Cyclic Prefix in OFDM: Guard Interval, ISI, Overhead and Length Choice

A cyclic prefix is a copy of the last part of an OFDM symbol placed in front of that symbol before transmission. It acts as a guard interval against multipath echoes, and it makes the channel look circular to the receiver’s FFT, so inter-symbol interference disappears and each subcarrier can be corrected with one complex division. The cost is a small loss of time and energy, typically about 7% in cellular systems.

This article concentrates on the prefix itself. For subcarriers, the transmitter chain and PAPR, see Understanding OFDM: A Comprehensive Overview.

What Is a Cyclic Prefix in OFDM?

An OFDM transmitter produces a block of N samples with an inverse FFT, lasting the useful symbol time T_u. It then copies the last N_cp samples to the front. The transmitted OFDM symbol duration is:

T_sym = T_u + T_cp

The prefix is not new data. It repeats the waveform so that the signal looks periodic across a slightly longer interval, and that periodicity is what the receiver exploits.

Cyclic prefix in OFDM as a guard interval, with direct path and delayed echo both inside the FFT window — Figure 1: Schematic of a cyclic prefix. The direct path and an echo delayed by less than T_cp both supply a complete, cyclically shifted symbol to the FFT window.

Guard Interval, ISI and OFDM Multipath

In a multipath channel, echoes arrive late by up to the maximum excess delay τ_max. Without protection, the end of one symbol would run into the start of the next, which is inter-symbol interference (ISI). An empty guard interval would stop that, but it would leave a second problem: an echo arriving partway through the FFT window would not contain a whole number of cycles of each subcarrier, so subcarriers would no longer be orthogonal. This is inter-carrier interference (ICI).

The cyclic prefix fixes both problems at once. As long as τ_max ≤ T_cp:

The tail of the previous symbol ends within the prefix, so ISI never reaches the FFT window.
Every echo inside the window is a complete, cyclically shifted copy of the symbol, so each subcarrier still has an integer number of cycles and ICI does not arise.

How the Cyclic Prefix Turns Linear Convolution into Circular Convolution

A channel performs linear convolution, but the DFT property Y_k = H_kX_k applies only to circular convolution. The prefix bridges the two, and a short example shows how.

Take the block x = [1, 2, 3, 4] and a two-path channel h = [1, 0.5]. With a one-sample prefix, the transmitter sends [4, 1, 2, 3, 4].

Channel output: [4, 3, 2.5, 4, 5.5, 2].
After the receiver discards the first sample and takes the next four: [3, 2.5, 4, 5.5].
Without a prefix, the same four samples would be [1, 2.5, 4, 5.5], with the first one wrong.

The block [3, 2.5, 4, 5.5] is exactly the circular convolution of x and h, and a numerical check confirms that its DFT equals the DFT of x multiplied by the DFT of h, term by term. The channel has become a single complex gain on each subcarrier, which is why a one-tap equalizer is enough.

Cyclic Prefix Removal at the Receiver

Cyclic prefix elimination is a simple step. After timing synchronization, the receiver throws away N_cp samples and passes the next N samples to the FFT. Everything corrupted by the previous symbol goes with them.

The discarded samples are still useful before that point. Because the prefix repeats the symbol tail, correlating the received signal with itself at a lag of N samples gives a peak at the symbol boundary, a common basis for blind timing and coarse frequency estimation.

The FFT window also need not start at the exact boundary. Any start point in the part of the prefix free of echoes gives a clean result, because a timing shift of d samples only adds a phase ramp of 2πkd/N across subcarrier k, which the equalizer removes along with the channel.

When the Prefix Is Too Short

If an echo outlasts the prefix by Δ samples, two errors appear in equal measure: part of the previous symbol enters the window, and part of the current symbol is missing. For a single echo of relative amplitude a, a simple estimate of the interference power after one-tap equalization, relative to the signal, is:

P_err ≈ (2a²Δ/N) / (1 − a²)

Figure 2 shows a simulation with N = 256, QPSK on every subcarrier, one echo at −6 dB (a = 0.5) and no noise. Inside the prefix the error is at numerical zero. One sample beyond a 16-sample prefix it is already −25.8 dB, and at 8 samples beyond it is −16.8 dB, or an EVM of about 14.5%. The formula predicts −25.8 dB and −16.8 dB for those two cases. The transition is abrupt, so a prefix slightly too short does noticeably more harm than one slightly too long.

Simulated OFDM residual error against echo delay for cyclic prefix lengths of 8, 16 and 32 samples, rising once the echo exceeds the prefix — Figure 2: Simulated residual error for three prefix lengths. Error stays at numerical zero until the echo delay passes the prefix, then rises steeply.

Cyclic Prefix Overhead

The prefix takes time and transmit energy that carry no extra information. Its overhead has two common measures, so quoted figures can differ slightly:

Prefix relative to useful time: T_cp / T_u.
Share of total symbol time: T_cp / (T_u + T_cp).

