaling factor (1-2) rather than exact equality. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the prior work supplies the i.i.d. base while the correlated extension and its bounded approximation are derived and simulated independently. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The analysis rests on standard wireless channel and noise models plus one new approximation introduced for tractability; no free parameters are explicitly fitted beyond the bounded scaling factor.

free parameters (1)
  • scaling factor = 1 to 2
    Appears in the closed-form expressions and is stated to lie between 1 and 2; its exact value is not derived but bounded.
axioms (2)
  • domain assumption Channels follow correlated Rayleigh fading (complex Gaussian with given covariance matrix)
    Invoked to model the spatially correlated fading environment.
  • domain assumption Additive white Gaussian noise at the receiver
    Standard noise model used throughout the SEP derivation.
invented entities (1)
  • asymptotically equivalent MRC combiner no independent evidence
    purpose: To obtain tractable closed-form SEP expressions under correlation
    New construct introduced to replace exact MRC; no independent evidence outside the paper's simulations is provided.

pith-pipeline@v0.9.0 · 5481 in / 1495 out tokens · 49736 ms · 2026-05-07T11:14:34.358333+00:00 · methodology

discussion (0)

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Reference graph

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