Exercises

$\lambda = 30$ mm, $D = 64 \times 30/2 = 960$ mm $= 0.96$ m.

$$d_R = \frac{2 \times 0.96^2}{0.030} = \frac{1.843}{0.030} = 61.4\;\text{m}$$

At $d = 50$ m: $50 < 61.4$, so the user is in the near field.

$\lambda = 10.7$ mm, $D = 256 \times 10.7/2 = 1370$ mm $= 1.37$ m.

$$d_R = \frac{2 \times 1.37^2}{0.0107} = \frac{3.754}{0.0107} = 350.8\;\text{m}$$

At $d = 50$ m: $50 \ll 350.8$, so the user is deep in the near field.

$\lambda = 2.14$ mm, $D = 512 \times 2.14/2 = 548$ mm $= 0.548$ m.

$$d_R = \frac{2 \times 0.548^2}{0.00214} = \frac{0.600}{0.00214} = 280.5\;\text{m}$$

At $d = 50$ m: $50 < 280.5$, so the user is in the near field.

In all three cases, $d = 50$ m falls within the near field — confirming that near-field operation is pervasive at 6G array sizes. $\blacksquare$

Total cancellation: $30 + 45 + 35 = 110$ dB.

Residual SI: $23 - 110 = -87$ dBm.

$N_0 = -174 + 10\log_{10}(20 \times 10^6) + 5 = -174 + 73.0 + 5 = -96.0\;\text{dBm}$$

$\text{SI/Noise} = -87 - (-96) = 9\;\text{dB}$$

The residual SI is 9 dB above the thermal noise floor.

To bring residual SI to the noise floor, we need 9 dB more cancellation. If this is achieved in the analog domain:

Required analog cancellation: $45 + 9 = 54$ dB.

Alternatively, the digital domain could be improved from 35 to 44 dB, or a combination of both. $\blacksquare$

$\text{MSE} = \frac{1}{K^2 \cdot \text{SNR}} = \frac{1}{K^2 \times 100} = \frac{10^{-2}}{K^2}$$

$K$	MSE
10	$10^{-2}/100 = 10^{-4}$
50	$10^{-2}/2500 = 4 \times 10^{-6}$
100	$10^{-2}/10^4 = 10^{-6}$
500	$10^{-2}/2.5 \times 10^5 = 4 \times 10^{-8}$

The MSE decreases rapidly with $K$ — adding more devices actually improves the estimation quality (noise averaging).

$\frac{10^{-2}}{K^2} < 10^{-4} \implies K^2 > 100 \implies K > 10$$At least 11 devices are needed for MSE$< 10^{-4}$ at 20 dB SNR.

AirComp: 1 time slot (all devices transmit simultaneously).

TDMA: 100 time slots (one per device).

AirComp achieves a $100\times$ latency reduction — the fundamental advantage for federated learning and distributed sensing. $\blacksquare$

$\Delta R = \frac{3 \times 10^8}{2 \times 200 \times 10^6} = 0.75\;\text{m}$$

$R_{\text{max}} = \frac{3 \times 10^8}{2 \times 60 \times 10^3} = 2500\;\text{m}$$

$T_f = 64 \times 17.84 \times 10^{-6} = 1.142$ ms.

$$\Delta v = \frac{3 \times 10^8}{2 \times 1.142 \times 10^{-3} \times 10 \times 10^9} = 13.1\;\text{m/s} = 47.3\;\text{km/h}$$

Pedestrians at 5 km/h ($= 1.39$ m/s): below $\Delta v = 13.1$ m/s, so pedestrians cannot be resolved in velocity. They can still be detected (they appear in the zero-Doppler bin) but their speed cannot be measured.

Vehicles at 50 km/h ($= 13.9$ m/s): just above $\Delta v$, so vehicles can be resolved.

To detect pedestrians, the sensing frame must be longer: $N_{\text{sym}} \geq 64 \times 13.1/1.39 \approx 603$ symbols, or about 10.8 ms. $\blacksquare$

$D = 128 \times 10.7/2 = 684.8$ mm $= 0.685$ m.

$$d_R = \frac{2 \times 0.685^2}{0.0107} = \frac{0.938}{0.0107} = 87.7\;\text{m}$$

Since $r_0 = 15$ m $< d_R = 87.7$ m, the user is in the near field.

