Exercises

$16^2 = 256$ candidates.

Plug the MMSE matrix into the signal-plus-interference expression for stream $k$; the result is the standard identity linking diagonal entries of the MMSE matrix to the per-stream SINR.

ZF minimizes interference alone, so at low SNR it inverts a nearly-singular $\mathbf{H}^{H}\mathbf{H}$ and amplifies the noise. MMSE regularizes by $\sigma^2\mathbf{I}$ and trades residual interference against noise.

$\lambda_1 = \min_{\mathbf{v}\in\Lambda\setminus\{0\}}\|\mathbf{v}\|$, the length of the shortest nonzero lattice vector.

The first-decoded stream is not protected by any prior cancellations, so its reliability dominates the subsequent cancellation errors. Picking the strongest first minimizes the chance of an early mistake that contaminates all later streams.

The signal on the $k$-th output is $x_k$; the noise is $\mathbf{e}_k^T (\mathbf{H}^{H}\mathbf{H})^{-1}\mathbf{H}^{H}\mathbf{w}$.

The noise variance is $\sigma^2\cdot[(\mathbf{H}^{H}\mathbf{H})^{-1}]_{kk}$. Hence $\text{SINR}_k^{\text{ZF}} = 1/(\sigma^2[(\mathbf{H}^{H}\mathbf{H})^{-1}]_{kk})$.

$I(\mathbf{x};\mathbf{y}) = \sum_{k=1}^{n_t} I(x_k;\mathbf{y}|x_{1:k-1})$.

At stage $k$ with perfect cancellation of $x_{1:k-1}$, the MMSE-SIC receiver achieves rate $\log(1+\text{SINR}_k^{\text{MMSE-SIC}})$.

Summing gives $\log\det(\mathbf{I}+\text{SNR}\,\mathbf{H}\mathbf{H}^{H})$, exactly the Gaussian MIMO capacity.

$\tilde{\mathbf{y}} = \mathbf{Q}^H\mathbf{y} = \mathbf{R}\mathbf{x} + \tilde{\mathbf{w}}$ where $\mathbf{R}$ is upper triangular.

Partial distance at level $k$: $d_k = d_{k+1} + |\tilde{y}_k - \sum_{j\geq k} R_{kj} x_j|^2 \leq r^2$. Prune the branch whenever $d_k > r^2$.

$\mu = \langle \mathbf{b}_2,\mathbf{b}_1\rangle/\|\mathbf{b}_1\|^2 = 12/16 = 0.75$; round to 1. Replace $\mathbf{b}_2 \leftarrow \mathbf{b}_2 - \mathbf{b}_1 = (-1,1)^T$.

$\|\mathbf{b}_2\|^2 = 2 < \delta\|\mathbf{b}_1\|^2 = 12$, so swap: the new basis becomes $(\mathbf{b}_2,\mathbf{b}_1) = ((-1,1)^T, (4,0)^T)$.

The two detectors coincide as $\sigma^2\to 0$, i.e., in the noise-free (infinite-SNR) limit.

Plain ZF's performance is limited by the smallest singular value of $\mathbf{H}$, which scales poorly in $n_t,n_r$.

LLL reduces the basis of the lattice spanned by $\mathbf{H}$, bounding the smallest basis vector by a function of $\lambda_1$. The effective condition number becomes O(1), restoring full receive diversity $n_r$ (Jaldén-Elia 2010).

At high SNR the received vector is close to the true lattice point; the ZF rounding finds that point with high probability, so the initial search radius is tiny and few branches survive pruning.

With MMSE-SIC, SINR of the first (ordered) stream is $\|\mathbf{H}_{1}\|^2/(1 + \text{SNR}\|\mathbf{H}_{2}\|^2/\ntn{something})$ — exact algebra is a 1-page calculation (see Tse-Viswanath Ch.8). Alternatively compute $\mathbb{E}[\log\det(\mathbf{I}+\text{SNR}\,\mathbf{H}\mathbf{H}^{H})]$.

For i.i.d. Rayleigh $2\times 2$: $\mathbb{E}[C] = 2\cdot e^{1/\text{SNR}}[-\text{Ei}(-1/\text{SNR})]$ (after rescaling). At $\text{SNR}=10$ dB this evaluates to $\approx 5.7$ bits/s/Hz.

If the initial radius is chosen as $\|\mathbf{y}-\mathbf{H}\mathbf{x}_0\|$ for any feasible $\mathbf{x}_0$ (e.g., the Babai point), then $r \geq \|\mathbf{y}-\mathbf{H}\hat{\mathbf{x}}_{\text{ML}}\|$ since $\hat{\mathbf{x}}_{\text{ML}}$ minimizes.

Hence $\hat{\mathbf{x}}_{\text{ML}}$ is in the search set, and the algorithm returns the minimum over a superset of $\{\hat{\mathbf{x}}_{\text{ML}}\}$ — which is $\hat{\mathbf{x}}_{\text{ML}}$ itself.

Detect on the reduced basis $\mathbf{G} = \mathbf{H}\mathbf{T}$ with integer coefficients, then map back via $\mathbf{T}^{-1}$. Since $\mathbf{T}$ is unimodular, the mapping preserves the integer lattice.

LLL-aided detection's error exponent scales as $n_r$ (full diversity), while plain MMSE-SIC scales as $n_r - n_t + 1 = 1$. At 12 dB the curves diverge by several dB; the gap increases proportionally to SNR on a log-log BER plot.