Signal Processing

Cyclic Prefix

Q: How long should the cyclic prefix be relative to the channel delay spread?

The cyclic prefix duration must equal or exceed the maximum excess delay of the channel impulse response. If the longest multipath echo arrives within the CP window, all delayed copies of the previous useful symbol fall inside the guard and are discarded by the receiver, leaving the FFT window free of intersymbol interference. In LTE the normal CP is about 4.7 microseconds, which covers roughly 1.4 km of differential path length and is sized for typical urban and suburban delay spreads. The extended CP of 16.67 microseconds is reserved for large cells and heavy multipath such as single-frequency broadcast networks where echoes from distant transmitters can exceed 5 km. If the delay spread exceeds the CP, the orthogonality between subcarriers breaks down and both intersymbol and intercarrier interference appear as an irreducible error floor.

Q: Why does the cyclic prefix make the channel a circular convolution instead of a linear one?

A linear time-invariant channel performs linear convolution between the transmitted samples and the channel impulse response, which mixes energy from one OFDM symbol into the next. By copying the last L samples of the N-sample symbol to the front, the transmitted block becomes periodic over the FFT window. When the multipath spread is shorter than the CP, the received useful portion looks exactly as if the symbol had been convolved circularly with the channel. Circular convolution in time corresponds to element-wise multiplication in the DFT domain, so each subcarrier simply experiences a single complex gain H[k]. This is what allows a one-tap frequency-domain equalizer to undo the channel by dividing each subcarrier by its estimated H[k].

Q: What is the spectral and SNR cost of the cyclic prefix?

The CP carries no new information, so it consumes both bandwidth and transmit energy. The overhead fraction is L divided by (N plus L), where L is the CP length and N is the FFT size. For LTE with N equal to 2048 and a normal CP averaging about 144 samples, the overhead is roughly 7 percent, which reduces throughput and costs about 0.3 dB of SNR. Extended CP at 512 samples raises the overhead to 20 percent. 5G NR keeps the CP overhead near 7 percent across all numerologies by scaling the CP length with the subcarrier spacing, so a 30 kHz numerology has a CP near 2.3 microseconds and a 120 kHz numerology near 0.59 microseconds. Choosing the CP is a trade between robustness to delay spread and the efficiency penalty.

/SY-klik PREE-fiks/ (CP)

Copying the last samples of each OFDM symbol and prepending them to the front creates a periodic guard interval that absorbs channel echoes. When this guard is at least as long as the channel delay spread, every multipath copy of the previous symbol arrives inside the discarded guard window before the receiver's FFT window opens, so intersymbol interference vanishes. Inside the window the channel behaves as a circular convolution, which collapses to a single complex multiplier per subcarrier and lets a one-tap frequency-domain equalizer flatten the response. In LTE the normal CP runs about 4.7 μs (144 samples at a 30.72 MHz clock), the extended CP 16.67 μs, while 5G NR scales the CP with numerology to hold overhead near 7%. The price is that guard samples carry no payload, costing roughly L / (N + L) in spectral efficiency and about 0.3 dB of SNR.

Category: Signal Processing

LTE Normal CP: ≈ 4.7 μs

Typical Overhead: ≈ 7%

How the Cyclic Prefix Defeats Multipath in OFDM

In a multicarrier system the transmitter builds each symbol by inverse-FFT of N modulated subcarriers, producing N time samples. A dispersive radio channel smears each transmitted sample across several sample periods, so the tail of one symbol bleeds into the head of the next. Two distinct problems follow: intersymbol interference (ISI) between consecutive symbols, and intercarrier interference (ICI) because the FFT window no longer sees an integer number of cycles of every subcarrier. The cyclic prefix solves both at once by inserting a guard that is a verbatim copy of the symbol's own tail, so the waveform looks periodic across the entire FFT observation window.

Because the guard reproduces the symbol's ending samples, the receiver can discard the corrupted CP region and still capture a clean block in which the linear convolution of the channel is indistinguishable from a circular convolution. That equivalence is the whole point. Circular convolution in the time domain maps to element-wise multiplication in the frequency domain, so after the FFT each subcarrier k simply carries the transmitted symbol scaled by the channel frequency response H[k]. Recovery reduces to dividing by an estimated H[k], a single complex tap per subcarrier rather than a long time-domain filter.

