# VLSF Decoding with Reliability Guarantees over Correlated Noncoherent Fading Channels

Guodong Sun, Samir M. Perlaza, Philippe Mary, Jean-Marie Gorce

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Guodong Sun, Samir M. Perlaza, Philippe Mary, Jean-Marie Gorce. VLSF Decoding with Reliability Guarantees over Correlated Noncoherent Fading Channels. RR-9609, Inria. 2026. hal-05471625

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informatics mathematics
Inria

VLSF Decoding with
Reliability Guarantees
over Correlated
Noncoherent Fading
Channels

Guodong Sun, Samir M. Perlaza, Philippe Mary, and
Jean-Marie Gorce

RESEARCH
REPORT

Nº 9609

January 2026

Project-Team NEO
<figure><img src="image_3.png" /></figure>
<figure><img src="image_4.png" /></figure>

# VLSF Decoding with Reliability Guarantees over Correlated Noncoherent Fading Channels

Guodong Sun, Samir M. Perlaza, Philippe Mary, and Jean-Marie Gorce

Project-Team NEO

Research Report n° 9609 — January 2026 — 19 pages

**Abstract:** This paper studies reliability-guaranteed decoding for variable-length stop-feedback (VLSF) codes over correlated noncoherent fading channels. The decoding rule is based on the evolution of the information density associated with a channel input–output realization. Due to channel memory, exact evaluation of this information density is intractable. To enable a constructive decoding, computable finite-blocklength lower and upper bounds on the information density that hold uniformly over time along each input–output sequence are derived. The lower bound enables a stopping-time analysis for VLSF decoding and has an operational meaning, while the upper bound quantifies the relaxation gap. Moreover, the relaxation gap between the bounds is explicitly characterized. For Gaussian signaling, the stopping-time distribution and the impact of fading correlation on decoding performance are numerically studied.

**Key-words:** Variable-length stop-feedback code, noncoherent channel, time-correlated fading, Hölder inequality, Rényi divergence.

RESEARCH CENTRE
SOPHIA ANTIPOLIS – MÉDITERRANÉE

2004 route des Lucioles - BP 93
06902 Sophia Antipolis Cedex

<footnote>Guodong Sun and Samir M. Perlaza are with INRIA, Centre Inria d'Université Côte d'Azur, Sophia Antipolis 06902, France. Samir M. Perlaza is also with the GAATI Laboratory at the Université de la Polynésie Française, Faaa 98702, French Polynesia; and also with the Electrical and Computer Engineering Department at Princeton University, Princeton N.J. 08544, USA. Philippe Mary is with Univ. Rennes, INSA, CNRS, IETR UMR 6164 F-35000, Rennes, France. Jean-Marie Gorce is with Inria, INSA Lyon, CITI Laboratory, UR3720, Villeurbanne, France.</footnote>

<footnote>This work is supported in part by the French National Agency for Research under the France 2030 program through the PEPR project Réseaux du Futur (ANR-22-PEFT-0010) and the RIS3 project (ANR-23-CMAS-0023).</footnote>
# Décodage VLSF avec garanties de fiabilité sur des canaux à évanouissement non cohérents corrélés.

**Résumé** : Ce rapport étudie le décodage avec garantie de fiabilité pour des codes à longueur variable avec retour d'arrêt (VLSF) sur des canaux à évanouissement non cohérents corrélés. La règle de décodage est fondée sur l'évolution de la densité d'information associée à une réalisation entrée-sortie du canal. En raison de la mémoire du canal, l'évaluation exacte de cette densité d'information est inabordable. Afin de permettre un décodage constructif, des bornes inférieure et supérieure calculables à longueur finie sur la densité d'information, valables uniformément dans le temps le long de chaque séquence entrée-sortie, sont dérivées. La borne inférieure permet une analyse en temps d'arrêt du décodage VLSF, tandis que la borne supérieure quantifie l'écart de relaxation. De plus, l'écart de relaxation entre ces bornes est caractérisé explicitement. Pour une signalisation gaussienne, la distribution du temps d'arrêt et l'impact de la corrélation de l'évanouissement sur les performances de décodage sont caractérisés numériquement.

**Mots-clés** : Codes à longueur variable avec retour d'arrêt, canal non cohérent, évanouissement corrélé dans le temps, inégalité de Hölder, divergence de Rényi.
# Contents

<table><tr><td>1</td><td>Introduction</td><td>4</td></tr><tr><td>2</td><td>VLSF Decoder and Problem Formulation</td><td>5</td></tr><tr><td>3</td><td>Main Results</td><td>7</td></tr><tr><td>3.1</td><td>Information density lower bound</td><td>8</td></tr><tr><td>3.2</td><td>Specialization to i.i.d. Gaussian reference fading and Gaussian signaling</td><td>9</td></tr><tr><td>3.3</td><td>Information density upper bound</td><td>11</td></tr><tr><td>4</td><td>Numerical Results</td><td>11</td></tr><tr><td>5</td><td>Conclusion</td><td>14</td></tr><tr><td></td><td>Appendices</td><td>14</td></tr><tr><td>A</td><td>Proof of Theorem 2.1</td><td>14</td></tr><tr><td>B</td><td>Proof of Lemma 3.1</td><td>16</td></tr><tr><td></td><td>References</td><td>18</td></tr></table>
# 1 Introduction

Variable-length stop-feedback (VLSF) coding has been extensively studied for coherent memoryless channels, for which optimal decoding rules and sharp non-asymptotic performance characterizations are available [1–5]. At finite block-lengths, however, channel estimation incurs a non-negligible blocklength overhead, particularly in fading environments in which the channel evolves rapidly and experiences symbol-wise temporal correlation. Such scenarios lead to non-coherent channels with fading memory, for which existing VLSF analyses do not directly apply.

