Cyclic prefix
Prefixing a symbol with its end to combat multipath interference.
A cyclic prefix is created by taking the end of a transmitted symbol and copying it to the front. This technique serves two main purposes. First, it acts as a guard interval that prevents interference from the preceding symbol. Second, by repeating the symbol’s tail, it makes the linear convolution caused by a frequency-selective multipath channel behave like a circular convolution. That circular convolution can then be converted into the frequency domain using a discrete Fourier transform, which simplifies tasks like channel estimation and equalization. For this to work, the cyclic prefix must be at least as long as the multipath channel’s impulse response. While traditionally used in OFDM systems, it is now also applied in single-carrier systems to improve resistance to multipath.
The principle relies on the fact that sinusoidal signals are eigenfunctions of linear, time-invariant systems. In theory, an infinitely long sinusoid would remain an eigenfunction after passing through such a channel, but real signals are always time-limited. By prefixing the symbol’s end to its beginning, the channel’s linear convolution is made to appear circular, preserving the sinusoid’s eigenfunction property in the part of the symbol after the prefix.
In OFDM, the cyclic prefix simplifies channel estimation. For a system with multiple subcarriers, the message symbols are first transformed using an inverse discrete Fourier transform, then a cyclic prefix is added. When this OFDM symbol passes through a multipath channel, the received signal is a convolution. If the cyclic prefix is long enough, the convolution becomes circular for the samples after the prefix. Taking the discrete Fourier transform of those samples yields the transmitted symbols multiplied by the channel’s frequency response at each subcarrier. This converts the multipath channel into a set of orthogonal scalar subchannels, making receiver design straightforward. Channel estimation then only requires knowing the scalar coefficients for each subcarrier, and equalization is simply a multiplication by their inverses.
- Purpose
- Provides a guard interval and enables circular convolution modeling
- Minimum length
- At least equal to the length of the multipath channel
- Associated systems
- Traditionally OFDM, now also used in single carrier systems
Lore & Background
The concept of a cyclic prefix is traditionally associated with OFDM systems, but it is now also used in single carrier systems to improve robustness to multipath propagation. The principle relies on the fact that sinusoidal signals are eigenfunctions of linear and time-invariant systems; a sinusoid of infinite duration would be an eigenfunction, but real signals are time-limited. Prefixing the end of the symbol to the beginning makes the linear convolution of the channel appear as though it were circular convolution, preserving this property in the part of the symbol after the cyclic prefix.
In OFDM, cyclic prefixes combat multipath by making channel estimation easy. An OFDM symbol is constructed by taking the inverse discrete Fourier transform of the message symbol, followed by cyclic prefixing. When the cyclic prefix length is at least equal to the channel impulse response length, the convolution becomes circular convolution at the relevant sample values. Taking the discrete Fourier transform of these values converts the multipath channel into scalar orthogonal sub-channels in the frequency domain, simplifying receiver design considerably.
Reader's Guide
The cyclic prefix is significant because it transforms a multipath channel into scalar orthogonal sub-channels in the frequency domain, thereby simplifying receiver design considerably. Channel estimation is simplified, as only estimates of the scalar coefficients for each subcarrier are needed for the duration in which the channel does not vary significantly. Multiplying the received demodulated symbols by the inverse of these coefficients yields estimates of the transmitted symbols. This technique is fundamental to OFDM systems and has been extended to single carrier systems to improve robustness to multipath propagation. The cyclic prefix must have a length at least equal to the length of the multipath channel to serve its objectives. Its legacy lies in enabling efficient frequency-domain processing for channel estimation and equalization in modern wireless communications.
Did You Know?
- The cyclic prefix is the prefixing of a symbol with a repetition of its end.
- It allows linear convolution of a frequency-selective multipath channel to be modeled as circular convolution.
Frequently Asked Questions
Who is Cyclic prefix?
Cyclic prefix is a signal-processing technique that duplicates the tail of a transmitted symbol and prepends it to the front of that same symbol. It is a core building block of OFDM and has since been adopted in single-carrier systems as well.
What are Cyclic prefix's powers and role?
It fulfills two jobs simultaneously: it acts as a guard interval that blocks inter-symbol interference from the preceding symbol, and it converts the linear convolution of a frequency-selective multipath channel into a circular convolution. That circular behavior is what unlocks frequency-domain processing via the DFT.
Why is Cyclic prefix important to the modulation community?
Without it, multipath propagation would smear symbols into one another and force engineers to rely on heavy time-domain equalizers. By guaranteeing circular convolution, it lets the DFT serve as an efficient bridge between time and frequency domains, keeping the whole OFDM pipeline tractable.
What is Cyclic prefix's minimum length requirement?
Its duration must be at least as long as the multipath channel's delay spread. If it falls short, the guard interval no longer absorbs all echoes and the circular-convolution assumption breaks, undermining both interference protection and DFT-based equalization.
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