Exercises

$\mathcal{H}_{\mathrm{DD}}[\ell, k] = \sum_{i=1}^{P} a_i \mathbf{a}_r(\theta_i) \mathbf{a}_t(\phi_i)^H \delta[\ell - \ell_i, k - k_i] e^{-j2\pi\nu_i \ell_i/(MN)}$

Each term is rank-one in the spatial dimensions. Sum of $P$ rank-one terms has rank $\leq P$.

The channel is a tensor in $(\tau, \nu, \theta, \phi)$-space with $P$ point masses.

$\mathbf{F}_{BS}$ is the $N \times N$ DFT matrix. Entry $[\mathbf{F}_{BS}]_{n, b} = e^{j2\pi nb/N}/\sqrt{N}$.

Converts antenna indices to beam indices. A path at angle $\theta_b = \sin^{-1}(2b/N - 1)$ maps to a point at beam index $b$ in the beamspace channel.

Beamspace-DD channel has at most $P$ nonzero entries out of $MN N_r N_t$ — ultra-sparse.

Averaged channel $\bar{\mathbf{H}} = \sum_i a_i \mathbf{a}_r(\theta_i) \mathbf{a}_t(\phi_i)^H$ has column space $\subseteq \mathrm{span}\{\mathbf{a}_t(\phi_i)\}$.

Right singular vectors of $\bar{\mathbf{H}}$ span the same subspace. Optimal precoder is a linear combination of them.

Optimal precoder lies in $\mathrm{span}\{\mathbf{a}_t(\phi_i)\}$ — an at-most-$P$-dimensional subspace. Hybrid beamforming with $N_{RF} = P$ RF chains suffices.

$2 \cdot MN \cdot 16 \cdot 4 = 128 MN$. For $MN = 1024$: $131{,}072$ reals.

$6P = 36$ reals.

Ratio: $131{,}072 / 36 \approx 3{,}600$. Sample complexity reduction: $\sim 3{,}600\times$ for channel estimation.

Per-subcarrier ICI: $\sigma_{ICI}^2 = (\nu/\Delta f)^2 \pi^2/3 \cdot P_s$, where $P_s$ is signal power.

$\mathrm{SINR} = P_s / (\sigma_w^2 + \sigma_{ICI}^2) = \text{SNR}/(1 + (\nu/\Delta f)^2 \pi^2/3 \cdot \text{SNR})$.

As $\text{SNR} \to \infty$: SINR → $3/(\pi^2 (\nu/\Delta f)^2)$ — finite ceiling. At $\nu/\Delta f = 0.1$: ceiling $\approx 30$ (15 dB). Cannot exceed regardless of Tx power.

$d^*(0) = 4 \cdot 4 \cdot 8 = 128$. Full diversity.

$d^*(2) = 2 \cdot 2 \cdot 8 = 32$. Diversity 32.

MIMO-OFDM: $d^*(0) = 16$, $d^*(2) = 4$. MIMO-OTFS multiplies by $P = 8$ at both points.

Mixing old and new: $\mu^{(t+1)} = \alpha \mu^{(t)} + (1-\alpha) \tilde\mu^{(t+1)}$. $\alpha$ large: slow change, stable. $\alpha$ small: fast response but oscillates on dense channels.

MP iteration is (under conditions) a contraction with factor $\rho < 1$. Damping multiplies $\rho$ by $(1 - \alpha)$ or $\alpha \rho + (1 - \alpha)$. For $\alpha = 0.9$: contraction factor $\leq 0.9 \rho + 0.1 < 1$. Convergence guaranteed.

Dense channels have strong inter-symbol correlation → raw MP iteration may have $\rho \geq 1$ (non-contractive, divergent). Damping with $\alpha = 0.9$ brings effective contraction back to $< 1$.

$N_{\text{total}} = MN N_t N_r = 2048 \cdot 64 \cdot 64 = 8.4 \times 10^6$.

$\log(N_{\text{total}}/P) = \log(8.4 \times 10^5) \approx 14$.

$m \geq 10 \cdot 14 = 140$. Comfortably scales for OMP/AMP.

Dense estimation needs $m \geq N_t \cdot MN = 1.3 \times 10^5$ pilots. CS reduces by 1000×.

$\mathcal{H}_{\mathrm{DD}} = \sum_{i=1}^{P} a_i (\mathbf{a}_r(\theta_i) \otimes \mathbf{a}_t(\phi_i)^* \otimes \mathbf{e}_{\ell_i, k_i})$, where $\mathbf{e}_{\ell, k}$ is the indicator on DD cell $(\ell, k)$.

Sum of $P$ rank-one tensors: CP-rank $\leq P$. Equality holds when paths are distinct in at least one dimension.

Kruskal's condition: CP decomposition is unique when $k_1 + k_2 + k_3 \geq 2P + 2$, where $k_i$ are Kruskal ranks of the mode factors. For MIMO-OTFS with paths at distinct angles and distinct DD cells: Kruskal ranks = $P$, condition $3P \geq 2P + 2$, i.e. $P \geq 2$. Generically satisfied.

Unique CP decomposition enables path-based channel estimation and separation. Foundation for off-grid methods.

After DD-domain precoder/detector, the effective channel matrix is $\mathbf{H}_{\text{eff}}$ of shape $N_r \times N_t$.

$C = \mathbb{E}[\log\det(\mathbf{I} + \frac{\text{SNR}}{N_t} \mathbf{H}_{\text{eff}} \mathbf{H}_{\text{eff}}^H)]$.

