Exercises

$\lambda = 3\times 10^8 / 28\times 10^9 \approx 1.07\times 10^{-2}$ m, so $\lambda^{2} \approx 1.14\times 10^{-4}$ m$^2$. Noise: $-174 + 80 + 7 = -87$ dBm $\approx 2 \times 10^{-12}$ W.

$N_t^{2} P_t \sigma \lambda^{2} = 4096 \cdot 1 \cdot 10 \cdot 1.14\times 10^{-4} \approx 4.67$ (with $P_t = 30$ dBm = 1 W).

$(4\pi)^3 r^4 \sigma^2 = 1984 \cdot (150)^4 \cdot 2\times 10^{-12} = 1984 \cdot 5.06\times 10^8 \cdot 2\times 10^{-12} \approx 2.01\times 10^0 = 2.01$.

$\rho \approx 4.67 / 2.01 \approx 2.33 \approx 3.67$ dB per pulse. With pulse compression over 100 MHz $\cdot$ 1 ms = $10^5$ $\to$ +50 dB, integrated SNR $\approx 54$ dB. $\blacksquare$

$128 - 12 = 116$ DoF remain in the null space of the $12$ user channels.

The sensing beam has an effective aperture of 116 elements instead of 128. Array gain is $10\log_{10}(116/128) \approx -0.43$ dB — a negligible loss from the user nulling constraints. This is the quantitative statement of "massive MIMO is the natural ISAC platform": user constraints cost almost nothing in sensing gain when $N_t \gg K$. $\blacksquare$

$P(\theta) = \mathbf{a}^H \mathbf{R}_x \mathbf{a} = \text{tr}(\mathbf{a}\mathbf{a}^H \mathbf{R}_x) = \text{tr}(\mathbf{A}_\theta \mathbf{R}_x)$, which is linear in $\mathbf{R}_x$ (trace of a linear map).

$\|\mathbf{p}_d - \mathbf{p}(\mathbf{R}_x)\|^2 = \sum_{i=1}^{M}(P_d(\theta_i) - \text{tr}(\mathbf{A}_{\theta_i}\mathbf{R}_x))^2$. Each term is a squared affine function of $\mathbf{R}_x$; the sum is a convex quadratic.

Introduce slack variables $s_i \geq (P_d(\theta_i) - \text{tr}(\mathbf{A}_i\mathbf{R}_x))^2$, or equivalently $|P_d(\theta_i) - \text{tr}(\mathbf{A}_i\mathbf{R}_x)| \leq t_i$ with $\sum t_i^2$ bounded. Combined with the PSD constraint $\mathbf{R}_x \succeq 0$ and linear SINR / power constraints, this is a standard SDP solvable in polynomial time. $\blacksquare$

$1 - P_d^{(\ell)} = 1 - 0.8 = 0.2$.

Under independent Swerling-I: $1 - P_d(4) = (0.2)^4 = 0.0016$, so $P_d(4) = 0.9984$.

The miss rate went from $0.2 \to 0.0016$, i.e., a factor $125\times$ reduction. This is exactly the diversity-4 scaling of the cell-free ISAC macro-diversity theorem: each additional AP contributes one order of SNR decay in the miss probability. $\blacksquare$

If $D_1 < D_2$, then any $\mathbf{R}_x$ feasible for the $D_1$ constraint is also feasible for the $D_2$ constraint (relaxation). The maximum rate over a larger feasible set is at least as large. Hence $\mathcal{C}(D_2) \geq \mathcal{C}(D_1)$ — the curve is non-decreasing in $D$, equivalently non-increasing in the sensing accuracy (reciprocal of MSE).

Let $\mathbf{R}_1, \mathbf{R}_2$ achieve $\mathcal{C}(D_1), \mathcal{C}(D_2)$. By concavity of $\log\det$, the convex combination $\mathbf{R}_\lambda = \lambda\mathbf{R}_1 + (1-\lambda)\mathbf{R}_2$ satisfies $\log\det(\mathbf{I} + \mathbf{H}\mathbf{R}_\lambda\mathbf{H}^H/\sigma^2) \geq \lambda \log\det(\cdot_1) + (1-\lambda)\log\det(\cdot_2)$. The distortion $\text{tr}(\mathbf{J}^{-1}(\mathbf{R}_\lambda))$ is convex in $\mathbf{R}_\lambda$, so it is at most $\lambda D_1 + (1-\lambda)D_2$. Hence $\mathcal{C}(\lambda D_1 + (1-\lambda)D_2) \geq \lambda\mathcal{C}(D_1) + (1-\lambda)\mathcal{C}(D_2)$ — the defining property of concavity.

