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arxiv: 2601.03387 · v2 · submitted 2026-01-06 · 📡 eess.SP

SEP Analysis of a Low-Resolution SIMO System with M-PSK over Fading Channels

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

Pith reviewed 2026-05-16 16:34 UTC · model grok-4.3

classification 📡 eess.SP
keywords symbol error probabilityphase quantizationSIMO systemsM-PSK modulationRayleigh fadingdiversity gainmaximum ratio combininglimited CSI
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The pith

Exact SEP expressions for QPSK phase-quantized SIMO-MRC follow from a duality to reciprocal MISO systems.

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

The paper derives exact average symbol error probability expressions for phase-quantized single-input multiple-output systems using M-ary phase-shift keying over Rayleigh fading. For quadrature PSK with n-bit phase quantization and maximum ratio combining, closed-form SEP formulas are obtained along with high-SNR diversity and coding gains. A duality is identified that maps these SIMO setups to phase-quantized multiple-input single-output systems with maximum ratio transmission when modulation order, quantization bits, antenna counts, and CSI conditions are reciprocal. This duality directly supplies the diversity order for arbitrary M-PSK. The primary results assume perfect channel state information at the receiver; when each antenna supplies only two bits of phase information the diversity order is halved in general.

Core claim

By leveraging a novel method, exact SEP expressions are derived for a QPSK-modulated n-bit phase-quantized SIMO system with maximum ratio combining, together with high-SNR characterizations in terms of diversity and coding gains. For a QPSK-modulated 2-bit phase-quantized SIMO system with selection combining the diversity and coding gains are obtained for an arbitrary number of receive antennas. The method reveals a duality between the SIMO-MRC system and a phase-quantized MISO system with maximum ratio transmission under reciprocal conditions; this duality yields the diversity order of a general M-PSK-modulated n-bit phase-quantized SIMO-MRC system and extends the results to the MISO case.

What carries the argument

Duality between an n-bit phase-quantized SIMO-MRC system and its reciprocal phase-quantized MISO-MRT counterpart, which maps parameters symmetrically to obtain SEP and diversity.

If this is right

  • Exact closed-form SEP holds for QPSK with arbitrary n-bit phase quantization under MRC.
  • Diversity order equals the number of receive antennas at high SNR with perfect CSIR.
  • Diversity and coding gains are available for 2-bit quantized selection combining with any number of antennas.
  • Diversity order for general M-PSK follows immediately from the SIMO-MRC/MISO-MRT duality.
  • Diversity order is halved when each receive antenna supplies only two bits of phase CSI.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same duality may permit SEP analysis for other linear combiners or for M-PSK with non-uniform phase quantization.
  • In large-array regimes the halved diversity under two-bit CSI implies a quantifiable rate penalty that can be traded against feedback overhead.
  • The high-SNR characterizations supply simple design rules for choosing quantization bits to meet target diversity in Rayleigh fading.

Load-bearing premise

The derivations assume perfect channel state information is available at the receiver.

What would settle it

Monte Carlo simulation of symbol error rate versus SNR for a four-antenna QPSK system with three-bit phase quantization; if the high-SNR slope on a log-log plot deviates from minus four, the claimed diversity order is falsified.

Figures

Figures reproduced from arXiv: 2601.03387 by Amila Ravinath, Antti T\"olli, Bikshapathi Gouda, Italo Atzeni, Minhua Ding.

