CP-OFDM
Why the Cyclic Prefix Makes Wideband OFDM Practical
Ordinary OFDM divides a wide channel into a large set of orthogonal subcarriers, each modulated at a low symbol rate so that the per-subcarrier symbol is long compared with the channel delay spread. That alone is not enough: when a symbol propagates through a multipath channel, energy from the previous symbol still arrives during the start of the current one, breaking orthogonality and producing inter-symbol interference. CP-OFDM solves both problems at once by copying the last portion of each time-domain symbol and pasting it in front. As long as that cyclic prefix is longer than the maximum excess delay, every delayed echo lands inside the prefix window and is discarded before the receiver runs its FFT.
The deeper reason the prefix is a copy of the tail, rather than just zeros, is mathematical. Linear convolution of the transmitted symbol with the channel impulse response becomes equivalent to circular convolution over the useful symbol interval, and circular convolution in time is multiplication in frequency. After the receiver strips the prefix and transforms the block, each subcarrier is simply scaled by one complex channel coefficient. Equalization reduces to a single complex division per subcarrier instead of the long adaptive time-domain filters that single-carrier systems require, which is why CP-OFDM scales gracefully to the 100 MHz and 400 MHz channel bandwidths used in 5G NR.
The trade-offs are spectral efficiency and envelope dynamics. The prefix carries no new data, so a normal 5G NR cyclic prefix consumes roughly 7 percent of each symbol; an extended prefix used for very large delay spreads costs about 25 percent. Because the transmitted symbol is the sum of many independent subcarriers, the envelope occasionally peaks far above its average, giving a high PAPR that forces the power amplifier into back-off. That linearity penalty is acceptable at a base station but punishing for a battery-powered handset, which is why the NR uplink switches to a precoded variant.
Governing Relationships
TCP ≥ τmax (prefix duration must exceed channel max excess delay)
Per-subcarrier channel after FFT:
Y(k) = H(k) × X(k) + N(k) → X̂(k) = Y(k) / H(k)
Subcarrier spacing and symbol time:
Δf = 1 / Tu and Tsym = Tu + TCP
Cyclic-prefix overhead:
ηCP = TCP / (Tu + TCP)
Where Tu = useful symbol time, Δf = subcarrier spacing, H(k) = channel response on subcarrier k, τmax = maximum excess delay. Example: NR Δf = 30 kHz → Tu ≈ 33.3 μs, normal TCP ≈ 2.34 μs → ηCP ≈ 7%.
CP-OFDM Across Wireless Standards
| System | Subcarrier Spacing | FFT Size (typ.) | Cyclic Prefix | Link / Role |
|---|---|---|---|---|
| 5G NR (FR1) | 15 / 30 / 60 kHz | 2048 to 4096 | ~4.7 / 2.34 / 1.17 μs | Downlink CP-OFDM |
| 5G NR (FR2 mmWave) | 120 / 240 kHz | 1024 to 4096 | ~0.59 / 0.29 μs | Downlink CP-OFDM |
| LTE | 15 kHz | 2048 | 4.7 μs normal / 16.7 μs ext. | Downlink CP-OFDM |
| Wi-Fi 802.11ac/ax | 312.5 / 78.125 kHz | 256 to 1024 | 800 / 400 ns GI | Both directions |
| 5G NR uplink | 15 to 120 kHz | 1024 to 4096 | same as DL | DFT-s-OFDM (low PAPR) |
Frequently Asked Questions
How long should the cyclic prefix be relative to the channel delay spread?
The cyclic prefix must be at least as long as the channel's maximum excess delay, or the previous symbol's tail leaks into the FFT window and breaks subcarrier orthogonality. In 5G NR at 30 kHz spacing the normal prefix is about 2.34 μs, covering urban macro delay spreads near 1 to 2 μs. Wi-Fi 802.11ac uses an 800 ns guard interval, or 400 ns short GI. A longer prefix tolerates more delay spread but cuts spectral efficiency; a normal NR prefix costs roughly 7% of symbol time.
Why does CP-OFDM have a high peak-to-average power ratio?
A symbol is the IFFT sum of hundreds or thousands of independent subcarriers. When many align in phase, their amplitudes add and the envelope spikes well above average, giving a PAPR near 10 to 13 dB for a 1024-point FFT. That forces large amplifier back-off and lowers efficiency, so NR keeps CP-OFDM on the downlink but moves to DFT-spread-OFDM on the uplink, trimming PAPR by 2 to 4 dB to spare the handset amplifier.
How does the receiver use the cyclic prefix to equalize the channel?
Because the prefix is a copy of the symbol tail, linear convolution with the channel looks like circular convolution over the useful window, which is multiplication in the frequency domain. After discarding the prefix and taking the FFT, each subcarrier k is scaled by one complex coefficient H(k). The receiver estimates H(k) from reference subcarriers and corrects with a single complex division, a one-tap equalizer, instead of the long time-domain equalizers single-carrier systems need.