Ferkans — Interactive Telecom Tutor

Fading: The Wireless Reality

All the MAC models we have studied so far assume a fixed channel. In wireless systems, the channel varies over time due to user mobility, scattering, and multipath propagation. The fading MAC captures this reality by modeling the channel gains as random processes.

The key question becomes: how does channel state information (CSI) affect the capacity region? We will see that:

With CSIR only (channel known at receiver), the capacity region averages over the fading states.
With CSIT (channel known at transmitter too), the users can adapt their powers and rates to the fading state, leading to opportunistic scheduling: at any given time, serve the user with the best channel.

Definition:
The Ergodic Fading MAC

The two-user ergodic fading MAC is

$Y_i = h_{1,i} X_{1,i} + h_{2,i} X_{2,i} + Z_{i},$

where $h_{k,i} \in \mathbb{C}$ is the fading coefficient of user $k$ at time $i$ , and $Z_{i} \sim \mathcal{CN}(0, 1)$ . The fading process $\{(h_{1,i}, h_{2,i})\}_i$ is stationary and ergodic with known joint distribution.

Average power constraints:

$\frac{1}{n}\sum_{i=1}^n \mathbb{E}[|X_{k,i}|^2] \leq P_k, \quad k = 1, 2.$

We consider two CSI models:

CSIR: Both $h_{1,i}$ and $h_{2,i}$ are known to the receiver (but not to the transmitters) at each time $i$ .
CSIT + CSIR: The fading state $(h_{1,i}, h_{2,i})$ is known to all parties (both transmitters and the receiver) at time $i$ .

The ergodicity assumption means that over a long block, the fading process visits all states according to its stationary distribution. This allows us to compute capacity via expectations over the fading.

Theorem: Capacity Region of the Fading MAC with CSIR

The capacity region of the two-user ergodic fading MAC with CSIR only and average power constraints $P_1, P_2$ is the set of all rate pairs satisfying:

$R_{1} \leq \mathbb{E}\!\left[\log\!\left(1 + |h_1|^2 P_1\right)\right],$

$R_{2} \leq \mathbb{E}\!\left[\log\!\left(1 + |h_2|^2 P_2\right)\right],$

$R_{1} + R_{2} \leq \mathbb{E}\!\left[\log\!\left(1 + |h_1|^2 P_1 + |h_2|^2 P_2\right)\right].$

Without CSIT, the transmitters cannot adapt to the fading. Each user transmits at constant power $P_k$ regardless of the channel state. The capacity region is the ergodic average of the instantaneous Gaussian MAC regions. The pentagon shape is preserved, but the mutual information quantities are replaced by expectations over the fading.

Proof

Reduction to fixed-channel MAC

With CSIR, the receiver knows the fading state $\mathbf{h}_i = (h_{1,i}, h_{2,i})$ at each time $i$ . Without CSIT, the transmitters use constant power. By ergodicity, the channel statistics are captured by the stationary distribution of $\mathbf{h}$ .

At each time $i$ , the instantaneous channel is a Gaussian MAC with channel gains $h_{1,i}, h_{2,i}$ . The mutual information quantities for the ergodic channel are the expectations of the instantaneous quantities:

$I(X_1; Y | X_2, \mathbf{h}) = \mathbb{E}_{\mathbf{h}}\!\left[\log(1 + |h_1|^2 P_1)\right],$

where the expectation is over the stationary distribution of $\mathbf{h}$ .

Converse and achievability

The converse follows from the standard MAC converse, conditioning on the fading state (which is known at the receiver). The achievability uses random Gaussian codebooks with constant power and joint typicality decoding, where the typicality criterion accounts for the fading statistics. $\blacksquare$

Theorem: Capacity Region of the Fading MAC with CSIT

The capacity region of the two-user ergodic fading MAC with CSIT and CSIR, and average power constraints $P_1, P_2$ , is the set of all rate pairs in the closure of the convex hull of:

$R_{1} \leq \mathbb{E}\!\left[\log\!\left(1 + |h_1|^2 P_1(\mathbf{h})\right)\right],$

$R_{2} \leq \mathbb{E}\!\left[\log\!\left(1 + |h_2|^2 P_2(\mathbf{h})\right)\right],$

$R_{1} + R_{2} \leq \mathbb{E}\!\left[\log\!\left(1 + |h_1|^2 P_1(\mathbf{h}) + |h_2|^2 P_2(\mathbf{h})\right)\right],$

over all power allocation policies $P_k(\mathbf{h}) \geq 0$ satisfying $\mathbb{E}[P_k(\mathbf{h})] \leq P_k$ for $k = 1, 2$ , where $\mathbf{h} = (h_1, h_2)$ is the fading state.

With CSIT, the transmitters adapt their power to the instantaneous fading state. The optimization is over the power policies $P_k(\mathbf{h})$ , which depend on the fading realization. The additional flexibility of power adaptation strictly enlarges the capacity region compared to CSIR-only (where $P_k(\mathbf{h}) = P_k$ is forced).

