Exercises
ex-ch18-01-fractional-bw
beginner.
Bandwidth
GHz (5% fractional).
Narrowband regime
Narrowband holds for signal MHz. Beyond this, Theorem 18.1 losses at band edges start degrading performance. 5G 400 MHz signals at 28 GHz are already at the edge β TTD or per-subcarrier design helps.
ex-ch18-02-ttd-savings
beginner.
Narrowband edge
Edge offset = MHz. ( dB).
TTD
TTD: flat response β ( dB).
Gain
TTD recovers dB at edge, about dB averaged across band.
ex-ch18-03-aging-ped
beginnerDoppler , then .
Doppler
Hz.
Aging
rad β not small. Must use exact Bessel: , . Essentially decorrelated!
Interpretation
Even pedestrian mobility at mmWave is not negligible for 10 ms control-loop latency. Need ms for adequate performance.
ex-ch18-04-signaling-budget
beginner.
Calculation
= 3.07 Mbps.
Feasibility
Within typical 5G PDCCH capacity (50-100 Mbps). At higher refresh ( kHz = typical target), Mbps β getting tight, would want compressed encoding.
ex-ch18-05-capacity-scaling
beginnerSubstitute and solve.
Solve for C_0
β .
Predict C(256)
bits/s/Hz. Adds 4 bits/s/Hz from more elements β the scaling at work.
ex-ch18-06-coherence-cond
Intermediateβ .
Derivation
β .
Examples
Hz: ms. Hz: ms. kHz: β challenging.
Plot
Log-log plot: scales as . The plot is a straight line with slope .
ex-ch18-07-ris-for-v2v
Intermediatefor slow-fade model.
Coherence
Hz. .
RIS feasibility
Need . Standard 1 kHz RIS has ms β infeasible without sub-carrier-latency control.
Alternatives
(i) Use RIS for slow-varying components only (angular spread). (ii) Use C-V2X sidelink (no RIS). (iii) Drop to sub-6 GHz where is much larger (hundreds of ms).
ex-ch18-08-ttd-design
IntermediateMax delay = (aperture Γ sin ΞΈ)/c.
Aperture
mm in one dimension, mm in other. Max aperture = 172 mm.
Delay
ps.
Design
Need ps tunable delay per element β achievable with switched transmission-line segments but costly.
ex-ch18-09-3gpp-timeline
IntermediateAdd 24 months per release; commercial usually 12-18 months after spec.
Timeline
Rel-18: 2024. Rel-19: 2025 (study item for RIS). Rel-20: late 2025/early 2026 (spec complete). Rel-21: 2027.
Commercial
Rel-20 standardization late 2025/early 2026. Vendor implementation: 12 months. First commercial deployments: 2027. Scale: 2028-2029.
ex-ch18-10-ris-in-6g-vision
IntermediatePillars: peak rate, connection density, energy efficiency, latency, AI-native, integrated sensing.
Addressed
- Peak rate: RIS aperture β higher capacity (yes).
- Connection density: RIS coverage fill β more simultaneous connections (yes).
- Energy efficiency: passive RIS β lower power per dB gain (yes).
- AI-native: RIS control is ML-driven (yes).
- Integrated sensing: RIS-ISAC (yes, Chapter 13).
Not addressed
- Latency: RIS control adds latency (only partially helpful). RIS is central to 5 of the 6 pillars β a principal 6G technology.
ex-ch18-11-wideband-loss-formula
AdvancedUse independence of , and for uniform .
Expansion
.
Expectation
where . Since has a triangular distribution on (sum of two i.i.d. Uniform): .
Result
. Agrees with Theorem 18.1.
ex-ch18-12-dof-analysis
AdvancedCapacity = ; SNR scales as .
SNR scaling
Coherent combining at the RIS adds signals in amplitude β power gain. At high SNR: .
Two sources
One comes from the RIS-side coherent combining; the other comes from the BS-side transmit beamforming (which scales as of effective antennas if the BS has antennas matched to RIS rank).
Additive
In log-domain, multiplication becomes addition: SNR scales multiplicatively with the two factors, capacity scales additively. Same reason MIMO capacity scales with added to SNR.
ex-ch18-13-imperfect-csi
AdvancedThe estimation error acts as additional noise; use achievable rate formula with effective SNR.
Effective SNR
Signal: . Estimation-error noise: . Effective SNR: .
Rate bound
. Note the noise term grows with power β causing a ceiling effect similar to aging.
Open question
The TIGHT capacity (upper and lower bounds matching) under arbitrary distribution is still open for RIS channels.
ex-ch18-14-mimo-ris-dof
AdvancedThe rank of the cascaded channel is at most .
Effective rank
Cascade: . DoF = rank of effective channel = .
Practical limit
If and : DoF = 16. The RIS provides aperture gain but does NOT add spatial multiplexing beyond the minimum of .
Open work
Tighter analysis under near-field, correlation, and finite- blocklength regimes is ongoing research.
ex-ch18-15-book-reflection
AdvancedNo single right answer; judge by clarity and defense.
Example answer
(1) Product path loss drives placement (Ch 1, 17): the gain is real, but only if the RIS is close to an endpoint. This is counterintuitive from point-to-point thinking and shapes deployment economics.
(2) Joint active-passive beamforming is non-convex but tractable (Ch 5, 6): the unit-modulus constraint on looks intimidating, but AO + SDR + manifold optimization reach provably near-optimal solutions. The algorithms lesson generalizes to many other wireless problems.
(3) The 4-6 dB theory-practice gap is persistent but manageable (Ch 16): hardware calibration, thermal drift, and 2-bit quantization collectively cost 4-6 dB relative to ideal. The algorithmic response is robustness: design algorithms that remain near-optimal under realistic imperfections. This matches the CommIT Group's research thread on robust optimization.