For noncoherent fading channels, most established results rely on asymptotic analyses [6–9]. These works characterize the double-logarithmic growth of capacity with respect to signal-to-noise ratio (SNR). In particular, Lapidoth and Moser introduced the notion of fading numbers that can quantify the impact of fading memory in the high-SNR regime [6, 7]. While these results provide fundamental insights into ergodic capacity and high-SNR scaling laws, their expectation-based formulation makes them unsuitable for constructing finite blocklength decoding rules or stop criteria for variable-length codes.

In the finite-blocklength regime, performance is characterized by the information density induced by a given channel input and output realization [10, 11]. In particular, stopping rules for variable-length decoding depend on the cumulative information density evaluated along such an input–output sequence [1]. To maintain analytical tractability, most existing non-asymptotic studies of non-coherent channels focus on fixed blocklength models with block-fading [12] or quasi-static fading [13]. These assumptions simplify temporal correlation by assuming blockwise-constant fading without symbol-wise channel correlation. As a result, a finite-blocklength analysis of information density evolution that holds uniformly over time for each input-output realization remains missing for time-correlated noncoherent channels. This is analogous to the asymptotic case investigated in [6, 7].

From an information-theoretical perspective, short packet communication over fast fading channels with temporal correlation presents two fundamental challenges: the non-negligible blocklength cost of channel estimation; and the need for reliability guarantees in the non-asymptotic regime. While VLSF codes address the latter in coherent, memoryless settings, their extension to noncoherent channels with memory is nontrivial due to the intractability of the information density evaluation.

This paper addresses these challenges, and the main contributions are summarized as follows:

* A computable finite-blocklength lower bound on the information density associated with each input-output realization is derived, which holds uniformly over time. The bound enables a reliability-guaranteed stopping rule for VLSF decoding over noncoherent fading channels with memory.

* The proposed lower bound admits a structural decomposition into a memoryless noncoherent envelope term and a penalty term capturing the effect
of relaxing the channel correlation. The penalty term is formulated as a change-of-measure cost, which is independent of the channel input. A complementary upper bound is also derived to quantify the relaxation gap.

* For Gaussian signaling, we numerically characterize the stopping-time distribution and investigate the impact of channel correlation on the performance of VLSF codes.

The paper is organized as follows. Section 2 introduces the noncoherent channel model with memory and formulates the problem. Section 3 derives computable uniform lower and upper bounds on the information density induced by channel input-output. Section 4 presents numerical results illustrating the behavior of the proposed bound and its applications. Finally, section 5 concludes the paper and discusses possible future work.

## 2 VLSF Decoder and Problem Formulation

Consider a discrete-time real-valued fading channel with additive Gaussian noise. At each time index $k \in \mathbb{N}_{>0}$, the channel input $X_k \in \mathcal{X}$ induces an output $Y_k \in \mathcal{Y}$, with $\mathcal{X}, \mathcal{Y} \subseteq \mathbb{R}$, according to

$$Y_k = H_k X_k + Z_k, \quad (1)$$

where $Z_k \sim \mathcal{N}(0, \sigma_z^2)$, with $k \in \mathbb{N}_{>0}$, are independent and identically distributed (i.i.d.). Throughout this work, $\mathcal{N}(\mu, \sigma^2)$ denotes the Gaussian probability measure with mean $\mu$ and variance $\sigma^2$. The fading process $\{H_k\}_{k>0}$ is modeled as a first-order Gauss-Markov process

$$H_k = \rho H_{k-1} + \sqrt{1 - \rho^2} W_k, \quad (2)$$

where $\rho \in [0, 1]$ is the temporal correlation coefficient. $W_k \sim \mathcal{N}(0, 1)$, with $k \in \mathbb{N}_{>0}$, are i.i.d. The initialization satisfies $H_0 \sim \mathcal{N}(0, 1)$, independent of $\{W_k\}_{k>0}$, which ensures the fading process to be stationary with marginal distribution $H_k \sim \mathcal{N}(0, 1)$, with $k \in \mathbb{N}_{>0}$. This work considers a noncoherent setting in which neither the transmitter nor the receiver has instantaneous channel state information (CSI). Instead, both know the distribution of the Gauss-Markov fading process in (2).

For any blocklength $n \in \mathbb{N}_{>0}$, the fading vector $H^n \triangleq (H_1, \dots, H_n)$ is jointly Gaussian with distribution

$$P_{H^n} = \mathcal{N}(0, \Sigma_{H^n}), \quad (3)$$

where $\Sigma_{H^n}$ is a Toeplitz matrix with entries $(\Sigma_{H^n})_{ij} = \rho^{|i-j|}$. The matrix is positive semidefinite for $\rho \in [0, 1]$.

For any $n \in \mathbb{N}_{>0}$, denote the input and output sequences by

$$x^n \triangleq (x_1, \dots, x_n) \text{ and } y^n \triangleq (y_1, \dots, y_n). \quad (4)$$
Assume a random coding ensemble in which every symbol of the input vector $X^n$ is generated i.i.d. according to $P_X$. The codebook $C \triangleq \{x_m^\infty : m \in \{1, \dots, M\}\}$ consists of $M$ codewords, in which each codeword $x_m^\infty \triangleq (x_{m,1}, \dots)$ is drawn independently according to the product measure $P_X^\infty \triangleq \prod_{k=1}^\infty P_X$. For each codeword $x_m^\infty$, denote the prefix by $x_m^n \triangleq (x_{m,1}, \dots, x_{m,n})$. Consider a message $W$ is uniformly chosen over $\{1, \dots, M\}$, and when $W = m$ is transmitted, the channel input at time $k$ is $X_k = x_{m,k}$.