Figure 1
Figure 1. Figure 1: Average SEP versus ρ for a QPSK-modulated 2-bit phase￾quantized SIMO-MRC system with Nr ∈ {1, 4, 8, 16}. The corre￾sponding SEP bound is specified by (25). which leads to fU (u) = c1u c2 + O(u c2 ), (26) for u → 0 +, with c1 ≜ 1 2 1 (Nr − 1)! 2Nr π  Nr 2 , c2 ≜ Nr − 2 2 . (27) Based on (26)–(27), and using the results in [12, Prop. 1] (see Appendix I), we obtain (25) [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 2
Figure 2. Figure 2: Average SEP versus ρ for a QPSK-modulated 3-bit phase￾quantized SIMO-MRC system with Nr ∈ {1, 2, 3, 4}. The correspond￾ing SEP bound is specified by (31). i.i.d. Rayleigh fading, the diversity and coding gains are given by G n≥3 d,MRC = Nr , (31a) G n≥3 c,MRC = (Nr !) 1 Nr Nr π 2 n−1 cot π 2 n−1 , (31b) respectively. Proof: See Appendix III. In the limit of n → ∞, the phase error in (17) vanishes since eθi… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the pdf of Zi in (21a) with n = 1, 2, 3 and n → ∞. n = 1, 2 and for n ≥ 3 satisfy G n=1 d,MRC = 0, Gn=2 d,MRC = Nr/2, Gn≥3 d,MRC = Nr , (35) revealing two distinct phase transitions around n = m = 2. Similar phenomena have been reported in [10], [11], [17] for phase-quantized SIMO-SC and MISO-MRT systems. These transitions originate from abrupt changes in the key statistics of the SEP expressi… view at source ↗
Figure 6
Figure 6. Figure 6: Average SEP versus ρ using SC with Nr ∈ {1, 2, 4, 8}. The corresponding SEP bound is specified by (47). given by PSC QPSK → E ( Q r ρmax i∈[Nr] |hi | 2(1 − |sin 2eθi |) !), ρ → ∞. (46) Using (46), we derive the corresponding diversity gain Gn=2 d,SC and coding gain Gn=2 c,SC as given in the following proposition. Proposition 3. The diversity and coding gains of a QPSK￾modulated 2-bit phase-quantized SIMO-S… view at source ↗
Figure 7
Figure 7. Figure 7: MISO-SIMO duality for Nr = Nt ∈ {1, 4, 8, 16} with 8-PSK and n = 3. the receiver, the detected symbol is obtained using sˆ ≜ Qm(y). The SEP corresponding the above MISO system is PMISO = E{IEMISO (h, we)|s}, (50) with E MISO ≜ {(h, we) : Qm(y) ̸= s}. (51) Proposition 4. Assume Nt = Nr . When the number of bits used for quantization and the modulation order are identical, a MISO system with low-resolution D… view at source ↗
Figure 8
Figure 8. Figure 8: SEP with limited CSI for Nr = 1, 3, 5, 7 using the majority￾decision rule. The high-SNR characterization follows (58a)–(58b). and then take the majority decision separately for the real and imaginary parts as sˆ LCSI MD ≜ Q2 X i sˆ LCSI i ! = Q2 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
read the original abstract

In this paper, the average symbol error probability (SEP) of a phase-quantized single-input multiple-output (SIMO) system with M-ary phase-shift keying (PSK) modulation is analyzed under Rayleigh fading and additive white Gaussian noise. By leveraging a novel method, we derive exact SEP expressions for a quadrature PSK (QPSK)-modulated n-bit phase-quantized SIMO system with maximum ratio combining (SIMO-MRC), along with the corresponding high signal-to-noise ratio (SNR) characterizations in terms of diversity and coding gains. For a QPSK-modulated 2-bit phase-quantized SIMO system with selection combining, the diversity and coding gains are further obtained for an arbitrary number of receive antennas, complementing existing results. Interestingly, the proposed method also reveals a duality between a SIMO-MRC system and a phase-quantized multiple-input single-output (MISO) system with maximum ratio transmission, when the modulation order, phase-quantization resolution, antenna configuration, and the channel state information (CSI) conditions are reciprocal. This duality enables direct inference to obtain the diversity of a general M-PSK-modulated n-bit phase-quantized SIMO-MRC system, and extends the results to its MISO counterpart. All the above results have been obtained assuming perfect CSI at the receiver (CSIR). Finally, the SEP analysis of a QPSK-modulated 2-bit phase-quantized SIMO system is extended to the limited CSIR case, where the CSI at each receive antenna is represented by only 2 bits of channel phase information. In this scenario, the diversity gain is shown to be further halved in general.

