Exercises

(a) From the LTE bandwidth-to-RB mapping table: 5 MHz $\to$ 25 RBs.

(b) $25 \times 12 = 300$ subcarriers (plus DC subcarrier, not used). FFT size: 512.

(c) Per subframe (1 ms = 14 symbols with normal CP): $300 \times 14 = 4{,}200$ REs. $\blacksquare$

(a) Worst-case PAPR for 12 subcarriers: $\text{PAPR}_{\max} = 10\log_{10}(12) = 10.8$ dB. Typical (99.9th percentile) for 16-QAM: $\sim$7--8 dB.

(b) SC-FDMA with DFT precoding: the signal envelope is single-carrier-like. For 16-QAM: $\text{PAPR}_{\text{SC-FDMA}} \approx 3.5$--$4.5$ dB (similar to single-carrier 16-QAM). Gain: $\sim$3--4 dB.

(c) With $P_{1\text{dB}} = 23$ dBm and $\sim$2 dB margin: OFDMA: $P_{\text{avg}} = 23 - 8 - 2 = 13$ dBm. SC-FDMA: $P_{\text{avg}} = 23 - 4.5 - 2 = 16.5$ dBm.

SC-FDMA achieves 3.5 dB higher average power, directly improving uplink coverage and cell-edge performance. $\blacksquare$

(a) PDSCH — Physical Downlink Shared Channel. (b) PDCCH — Physical Downlink Control Channel. (c) PBCH — Physical Broadcast Channel. (d) PUCCH — Physical Uplink Control Channel. (e) SRS — Sounding Reference Signal (not a channel per se, but a reference signal on the uplink). $\blacksquare$

(a) Per RB per slot (84 REs): 1 port: 4 REs $\to 4/84 = 4.8$%. 2 ports: 8 REs $\to 8/84 = 9.5$%. 4 ports: 16 REs (including port 2/3 positions) $\to 16/84 = 19$%.

(b) With massive MIMO (e.g., 32 ports), always-on CRS would require 32 orthogonal pilot patterns $\to$ enormous overhead ($> 50$%). The CRS design fundamentally does not scale beyond 4 ports.

(c) NR uses configurable DM-RS (only when data is transmitted, only on allocated RBs) and configurable CSI-RS (only when CSI measurement is needed). This decouples overhead from antenna count: 32-port CSI-RS is configured infrequently (e.g., every 5 ms), while DM-RS uses only 1--2 ports for demodulation. Total overhead: $\sim$10--15% regardless of antenna count. $\blacksquare$

Each step doubles the SCS, halves the symbol duration, halves the slot duration, and doubles the slots per frame. $\blacksquare$

(a) Usable BW: $\sim$98 MHz. Subcarriers: $98{,}000/30 = 3{,}267$. RBs: $\lfloor 3267/12 \rfloor = 272$ (3GPP specifies 273).

(b) REs per slot: $273 \times 12 \times 14 = 45{,}864$. Data REs: $45{,}864 \times (1-0.18) = 37{,}609$. Bits per RE: $4 \times 6 \times 0.65 = 15.6$ bits. Per slot: $37{,}609 \times 15.6 = 586{,}700$ bits. Per second: $586{,}700 \times 2{,}000 = 1.17$ Gbps.

(c) LTE 20 MHz, 2x2: 100 RBs, 2 layers, same MCS. Data REs per subframe: $100 \times 12 \times 14 \times 0.75 = 12{,}600$. Bits: $12{,}600 \times 2 \times 6 \times 0.65 = 98{,}280$. Throughput: $98{,}280 \times 1{,}000 = 98.3$ Mbps.

NR/LTE ratio: $1{,}170/98.3 = 11.9\times$. $\blacksquare$

(a) BWP 1: $24 \times 12 \times 30 = 8.64$ MHz $\approx 10$ MHz. BWP 2: $273 \times 12 \times 30 = 98.28$ MHz $\approx 100$ MHz.

(b) RF power (ADC, LNA bandwidth) scales roughly linearly: $P_1/P_2 \approx 10/100 = 0.1$ ($90$% reduction on BWP 1).

(c) Average power: $0.8 \times P_1 + 0.2 \times P_2 = 0.8 \times 0.1 P_2 + 0.2 P_2 = 0.28 P_2$.

Overall reduction: $72$%. BWP switching is a major UE power saving mechanism in NR. $\blacksquare$

(a) $120^{\circ}/8 = 15^{\circ}$ per beam.

(b) Worst case: UE arrives just after SSB burst. Wait up to 20 ms (SSB periodicity) + RACH $\approx$ 5 ms. Total: $\sim$25 ms.

(c) $30 \times 1000/3600 = 8.33$ m/s. In 20 ms: $8.33 \times 0.02 = 0.167$ m. Negligible — the beam does not need to be updated every SSB period at walking/driving speeds. $\blacksquare$

(a) Wideband PMI ($i_1$): $\sim$4 bits (beam group from oversampled DFT). Subband PMI ($i_2$): 2 bits $\times$ 13 subbands $= 26$ bits. RI: 2 bits. CQI: 4 bits (WB) + $2 \times 13 = 26$ bits (SB). Total: $\sim$62 bits.