Concave + non-increasing: increasing the distortion budget gives diminishing returns in rate. Near the sensing-tight end, each unit of extra MSE budget buys a lot of rate; near the capacity end, extra MSE buys almost nothing. The operator chooses the operating point by the slope, which equals the Lagrange multiplier $\mu$. $\blacksquare$

$B = 256 \cdot 120 \text{ kHz} = 30.72$ MHz. $\beta_{\text{rms}}^2 \approx B^2/12 = (3.072\times 10^7)^2 / 12 \approx 7.87\times 10^{13}$.

$\rho = 10^3$ (30 dB linear). $\text{CRB}(\tau) = 1/(8\pi^2 \rho \beta_{\text{rms}}^2) \approx 1/(8\pi^2 \cdot 10^3 \cdot 7.87\times 10^{13}) \approx 1/(6.22\times 10^{18}) \approx 1.6\times 10^{-19}$ s$^2$.

$\sigma_r = (c/2)\sqrt{\text{CRB}(\tau)} = 1.5\times 10^8 \sqrt{1.6\times 10^{-19}} = 1.5\times 10^8 \cdot 4\times 10^{-10} = 0.06$ m. Sub-wavelength range precision at 30 dB integrated SNR — consistent with the 5G NR OFDM-ISAC benchmark in Section 24.5. $\blacksquare$

The three directions $\theta \in \{-30, 0, 30\}^\circ$ are not collinear in the steering-vector space: $\mathbf{a}(0)$ is not orthogonal to either $\mathbf{a}(-30)$ or $\mathbf{a}(30)$. The projection of $\mathbf{a}(0)$ onto the subspace orthogonal to both user directions is not parallel to the original vector — the sensing beam shape is distorted by the nulling constraints.

With user SINR constraints, the SDR finds the best sensing beampattern consistent with the two nulls. As the target SINR requirement is raised, the achievable sensing beampattern gain at $\theta_t$ decreases — the residual spatial DoF after nulling is 14, but the usable sensing gain toward $0^\circ$ is lower than the $16$-element maximum because the null-space projection of $\mathbf{a}(0)$ has smaller norm than $\sqrt{16}$.

This is a point on the interior of the capacity-distortion curve: no free lunch, but the tradeoff is graceful because $N_t = 16 \gg K = 2$. The Lagrange multiplier $\mu$ smoothly slides between full-comm and full-sensing solutions. $\blacksquare$

Let $\alpha_\ell = \alpha$ for all $\ell$ (same realization for every AP). Then every AP observes $\rho_\ell = |\alpha|^2 g_\ell$ for some deterministic geometric factor $g_\ell$. The total SNR is $\rho_{\text{tot}} = |\alpha|^2 \sum_\ell g_\ell = |\alpha|^2 G$.

The detector outage probability is $\Pr(\rho_{\text{tot}} < \tau) = \Pr(|\alpha|^2 < \tau/G) = 1 - e^{-\tau/(G\bar\rho)}$. At high SNR this scales as $(\tau/(G\bar\rho))^1$ — diversity order 1, not $L$.

Under i.i.d. $\alpha_\ell$, the joint miss probability factorizes to $(1 - P_d^{(\ell)})^L$, which at high SNR is $(\tau/\bar\rho)^L$ — diversity order $L$. The physical requirement for independence is angular separation between AP viewing directions larger than the target's angular Markov decorrelation scale.

Cell-free ISAC deployments must space APs wide enough that different APs view the target from independent scattering realizations — otherwise the macro-diversity gain is lost. Rule of thumb: AP separation $\gg$ target extent projected onto the target-to-AP line. For typical vehicle targets and 100 m ranges, this translates to AP spacing on the order of 10+ meters. $\blacksquare$

$\Delta\tau = 1/B = 10$ ns. $\Delta\nu = 1/(NT_{\text{sym}}) = 1/(128 \cdot 10\,\mu\text{s}) = 781.25$ Hz.

Delay: $|\tau| \leq 128 \cdot 10 / 2 = 640$ ns $\to |r| \leq c\tau/2 \approx 96$ m. Doppler: $|\nu| \leq 64 \cdot 781.25 / 2 = 25000$ Hz. At 28 GHz carrier, $v_{\max} = c\nu_{\max}/(2 f_c) = 3\times 10^8 \cdot 25000/(2\cdot 28\times 10^9) \approx 134$ m/s $\approx 482$ km/h.

Range resolution: $c\Delta\tau/2 = 1.5$ m. Velocity resolution: $c\Delta\nu/(2 f_c) \approx 4.2$ m/s $\approx 15$ km/h.