For a fixed input sequence $x^n$ and fading realization $h^n$, the conditional distribution of the channel output sequence $Y^n$ follows from (1) and is given by

$$P_{Y^n|X^n=x^n, H^n=h^n} = \mathcal{N}(\mathrm{diag}(x^n)h^n, \sigma_z^2 I_n), \quad (5)$$

where $I_n$ denotes the identity matrix of size $n$. Averaging over the jointly Gaussian fading vector $H^n$, which is independent of $X^n$, the conditional output distribution given the input sequence $x^n$ is

$$P_{Y^n|X^n=x^n} = \mathcal{N}(0, \mathrm{diag}(x^n)\Sigma_{H^n}\mathrm{diag}(x^n) + \sigma_z^2 I_n). \quad (6)$$

The marginal output distribution $P_{Y^n}$ is obtained by averaging over the random coding input and channel fading. Let $\mathcal{B}(\mathcal{Y}^n)$ denote the Borel $\sigma$-algebra on $\mathcal{Y}^n$. For any measurable set $A \in \mathcal{B}(\mathcal{Y}^n)$,

$$P_{Y^n}(A) = \iint P_{Y^n|X^n=x^n, H^n=h^n}(A) dP_{X^n}(x^n) dP_{H^n}(h^n) \quad (7)$$

$$= \iint P_{Y^n|H^n=h^n}(A) dP_{H^n}(h^n). \quad (8)$$

Both measures in (6) and (8) admit everywhere positive densities on $\mathbb{R}^n$ for all $n > 0$, hence $P_{Y^n|X^n=x^n} \ll P_{Y^n}$. The information density associated with a given input-output realization $(x^n, y^n)$ is defined as

$$\iota(x^n; y^n) \triangleq \log \frac{dP_{Y^n|X^n=x^n}}{dP_{Y^n}}(y^n). \quad (9)$$

Consider a threshold-based VLSF decoding rule with a guaranteed reliability $1 - \epsilon$, for some $\epsilon > 0$. The decoder sequentially observes the channel outputs $\{y_k\}_{k>0}$ and for each codeword $m$ computes $\iota(x_m^n; y^n)$ for every $n \in \mathbb{N}_{>0}$. The stopping time is defined as

$$\tau = \inf \{ n \in \mathbb{N}_{>0} : \exists \bar{m} \in \{1, \dots, M\}, \iota(x_m^n; y^n) \ge \gamma \}. \quad (10)$$

If at time $\tau$, there is a unique $\bar{m}$ such that $\iota(x_m^\tau; y^\tau) \ge \gamma$, the decoder outputs $\hat{W} = \bar{m}$. Otherwise, an error is declared. The decoding error probability is defined as $\mathbb{P}[W \ne \hat{W}]$, where the probability is taken over the random codebook, message, fading process, and noise.

**Theorem 2.1.** Let $\epsilon > 0$ and $M \in \mathbb{N}_{>0}$. For any threshold $\gamma \ge \log \frac{M-1}{\epsilon}$, the threshold-based VLSF decoding rule satisfies

$$\mathbb{P}[W \ne \hat{W}] \le \epsilon.$$
Proof: See Appendix A.

**Remark 1.** Threshold-based VLSF decoding rules have been previously analyzed for coherent memoryless channels in [1] and [4]. Theorem 2.1 extends this reliability guarantee to the noncoherent setting with memory.

To evaluate the information density induced by $(x^n, y^n)$ in (9), observe that the measure $P_{Y^n|X^n=x^n}$ in (6) admits a closed-form density and is thus analytically tractable. In contrast, the measure $P_{Y^n}$ in (7) is a high-dimensional Gaussian mixture induced jointly by the fading process and the randomness of the channel input. In general, this mixture is analytically intractable in the presence of channel memory. This analytical intractability motivates the development of computable upper and lower bounds on the information density in (9) in the sequel.

## 3 Main Results

Since the exact information density induced by the input-output realization is generally intractable, we base the decoding rule in Theorem 2.1 on computable bounds that hold uniformly over time. In particular, the lower bound plays a direct operational role. Whenever the lower bound exceeds the decoding threshold, the actual information density is guaranteed to exceed the same threshold. Hence, the decoding reliability constraint is guaranteed. The upper bound, while not directly involved in the threshold-based stopping rule, serves a complementary purpose. It enables a quantitative assessment of the lower bound by characterizing the gap between the two bounds, thereby qualifying the conservativeness and cost of the proposed bounding approach.

Let $\mathcal{Q}$ be the class of probability measures on $\mathbb{R}^n$ such that, for every $Q_{H^n} \in \mathcal{Q}$, $P_{H^n}$ is absolutely continuous with respect to (w.r.t.) $Q_{H^n}$, i.e. $P_{H^n} \ll Q_{H^n}$, with $P_{H^n}$ in (3). For any such $Q_{H^n}$, define the likelihood ratio

$$L(h^n) \triangleq \frac{dP_{H^n}}{dQ_{H^n}}(h^n). \qquad (11)$$

Throughout this section, we assume that $P_{Y^n|H^n=h^n}$, $P_{Y^n|X^n=x^n,H^n=h^n}$, and $P_{Y^n|X^n=x^n}$ admit densities with respect to Lebesgue measure $\mathbb{R}^n$, denoted by $f_{Y^n|H^n}(y^n|h^n)$, $f_{Y^n|X^n,H^n}(y^n|x^n, h^n)$, and $f_{Y^n|X^n}(y^n|x^n)$, respectively. The marginal output in (8) admits a density of the form

$$f_{Y^n}(y^n) = \int f_{Y^n|H^n}(y^n|h^n)L(h^n)dQ_{H^n}(h^n). \qquad (12)$$

The results in this section provide bounds on the information density induced by every input-output realization $(x^n, y^n)$ for all $n$ via a change of measure on the fading process.
## 3.1 Information density lower bound

The following theorem provides a lower bound on the actual information density in (9).

