| Beamforming Codebook - Compressed Feedback | IEEE 802.11ac
§22.4.7.5
IEEE 802.11ax
§27.3.11 | Compressed BF feedback uses Givens rotation angles to represent the unitary steering matrix V.
For each layer l: V = G(i,j,φ)×G(i,j,ψ)×D
where G(i,j,φ) is Givens rotation matrix.
Angle quantization:
• φ angles: Nφ = {2,4,6} bits per angle (Ng=4,2,1 subcarrier grouping)
• ψ angles: Nψ = {1,2,4} bits per angle
Feedback size = Nc × Nb × (φ_count + ψ_count)
where Nc = number of subcarrier groups, Nb = bits/angle
For 4×4 MIMO, 4 streams, 80 MHz:
• φ angles per column: 3+2+1 = 6 per stream
• ψ angles per column: 3+2+1 = 6 per stream
• Total angles: 4 streams × 12 = 48 angles
• At Nφ=6, Nψ=4: 48×10 = 480 bits per subcarrier group | Beamforming feedback is sent in a Compressed Beamforming Report IE (Action frame Category 21 VHT or Category 30 HE). AP uses V matrix to compute steering matrix W = V* for transmit beamforming.
Feedback overhead: high NSS + narrow grouping = large report. In MU-MIMO with 8 STAs, aggregate BF feedback can consume significant airtime. | 802.11ac: Ng=4 → 52 subcarrier groups per 80 MHz
802.11ax: Ng=4 → 60 subcarrier groups per 80 MHz
Typical report size: 200-3000 bytes
Sounding interval: 100 ms - 5 s (vendor-tunable) |
| MIMO Capacity - Foschini-Telatar | Foschini & Gans 1998
Telatar 1999
IEEE 802.11 TGn eval | C = Σᵢ₌₁ᵣ log₂(1 + λᵢ · SNR/Nₜ) [bits/s/Hz]
where:
• H = Nr×Nt channel matrix (complex Gaussian)
• λᵢ = eigenvalues of HH† (H times H conjugate transpose)
• r = rank(H) = min(Nr, Nt) in full-rank channel
• SNR = total transmit SNR
• Nₜ = number of transmit antennas
Waterfilling interpretation:
Optimal power allocation: Pᵢ = (µ - N₀/λᵢ)⁺
where µ is the water level (chosen to satisfy total power constraint)
Equal power (no CSI at TX): Pᵢ = SNR/Nₜ for all i
= sub-optimal but standard for Wi-Fi (STA doesn't know channel at TX by default) | MIMO multiplexing gain: r independent streams
MIMO diversity gain: Nr × Nt (maximum)
Tradeoff: D + M ≤ Nr × Nt (diversity-multiplexing tradeoff)
Wi-Fi uses spatial multiplexing (equal power, no waterfilling from TX side). CSI is at RX side (STA) via BF feedback.
Correlated channel: rank < min(Nr,Nt) → capacity loss.
LOS environment (K>0): dominant eigenvalue, others small → MIMO gain collapses. | 2×2 MIMO at SNR=20dB:
Independent: C ≈ 2×log₂(1+10) = 6.9 bps/Hz
Correlated (ρ=0.9): C ≈ 5.2 bps/Hz (25% loss)
4×4 MIMO at SNR=20dB:
Full rank: C ≈ 4×3.46 = 13.8 bps/Hz
Rank-2 (corridor LOS): C ≈ 6.9 bps/Hz |
| Cyclic Prefix - ISI/ICI Analysis | IEEE 802.11-2020
§17.3.2
Proakis Ch.8 | ISI condition: TCP ≥ τmax (CP duration ≥ maximum delay spread)
• If violated: delayed copies of previous symbol arrive during current symbol → ISI
OFDM orthogonality: maintained if channel is constant during symbol duration AND CP > τmax
ICI from Doppler: ICI ≈ (fD/Δf)² × (-1/12)
where fD = Doppler spread, Δf = subcarrier spacing
CP overhead efficiency:
η = Tdata / (Tdata + TCP)
Per standard:
• 802.11a/n/ac: η = 3.2/(3.2+0.8) = 80% (LGI)
• 802.11a/n/ac: η = 3.2/(3.2+0.4) = 88.9% (SGI)
• 802.11ax: η = 12.8/(12.8+0.8) = 94.1% (0.8µs GI)
• 802.11ax: η = 12.8/(12.8+1.6) = 88.9% (1.6µs GI)
• 802.11ax: η = 12.8/(12.8+3.2) = 80.0% (3.2µs GI) | CP selection for 802.11ax deployment:
• Indoor office (στ<100ns): 0.8µs GI → η=94.1%
• Large venue (στ<200ns): 1.6µs GI → η=88.9%
• Stadium/outdoor (στ<500ns): 3.2µs GI → η=80.0%
SGI risk: if actual delay spread > 400ns and SGI (0.4µs) used → ISI → BER degradation → automatic fallback to LGI.
