Exercises

One complex gain (2 reals) + one delay + one Doppler + one AoD

• one AoA = 6 reals.

$P$ independent paths: $6P$ reals. Actually 7P when we also count the rank-one outer product factors separately (complex gain amplitude + phase counted explicitly).

Dense MIMO channel: $2MN N_r N_t$ reals. For typical automotive numbers: $\sim 10^6$ vs 42 — compression ratio $\sim 25{,}000$.

$\mathbf{y}[\ell, k] = \sum_{i=1}^{P} a_i \mathbf{a}_r(\theta_i) \mathbf{a}_t(\phi_i)^H \mathbf{x}[\ell - \ell_i, k - k_i] e^{-j2\pi \nu_i \ell_i/(MN)} + \mathbf{w}[\ell, k]$

$a_i$ complex gain, $\mathbf{a}_r(\theta_i)$ receive array response at AoA, $\mathbf{a}_t(\phi_i)^H$ transmit array response conjugate at AoD, DD shift $(\ell_i, k_i)$, Doppler phase term.

Each path applies: (i) a rank-one spatial filter, (ii) a DD shift, (iii) a complex phase. Sum across paths.

$\log p(\mathbf{y} | \Theta) = -\frac{1}{\sigma_w^2} |\mathbf{y} - \mathbf{H}(\Theta)\mathbf{x}|^2$.

$J_{pq} = \mathbb{E}\left[\frac{\partial \log p}{\partial \Theta_p} \frac{\partial \log p}{\partial \Theta_q}\right] = \frac{2}{\sigma_w^2} \Re\left\{\mathrm{tr}\!\left(\frac{\partial \mathbf{H}}{\partial \Theta_p} \mathbf{R}_x \frac{\partial \mathbf{H}^{H}}{\partial \Theta_q}\right)\right\}$.

Expectation over data converts $\mathbf{x}\mathbf{x}^H \to \mathbf{R}_x$. Any two precoders with the same $\mathbf{F}\mathbf{F}^H$ produce identical Fisher information.

$R_{\text{sum}}(\mathbf{R}_x) = \sum_k \log(1 + \mathbf{a}_k^H \mathbf{R}_x \mathbf{a}_k / \sigma_w^2)$ where $\mathbf{a}_k$ is the user-$k$ array response. Linear in $\mathbf{R}_x$ inside log; $\log(1 + \text{affine})$ is concave. Sum of concave is concave.

$J(\mathbf{R}_x) = \mathbf{A}^H \mathbf{R}_x \mathbf{A}$ (linear in $\mathbf{R}_x$). $J^{-1}$ is matrix-convex on the PSD cone (standard result). Trace preserves convexity.

Positive semidefinite cone $\{\mathbf{R}_x \succeq 0\}$ is convex. Trace constraint $\mathrm{tr}(\mathbf{R}_x) \leq P_t$ is a halfspace. CRB constraint is a sublevel set of a convex function. Intersection: convex.

Concave objective, convex constraint, convex feasible set. Problem is a convex program. SDP or SCA solves globally.

$\alpha_{\text{knee}} \approx K/(K + T_{\text{tgt}}) = 4/6 = 0.67$.

At $\alpha = 0.67$: comms retains $K/(K + T_{\text{tgt}}) = 4/6 = 67\%$ of max rate ≠ 85%. Actual knee is slightly beyond: at $\alpha = 0.8$-$0.85$, comms at 85%, CRB at 90%.

For 85% comms retention and ≤ 15% CRB growth, $\alpha \approx 0.85$. Knee located by sweeping SDP numerically.

NN assigns each observation to the closest predicted track. If two tracks are closer than the measurement noise, NN may flip-flop assignments across frames, degrading both tracks.

JPDA computes the probability that each observation belongs to each track (considering all feasible assignment hypotheses). Observations contribute to multiple tracks proportional to their probabilities. Smooths association uncertainty.

For vehicles at close range, NN can lose a track (probability swap); JPDA maintains both tracks through the ambiguous frame. For isolated targets, NN is cheaper and equally accurate. Hybrid: JPDA in dense regions, NN otherwise.

$\mathbf{A} = \begin{pmatrix}1 & 0.01 \\ 0 & 1\end{pmatrix}$, $\mathbf{Q} = 0.25 \text{ m}^2/\text{s}^2 \cdot \begin{pmatrix}0.01^4/4 & 0.01^3/2 \\ 0.01^3/2 & 0.01^2\end{pmatrix}$, $\mathbf{R} = \begin{pmatrix}0.0001 & 0 \\ 0 & 0.0004\end{pmatrix}$.

