Ferkans — Interactive Telecom Tutor

From Single-User to Multi-User MIMO

In Chapters 15 and 16 we studied a single transmitter communicating with a single receiver over a MIMO channel. Real wireless systems, however, serve many users simultaneously. The uplink of a cellular system, where $K$ single-antenna users transmit to a base station with $N_r$ antennas, is the archetypal multiple-access channel (MAC). This section develops the MAC capacity region and shows that successive interference cancellation (SIC) at the receiver achieves every point on the capacity boundary — a result that underpins all modern uplink receiver design.

Definition:
K-User MIMO Multiple-Access Channel

The $K$ -user MIMO MAC models the uplink where $K$ users, each with a single antenna, transmit to a base station with $N_r$ receive antennas. The received signal is:

$\mathbf{y} = \sum_{k=1}^{K} \mathbf{h}_k x_k + \mathbf{n}$

where $\mathbf{h}_k \in \mathbb{C}^{N_r}$ is the channel from user $k$ , $x_k$ is the transmitted symbol with power constraint $\mathbb{E}[|x_k|^2] \leq P_k$ , and $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I}_{N_r})$ .

In compact form with $\mathbf{H} = [\mathbf{h}_1, \ldots, \mathbf{h}_K] \in \mathbb{C}^{N_r \times K}$ and $\mathbf{x} = [x_1, \ldots, x_K]^T$ :

$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$

When the users have multiple antennas (user $k$ has $N_{t,k}$ antennas), the scalar $x_k$ generalises to a vector $\mathbf{x}_k \in \mathbb{C}^{N_{t,k}}$ with covariance constraint $\text{tr}(\mathbf{Q}_k) \leq P_k$ . The capacity region has the same polymatroid structure but optimisation is over covariance matrices rather than scalar powers.

Definition:
MAC Capacity Region

The MAC capacity region is the closure of the set of achievable rate tuples $(R_1, \ldots, R_K)$ satisfying:

$\sum_{k \in \mathcal{S}} R_k \leq \log_2 \det\!\left( \mathbf{I}_{N_r} + \frac{1}{\sigma^2}\sum_{k \in \mathcal{S}} P_k \mathbf{h}_k \mathbf{h}_k^H \right) \quad \forall\; \mathcal{S} \subseteq \{1, \ldots, K\}$

For $K = 2$ users this defines a pentagon in the $(R_1, R_2)$ plane, bounded by:

$R_1 \leq \log_2\!\left(1 + \frac{P_1 \|\mathbf{h}_1\|^2}{\sigma^2}\right)$

$R_2 \leq \log_2\!\left(1 + \frac{P_2 \|\mathbf{h}_2\|^2}{\sigma^2}\right)$

$R_1 + R_2 \leq \log_2 \det\!\left(\mathbf{I} + \frac{1}{\sigma^2}(P_1\mathbf{h}_1\mathbf{h}_1^H + P_2\mathbf{h}_2\mathbf{h}_2^H)\right)$

The capacity region has a polymatroid structure: it is a contra-polymatroid determined by the submodular set function $f(\mathcal{S}) = I(\mathbf{x}_\mathcal{S}; \mathbf{y} \mid \mathbf{x}_{\mathcal{S}^c})$ . This structure is fundamental to understanding rate splitting and time-sharing strategies.

Theorem: MAC Capacity Achievement via Successive Interference Cancellation

Every point on the boundary of the $K$ -user MIMO MAC capacity region is achievable by successive interference cancellation (SIC) with Gaussian codebooks. Specifically, for any permutation $\pi$ of $\{1, \ldots, K\}$ , decoding user $\pi(k)$ while treating users $\{\pi(k+1), \ldots, \pi(K)\}$ as noise (and having already cancelled users $\{\pi(1), \ldots, \pi(k-1)\}$ ) achieves the corner point:

$R_{\pi(k)} = \log_2 \det\!\left(\mathbf{I} + \frac{P_{\pi(k)}\mathbf{h}_{\pi(k)}\mathbf{h}_{\pi(k)}^H} {\sigma^2\mathbf{I} + \sum_{j=k+1}^{K} P_{\pi(j)}\mathbf{h}_{\pi(j)}\mathbf{h}_{\pi(j)}^H}\right)$

The entire capacity region is the convex hull of all $K!$ corner points, achievable by time-sharing between SIC decoding orders.

SIC peels off users one at a time: the first decoded user sees all others as interference, but the last decoded user enjoys an interference-free channel. Each decoding order yields a different rate allocation (corner point), and time-sharing between orders traces the entire boundary. The key insight is that — unlike a heuristic — SIC with optimal Gaussian codes loses nothing: it achieves the information-theoretic limit.