This OTFS-ISAC configuration covers $\pm 96$ m unambiguous range with 1.5 m resolution, and $\pm 134$ m/s unambiguous velocity with 4 m/s resolution — matching or exceeding typical automotive radar specifications. The "waveform budget" $M/(2BT_{\text{sym}}) = 64/(2\cdot 100\cdot 10) = 0.032$ indicates the fraction of delay-Doppler area used per frame. $\blacksquare$

$L = \log_2\det(\mathbf{I} + \tfrac{1}{\sigma^2}\mathbf{H}\mathbf{R}_x\mathbf{H}^H) - \lambda(\text{tr}(\mathbf{R}_x) - P_t) - \mu(\text{tr}(\mathbf{J}^{-1}(\mathbf{R}_x)) - D)$.

$\partial L/\partial\mathbf{R}_x = \tfrac{1}{\ln 2}\tfrac{\mathbf{H}^H\mathbf{H}}{\sigma^2} (\mathbf{I} + \tfrac{\mathbf{H}\mathbf{R}_x\mathbf{H}^H}{\sigma^2})^{-1} + \mu \mathcal{G}(\mathbf{R}_x) - \lambda \mathbf{I} = 0$, where $\mathcal{G}(\mathbf{R}_x) = -\partial\text{tr}(\mathbf{J}^{-1})/\partial\mathbf{R}_x$ is PSD for $\mathbf{R}_x \succeq 0$.

Rearranging, the optimal covariance aligns with the eigenbasis of the augmented matrix $\mathbf{M} = \mathbf{H}^H\mathbf{H} + \mu\sigma^2\mathcal{G}$. On each eigenvalue $m_i$ of $\mathbf{M}$, the optimal power is $p_i = (1/\lambda - \sigma^2/m_i)^+$ — standard water-filling with a modified channel strength.

$\mu$ is the rate at which one bit of communication can be exchanged for one unit of sensing-MSE budget. At $\mu = 0$: pure water-filling on $\mathbf{H}^H\mathbf{H}$ (Shannon-optimal). As $\mu$ increases, eigenmass shifts from purely communication-rich eigendirections toward sensing-rich directions — the "dither" of the deterministic-random tradeoff. $\blacksquare$

$\nu = 2 \cdot (200/3.6) \cdot 77\times 10^9 / (3\times 10^8) \approx 28.5$ kHz.

For $\Delta f = 240$ kHz, $\nu/\Delta f \approx 0.12$. ICI scales as $(\nu/\Delta f)^2 \approx 1.4\%$ of signal power — 18 dB below signal. Tolerable for demodulation, but degrades sensing by smearing the range-Doppler peak.

In OTFS, 28.5 kHz is just a data-grid index in the Doppler dimension. No ICI, no Doppler smearing — the target appears as a clean shifted peak. Ranking: OTFS $\approx$ FMCW $>$ OFDM.

FMCW handles Doppler via the range-Doppler FFT step without any ICI issue, but cannot carry a comm payload. The sensing performance is effectively tied with OTFS; the difference is in the comm half of the ISAC system.

OTFS wins on the joint ISAC metric: full comm rate plus OFDM-beating sensing robustness. FMCW ties on sensing but loses on comm. OFDM is the legacy-compatible fallback. $\blacksquare$

$R(\epsilon) = \log_2\det(\mathbf{I} + \tfrac{1}{\sigma^2}\mathbf{H}\mathbf{R}_x(\epsilon)\mathbf{H}^H)$ with $\mathbf{R}_x$ linear in $\epsilon$. By concavity of $\log\det$ in its argument, $R(\epsilon)$ is concave in $\epsilon$.

The CRB $\text{tr}(\mathbf{J}^{-1}(\mathbf{R}_x))$ is convex in $\mathbf{R}_x$ (matrix fractional programming), hence convex in $\epsilon$ as a composition with an affine map.

Varying $\epsilon \in [0,1]$ traces a curve in the (CRB, Rate) plane from $(\text{CRB}_c, R_c)$ to $(\text{CRB}_s, R_s)$. This curve lies below or on the true Pareto boundary $\mathcal{C}(D)$ because only a one-dimensional family of covariances is explored, not the full feasible set.

Linear dithering of a comm-only and a sensing-only beam is a simple, computable heuristic, but it is suboptimal whenever the fully-optimal $\mathbf{R}_x^\star$ does not lie on the segment $[\mathbf{R}_c, \mathbf{R}_s]$. The gap motivates the full SDP solution of Section 24.3. $\blacksquare$

16 APs on a $4\times 4$ grid over 500 m. AP spacing $\approx$ 167 m. Distances from the center range from ~118 m (closest 4 APs) to ~353 m (corner APs).

