One Fast Stream, or a Thousand Slow Ones
M8-L3 measured the damage multipath does and then promised OFDM would fix it. This is that lesson — and the fix turns multipath from a hazard into one complex multiply.
Five Symbols of Interference
- Urban delay spread στ ≈ 1 µs (M8-L3)
- 5 MBd single carrier → Ts = 1/5×10⁶ = 200 ns (M6-L2)
- 1 µs / 200 ns = 5 symbols of ISI → a 5-tap equaliser
- At 50 MBd it is 50 symbols. That is the wall
Split the stream into N slow subcarriers instead. At N = 128 each symbol lasts 25.6 µs, so the echo is 3.9% of one symbol, not five whole ones.
Orthogonality Costs Only Exactness
At Δf = 15 kHz and Tu = 66.67 µs, subcarrier 1200 fits exactly 1200.0 cycles. Their sinc spectra overlap heavily — and each is zero at every other centre.
The IFFT Is the Modulator
- N oscillators and N filters, or one transform
- Direct sum, N = 2048: N² = 4,194,304 multiplies
- Radix-2 FFT: (N/2)log₂N = 1024 × 11 = 11,264
- Ratio 372×, for a bit-identical answer
- Chang 1966, Weinstein 1971, CP 1980 — shipped 1995
OFDM’s adoption date was set by Moore’s law, not by insight. Nobody doubted the mathematics; nobody could afford the arithmetic.
Paste the Tail on the Front
An echo inside the prefix becomes a cyclic shift, and a cyclic shift is a phase rotation. So Yk = HkXk: one complex multiply per subcarrier replaces the equaliser. LTE’s 4.69 µs covers 1.4 km of excess path.
Symbol, Prefix, Echo
The prefix has to cover the last echo, not the RMS spread στ. For an exponential delay profile truncated 20 dB down that is τmax = 4.6 στ — computed from that model, and the 20 dB is the one chosen number. Real profiles agree: ITU-R M.1225 Vehicular A has στ = 370 ns with its last tap at 2510 ns, 6.8×.
LTE and 802.11a
- LTE: Δf 15 kHz → Tu = 1/15000 = 66.67 µs
- CP 4.69 µs → 4.69/71.36 = 6.6% overhead
- WiFi: 20×10⁶/64 = 312.5 kHz → Tu = 3.2 µs
- CP 0.8 µs → symbol 4.0 µs, 0.8/4.0 = 20%
- Only 48 of 64 subcarriers carry data: 48/64 = 75%
Check: 48 / 4.0 µs × 6 bits × 3/4 = 54 Mbit/s ✓ — 802.11a’s top rate, rebuilt from the numerology.
What OFDM Costs
The bound is N; practice is 8 to 12 dB — still enough that LTE’s uplink uses SC-FDMA to spare the handset’s amplifier (M5-L4). Add frequency-offset sensitivity and 6.6–20% of prefix.
A Two-Dimensional Grid
- Give each user a rectangle of subcarriers × symbols
- FDMA and TDMA at once — no guard bands, no guard times
- LTE resource block: 12 × 15 kHz = 180 kHz, 0.5 ms
- Scheduler gives each user the subcarriers where they are strong
What you learned
- N slow subcarriers turn 5 symbols of ISI into a fraction of one
- Δf = 1/Tu makes overlapping spectra separable
- The IFFT is the modulator — 372× cheaper at N = 2048
- CP > στ, and 4.69/71.36 = 6.6% is what it costs LTE
- OFDMA schedules the grid — and beats FDMA, TDMA and CDMA