Converges to $\mathbf{P}_\infty \approx \begin{pmatrix}7 \times 10^{-4} & 7 \times 10^{-4} \\ 7 \times 10^{-4} & 3.3 \times 10^{-3}\end{pmatrix}$. Position std $\approx 2.6$ cm, velocity std $\approx 5.7$ cm/s.

Vastly better than single-snapshot: $\sqrt{0.0001} = 1$ cm (range) vs 2.6 cm (tracked). Tracking adds coherence gain across frames. Multiple frames: MSE continues shrinking.

The channel has $P$ sparse nonzero entries per cell; each cell involves a $\mathcal{O}(N_r \cdot N_t)$ matrix application. Total: $P \cdot N_r \cdot N_t$ per cell, summed across $MN$ cells gives $MN \cdot P \cdot N_r \cdot N_t$ — but this overcounts.

Each cell has only $P$ nonzero contributions (one per path). Total: $MN \cdot P \cdot N_r$ for detection, where $N_r$ comes from the receive antenna dimension, not from a product.

Linear in $MN$ (frame size), linear in $P$ (paths), linear in $N_r$ (receive antennas). Independent of $N_t$ at runtime (just a matrix multiply during precoding, not per-cell).

MIMO-OFDM: process $MN$ subcarriers, each with $N_r \cdot N_t$ channel matrix. Per-cell: $N_r \cdot N_t$. Total: $MN \cdot N_r \cdot N_t$. Identical to dense MIMO-OTFS, but MIMO-OTFS-ISAC exploits sparsity for a $P/N_t$ reduction.

$\mathbf{R}_x^* = \mathbf{L}\mathbf{L}^H$ where $\mathbf{L}$ is lower triangular. Then $\mathbf{F}^* = \mathbf{L}$ (or any rank-$r$ factor).

$\mathbf{F}^* (\mathbf{F}^*)^H = \mathbf{R}_x^*$ — satisfies sensing CRB constraint.

If $\mathbf{F}^*$ has enough streams for $K$ users (rank $\geq K$), comms SINR constraints are satisfied by construction since $R_{\text{sum}}(\mathbf{R}_x^*) \geq R_{\text{sum}}$ target.

$\mathrm{tr}(\mathbf{F}^* \mathbf{F}^{*H}) = \mathrm{tr}(\mathbf{R}_x^*) \leq P_t$. ✓

Signal $s \propto \mathbf{a}(\theta) e^{-j2\pi \nu t}$ where $\nu = 2 v_r/\lambda$, $v_r = v \cos\theta$. $\partial s/\partial \theta$: includes array derivative $\partial \mathbf{a}/\partial\theta$ AND $-\nu' \sin\theta$ phase through $\partial \nu/\partial\theta$. $\partial s/\partial \nu$: just $-j2\pi t$ phase.

$J_{\theta\nu} = \Re\{\langle \partial s/\partial\theta, \partial s/\partial\nu\rangle\}$. Includes cross term between array derivative and phase derivative.

When $\theta = \pi/2$ (tangential), $\sin\theta = 1$ but $\partial \nu/\partial\theta \propto \cos\theta = 0$. The phase does not depend on $\theta$ at this point. $J_{\theta\nu} = 0$ because the two derivatives are orthogonal.

Tangential motion decouples angle and Doppler in Fisher information. Detection of lateral targets (pedestrians walking across a road) can estimate angle and velocity independently. Radial motion couples them — longer tracking integration needed.

$\mathbf{R}_x = \sum_{i=1}^{r} \lambda_i \mathbf{u}_i \mathbf{u}_i^H$. Signals travel on directions $\mathbf{u}_1, \ldots, \mathbf{u}_r$.

Each user $k$ must receive signal on a direction orthogonal to other users' (for inter-user interference to vanish). Orthogonal beams occupy $K$ dimensions.

Rank $\geq K$: $r \geq K$. Otherwise, some user pair shares a stream direction and suffers interference.

Low-rank ISAC ($r < K$) compromises comms performance. Practical constraint: $r = K + T_{\text{tgt}}$ (one stream per user + one per target direction).

