Ferkans — Interactive Telecom Tutor

From Single-User to Multi-User Finite Blocklength

The normal approximation and PPV bounds were originally developed for point-to-point channels. But the most exciting applications of finite-blocklength theory are in multi-user settings: URLLC coexists with eMBB on the same resources, multiple URLLC users share the uplink in grant-free access, and massive IoT devices transmit short packets simultaneously.

Extending finite-blocklength analysis to multi-user channels introduces new challenges: the second-order characterization involves a dispersion matrix (not just a scalar), and the normal approximation becomes a multi-dimensional integral. We present the key results for the MAC and BC, which form the theoretical foundation for grant-free access and URLLC scheduling.

Definition:
Second-Order Rate Region of the MAC

For a two-user MAC $p_{Y|X_1,X_2}$ with capacity region $\mathcal{C}$ , the second-order rate region at blocklength $n$ and error probability $\epsilon$ is:

$\mathcal{R}(n, \epsilon) \approx \mathcal{C} - \frac{1}{\sqrt{n}}\, \mathcal{S}(\epsilon)$

where $\mathcal{S}(\epsilon)$ is determined by the dispersion matrix $\mathbf{V}$ :

$\mathbf{V} = \text{Cov}\!\begin{bmatrix}\iota(X_1; Y | X_2) \\ \iota(X_2; Y | X_1) \\ \iota(X_1, X_2; Y)\end{bmatrix}$

and the set $\mathcal{S}(\epsilon)$ is defined through the multivariate normal distribution:

$\mathcal{S}(\epsilon) = \{\mathbf{s} \in \mathbb{R}^3 : \Pr[\mathbf{V}^{1/2}\mathbf{Z} \le \mathbf{s}] \ge 1 - \epsilon,\; \mathbf{Z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_3)\}.$

The MAC dispersion is a $3 \times 3$ matrix (for two users), not a scalar. This reflects the fact that the three MAC constraints (individual rates and sum rate) can have different variances and correlations. The second-order rate region is the asymptotic capacity region "shrunk" by an amount that depends on the dispersion matrix and the target error probability.

Theorem: Finite-Blocklength Gaussian MAC

For the two-user Gaussian MAC $Y = X_1 + X_2 + Z$ with $Z \sim \mathcal{N}(0, \sigma^2)$ and power constraints $P_1, P_2$ , the maximum sum rate at blocklength $n$ and per-user error probability $\epsilon$ satisfies:

$R_1 + R_2 \le \frac{1}{2}\log\!\left(1 + \frac{P_1 + P_2}{\sigma^2}\right) - \sqrt{\frac{V_{\text{sum}}}{n}}\, Q^{-1}(\epsilon) + O\!\left(\frac{\log n}{n}\right)$

where $V_{\text{sum}}$ is the sum-rate dispersion:

$V_{\text{sum}} = \text{Var}[\iota(X_1, X_2; Y)] = \frac{(P_1+P_2)(P_1+P_2+2\sigma^2)}{2(P_1+P_2+\sigma^2)^2}$

(in nats $^2$ ), which equals the point-to-point AWGN dispersion at the sum SNR $(P_1+P_2)/\sigma^2$ .

The sum-rate dispersion of the MAC is the same as the point-to-point dispersion at the sum power. This makes sense: from the sum-rate perspective, the MAC is indistinguishable from a single user with the combined power. The finite-blocklength penalty applies to each face of the MAC capacity region independently, each with its own dispersion.

Proof

Sum-rate information density

The sum-rate information density is $\iota(X_1, X_2; Y) = \log\frac{p_{Y|X_1,X_2}(Y|X_1,X_2)}{p_Y(Y)}$ . For the Gaussian MAC with Gaussian inputs, this equals the point-to-point information density at the sum power, since the decoder treats both users jointly.

Variance computation

Since $\iota(X_1, X_2; Y)$ is identical in distribution to the point-to-point AWGN information density at SNR $= (P_1+P_2)/\sigma^2$ , its variance is the AWGN dispersion at that SNR.

Normal approximation

By the CLT for sums of i.i.d. variables, the cumulative sum-rate information density concentrates around $n \times \frac{1}{2}\log(1 + (P_1+P_2)/\sigma^2)$ with standard deviation $\sqrt{nV_{\text{sum}}}$ . The normal approximation follows as in the point-to-point case.

Definition:
Grant-Free Random Access and Finite Blocklength

In grant-free random access, a potentially large number $K_{\text{total}}$ of devices share a common channel. In each time slot, a random subset of $K_a$ devices become active and transmit short packets of $B$ bits at blocklength $n$ , without prior scheduling.

The system operates in the many-access regime: $K_a$ grows with $n$ (unlike the fixed- $K$ MAC). The key metrics are:

Per-user probability of error (PUPE): $\epsilon = \frac{1}{K_a}\sum_{k=1}^{K_a} P_e^{(k)}$
Energy efficiency: $E_b/N_0$ required per information bit
Spectral efficiency: Total throughput $K_a \cdot B / n$ bits per channel use

The information-theoretic analysis shows that the per-user energy-per-bit requirement $E_b/N_0$ increases logarithmically with the number of active users, reflecting a many-access penalty absent in the asymptotic MAC.

Grant-free random access

An uplink access scheme where devices transmit without prior scheduling grants, using contention-based protocols. Critical for massive MTC where the scheduling overhead would exceed the data payload.

Ultra-reliable low-latency communication (URLLC)

A 5G service category targeting packet error rates below $10^{-5}$ at user-plane latencies of 1 ms, requiring finite-blocklength code design.

Finite-Blocklength MAC Rate Region

Visualize how the MAC rate region shrinks at finite blocklength. Compare the asymptotic capacity region with the second-order rate region for different blocklengths and reliability targets.