Position of element $n$: $p_n = (n - 64.5) \times 5.35$ mm.

Distance to focal point at broadside: $d_n = \sqrt{r_0^2 + p_n^2} \approx r_0 + p_n^2/(2r_0)$.

Steering vector entry: $$a_n(r_0, 0) = \frac{1}{\sqrt{N}} \exp\!\left(-j\frac{2\pi}{\lambda}\left(r_0 + \frac{p_n^2}{2r_0}\right)\right)$$

The quadratic phase term $p_n^2/(2r_0)$ is the near-field correction compared to the far-field (constant phase at broadside).

$\Delta r = \frac{2\lambda r_0^2}{D^2} = \frac{2 \times 0.0107 \times 15^2}{0.685^2} = \frac{4.815}{0.469} = 10.3\;\text{m}$$

Range separation: $|r_1 - r_2| = |12 - 18| = 6$ m.

Since $6 < \Delta r = 10.3$ m, the two users cannot be cleanly separated by range-domain beamfocusing alone.

To resolve them, we would need either a larger array (doubling $D$ reduces $\Delta r$ by $4\times$ to 2.6 m) or a higher frequency (doubling $f$ halves $\lambda$ and thus halves $\Delta r$). With $N = 256$ at the same frequency, $\Delta r = 2.6$ m $< 6$ m, and the users can be separated. $\blacksquare$

$N_0 = -174 + 10\log_{10}(10^8) + 6 = -174 + 80 + 6 = -88\;\text{dBm}$$

DL SNR: $24 - 90 - (-88) = 22$ dB.

UL SNR: $24 - 95 - (-88) = 17$ dB.

HD sum rate (each direction uses $B/2 = 50$ MHz):

$$R_{\text{HD}} = 50 \times \log_2(1 + 10^{2.2}) + 50 \times \log_2(1 + 10^{1.7})$$ $$= 50 \times 7.37 + 50 \times 5.71 = 368.5 + 285.5 = 654.0\;\text{Mbps}$$

Residual SI: $P_{\text{SI,res}} = 24 - 110 = -86$ dBm.

DL SINR: $P_{\text{DL,rx}} = 24 - 90 = -66$ dBm.

Interference + noise: $-86$ dBm (SI) and $-88$ dBm (thermal).

Total interference + noise (linear sum): $10^{-8.6} + 10^{-8.8} = 2.512 \times 10^{-9} + 1.585 \times 10^{-9} = 4.097 \times 10^{-9}$ $= -83.9$ dBm.

DL SINR: $-66 - (-83.9) = 17.9$ dB.

UL SNR: unchanged at 17 dB (UL Rx has no SI in this model).

FD uses full $B = 100$ MHz in both directions:

$$R_{\text{FD}} = 100 \times \log_2(1 + 10^{1.79}) + 100 \times \log_2(1 + 10^{1.7})$$ $$= 100 \times 6.01 + 100 \times 5.71 = 601.0 + 571.0 = 1172.0\;\text{Mbps}$$

$$\rho = \frac{1172}{654} = 1.79$$

The FD system achieves 79% more throughput than HD — close to the theoretical $2\times$ limit, because the 110 dB SI cancellation keeps residual SI only slightly above the noise floor. $\blacksquare$

$\mathbb{E}[\min_k |h_k|^2] = 1/50 = 0.02$.

$p_0 = P \cdot 0.02 = 100 \times 0.02 = 2$ mW $= 3$ dBm (where $P = 100$ mW).

Expected MSE: $$\text{MSE} = \frac{\sigma^2}{K^2 p_0} = \frac{10^{-9} \times 10^{-3}}{50^2 \times 2 \times 10^{-3}} = \frac{10^{-12}}{5} = 2 \times 10^{-13}$$

Wait — let us keep consistent units. $\sigma^2 = 10^{-90/10}$ mW $= 10^{-9}$ mW $= 10^{-12}$ W. $p_0 = 10^{3/10}$ mW $= 2 \times 10^{-3}$ W.

$$\text{MSE} = \frac{10^{-12}}{2500 \times 2 \times 10^{-3}} = \frac{10^{-12}}{5} = 2 \times 10^{-13}$$

$\Pr(|h_k|^2 < 0.05) = 1 - e^{-0.05} \approx 0.0488$.