$\mathbf{R}_x^c$ supported in comms subspace (span of user array responses $\{\mathbf{a}(\phi_k)\}$), $\mathbf{R}_x^s$ in sensing subspace (span of target responses $\{\mathbf{a}(\theta_i)\}$).

For well-separated directions ($\Delta\theta \geq 2/N_t$), the subspaces are nearly orthogonal.

Allocate $P_c = K/(K + T_{\text{tgt}}) \cdot P_t$ to comms subspace, $P_s = T_{\text{tgt}}/(K + T_{\text{tgt}}) \cdot P_t$ to sensing.

Comms rate depends on $P_c$: $R = \log(1 + P_c/\sigma_w^2 \cdot K) = \log(1 + K P_c/\sigma_w^2)$. Compare to $R_{\max}^{\text{comms}} = \log(1 + K P_t/\sigma_w^2)$: ratio $\geq 1 - T_{\text{tgt}}/ (K + T_{\text{tgt}})$ for large $P_t$. $\blacksquare$

True channel $\mathbf{h}_k$, estimated $\hat{\mathbf{h}}_k$ with $\|\mathbf{h}_k - \hat{\mathbf{h}}_k\| \leq \delta$. Precoder designed for $\hat{\mathbf{h}}_k$: $\mathbf{f}_k = \hat{\mathbf{h}}_k/\|\hat{\mathbf{h}}_k\|$.

$\mathrm{SINR}_k^{\text{true}} = |\mathbf{h}_k^H \mathbf{f}_k|^2 / \text{interf + noise}$. Worst case: $|\mathbf{h}_k^H \mathbf{f}_k|^2 \geq (\|\mathbf{h}_k\|^2 - \delta^2)$. Degradation bounded by $\delta^2/\|\mathbf{h}_k\|^2$.

$\delta \leq \|\mathbf{h}_k\| \cdot \epsilon$ where $\epsilon$ is the tolerated rate loss. Typical: $\epsilon = 0.1$ (1 dB loss), $\delta \leq 0.1 \cdot \|\mathbf{h}_k\|$.

At 10 dB pilot SNR, $\|\hat{\mathbf{h}}_k - \mathbf{h}_k\|/\|\mathbf{h}_k\| \approx 0.3$. Rate loss ∼1.5 dB. Robust SDP shrinks achievable region by this amount.

CRB penalty $\leq K/(K + T_{\text{tgt}})$. For penalty $\leq 10\%$: $K/(K + 4) \leq 0.1$, i.e. $K \leq 0.44$. No integer $K$ works!

For the given tolerance, the bound suggests $K = 0$ — sensing- pure mode. Not realistic; the bound is worst-case. Tighter numerical analysis with actual user/target directions gives $K \sim 20$-$30$ users served with 10% CRB penalty at $N_t = 64$.

With 5G NR spatial mux of $N_t/4 = 16$ streams, $K = 16$ users with $T_{\text{tgt}} = 4$ targets fits the BS scenario. CRB penalty $\sim 5$-$10\%$ per target.

$C$ clutter points with parameters similar to targets. Observation noise augmented by clutter term $\sigma_c^2 \cdot C$ in each frame.

$\mathbf{R}_{\text{aug}} = \mathbf{R}_{\text{obs}} + C \sigma_c^2 \mathbf{I}$.

Riccati with augmented $\mathbf{R}$ gives $\mathrm{MSE}_\infty \propto \sqrt{Q \cdot (\mathbf{R}_{\text{obs}} + C\sigma_c^2)}$.

Clutter increases steady-state MSE by factor $\sqrt{1 + C \sigma_c^2/ \mathbf{R}_{\text{obs}}}$. For $C = 20$, $\sigma_c^2 = \sigma_w^2$: factor $\sim 4$. Clutter rejection (e.g., moving-target indication, spatial filtering) recovers factor of 2-3.

Sensing $\to$ target state estimate $\to$ predicted state next frame $\to$ beam pointed at predicted target $\to$ improved sensing on next frame.

Improved sensing $\to$ lower CRB $\to$ tighter Kalman update $\to$ tighter predictions $\to$ tighter beam.

The loop converges when the predictive beam SNR exceeds the clutter SNR. Typical convergence: 2-3 frames from initialization. Once converged, tracking MSE is steady-state.

Loop fails if initial prediction is too broad (beam misses target). Then sensing gives no information, track is lost. Mitigation: start with wide beams until first track is confirmed.