**Theorem 3.1** (Information density lower bound). Let $Q_{H^n} \in \mathcal{Q}$ be a reference measure. Fix $r > 1$, and let $s \triangleq \frac{r}{r-1}$ denote its Hölder conjugate. Then, given a pair $(x^n, y^n) \in \mathcal{X}^n \times \mathcal{Y}^n$,

$$ \begin{aligned} \iota(x^n; y^n) \geq \log f_{Y^n|X^n}(y^n|x^n) & - \frac{r-1}{r} D_r(P_{H^n}\|Q_{H^n}) \\ & - \frac{1}{s} \log \mathbb{E}_{Q_{H^n}} [f_{Y^n|H^n}(y^n|H^n)^s], \end{aligned} \quad (13) $$

where $D_r(P_{H^n}\|Q_{H^n})$ is the Rényi divergence of order $r$.

*Proof:* Since $P_{H^n} \ll Q_{H^n}$, the Radon-Nikodym derivative $\frac{dP_{H^n}}{dQ_{H^n}}$ exists. Fix $r > 1$ and $s = \frac{r}{r-1}$ so that $\frac{1}{r} + \frac{1}{s} = 1$. Applying Hölder's inequality [14, Theorem 2.11.1] to (12) gives,

$$ \begin{aligned} f_{Y^n}(y^n) &= \int f_{Y^n|H^n}(y^n|h^n)L(h^n)dQ_{H^n}(h^n) \\ &\leq \left( \int L(h^n)^r dQ_{H^n}(h^n) \right)^{\frac{1}{r}} \left( \int f_{Y^n|H^n}(y^n|h^n)^s dQ_{H^n}(h^n) \right)^{\frac{1}{s}}. \end{aligned} \quad (14) $$

The first factor can be expressed in terms of the Rényi divergence of order $r$,

$$ \mathbb{E}_{Q_{H^n}} [L(h^n)^r] = \exp \left( (r-1) D_r(P_{H^n}\|Q_{H^n}) \right), \quad (15) $$

where

$$ D_r(P_{H^n}\|Q_{H^n}) \triangleq \frac{1}{r-1} \log \int \left( \frac{dP_{H^n}}{dQ_{H^n}}(h^n) \right)^r dQ_{H^n}(h^n). \quad (16) $$

The information density in (9) can be expressed as

$$ \iota(x^n; y^n) = \log \frac{dP_{Y^n|X^n=x^n}}{dP_{Y^n}}(y^n) = \log f_{Y^n|X^n}(y^n|x^n) - \log f_{Y^n}(y^n), \quad (17) $$

where the last equation holds whenever $P_{Y^n|X^n=x^n}$ and $P_{Y^n}$ are absolutely continuous w.r.t. Lebesgue measure $\lambda^n$. Combining (14), (15), and (17) completes the proof. ■

**Remark 2.** The inequality in (14) holds for every $y^n$ and for all $n$. As a consequence, the information density lower bound in (13) is directly applicable to the threshold-based VSLF decoding rule. The change-of-measure factor given in (15) $\exp((r-1)D_r(P_{H^n}\|Q_{H^n}))$ depends only on the pair $(P_{H^n}, Q_{H^n})$ and the order $r$. It is therefore a deterministic constant that does not affect the sample-path fluctuations of the information density induced by $(x^n, y^n)$, but only induces a constant offset.

Below, we refer to the term $\frac{r-1}{r} D_r(P_{H^n}\|Q_{H^n})$ as the Rényi penalty. The remaining term in the Hölder inequality $\frac{1}{s} \log \mathbb{E}_{Q_{H^n}} [f_{Y^n|H^n}(y^n|H^n)^s]$ is called the Hölder envelope.
## 3.2 Specialization to i.i.d. Gaussian reference fading and Gaussian signaling

We adopt an i.i.d. Gaussian reference fading measure, which enables closed-form evaluation of the Rényi change-of-measure penalty and a tractable expression for the Hölder envelope in (13). Let $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$, where $\sigma_h^2$ is a design parameter.

**Lemma 3.1 (Rényi penalty).** Let $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$ and $P_{H^n} = \mathcal{N}(0, \Sigma_{H^n})$ in (3). The Rényi penalty in (15) admits an explicit form

$$ \mathbb{E}_{Q_{H^n}} [L(h^n)^r] = \left( \frac{\sigma_h^{2n}}{|\Sigma_{H^n}|} \right)^{\frac{r}{2}} \det \left( I_n + r \sigma_h^2 (\Sigma_{H^n}^{-1} - \frac{1}{\sigma_h^2} I_n) \right)^{-\frac{1}{2}}. \quad (18) $$

Moreover, $\mathbb{E}_{Q_{H^n}} [L(h^n)^r] < \infty$ if

$$ \sigma_h^2 > \frac{r-1}{r} \frac{1+\rho}{1-\rho}. \quad (19) $$

If (19) holds, then

$$ \lim_{n \to \infty} \frac{1}{n} \log \mathbb{E}_{Q_{H^n}} [L(h^n)^r] = \frac{1}{2} \int_{-\pi}^{\pi} \log \frac{r \sigma_h^2 S_H(\omega)}{r \sigma_h^2 - (r-1)S_H(\omega)} \frac{d\omega}{2\pi}, \quad (20) $$

where

$$ S_H(\omega) = \frac{1 - \rho^2}{1 + \rho^2 - 2\rho \cos \omega}. \quad (21) $$

*Proof:* See Appendix B.

<figure><img src="image_5.png" /></figure>

We now specialize in Gaussian signaling for its tractability. Let $\phi_v(y) \triangleq \frac{1}{\sqrt{2\pi v}} e^{-y^2/(2v)}$ denote the probability density function of a zero-mean Gaussian random variable with variance $v$.