Expected number excluded: $50 \times 0.0488 \approx 2.4$ devices.

$K_a \approx 47.6$. The new minimum is over $K_a$ devices conditioned on $|h_k|^2 \geq 0.05$. The conditional minimum has expected value approximately $\gamma_{\text{th}} + 1/K_a = 0.05 + 0.021 = 0.071$.

New $p_0 = P \times 0.071 = 7.1$ mW $= 8.5$ dBm.

Truncated MSE: $$\text{MSE}_{\text{trunc}} = \frac{10^{-12}}{47.6^2 \times 7.1 \times 10^{-3}} = \frac{10^{-12}}{16.1} = 6.2 \times 10^{-14}$$

Improvement: $\text{MSE}_{\text{full}} / \text{MSE}_{\text{trunc}} = 2 \times 10^{-13} / 6.2 \times 10^{-14} \approx 3.2\times$ ($5.1$ dB).

Truncation improves the noise-limited MSE by $\sim$3$\times$ at the cost of a small bias from excluding $\sim$2.4 devices (4.8% of the population). $\blacksquare$

The radial velocity component toward the UE at elevation 45°: $v_r = v_{\text{sat}} \cos(\theta_{\text{el}}) = 7600 \times \cos(45°) = 7600 \times 0.707 = 5373$ m/s.

$$f_d = \frac{v_r}{c} f_0 = \frac{5373}{3 \times 10^8} \times 2 \times 10^9 = 35.8\;\text{kHz}$$

$\frac{f_d}{\Delta f} = \frac{35.8}{15} = 2.39$$The Doppler shift is$2.4\times$ the subcarrier spacing! Without pre-compensation, OFDM demodulation would fail due to massive inter-carrier interference.

$d_{\text{slant}} = \frac{h}{\sin(45°)} = \frac{550}{0.707} = 778\;\text{km}KATEXPLACEHOLDER0END\tau = \frac{778 \times 10^3}{3 \times 10^8} = 2.59\;\text{ms}$$Round-trip delay:$2\tau = 5.18$ ms.

Doppler pre-compensation: The satellite knows its own ephemeris (orbital position and velocity) precisely. It pre-shifts the downlink carrier by $-f_d$ so the UE receives the signal at the nominal carrier frequency. A small residual Doppler (due to UE position uncertainty) remains, but it is within the SCS tolerance.

Timing advance: The UE must transmit its uplink signal $2\tau \approx 5.2$ ms before the scheduled Rx time at the satellite. This is communicated via the timing advance command, which in NTN can be partially derived from GNSS position and satellite ephemeris data (autonomous TA). $\blacksquare$

The cluster subtends $\Delta\theta = 5°$ from the array centre. At the array, this corresponds to a spatial segment of approximately:

$D_{\text{vis}} \approx d_c \times \tan(\Delta\theta) \approx 30 \times \tan(5°) = 30 \times 0.0875 = 2.62$ m.

Number of elements in visibility region: $N_{\text{vis}} = D_{\text{vis}} / (\lambda/2) = 2.62 / 0.0429 = 61$ elements.

So only about $61/256 = 24\%$ of the array sees this cluster.

The 8 clusters span $120° / 8 = 15°$ per cluster. Each cluster is visible from $\sim$61 elements ($\sim$24% of the array).

Each element sees clusters within its angular cone. For a centrally located element, approximately $61/256 \times 8 \approx 1.9$ clusters are visible.

More carefully: each element has an angular view that covers several cluster positions. A typical element at the array centre sees all clusters within $\pm 60°$, so $\sim$8 clusters. But an element near the array edge has a shifted angular perspective and may see only $\sim$4--6 clusters, while losing visibility of those behind obstacles at extreme angles.

The key point is that edge elements see fewer clusters than centre elements, breaking spatial stationarity.

If ZF precoding is computed using the full $256 \times K$ channel matrix, it implicitly assumes every element contributes to every user's signal. But if element $n$ has zero channel gain to a cluster relevant for user $k$ (because the cluster is outside element $n$'s visibility region), the ZF precoder may assign nonzero weight to element $n$ for user $k$ — wasting power and potentially creating unintended interference.

