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Average Finite-Blocklength Packet Error Rate over Nakagami- $m$ Fading via a Logistic--Lerch Approximation

This paper proposes a closed-form approximation for the average finite-blocklength packet error rate over Nakagami- $m$ fading channels by modeling the conditional error curve with a logistic function, which reduces the integration to a single Lerch-transcendent term that accurately interpolates between finite-blocklength and outage limits while enabling quality-of-service-aware rate selection.

Original authors: Aamir Mahmood

Published 2026-08-25

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Aamir Mahmood

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the invisible world of wireless communication, data travels not as a steady stream, but as a series of tiny, fragile packets. For decades, engineers designed systems assuming these packets were long enough that random noise would average out, allowing for near-perfect reliability. But the modern world demands something different: machines that talk to each other in split seconds, sending small bursts of information with extreme urgency. This is the realm of short-packet communication, where the old rules no longer apply. When a message is very short, the random fluctuations of the air itself—the fading that occurs as signals bounce off buildings and trees—can cause a packet to fail completely. To build reliable networks for these critical tasks, engineers must predict exactly how often these short messages will fail, a calculation that has proven notoriously difficult to solve with a simple formula.

The challenge lies in the shape of the failure curve. In a perfect, static environment, the chance of a message failing drops off like a steep cliff as the signal gets stronger. This "waterfall" is sharp and precise. However, in the real world, the signal strength is constantly shifting due to fading. To know the true reliability of a link, one must average that sharp cliff over all the possible variations of the fading. For general types of fading, this averaging process is mathematically intractable; the equations become so complex that they cannot be solved directly, forcing engineers to rely on rough guesses or slow, computer-heavy simulations.

A researcher has now found a way to cut through this complexity. They discovered that the sharp cliff of the failure curve can be closely mimicked by a smooth, S-shaped curve known as a logistic function. By swapping the difficult-to-calculate cliff for this smooth curve, they transformed the impossible averaging problem into a solvable one. The result is a single, compact mathematical expression that predicts the average failure rate for short packets over a wide range of fading conditions. This formula works for any level of signal distortion, from mild to severe, and remains accurate even when the packet length is short, a regime where previous methods often failed.

The researcher tested their new formula against the standard, highly accurate but computationally heavy methods used today. In simulations covering a broad range of signal strengths and packet sizes, their new approach matched the complex numerical results with an error of less than one percent. This level of precision is significant because older, simpler methods used by engineers often deviated by several percent, especially when the signal environment was harsh. The new formula is not just a theoretical curiosity; it is a practical tool that allows engineers to calculate reliability instantly without running time-consuming simulations.

Because the formula is so accurate and easy to use, it enables a new way to manage network traffic. Engineers can now use it to dynamically adjust the speed at which data is sent, balancing the need for speed against the risk of failure. If a connection is shaky, the system can automatically slow down to ensure the message gets through; if the connection is strong, it can speed up. This ability to adapt in real time is crucial for applications like autonomous vehicles or remote surgery, where a delay or a lost message could have serious consequences. The work demonstrates that by finding the right mathematical shape to represent a physical phenomenon, a problem that once required heavy computation can be solved with a simple, elegant expression.

Technical Summary: Average Finite-Blocklength Packet Error Rate over Nakagami-m Fading

Problem Statement
In short-packet communication systems, such as those required for Ultra-Reliable Low-Latency Communication (URLLC) and machine-type services, the long-codeword assumption no longer holds. Consequently, the Packet Error Rate (PER) must be characterized in the finite-blocklength (FBL) regime. While the conditional block-error probability over an Additive White Gaussian Noise (AWGN) channel is well-approximated by the normal approximation (a sharp sigmoidal "waterfall" function), evaluating the average PER over a fading channel requires integrating this non-linear function against the fading distribution.