**Lemma 3.2 (Hölder envelope under Gaussian signaling).** Assume Gaussian signaling $X_k \sim \mathcal{N}(0, P_0)$ for all $k > 0$, independent of the fading and noise processes. Then the AWGN channel induces

$$ P_{Y_k|H_k=h_k} = \mathcal{N}(0, \sigma_z^2 + P_0 h_k^2), \quad \forall k > 0. \quad (22) $$

Let $f_{Y^n|H^n}(y^n|h^n)$ denote the conditional output density. Then,

$$ f_{Y^n|H^n}(y^n|h^n) = \prod_{k=1}^{n} \phi_{v_k}(y_k). \quad (23) $$

Fix $s > 1$ and define $c_{s,v} \triangleq \frac{1}{\sqrt{s(2\pi v)^{s-1}}}$. Then the $s$-th power of (23) is

$$ (f_{Y^n|H^n}(y^n|h^n))^s = \prod_{k=1}^{n} c_{s,v_k} \phi_{\frac{v_k}{s}}(y_k). \quad (24) $$
Moreover, under the reference fading measure $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$, the Hölder envelope term in (13) factorizes as

$$ \int f_{Y^n|H^n}(y^n|h^n)^s dQ_{H^n}(h^n) = \prod_{k=1}^{n} \int c_{s,v_k} \phi_{\frac{v_k}{s}}(y_k) \phi_{\sigma_h^2}(h_k) dh_k, \quad (25) $$

where each scalar integral is finite and computable.

*Proof:* The product form (23) follows from the conditional independence of the channel output given $H^n$. A Gaussian density $\phi_v(y)$ raised to power $s$ is given by

$$ \begin{aligned} \left(\phi_v(y)\right)^s &= \left(\frac{1}{\sqrt{2\pi v}}\right)^s e^{-sy^2/(2v)} = \frac{1}{\sqrt{s(2\pi v)^{s-1}}} \left(\frac{1}{\sqrt{2\pi v/s}}\right) e^{-y^2/(\frac{2v}{s})} \\ &= c_{s,v} \phi_v(y), \end{aligned} \quad (26) $$

where $c_{s,v} = \frac{1}{\sqrt{s(2\pi v)^{s-1}}}$. Under $Q_{H^n}$, the independence of the components of $H^n$ yields the factorization in (25). ■

The expression (25) factorizes the $n$-dimensional integral into a product of $n$ one-dimensional Gaussian integrals. Consequently, the Hölder envelope term can be computed with linear complexity in $n$, making the information density lower bound numerically tractable at finite blocklength.

**Corollary 3.1 (A computable lower bound).** Consider the noncoherent fading channel in (1) with fading process $H^n$ defined in (3). Assume $P_X = \mathcal{N}(0, P_0)$, $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$, a Hölder exponent $r > 1$ with conjugate $s = \frac{r}{r-1}$ satisfying $\sigma_h^2 \ge \frac{r-1}{r} \frac{1+\rho}{1-\rho}$. Then, for every input-output realization $(x^n, y^n)$, the information density satisfies

$$ \iota(x^n; y^n) \ge \psi(x^n, y^n), \quad (27) $$

with

$$ \psi(x^n, y^n) \triangleq \log f_{Y^n|X^n}(y^n|x^n) - \frac{1}{s} \log q_s(y^n) - \frac{r-1}{r} D_r(P_{H^n}\|Q_{H^n}), \quad (28) $$

where the computable envelope term is

$$ q_s(y^n) \triangleq \prod_{k=1}^{n} \int c_{s,v_k} \phi_{\frac{v_k}{s}}(y_k) \phi_{\sigma_h^2}(h_k) dh_k, \quad (29) $$

with $v_k = \sigma_z^2 + P_0 h_k^2$.

*Proof:* The result follows by combining Theorem 3.1 with Lemmas 3.1 and 3.2. ■

This bound provides a stopping metric of the form $\log f_{Y^n|X^n}(y^n|x^n) - \frac{1}{s} \log q_s(y^n)$, with a deterministic penalty $\frac{r-1}{r} D_r(P_{H^n}\|Q_{H^n})$. This structure is directly compatible with VLSF decoding rules, where decoding is triggered once the lower bound exceeds a prescribed threshold. Note that the tightness of the lower bound is determined by the parameters $(r, \sigma_h^2)$, which control the tradeoff between the Hölder envelope and the Rényi penalty.
## 3.3 Information density upper bound

The following theorem provides an upper bound on the actual information density in (9).

**Theorem 3.2 (Information density upper bound).** Let $Q_{H^n} \in \mathcal{Q}$, and the likelihood ratio $L(h^n)$ in (11). Then, for every realization $(x^n, y^n)$,

$$ \iota(x^n; y^n) \le \varphi(x^n, y^n), \qquad (30) $$

where

$$ \varphi(x^n, y^n) \triangleq \log f_{Y^n|X^n}(y^n|x^n) - \mathbb{E}_{Q_{H^n}}[\log f_{Y^n|H^n}(y^n|H^n)] + D(P_{H^n}\|Q_{H^n}), \qquad (31) $$

where $D(P_{H^n}\|Q_{H^n})$ denotes the Kullback-Leibler (KL) divergence.

*Proof:* Taking a logarithm to (12) and applying Jensen's inequality to the concave function $\log(\cdot)$ give

$$ \begin{aligned} \log f_{Y^n}(y^n) &= \log \mathbb{E}_{Q_{H^n}} [f_{Y^n|H^n}(y^n|H^n)L(H^n)] \\ &\ge \mathbb{E}_{Q_{H^n}} [\log f_{Y^n|H^n}(y^n|H^n)L(H^n)] \\ &= \mathbb{E}_{Q_{H^n}} [\log f_{Y^n|H^n}(y^n|H^n)] + \mathbb{E}_{Q_{H^n}} [\log (L(H^n))] \\ &= \mathbb{E}_{Q_{H^n}} [\log f_{Y^n|H^n}(y^n|H^n)] - D(P_{H^n}\|Q_{H^n}), \end{aligned} \qquad (32) $$

where in the last equality the second term is $-D(P_{H^n}\|Q_{H^n})$ by definition of the KL divergence. Substituting the bound of $\log f_{Y^n}(y^n)$ into the information density completes the proof. ■

**Remark 3.** Specializing to $P_X = \mathcal{N}(0, P_0)$ and $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$ enables closed-form expressions for both the envelope term $\mathbb{E}_{Q_{H^n}}[\log f_{Y^n|H^n}(y^n|H^n)]$ and the KL penalty $D(P_{H^n}\|Q_{H^n})$, resulting in a tractable upper bound.

