Talk:Cyclic prefix

Technical inaccuracy
This article is, at best, misleading. All three comments below are right and have been integrated to the main page. Mel aad (talk) 16:41, 22 March 2010 (UTC)

The cyclic prefix (CP) is not "to allow multipath to settle before the main data arrives at the receiver". It serves two purposes:
 * As a guard interval, it eliminates inter-symbol interference from the previous OFDM symbol.
 * Makes the linear convolution of a frequency-selective channel look like circular convolution, so that sub-carrier orthogonality is maintained (so no inter-carrier interference), allowing frequency-domain channel estimation and equalisation to be performed.

Frequencies do not "become"' orthogonal to one another.

The length of the CP is not "often equal to the guard interval", it is the guard interval. If the guard interval were longer than the CP, the circular convolution principle would break down, and the CP would be useless. Oli Filth 00:41, 1 September 2006 (UTC)

All three comments above are right and being integrated to the main page. Mel aad (talk) 16:14, 22 March 2010 (UTC)

I personally would avoid writing something like 'makes the linear convolution look like circular convolution'. That is somewhat vague. It would be much more meaningful by saying something like adding a cyclic prefix allows the modified data sequence (that now contains the CP on the front end) to propagate through the channel (by means of the usual linear convolution style), and the received sequence (at the receiver side) will contain (within a section of the total received sequence) a pattern that matches the 'circular convolution' between the raw data set (without the CP added) and the channel impulse response. And the length of the circular convolution (eg Mod N circular convolution) is assumed to be the length of the raw data set. And, importantly - if any old arbitrary sequence were attached to the front end of the original raw data set (other than a CP of suitable length), then the result of the linear convolution will not contain (within a portion of the total received sequence) a pattern that matches the circular convolution result. If only sources attempt to explain things properly - rather than leave people scratching their heads - then it would be great. But that's the life story of so many documents - it's useless if things aren't explained properly. KorgBoy (talk) 00:29, 14 December 2018 (UTC)

Some math to clarify the CP use
I have tried to add some rudimentary stuff. Please note that it may not be clear, or consistent with the pattern of math symbol use. If so, please correct it or bring it to my notice. Thanks. Kumar Appaiah 14:45, 11 March 2007 (UTC)


 * Added a few more things in a hurry, forgot to even label the changes! Sorry. I think I'll stop now, and let the experts take over. Comments would be appreciated! Kumar Appaiah 15:10, 11 March 2007 (UTC)

Please let me add to your comments...

The cyclic prefix may be viewed in more general terms. Transmitting a periodic waveform makes linear and circular convolution the same. This leads to frequency-domain processing at the receiver possible with the DFT. This isn't unique to OFDM. Indeed, an active area of research is single carrier with frequency-domain equalization (SC-FDE). I don't have time right now to contribute SC-FDE to Wikipedia, but someone will, I'm sure, in the future. -- sct 2007-05-17

clarification!
Can someone clarify if the cyclic-prefix is equivalent or is the same as the symbol extension.

This symbol extension is specified in the standard as ¼, 1/8, 1/16 or 1/32 of the symbol time to provide flexibility for different environments, but the WiMAX Forum specifies as a mandatory implementation the 1/8 fraction. The maximum multipath spread distances supported for symbol duration of 102.86 μs are shown in the table below.

In that case, according to my understanding, then the cyclic prefix can be used to determine the distances a signal 'should' be expected to optimally cover.Kendirangu (talk) 17:10, 10 March 2008 (UTC)

How does a receiver recognize cyclic prefix samples?
I am pondering this article from an OFDM perspective. The article states "the receiver is typically configured to discard the cyclic prefix samples". To me this statement begs the question - how does a receiver recognize cyclic prefix samples? Please correct any of the following suppositions:

1) The receiver samples for a period equal to the inverse of the symbol rate (which is constant regardless of sub-carrier frequency).

2) A symbol transmitted on an arbitrary sub-channel (call it S) will be delayed by some amount of time within the delay spread before the sampling receiver grabs it.

3) The delay spread is arbitrary depending on channel conditions, multipath, distance between transmitter and receiver, etc.

4) Some of what the receiver samples as S will be cyclic prefix (unless the delay from the transmitter to the receiver for S equals the guard interval in which case the receiver misses the cyclic prefix altogether).

5) The length of the delay of S from the transmitter to the receiver we do not know, we only know that it is somewhere within the delay spread (which is hopefully less than the guard interval).

I am not sure how the delay spread is determined or if it necessary to know it to any precision. But, if all of my above suppositions are correct, how can the receiver know what to discard as cyclic prefix samples unless said receiver somehow knows what the delay (from transmission to reception) of S was to some reasonable precision? If the receiver does know this delay, how does it know it? Does the transmitter occasionally send a pilot at the same sub-carrier frequency as S and the receiver use this pilot to estimate the delay? If so, what if that delay changes between when the last pilot is received and the next symbol is received. To me it all comes down to knowing what that delay is, otherwise how could the receiver know (if S were a QPSK symbol for example) if the symbol val is 00, 01, 10, or 11 since knowing that is all based on phase relative to the sub-carrier for S - phase which is offset by said time delay?