For general Nakagami- $m$ fading, this integration is analytically intractable. While closed-form results exist for Rayleigh fading, accurate ones for general Nakagami- $m$ remain scarce; existing tractable approaches (such as piecewise-linear approximations or outage limits) are tight only at low fading diversity and degrade significantly as the fading parameter $m$ increases. Furthermore, existing methods often rely on numerical integration or provide bounds rather than a direct approximation of the fading-averaged PER itself, making them computationally expensive or insufficiently accurate for optimization tasks in high-reliability regimes.

Methodology
The paper proposes a new closed-form approximation strategy that decouples the error characterization from the fading integration:

Sigmoid Approximation: Instead of integrating the normal approximation directly, the author approximates the conditional FBL error curve (the AWGN waterfall) with a tractable logistic function. This choice is motivated by the symmetry of the FBL waterfall, which contrasts with the skewed Gumbel distribution used for uncoded schemes.
Parameter Matching: The location ( $a$ ) and scale ( $b$ ) parameters of the logistic function are derived by matching the value and slope of the logistic curve to the normal approximation at the rate threshold $\gamma^* = 2^R - 1$ .
Analytical Integration: The author substitutes this logistic surrogate into the expectation integral over the Nakagami- $m$ probability density function (a Gamma distribution). Through variable substitution, this integral is shown to map directly to the integral representation of the Lerch transcendent function.

Key Contributions

Single-Term Closed-Form Solution: The paper derives a compact expression for the average FBL PER over Nakagami- $m$ fading (Equation 5) expressed as a single Lerch transcendent term. This result is valid for all real fading parameters $m > 0$ , including non-integer values and the sub-Rayleigh regime ( $1/2 \le m < 1$ ), which previous integer-only derivations excluded.
Regime Interpolation: The derived formula naturally interpolates between the finite-blocklength waterfall behavior (at moderate blocklengths) and the classical outage probability limit (as blocklength $N \to \infty$ ).
QoS-Aware Rate Selection: By composing the closed-form PER into an effective capacity objective, the author formulates a Quality-of-Service (QoS) aware rate-selection rule. This allows for the direct optimization of transmission rates to balance payload and reliability without resorting to nested numerical integrations.
Regime Selection Strategy: The paper identifies that while the logistic approximation is superior for typical FBL regimes ( $N \gtrsim 128$ ), the Gumbel approximation remains preferable for ultra-short, low-rate packets, providing a complementary framework for different operating points.

Results and Accuracy
Numerical validation against the standard normal-approximation integral (computed via numerical quadrature) demonstrates the following:

High Accuracy: The proposed closed form matches the numerical integration to within approximately 1% uniformly across the nominal FBL operating region ( $N \gtrsim 100$ , $R \gtrsim 0.75$ ) for various $m$ values.
Superiority over Baselines: At high fading diversity (e.g., $m=3$ or $m=4$ ), the proposed logistic-Lerch approximation significantly outperforms existing baselines. Piecewise-linear approximations and outage limits exhibit errors ranging from 6% to over 16% in high-diversity scenarios, whereas the proposed method remains near 1%.
Computational Efficiency: The evaluation of the Lerch transcendent is computationally lightweight compared to adaptive numerical integration of the original error curve, making it suitable for real-time rate adaptation and cross-layer optimization.

Significance
The paper proposes a direct, accurate, and closed-form approximation of the fading-averaged FBL PER for general Nakagami- $m$ channels, addressing the scarcity of such results compared to Rayleigh fading. By replacing the intractable integral with a single special function, the method enables precise reliability assessment and efficient resource allocation in short-packet communications. The approach bridges the gap between theoretical bounds and practical system design, offering a tool that is both analytically rigorous and computationally efficient for QoS-constrained URLLC applications. The author notes that while the method relies on the specific structure of the Gamma distribution (Nakagami- $m$ ), extending this to composite fading models would require different special-function reductions.

Technical Summary: Average Finite-Blocklength Packet Error Rate over Nakagami-m Fading

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