5G NR Cyclic Prefix: The Insurance Premium Every OFDM Symbol Pays
What the cyclic prefix does, how long it is at each numerology, why two symbols per half-subframe are slightly longer, and why the extended CP exists only at 60 kHz.
The mental model: a run-up before the jump
Radio signals reach the receiver by many paths of different lengths, so copies of each OFDM symbol arrive spread over a few microseconds — the delay spread. Without protection, the tail of one symbol's late copies would land on the head of the next symbol: inter-symbol interference (ISI).
The cyclic prefix (CP) is the fix: copy the last few microseconds of each symbol and paste them in front of it. Now the receiver waits out the "run-up" and only starts its FFT once all the echoes of the previous symbol have died away and all the echoes of the current symbol have arrived. As long as the delay spread is shorter than the CP, two good things happen:
- No ISI — the previous symbol never reaches the FFT window.
- The channel acts as a circular convolution over the FFT window, so every subcarrier sees a single complex gain and can be equalised with one multiplication. This is the property that makes OFDM cheap to receive.
The CP is pure overhead — it carries no new information — so you want it just long enough for the channel and no longer.
How long it is
NR defines the CP as an integer number of basic time units T_c (≈ 0.509 ns), scaled by κ = 64 and by the numerology (TS 38.211 §5.3.1):
Normal CP: N_CP = 144 · κ · 2^−μ (+ 16 · κ for the first symbol of each 0.5 ms)
Extended CP: N_CP = 512 · κ · 2^−μ (μ = 2 only)
In microseconds:
| μ | SCS | Useful symbol | Normal CP (regular) | Normal CP (long symbol) | Extended CP |
|---|---|---|---|---|---|
| 0 | 15 kHz | 66.67 µs | 4.69 µs | 5.21 µs | — |
| 1 | 30 kHz | 33.33 µs | 2.34 µs | 2.86 µs | — |
| 2 | 60 kHz | 16.67 µs | 1.17 µs | 1.69 µs | 4.17 µs |
| 3 | 120 kHz | 8.33 µs | 0.59 µs | 1.11 µs | — |
| 4 | 240 kHz | 4.17 µs | 0.29 µs | 0.81 µs | — |
Two things to see in the table:
- The normal CP is always about 7 % of the useful symbol (144/2048). Overhead is constant across numerologies; absolute protection halves with each step of μ.
- The extended CP at 60 kHz is 25 % of the symbol (512/2048), so it costs 12 symbols per slot instead of 14.
The two "long" symbols: why 16κ extra
Look at the 15 kHz row: 14 symbols × (66.67 + 4.69) µs = 999 µs, not 1000. The missing microsecond is absorbed by making the first symbol of each 0.5 ms half-subframe 16κ·T_c ≈ 0.52 µs longer. Every numerology does the same (symbols 0 and 7·2^μ of each subframe), which guarantees:
- each numerology has an integer number of symbols in exactly 0.5 ms, and
- symbol boundaries of all numerologies coincide at every 0.5 ms — the meshing-gears property from the numerology article.
The receiver knows which symbols are long, so it is transparent to everything above the PHY. But it is why "CP length" is not a single number.
Why the extended CP exists only at 60 kHz
An extended CP would be useful wherever the delay spread exceeds the normal CP. In practice:
- At 15 and 30 kHz the normal CP (4.7 / 2.3 µs) already covers typical macro-cell delay spreads, so an extended CP would waste 18 % more overhead for little gain. LTE had an extended CP at 15 kHz mainly for MBSFN, which NR did not carry over.
- At 120 kHz and above the normal CP is 0.6 µs — but these numerologies are only used in small mmWave cells with short echoes, and 512κ·2^−3 would still be only 1.4 µs; not worth 25 % overhead.
- 60 kHz is the crossover: a numerology fast enough for URLLC-style latency, yet possibly deployed in FR1 macro cells (or FR2 large cells) where 1.2 µs is too short. The extended CP (4.2 µs — nearly the 15 kHz normal CP) lets 60 kHz be used there.
So the extended CP is a tool for "I want short slots but I still have long echoes". Rel-15 UEs are not required to support it; many deployments never use it.
A worked check: does the CP cover my cell?
Rule of thumb: delay spread ≈ (longest significant path − shortest path) / c. A 1 km path difference ≈ 3.3 µs. Urban macro RMS delay spreads are ~0.3–1 µs with tails to a few µs; indoor/mmWave ~50–100 ns.
