Ferkans — Interactive Telecom Tutor

Why Duality Matters

Computing the DPC capacity region of the MIMO BC directly is hard: it involves optimizing over covariance matrices $\mathbf{K}_1, \ldots, \mathbf{K}_K$ and all $K!$ encoding orders, with a non-convex objective. The remarkable observation, discovered independently by Vishwanath, Jindal, and Goldsmith (2003) and by Viswanath and Tse (2003), is that the MIMO BC capacity region is dual to the MIMO MAC capacity region with the same total power.

The point is that the MAC capacity region is much easier to compute — it is a convex optimization (maximizing a concave function over a convex constraint set). The duality transformation lets us solve the easy MAC problem and then map the solution to the BC, avoiding the non-convex BC optimization entirely.

Definition:
The Dual MIMO MAC

Given a $K$ -user MIMO BC with channels $\{\mathbf{H}_{k}\}$ and total power $P$ , the dual MAC is the $K$ -user MIMO multiple access channel: $\mathbf{y} = \sum_{k=1}^K \mathbf{H}_{k}^{H} \mathbf{x}_k + \mathbf{z}$ where:

User $k$ transmits $\mathbf{x}_k$ with individual power constraint $P_k$ and $\sum_k P_k \leq P$
$\mathbf{H}_{k}^{H}$ (the Hermitian transpose of the BC channel) is the MAC channel
$\mathbf{z} \sim \mathcal{CN}(\mathbf{0}, \mathbf{I})$ is unit-variance noise

Notice the reversal: the BC downlink channel $\mathbf{H}_{k}$ becomes the MAC uplink channel $\mathbf{H}_{k}^{H}$ , and the roles of transmitter and receiver are swapped.

Theorem: MAC-BC Duality

The capacity region of the $K$ -user Gaussian MIMO broadcast channel with channels $\{\mathbf{H}_{k}\}_{k=1}^K$ and total power constraint $P$ equals the capacity region of the dual $K$ -user Gaussian MIMO MAC with channels $\{\mathbf{H}_{k}^{H}\}_{k=1}^K$ and sum power constraint $\sum_k P_k \leq P$ : $C_{\text{BC}}(P) = C_{\text{MAC}}^{\Sigma}(P)$

where $C_{\text{MAC}}^{\Sigma}(P)$ denotes the MAC capacity region under a sum (rather than individual) power constraint.

Intuitively, the BC and MAC represent two sides of the same coin: the BC is the downlink (one transmitter, multiple receivers) and the MAC is the uplink (multiple transmitters, one receiver) of the same physical channel. The duality says that the same set of rate tuples is achievable in both directions, provided the total power budget is the same.

The mathematical reason is that DPC in the BC and successive cancellation in the MAC are "mirror" operations: DPC pre-cancels interference at the encoder, while SIC post-cancels interference at the decoder. The same mutual information expressions arise from both, up to a power reallocation.

Proof

MAC capacity region with sum power

The MAC capacity region with sum power $P$ is: $C_{\text{MAC}}^{\Sigma}(P) = \bigcup_{\substack{\mathbf{K}_k \succeq 0 \\ \sum_k \text{tr}(\mathbf{K}_k) \leq P}} \left\{ (R_{1}, \ldots, R_{K}) : \sum_{k \in S} R_{k} \leq \log\det\left(\mathbf{I} + \sum_{k \in S} \mathbf{H}_{k}^{H} \mathbf{K}_k \mathbf{H}_{k}\right) \; \forall S \subseteq [K] \right\}$ This is a convex optimization: the rate constraints are concave in $\{\mathbf{K}_k\}$ .

DPC rate equals MAC corner point rate

For a fixed encoding order $\pi$ and covariances $\{\mathbf{K}_k^{\text{BC}}\}$ in the BC, the DPC rate for user $\pi(k)$ can be written as: $R_{\pi(k)}^{\text{BC}} = \log\det\left(\mathbf{I} + \mathbf{H}_{\pi(k)} \left(\sum_{j \geq k} \mathbf{K}_{\pi(j)}^{\text{BC}}\right) \mathbf{H}_{\pi(k)}^{H}\right) - \log\det\left(\mathbf{I} + \mathbf{H}_{\pi(k)} \left(\sum_{j > k} \mathbf{K}_{\pi(j)}^{\text{BC}}\right) \mathbf{H}_{\pi(k)}^{H}\right)$ This is a telescoping difference, identical in structure to the SIC rate at the corresponding corner point of the dual MAC with order $\bar{\pi}$ (the reverse of $\pi$ ).

Power allocation transformation

The BC covariances $\{\mathbf{K}_k^{\text{BC}}\}$ can be mapped to MAC covariances $\{\mathbf{K}_k^{\text{MAC}}\}$ with the same sum power, such that the rate tuple is preserved. The mapping is obtained by solving a system of matrix equations (Vishwanath-Jindal-Goldsmith transformation): $\mathbf{K}_k^{\text{MAC}} = \mathbf{F}_k \mathbf{K}_k^{\text{BC}} \mathbf{F}_k^H$ where $\mathbf{F}_k$ depends on the channel matrices and the other covariances. The total power is preserved: $\sum_k \text{tr}(\mathbf{K}_k^{\text{MAC}}) = \sum_k \text{tr}(\mathbf{K}_k^{\text{BC}})$ .