Sub-array-based processing partitions the array into overlapping segments, estimates per-segment channels, and applies precoding only within each user's effective visibility region. This approach sacrifices some array gain but avoids the mismatch inherent in full-array processing with non-stationary channels. $\blacksquare$

Expanding the exponent: $$-\frac{\pi}{\lambda d_0}(p_n^r - p_m^t)^2 = -\frac{\pi(p_n^r)^2}{\lambda d_0} + \frac{2\pi p_n^r p_m^t}{\lambda d_0} - \frac{\pi(p_m^t)^2}{\lambda d_0}$$

The first and third terms are diagonal phases (element-dependent but independent of the other array), and the cross-term $2\pi p_n^r p_m^t/(\lambda d_0)$ gives the matrix its rank.

The cross-term has the form of a DFT matrix with "frequencies" $p_m^t/(\lambda d_0)$. The effective bandwidth of $p_m^t$ is $D_t/(\lambda d_0)$, and the sampling interval of $p_n^r$ is $\lambda/2$. By the Nyquist criterion, the number of resolvable frequencies is:

$$\nu = \frac{D_t}{\lambda d_0} \times \frac{D_r}{1} \times \frac{1}{2/\lambda} + 1 = \frac{D_t D_r}{4\lambda d_0} + 1$$

(The factor of 4 arises from the $\lambda/2$ spacing on both arrays.)

$\lambda = 10.7$ mm, $D_t = D_r = 256 \times 10.7/2 = 1370$ mm $= 1.37$ m.

$$\nu(d_0) = \frac{1.37 \times 1.37}{4 \times 0.0107 \times d_0} + 1 = \frac{1.877}{0.0428 \times d_0} + 1 = \frac{43.85}{d_0} + 1$$

$d_0$ (m)	$\nu$	Effective rank
5	9.77	$\approx 10$ streams
10	5.39	$\approx 5$ streams
20	3.19	$\approx 3$ streams
50	1.88	$\approx 2$ streams

$4 = \frac{43.85}{d_0} + 1 \implies d_0 = \frac{43.85}{3} = 14.6\;\text{m}$$At$d_0 = 14.6$ m, the LOS near-field channel supports 4 streams — comparable to a rich-scattering NLOS channel, but achieved purely through spherical-wave geometry.

Near-field LOS spatial multiplexing is only useful at short range (tens of metres for 256-element arrays at 28 GHz). At typical macro-cell distances ($d_0 > 50$ m), the rank collapses to $\sim$2.

However, for indoor deployments, access points (FR3/sub-THz with $N \geq 512$), and backhaul links, near-field MIMO offers significant multiplexing gain in pure LOS — a fundamentally new capability compared to far-field MIMO. $\blacksquare$

$\mathbf{H}_{\text{SI}} \in \mathbb{C}^{32 \times 32}$ with rank 32.

$\dim(\text{null}(\mathbf{H}_{\text{SI}})) = 32 - 32 = 0$.

The null space is trivial — there is no direction in $\mathbb{C}^{32}$ from which the SI is zero. Complete spatial nulling of a full-rank SI channel is impossible with $M_t = M_r$.

With uniformly distributed singular values in $[0.1, 10]$: $\sigma_i \approx 0.1 + (10 - 0.1)(i-1)/31$ for $i = 1, \ldots, 32$.

The 4 weakest singular values: $\sigma_1 \approx 0.1$, $\sigma_2 \approx 0.42$, $\sigma_3 \approx 0.74$, $\sigma_4 \approx 1.06$.

Residual SI power fraction (using the 4 weakest): $$\frac{\sum_{i=1}^{4}\sigma_i^2}{\sum_{i=1}^{32}\sigma_i^2} = \frac{0.01 + 0.18 + 0.55 + 1.12}{0.01 + 0.18 + \cdots + 100}$$

Total power: $\sum_{i=1}^{32} \sigma_i^2 = \sum_{i=1}^{32} (0.1 + 0.319(i-1))^2$. Using the approximation: mean $\sigma \approx 5.05$, $\sum \sigma_i^2 \approx 32 \times (5.05^2 + \text{var}/12) \approx 1086$.