The rate falls by the factor T_u / (T_u + T_cp), and the SNR penalty is:

Loss = 10 log₁₀(1 + T_cp/T_u) dB

The LTE normal prefix has a ratio of 7.0% and uses 6.6% of total symbol time. A 25% prefix drops efficiency to 80% and costs 0.97 dB.

Cyclic prefix overhead curve of time efficiency against prefix length, marking LTE normal CP, 802.11n short GI and LTE extended CP — Figure 3: Time efficiency against cyclic prefix length, with common systems marked.

Choosing the Cyclic Prefix Length

Cyclic prefix length is a compromise between covering the channel and wasting symbol time. A practical procedure is:

Estimate the maximum excess delay. Radio paths add about 3.3 µs per km of extra length, or 300 m per µs.
Add margin for timing error and for transmit and receive filter delay, both of which use up the prefix.
Set the symbol long enough to keep overhead low, commonly with T_u at least several times T_cp.
Keep the subcarrier spacing well above the Doppler shift. This is the limit on making T_u longer, since long symbols mean narrow subcarriers.

Worked example: sizing a prefix

Take N = 1024 at a sampling rate of 15.36 MHz, with 600 data subcarriers carrying 16-QAM, and a channel with a 3 µs maximum excess delay.

Sample period: 1/15.36 MHz = 65.1 ns.
Subcarrier spacing: 15.36 MHz / 1024 = 15 kHz, so T_u = 66.67 µs.
Minimum prefix: 3 µs × 15.36 MHz = 46.08, so at least 47 samples.
Chosen prefix: 72 samples, or 4.69 µs, covering a path difference of 1.41 km.
Spare margin: 4.69 − 3 = 1.69 µs, about 26 samples for timing and filters.
Symbol duration: (1024 + 72) / 15.36 MHz = 71.35 µs.
Overhead: 72 / 1096 = 6.57%, an SNR loss of 0.30 dB.
Raw data rate: 600 × 4 bits / 71.35 µs = 33.64 Mbit/s, against 36.00 Mbit/s with no prefix.
Doppler check at 3 GHz and 120 km/h: f_d = 333 Hz, which is 2.2% of the subcarrier spacing and acceptable.

The bare minimum of 47 samples would give 34.42 Mbit/s, but only 0.06 µs of margin. The extra 0.78 Mbit/s is rarely worth the risk, given how sharply Figure 2 rises.

Cyclic Prefix in LTE, 5G NR and Wi-Fi

Real systems choose values to suit their environment. The figures below are typical and approximate.

System	Useful symbol T_u	Prefix T_cp	T_cp/T_u	Path difference covered
LTE and NR normal, 15 kHz	66.67 µs	4.69 µs	7.0%	1.4 km
NR normal, 30 kHz	33.33 µs	2.34 µs	7.0%	703 m
NR normal, 60 kHz	16.67 µs	1.17 µs	7.0%	352 m
NR normal, 120 kHz	8.33 µs	0.59 µs	7.0%	176 m
LTE extended, 15 kHz	66.67 µs	16.67 µs	25%	5 km
802.11a/g/n long GI	3.2 µs	0.8 µs	25%	240 m
802.11n short GI	3.2 µs	0.4 µs	12.5%	120 m

The first symbol of each LTE slot has a slightly longer prefix of 5.2 µs, which makes seven symbols fill the 0.5 ms slot exactly; averaged over a slot, the overhead is 6.7%. Extended cyclic prefix suits large cells and single-frequency broadcast, where signals from distant sites arrive as long echoes. Wi-Fi uses short prefixes because indoor delay spreads are small. Wi-Fi 6 lengthens the useful symbol to 12.8 µs and offers guard intervals of 0.8, 1.6 and 3.2 µs.

Frequently asked questions

Why is a cyclic prefix used instead of an empty guard interval?

Empty time would stop ISI, but late echoes would then contain a partial symbol inside the FFT window, which breaks orthogonality and causes ICI. Copying the symbol tail keeps every echo cyclic, so the FFT still sees whole cycles of each subcarrier.

What happens if the cyclic prefix is shorter than the delay spread?

Energy from the previous symbol enters the FFT window, and orthogonality is partly lost. The result is an error floor that more transmit power cannot remove, because the interference scales with the signal.

Does a cyclic prefix reduce data rate?

Yes. The rate falls to T_u / (T_u + T_cp) of its no-prefix value, a loss of about 6.6% for the LTE normal prefix. The receiver also loses the same small fraction of signal energy, 0.3 dB in that case.

Is a cyclic prefix always needed in OFDM?

No, although it is the usual choice. Zero-padded OFDM leaves the interval empty and handles the tail with extra receiver processing, and other variants use known training sequences. Single-carrier frequency domain equalization, such as the DFT-spread OFDM uplink in LTE, relies on a cyclic prefix for the same reason.

Introduction to OFDM and Cyclic Prefix