Ferkans — Interactive Telecom Tutor

From Fading Enemy to Fading Friend

In single-user communications (Chapters 10--11), fading is an impairment that reduces capacity and demands diversity to combat. A striking paradigm shift occurs in multiuser systems: when the base station can choose which user to serve in each time slot, fading becomes a source of diversity gain. At any instant, it is likely that at least one user among $K$ experiences a channel peak. By scheduling that user, the system harvests a gain that grows with $K$ — even though no single user's channel improves. This multiuser diversity principle underlies every modern cellular scheduler and fundamentally changed how engineers think about fading channels.

Definition:
Multiuser Diversity and Opportunistic Scheduling

Consider a downlink system with $K$ single-antenna users. In each time slot $t$ , user $k$ has an instantaneous channel gain $|h_k[t]|^2$ , where $\{h_k[t]\}$ are i.i.d. across users and ergodic across time. The instantaneous rate available to user $k$ is:

$R_k[t] = \log_2\!\left(1 + \text{SNR}\,|h_k[t]|^2\right)$

An opportunistic (max-rate) scheduler serves the user with the highest instantaneous rate in each slot:

$k^{\star}[t] = \arg\max_{1 \leq k \leq K} R_k[t]$

Multiuser diversity refers to the phenomenon that the effective channel quality $\max_k |h_k[t]|^2$ improves with $K$ , yielding a sum-rate gain even when each user's marginal distribution is unchanged.

Multiuser diversity is a form of selection diversity across users rather than across antennas or time slots. It requires (i) independent fading across users, (ii) channel-state information at the scheduler, and (iii) sufficient traffic backlog so that every user always has data to send or receive.

Theorem: Capacity Scaling with Multiuser Diversity

Let $K$ users have i.i.d. Rayleigh fading channels with $|h_k|^2 \sim \text{Exp}(1)$ . The sum-rate capacity of the opportunistic scheduler satisfies:

$C_{\text{sum}}(K) = \mathbb{E}\!\left[\log_2\!\left(1 + \text{SNR} \cdot \max_{1 \leq k \leq K} |h_k|^2\right)\right] \sim \log_2(\text{SNR} \cdot \ln K)$

as $K \to \infty$ . More precisely, the expected maximum channel gain scales as:

$\mathbb{E}\!\left[\max_{1 \leq k \leq K} |h_k|^2\right] = \ln K + \gamma_{\text{EM}} + o(1)$

where $\gamma_{\text{EM}} \approx 0.5772$ is the Euler--Mascheroni constant. The capacity thus grows double-logarithmically in the number of users: $C_{\text{sum}} \approx \log_2(\text{SNR}) + \log_2(\ln K)$ .

The maximum of $K$ i.i.d. exponential random variables grows as $\ln K$ , which is an extreme-value result (Gumbel distribution). Taking $\log_2$ of $1 + \text{SNR} \cdot \ln K$ yields a $\log_2(\ln K)$ growth term — hence the "double logarithm." The gain is real but slow: doubling $K$ from 50 to 100 adds only about $\log_2(\ln 100 / \ln 50) \approx 0.18$ bits/s/Hz.

Proof

CDF of the maximum

Let $X_k = |h_k|^2 \sim \text{Exp}(1)$ with CDF $F(x) = 1 - e^{-x}$ . The CDF of $M_K = \max_{1 \leq k \leq K} X_k$ is:

$F_{M_K}(x) = (1 - e^{-x})^K$

The PDF is:

$f_{M_K}(x) = K e^{-x}(1 - e^{-x})^{K-1}$

Expected maximum via Gumbel approximation

For large $K$ , the maximum of $K$ i.i.d. exponential random variables converges (after centering and scaling) to a Gumbel distribution. The centering sequence is $a_K = \ln K$ with scale $b_K = 1$ :

$\Pr[M_K - \ln K \leq x] \to \exp(-e^{-x})$

Therefore $\mathbb{E}[M_K] = \ln K + \gamma_{\text{EM}} + o(1)$ , where $\gamma_{\text{EM}} = \lim_{n\to\infty}(\sum_{k=1}^n 1/k - \ln n) \approx 0.5772$ is the Euler--Mascheroni constant.

Sum-rate scaling

The sum rate equals:

$C_{\text{sum}}(K) = \mathbb{E}[\log_2(1 + \text{SNR} \cdot M_K)]$

By Jensen's inequality (applied in the appropriate direction) and the concentration of $M_K$ around $\ln K$ :

$C_{\text{sum}}(K) \approx \log_2(1 + \text{SNR}\,\ln K) \sim \log_2(\text{SNR}) + \log_2(\ln K)$

as $K \to \infty$ . The scaling is $\Theta(\log_2 \ln K)$ , confirming the double-logarithmic growth. $\blacksquare$

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Multiuser Diversity Gain

Observe how the sum-rate capacity grows with the number of users $K$ . The plot shows the expected rate of the opportunistic scheduler $\mathbb{E}[\log_2(1 + \text{SNR} \cdot \max_k |h_k|^2)]$ as a function of $K$ , along with the single-user ergodic capacity baseline. Increase the SNR to see how the absolute gain changes while the relative gain (in dB) remains similar.

Parameters

K_{\max}

(maximum number of users)50

SNR (dB)10

Example: Quantifying Multiuser Diversity Gain

A base station serves $K = 20$ users with i.i.d. Rayleigh fading and average SNR = 10 dB per user.

