Exercises

(a) FFT size: 256 (compared to 64 in 802.11a/n/ac for 20 MHz).

(b) 234 data subcarriers (out of 256 total: 8 pilot + 3 DC null

• 11 guard null = 22 non-data).

(c) $\Delta f = 20 \text{ MHz} / 256 = 78.125$ kHz.

(d) $T_u = 1/\Delta f = 1/78125 = 12.8$ $\mu$s.

Compare to 802.11a: $T_u = 1/312500 = 3.2$ $\mu$s. The $4\times$ longer symbol improves multipath tolerance. $\blacksquare$

(a) $N_{\mathrm{SD}} = 48$, $T_u = 3.2$ $\mu$s, $T_{\mathrm{GI}} = 0.8$ $\mu$s. $R = 48 \times 6 \times 0.75 / 4.0 \times 10^{-6} = 54$ Mbps.

(b) $N_{\mathrm{SD}} = 108$ (40 MHz), $T_{\mathrm{GI}} = 0.4$ $\mu$s. $R = 108 \times 6 \times 5/6 / 3.6 \times 10^{-6} = 150$ Mbps.

(c) $N_{\mathrm{SD}} = 234$ (80 MHz), $T_{\mathrm{GI}} = 0.4$ $\mu$s. $R = 234 \times 8 \times 5/6 / 3.6 \times 10^{-6} = 433$ Mbps.

(d) $N_{\mathrm{SD}} = 1960$ (160 MHz), $T_{\mathrm{GI}} = 0.8$ $\mu$s. $R = 1960 \times 10 \times 5/6 / 13.6 \times 10^{-6} = 1201$ Mbps.

Peak rate growth: $54 \to 150 \to 433 \to 1201$ Mbps (per spatial stream). $\blacksquare$

(a) The 3.2 $\mu$s GI tolerates multipath delays up to 3.2 $\mu$s, corresponding to a path length difference of $3.2 \times 10^{-6} \times 3 \times 10^8 = 960$ m.

(b) Typical indoor RMS delay spread: 20--100 ns. Rule of thumb: GI $\geq 4\tau_{\mathrm{rms}}$. Maximum $\tau_{\mathrm{rms}} = 3200/4 = 800$ ns. This covers even large indoor venues and some outdoor scenarios.

(c) 802.11a: GI = 0.8 $\mu$s $\to$ max $\tau_{\mathrm{rms}} = 200$ ns. 802.11ax: GI = 3.2 $\mu$s $\to$ max $\tau_{\mathrm{rms}} = 800$ ns.

802.11ax provides $4\times$ more delay spread tolerance, enabling outdoor and large-venue operation. $\blacksquare$

(a) OFDM: 802.11a (1999). (b) 40 MHz bonding: 802.11n (2009). (c) DL MU-MIMO: 802.11ac (2013). (d) OFDMA: 802.11ax (2020). (e) 320 MHz: 802.11be (2024). (f) MLO: 802.11be (2024). $\blacksquare$

Data duration: $40 + 1500 \times 8 / (86 \times 10^6) = 40 + 140 = 180$ $\mu$s. $T_s = 180 + 16 + 44 + 34 = 274$ $\mu$s. $T_c \approx 180 + 44 + 34 = 258$ $\mu$s.

Iteration converges to $\tau^* \approx 0.065$, $p^* \approx 0.235$.

$P_{\mathrm{tr}} = 1-(1-0.065)^5 = 0.286$. $P_s = 5 \times 0.065 \times (1-0.065)^4 / 0.286 = 0.870$.

$S = \frac{0.286 \times 0.870 \times 12000}{0.714 \times 9 + 0.286 \times 0.870 \times 274 + 0.286 \times 0.130 \times 258}$

$= 2985 / (6.4 + 68.1 + 9.6) = 2985 / 84.1 = 35.5$ Mbps.

MAC efficiency: $35.5/86 = 41.3$%.

