Ferkans — Interactive Telecom Tutor

From Sub-Optimal to Capacity-Achieving

In Section 16.3 we saw that MMSE-OSIC is the best practical MIMO receiver. A remarkable result from information theory is that MMSE-SIC is not merely a good heuristic — with perfect cancellation and appropriate rate allocation, it achieves the MIMO channel capacity. This section proves this fundamental result and reveals its connection to the chain rule of mutual information and decision-feedback architectures.

Theorem: MMSE-SIC Achieves MIMO Capacity

Consider the $N_r \times N_t$ MIMO channel $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$ with i.i.d. Gaussian inputs $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \frac{P}{N_t}\mathbf{I})$ . An MMSE-SIC receiver that sequentially decodes streams $x_{\pi(1)}, x_{\pi(2)}, \ldots, x_{\pi(N_t)}$ (for any permutation $\pi$ ), where each stream is decoded at the rate:

$R_{\pi(k)} = \log_2\!\left(1 + \text{SINR}_{\pi(k)}^{\text{MMSE}}\right)$

and then perfectly cancelled, achieves the total sum rate:

$R_{\text{total}} = \sum_{k=1}^{N_t} R_{\pi(k)} = \log_2 \det\!\left(\mathbf{I}_{N_r} + \frac{P}{N_t\sigma^2} \mathbf{H}\mathbf{H}^{H}\right) = C_{\text{MIMO}}$

That is, the sum of individual MMSE-SIC rates equals the MIMO channel capacity, regardless of the decoding order $\pi$ .

This is the MIMO analogue of the successive decoding result for the multiple-access channel (MAC). Each stream sees a point-to-point AWGN channel after MMSE suppression of the remaining streams. The key insight is that MMSE estimation preserves mutual information: the information about $x_k$ contained in $\mathbf{y}$ after subtracting the previously decoded streams is exactly captured by the MMSE filter output.

Proof

Chain rule decomposition

The mutual information between $\mathbf{x}$ and $\mathbf{y}$ can be decomposed via the chain rule:

$I(\mathbf{x}; \mathbf{y}) = \sum_{k=1}^{N_t} I(x_{\pi(k)}; \mathbf{y} \mid x_{\pi(1)}, \ldots, x_{\pi(k-1)})$

Each term is the information in stream $\pi(k)$ given that streams $\pi(1), \ldots, \pi(k-1)$ have been perfectly decoded and cancelled.

MMSE-SIC rate equals conditional mutual information

After cancelling streams $\pi(1), \ldots, \pi(k-1)$ , the effective channel for stream $\pi(k)$ is:

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

The MMSE filter for stream $\pi(k)$ yields SINR:

$\text{SINR}_{\pi(k)} = \frac{P}{N_t}\mathbf{h}_{\pi(k)}^H \left(\sum_{j=k+1}^{N_t}\frac{P}{N_t}\mathbf{h}_{\pi(j)}\mathbf{h}_{\pi(j)}^H + \sigma^2\mathbf{I}\right)^{-1}\mathbf{h}_{\pi(k)}$

For Gaussian inputs, $I(x_{\pi(k)}; \mathbf{y}_k) = \log_2(1 + \text{SINR}_{\pi(k)})$ .

Telescoping product yields capacity

Using the matrix determinant lemma iteratively:

$\prod_{k=1}^{N_t}(1 + \text{SINR}_{\pi(k)}) = \frac{\det(\mathbf{R}_{0})}{\det(\sigma^2\mathbf{I})} = \det\!\left(\mathbf{I} + \frac{P}{N_t\sigma^2}\mathbf{H}\mathbf{H}^{H}\right)$

where $\mathbf{R}_{0} = \frac{P}{N_t}\mathbf{H}\mathbf{H}^{H} + \sigma^2\mathbf{I}$ . Taking $\log_2$ :

$\sum_{k=1}^{N_t}\log_2(1 + \text{SINR}_{\pi(k)}) = C_{\text{MIMO}}$

Since the determinant is invariant to the ordering $\pi$ , the sum rate is the same for any decoding order. $\blacksquare$

Definition:
Connection to Decision-Feedback Equalisation

MMSE-SIC is the MIMO generalisation of decision-feedback equalisation (DFE). The MMSE feedforward filter suppresses interference from not-yet-detected streams (analogous to the feedforward section of a DFE), and the SIC step subtracts already-detected streams (analogous to the feedback section).

Specifically, the QR decomposition of the augmented channel:

$\begin{bmatrix} \mathbf{H} \\ \sigma\mathbf{I} \end{bmatrix} = \mathbf{Q}\mathbf{R}$

yields the triangular structure that defines the MMSE-DFE:

$\mathbf{R}$ is upper triangular — the feedforward filter naturally orders the detection.
Back-substitution through $\mathbf{R}$ is exactly SIC.

