Classes of Full-Duplex Channels with Capacity Achieved Without Adaptation

Anas Chaaban; Daewon Seo; Lav R. Varshney; Mohamed-Slim Alouini

REVIEW 3 major objections 5 minor 36 references

Classes of Full-Duplex Channels with Capacity Achieved Without Adaptation

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Pith's one-line read Non-adaptive transmission achieves the Shannon capacity region for injective semi-deterministic two-way channels and is asymptotically optimal for the Poisson two-way channel at high dark current.

desk verdict Solid ISD capacity results, but the Poisson half has a load-bearing inconsistency: Theorem 5's one-way capacity is contradicted by the proof of Theorem 6. read the letter →

arxiv 1908.04327 v2 pith:PLVQLFWN submitted 2019-08-12 cs.IT math.IT

Daewon Seo , Anas Chaaban , Lav R. Varshney , Mohamed-Slim Alouini This is my paper · ORCID

classification cs.ITmath.IT

keywords channelsachievedadaptationcapacitychannelclassesfull-duplexnon-adaptive

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

In a two-way channel, two terminals transmit and receive at the same time, and each terminal can in principle adapt its future transmissions based on what it has received so far. Shannon gave an inner bound on the capacity region, achieved by non-adaptive coding where each terminal's transmission depends only on its own message, and an outer bound that allows adaptive coding. For most channels these bounds differ, meaning adaptation buys rate. This paper gives two classes of channels where they coincide, so adaptation is useless.

The first class is the injective semi-deterministic two-way channel: each output is obtained by passing the other terminal's input, plus independent noise, through injective functions. This covers additive channels with exponential, Cauchy, or Gaussian noise, multiplicative channels, and some input-dependent noise models. For all of these, the capacity region is a rectangle and the paper gives closed-form rate expressions. The second class is the continuous-time Poisson channel, used for optical communication, where the receiver counts photons. Using Wyner's discretization, the authors reduce it to a binary channel and prove that as dark current grows, the gap between the non-adaptive inner bound and the one-way outer bound shrinks like the inverse square of the dark current rate, so non-adaptive coding is asymptotically optimal.

The core mathematics is largely clean and the appendix algebra checks out. The main blemishes are textual: the stated capacity regions in Theorems 3 and 4 contain index and notation errors that contradict the proofs, and the discretization lemma for the Poisson channel is given only as a proof sketch.

Extended reading notes

Core claim

For injective semi-deterministic two-way channels (Definition 1), Shannon's inner and outer bounds coincide, so the capacity region is achieved without adaptation and equals the rectangle R_i ≤ max_{p_{X_i}} [H(g_j(X_i,Z_j)) - H(Z_j)] (Theorem 2, Corollary 1). For the Poisson two-way channel, the non-adaptive inner bound meets the one-way outer bound asymptotically as dark current grows: the gap is O(s^-2) = O(λ_0^-2) (Theorem 6). If correct, non-adaptive coding achieves full capacity on these families.

Load-bearing premise

The Poisson TWC result rests entirely on Lemma 1 (Section V-B): Wyner's one-way discretization, which replaces the continuous-time Poisson channel by a discrete-time binary channel with transition probabilities α, β, γ, is asserted to carry over to the two-way channel 'without significant changes.' This is presented as a proof sketch. If the two interacting waveforms cannot be independently discretized without disturbing the two-way capacity, or if the genie outer bound in Proposition 1 is not tight in the relevant limit, the asymptotic optimality claim loses its foundation. The ISD results rest on a different premise: injectivity of f_i and g_i, which by the paper's own note excludes the binary multiplier channel where Shannon's bounds do not coincide.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I agree with the reader's positive assessment of the ISD part, but the reader missed a sharper problem in the Poisson section. The novel ISD material is clean: Theorem 1 gives a simple sufficient condition, Theorem 2 and Corollary 1 prove capacity without adaptation for injective semi-deterministic channels, and the examples (exponential, Cauchy, multiplicative, input-dependent Gaussian) are genuinely useful. The Poisson TWC is a good problem choice, and the appendix's O(s^-2) gap calculation is correct—but only if you use the right one-way capacity.