(a) Compute the single-user ergodic capacity. (b) Compute the expected maximum channel gain with $K = 20$ . (c) Estimate the sum-rate capacity of the opportunistic scheduler. (d) What is the multiuser diversity gain in bits/s/Hz?

Solution

Single-user ergodic capacity

(a) For Rayleigh fading with $|h|^2 \sim \text{Exp}(1)$ and SNR = 10 (linear):

$C_1 = \mathbb{E}[\log_2(1 + 10|h|^2)] = \int_0^{\infty} \log_2(1 + 10x)\,e^{-x}\,dx$

Evaluating numerically (or via the closed-form involving $e^{1/10}\text{E}_1(1/10)$ ):

$C_1 \approx 2.51 \;\text{bits/s/Hz}$

Expected maximum channel gain

(b) $\mathbb{E}[\max_{k=1}^{20} |h_k|^2] = \sum_{k=1}^{20} \frac{1}{k} = H_{20} \approx 3.60$

where $H_K = \sum_{k=1}^{K} 1/k$ is the $K$ -th harmonic number. (The exact formula for exponential order statistics gives $\mathbb{E}[X_{(K)}] = H_K$ .)

Sum-rate estimate

(c) $C_{\text{sum}}(20) \approx \log_2(1 + 10 \times 3.60) = \log_2(37) \approx 5.21$ bits/s/Hz.

A more careful computation accounting for the full distribution of $M_{20}$ yields approximately 4.86 bits/s/Hz.

Multiuser diversity gain

(d) Gain $= C_{\text{sum}}(20) - C_1 \approx 4.86 - 2.51 = 2.35$ bits/s/Hz.

This is nearly double the single-user capacity, obtained purely by selecting the best user in each slot. $\blacksquare$

Quick Check

How does the sum-rate capacity of an opportunistic scheduler scale with the number of users $K$ under i.i.d. Rayleigh fading?

Linearly: the capacity scales as $K$

Logarithmically: the capacity scales as $\log K$

Double-logarithmically: the capacity scales as $\log(\log K)$

It does not grow at all: adding users does not help

Correction:

Double-logarithmically: the capacity scales as

\log(\log K)

The maximum of $K$ i.i.d. exponential random variables grows as $\ln K$ (Gumbel extreme-value scaling). The rate is $\log_2(1 + \text{SNR}\,\ln K) \sim \log_2(\ln K)$ , which is $\log(\log K)$ .

Historical Note: The Discovery of Multiuser Diversity

1995--2002

The idea that fading can help rather than hurt in multiuser systems was first formalised by Knopp and Humblet (1995), who showed that the sum capacity of a fading multiple-access channel is achieved by transmitting only from the user with the best channel in each slot. This counterintuitive result — that optimal scheduling is "unfair" by nature — stimulated intense research. Viswanath, Tse, and Laroia (2002) extended the concept to the downlink and coined the term "multiuser diversity," establishing the $\log(\log K)$ scaling law. Their work also introduced proportional fair scheduling as a practical compromise between throughput and fairness, directly influencing the design of the Qualcomm HDR (1xEV-DO) system and subsequently 3GPP LTE.

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Key Takeaway

Fading is a friend in multiuser systems. While single-user communications must combat fading through diversity and coding, a multiuser scheduler exploits fading by serving users at their channel peaks. The sum-rate gain scales as $\log_2(\ln K)$ with the number of users — slow but unbounded. This paradigm shift from "fading as enemy" to "fading as resource" underlies every modern cellular scheduler from 3G HSDPA through 5G NR.

Common Mistake: Multiuser Diversity Requires Timely CSI

Mistake:

Assuming the $\log(\log K)$ scaling is free — that adding more users automatically improves throughput without additional overhead.

Correction:

To exploit multiuser diversity, the scheduler must know each user's instantaneous channel quality. This requires CQI feedback from all $K$ users every scheduling interval. The feedback overhead scales as $O(K)$ per slot, consuming uplink resources that could otherwise carry data.

At very large $K$ , the feedback overhead can negate the multiuser diversity gain. In LTE/NR, CQI reporting is typically periodic (every 5--40 ms) or aperiodic (triggered on demand), and only a subset of users may report in each slot. The scheduler must then rely on extrapolated or outdated CSI, degrading the scheduling gain.

Multiuser Diversity

A form of diversity that arises in multiuser systems when independent fading creates channel peaks for different users at different times. An opportunistic scheduler exploits multiuser diversity by serving the user with the strongest channel in each time slot, yielding a sum-rate gain that scales as $\log(\log K)$ .

Opportunistic Scheduling

A scheduling policy that selects users for transmission based on their instantaneous channel conditions, serving users when their channels are near their peaks. The max-rate scheduler is the simplest form; proportional fair scheduling adds fairness by normalising each user's rate by its own average throughput.

Multiuser Diversity

From Fading Enemy to Fading Friend

Definition: Multiuser Diversity and Opportunistic Scheduling

Theorem: Capacity Scaling with Multiuser Diversity

CDF of the maximum

Expected maximum via Gumbel approximation

Sum-rate scaling

Multiuser Diversity Gain

Parameters

Example: Quantifying Multiuser Diversity Gain

Single-user ergodic capacity

Expected maximum channel gain

Sum-rate estimate

Multiuser diversity gain

Quick Check

Historical Note: The Discovery of Multiuser Diversity

Key Takeaway

Common Mistake: Multiuser Diversity Requires Timely CSI

Multiuser Diversity

Opportunistic Scheduling

Definition:
Multiuser Diversity and Opportunistic Scheduling