Ferkans — Interactive Telecom Tutor

From Scalar to Vector: The MIMO MAC

Modern base stations have multiple antennas, and so do user devices. The natural extension of the Gaussian MAC to the multi-antenna setting is the MIMO MAC (or vector Gaussian MAC), where each user transmits a vector and the receiver collects a vector observation. This model captures the uplink of multi-user MIMO (MU-MIMO) systems, which are the backbone of 4G LTE and 5G NR.

The beautiful result is that the capacity region of the MIMO MAC is achieved by Gaussian inputs and successive decoding, just as in the scalar case. The key difference is that the capacity region is now parameterized by the users' input covariance matrices (not just their scalar powers), and optimization over these matrices involves matrix-valued water-filling.

Definition:
The Two-User MIMO MAC

The two-user MIMO MAC is defined by

$\mathbf{y} = \mathbf{H}_{1} \mathbf{x}_1 + \mathbf{H}_{2} \mathbf{x}_2 + \mathbf{z},$

where:

$\mathbf{x}_k \in \mathbb{C}^{n_k}$ is the transmitted signal of user $k$ ( $k = 1, 2$ ), with $n_k$ transmit antennas.
$\mathbf{H}_{k} \in \mathbb{C}^{m \times n_k}$ is the channel matrix for user $k$ , assumed fixed and known to all parties.
$\mathbf{y} \in \mathbb{C}^{m}$ is the received signal at the base station with $m$ receive antennas.
$\mathbf{z} \sim \mathcal{CN}(\mathbf{0}, \mathbf{I}_m)$ is i.i.d. circularly symmetric complex Gaussian noise.

Each user $k$ has a covariance constraint:

$\mathbb{E}[\mathbf{x}_k \mathbf{x}_k^H] \preceq \mathbf{K}_k, \quad \text{tr}(\mathbf{K}_k) \leq P_k,$

where $\mathbf{K}_k \succeq \mathbf{0}$ is the input covariance matrix.

The noise covariance is $\mathbf{I}$ without loss of generality — any colored noise $\mathbf{z} \sim \mathcal{CN}(\mathbf{0}, \mathbf{K}_z)$ can be whitened by pre-multiplying the received signal by $\mathbf{K}_z^{-1/2}$ .

MIMO MAC (Vector Gaussian MAC)

The multiple access channel where each user has multiple transmit antennas and the receiver has multiple receive antennas. The capacity region is parameterized by the input covariance matrices of each user.

Theorem: Capacity Region of the Two-User MIMO MAC

The capacity region of the two-user MIMO MAC is the closure of the convex hull of all rate pairs $(R_{1}, R_{2})$ satisfying

$R_{1} \leq \log\det\!\left(\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} (\mathbf{I} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H})^{-1}\right),$

$R_{2} \leq \log\det\!\left(\mathbf{I} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H} (\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H})^{-1}\right),$

$R_{1} + R_{2} \leq \log\det\!\left(\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H}\right),$

for some $\mathbf{K}_1 \succeq \mathbf{0}$ , $\mathbf{K}_2 \succeq \mathbf{0}$ with $\text{tr}(\mathbf{K}_k) \leq P_k$ .

This is the matrix generalization of the scalar Gaussian MAC region. The log-det expressions replace the scalar $\frac{1}{2}\log(1+\text{SNR})$ terms. The conditional mutual information $I(X_1; Y | X_2)$ becomes the capacity of a single-user MIMO channel $\mathbf{H}_{1}$ with colored noise $\mathbf{H}_{2} \mathbf{x}_2 + \mathbf{z}$ (interference from user 2 plus thermal noise).

Unlike the scalar case, the optimal covariance matrices $\mathbf{K}_1^*$ and $\mathbf{K}_2^*$ depend on which point on the boundary of the capacity region we target. The convex hull over covariance matrices enlarges the region, and time-sharing may be needed.

Proof

Achievability with Gaussian inputs

Fix covariance matrices $\mathbf{K}_1, \mathbf{K}_2$ . Generate Gaussian codebooks: $\mathbf{x}_k^{(i)} \sim \mathcal{CN}(\mathbf{0}, \mathbf{K}_k)$ for user $k$ . The mutual information quantities are:

$I(\mathbf{x}_1; \mathbf{y} | \mathbf{x}_2) = h(\mathbf{y} | \mathbf{x}_2) - h(\mathbf{y} | \mathbf{x}_1, \mathbf{x}_2)$ $= \log\det(\pi e(\mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} + \mathbf{I})) - \log\det(\pi e \mathbf{I})$ ... but we need to include user 2's contribution in the noise:

$h(\mathbf{y}|\mathbf{x}_2) = \log\det(\pi e(\mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} + \mathbf{I}))$ .