Trade-off: longer CP = better multipath robustness but lower throughput efficiency. | Environment → στ → recommended CP:
Home: 30-50ns → 0.8µs ✓
Office: 50-100ns → 0.8µs ✓
Warehouse: 100-200ns → 1.6µs ✓
Stadium: 200-500ns → 3.2µs ✓
Outdoor >500m: >500ns → 3.2µs (max in ax) |
| OFDM Pilot Subcarriers & Channel Estimation | IEEE 802.11-2020
§17.3.5.9 (OFDM pilots)
§19.3.9 (HT LTF)
§27.3.11 (HE LTF) | Pilot subcarrier positions (20 MHz):
• 802.11a/g: ±7, ±21 (4 pilots)
• 802.11n HT: ±7, ±21, ±21, ±28 → varies by BW
• 802.11ax HE (20 MHz): ±13, ±50, ±75, ±89, ±114, ±139, ±164, ±178 (8 pilots)
Channel estimation methods:
LS (Least Squares): Ĥ = Y/X where Y=received, X=known pilot
• Simple, noise-amplifying. Used for initial estimate from LTF.
MMSE (Min Mean Square Error): Ĥ = RHH(RHH + σ²ᵥRXX⁻¹)⁻¹ŶLS
• Better noise averaging. Requires noise variance estimate σ²ᵥ.
Common Phase Error (CPE) correction:
• Pilot phase drift per OFDM symbol due to phase noise / freq offset
• CPE = ∠(Σₖ Yk · Xk*) where k=pilot indices
• Corrected by rotating all subcarriers by -CPE per symbol | LTF (Long Training Field) design:
HT-LTF: 64-pt IFFT, BPSK, same subcarrier as HT-SIG
VHT-LTF: 64/128/256/512-pt based on BW
HE-LTF: 3 sizes (1x, 2x, 4x symbol duration)
• 1x (4µs): fast, less averaging
• 2x (8µs): default in most HE configurations
• 4x (16µs): maximum averaging, MU-MIMO sounding
Number of LTF symbols = Nsts (spatial streams). More streams → more LTF overhead per PPDU. | Pilot spacing: every ~30-40 subcarriers allows channel interpolation between pilots. 802.11ax: 8 pilots in 234 data+pilot tones = 1 pilot per ~30 tones.
CPE correction error: good implementations achieve <0.5° residual phase error per symbol. |
| Diversity & Combining Gain | Proakis 5th Ed.
Ch.14
IEEE 802.11-2020
§19 (STBC) | Combining methods and SNR gain:
MRC (Maximal Ratio Combining):
• SNR_out = Σᵢ SNRᵢ = N × SNRavg (linear sum)
• dB gain: 10·log₁₀(N) dB over single branch
• Optimal when noise is AWGN. Requires coherent phase reference.