Example: MMSE-SIC Rate Calculation for a 2x2 Channel

Consider a $2 \times 2$ channel with:

$\mathbf{H} = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}$

$P/N_t = 1$ , $\sigma^2 = 1$ . Compute the individual stream rates under MMSE-SIC (decoding stream 2 first, then stream 1) and verify they sum to the MIMO capacity.

Solution

MIMO capacity

$\mathbf{H}\mathbf{H}^{H} = \begin{bmatrix} 5 & 1 \\ 1 & 1 \end{bmatrix}KATEXPLACEHOLDER0ENDC = \log_2\det\!\left(\mathbf{I}_2 + \mathbf{H}\mathbf{H}^{H}\right) = \log_2\det\begin{bmatrix} 6 & 1 \\ 1 & 2 \end{bmatrix} = \log_2(12 - 1) = \log_2(11) \approx 3.459\;\text{bits/s/Hz}$ $

Stream 2 rate (decoded first)

With both streams present, the MMSE SINR for stream 2 is:

$\text{SINR}_2 = \mathbf{h}_2^H(\mathbf{h}_1\mathbf{h}_1^H + \sigma^2\mathbf{I})^{-1}\mathbf{h}_2$

$\mathbf{h}_1 = [2, 0]^T$ , $\mathbf{h}_2 = [1, 1]^T$

$\mathbf{h}_1\mathbf{h}_1^H + \mathbf{I} = \begin{bmatrix} 5 & 0 \\ 0 & 1 \end{bmatrix}$

$\text{SINR}_2 = [1, 1]\begin{bmatrix} 1/5 & 0 \\ 0 & 1 \end{bmatrix}[1, 1]^T = 1/5 + 1 = 1.2$

$R_2 = \log_2(1 + 1.2) = \log_2(2.2) \approx 1.138\;\text{bits/s/Hz}$

Stream 1 rate (decoded second, after cancelling stream 2)

After perfectly cancelling stream 2, stream 1 sees a clean channel $\mathbf{h}_1 = [2, 0]^T$ with noise only:

$\text{SINR}_1 = \frac{\|\mathbf{h}_1\|^2}{\sigma^2} = \frac{4}{1} = 4$

$R_1 = \log_2(1 + 4) = \log_2(5) \approx 2.322\;\text{bits/s/Hz}$

Verification

$R_1 + R_2 = 2.322 + 1.138 = 3.459 = C_{\text{MIMO}}$ $The MMSE-SIC sum rate matches the MIMO capacity exactly, confirming the theorem.$ \blacksquare$

Quick Check

In MMSE-SIC, why does the sum rate equal the MIMO capacity regardless of the decoding order?

Because ML detection is used for each layer

Because the determinant $\det(\mathbf{I} + \frac{P}{N_t\sigma^2}\mathbf{H}\mathbf{H}^{H})$ decomposes as a product of $(1 + \text{SINR}_k)$ terms that is independent of the ordering

Because all layers have the same SINR

Because the channel is i.i.d. Rayleigh

Correction:

Because the determinant

\det(\mathbf{I} + \frac{P}{N_t\sigma^2}\mathbf{H}\mathbf{H}^{H})

decomposes as a product of

(1 + \text{SINR}_k)

terms that is independent of the ordering

The matrix determinant lemma applied iteratively shows that the product $\prod(1 + \text{SINR}_k)$ telescopes to the same determinant for any permutation. This is essentially the chain rule of mutual information.

Common Mistake: Ignoring Error Propagation in SIC

Mistake:

Assuming that MMSE-SIC achieves capacity in practice. The capacity-achieving result requires perfect cancellation (correctly decoded streams are subtracted exactly).

Correction:

In practice, decoding errors in early layers propagate to subsequent layers, degrading performance. At finite block lengths and without capacity-achieving codes on each layer, the actual sum rate falls short of $C_{\text{MIMO}}$ .

Mitigations include: (1) using powerful channel codes (turbo, LDPC) per layer, (2) optimal ordering (detect strongest layer first), and (3) iterative detection with soft cancellation. The gap between practical MMSE-OSIC and capacity motivates the study of iterative MIMO receivers (turbo-MIMO).

Why This Matters: Successive Decoding and the MAC Capacity Region

The MMSE-SIC capacity result is the MIMO instantiation of a general principle from multi-user information theory: successive decoding achieves the capacity region of the multiple-access channel (MAC). The ITA book develops this theory in depth, covering the MAC capacity region, the duality between MAC and broadcast channels (BC-MAC duality), and how DPC at the transmitter achieves the BC capacity region. Understanding the information-theoretic foundations of successive decoding is essential for multi-user MIMO system design.

MMSE-SIC

A MIMO receiver that combines MMSE linear filtering with successive interference cancellation; with Gaussian codebooks and perfect cancellation, it achieves the MIMO channel capacity.

Chain Rule of Mutual Information

The decomposition $I(\mathbf{x};\mathbf{y}) = \sum_k I(x_k; \mathbf{y} \mid x_1,\ldots,x_{k-1})$ that underlies the successive decoding interpretation of MMSE-SIC.

Related: MMSE-SIC

MMSE-SIC Optimality