Iteration converges to $\tau^* \approx 0.032$, $p^* \approx 0.466$.

$P_{\mathrm{tr}} = 1-(1-0.032)^{20} = 0.479$. $P_s = 20 \times 0.032 \times (1-0.032)^{19} / 0.479 = 0.721$.

$S = \frac{0.479 \times 0.721 \times 12000}{0.521 \times 9 + 0.479 \times 0.721 \times 274 + 0.479 \times 0.279 \times 258}$

$= 4142 / (4.7 + 94.6 + 34.5) = 4142 / 133.8 = 31.0$ Mbps.

MAC efficiency: $31.0/86 = 36.0$%.

Throughput drops by 13% as stations increase $4\times$. $\blacksquare$

(a) RTS: $20 \times 8 / (6 \times 10^6) + 20 = 46.7$ $\mu$s (preamble + data). CTS: $14 \times 8 / (6 \times 10^6) + 20 = 38.7$ $\mu$s. Total overhead: $46.7 + 16 + 38.7 + 16 = 117.4$ $\mu$s.

(b) Without RTS/CTS: $T = 34 + 72 + 180 + 16 + 44 + 34 = 380$ $\mu$s. Throughput: $12000/380 = 31.6$ Mbps.

With RTS/CTS: $T = 34 + 72 + 117.4 + 180 + 16 + 44 + 34 = 497.4$ $\mu$s. Throughput: $12000/497.4 = 24.1$ Mbps.

RTS/CTS costs 24% throughput when there are no hidden nodes.

(c) With 10% collision probability on data frames: Without RTS/CTS: effective time per success = $T_{\mathrm{basic}} / (1-p_c) = 380/0.9 = 422$ $\mu$s. With RTS/CTS: collisions only on short RTS: $T_{\mathrm{rts}} / (1-p_c) + T_{\mathrm{data}} = 117.4/0.9 + 296 = 426.4$ $\mu$s.

Break-even: approximately 1500 bytes at 10% collision rate. For larger frames or higher collision rates, RTS/CTS is beneficial. $\blacksquare$

(a) Each voice user: $64 \times 8 / 0.02 = 25.6$ kbps. Total voice: $5 \times 25.6 = 128$ kbps.

Voice transmission probability per slot is very low ($\ll 0.01$) because voice packets are small and infrequent. AC_VO wins contention almost always against AC_BE due to shorter AIFS (AIFS_VO = 16 + 2 $\times$ 9 = 34 $\mu$s vs. AIFS_BE = 16 + 3 $\times$ 9 = 43 $\mu$s) and smaller $W_{\min}$ (4 vs. 16).

Collision probability for voice $< 1$% (voice packets rarely coincide, and voice wins over BE).

(b) Yes, easily. Total voice load is 128 kbps out of $\sim$35 Mbps available capacity. Voice uses $< 0.4$% of the channel, and EDCA priority ensures voice packets experience minimal delay.

(c) BE throughput $\approx 35 - 0.128 \approx 34.9$ Mbps. Voice has negligible impact on BE throughput. The EDCA priority differentiation is meaningful only when voice traffic is bursty or when latency (not throughput) matters. $\blacksquare$

(a) The vulnerable period is $2T_D$ (C must not start during A's frame or start a frame that overlaps with A's frame). If C starts uniformly in $[0, 2T_D]$: Collision occurs if C starts within $T_D$ of A's start. $P_{\mathrm{collision}} = T_D / (2T_D) = 0.5$.

(b) If C hears B's CTS, it defers for the entire duration indicated in the NAV. The only vulnerable period is the RTS duration ($\sim$47 $\mu$s): $P_{\mathrm{collision}} = 47 / (2 \times 500) = 0.047$.

RTS/CTS reduces collision probability by $\sim 10\times$.

(c) Without RTS/CTS: effective throughput $\sim (1 - 0.5) \times S = 0.5S$. With RTS/CTS: throughput $\sim 0.953 \times S' \approx 0.75S$ (accounting for RTS/CTS overhead of $\sim$25%). No hidden node: throughput $\approx S$.