**Remark 4.** Theorem 3.2 can be interpreted as an analogue of the capacity upper bound of Lapidoth and Moser in [6] valid for every input-output $(x^n, y^n)$ and for all $n$, which is obtained by introducing a reference output distribution induced by the true input and the reference fading channel.

## 4 Numerical Results

This section evaluates the VLSF stopping rule induced by the proposed com-
putable bounds derived in Section 3 under an i.i.d. Gaussian signaling $X_k \sim \mathcal{N}(0, P_0)$ and an i.i.d. Gaussian reference measure $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$. The design parameter, i.e., the Hölder exponent $r$ and the reference variance $\sigma_h^2$ in Corollary 3.1, are optimized via numerical search.

Figure 1 illustrates the behavior of the VLSF decoding rule induced by the information density lower bound $\psi(x_m^n, y^n)$ in (28). The upper subfigure shows four representative realizations up to the threshold crossing, while the
<figure><img src="image_6.png" /><figcaption>Figure 1: Top: four realizations of the information density lower bound $\psi(x_m^n, y^n)$ that trigger the stopping event. Bottom: the corresponding empirical stopping-time distribution of the VLSF decoder.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 2: For $n = 50$, the lower bound $\psi(x^n, y^n)$ in (28) from Corollary 3.1 is shown by the blue solid curves, while the upper bound $\varphi(x^n, y^n)$ in (31) is shown by the orange dashed curves. The optimal lower bound, obtained via a search over the free parameters $(r, \sigma_h^2)$, is highlighted in red.</figcaption></figure>

lower subfigure depicts the corresponding empirical stopping-time distribution. The simulations are performed for a correlation coefficient $\rho = 0.3$, an SNR of $\frac{F_0}{\sigma_z^2} = 100$, a message size of $\log_2 M = 10$ bits, and a target error probability $\epsilon = 10^{-3}$. The choice $\rho = 0.3$ is due to the use of the i.i.d. Gaussian reference measure $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$, ensuring that the Rényi divergence w.r.t. the true fading measure does not become excessively large. Figure 1 illustrates the advantage of VLSF codes: transmission terminates once the accumulated information density exceeds the decision threshold, whereas fixed blocklength codes must continue until the end of the block. Moreover, the stopping-time distribution characterizes the decoding delay, enabling the design of time-sensitive services with probabilistic delay guarantees under reliability constraints.

Figure 2 presents the average upper and lower bounds on the information density for a blocklength of $n = 50$ under the above setting. This analysis is motivated by the dependence of the lower bound in Corollary 3.1 on the design parameters, namely the Rényi order $r$ and the reference variance $\sigma_h^2$, as well as the dependence of the upper bound in Theorem 3.2 on $\sigma_h^2$. As shown in Figure 2, the tightness of both bounds is determined by appropriately optimizing over the
parameter pair $(r, \sigma_h^2)$. The current optimization is performed via an exhaustive search over all pairs $(x_m^n, y^n)$ and over all $n$. The gap between the upper and lower bounds is driven by the change-of-measure penalties $\frac{r-1}{r}D_r(P_{H^n}\|Q_{H^n})$ and $D(P_{H^n}\|Q_{H^n})$, which grow approximately linear in $n$. This gap quantifies the cost of having an explicit decoding rule with decoding reliability guarantee in this non-asymptotic regime. Moreover, these observations motivate future work on systematic methods for identifying optimal parameters that yield the tightest possible bounds.

# 5 Conclusion

This work presents the first finite-blocklength analysis yielding bounds on the information density that hold uniformly over time along each input-output sequence for time-correlated noncoherent fading channels. The proposed lower bound directly enables the design of stopping rules with reliability guarantees, making it suitable for VLSF decoding in the non-asymptotic regime. Promising directions for future research include investigating alternative signaling schemes and reference measures to obtain tighter lower bounds across different channel correlation regimes, as well as developing optimization methods for the free parameters $(r, \sigma_h^2)$.

# Appendices

## A Proof of Theorem 2.1

Under the random coding ensemble, each codeword $x_m^n$ is generated independently according to the product distribution $P_X^n$. The information density threshold rule corresponds to a likelihood-ratio test between the hypotheses

$$ (x^n, y^n) \sim \begin{cases} \mathcal{H}_0, & P_X^n P_{Y^n|X^n}, \\ \mathcal{H}_1, & P_X^n P_{Y^n}, \end{cases} \quad (33) $$

where $P_{Y^n}$ denotes the output distribution induced by $P_X^n$. Under $\mathcal{H}_1$, the sequences $X^n$ and $Y^n$ are independent.

The information density $\iota(x^n; y^n)$ defined in (9) is the log-likelihood ratio associated with this test. Define the corresponding likelihood ratio process

$$ \mathcal{L}_n(X^n; Y^n) \triangleq \exp(\iota(X^n; Y^n)) = \frac{dP_{Y^n|X^n}}{dP_{Y^n}}(Y^n). \quad (34) $$

Let $\{\mathcal{F}_n\}_{n \ge 1}$ denote the natural filtration generated by $(X^n; Y^n)$. Since $\mathcal{F}_{n-1} \subseteq \mathcal{F}_n$, the Radon-Nikodym derivative defined in (34) can be decomposed with respect to this filtration. Conditioning on $\mathcal{F}_{n-1}$ (equivalently, on the random
variables $(X^{n-1}, Y^{n-1})$, which are $\mathcal{F}_{n-1}$-measurable), the chain rule for Radon-Nikodym derivative gives

$$ \frac{dP_{Y^n|X^n}}{dP_{Y^n}}(Y^n) = \frac{dP_{Y^{n-1}|X^{n-1}}(Y^{n-1})}{dP_{Y^{n-1}}(Y^{n-1})} \cdot \frac{dP_{Y_n|X_n,Y^{n-1}}(Y_n)}{dP_{Y_n|Y^{n-1}}(Y_n)} \quad (35) $$