The problem is that Theorem 5 states the one-way capacity with π0(s) = ((1+s)^(1+s)/s^s)e^{-s}, which decays like e·s·e^{-s} as s→∞. That would make C_OWC(σ) collapse exponentially. But the paragraph before Theorem 6 and the appendix silently use π*_i = min(σ_i, 1/2), which is the correct large-s optimizer of the binary channel expression. These are not asymptotically equivalent: for s=10, σ=0.3, the printed π0 gives a one-way rate around 0.003A, while min(σ,1/2)=0.3 gives about 0.011A. So the proof's comparison is not with the outer bound stated in Theorem 5. The fix is likely small—the stationarity condition gives π0(s) = e^{-1}(1+s)^(1+s)/s^s - s, which tends to 1/2—but as written, Theorem 6 is not proven.

Other soft spots: Lemma 1 is only a proof sketch, and carrying Wyner's discretization to the two-way channel deserves more than 'without significant changes.' Theorem 3 swaps indices against its own proof, Theorem 4 uses undefined m_i, and Section IV-C writes an equality for h(Y2|X2) that only holds under independence. These are minor by comparison, and they don't affect the ISD part.

The Poisson section, as printed, is internally contradictory in its load-bearing step. Still, this is a paper worth engaging: the ISD results stand, and the Poisson result is probably true with a corrected Theorem 5. I would not desk-reject it, but I would send it back for major revision and ask for a corrected one-way capacity statement and a proper proof of Lemma 1. A serious referee can fix these without starting over.

Referee Report

3 major / 5 minor

Summary. The paper studies two-way channels (TWCs) and asks when Shannon's inner bound, which is achievable by non-adaptive coding, coincides with the general outer bound. It introduces the class of injective semi-deterministic (ISD) TWCs, proves that for this class the Shannon inner and outer bounds coincide, and derives an explicit rectangular capacity region (Theorems 1-2, Corollary 1). Examples include multiplicative channels, additive channels with exponential and Cauchy noise, and an input-dependent Gaussian noise model. The second contribution is a continuous-time Poisson TWC. The authors discretize it into a binary TWC (Lemma 1), define Shannon inner and one-way outer bounds, and claim in Theorem 6 that the inner bound asymptotically meets the one-way outer bound as the dark current grows, with an O(s^{-2}) gap; the proof is an asymptotic expansion in the Appendix. The ISD part is clean and checkable. The Poisson part currently contains a serious internal inconsistency in the statement of the one-way capacity formula and its use in the asymptotic proof.

Significance. If the ISD results stand, they provide a broad and useful sufficient condition under which adaptation is unnecessary, together with closed-form capacity expressions for realistic full-duplex channel models; this part is a genuine contribution. The Poisson TWC result, if proved with a correct one-way capacity statement and a rigorous discretization, would be an interesting first continuous-time full-duplex capacity characterization and would show that adaptation is asymptotically useless in the high-dark-current regime. The paper contains no fitted parameters and the asymptotic expansions are explicit and testable. However, the Poisson claim is not currently supported as written because of the contradiction described below, so the significance of the paper depends on a fixable but nontrivial revision.

major comments (3)

[Section V-C, Theorem 5 and Appendix] The one-way capacity formula in Theorem 5 defines pi* = min(sigma, pi0(s)) with pi0(s) = ((1+s)^{1+s}/s^s)e^{-s}. For large s this pi0(s) decays as s e^{1-s}, so the printed C_OWC(sigma) would decay exponentially in s for every fixed sigma>0. The unconstrained maximizer of the displayed expression is actually ((1+s)^{1+s}/s^s)e^{-1} - s, which tends to 1/2; the printed pi0(s) also exceeds 1 for some s (for example at s=1 it is 4/e). Yet the paragraph immediately before Theorem 6 and the entire Appendix set pi*_i = min(sigma_i,1/2) and derive the gap A pi*_1(1-pi*_1)pi*_2/(2s^2) + O(s^{-3}). Thus the asymptotic proof is not comparing against the outer bound stated in Theorem 5. Theorem 6 needs to be re-derived from a corrected, consistently stated one-way capacity expression, with the optimizer used in the Appendix matching the optimizer in Theorem 5.
[Section V-B, Lemma 1] Lemma 1 is the bridge from the continuous-time Poisson TWC to the discrete-time binary channel in (6), but its proof is only a sketch asserting that Wyner's one-way discretization carries over 'without significant changes.' In a two-way channel the two transmitted waveforms interact in both received Poisson processes, and each terminal's encoder may in general depend on its past received signal; it is not automatic that both waveforms can simultaneously be made slot-constant 0/A and that slot-wise photon-count thresholding is optimal without changing the rate region. Both the inner bound Ri and the equivalent channel used in the asymptotic computation rely on this lemma. A complete proof of the two-way discretization, or a precise citation establishing it, is needed before the Poisson capacity claim is fully supported.
[Appendix, first paragraph] The proof says it is sufficient to show that at least one point of C lies within an O(1/lambda_0^2) ball centered at the outer-bound corner. For a general convex region this would not imply that the boundaries are close; however, the subsequent computation in fact gives both coordinates of the inner-bound rectangle for the fixed product distribution (p*_1,p*_2), and for fixed product inputs the inner bound contains the rectangle [0,I_1] x [0,I_2]. The proof should state this explicitly, since the boundary-gap conclusion follows from the rectangle, not from the single-point formulation as written.