$C \approx \min(N_t, N_r) \log \text{SNR} + \mathcal{O}(1)$. Same scaling as MIMO-OFDM.

MIMO-OFDM capacity: $\min(N_t, N_r) \log(1 + \frac{\mathrm{SINR}_{\text{eff}}}{N_t})$ where $\mathrm{SINR}_{\text{eff}}$ saturates at $1/(\pi^2/3 (\nu/\Delta f)^2)$. Below the saturation point: same as MIMO-OTFS. Above: MIMO-OTFS dominates.

$N_{RF} = P = 6$. Matches path count exactly.

$\mathbf{V}_{RF} = [\mathbf{a}_t(\phi_1), \ldots, \mathbf{a}_t(\phi_6)]/\sqrt{64}$. Constant-envelope phase shifters.

$\mathbf{V}_{BB} \in \mathbb{C}^{6 \times 4}$ handles the user-stream assignments. Can be computed by effective-channel SVD.

Fully digital: 64 RF chains. Hybrid: 6 RF chains. $\sim 10\times$ savings on power and cost.

$\mathcal{O}(MN \cdot P \cdot N_r)$ — independent of Doppler.

Without ICI: $\mathcal{O}(MN \cdot N_t \cdot N_r^2)$. With ICI correction: add $\mathcal{O}(MN \cdot N_t N_r (\nu T)^2)$.

$\frac{\text{MIMO-OTFS}}{\text{MIMO-OFDM}} \propto \frac{P \cdot N_r}{N_t N_r^2 (\nu T)^2} \sim \frac{1}{(\nu T)^2}$. Advantage grows quadratically with Doppler.

MIMO-OTFS wins when $(\nu T)^2 > c \cdot P/(N_t N_r)$ for some constant $c$. For $P = 8$, $N_t N_r = 64$: $\nu T > 0.35$ for some ratio. More typically, crossover is at $\nu T \sim 0.05$ (high mobility).

For diversity $d^*(r) = (N_t - r)(N_r - r) P$: $P_{\text{out}}(\gamma, r) \sim \gamma^{-d^*(r)}$.

$P_{\text{out}}(\gamma, r) \leq \beta$ iff $d^*(r) \geq -\log\beta / \log\gamma$.

Choose largest $r$ such that $d^*(r) \geq -\log\beta/\log\gamma$. E.g., $\beta = 10^{-5}$, $\gamma = 20$ dB (= 100): $d^* \geq 5/\log 100 = 2.5$. So $d^* \geq 3$ works. Given $N_t = N_r = 4$, $P = 8$: $r = 0$ gives $d^* = 128$, $r = 1$ gives 72, $r = 2$ gives 32. Choose $r = 2$.

Lookup table of $(\gamma, \beta) \to r$ refreshed every $\sim 10$ ms. Deployed in 5G NR link adaptation.

$\mathbf{a}(\theta_0) = [1, e^{j\pi \sin 32.7°}, \ldots, e^{j15\pi \sin 32.7°}]$. $\sin 32.7° \approx 0.54$.

$\mathbf{F}_{BS}^H \mathbf{a}(\theta_0)$: Dirichlet kernel peaked near beam index $b^* \approx N \sin \theta_0 / 2 = 16 \cdot 0.54 / 2 \approx 4.3$.

3-dB support: $\pm 1$ bin around $b^* = 4$. 20-dB support: $\pm 3$ bins. Total effective support: 5 beamspace bins.

Even a single path spreads over 3-5 bins in beamspace. For accurate representation: over-sample beamspace grid by $2\times$-$4\times$ or use off-grid methods (atomic-norm).

Low SINR $\sim 0$ dB. Rate adapt: $r = 0$, full diversity $N_t N_r P$. Rate: $\log(1 + \text{SNR}) \approx 1$ bit/s/Hz. Reliable: BER $\leq 10^{-6}$ at SNR = 5 dB.

High SINR $\sim 20$ dB. Rate adapt: $r = \min(N_t, N_r) = 4$, diversity $d^* = 0$. Rate: $4 \log(101) \approx 27$ bits/s/Hz. Less reliable: BER $\leq 10^{-3}$.

As user moves cell-center to cell-edge: $r$ decreases from 4 to 0, rate from 27 to 1 bits/s/Hz, BER from $10^{-3}$ to $10^{-6}$. Smooth interpolation.

Cell-edge users benefit most from MIMO-OTFS's $\times P$ diversity multiplier. Urban coverage uniformity improves markedly vs MIMO-OFDM.

MP iteration: $\mu^{(t+1)} = T(\mu^{(t)})$ where $T$ is the update operator. Linearized: $T'(\mu^*) = \mathbf{J}$. Spectral radius $\rho(\mathbf{J}) = \lambda_{\max}$.

For sparse channel: $\rho \sim P / \min(M, N)$. Fewer paths → more sparse → smaller $\rho$ → faster convergence.

$\mu^{(t+1)} = \alpha \mu^{(t)} + (1-\alpha) T(\mu^{(t)})$. Effective contraction: $\alpha + (1-\alpha) \rho$. Convergent iff $< 1$: $\rho < 1$ (or damping with $\alpha < 1$ always converges for $\rho \geq 1$).

$N_{\text{iter}} = \log(1/\epsilon) / |\log(\alpha + (1-\alpha)\rho)|$. For $P = 8$, $\rho = 0.2$, $\alpha = 0.7$: $|\log(0.76)| \approx 0.28$. $N_{\text{iter}} \approx 3.6 \log(1/\epsilon)$. At $\epsilon = 10^{-5}$: $\approx 18$ iterations. Practical budget.