Consequently, the likelihood process $L_n(X^n; Y^n)$ can be written in recursive form

$$ L_n(X^n; Y^n) = L_{n-1}(X^{n-1}; Y^{n-1}) \frac{dP_{Y_n|X_n,Y^{n-1}}(Y_n)}{dP_{Y_n|Y^{n-1}}(Y_n)}, \quad (36) $$

where the Radon-Nikodym derivative is well-defined under $P_{Y_n|Y^{n-1}}$. Taking conditional expectation under hypothesis $\mathcal{H}_1$ with respect to $\mathcal{F}_{n-1}$ gives

$$ \mathbb{E}_{\mathcal{H}_1}[L_n(X^n; Y^n)|\mathcal{F}_{n-1}] = L_{n-1}(X^{n-1}; Y^{n-1})\mathbb{E}_{\mathcal{H}_1}\left[\frac{dP_{Y_n|X_n,Y^{n-1}}(Y_n)}{dP_{Y_n|Y^{n-1}}(Y_n)}|\mathcal{F}_{n-1}\right] \quad (37) $$

Under $\mathcal{H}_1$, the random variables $X_n$ and $Y_n$ are conditionally independent given $Y^{n-1}$, and the conditional distribution of $Y_n$ given $\mathcal{F}_{n-1}$ is $P_{Y_n|Y^{n-1}}$. Since $Y^{n-1}$ is $\mathcal{F}_{n-1}$-measurable, the conditional expectation reduces to an integral with respect to this conditional measure:

$$ \mathbb{E}_{\mathcal{H}_1}\left[\frac{dP_{Y_n|X_n,Y^{n-1}}(Y_n)}{dP_{Y_n|Y^{n-1}}(Y_n)}|\mathcal{F}_{n-1}\right] = \int \frac{dP_{Y_n|X_n,Y^{n-1}}(y_n)}{dP_{Y_n|Y^{n-1}}(y_n)}(y_n)dP_{Y_n|Y^{n-1}}(y_n). \quad (38) $$

By the defining property of the Radon-Nikodym derivative [15], the integral (38) equals one almost surely. Consequently,

$$ \mathbb{E}_{\mathcal{H}_1}[L_n(X^n; Y^n)|\mathcal{F}_{n-1}] = L_{n-1}(X^{n-1}; Y^{n-1}). \quad (39) $$

Moreover,

$$ \mathbb{E}_{\mathcal{H}_1}[L_n(X^n; Y^n)] = \int \frac{dP_{Y^n|X^n}}{dP_{Y^n}}(y^n)dP_{Y^n}(y^n) = 1. \quad (40) $$

Hence, the likelihood-ratio process $L_n(X^n; Y^n)$ is a nonnegative martingale with unit mean under $\mathcal{H}_1$.

By Ville's inequality [16, 17], for $\gamma > 0$, under hypothesis $\mathcal{H}_1$

$$ \mathbb{P}_{\mathcal{H}_1}\left(\sup_{n>1} L_n(X^n, Y^n) \ge e^\gamma\right) \le e^{-\gamma}. \quad (41) $$

Equivalently, this inequality ensures that the information density of a given pair $(X^n, Y^n)$ generated by independent input and output distributions ever crosses the threshold $\gamma$ is at most $e^{-\gamma}$. That is

$$ \mathbb{P}_{\mathcal{H}_1}\left(\sup_{n>0} \iota(X^n, Y^n) \ge \gamma\right) \le e^{-\gamma}. \quad (42) $$
Consider now the decoding rule based on the information-density threshold $\gamma$. Conditioned on $W = m$ being sent, a decoding error occurs if at least one of the $M-1$ non-transmitted codewords crosses the threshold

$$ \mathbb{P}[W \neq \hat{W} | W = m] = \mathbb{P} \left[ \bigcup_{j \neq m} \{ \iota(x_j^n; Y^n) \geq \gamma \} \right]. \quad (43) $$

Applying the union bound gives

$$ \mathbb{P}[W \neq \hat{W} | W = m] \leq \sum_{j \neq m} \mathbb{P}[\{\iota(x_j^n; Y^n) \geq \gamma\}]. \quad (44) $$

Under the random coding ensemble, each non-transmitted codeword is independent of $Y^n$, consequently,

$$ \mathbb{P}[W \neq \hat{W} | W = m] \leq (M-1) \mathbb{P}_{H_1}[\{\iota(X^n; Y^n) \geq \gamma\}]. \quad (45) $$

The inequality in (41) implies that for every $n$ and for all $\gamma > 0$,

$$ \mathbb{P}_{H_1}[\{\iota(X^n; Y^n) \geq \gamma\}] \leq e^{-\gamma}. \quad (46) $$

Choosing $\gamma \geq \log \frac{M-1}{\epsilon}$ and assuming the message $W$ is chosen uniformly from the codebook, the average error probability satisfies

$$ \mathbb{P}[W \neq \hat{W}] = \frac{1}{M} \sum_{m=1}^{M} \mathbb{P}[\hat{W} \neq W | W = m] \leq \epsilon, \quad (47) $$

which concludes the theorem.