Capacity region equality

Since every BC rate tuple maps to a MAC rate tuple (and vice versa) with the same sum power, and the convex hulls of these rate tuples are the respective capacity regions, we conclude $C_{\text{BC}}(P) = C_{\text{MAC}}^{\Sigma}(P)$ .

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Example: Duality for the 2-User MISO BC

Consider a 2-user MISO BC ( $n_t = 2$ , single-antenna users) with channels $\mathbf{h}_1 = [1, 1]^T / \sqrt{2}$ and $\mathbf{h}_2 = [1, -1]^T / \sqrt{2}$ , total power $P = 10$ . Use MAC-BC duality to find the sum capacity.

Solution

Dual MAC formulation

The dual MAC has channels $\mathbf{h}_1^H = [1, 1]/\sqrt{2}$ and $\mathbf{h}_2^H = [1, -1]/\sqrt{2}$ , with sum power constraint $P_1 + P_2 \leq 10$ . The MAC sum capacity is: $C_{\text{sum}} = \max_{P_1 + P_2 \leq 10} \log\det\left(\mathbf{I} + P_1 \mathbf{h}_1 \mathbf{h}_1^H + P_2 \mathbf{h}_2 \mathbf{h}_2^H\right)$

Exploit channel orthogonality

Since $\mathbf{h}_1 \perp \mathbf{h}_2$ (they are orthogonal), the sum $P_1 \mathbf{h}_1 \mathbf{h}_1^H + P_2 \mathbf{h}_2 \mathbf{h}_2^H$ has eigenvalues $P_1/2$ and $P_2/2$ . Thus: $C_{\text{sum}} = \max_{P_1 + P_2 \leq 10} \left[\log(1 + P_1/2) + \log(1 + P_2/2)\right]$

Optimize power allocation

By the concavity of $\log$ , equal power $P_1 = P_2 = 5$ is optimal: $C_{\text{sum}} = 2\log(1 + 5/2) = 2\log(3.5) \approx 3.64 \text{ bits/use}$

By MAC-BC duality, this is also the sum capacity of the original MISO BC. The corresponding BC covariances are $\mathbf{K}_k = 5 \mathbf{h}_k \mathbf{h}_k^H$ (beamforming along the channel directions), which is also ZF precoding — as expected for orthogonal channels.

Computational Advantage of Duality

The MAC-BC duality is not just a theoretical curiosity — it is the workhorse behind most practical MIMO precoder design algorithms. The reason: optimizing over the MAC capacity region is a convex problem (the rate constraints involve $\log\det$ of a sum of positive semidefinite matrices, which is concave). The BC optimization, by contrast, involves $\log\det$ of a difference (the "chain rule" structure of DPC rates), which is non-convex.

In practice, one solves the MAC problem using standard convex optimization tools (interior point methods, or the iterative water-filling algorithm), then maps the solution to the BC via the power allocation transformation. This is the approach used in Section 16.5.

Common Mistake: Sum Power vs. Individual Power Constraints

Mistake:

Applying MAC-BC duality with individual per-user power constraints in the MAC.

Correction:

The duality holds with a sum power constraint $\sum_k P_k \leq P$ in the MAC, matching the single total power constraint $P$ in the BC. With individual power constraints in the MAC ( $P_k \leq P_k^{\max}$ for each $k$ ), the MAC and BC capacity regions are generally different. This is a common source of errors when applying duality to practical systems.

MAC vs. BC: A Side-by-Side Comparison

Aspect	Multiple Access Channel (MAC)	Broadcast Channel (BC)
Direction	Uplink: multiple Tx, one Rx	Downlink: one Tx, multiple Rx
Capacity achieving scheme	Joint typicality decoding or SIC	Dirty-paper coding (DPC)
Interference handling	Post-cancellation at decoder (SIC)	Pre-cancellation at encoder (DPC)
Optimization	Convex (concave rate expressions)	Non-convex (rate differences)
Practical schemes	MMSE-SIC, joint detection	ZF, MMSE, RZF precoding
Power constraint	Per-user or sum power	Total transmit power

Quick Check

In MAC-BC duality, what happens to the encoding order?

The SIC decoding order in the MAC is the reverse of the DPC encoding order in the BC

The orders are identical in both MAC and BC

The order does not matter because duality holds for all orderings simultaneously

Correction:

The SIC decoding order in the MAC is the reverse of the DPC encoding order in the BC

If the BC encodes in order $\\pi(1), \\ldots, \\pi(K)$ (with $\\pi(1)$ encoded first), then the dual MAC achieves the same rate tuple by decoding in order $\\pi(K), \\ldots, \\pi(1)$ (the last BC user is decoded first in the MAC).

Key Takeaway

MAC-BC duality is both a theoretical insight and a computational tool. It establishes that the MIMO broadcast channel (hard to optimize) and the MIMO multiple access channel (easy to optimize) have the same capacity region under a sum power constraint. The duality transformation maps MAC covariances to BC covariances at equal sum power, enabling efficient computation of the BC capacity region via convex optimization.

MAC-BC Duality

Why Duality Matters

Definition: The Dual MIMO MAC

Theorem: MAC-BC Duality

MAC capacity region with sum power

DPC rate equals MAC corner point rate

Power allocation transformation

Capacity region equality

Example: Duality for the 2-User MISO BC

Dual MAC formulation

Exploit channel orthogonality

Optimize power allocation

Computational Advantage of Duality

Common Mistake: Sum Power vs. Individual Power Constraints

MAC vs. BC: A Side-by-Side Comparison

Quick Check

Key Takeaway

Definition:
The Dual MIMO MAC