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

0 major / 3 minor

Summary. The paper derives exact average symbol error probability (SEP) expressions for a phase-quantized SIMO system employing M-PSK modulation over Rayleigh fading channels with AWGN. It focuses on QPSK with n-bit phase quantization and MRC combining, provides high-SNR asymptotic characterizations in terms of diversity and coding gains, establishes a duality with reciprocal MISO-MRT systems, and extends the analysis to a 2-bit limited-CSIR case where diversity gain is halved.

Significance. If the derivations are correct, the closed-form SEP expressions and diversity/coding gain results offer practical tools for performance evaluation of low-resolution phase-quantized receivers in fading environments. The reciprocity duality between SIMO-MRC and MISO-MRT configurations is a useful structural insight that enables extension to general M-PSK cases without re-derivation. The limited-CSIR extension addresses a realistic hardware constraint and shows the diversity penalty explicitly.

minor comments (3)
  1. [Limited CSIR extension] In the limited-CSIR analysis, explicitly state the mapping from continuous phase to the 2-bit representation and confirm that the diversity-halving result follows directly from the modified decision regions rather than an approximation.
  2. [Introduction] The introduction should include a brief comparison table or paragraph contrasting the proposed method with prior exact SEP derivations for quantized systems (e.g., those using moment-generating functions or integral representations) to substantiate the novelty claim.
  3. [High-SNR analysis] Ensure all high-SNR asymptotic expressions are accompanied by the precise conditions on n, M, and the number of antennas under which the diversity order holds, particularly when extending via the duality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The recognition of the utility of the closed-form SEP expressions, diversity/coding gain characterizations, and the SIMO-MISO duality is appreciated. No specific major comments were raised in the report.

read point-by-point responses

  1. Referee: The paper derives exact average symbol error probability (SEP) expressions for a phase-quantized SIMO system employing M-PSK modulation over Rayleigh fading channels with AWGN. It focuses on QPSK with n-bit phase quantization and MRC combining, provides high-SNR asymptotic characterizations in terms of diversity and coding gains, establishes a duality with reciprocal MISO-MRT systems, and extends the analysis to a 2-bit limited-CSIR case where diversity gain is halved.

    Authors: We appreciate the referee's accurate summary of the contributions. The exact SEP expressions for the QPSK n-bit phase-quantized SIMO-MRC system are obtained via a novel method that integrates the effects of phase quantization, Rayleigh fading, and AWGN. The high-SNR analysis yields explicit diversity and coding gains, while the duality follows from the reciprocity in modulation order, quantization resolution, antenna configuration, and CSI conditions, allowing direct extension to general M-PSK without re-derivation. The limited-CSIR extension for the 2-bit case explicitly shows the halved diversity gain. All derivations have been cross-verified with Monte Carlo simulations in the manuscript, confirming their correctness. No changes are needed. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives exact SEP expressions for QPSK-modulated n-bit phase-quantized SIMO-MRC systems using standard probabilistic models of Rayleigh fading and AWGN, along with asymptotic high-SNR analysis for diversity and coding gains. The claimed duality to MISO-MRT follows directly from the proposed method applied to reciprocal configurations, without reducing to self-referential definitions or fitted parameters renamed as predictions. The limited-CSIR extension is explicitly separated from the perfect-CSIR results and does not rely on self-citation chains or imported uniqueness theorems. All load-bearing steps remain grounded in external channel models and quantization effects, with no evidence that any central claim collapses to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The analysis rests on standard domain assumptions in wireless communications; no free parameters, new entities, or ad-hoc inventions are mentioned.

axioms (3)
  • domain assumption Rayleigh fading channel model
    Explicitly used for the fading environment in the abstract.
  • domain assumption Additive white Gaussian noise
    Standard noise model stated in the abstract.
  • domain assumption Perfect CSIR for primary derivations
    Explicitly stated assumption for the main SEP and duality results.

pith-pipeline@v0.9.0 · 5623 in / 1536 out tokens · 61231 ms · 2026-05-16T16:34:38.697490+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    eess.SP 2026-04 unverdicted novelty 4.0

    Closed-form high-SNR SEP expressions are derived for phase-quantized SIMO M-PSK systems over correlated Rayleigh fading, showing correlation degrades coding gain but preserves diversity when the covariance matrix is f...

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