Hidden node causes 25--50% throughput loss depending on mitigation. $\blacksquare$

(a) 802.11ac (312.5 kHz SCS): 20 MHz: 52/64 = 81.3% (48 data + 4 pilot). 40 MHz: 114/128 = 89.1%. 80 MHz: 242/256 = 94.5%. 160 MHz: 484/512 = 94.5%.

(b) 802.11ax (78.125 kHz SCS): 20 MHz: 242/256 = 94.5%. 40 MHz: 484/512 = 94.5%. 80 MHz: 996/1024 = 97.3%. 160 MHz: 1992/2048 = 97.3%.

(c) 802.11ax has better spectral efficiency at all bandwidths because the $4\times$ narrower SCS means the null subcarriers (DC, guard) consume a smaller fraction of the total.

At 20 MHz: 802.11ax gains $94.5/81.3 = 16$% more data subcarriers than 802.11ac. $\blacksquare$

(a) Per frame overhead: DIFS (34) + avg backoff (72) + preamble (40)

• data (50$\times$8/(86M) $\approx 5$ $\mu$s) + SIFS (16) + ACK (44) = 211 $\mu$s per frame.

100 users $\times$ 10 frames/s = 1000 frames/s. Channel time: $1000 \times 211 = 211{,}000$ $\mu$s/s = 21.1%.

With collisions ($n = 100$, high collision rate $\sim$40%): effective channel time $\approx 35$%.

(b) With 37 users per group on 80 MHz using 26-tone RUs: Need $\lceil 100/37 \rceil = 3$ multi-user transmissions per 100 ms.

Per MU transmission: trigger (40) + SIFS (16)

• data ($400 / (24 \times 4 \times 0.75) = 5.6$ symbols $\to$ 6 $\times$ 13.6 = 81.6 $\mu$s) + SIFS (16) + ACK (60) = 213.6 $\mu$s.

Channel time: $3 \times 213.6 \times 10 = 6{,}408$ $\mu$s/s = 0.64%.

(c) CSMA/CA: 35% of channel capacity for trivial traffic. OFDMA: 0.64% of channel capacity.

OFDMA is $55\times$ more efficient for this dense small-packet scenario. This demonstrates why 802.11ax was designed for dense environments. $\blacksquare$

(a) Received power at distance $d$: $P_r = 20 - 40 - 35\log_{10}(d)$ dBm.

Carrier sense at $-82$ dBm: $20 - 40 - 35\log_{10}(d) \geq -82$ $35\log_{10}(d) \leq 62$ $d \leq 10^{62/35} = 10^{1.77} = 58.9$ m.

In the 4$\times$5 grid with 5 m spacing, all 20 APs are within 59 m of each other. Each AP senses all 19 others.

(b) OBSS/PD = $-72$ dBm (10 dB higher than default): $20 - 40 - 35\log_{10}(d) \geq -72$ $35\log_{10}(d) \leq 52$ $d \leq 10^{52/35} = 10^{1.49} = 30.5$ m.

With 5 m spacing, APs within 30.5 m: approximately 12 APs (those within 6 grid positions). Each AP ignores $\sim$7 distant APs. With TX power reduced by 10 dB (to 10 dBm) per OBSS/PD rules, the sensing range further decreases.

(c) Without BSS coloring: each AP shares the channel with 19 others. Per-AP throughput $\approx S/20$.

With BSS coloring: effectively sharing with $\sim$12 APs. Per-AP throughput $\approx S/13$.

Improvement: $(S/13)/(S/20) = 1.54\times$ ($54$% gain). In practice, gains of $2$--$3\times$ are reported because BSS coloring also reduces collision probability. $\blacksquare$

(a) Per sensor: $50 \times 8 / 5 = 80$ bps. Total: $200 \times 80 = 16$ kbps. This is trivial for 802.11ax ($< 0.01$% of capacity).