At $r = 118$ m, $f_c = 28$ GHz, $\sigma = 10$ dBsm, $N_t = 32$, $P_t = 0.2$ W, $B = 100$ MHz, NF 7 dB: $\rho_{\text{near}} \approx N_t^{2} P_t\sigma\lambda^{2}/((4\pi)^3 r^4 \sigma^2) \approx 1024 \cdot 0.2 \cdot 10 \cdot 1.14\times 10^{-4}/(1984 \cdot 1.94\times 10^8 \cdot 2\times 10^{-12}) \approx 0.23 / 0.77 \approx 0.3 \approx -5.2$ dB per pulse. With 50 dB pulse compression gain, $\approx 45$ dB integrated.

Take the four closest APs dominant (~45 dB each). Noncoherent diversity gives roughly $P_d \approx 1 - (1 - P_d^{(1)})^4$. With $P_d^{(1)} \approx 0.99$ at 45 dB, $P_d(4) \approx 1 - 10^{-8} \approx 1$.

At 10 dBsm RCS and 100+ m range, the dominant near-APs already achieve near-certain detection per pulse; the diversity from the far APs is insurance against fading. Cell-free ISAC delivers reliable detection throughout the area, with the limit set by the coverage of the farthest target cells rather than by the center geometry. $\blacksquare$

Under an idealized doubly-flat channel (no Doppler, no delay spread beyond the CP), the two waveforms are unitary equivalent and deliver identical Shannon capacity. On the sensing side, too, the ambiguity function is the autocorrelation of the transmitted energy distribution — the same under unitary transformation. So at zero Doppler, OFDM and OTFS have the same capacity-distortion function.

At nonzero Doppler: OFDM suffers ICI, which reduces the effective channel capacity AND biases the Fisher information matrix on the radar side (the target response blurs into multiple subcarriers). OTFS handles Doppler as a data-grid shift with no ICI, so its effective channel retains full rank and its Fisher information is unaffected.

At vehicular Doppler, OTFS has a strictly larger capacity-distortion region than OFDM — both higher rate and lower sensing MSE for any fixed operating point. The student's argument fails because the "same resource grid" assumption masks the different effective channels under Doppler. $\blacksquare$

A 6G ISAC BS consists of $L = 16$ hybrid-beamforming APs ($N_t = 64$ each, $N_{\text{RF}} = 4$) connected by a fiber fronthaul to a CPU. Each AP transmits an OTFS waveform designed jointly by the CPU using the SDR of Section 24.3 with per-user SINR constraints and a desired sensing beampattern over the monitored region.

Per OTFS frame: CPU solves the joint ISAC SDR, broadcasts precoder coefficients to APs, APs transmit the beamformed OTFS frame. Receive side: each AP match-filters echoes to the reference waveform, compresses to fronthaul budget, forwards to CPU. CPU fuses echoes coherently (or noncoherently at mmWave) for multistatic target detection.

1. Joint resource allocation — comm scheduling and sensing dwell budget over the same space-time-frequency grid. Neither the comm literature (Ch. 4, 12) nor the radar literature handle this fully; it is the central open question.
2. Cell-free OTFS synchronization — coherent fusion across APs at mmWave requires sub-picosecond sync. Ch. 13 gives the comm-side requirements; Ch. 24 adds the sensing-side.
3. Fronthaul compression under joint comm+sense loads — Ch. 14 handles comm-only compression; extending to joint load is an active research area (Liu–Wan–Caire).
4. AI/ML integration — joint comm+sense neural precoders, learned beampattern synthesis, and end-to-end training. This is the subject of Chapter 25.
5. Full-duplex or bistatic? — Engineering Note 24.3 argues bistatic cell-free ISAC sidesteps the self-interference problem, but monostatic single-BS ISAC is simpler to deploy. The tradeoff is deployment-specific.