Proof

Structure of optimal power allocation

The sum-rate maximizing power allocation is found by solving:

$\max_{P_1(\mathbf{h}), P_2(\mathbf{h})} \mathbb{E}\!\left[\log(1 + |h_1|^2 P_1(\mathbf{h}) + |h_2|^2 P_2(\mathbf{h}))\right]$

subject to $\mathbb{E}[P_k(\mathbf{h})] \leq P_k$ .

By the KKT conditions, the optimal allocation has a water-filling structure over the fading states. The effective channel gain at state $\mathbf{h}$ is $|h_1|^2 + |h_2|^2$ (when both users cooperate in terms of the sum-rate objective), and the water-filling solution is:

$P_1^*(\mathbf{h}) + P_2^*(\mathbf{h}) = \left[\nu - \frac{1}{\max(|h_1|^2, |h_2|^2)}\right]_+,$

with the optimal strategy allocating all power to the user with the stronger channel at each fading state. This is opportunistic scheduling.

Convexity and time-sharing

Different power policies achieve different pentagons. The full capacity region is the convex hull over all feasible power policies. Time-sharing between different policies is equivalent to using a different policy in each fading state, which is already captured by the optimization over $P_k(\mathbf{h})$ . $\blacksquare$

Definition:
Opportunistic Scheduling

Opportunistic scheduling is a resource allocation strategy for the fading MAC where, at each time slot, only the user with the best instantaneous channel is allowed to transmit. Formally:

$\text{At time } i: \quad k^*(i) = \arg\max_k |h_{k,i}|^2.$

User $k^*(i)$ transmits with power $P_{k^*}(\mathbf{h})$ determined by water-filling over the fading states; all other users remain silent.

At high SNR, this strategy is approximately sum-rate optimal for the fading MAC with CSIT.

Opportunistic scheduling exploits multiuser diversity: the more users in the system, the higher the probability that at least one user has a very good channel at any given time. This "selection diversity" effect makes the throughput grow as $\log\log K$ with the number of users $K$ (for i.i.d. Rayleigh fading), providing a logarithmic multiuser diversity gain.

Multiuser Diversity

The phenomenon where the overall throughput of a multiuser system increases with the number of users because the scheduler can always select a user with a favorable channel. The gain scales as $\log\log K$ for $K$ i.i.d. Rayleigh fading users.

Related: Multiple Access Channel (MAC)

Example: Sum Rate of the Rayleigh Fading MAC

Consider a two-user fading MAC with i.i.d. Rayleigh fading: $|h_k|^2 \sim \text{Exp}(1)$ for $k = 1, 2$ , independently across users and time. The average power constraint is $P$ per user and noise variance is 1.

Compare the ergodic sum rate with: (a) CSIR only (constant power), (b) CSIT with opportunistic scheduling.

Solution

CSIR only

With constant power $P$ per user:

$C_{\text{sum}}^{\text{CSIR}} = \mathbb{E}\!\left[\log(1 + |h_1|^2 P + |h_2|^2 P)\right] = \mathbb{E}[\log(1 + PG)],$

where $G = |h_1|^2 + |h_2|^2 \sim \text{Gamma}(2, 1)$ . The integral can be evaluated numerically. For $P = 10$ (10 dB): $C_{\text{sum}}^{\text{CSIR}} \approx 2.86$ bits.

CSIT with opportunistic scheduling

With opportunistic scheduling, the effective channel gain is $G^* = \max(|h_1|^2, |h_2|^2)$ . The sum rate with optimal power allocation (water-filling over $G^*$ ) is:

$C_{\text{sum}}^{\text{CSIT}} = \max_{P(g) \geq 0,\; \mathbb{E}[P(G^*)] \leq 2P} \mathbb{E}[\log(1 + G^* P(G^*))].$

The water-filling solution is $P^*(g) = [\nu - 1/g]_+$ with $\nu$ chosen to satisfy the average power constraint.

For $P = 10$ (10 dB), numerical evaluation gives $C_{\text{sum}}^{\text{CSIT}} \approx 3.21$ bits, a gain of about $12\%$ over CSIR.

Multiuser diversity gain

The CDF of $G^* = \max(|h_1|^2, |h_2|^2)$ is $(1 - e^{-g})^2$ . Its mean is $\mathbb{E}[G^*] = 1 + 1/2 = 3/2$ , while each individual $|h_k|^2$ has mean 1. The selection gain provides a $50\%$ increase in average effective SNR. With $K$ users, $\mathbb{E}[\max_k |h_k|^2] = \sum_{j=1}^K 1/j \approx \ln K$ , giving the $\log K$ scaling of multiuser diversity.

Fading MAC: CSIR vs CSIT with Opportunistic Scheduling

Compare the ergodic sum rate of the fading MAC under different CSI assumptions. The plot shows the sum rate vs average SNR for CSIR-only (constant power), CSIT with opportunistic scheduling, and TDMA with round-robin scheduling.