Parameters

User 1 SNR (dB)10

User 2 SNR (dB)10

Blocklength

n

500

Error probability

\epsilon

0.001

Example: Energy Efficiency of Short-Packet mMTC

Consider a massive IoT system where $K_a = 100$ devices each transmit $B = 100$ information bits at blocklength $n = 200$ over an AWGN channel. The target per-user error probability is $\epsilon = 10^{-3}$ .

(a) What is the required energy-per-bit $E_b/N_0$ per user?

(b) Compare with the Shannon limit $E_b/N_0|_{\min} = \ln 2 = -1.59$ dB.

Solution

Per-user coding rate

Each user's coding rate is $R = B/n = 100/200 = 0.5$ bits/channel use.

Required SNR from normal approximation

From $R = C(\text{SNR}) - \sqrt{V(\text{SNR})/n}\,Q^{-1}(\epsilon)$ : $0.5 = \frac{1}{2}\log(1+\text{SNR}) - \sqrt{V(\text{SNR})/200} \times 3.09$ .

This is an implicit equation in $\text{SNR}$ . Solving numerically (iterate on $\text{SNR}$ ): $\text{SNR} \approx 1.8$ (2.55 dB).

Energy-per-bit

$E_b/N_0 = \text{SNR}/R = 1.8/0.5 = 3.6$ (5.56 dB).

The Shannon limit for $R = 0.5$ is $E_b/N_0 = (2^{2R}-1)/(2R) = 1/1 = 1$ (0 dB). The finite-blocklength penalty is $5.56 - 0 = 5.56$ dB.

Note: this analysis ignores the multi-access interference from the other 99 users, which would further increase the required $E_b/N_0$ .

Theorem: Finite-Blocklength Broadcast Channel

For the degraded Gaussian BC with two users, the second-order sum-rate expansion at blocklength $n$ and per-user error probability $\epsilon$ is:

$R_1 + R_2 \le \frac{1}{2}\log\!\left(1 + \frac{P}{\sigma^2_{2}}\right) - \sqrt{\frac{V_{\text{BC}}}{n}}\, Q^{-1}(\epsilon) + O\!\left(\frac{\log n}{n}\right)$

where $V_{\text{BC}}$ is the BC dispersion and $\sigma^2_{2}$ is the noise variance of the weaker user.

The key finding is that superposition coding incurs no second-order penalty: the BC dispersion equals the point-to-point dispersion at the stronger user's SNR, just as the BC sum-capacity equals the point-to-point capacity.

This is a reassuring result for system design: superposition coding (the capacity-achieving strategy for the degraded BC) remains near-optimal even at finite blocklength. The dispersion penalty is the same as if we were communicating only to the stronger user. The weaker user experiences a different (potentially larger) dispersion, but the sum rate is governed by the stronger user's dispersion.

Proof

Achievability via superposition

The achievability follows from the superposition coding scheme analyzed at finite blocklength. User 1 (stronger) decodes its message after SIC of user 2's message. The error probability is bounded by the union of two events: failure to decode user 2, and failure to decode user 1 after SIC.

Converse via meta-converse

The converse applies the meta-converse to the sum rate, using the degradedness of the channel to bound the total information flow. The resulting dispersion matches the achievability, establishing the second-order optimality of superposition coding.

⚠️Engineering Note

Finite-Blocklength Limits for Massive MTC

In massive MTC (mMTC), thousands of IoT devices transmit short packets ( $B \sim 20$ - $100$ bits) infrequently. The finite-blocklength analysis reveals fundamental limits:

Minimum $E_b/N_0$ grows with the number of active users $K_a$ . For $K_a = 300$ and $B = 100$ bits at $\epsilon = 10^{-3}$ , the required $E_b/N_0$ is 8-10 dB above the Shannon limit.
Pilot overhead dominates at short blocklengths. With $n = 200$ and $K_a = 100$ users, orthogonal pilots require $\tau_p \ge 100$ symbols, leaving only 100 for data. Non-orthogonal pilot schemes (compressed sensing) are essential.
Activity detection must be performed jointly with decoding, adding another layer of complexity absent in the asymptotic theory.

Practical Constraints

•
3GPP mMTC target: 1 million devices per km^2
•
Typical IoT payload: 20-100 bytes
•
Battery lifetime requirement: 10 years on a coin cell

Common Mistake: Ignoring the Multi-Access Penalty at Finite Blocklength

Mistake:

Designing a grant-free system assuming each user achieves the point-to-point finite-blocklength rate, ignoring the interference from other active users.

Correction:

In the many-access regime, the per-user energy-per-bit requirement grows as $\Theta(\log K_a)$ : each additional user imposes a logarithmic penalty on all others. This is much worse than the asymptotic MAC, where the sum capacity is the same as the point-to-point capacity at the sum power. The finite-blocklength analysis shows that short-packet multi-access communication is fundamentally harder than long-packet communication, even beyond the $\sqrt{V/n}$ penalty.

Key Takeaway

Multi-user finite blocklength: The MAC second-order rate region is the asymptotic capacity region shrunk by $O(1/\sqrt{n})$ , with the shrinkage governed by a dispersion matrix. Superposition coding remains second-order optimal for the degraded BC. For massive MTC, the many-access regime introduces a $\Theta(\log K_a)$ energy penalty per user, fundamentally limiting the scalability of short-packet random access.

Quick Check

For the Gaussian MAC, the sum-rate dispersion equals the point-to-point dispersion at the sum SNR. What does this imply about the finite-blocklength sum-rate penalty?

The MAC sum rate has the same finite-blocklength penalty as a single user with the combined power

Each user has half the finite-blocklength penalty of a single user

The MAC has no finite-blocklength penalty for the sum rate