Ch. 1–6: single-BS massive MIMO foundations. Ch. 11–15: cell-free architecture, distributed processing. Ch. 17: near-field considerations for dense deployments. Ch. 20: hybrid beamforming for cost-efficient large arrays. Ch. 24: the ISAC integration itself. Ch. 25: the ML layer that learns the joint resource allocation. Ch. 22: the 3GPP NR context that ensures backward compatibility. $\blacksquare$

1. Coexisting: Wi-Fi and automotive radar share the 5 GHz band but treat each other as interference.
2. Cooperating: Vehicle-to-Everything (V2X) systems exchange sensing results to improve comm link adaptation.
3. Integrated: 5G NR downlink signal itself is used as a radar probe — the comm waveform and sensing waveform are the same. Massive MIMO ISAC is the canonical example. $\blacksquare$

The FIM is $\mathbf{J}(\mathbf{R}_x) = \frac{1}{\sigma^2} \sum_i \mathbf{A}_i\mathbf{R}_x\mathbf{A}_i^H$, linear in $\mathbf{R}_x$. Scaling $\mathbf{R}_x \mapsto c\mathbf{R}_x$ scales $\mathbf{J} \mapsto c\,\mathbf{J}$ and thus $\mathbf{J}^{-1} \mapsto c^{-1}\mathbf{J}^{-1}$.

$\text{tr}(\mathbf{J}^{-1}) \mapsto c^{-1}\text{tr}(\mathbf{J}^{-1})$, which is non-increasing in $c \geq 1$. Hence CRB decreases as power increases, as expected.

Without a power constraint, the optimization would drive $P_t \to \infty$ and both the rate and the CRB improve without bound — the problem would be trivial. The power constraint turns it into a meaningful tradeoff. Operationally, the constraint models hardware power-amplifier limits, regulatory emission masks, and energy budgets. $\blacksquare$

Transmit beam from $N_t$ elements gives $N_t$-fold power concentration at the target. The returned echo is coherently combined across $N_t$ receive elements, providing a second $N_t$-fold SNR gain. Total: $N_t^{2}$.

The transmit beamforming gain is still $N_t$ (at the transmitting AP). The single receiving AP, however, has only $N_t$ elements and can coherently combine once: receive gain $N_t$. Product: $N_t^{2}$ — same as monostatic, provided both APs are full massive-MIMO arrays.

With only a single-element receive AP, the receive gain is 1, so the total is only $N_t$. Loss factor $N_t$.

With $L$ receive APs, each contributing coherent gain $N_t$, the total gain is $N_t^{2} \cdot L$ (if fully coherent across APs) or $N_t^{2} \cdot (1 + (L-1)\epsilon)$ with incoherent fusion. Massive MIMO per-AP plus $L$-AP fusion gives a gain that exceeds monostatic at modest $L$. This is the quantitative reason cell-free is an ISAC architecture of choice. $\blacksquare$

$\sigma_r^2 = (c/2)^2/(8\pi^2 \rho \beta_{\text{rms}}^2)$. For $B = 100$ MHz, $\beta_{\text{rms}}^2 \approx 8.3\times 10^{14}$, so $\sigma_r^2 \approx 0.34/\rho$ (meters squared).

Want $\sigma_r = 0.01$ m $\Rightarrow \sigma_r^2 = 10^{-4}$. $\rho = 0.34/10^{-4} = 3400 \approx 35.3$ dB.

From the radar equation, at $r = 100$ m, $\sigma = 10$ dBsm, $N_t = 64$, the integrated SNR at 30 dBm $P_t$ with 100 MHz bandwidth plus pulse compression is $\sim 70$ dB — well above the 35 dB needed. Sub-centimeter range accuracy is easily achievable.

Want $\sigma_r = 10^{-3}$ m: $\sigma_r^2 = 10^{-6}$, requiring $\rho = 3.4\times 10^5 \approx 55.3$ dB. Still well within the 70 dB budget. Sub-mm range accuracy is feasible with massive MIMO plus pulse compression, at least for line-of-sight targets in low-clutter environments. $\blacksquare$

For i.i.d. Gaussian $\mathbf{h}_k$ with unit norm expected, $\mathbb{E}[\mathbf{P}_{\perp}] = (1 - K/N_t)\mathbf{I}$ in expectation, because the orthogonal complement has effective dimension $N_t - K$.

$\mathbb{E}[\mathbf{a}^H(\theta)\mathbf{R}_s\mathbf{a}(\theta)] \approx (1 - K/N_t)^2 \mathbf{a}^H(\theta)\mathbf{R}_0\mathbf{a}(\theta)$ at the target angle $\theta = \theta_t$, $= (1 - K/N_t)^2 P_t N_t$.

Each user constraint costs a factor $(1 - 1/N_t)$ in sensing beamforming gain, squared because the projector operates on both sides. For $N_t = 64$ and $K = 8$, the loss is $(1 - 1/8)^2 \approx 0.77$, about 1.1 dB below the unconstrained case. This quantifies the "free lunch" claim of the remark on null-forming in Section 24.3. $\blacksquare$