# How is symbol synchronization with OFDM done?

I'm trying to understand how symbol synchronization is done in OFDM using pilot tones, cyclic prefixes, or any other technique.

I've read the following answers which provide some explanation, but I still don't totally understand it.

How to demodulate an OFDM signal

How to estimate the number of taps needed for subsequent channel estimation algorithms?

Specific questions:

1) How is a pilot tone found? What makes it different than the regular data on a sub-carrier? How can it be used to determine symbol starts and ends?

2) If I understand the answers above correctly, a cyclic prefix can be used to find the symbol start/end because it will auto-correlate with some delay. However, the cyclic-prefix exists in order to "absorb" ISI. So if the prefix has been munged with ISI, then how can this auto-correlation be successful?

• Tere is any formula to find the pilot signal loation?? or How can we know where we should place the pilot signal?? – user16047 Jun 1 '15 at 6:59

Regarding you're general question about how symbol sychronization is done in OFDM systems:

1. One of the most popular and frequently used techniques is the transmission of one or several pilot symbols that are known in the receiver. A pilot symbol is a complete OFDM symbol where the value of each subcarrier is predefined and known in transmitter and receiver. It is repeated with a certain rate that depends on how fast the channel changes. The received signal is correlated with the pilot symbol to detect the OFDM symbol start. It can also be used for channel estimation. Schmidl and Cox have introduced in [1] a pilot symbol based technique where the pilot symbol has a special symmetry so that the pilot symbol need not to be known at the receiver.

2. As Jason R has noted in his comment, although it is not its initial purpose, the cyclic prefix can also be used for symbol synchronization because its a known repetition of some part of the received signal that can be detected through autocorrelation. It is especially well-suited for fast-changing channels, because the delay time can be updated on a per-symbol basis. Additionaly, it does not add any additional overhead. However, it is more sensitive to noise [2] and presumably also to ISI.
Edit: The maximum delay that can be detected by this method is the lenght of one OFDM symbol. It's therefore only suited for fine synchronization.

3. There are some more "exotic" techniques. In one of these, for instance, the N-DFT (N = number of subcarriers) of time-shifted versions of the received signal is calculated. If you apply the DFT to the wrong time window, the resulting constellation diagram will be a mess. If you got the correct time window, the constallation digaram shows distinct constellation points. This can be detected by calculating the standard deviation of the DFT output. This method implies a high computational cost.

Regarding you specific questions

How is a pilot tone found? What makes it different than the regular data on a sub-carrier? How can it be used to determine symbol starts and ends?

Once you have synchronized the received signal the pilot tones are at predefined bins of the DFT. When designing the system the location of pilot tones in the spectrum is fixed. There are more complex schemes, where the location of the pilot tones changes in a predefined pattern to get a good approximation of the channel in both frequency and time domain. Pilot tones cannot be used for synchronization, because the received signal has first to be synchronized before you can even extract the pilot tones in frequency domain. Assume that a wrong time window is used: ortogonality of subcarriers will be lost and the result of DFT is some mixture of two consecutive OFDM symbols. This is a nonlinear effect and the pilot symbols cannot be extracted from this mixture. Pilot tones are used for channel estimation and sometimes phase noise mitigation.
Edit: As Jim Clay has pointed out in his comments, fine synchronization through pilot tones is possible if a coarse value for the delay is known and the residual delay does not exceed the length of the cyclic prefix.

If I understand the answers above correctly, a cyclic prefix can be used to find the symbol start/end because it will auto-correlate with some delay. However, the cyclic-prefix exists in order to "absorb" ISI. So if the prefix has been munged with ISI, then how can this auto-correlation be successful?

Like all synchronisation techniques this method will suffer from noise and channel dispersion and consequently will only work to some extent of the beforementioned effects. Quantifying to which extent exactly it is still working would require some thorough research that somone has certainly already done.