minor comments (5)

[Section IV-B.2, Theorem 4] Theorem 4 uses m1 and m2 in the capacity expressions, but the Cauchy model is parameterized by dispersions gamma_1 and gamma_2 and no m_i is defined in that section; the theorem should read log(A1/gamma_2) and log(A2/gamma_1), consistent with the proof.
[Section V-B, Definition 2] Definition 2 calls Ri the 'Shannon inner bound' but omits the convex-hull operation that appears in the discrete-time inner bound in Section II-B and that is needed for time sharing between product distributions; the definition and Proposition 1 should be aligned on this point.
[Section V-C, Theorem 5] After correcting pi0(s), the notation pi* should be unified: Theorem 5 uses it for the one-way optimizer, while the Appendix uses pi*_i = min(sigma_i,1/2) without noting that this is the large-s limit of the corrected optimizer; a single consistent definition would remove the current ambiguity.
[Section V-A] The sentence 'Generalization does not significantly change results of this paper' about unequal peak powers/dark currents is unsupported; either give the straightforward generalization or remove the claim.
[Appendix] The expansion assumes sigma_i>0, since pi*_i appears in a denominator later as a factor in the leading term; this should be stated as a standing assumption in Theorem 6.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: ISD capacity follows from Shannon bounds; Poisson asymptotic gap is an analytic comparison to an external one-way upper bound.

full rationale

The paper's derivations are not circular. The ISD capacity result (Theorem 2 and Corollary 1) is obtained by checking Shannon's conditions (C1) and (C2) from the stated structural assumptions; the capacity expression is then a direct evaluation of mutual information, not a quantity fitted to the target. The examples (multiplicative, exponential, Cauchy, input-dependent Gaussian noise) are applications of the same theorem, and while one sub-result is deferred to the authors' [1], that citation is not load-bearing because Theorem 2 supplies the capacity region independently. For the Poisson TWC, Theorem 6 is an asymptotic comparison between Shannon's inner bound at a product input and the one-way capacity used as an outer bound; the outer bound is imported from external work [24]-[26], and the gap is computed analytically without fitting the O(s^-2) rate to data. Lemma 1 is a proof sketch carrying over Wyner's discretization, but carrying over an external result is not circular. I note, as a correctness issue rather than a circularity issue, that the appendix sets pi*_i = min(sigma_i,1/2) while Theorem 5 prints pi0(s) = ((1+s)^(1+s)/s^s)e^(-s), which appear inconsistent; a referee should correct this, but it does not make the derivation circular because the theorem is not being used to define its own outer bound. Self-citations [1], [2], and [20] are conference versions or surveys and do not carry the central argument.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: all constants (A, σ, m_i, γ_i, λ_0) come from the channel models or are derived; π*_i = min(σ_i, 1/2) is inherited from the external one-way Poisson capacity, not fitted. No invented entities: the paper postulates no new physical objects. The genuine epistemic burden sits in the domain assumptions: the unproven extension of Wyner's discretization to the two-way channel (Lemma 1) and the injectivity condition that defines the ISD class.

assumptions (6)

standard math Shannon's inner and outer bounds (with convex hull) sandwich the TWC capacity region
Section II-B: every capacity statement in the paper is framed by Shannon's inner bound Ri (product inputs, non-adaptive) and outer bound Ro; standard since Shannon 1961.
domain assumption Wyner's discretization reduces the continuous-time Poisson channel to an i.i.d. discrete-time binary channel without changing capacity
Lemma 1, Section V-B: the continuous-time Poisson TWC is replaced by the discrete-time binary channel with probabilities α, β, γ; given as a proof sketch and load-bearing for Theorem 6.
standard math One-way Poisson channel capacity formula (Theorem 5)
Cited from Davis [24] and Wyner [25], [26]; used as the outer bound R_OWC_o in the Poisson TWC proof.
domain assumption Injectivity of f_i and g_i in the ISD definition
Definition 1, Section III-B: f_i injective in T_i and g_i injective in Z_i; failure (e.g., the binary multiplier channel with 0 in the alphabet) removes the channel from the class.
domain assumption Noise variables Z_1, Z_2 are independent of the inputs
Definition 1: input-independent noise is used in Theorem 2's proof to obtain H(Y_1|X_1,X_2) = H(Z_1).
standard math Exponential and Cauchy one-way saddle-point capacity results
Theorems 3 and 4 rely on one-way capacity-achieving input distributions from [31]-[34] (Verdu, Anantharam-Verdue, Coleman-Raginsky, Fahs-Abou-Faycal).

