Exercises

$T_{\text{OFDM}}(\mu) = 1/(15\,\text{kHz}\cdot 2^{\mu})$ gives $66.7, 33.3, 16.7, 8.33, 4.17$ µs for $\mu = 0,\ldots,4$.

$T_{\text{slot}}(\mu) = 14 \cdot T_{\text{OFDM}}(\mu) = 1/2^{\mu}$ ms, giving $1.0, 0.5, 0.25, 0.125, 0.0625$ ms. This confirms the rule "slot $= 1\,\text{ms}/2^{\mu}$."

$\rho_{\text{SRS}}^{(k)} = \frac{1}{14 \cdot 4 \cdot 10} \approx 0.179\%$ per UE.

$\rho_{\text{SRS}} = 8 \cdot 0.179\% \approx 1.43\%$ without cyclic shifts. With cyclic-shift multiplexing (factor 4), the 8 UEs fit in 2 comb groups, dropping the total to $\approx 0.36\%$.

The fact that SRS overhead depends on $K$ rather than $N_t$ is the fundamental enabler of TDD massive MIMO. $\blacksquare$

At $\mu = 4$, $T_{\text{OFDM}} = 1/240\,\text{kHz} \approx 4.17\,\mu s$. $T_{\text{SSB}} = 4 T_{\text{OFDM}} \approx 16.7\,\mu s$.

$64 \cdot 16.7\,\mu s \approx 1.07$ ms. The sweep fits easily in the 5-ms SS burst window. $\blacksquare$

$N_1 N_2 O_1 O_2 = 4 \cdot 4 \cdot 4 \cdot 4 = 256$ beams. Beam index payload: $\log_2 256 = 8$ bits.

Adding the 2-bit QPSK co-phasing: $8 + 2 = 10$ bits for the Type I PMI field (excluding RI and CQI overhead). $\blacksquare$

$(1 - 4/84) \approx 95.2\%$ — a 4.8% rate loss from the SRS overhead. All 84 symbols remain in the downlink data budget after allocating 4 of them to uplink SRS.

The 4 NZP-CSI-RS symbols cost the same 4.8%. The feedback UCI (32 bits on PUCCH) costs another $\sim 1\%$ of uplink capacity. Total pre-log $\approx (1 - 0.048)(1 - 0.01) \approx 94.3\%$.

The pre-log difference is about 1% in favor of TDD. The real difference is in the CSI quality, not the pre-log: TDD has a raw channel estimate, while FDD has a quantized codebook approximation. Section 22.3 quantifies the latter loss as 1-3 dB. $\blacksquare$

Beam indices (wideband, $L = 3$): $\log_2 \binom{128}{3} \approx 18$ bits. Per-subband coefficients: $3 \cdot (3+3) = 18$ bits per subband, $18 \cdot 13 = 234$ bits total. Grand total: $\approx 252$ bits.

Same beam indices ($\approx 18$ bits). Per-basis coefficients: $3 \cdot 6 = 18$ bits per basis, $6 \cdot 18 = 108$ bits total. Basis selection from the size-13 DCT grid: $\log_2 \binom{13}{6} \approx 11$ bits. Grand total: $\approx 137$ bits.

$252/137 \approx 1.84\times$ compression from the Rel-17 frequency-DCT compression. This matches the 40-60% compression range reported in Rel-17 literature. $\blacksquare$

$(1+\text{SNR})^2 = 1 + 2\text{SNR}$, which gives $1 + 2\text{SNR} + \text{SNR}^{2} = 1 + 2\text{SNR}$.

$\text{SNR}^{2} = 0 \Rightarrow \text{SNR} = 0$. The two rates are equal only at $\text{SNR} = 0$. For any $\text{SNR} > 0$, NCJT's two-layer rate strictly exceeds CJT's one-layer rate (see the monotonicity proof in Theorem 22.5).

This assumes the UE has at least 2 receive antennas. For a single-antenna UE, NCJT is infeasible and CJT is the only option — the crossover comparison does not apply. $\blacksquare$

$v = 60$ km/h $\approx 16.7$ m/s; $\dot\theta = 16.7/100 = 0.167$ rad/s $\approx 9.6$ deg/s.