Parameters

K

(number of users)2

Max SNR (dB)20

Fading distribution

Historical Note: Knopp and Humblet: The Birth of Opportunistic Communications

1995

The idea that fading can be exploited rather than merely combated was formalized by Knopp and Humblet in their 1995 paper "Information Capacity and Power Control in Single-Cell Multiuser Communications." They showed that for the fading MAC with CSIT, the sum-rate-optimal strategy is to transmit only to the user with the best channel at each time — a result that was initially counterintuitive.

This paper, along with the work of Tse and Hanly on the downlink, launched the field of opportunistic communication. The practical realization came with the Qualcomm HDR (High Data Rate) system (later standardized as 1xEV-DO), which implemented a proportional-fair scheduler that exploits multiuser diversity while maintaining some fairness among users.

The key insight — that channel variability is a resource, not just an impairment — fundamentally changed how wireless systems are designed. Modern 4G and 5G schedulers (proportional fair, max-throughput, round-robin) are all descendants of this information-theoretic insight.

Why This Matters: Multiuser Diversity in 4G/5G Scheduling

The fading MAC theory directly motivates the scheduling algorithms used in cellular systems:

Max-throughput scheduler: Serves the user with the best instantaneous channel (pure opportunistic scheduling). Maximizes sum rate but can starve weak users.
Proportional-fair scheduler: Serves the user with the highest ratio of instantaneous rate to average rate. Balances throughput and fairness, exploiting multiuser diversity.
Round-robin scheduler: Serves users in turn regardless of channel state. Does not exploit multiuser diversity.

In 5G NR, the gNB scheduler uses CQI (Channel Quality Indicator) feedback to estimate each user's instantaneous channel quality and allocate resources accordingly. The scheduling gain from multiuser diversity is one of the reasons why 5G NR supports up to 64 simultaneously scheduled users per cell.

Common Mistake: Opportunistic Scheduling and Fairness

Mistake:

Assuming that opportunistic scheduling (always serving the best user) is optimal in all senses. While it maximizes the sum rate, it can completely starve weak users in asymmetric scenarios.

Correction:

Sum-rate optimality and fairness are conflicting objectives. In practice, systems use weighted sum-rate maximization or proportional-fair scheduling to balance throughput and user fairness. The information-theoretic capacity region characterizes all achievable rate tuples — the choice of operating point within the region is a system design decision that depends on fairness criteria and quality-of-service requirements.

🎓CommIT Contribution(2022)

Coded Random Access for the Many-User MAC

K.-H. Ngo, A. Lancho, G. Durisi, G. Caire — IEEE Trans. Inform. Theory, vol. 68, no. 11

This work by the CommIT group addresses the modern challenge of massive random access in IoT networks, where a large number of devices ( $K$ potentially in the thousands) sporadically access the channel. The classical $K$ -user MAC framework becomes intractable because $K$ is unknown and the $2^K - 1$ constraints are infeasible to evaluate.

The paper proposes a coded random access scheme based on the unsourced MAC model (Polyanskiy, 2017), where the goal is to recover the set of transmitted messages without identifying which user sent which message. The scheme achieves near-optimal energy efficiency in the many-user regime, connecting the classical MAC capacity theory to practical grant-free access protocols for 5G NR and beyond.

random-accessunsourced-MACIoTmassive-connectivityView Paper →

⚠️Engineering Note

CSI Acquisition Overhead in Fading MACs

The fading MAC capacity results assume perfect CSI at the receiver (CSIR) and sometimes at the transmitters (CSIT). In practice, CSI must be estimated from pilot symbols, which consumes resources:

CSIR: Pilot symbols are transmitted by each user, consuming a fraction of the coherence block. With $K$ users and coherence block length $T_c$ , at least $K$ pilot symbols are needed per block, leaving $T_c - K$ symbols for data. When $K$ approaches $T_c$ , the "pilot contamination" effect severely limits performance.
CSIT: Requires feedback from the receiver to the transmitters. In TDD systems, channel reciprocity provides CSIT from uplink pilots. In FDD systems, explicit feedback is needed, consuming downlink resources.

The net effect is that the ideal capacity with perfect CSI is an upper bound on achievable rates. The gap can be significant when the number of users is large relative to the coherence block.

Practical Constraints

•
Pilot overhead scales linearly with the number of users
•
Coherence block length limits the number of orthogonal pilots
•
Channel estimation errors create residual interference
•
CSIT requires feedback bandwidth or channel reciprocity (TDD)

Key Takeaway

The fading MAC capacity region depends critically on CSI availability. With CSIR only, users transmit at constant power and the capacity region is the ergodic average of the fixed-channel region. With CSIT, opportunistic scheduling — transmitting only to the user with the best channel — is sum-rate optimal and exploits multiuser diversity. The throughput gain from multiuser diversity scales as $\log\log K$ with the number of users, motivating the channel-aware schedulers used in modern 4G/5G systems.

The Fading MAC