OFDM, the Cyclic Prefix & 5G Numerology
"How to make a brutal channel behave like a spreadsheet."
In this episode
Multipath makes a wideband single-carrier channel painful in two ways: intersymbol interference and frequency-selective fading that demands an expensive time-domain equalizer. OFDM's move is to split the band into many narrow, orthogonal subcarriers — each sees a flat channel, one complex number. The IFFT is the modulator and the FFT is the demodulator; orthogonality falls out of spacing the subcarriers exactly 1/T apart. The star of the episode is the cyclic prefix, which does two distinct jobs people constantly conflate: it's a guard interval that absorbs the channel's delay spread (killing ISI), and it turns the channel's linear convolution into circular convolution — which is exactly what makes the channel diagonal in the FFT domain, so equalization collapses to one division per subcarrier. The price is PAPR. Then 5G NR makes the whole waveform reconfigurable through numerology.
Lesson map
- ObjectiveUnderstand why OFDM diagonalizes a channel, what the cyclic prefix really does, and what 5G changed.
- The refresherMultipath → delay spread → ISI and frequency-selective fading; the cost of wideband single-carrier equalization.
- What changedDivide-and-conquer the spectrum into narrowband orthogonal subcarriers, all built with one FFT.
- The mechanism — IFFT/FFTThe transform as modulator/demodulator; orthogonality from Δf = 1/T.
- The cyclic prefix's two jobsGuard interval vs. delay spread, and linear → circular convolution → one-tap equalization. Plus the PAPR tax.
- 5G numerology + self-testSCS = 15·2^μ kHz and the symbol/CP/latency/Doppler trade-off.
Key concepts
Self-test
1 · Pick a numerology from a delay spread
For an outdoor cell with a delay spread of a couple of microseconds, the cyclic prefix must exceed that delay spread. Work out how long the prefix has to be, then find which numerology gives a symbol period long enough that a sensible fractional CP covers it. (You'll find narrow 15/30 kHz fits and 120 kHz doesn't.) Then compute the CP overhead as the prefix-to-symbol ratio.
2 · Why cyclic, not blank?
Explain why the prefix must be a cyclic copy rather than a blank guard interval. The answer: a blank guard kills ISI but leaves you with linear convolution; the cyclic copy makes the channel act as a circular convolution — therefore diagonal in the FFT domain — enabling one-tap equalization.
Warm up in CircuitQuEEst
The matching undergraduate foundations are quests in CircuitQuEEst, the free, offline Android app for learning EE (download the APK). Play these first, then come back for the graduate-level version:
- Signals & Systems — Fourier Transform: the basis OFDM is built on.
- Digital Signal Processing — The Discrete Fourier Transform (DFT/FFT): here the IFFT is the modulator and the FFT the demodulator.
- Communication Systems — Digital Modulation: the QAM symbols that ride on each subcarrier.
- IoT & Wireless — Wi-Fi and Bluetooth/BLE: Wi-Fi is OFDM underneath.
Go deeper
- 3GPP 5G NR (Rel-15+) — scalable numerology, SCS = 15·2^μ kHz (μ = 0…3 for data in Rel-15/16; 240 kHz for sync signals; 480/960 kHz added in Rel-17).
- CP-OFDM & DFT-s-OFDM — downlink/uplink waveforms; the DFT-spread variant tames uplink PAPR for coverage.
- LTE (fixed 15 kHz SCS) — the contrast that makes NR's flexibility visible.
- DFT diagonalization identity — circular convolution in time = element-wise product of DFTs (the math behind one-tap equalization).
For the full numerology table and bandwidth-part details, see 3GPP TS 38.211.