Wait — given $\mathbf{x}_2$ , the effective channel is $\mathbf{y} - \mathbf{H}_{2} \mathbf{x}_2 = \mathbf{H}_{1} \mathbf{x}_1 + \mathbf{z}$ , so $h(\mathbf{y}|\mathbf{x}_2) = \log\det(\pi e (\mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} + \mathbf{I}))$ .

Therefore:

$I(\mathbf{x}_1; \mathbf{y}|\mathbf{x}_2) = \log\det(\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H}).$

For the unconditional mutual information: $I(\mathbf{x}_1; \mathbf{y}) = \log\det(\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} (\mathbf{I} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H})^{-1})$ , where $\mathbf{I} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H}$ is the noise-plus-interference covariance.

Converse

The converse follows the same structure as the scalar case, using the fact that Gaussian inputs maximize the mutual information for a Gaussian channel with fixed input covariance constraint. This is the vector generalization of the "Gaussian maximizes entropy" argument: for any input with covariance $\mathbf{K}_k$ , the Gaussian input achieves the highest $I(\mathbf{x}_k; \mathbf{y} | \mathbf{x}_j)$ .

Convex hull

For different choices of $(\mathbf{K}_1, \mathbf{K}_2)$ , we get different pentagons. The full capacity region is the convex hull of the union over all feasible covariance pairs. Unlike the scalar case, the optimal $\mathbf{K}_k$ varies along the boundary, so the convex hull is essential. $\blacksquare$

,

Successive Decoding for the MIMO MAC

Just as in the scalar case, SIC achieves the corner points of the MIMO MAC capacity region. The SIC procedure for the MIMO MAC is:

Decode user 2 first: Treat $\mathbf{H}_{1} \mathbf{x}_1$ as colored Gaussian noise. The effective noise covariance is $\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H}$ . User 2's achievable rate is:

$R_{2} \leq \log\det\!\left(\mathbf{I} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H} (\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H})^{-1}\right).$
Subtract user 2 and decode user 1: After removing $\mathbf{H}_{2} \hat{\mathbf{x}}_2$ , user 1 sees a clean MIMO channel:

$R_{1} \leq \log\det\!\left(\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H}\right).$

The sum of these rates equals the sum capacity $\log\det(\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H})$ , which can be verified using the matrix identity $\det(\mathbf{A}\mathbf{B}) = \det(\mathbf{A})\det(\mathbf{B})$ .

Example: Two-User MIMO MAC with Single-Antenna Users

Consider a MIMO MAC with $m = 2$ receive antennas at the base station and two single-antenna users ( $n_1 = n_2 = 1$ ). The channel vectors are

$\mathbf{h}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \mathbf{h}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix},$

with power constraints $P_1 = P_2 = P$ and unit noise. Find the capacity region.

Solution

Compute the mutual information quantities

Since the users have single antennas, $\mathbf{K}_k = P_k$ (scalar). The channel matrices are column vectors $\mathbf{H}_{k} = \mathbf{h}_k$ .

$\mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} = P \mathbf{h}_1 \mathbf{h}_1^H = P \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},$ $\mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H} = P \mathbf{h}_2 \mathbf{h}_2^H = P \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$

Sum rate

$I(\mathbf{x}_1, \mathbf{x}_2; \mathbf{y}) = \log\det\!\left(\begin{pmatrix} 1+P & 0 \\ 0 & 1+P \end{pmatrix}\right) = 2\log(1+P).$ $

Individual rates

$I(\mathbf{x}_1; \mathbf{y} | \mathbf{x}_2) = \log\det(\mathbf{I} + P \mathbf{h}_1 \mathbf{h}_1^H) = \log(1+P).$ $Similarly,$ I(\mathbf{x}_2; \mathbf{y} | \mathbf{x}_1) = \log(1+P)$.

Capacity region

The sum rate $2\log(1+P) = \log(1+P) + \log(1+P) = A + B$ , so $C = A + B$ . This means the sum-rate constraint is inactive, and the capacity region is the rectangle $[0, \log(1+P)] \times [0, \log(1+P)]$ .

Intuitively, the two users' channels are orthogonal ( $\mathbf{h}_1 \perp \mathbf{h}_2$ ), so they create no interference for each other. Each user independently achieves its single-user capacity. The MAC region degenerates to a rectangle — there is no penalty for simultaneous access.