Residual fraction: $1.86/1086 \approx 0.00171 = -27.7$ dB.

So the spatial suppression provides approximately 28 dB of SI reduction.

With propagation isolation (25 dB) + spatial suppression (28 dB) = 53 dB (rounding to 55 dB as stated).

Required analog + digital: $110 - 55 = 55$ dB.

Configuration	Prop.	Spatial	Analog + Digital needed
Without MIMO	25 dB	0 dB	85 dB
With MIMO spatial	25 dB	28 dB	55 dB

Massive MIMO spatial suppression reduces the analog + digital cancellation requirement by 30 dB. This is a dramatic relaxation: 55 dB of analog + digital cancellation is achievable with modest hardware (e.g., 35 dB analog + 20 dB digital), whereas 85 dB requires state-of-the-art wideband cancellers and nonlinear digital estimation. $\blacksquare$

$\sigma_{\text{eff}}^2 = 1.0 + 100 = 101$.

The AirComp noise ($d \cdot \sigma_{\text{AC}}^2 = 100$) dominates the SGD noise ($\sigma_{\text{SGD}}^2 = 1.0$) by a factor of 100. This means the communication channel is the bottleneck, not the statistical gradient noise.

$\frac{\alpha L \sigma_{\text{eff}}^2}{2\mu} = \frac{0.01 \times L/\mu \times \sigma_{\text{eff}}^2}{2} = \frac{0.01 \times 100 \times 101}{2} = 50.5$$This optimality gap may be unacceptable depending on the scale of$f^*$. For comparison, with perfect communication ($\sigma_{\text{AC}}^2 = 0$): gap$= 0.01 \times 100 \times 1.0 / 2 = 0.5$. AirComp noise increases the gap by$100\times$.

To make $d \cdot \sigma_{\text{AC}}^2 < 0.1$:

$10^6 \cdot \sigma_{\text{AC}}^2 < 0.1 \implies \sigma_{\text{AC}}^2 < 10^{-7}$.

Current: $\sigma_{\text{AC}}^2 = 10^{-4}$. Need $10^3\times$ reduction.

Since $\sigma_{\text{AC}}^2 = 1/(K^2 \cdot \text{SNR})$, at fixed $K = 100$: SNR must increase by $10^3 = 30$ dB, from 20 dB to 50 dB.

This is impractical — 50 dB SNR at the device is unrealistic.

At fixed SNR $= 20$ dB $= 100$ (linear):

$10^6 / (K^2 \times 100) < 0.1 \implies K^2 > 10^6 / 10 = 10^5$ $\implies K > 316$.

So $K \geq 317$ devices (vs current 100) would make AirComp noise negligible. This is a $3.2\times$ increase in the number of participating devices — much more feasible than a 30 dB SNR improvement.

This highlights a remarkable property of AirComp: scaling up the number of devices improves both the statistical quality (more data) and the communication quality (noise averaging) of the gradient estimate. $\blacksquare$

$G(\theta_s) = p_s \cdot |\mathbf{a}^H(\theta_s) \mathbf{f}_s|^2 = p_s \cdot \frac{|\mathbf{a}^H\mathbf{a}|^2}{N_t} = p_s \cdot N_t = 64\, p_s$$With$p_s$in watts,$G(\theta_s) = 64 p_s$ watts EIRP toward the target.

$G_{\text{ZF}} = (64 - 2)/64 = 62/64 = 0.969$ ($-0.14$ dB).

$|h_k|^2 = 10^{-9}$ (linear, from $-90$ dB path loss). $\sigma^2 = 10^{-8.8/1}$ ... let us compute: $\sigma^2 = 10^{-88/10}$ mW $= 10^{-11.8}$ W $= 1.585 \times 10^{-12}$ W.

Wait — $\sigma^2 = -88$ dBm $= 10^{(-88-30)/10}$ W $= 10^{-11.8}$ W.

Per-user rate: $$R_k = 100 \times \log_2\!\left(1 + \frac{p_k \times 10^{-9} \times 0.969}{1.585 \times 10^{-12}}\right)$$ $$= 100 \times \log_2(1 + 611.4 \, p_k)\;\text{MHz} \times \text{bps/Hz}$$

where $p_k$ is in watts.