Proof

Achievability of corner points

Fix a decoding order $\pi$ . User $\pi(k)$ is decoded treating users $\{\pi(k+1), \ldots, \pi(K)\}$ as Gaussian noise. After perfect cancellation of users $\{\pi(1), \ldots, \pi(k-1)\}$ , the effective channel for user $\pi(k)$ is:

$\mathbf{y}_k = \mathbf{h}_{\pi(k)} x_{\pi(k)} + \sum_{j=k+1}^{K}\mathbf{h}_{\pi(j)} x_{\pi(j)} + \mathbf{n}$

The interference-plus-noise covariance is:

$\mathbf{R}_{k} = \sigma^2\mathbf{I} + \sum_{j=k+1}^{K} P_{\pi(j)}\mathbf{h}_{\pi(j)}\mathbf{h}_{\pi(j)}^H$

The achievable rate for user $\pi(k)$ is:

$R_{\pi(k)} = \log_2\!\left(1 + P_{\pi(k)}\mathbf{h}_{\pi(k)}^H \mathbf{R}_{k}^{-1} \mathbf{h}_{\pi(k)}\right)$

Sum rate equals MAC sum capacity

By the matrix determinant lemma applied iteratively (the same telescoping argument as MMSE-SIC in Chapter 16):

$\sum_{k=1}^{K} R_{\pi(k)} = \log_2 \det\!\left( \mathbf{I} + \frac{1}{\sigma^2}\sum_{k=1}^{K} P_k \mathbf{h}_k\mathbf{h}_k^H\right)$

This equals the sum-rate bound with $\mathcal{S} = \{1, \ldots, K\}$ , confirming that SIC achieves the sum capacity for every decoding order.

Converse (information-theoretic outer bound)

For any subset $\mathcal{S}$ , Fano's inequality gives:

$\sum_{k \in \mathcal{S}} R_k \leq I(\mathbf{x}_\mathcal{S}; \mathbf{y} \mid \mathbf{x}_{\mathcal{S}^c}) + \epsilon_n$

Since Gaussian inputs maximise mutual information under a power constraint, $I(\mathbf{x}_\mathcal{S}; \mathbf{y} \mid \mathbf{x}_{\mathcal{S}^c}) \leq \log_2 \det(\mathbf{I} + \frac{1}{\sigma^2}\sum_{k \in \mathcal{S}} P_k \mathbf{h}_k\mathbf{h}_k^H)$ , establishing the outer bound. $\blacksquare$

MAC Capacity Region Animation

Watch the 2-user MAC capacity pentagon being constructed step by step: individual rate bounds, sum-rate constraint, and the SIC corner points on the dominant face. Each decoding order yields a different corner point; time-sharing traces the boundary.

The MAC capacity region for 2 users with

P_1 = P_2 = 10

,

\|\mathbf{h}_1\|^2 = 2

,

\|\mathbf{h}_2\|^2 = 1.5

. The dominant face connects the two SIC corner points.

MAC Capacity Region for the 2-User SIMO Channel

Visualise the pentagon-shaped MAC capacity region for two users transmitting to a multi-antenna base station. Adjust user powers and the number of BS antennas to see how the region expands. The corner points achieved by different SIC decoding orders are highlighted.

Parameters

P_1

(dB)10

P_2

(dB)10

N_r

(BS antennas)4

Example: Computing the MAC Capacity Region for 2-User SIMO

Consider a 2-user SIMO MAC where the base station has $N_r = 2$ antennas. The channels are $\mathbf{h}_1 = [1, j]^T$ and $\mathbf{h}_2 = [1, -j]^T$ , with $P_1 = P_2 = 10$ (linear) and $\sigma^2 = 1$ .

(a) Compute the individual rate bounds. (b) Compute the sum-rate bound. (c) Find the two corner points of the dominant face.

Solution

Individual rate bounds

$\|\mathbf{h}_1\|^2 = |1|^2 + |j|^2 = 2$ , similarly $\|\mathbf{h}_2\|^2 = 2$ .