• Used in 802.11 receive MIMO (each antenna = 1 branch)
EGC (Equal Gain Combining):
• SNR_out ≈ (N + N(N-1)·π/4) × SNRavg/N for Rayleigh
• ~0.5-1 dB worse than MRC
• Used when channel phase estimation is noisy
SC (Selection Combining):
• Selects branch with highest instantaneous SNR
• SNR gain vs single: G = Σₖ₌₁ᴺ 1/k (harmonic series)
• N=2: G=1.5 (1.76 dB). N=4: G=2.08 (3.18 dB)
• Used in antenna selection (low-cost receivers)
STBC (Space-Time Block Coding):
• Alamouti 2×1: SNR = 10·log₁₀(2) = 3 dB gain (same as MRC 2-branch)
• No TX CSI needed. Tx diversity.
• 2×2 STBC: 6 dB SNR gain over SISO in Rayleigh fading
Outage Probability with diversity (Rayleigh):
Pout(N) = [1 - exp(-γ/γ̄)]^N ≈ (γ/γ̄)^N for low SNR
Diversity gain = N (slope of outage curve in log scale) | 802.11 diversity usage:
RX diversity: MRC across receive antennas (transparent to protocol)
TX diversity: STBC or CSD (Cyclic Shift Diversity)
• CSD: mandatory for multi-antenna non-BF TX
• STBC: optional, signaled in HT-SIG/VHT-SIG STBC field
Antenna selection (ASEL): optional 802.11n feature. Controller selects best subset of antennas. Feedback: ASEL Report in HT Control field. | Single RX: reference
2-branch MRC: +3 dB (Rayleigh)
4-branch MRC: +6 dB
2×1 STBC (Alamouti): +3 dB TX diversity
2×2 STBC: +6 dB (TX + RX combined)
Selection combining (N=2): +1.76 dB |
| Eb/N0 to SNR - Full Derivation | IEEE 802.11-2020
§17.3.10
Proakis Appendix B | SNR (dB) = Eb/N0 (dB) + 10·log₁₀(Rb/B)
where:
• Rb = bit rate (bps)
• B = noise bandwidth (Hz) ≈ channel BW
• Eb = energy per information bit
• N0 = single-sided noise spectral density
Receiver sensitivity formula derivation:
Noise floor (dBm) = kTB = -174 dBm/Hz + 10log(B)
→ 20 MHz: -174+73 = -101 dBm
→ 40 MHz: -174+76 = -98 dBm
→ 80 MHz: -174+79 = -95 dBm
Sensitivity = Noise floor + NF + SNRmin
= -174 + 10log(B) + NF + SNRmin
Example: 802.11ax MCS 11 (1024-QAM 5/6), 20 MHz:
• Noise floor = -101 dBm
• NF (client) = 8 dB
• SNRmin = 33 dB (at BER=10⁻⁴ after LDPC coding gain)
• Sensitivity = -101+8+33 = -60 dBm
• IEEE spec (Table 27-82): -57 dBm (matches ±3 dB) | Spectral efficiency per MCS:
SE = Rb/B = MCS_data_rate / channel_BW (b/s/Hz)
Comparison table (20 MHz):
• MCS 0 (BPSK 1/2): SE = 8.6/20 = 0.43 b/s/Hz
• MCS 7 (64-QAM 5/6): SE = 86/20 = 4.3 b/s/Hz
• MCS 11 (1024-QAM 5/6): SE = 143.4/20 = 7.17 b/s/Hz
• MCS 13 (4096-QAM 5/6): SE = 172.1/20 = 8.6 b/s/Hz
• Shannon limit @33dB SNR: log₂(1+2000)≈10.97 b/s/Hz
→ 802.11be MCS 13 achieves 78% of Shannon capacity at 33 dB SNR | 802.11 coding gain (LDPC vs BCC):
LDPC vs BCC: ~2 dB gain at BER=10⁻³
Turbo codes theoretical: ~0.5 dB from Shannon (not used in Wi-Fi)
LDPC used in 802.11n (optional), 802.11ac (optional), 802.11ax (mandatory high MCS), 802.11be (mandatory) |