(b) $L = 4$ beam indices: $4 \times \log_2(N_1 O_1 N_2 O_2) \approx 4 \times 6 = 24$ bits. Wideband amplitudes: $4 \times 3 = 12$ bits. Subband coefficients: $4 \times 4 \times 13 = 208$ bits (phase $\times$ beams $\times$ subbands). Total: $\sim$246 bits.

(c) At 10 dB SNR, 32 ports, 4 users (from literature): Type I: $\sim$18 bps/Hz (MU-MIMO). Type II ($L = 4$): $\sim$28 bps/Hz. Ideal CSI: $\sim$32 bps/Hz.

(d) Type I: $18 \times 10^6 / (62 \times 4 \times 100) \approx 726$ throughput bits per feedback bit (100 Hz report rate, 4 users). Type II: $28 \times 10^6 / (246 \times 4 \times 100) \approx 285$.

Type I has better feedback efficiency, but Type II delivers 55% higher absolute throughput. $\blacksquare$

(a) Slot = 0.5 ms = 14 symbols. In 1 ms: 28 symbols.

(b) 256 data bits at rate 1/5: 1280 coded bits. QPSK: 2 bits/RE. REs needed: 640. Per 2-symbol mini-slot: $\sim$12 data REs per RB (2 symbols $\times$ 12 subcarriers, minus DM-RS). RBs: $\lceil 640/12 \rceil = 54$ RBs.

(c) Mini-slot: $2 \times 35.7 = 71.4$ $\mu$s. Processing (N2): $\sim$10 symbols $= 357$ $\mu$s. One round trip: $71.4 + 357 + 71.4 + 357 \approx 857$ $\mu$s. Only 1 retransmission fits within 1 ms.

(d) With chase combining and BLER = $10^{-2}$ per transmission: After 2 transmissions: BLER$_{\text{eff}} \approx 10^{-2} \times 10^{-1.5} \approx 3 \times 10^{-4}$ (accounting for combining gain reducing the second BLER).

Not sufficient for $10^{-5}$. Need BLER $< 10^{-3}$ per transmission (lower MCS) or more repetitions. $\blacksquare$

(a) eMBB: Peak data rate (20 Gbps DL) and spectral efficiency (30 bps/Hz) are the primary KPIs.

(b) URLLC: User-plane latency (1 ms) and reliability ($10^{-5}$ BLER) are the primary KPIs.

(c) mMTC: Connection density ($10^6$ devices/km$^2$) and battery life (10 years) are the primary KPIs. $\blacksquare$

(a) $C = \log_2(1 + 10^{0.5}) = \log_2(4.16) = 2.06$ bits/c.u.

(b) $\text{SNR} = 3.16$. $V = 1 - 1/(1+3.16)^2 = 1 - 0.058 = 0.942$. In bits$^2$: $V_{\text{bits}} = 0.942/(\ln 2)^2 = 1.959$. $Q^{-1}(10^{-5}) = 4.265$.

$R^{\star} = 2.06 - \sqrt{1.959/200} \times 4.265 = 2.06 - 0.099 \times 4.265 = 2.06 - 0.422 = 1.64$ bits/c.u.

(c) $Q^{-1}(10^{-1}) = 1.282$. $R^{\star} = 2.06 - 0.099 \times 1.282 = 2.06 - 0.127 = 1.93$ bits/c.u.

Rate loss from reliability: $1.93 - 1.64 = 0.29$ bits/c.u.

(d) To achieve 1.64 bits/c.u. with $\epsilon \to 0$: $2^{1.64} - 1 = 2.12$. $10\log_{10}(2.12) = 3.3$ dB. Shannon needs $5$ dB for 2.06 bits.

SNR penalty $= 5.0 - 3.3 = 1.7$ dB... more precisely: to achieve 1.93 bits Shannon needs $10\log_{10}(2^{1.93}-1) = 4.6$ dB. So from $\epsilon = 10^{-1}$ to $10^{-5}$, we lose 0.29 bits, equivalent to $\sim 1.8$ dB SNR penalty at this operating point. $\blacksquare$

NR's $20\times$ bandwidth advantage and $7\times$ finer scheduling granularity are the most impactful changes. $\blacksquare$

(a) Option A: 5 separate PDCCH regions, 5 sets of CRS. Overhead: $\sim$20--25% per carrier $\times$ 5. Option B: 1 CORESET, shared DM-RS. Overhead: $\sim$14%. Option C: 2 CORESETs, but NR CA is more efficient than LTE CA. Overhead: $\sim$15%.

(b) Option B has the best spectral efficiency:

• Fewest guard bands (single carrier).
• Lowest control overhead.
• Widest scheduling bandwidth (better frequency diversity and more flexible scheduling).

