The fundamental problem — multipath
In wireless communication, a transmitted signal does not travel along a single direct path. It reflects off buildings, the ground, and other objects. Multiple copies of the same signal arrive at the receiver at slightly different times — this is called multipath propagation.
The time difference between the first and last arriving copy is the delay spread. In a typical urban environment this is 1–5 μs. In large cells it can reach 10–20 μs. When a single wideband carrier is used, delayed copies overlap with the current symbol and corrupt it — this is Inter-Symbol Interference (ISI).
The OFDM solution
Instead of one wideband carrier, OFDM splits the bandwidth into hundreds of narrow subcarriers, each carrying a low-rate signal. A subcarrier narrow enough has a symbol duration much longer than the delay spread — it experiences flat fading that a single-tap equalizer handles trivially.
For 5G NR with 30 kHz SCS, symbol duration = 33.33 μs — roughly 10–30× longer than typical urban delay spread.
Orthogonality — why subcarriers don't interfere
With hundreds of overlapping subcarriers, why don't they interfere? Two sinusoids at frequencies f₁ and f₂ are orthogonal if their inner product over one symbol period T equals zero. This holds when subcarrier spacing Δf = 1/T exactly.
// Inner product of subcarriers k and m over symbol period T: (1/T) × ∫₀ᵀ e^(j2π·k·Δf·t) × e^(−j2π·m·Δf·t) dt = { 1 if k = m (same subcarrier) { 0 if k ≠ m (orthogonal — zero crosstalk) ✓ // This holds ONLY when Δf = 1/T // For 30 kHz SCS: T = 1/30,000 = 33.33 μs ← exactly one symbol // At FFT receiver: each bin captures ONLY its own subcarrier
Cyclic prefix — eliminating ISI between symbols
Even with long symbols, multipath causes the end of one OFDM symbol to overlap with the next. The cyclic prefix (CP) fixes this: copy the last N_CP samples of each symbol and prepend them as a guard interval. If CP duration > delay spread, no ISI occurs. The receiver simply discards the CP before the FFT.
s(t) = Σₖ aₖ · e^(j2π(k+½)Δf(t−N_u·κ·Tc)) −N_CP·κ·Tc ≤ t < N_u·κ·Tc
where N_CP is the cyclic prefix length and N_u is the FFT size.
OFDM parameters — 30 kHz SCS, 100 MHz bandwidth
// Subcarrier spacing and symbol duration Δf = 30,000 Hz T_useful = 1/Δf = 33.333 μs // Cyclic prefix (TS 38.211 Table 5.3.1-1) N_CP_first = 160 samples → 2.604 μs (symbol 0 of each slot) N_CP_normal = 144 samples → 2.344 μs (symbols 1–13) // FFT size and sampling rate N_FFT = 4096 Sampling rate = N_FFT × Δf × 2 = 122.88 Msps // Active subcarriers in 100 MHz N_active = 273 RBs × 12 = 3276 subcarriers Active BW = 3276 × 30 kHz = 98.28 MHz // One-tap equalization: Y[k] = H[k] × X[k] + N[k] X̂[k] = Y[k] / H[k] ← trivially simple, regardless of bandwidth