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Pith. "Pith review of Classes of Full-Duplex Channels with Capacity Achieved Without Adaptation." pith.science (2026). https://pith.science/paper/PLVQLFWN

@misc{pith2026190804327,
  author       = {Pith},
  title        = {Pith review of: Classes of Full-Duplex Channels with Capacity Achieved Without Adaptation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PLVQLFWN}},
  note         = {Machine review of arXiv:1908.04327}
}

read the original abstract

Full-duplex communication allows a terminal to transmit and receive signals simultaneously, and hence, it is helpful in general to adapt transmissions to received signals. However, this often requires unaffordable complexity. This work focuses on simple non-adaptive transmission, and provides two classes of channels for which Shannon's information capacity regions are achieved without adaptation. The first is the injective semi-deterministic two-way channel that includes additive channels with various types of noises modeling wireless, coaxial cable, and other settings. The other is the Poisson two-way channel, for which we show that non-adaptive transmission is asymptotically optimal in the high dark current regime.

Figures

Figures reproduced from arXiv: 1908.04327 by the authors.

**Figure 1.** A memoryless two-way channel model. explicit capacity formulas for such ISD channel family are available and such non-adaptive transmission is insightful for the design of full-duplex communication schemes over realistic channel models arising from important communication systems. Another contribution concerns the asymptotic capacity region of the Poisson TWC, a member of continuoustime channels for which obtainin… view at source ↗

**Figure 2.** A semi-deterministic two-way channel. With conditions that [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Inner and outer bounds for the Poisson TWC with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Two way communication over exponential family type channels,

L. R. Varshney, “Two way communication over exponential family type channels,” in Proc. 2013 IEEE Int. Symp. Inf. Theory , Jul. 2013, pp. 2795–2799

work page 2013

[2] [2]

The capacity of injective semi-deterministic two-way channels,

A. Chaaban, L. R. Varshney, and M.-S. Alouini, “The capacity of injective semi-deterministic two-way channels,” in Proc. 2017 IEEE Int. Symp. Inf. Theory , Jun. 2017, pp. 431–435

work page 2017

[3] [3]

Achieving single channel, full duplex wireless communication,

J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, “Achieving single channel, full duplex wireless communication,” inProc. 16th Annu. Int. Conf. Mobile Comput. Netw. (MobiCom’10) , Sep. 2010, pp. 1–12

work page 2010

[4] [4]

Practical, real-time, full duplex wireless,

M. Jain, J. I. Choi, T. M. Kim, D. Bharadia, S. Seth, K. Srinivasan, P. Levis, S. Katti, and P. Sinha, “Practical, real-time, full duplex wireless,” in Proc. 17th Annu. Int. Conf. Mobile Comput. Netw. (Mobi- Com’11), Sep. 2011, pp. 301–312

work page 2011

[5] [5]

Full-duplex wireless: Design, implementation and charac- terization,

M. Duarte, “Full-duplex wireless: Design, implementation and charac- terization,” Ph.D. dissertation, Rice University, Houston, TX, Apr. 2012

work page 2012

[6] [6]

In-band full-duplex wireless: Challenges and opportuni- ties,

A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges and opportuni- ties,” IEEE J. Sel. Areas Commun. , vol. 32, no. 9, pp. 1637–1652, Sep. 2014

2014

[7] [7]

Introducing full duplex in hybrid ﬁber coaxial networks,

W. Coomans, H. Chow, and J. Maes, “Introducing full duplex in hybrid ﬁber coaxial networks,” IEEE Commun. Stand. Mag. , vol. 2, no. 1, pp. 74–79, Mar. 2018. 9

work page 2018

[8] [8]

Full duplex DOCSIS: Opportunities and challenges,

B. Berscheid and C. Howlett, “Full duplex DOCSIS: Opportunities and challenges,” IEEE Commun. Mag., vol. 57, no. 8, pp. 28–33, Aug. 2019

work page 2019