$\Delta\theta = 0.167\,\text{rad/s} \cdot 5\,\text{ms} = 0.83$ mrad $\approx 0.048$ deg. This is much smaller than a 5-degree narrow beam.

The beam selected at the start of the SSB sweep is still within 1% of a full beamwidth at the end, so it remains valid for the first data transmission. Beam tracking will update it at the next CSI-RS measurement (P2 refinement), keeping the error bounded. $\blacksquare$

$\tau_p \text{SNR}_{k} = 8 \cdot 10 = 80$. $\gamma_k = 80/81 \approx 0.988$.

$\text{SINR} \approx (64-8) \cdot 10 \cdot 0.988 \approx 553 \Rightarrow 27.4$ dB.

Per-user: $\log_2(1+553) \approx 9.12$ bits/s/Hz. Summed over 8 users: $\approx 73$ bits/s/Hz. Applying the pre-log factor $1 - 8/84 \approx 0.905$: $\approx 66$ bits/s/Hz net.

A 64T64R commercial deployment would measure $\approx 40$-$50$ bits/s/Hz at this load — about 65-75% of the UatF bound, which matches the Section 22.6 field-trial gap. $\blacksquare$

The BS applies a fixed (slowly updated) $64 \times 32$ pre-beamforming matrix $\mathbf{M}$ based on long-term channel statistics. The UE sees only the effective $32$-dimensional channel $\mathbf{M}^H \mathbf{H}$.

The loss depends on how well $\mathbf{M}$ captures the channel covariance principal eigenspace. For one-ring channels with modest angular spread, a DFT-based $\mathbf{M}$ captures 90-95% of the channel energy, implying a residual of 0.3-0.5 dB SINR. For rich-scattering channels, the loss is larger (up to 1-2 dB).

This is exactly the JSDM two-stage precoder of Chapter 7, instantiated in the NR codebook framework. The NR standard allows any $\mathbf{M}$ the operator chooses — including the JSDM choice based on long-term covariance. $\blacksquare$

Per RB per period: $168 \cdot T_{\text{CSI}}$ REs. CSI-RS RE usage: $N_p^{\text{CSI-RS}} \cdot d_{\text{RE}}$ REs, independent of $N_t$.

$\rho_{\text{CSI-RS}} = N_p^{\text{CSI-RS}} d_{\text{RE}} / (168 T_{\text{CSI}})$. The physical antenna count $N_t$ does not appear in the formula.

Operators can deploy $N_t = 256$ physical antennas while paying the same CSI-RS overhead as a $N_t = 32$ cell, as long as the pre-beamforming matrix is good enough to preserve the useful channel energy in 32 ports. This is how commercial Rel-15 cells achieve massive MIMO without breaking the 32-port CSI-RS limit. $\blacksquare$

For $\rho = 0$, post-combining BLER is $p^2$. The gain in "effective SNR" at fixed BLER target is 3 dB (one bit per doubling).

With correlation $\rho$, the joint error probability is approximately $p^{2-\rho}$ at low $p$ (first-order approximation using the Gaussian-copula expansion). For $\rho = 1$ (fully correlated), this reduces to $p$ — no combining gain.

The combining gain in dB is $\text{Gain}(\rho) = 10\log_{10} (p^{2-\rho}/p)^{-1} = (1-\rho)\cdot 10\log_{10}(1/p)$. For $p = 10^{-3}$, independent gain is 30 dB; 50% of that is 15 dB at $\rho = 0.5$.

The 2-TRP repetition gain drops to half at $\rho = 0.5$. Real deployments with geographically separated TRPs typically achieve $\rho \in [0.1, 0.3]$, preserving 70-90% of the independent combining gain. $\blacksquare$

$2\pi f_D \Delta t = 2\pi \cdot 50 \cdot 0.01 = \pi \approx 3.14$. $J_0(3.14) \approx -0.30$, so $\alpha^2 \approx 0.09$. Wait — this is the fully-aged limit; at $\Delta t$ comparable to $1/f_D$, ZF has essentially lost its advantage.

Use a more realistic $\Delta t = 2$ ms instead. Then $2\pi f_D \Delta t = 0.63$ rad and $\alpha \approx J_0(0.63) \approx 0.90$, $\alpha^2 \approx 0.81$.

