OFDM and multicarrier

Overview

Orthogonal frequency-division multiplexing (OFDM) splits a wide channel into many narrow subcarriers and sends a slow stream of symbols on each. LTE, 5G NR and Wi-Fi all use it. The reason is multipath. A single-carrier signal 100 MHz wide sees a delay spread of a microsecond as many symbols of intersymbol interference, which needs a long equaliser. Split into thousands of subcarriers, each sees an almost flat channel that one complex multiplication can correct.

This hub explains how OFDM works, what the cyclic prefix buys, how 5G NR scales its numerology across bands, and the costs that come with OFDM, chiefly its peak-to-average power ratio and its sensitivity to frequency errors. It is for engineers who need to read a resource grid, size an FFT, or decide between CP-OFDM and DFT-s-OFDM.

Key ideas

Subcarriers and the FFT

OFDM places subcarriers exactly 1/T apart, where T is the useful symbol duration. Over one symbol each subcarrier completes a whole number of cycles, so they are orthogonal even though their spectra overlap. Weinstein and Ebert showed in 1971 that the modulator and demodulator can be built with the discrete Fourier transform. The transmitter places one QAM symbol on each subcarrier and takes an inverse FFT. The receiver takes an FFT and reads the symbols back. Unused subcarriers at the band edges and a null at DC give guard bands and avoid the receiver’s DC offset.

The cyclic prefix

Each OFDM symbol is preceded by a cyclic prefix (CP), a copy of its last few samples. If the channel’s delay spread is shorter than the CP, the echoes of one symbol die out inside the next symbol’s prefix, which the receiver discards. The channel’s linear convolution then acts as a circular one on the FFT window, so each subcarrier sees only a single complex gain. Equalisation becomes one division per subcarrier. The price is overhead: the CP carries no new data. It also gives some tolerance to timing error, because any FFT window that starts inside the clean part of the CP still works, with a phase ramp across subcarriers that the channel estimate absorbs.

NR numerology

LTE uses a fixed 15 kHz subcarrier spacing for its normal unicast carriers. 5G NR makes the spacing a parameter. In 3GPP TS 38.211 (Release 19 version) the subcarrier spacing is 15 × 2^μ kHz for numerology μ from 0 to 6, giving 15, 30, 60, 120, 240, 480 and 960 kHz. A slot holds 14 OFDM symbols with normal CP, a 1 ms subframe holds 2^μ slots, and a frame lasts 10 ms. The normal CP shrinks with the symbol: about 4.69 µs at 15 kHz, 2.34 µs at 30 kHz and 0.59 µs at 120 kHz. Only the 60 kHz numerology also offers an extended CP. The two largest spacings, 480 kHz and 960 kHz, appear in band n263 (57 GHz to 71 GHz), and the extension of NR to 71 GHz was introduced in Release 17.

The trade-off is between delay spread and everything else. Narrow spacing gives long symbols and a long CP, which suits large macro cells at low frequencies. Wide spacing gives short symbols that tolerate Doppler and phase noise, and short slots for low latency, which suits millimetre-wave bands where cells are small and oscillators noisy. A carrier can be split into bandwidth parts with different numerologies. The 5G NR hub covers the resource grid in full.

OFDMA, DFT-s-OFDM and PAPR

OFDMA gives different users different sets of subcarriers and symbols. In NR and LTE the scheduler allocates resource blocks of 12 subcarriers, and can place each user where its channel is strong.

OFDM has a high peak-to-average power ratio. The sum of many independently modulated subcarriers approaches a Gaussian signal with occasional large peaks. The power amplifier must back off to keep those peaks linear, which costs efficiency and battery life. LTE answered this in the uplink with SC-FDMA, in which a DFT spreads the symbols before the OFDM IFFT and the result behaves like a single-carrier signal with lower PAPR. NR allows both: CP-OFDM for high rates and MIMO, and DFT-s-OFDM (transform precoding) for coverage-limited uplinks. TS 38.300 limits DFT-s-OFDM to one MIMO layer and allows dynamic switching between the two waveforms for PUSCH. Transmitters also reduce PAPR with clipping and filtering, tone reservation and digital predistortion of the amplifier.

OFDM’s other weakness is frequency error. A carrier frequency offset that is a small fraction of the subcarrier spacing already breaks orthogonality and leaks energy between subcarriers. The synchronisation hub covers how receivers correct it.

Where to start

Read /learn/beginner/08-ofdm-from-scratch/, which builds a transmitter and receiver in NumPy, after /learn/beginner/02-complex-baseband-and-iq/ and /learn/beginner/07-fading-channels/. The project /projects/ofdm-link-channel-estimation/ adds pilots and channel estimation. Keep /reference/ofdm-and-nr-numerology/ open. Intermediate lessons on the NR resource grid are in preparation.

Tools and platforms

Common mistakes

• Forgetting the null at DC, or the FFT shift between subcarrier index and FFT bin order.
• Placing the FFT window late, inside the next symbol, instead of early inside the CP.
• Computing PAPR from a signal sampled only at the symbol rate. Oversample first, or the peaks between samples are missed.
• Assuming every NR numerology is allowed in every band. The band tables in TS 38.101-1 and TS 38.101-2 set which spacings apply.
• Ignoring CP overhead and reference signals when estimating throughput from subcarrier count.

Going deeper

• S. B. Weinstein and P. M. Ebert, Data transmission by frequency-division multiplexing using the discrete Fourier transform, IEEE Transactions on Communication Technology 19(5), 1971.
• H. G. Myung, J. Lim and D. J. Goodman, Single carrier FDMA for uplink wireless transmission, IEEE Vehicular Technology Magazine 1(3), 2006.
• S. H. Han and J. H. Lee, An overview of peak-to-average power ratio reduction techniques for multicarrier transmission, IEEE Wireless Communications 12(2), 2005.
• P. Banelli et al., Modulation formats and waveforms for 5G networks: who will be the heir of OFDM?, IEEE Signal Processing Magazine 31(6), 2014.
• 3GPP TS 38.211 (NR physical channels and modulation, including numerologies and the frame structure) and TS 36.211 (the LTE equivalent).