Sensing SINR: $\gamma_s = G(\theta_s) / (\text{PL}_{\text{linear}} \times \sigma^2)$.

$\text{PL} = 130$ dB $= 10^{13}$ (linear).

$$10 = \frac{64 \, p_s}{10^{13} \times 1.585 \times 10^{-12}} = \frac{64\, p_s}{15.85}$$

$$p_s = \frac{10 \times 15.85}{64} = 2.477\;\text{W}$$

But $P = 1$ W total! The sensing requirement exceeds the power budget.

With $P = 1$ W, maximum achievable sensing SINR: $\gamma_{s,\text{max}} = 64 \times 1 / 15.85 = 4.04 = 6.1$ dB.

So $\gamma_s = 10$ dB is infeasible. Let us lower the target to $\gamma_s = 3$ dB ($= 2$): $p_s = 2 \times 15.85 / 64 = 0.495$ W.

With $p_s = 0.495$ W and $p_1 = p_2 = (1 - 0.495)/2 = 0.253$ W:

$$R_k = 100 \times \log_2(1 + 611.4 \times 0.253) = 100 \times \log_2(155.7) = 100 \times 7.28 = 728\;\text{Mbps}$$

Sum rate: $R_{\text{sum}} = 2 \times 728 = 1456$ Mbps.

Without sensing ($p_s = 0$, $p_k = 0.5$ W): $R_k = 100 \times \log_2(1 + 305.7) = 100 \times 8.26 = 826$ Mbps. $R_{\text{sum}} = 1652$ Mbps.

Communication rate loss due to sensing: $1652 - 1456 = 196$ Mbps (12%).

The ISAC trade-off: allocating $\sim$50% of the power to sensing costs only $\sim$12% of the sum rate (because $\log_2$ is concave), while achieving 3 dB sensing SINR. This favourable trade-off is a key motivation for ISAC — sensing "piggybacks" on communication with modest rate sacrifice. $\blacksquare$

$\eta = \frac{R_{\text{min}}}{B} = \frac{2 \times 10^9}{10^9} = 2.0\;\text{bps/Hz}$$With practical coding ($\sim$80$\eta_{\text{raw}} = 2.5$ bps/Hz.

$\lambda = c/f = 20$ mm. FSPL anchor: $32.4 + 20\log_{10}(15) = 32.4 + 23.5 = 55.9$ dB.

$$PL = 55.9 + 10 \times 1.8 \times \log_{10}(28) = 55.9 + 18 \times 1.447 = 55.9 + 26.0 = 81.9\;\text{dB}$$

Noise floor: $N_0 = -174 + 10\log_{10}(10^9) + 6 = -174 + 90 + 6 = -78$ dBm.

Required per-user SNR: $2^{2.5} - 1 = 4.66 = 6.7$ dB. With 10 dB margin: 16.7 dB.

With $P_t = 24$ dBm (typical indoor AP):

$\text{SNR} = P_t + G_{\text{BF}} - PL - N_0 \geq 16.7$ dB

$24 + G_{\text{BF}} - 81.9 + 78 \geq 16.7$

$G_{\text{BF}} \geq 16.7 - 20.1 = -3.4$ dB.

Even without beamforming gain, the link closes! But we need multi-user capability. With ZF and $K = 8$ users, effective gain per user: $G_{\text{eff}} = 10\log_{10}((N-8)/N) + 10\log_{10}(N)$.

For $N = 64$: $G_{\text{eff}} = 10\log_{10}(56/64) + 18.1 = -0.6 + 18.1 = 17.5$ dBi. SNR $= 24 + 17.5 - 81.9 + 78 = 37.6$ dB. Far exceeds the requirement.

Even $N = 32$ gives: $G_{\text{eff}} = 10\log_{10}(24/32) + 15.1 = -1.25 + 15.1 = 13.85$ dBi. SNR $= 24 + 13.85 - 81.9 + 78 = 33.95$ dB. Still sufficient.

Choose $N = 64$ (an $8 \times 8$ UPA) for comfortable margin and near-field benefits.