Orthogonal Channels Eliminate the MAC Penalty

The previous example illustrates a profound point: when the base station has enough antennas to spatially separate the users (i.e., the channel vectors are orthogonal), the MAC penalty vanishes entirely. Each user achieves its interference-free capacity, and the capacity region is a rectangle.

This is the information-theoretic reason why massive MIMO (many antennas at the base station) is so powerful for uplink: as the number of base station antennas grows, the users' channel vectors become approximately orthogonal (by the law of large numbers), and the MAC capacity region approaches the interference-free rectangle.

MIMO MAC Capacity Region

Visualize the capacity region of a two-user MIMO MAC. Adjust the channel vectors and power levels to see how the region changes. When the channels are orthogonal, the region is a rectangle; when they are parallel, the scalar MAC penalty is maximized.

Parameters

\angle \mathbf{h}_1

(degrees)0

Angle of user 1 channel vector

\angle \mathbf{h}_2

(degrees)90

Angle of user 2 channel vector

P_1

5

P_2

5

Definition:
Sum Capacity of the MIMO MAC

The sum capacity of the MIMO MAC is

$C_{\text{sum}} = \max_{\substack{\mathbf{K}_1 \succeq 0, \mathbf{K}_2 \succeq 0 \\ \text{tr}(\mathbf{K}_k) \leq P_k}} \log\det\!\left(\mathbf{I} + \mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H} + \mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H}\right).$

This is equivalent to a single-user MIMO channel with "super-channel" $\mathbf{H} = [\mathbf{H}_{1} \; \mathbf{H}_{2}]$ and block-diagonal covariance constraint $\text{diag}(\mathbf{K}_1, \mathbf{K}_2)$ . The optimization over $(\mathbf{K}_1, \mathbf{K}_2)$ is a convex problem that can be solved by iterative water-filling: alternately optimize $\mathbf{K}_1$ (treating $\mathbf{H}_{2} \mathbf{K}_2 \mathbf{H}_{2}^{H}$ as noise) and $\mathbf{K}_2$ (treating $\mathbf{H}_{1} \mathbf{K}_1 \mathbf{H}_{1}^{H}$ as noise) until convergence.

The iterative water-filling algorithm converges to the global optimum because the sum-rate is a concave function of $(\mathbf{K}_1, \mathbf{K}_2)$ . This is a consequence of the concavity of $\log\det$ on the cone of positive semidefinite matrices.

Why This Matters: Uplink MU-MIMO: From Theory to 5G

The MIMO MAC capacity region directly describes the fundamental limits of the uplink of multi-user MIMO (MU-MIMO) systems. In 5G NR:

The base station (gNB) has $m = 64$ to $256$ antennas (massive MIMO).
Multiple UEs transmit simultaneously on the same time-frequency resource.
The gNB uses MMSE-SIC or MMSE detection to separate the users.

The MIMO MAC theory predicts that as $m \to \infty$ , the users' channels become nearly orthogonal and each user approaches its interference-free capacity. This "channel hardening" effect is the key enabler of massive MIMO systems, where hundreds of base station antennas serve tens of users with near-zero inter-user interference.

The iterative water-filling algorithm for computing the sum capacity has practical counterparts in uplink power control and rate adaptation algorithms used in 5G NR.

Common Mistake: Correlated Inputs in the MIMO MAC

Mistake:

Attempting to optimize the MIMO MAC capacity region over correlated inputs $p(\mathbf{x}_1, \mathbf{x}_2)$ by allowing the input covariance matrix to have off-diagonal blocks (cross-covariance between users).

Correction:

The encoders are independent, so the inputs must be independent: $p(\mathbf{x}_1, \mathbf{x}_2) = p(\mathbf{x}_1)p(\mathbf{x}_2)$ . The joint covariance matrix is always block-diagonal: $\text{diag}(\mathbf{K}_1, \mathbf{K}_2)$ . Any off-diagonal terms would require encoder cooperation, which is not available in the MAC.

Key Takeaway

The MIMO MAC capacity region is achieved by Gaussian inputs and SIC decoding. When the channel vectors are orthogonal, the capacity region is a rectangle (no multi-access penalty). This explains why massive MIMO — with many base station antennas creating near-orthogonal channels — achieves near-optimal uplink performance. The sum capacity is computed via iterative water-filling over the users' covariance matrices.

The Vector Gaussian MAC (MIMO MAC)