## B Proof of Lemma 3.1

**Closed form of the Rényi divergence** The Rényi divergence between two Gaussian measures admits a closed-form expression. Raising $L(h^n)$ in (11) to the r-th power gives

$$ (L(h^n))^r = \left( \frac{\sigma_h^{2n}}{|\Sigma_{H^n}|} \right)^{\frac{r}{2}} \exp \left( -\frac{r}{2} (h^n)^\mathrm{T} \left( \Sigma_{H^n}^{-1} - \frac{1}{\sigma_h^2} I_n \right) h^n \right). \quad (48) $$

Taking expectation with respect to $Q_{H^n}$ yields

$$ \mathbb{E}_{Q_{H^n}} \left[ \left( L(H^n) \right)^r \right] = \int \left( L(h^n) \right)^r dQ(h^n) \quad (49) $$

$$ = \left( \frac{1}{\sqrt{2\pi}} \right)^n \det(\Sigma_{H^n})^{-r/2} \det(\sigma_h^2 I_n)^{(r-1)/2} \quad (50) $$

$$ \int_{\mathbb{R}^n} \exp \left( -\frac{1}{2} (h^n)^\mathrm{T} \left( r \Sigma_{H^n}^{-1} + (1-r) \frac{1}{\sigma_h^2} I_n \right) (h^n) \right) dh^n. \quad (51) $$
This integral in (51) is a standard Gaussian integral provided that the matrix $r\Sigma_{H^n}^{-1} + (1-r)\frac{1}{\sigma_h^2}I_n$ is positive definite. Under this condition, the integral is given by

$$ \int_{\mathbb{R}^n} \exp\left(-\frac{1}{2}(h^n)^T (r\Sigma_{H^n}^{-1} + (1-r)\frac{1}{\sigma_h^2}I_n)(h^n)\right)dh^n = \frac{(\sqrt{2\pi})^n}{\sqrt{\det(r\Sigma_{H^n}^{-1} + (1-r)\frac{1}{\sigma_h^2}I_n)}}. \quad (52) $$

Hence, the Rényi divergence is finite only for values of $r > 1$ such that $r\Sigma_{H^n}^{-1} + (1-r)\frac{1}{\sigma_h^2}I_n$ is positive definite. Combining (50) and (52) yields (18).

**Feasible range of $(r, \sigma_h^2)$** The above closed form is valid under the positive definiteness condition

$$ r\Sigma_{H^n}^{-1} + (1-r)\frac{1}{\sigma_h^2}I_n > 0, \quad (53) $$

which is equivalent to

$$ \sigma_h^2 I_n \ge \frac{r-1}{r} \Sigma_{H^n}. \quad (54) $$

Since $Q_{H^n} = \mathcal{N}(0, \sigma_h^2 I_n)$, the finiteness condition becomes

$$ \sigma_h^2 > \frac{r-1}{r} \xi_{\max}(\Sigma_{H^n}), \quad (55) $$

where $\xi_{\max}(\Sigma_{H^n})$ denotes the largest eigenvalue of $\Sigma_{H^n}$.

For Gauss-Markov fading, $\Sigma_H$ is a Toeplitz matrix of the form $(\Sigma_H)_{ij} = \rho^{|i-j|}$, whose $k$-th eigenvalue for $k = 1, \dots, n$ admits the closed-form expression [18] [19, Theorem 4.1]

$$ \xi_k(\Sigma_H) = \frac{1 - \rho^2}{1 - 2\rho \cos \theta_k + \rho^2}, \quad (56) $$

with $\theta_k = \frac{k\pi}{n+1}$. The largest eigenvalue corresponds to the smallest angle $\theta_k$, i.e., $k = 1$, for which $\cos(\frac{\pi}{n+1}) < 1$ for any finite $n$. Hence,

$$ \xi_{\max}(\Sigma_H) < \frac{1 + \rho}{1 - \rho}. \quad (57) $$

Consequently, a sufficient condition for the finiteness of the Rényi divergence is

$$ \sigma_h^2 \ge \frac{r-1}{r} \frac{1+\rho}{1-\rho}. \quad (58) $$

Equivalently, since the Hölder conjugate $s = \frac{r}{r-1}$, the condition can also be written as

$$ s \ge \frac{1 + \rho}{(1 - \rho)\sigma_h^2}. \quad (59) $$
**Asymptotics** As $n \to \infty$, the per-symbol Rényi penalty in (20) is given by

$$ \lim_{n \to \infty} \frac{1}{n} \log \mathbb{E}_{Q_{H^n}} [L(H^n)^r] \qquad (60) $$

$$ = \lim_{n \to \infty} \frac{r}{2} \log \sigma_h^2 - \frac{r}{2n} \log |\Sigma_{H^n}| - \frac{1}{2n} \log \det \left( I_n + r\sigma_h^2 \left( \Sigma_{H^n}^{-1} - \frac{1}{\sigma_h^2} I_n \right) \right) \quad (61) $$

$$ = \lim_{n \to \infty} \frac{r}{2} \log \sigma_h^2 - \frac{r-1}{2n} \log |\Sigma_{H^n}| - \frac{1}{2} \log(r\sigma_h^2) - \frac{1}{2n} \log \det \left( I_n - \frac{r-1}{r\sigma_h^2} \Sigma_{H^n} \right) \quad (62) $$

$$ = \frac{r}{2} \log \sigma_h^2 - \frac{1}{2} \log(r\sigma_h^2) - \frac{r-1}{2} \int_{-\pi}^{\pi} \log S_H(\omega) \frac{d\omega}{2\pi} - \frac{1}{2} \int_{-\pi}^{\pi} \log \left(1 - \frac{r-1}{r\sigma_h^2} S_H(\omega)\right) \frac{d\omega}{2\pi} \quad (63) $$

$$ = \frac{1}{2} \int_{-\pi}^{\pi} \log \frac{r\sigma_h^2 S_H(\omega)}{r\sigma_h^2 - (r-1)S_H(\omega)} \frac{d\omega}{2\pi}, \quad (64) $$

where (63) follows from Szegö theorem applied to the Toeplitz covariance matrix $\Sigma_{H^n}$, i.e.,

$$ \lim_{n \to \infty} \frac{1}{n} \log |\Sigma_{H^n}| = \int_{-\pi}^{\pi} \log S_H(\omega) \frac{d\omega}{2\pi}, \quad (65) $$

where $S_H(\omega)$ is defined in (21).

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