$R_1 \leq \log_2(1 + 10 \times 2) = \log_2(21) \approx 4.39\;\text{bits/s/Hz}$

$R_2 \leq \log_2(1 + 10 \times 2) = \log_2(21) \approx 4.39\;\text{bits/s/Hz}$

Sum-rate bound

$\mathbf{R}_{\text{sum}} = P_1\mathbf{h}_1\mathbf{h}_1^H + P_2\mathbf{h}_2\mathbf{h}_2^H = 10\begin{bmatrix} 1 \\ j \end{bmatrix}[1, -j] + 10\begin{bmatrix} 1 \\ -j \end{bmatrix}[1, j]KATEXPLACEHOLDER0END= 10\begin{bmatrix} 1 & -j \\ j & 1 \end{bmatrix} + 10\begin{bmatrix} 1 & j \\ -j & 1 \end{bmatrix} = \begin{bmatrix} 20 & 0 \\ 0 & 20 \end{bmatrix}KATEXPLACEHOLDER1ENDR_1 + R_2 \leq \log_2\det\!\left(\begin{bmatrix} 21 & 0 \\ 0 & 21 \end{bmatrix}\right) = 2\log_2(21) \approx 8.78\;\text{bits/s/Hz}$ $Since the channels are orthogonal ($ \mathbf{h}_1^H\mathbf{h}_2 = 0$), the sum-rate bound equals the sum of individual bounds — no pentagon truncation occurs.

Corner points

SIC order 1 $\to$ 2 (decode user 1 first):

$R_1 = \log_2(1 + 10 \times \mathbf{h}_1^H(\mathbf{I} + 10\mathbf{h}_2\mathbf{h}_2^H)^{-1}\mathbf{h}_1)$

Since the channels are orthogonal: $\mathbf{h}_1^H(\mathbf{I} + 10\mathbf{h}_2\mathbf{h}_2^H)^{-1}\mathbf{h}_1 = \|\mathbf{h}_1\|^2 = 2$

So $R_1 = \log_2(21) \approx 4.39$ , $R_2 = \log_2(21) \approx 4.39$ .

Both corner points coincide at $(4.39, 4.39)$ because orthogonal channels create no mutual interference — the MAC region is a rectangle rather than a pentagon. $\blacksquare$

Quick Check

In a 2-user MAC with $N_r = 1$ (SISO), the sum capacity is $\log_2(1 + (P_1|h_1|^2 + P_2|h_2|^2)/\sigma^2)$ . Which statement is true about achieving this sum capacity?

Both users can simultaneously achieve their interference-free rates

SIC is required: one user is decoded first treating the other as noise, then the first user is cancelled

Time-division (TDMA) between the two users is sum-rate optimal

Joint ML decoding of both users is required; SIC is suboptimal

Correction:

SIC is required: one user is decoded first treating the other as noise, then the first user is cancelled

SIC achieves the sum capacity by first decoding the stronger user (treating the other as noise), cancelling it, then decoding the weaker user interference-free. Both SIC orderings achieve the same sum rate.

Common Mistake: Confusing the MAC with the Interference Channel

Mistake:

Assuming that the MAC model applies whenever multiple users transmit simultaneously. Students sometimes apply MAC capacity results to the interference channel, where each user has its own receiver.

Correction:

The MAC has a single receiver (the base station) that jointly processes all users' signals, enabling SIC. The interference channel (IC) has separate receivers, each decoding only its own message while treating other users' signals as interference. The IC capacity region is fundamentally different and generally unknown for more than 2 users. MAC capacity results do not apply to the IC.

Multiple-Access Channel (MAC)

A multi-user channel model where $K$ transmitters send independent messages to a single receiver. The MAC models the cellular uplink where multiple users transmit to a common base station.

Successive Interference Cancellation (SIC)

A receiver technique that decodes users sequentially: after decoding one user's message, its contribution is subtracted from the received signal before decoding the next user. SIC achieves the corner points of the MAC capacity region.

Successive Decoding

The information-theoretic term for SIC: decoding one user's codeword treating remaining users as noise, cancelling the decoded codeword, and repeating. Successive decoding with Gaussian codebooks achieves the MAC capacity region.

Multiple-Access Channel (MAC)

From Single-User to Multi-User MIMO

Definition: K-User MIMO Multiple-Access Channel

Definition: MAC Capacity Region

Theorem: MAC Capacity Achievement via Successive Interference Cancellation

Achievability of corner points

Sum rate equals MAC sum capacity

Converse (information-theoretic outer bound)

MAC Capacity Region Animation

MAC Capacity Region for the 2-User SIMO Channel

Parameters

Example: Computing the MAC Capacity Region for 2-User SIMO

Individual rate bounds

Sum-rate bound

Corner points

Quick Check

Common Mistake: Confusing the MAC with the Interference Channel

Multiple-Access Channel (MAC)

Successive Interference Cancellation (SIC)

Successive Decoding

Definition:
K-User MIMO Multiple-Access Channel

Definition:
MAC Capacity Region