The guard length is the central design parameter. It must exceed the maximum excess delay of the channel impulse response; otherwise late echoes spill past the CP boundary into the FFT window and the subcarrier orthogonality collapses, producing an irreducible error floor that no equalizer can remove. Sizing therefore follows the deployment: short CP for small cells and dense urban microcells with modest delay spread, extended CP for large rural cells and single-frequency broadcast networks where echoes from distant transmitters can span many kilometers.

Cyclic Prefix and Guard Interval Equations

Guard Sizing Condition:
T_CP ≥ τ_max  (CP duration ≥ maximum channel excess delay)

Total Symbol Duration:
T_sym = T_u + T_CP = (N + L) × T_s

Spectral / Energy Overhead:
η_CP = L / (N + L)  →  SNR loss ≈ −10·log₁₀(N / (N + L)) dB

Frequency-Domain Channel (per subcarrier):
Y[k] = H[k] × X[k] + W[k]  →  X̂[k] = Y[k] / H[k]

Where N = FFT size, L = CP length in samples, T_s = sample period, T_u = useful symbol time, τ_max = max excess delay, H[k] = channel response. Example: LTE 20 MHz, N = 2048, L ≈ 144 → η_CP ≈ 6.6%, SNR loss ≈ 0.30 dB.

Cyclic Prefix Configurations Across Standards

Standard / Mode	Subcarrier Spacing	Useful Symbol T_u	CP Duration	CP Overhead	Typical Use Case
LTE normal CP	15 kHz	66.7 μs	≈ 4.7 μs	≈ 7%	Urban / suburban macro
LTE extended CP	15 kHz	66.7 μs	16.67 μs	20%	Large cells, MBSFN
5G NR μ=0	15 kHz	66.7 μs	≈ 4.7 μs	≈ 7%	Sub-1 GHz wide-area
5G NR μ=1	30 kHz	33.3 μs	≈ 2.3 μs	≈ 7%	Mid-band (3.5 GHz)
5G NR μ=3	120 kHz	8.33 μs	≈ 0.59 μs	≈ 7%	mmWave (24 to 40 GHz)
802.11a/g	312.5 kHz	3.2 μs	0.8 μs	20%	Indoor WLAN

Common Questions

Frequently Asked Questions

How long should the cyclic prefix be relative to the channel delay spread?

The CP duration must equal or exceed the channel's maximum excess delay so every late echo of the previous symbol lands inside the guard and is discarded before the FFT window. LTE's normal CP of ≈ 4.7 μs covers roughly 1.4 km of differential path and suits urban and suburban delay spreads; the extended CP of 16.67 μs targets large cells and single-frequency broadcast where echoes can exceed 5 km. If delay spread overruns the CP, subcarrier orthogonality breaks and an irreducible ISI/ICI error floor appears.

Why does the cyclic prefix make the channel a circular convolution instead of a linear one?

A multipath channel performs linear convolution, which leaks energy from one symbol into the next. Prepending the last L samples of an N-sample symbol makes the transmitted block periodic over the FFT window, so when the spread is shorter than the CP the received useful portion looks exactly like a circular convolution with the channel. Circular convolution in time equals element-wise multiplication in the DFT domain, giving each subcarrier a single complex gain H[k] that a one-tap equalizer can divide out.

What is the spectral and SNR cost of the cyclic prefix?

The CP carries no payload, so it spends both bandwidth and transmit energy. The overhead fraction is L / (N + L). For LTE with N = 2048 and a normal CP averaging ≈ 144 samples the overhead is about 7% and the SNR penalty about 0.3 dB; extended CP at 512 samples raises overhead to 20%. 5G NR scales CP length with subcarrier spacing so overhead stays near 7% across numerologies, from ≈ 2.3 μs at 30 kHz to ≈ 0.59 μs at 120 kHz.

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