Anyone who might be able to better explain this, please respond. Thanks in advance. — Preceding unsigned comment added by BillyBarty68 (talk • contribs) 02:19, 10 December 2010 (UTC)


 * You are correct, in general, we don't know the absolute delay at the receiver. But that doesn't matter; all the receiver needs to do is ensure that it correctly "opens" its sampling window such that it picks up 1/f worth of waveform (where f is the sub-carrier frequency spacing).  It is able to do this because the transmitted waveform is typically interspersed with pilot symbols, which the receiver can correlate for and determine symbol timing.  Once it has identified the location of the pilots in time, it can re-align its sampling window to match.  This alignment will typically serve well enough for the next few data symbols, until the next pilot is transmitted.  In a highly mobile channel (where the delay and multipath vary quickly), more complicated approaches are required, such as interpolating between successive delay estimates, predictive delay tracking, or data-aided estimation (whereby the receiver effectively compares what it received against a resynthesised waveform based on what it's managed to demodulate).


 * In some systems, such as DAB, there are no pilots. The receiver will typically synchronise by using the fact that the cyclic prefix is a repetition of another part of the symbol, and hence can run a sliding auto-correlator and look for peaks.  This approach is less accurate, but typically doesn't matter because the simpler modulation scheme of DAB (DQPSK) doesn't require a high-performance receiver in order to demodulate adequately.  Oli Filth(talk&#124;contribs) 21:01, 10 December 2010 (UTC)


 * Oli_Filth, thank you for the reply. I think here is the nub of my confusion....  My information on OFDM (text from a professional education course on WiMAX) says the receiver uses pilots for frequency synchronization whereas you have indicated that the receiver will identify "the location of the pilots in time" (emphasis mine) and then "re-align its sampling window to match".  My information gives no indication that pilots are used for time synchronization - my information only shows a "time synchronization" black box in the receiver following the A-to-D conversion.  After time synchronization, the cyclic prefix is removed (makes sense - you wouldn't want to remove the cyclic prefix before time synchronization, right?), then the signal is converted into the frequency domain via an FFT, and then the "pilots are enlisted to maintain frequency synchronization".  So I think it comes down to my desire to know with more certainty what constitutes that "time synchronization" black box.  Your reply would suggest that the pilots would be used to do the time synchronization inside that black box whereas my information only says, "After the initial A-to-D conversion, the receiver is synchronized to the signal" - making no mention of pilots.  perhaps my information is simply incomplete and pilots are used for time as well as frequency synchronization.  If that is so, for me that opens up some more questions:


 * -Are pilots included in the transmitter's cyclic prefixing operation? Does it matter whether they are or not?


 * -If pilots are used for time synchronization, how might that look as far as FFT's, filtering, IFFT's, etc. inside the time synchronization black box?


 * Any further enlightenment would be greatly appreciated. :o) — Preceding unsigned comment added by BillyBarty68 (talk • contribs) 23:26, 13 December 2010 (UTC)


 * That's the main issue. Text book and online sources don't appear to truly 'properly' teach or show clearly what OFDM synchronisation actually involves. There are certainly articles and papers that discuss 'synchronisation'. But one will often find that the diagrams are not clearly explained, and information is cryptic. Sources that properly show a step-by-step example with truly useful guiding side-information appear to be rare, or don't exist. The first issue is, material about cyclic prefix is introduced, but nobody teaches anything like "ok....here is a complex-valued sequence with a cyclic pre-fix, and we now use the real part and imaginary part of each element of this set of sequential elements to modulate cos and sin carriers, for mixing up to a frequency transmittable by an antenna. Then, at the receiver....we need do this...followed by this....and this...". It's the same story with many systems. It's not that students or exponents aren't capable of understanding it all. The problem is that there are no sources that tell it like it really is. KorgBoy (talk) 20:16, 2 March 2018 (UTC)

I will also add that the question asked by Billy is an excellent one. Text books and sources don't appear to show what the incoming received sequence (in some form) is meant to look like. OFDM discussions involving cyclic prefix often involve confusing examples, such as x = [a b c d e x x x x x a b c d e] (with example prefix of 'a b c d e' with channel impulse response h = [p q], and leads people to believe that the received signal will just be linear convolution between x and h. But there's a catch. The values 'a', 'b', 'c' etc each represent a complex vector symbol, and the duration of each of those vector symbols is typically much longer than the impulse response 'h'. So things might make more sense in the discrete-time domain (when generating simulated received sequences) by choosing a sampling period that is much smaller than the rate that which 'a', 'b', 'c' etc get clocked out. That is, the channel impulse response 'h' is generally meant to be much shorter than the duration of 'a' or 'b' or 'c' etc. So everything becomes clear if we break down the sequence 'x' into finer resolution samples prior to carrying out linear convolution. KorgBoy (talk) 00:41, 14 December 2018 (UTC)

Error?
After convolution the correct range for m is one period of length N and

L-1 <= m  <= N - 1 + (L-1)  — Preceding unsigned comment added by 12.5.37.115 (talk) 17:49, 7 August 2015 (UTC)

Missing some on CP/GI length
Could someone add information on Cyclic Prefix/Guard Interval length to the article? I.e. why should you choose for what value of L. To minimize overhead, L should be as small as possible of course. And larger then N also seems illogical. But what then, and why? JHBonarius (talk) 10:42, 8 June 2016 (UTC)


 * Also, why is the channel impulse response represented by "L" points? The article has "X" in the frequency domain with N points (ie. 0 to N-1 index), and "H" with L points. So what's going on here regarding the X(k)H(k) product shown in the article? How can you multiply the two when they're different lengths? If they're going to say that the channel impulse response has "L" elements, then at least explain it, and comment on how to treat it - such as later assume that "H" is extended to N elements(where element 0 to L-1 are whatever they are supposed to be, and then the remaining elements up to index N-1 are assumed to be all zero. KorgBoy (talk) 19:53, 2 March 2018 (UTC)