- 30 kHz normal CP (2.34 µs): fine for urban macro; marginal in hilly rural with long reflections.
- 120 kHz normal CP (0.59 µs): fine for indoor and street-level mmWave.
- If you must run 60 kHz in a macro cell: extended CP (4.17 µs).
Common confusions
- "CP length is constant within a slot." Two symbols per subframe are ~0.5 µs longer.
- "Extended CP is available for any SCS." Only for 60 kHz, and it changes the slot to 12 symbols.
- "A longer CP improves throughput." It only prevents ISI; every microsecond of CP is a microsecond without data.
- "The symbol is 66.7 µs." With CP it is 71.35 µs (71.88 µs for the long symbol) at 15 kHz.
- "The CP fixes Doppler / phase noise." It addresses delay spread only; those are handled by SCS choice, TRS and PT-RS.
Spec pointers
- TS 38.211 §4.1 — T_c and κ; §5.3.1 — OFDM symbol and CP lengths (N_CP formula)
- TS 38.211 §4.2 — extended CP applicability (μ = 2)
- TS 38.101-1 / 38.104 — nothing on CP per se; deployment guidance is vendor/operator
Recap
The CP copies the symbol tail to its front so echoes never spill into the next symbol and each subcarrier can be equalised by a single multiply. Normal CP ≈ 7 % of the symbol at every numerology (4.69 µs at 15 kHz, halving with μ), with two symbols per subframe slightly longer to keep all numerologies aligned every 0.5 ms. The extended CP (≈ 25 %, 12 symbols per slot) exists only at 60 kHz, for the case of short slots with long echoes.
Interview questions
The questions below are drawn from this topic and phrased the way they tend to come up. Try each one out loud before revealing the answer.
18 questions
- A copy of the tail of an OFDM symbol prepended to its front, acting as a guard interval against multipath delay spread.
- Copying the tail makes the symbol appear periodic over the FFT window, which turns the channel's linear convolution into a circular one. That is what allows single-tap frequency-domain equalisation.
- Energy from the previous symbol leaks into the FFT window, causing inter-symbol interference, and orthogonality between subcarriers is lost, causing inter-carrier interference.
- In proportion to the symbol. Doubling the subcarrier spacing halves the symbol and halves the cyclic prefix, so 15 kHz gives about 4.7 µs and 120 kHz about 0.59 µs.
- The nominal cyclic prefix lengths do not divide the 1 ms subframe exactly. Two symbols get 16κ extra samples so the arithmetic closes precisely.
- The first symbol of each half-subframe — symbol 0, and the symbol at the 0.5 ms boundary — with the exact positions depending on numerology.
- Only at 60 kHz. It is the crossover spacing used in both FR1 and FR2, where the shortened symbol still meets significant delay spread and the extra robustness is worth the cost.
- Two symbols per slot — twelve instead of fourteen — which is roughly a 14% loss of capacity.
- Compare the CP duration against the expected delay spread, which for a macro cell scales with cell radius. A 1 km excess path length is about 3.3 µs, which already exceeds the 30 kHz cyclic prefix.
- About 7% of symbol time, essentially constant across numerologies because both the CP and the symbol scale together.
- No. It addresses delay spread in the time domain. Doppler causes frequency offset, which is handled by choosing a wider subcarrier spacing.
- Larger cells produce longer echo paths. The CP must exceed the delay spread, so cell size effectively sets a ceiling on subcarrier spacing.
- The CP must cover the difference in arrival time between the earliest and latest significant path. That difference grows with cell size, so 15 kHz suits macro cells and 120 kHz suits small cells.
- Yes. Cyclic prefix is part of the numerology configuration of a bandwidth part, so different BWPs on one carrier can differ.
- Because circular convolution in time is multiplication in frequency. With the CP intact, each subcarrier experiences a single complex gain that one multiplication corrects.
- T_c is the basic NR time unit of roughly 0.509 ns. Cyclic prefix lengths are specified as integer multiples of it, which is how the specification keeps all numerologies exactly commensurate.
- Move to a lower subcarrier spacing for a longer cyclic prefix, use extended cyclic prefix at 60 kHz, or reduce the effective delay spread through beamforming or a smaller cell.
- It costs about 7% of every symbol permanently, in exchange for turning a multipath channel into one that a single-tap equaliser can undo.
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