ZF useful term: $(N_t - K) \text{SNR} \alpha^2 \approx 56 \cdot 10 \cdot 0.81 \approx 454$. ZF residual interference: $K \text{SNR} (1-\alpha^2) \approx 8 \cdot 10 \cdot 0.19 \approx 15.2$.

$\text{SINR}_{\text{aged}} = 454/(1 + 15.2) \approx 28$, or $14.5$ dB. Fresh-CSI SINR was 27.4 dB (Exercise 9). The loss is about 13 dB. At $\Delta t = 10$ ms the loss is catastrophic — which is why CSI ageing matters so much at vehicular speeds. $\blacksquare$

Under the one-ring model with ULA and angular spread $\Delta$, the effective dimension is $L_{\text{eff}} \approx N_t \Delta / \pi$. For $L \leq L_{\text{eff}}$, the captured fraction grows linearly; for $L > L_{\text{eff}}$, it saturates.

$\mathbb{E}[\sin^2 \theta_L] \approx 1 - L/L_{\text{eff}}$ for $L \leq L_{\text{eff}}$. This captures the linear accumulation of principal components.

For $L \gg L_{\text{eff}}$, $\mathbb{E}[\sin^2 \theta_L] \sim L_{\text{eff}}/L \cdot (\text{basis mismatch constant})$. In practice, with $N_t = 32$ and $\Delta = 10^\circ$, $L_{\text{eff}} \approx 1.8$, which is why Type II with $L \in \{2, 3, 4\}$ is sufficient — larger $L$ yields diminishing returns. $\blacksquare$

$\text{SE}_{5} = \log_2(1 + 10^{-0.2}) = \log_2(1.63) \approx 0.71$ bits/s/Hz per UE.

With 20 UEs and full resource allocation, each UE gets $1/20$ of the cell's time-frequency resources on average (proportional fairness with equal-priority UEs).

$\text{Throughput}_5 = (100\,\text{MHz}) \cdot 0.71 \cdot (1/20) \approx 3.55$ Mb/s. This is the 5%-ile (cell-edge) user experience — roughly consistent with reported 2019 commercial measurements.

Even with a 64T64R array, the 5%-ile throughput is dominated by the per-user fair-share bottleneck, not by the beamforming gain. Increasing $N_t$ to 128 would roughly double the cell sum rate but the 5%-ile throughput scales much more slowly. $\blacksquare$

Naive: $\rho = N_p^{\text{CSI-RS}}/(168 T_{\text{CSI}})$, which for 32 ports at $T_{\text{CSI}} = 20$ is $\approx 0.95\%$ — already non-trivial.

Halving density uses code-domain multiplexing: two orthogonal CSI-RS sequences share the same RE pair via a length-2 Walsh code. The effective overhead is $\rho = N_p^{\text{CSI-RS}}/(2 \cdot 168 T_{\text{CSI}})$, which for 32 ports is $\approx 0.48\%$.

Halving density costs 3 dB of SNR on each CSI-RS sequence (because the pilot power is distributed over fewer REs of the same RB), which adds to the channel estimation error. For high port counts where the overhead would otherwise be prohibitive, the SNR loss is preferable. The NR standard provides this trade-off as a scheduler configuration parameter. $\blacksquare$

Allocate SRS in the last symbol of every 10th slot. At $\mu = 1$, this is $1$ SRS symbol per 5 ms, giving CSI refresh every 5 ms — well inside the 40+ ms coherence time at pedestrian speeds.

Allocate a 32-port NZP-CSI-RS every 80 slots ($40$ ms). The UE computes a Type II report and sends it in the next PUCCH instance. This provides a redundant CSI source in case SRS fails (UE in blocked state, uplink outage).

SRS: $1/(14 \cdot 10) \approx 0.71\%$ of REs. Type II CSI-RS: $32/(168 \cdot 80) \approx 0.24\%$ of REs. Total CSI overhead: $\approx 0.95\%$. Data fraction remains $> 99\%$.

With 5-ms CSI refresh, CSI ageing at $v = 3$ km/h pedestrian mobility adds about $0.1$ dB of SINR loss — within the 1-dB budget. For $v = 30$ km/h vehicular, the loss rises to about $0.8$ dB — still within budget. The scheduler can switch to more aggressive SRS cadence (every 5 slots) if it detects higher mobility via measurement reports. $\blacksquare$