(b) $\lceil 200/9 \rceil = 23$ groups. TWT interval: 5 s. Space groups evenly: $5000/23 \approx 217$ ms apart.

Each group's TWT service period: trigger (40 $\mu$s)

• SIFS (16) + UL data (2 OFDM symbols = 27.2 $\mu$s)
• SIFS (16) + ACK (60) = 159.2 $\mu$s. Add margin: 500 $\mu$s TWT service period.

(c) Each sensor is awake for $\sim$500 $\mu$s every 5 seconds. Duty cycle: $500 \times 10^{-6} / 5 = 10^{-4} = 0.01$%.

(d) $P_{\mathrm{avg}} = 0.0001 \times 100 + 0.9999 \times 0.1 = 0.01 + 0.10 = 0.11$ mW.

Compared to always-on (100 mW): $909\times$ power reduction. With a 1000 mAh battery at 3.3 V: battery life $= 3300 / 0.11 = 30{,}000$ hours $\approx 3.4$ years. $\blacksquare$

(a) Per stream: $3920 \times 12 \times 5/6 = 39{,}200$ data bits. Total (16 streams): $16 \times 39{,}200 = 627{,}200$ bits.

(b) Symbol duration: $12.8 + 0.8 = 13.6$ $\mu$s. $R_{\mathrm{peak}} = 627{,}200 / 13.6 \times 10^{-6} = 46.1$ Gbps.

(c) For 4096-QAM: required SNR $\approx 42$ dB per stream. With MIMO and 16 streams, the total required SNR is even higher (beamforming gain partially compensates).

(d) Free-space loss at 6 GHz: $L(d) = 20\log_{10}(4\pi d f/c) = 20\log_{10}(4\pi d \times 6 \times 10^9 / 3 \times 10^8)$.

Required $P_r \geq -174 + 10\log_{10}(320 \times 10^6) + 6 + 42$ $= -174 + 85 + 6 + 42 = -41$ dBm.

$20 - L(d) \geq -41 \Rightarrow L(d) \leq 61$ dB. $d \leq 10^{(61 - 47.8)/20} = 10^{0.66} = 4.6$ m.

4096-QAM is only usable within $\sim$5 m of the AP with line of sight. $\blacksquare$

(a) Link 1: mean = 3 ms. 99th percentile: $t_{99} = -3 \ln(0.01) = 13.8$ ms.

Link 2: mean = 0.5 ms. 99th percentile: $t_{99} = -0.5 \ln(0.01) = 2.3$ ms.

(b) Rate of minimum: $\lambda = 1/3 + 1/0.5 = 0.333 + 2.0 = 2.333$ ms$^{-1}$. Mean: $1/2.333 = 0.43$ ms. 99th percentile: $-\ln(0.01)/2.333 = 1.97$ ms.

(c) $< 5$ ms at 99th percentile: Link 1 alone: $t_{99} = 13.8$ ms — fails. Link 2 alone: $t_{99} = 2.3$ ms — passes. MLO: $t_{99} = 1.97$ ms — passes with large margin.

MLO reduces the 99th-percentile latency by $7\times$ compared to the congested link and provides robustness against single-link congestion. $\blacksquare$

$R_{\mathrm{peak}} = \frac{3920 \times 12 \times (5/6) \times 16}{13.6 \times 10^{-6}}$

Numerator: $3920 \times 12 \times 0.833 \times 16 = 627{,}000$ bits.

$R_{\mathrm{peak}} = 627{,}000 / 13.6 \times 10^{-6} \approx 46.1$ Gbps.

$R_{\mathrm{app}} = 0.65 \times 46.1 = 30.0$ Gbps.

$R_{\mathrm{user}} = 30.0 / 20 = 1.5$ Gbps per user.

This exceeds Gigabit Ethernet, demonstrating that Wi-Fi 7 can serve as a viable wired-replacement technology in enterprise environments. $\blacksquare$