(c) Options B and C (NR) support mini-slot scheduling ($\sim$71 $\mu$s at $\mu = 1$). Option A (LTE) has minimum TTI of 1 ms. NR is $\sim$14$\times$ faster. $\blacksquare$

(a) $2 \times 6 \times 0.93 \times 100 \times 12 \times 14 / 10^{-3} \times 0.80 = 150$ Mbps.

(b) $4 \times 6 \times 0.93 \times 500 \times 12 \times 14 / 10^{-3} \times 0.75 = 1.12$ Gbps. Driver: CA ($5\times$ BW) + 4$\times$4 MIMO ($2\times$ layers).

(c) $8 \times 8 \times 0.93 \times 264 \times 12 \times 14 / 125 \times 10^{-6} \times 0.86 \approx 12.9$ Gbps. Driver: mmWave BW ($400$ MHz) + 256-QAM + 8 layers.

(d) $4 \times 12.9 \approx 51.6$ Gbps. Driver: FR2 carrier aggregation.

(e)

Each generation delivers $\sim$5--10$\times$ through bandwidth and spatial multiplexing scaling. $\blacksquare$

(a) A neural network can directly map received pilots to optimal MCS, bypassing the CQI $\to$ SINR $\to$ BLER $\to$ MCS chain. Benefits: adapts to non-standard channels (hardware impairments, interference patterns) that the SINR-based model misses. Challenge: training data distribution shift across deployments.

(b) RIS creates controllable reflections to reach NLoS locations without deploying new active base stations. Unlike adding antennas, RIS is passive (no power amplifier, no backhaul), enabling deployment on building facades at low cost. Key use: coverage extension in mmWave NLoS.

(c) The OFDM waveform is already similar to a stepped-frequency radar. By processing the reflected signal from transmitted OFDM symbols, the gNB can estimate range ($\propto 1/\text{BW}$) and velocity ($\propto 1/T_{\text{frame}}$). NR's wide bandwidth at mmWave gives cm-level range resolution.

(d) Two main challenges:

1. Atmospheric absorption: peaks at 183 GHz (water vapor) and 325 GHz (water vapor) create $\sim$100 dB/km attenuation windows, limiting range to $\sim$10--50 m outdoor.
2. Device technology: PA output power at 300 GHz is limited to $\sim$0--5 dBm with current III-V semiconductors, requiring very large arrays ($> 1000$ elements) for useful link budgets. $\blacksquare$

(a) At $\mu = 3$: slot = 0.125 ms, 14 symbols/slot. In 5 ms: $5/0.125 = 40$ slots $= 560$ symbols. Each SSB: 4 symbols. Up to 64 SSBs fit (64 $\times$ 4 = 256 symbols, well within 560).

(b) Per SSB: $4 \times 240 = 960$ REs. Per burst set: $64 \times 960 = 61{,}440$ REs.

(c) Total REs in 5 ms: $264 \times 12 \times 560 = 1{,}774{,}080$ REs. SSB overhead: $61{,}440 / 1{,}774{,}080 = 3.5$%.

With 20 ms periodicity, the overhead per frame is $3.5/4 \approx 0.9$%. SSB overhead is modest even with 64 beams. $\blacksquare$

(a) $\mu = 1$ (30 kHz SCS): slot = 0.5 ms. Mini-slot: 4 symbols $\approx 143$ $\mu$s. Air interface budget: 1 ms $\to$ 2 mini-slot transmissions (1 initial + 1 retransmission for reliability). MCS: QPSK R = 1/3 for $10^{-3}$ BLER per transmission. After 2 transmissions: BLER $\approx 10^{-3} \times 10^{-2} = 10^{-5}$ (with HARQ combining gain).

(b) 2400 data bits per vehicle at R = 1/3: 7200 coded bits. QPSK: 3600 REs. Per 4-symbol mini-slot: $\sim$36 data REs/RB. RBs per vehicle: $\lceil 3600/36 \rceil = 100$ RBs per transmission. With 10 ms period and 0.5 ms slots: each vehicle needs 100 RBs in 1 of every 20 slots. Per slot (average): $50 \times 100 / 20 = 250$ RBs.

(c) Total UL RBs at $\mu = 1$, 100 MHz: 273 per slot. V2X: 250/273 $= 91.6$%.

This is too high. Solutions: (i) Use 16-QAM R = 1/2 reducing RBs by $3\times$ to 83 RBs/slot ($30$%); (ii) Use configured grants to reduce PDCCH overhead; (iii) Allocate a dedicated BWP for V2X.

(d) With 16-QAM R=1/2 V2X: 83 RBs consumed, 190 RBs remain. eMBB with 64-QAM R = 0.65, 4-layer MIMO: $190 \times 12 \times 14 \times 4 \times 6 \times 0.65 \times 0.85 / 0.5 \times 10^{-3} = 685$ Mbps.

The network slice can support both V2X and $\sim$700 Mbps eMBB on shared 100 MHz spectrum. $\blacksquare$