[1] Schmidl, T.M.; Cox, D.C.; , "Robust frequency and timing synchronization for OFDM," Communications, IEEE Transactions on , vol.45, no.12, pp.1613-1621, Dec 1997

[2] van de Beek, J.J.; Sandell, M.; Borjesson, P.O.; , "ML estimation of time and frequency offset in OFDM systems," Signal Processing, IEEE Transactions on , vol.45, no.7, pp.1800-1805, Jul 1997

• +1. You can also correct symbol-by-symbol if you use the phase offset in the pilot sub-carriers. – Jim Clay Feb 2 '13 at 17:27
• @JimClay I don't understand how this could work. The pilot subcarrier cannot be detected if the receiver is not already synchronized. And even if it could, how could you tell whether the phase change has been caused by time delay or by channel dispersion? Maybe I'm missing sth. here... – Deve Feb 2 '13 at 19:11
• You're right that you have to be synchronized enough to get your inverse FFT's worth of data within the boundaries of the symbol. I meant that the pilot tones can help fine tune the synchronization by indicating exactly where the data came from, relative to the beginning of the symbol. – Jim Clay Feb 2 '13 at 19:41
• The phase change that comes about from timing offset is different than the phase change that comes about from carrier offset. The timing offset phase change is proportional to the frequency bin, so for 802.11a the -7 bin will have the opposite phase change as the +7 bin. Likewise, the +21 bin will have 3 times the phase offset as the +7 bin. With carrier offset I believe that all of the bins have the same phase offset. Thus, by analyzing the phase offsets of the pilot tones you can determine both the time offset and the carrier offset. – Jim Clay Feb 2 '13 at 19:44
• I agree: time delay causes a linear phase shift, and frequency deviation causes a constant phase shift. So if coarse synchronization is assumed, pilot tones can be used for fine synchronisation. Thanks for the clarification! – Deve Feb 2 '13 at 20:47
How is a pilot tone found?


The location of pilot tones in terms of sub-carriers is defined by the signal protocol. For instance, in the case of 802.11a the pilot sub-carriers are -21, -7, 7, and 21.

What makes it different than the regular data on a sub-carrier?


It is different in that the receiver knows exactly what the pilot tone contains. There is no uncertainty other than noise and distortion brought on by carrier offset, symbol (timing) offset, channel effects (e.g. multi-path) , etc.

How can it be used to determine symbol starts and ends?


Circular shifts (sometimes called "barrel" shifts) produce phase offsets in FFT's. The cyclic prefix prepends the end of the symbol for the exact purpose of making a time shift a circular shift. Thus, when the inverse FFT is performed, any time offset will create a phase offset in all of the channels. Because we know exactly what the pilot tones should be, the phase offset (which corresponds to a time offset in the original symbol) can be detected and corrected.

If I understand the answers above correctly, a cyclic prefix can be used to find the
symbol start/end because it will auto-correlate with some delay.


Again, it is not an auto-correlation thing, it is that the inverse FFT translates the time shift into a phase shift that we can use the pilot channels to detect.

However, the cyclic-prefix exists in order to "absorb" ISI. So if the prefix has been
munged with ISI, then how can this auto-correlation be successful?


Without multi-path there is no ISI with OFDM signals. The only ISI they have to worry about is when there is a delayed multi-path signal that interferes with the primary signal. They intentionally make the cyclic prefix longer than any "normal" multi-path delay, so there is almost always an intact FFT's worth of uncorrupted data.

• Actually, autocorrelation can be used for timing recovery in OFDM systems. Since the cyclic prefix is just a repetition of the beginning of the symbol, and the distance between the beginning of the symbol and the cyclic prefix is known, you can calculate a sliding autocorrelation at the known cyclic prefix offset in order to detect the instant that the symbol starts. – Jason R Feb 2 '13 at 2:45
• That's a good point, though a multi-path signal would tend to mess that up. – Jim Clay Feb 2 '13 at 3:44
• "-21, -7, 7, and 21" are these FFT bin numbers relative to a center bin? So certain sub-carriers are used for pilot tones exclusively rather than data? – Dan Sandberg Feb 2 '13 at 10:12
• Still missing something -- it seems like if the pilot tones contain a sequence with good cross-correlation properties, you could find the symbol boundaries perfectly from that. So then why would you need to look at the phase change using the cyclic prefix? – Dan Sandberg Feb 2 '13 at 10:50
• Pilot tones and pilot symbols should not be confused here. A pilot tone is certain subcarrier that is modulated with a known value in each OFDM symbol. I doubt that it can be used for synchronization. A pilot symbol is a complete OFDM symbol with predefined content. It can be used for synchronization. – Deve Feb 2 '13 at 14:53

Synchronization is an important task in practical communication systems but it is not directly related to the theory of OFDM.

# Frame Synchronization

Practical communication systems (such as IEEE 802.11 or 802.3) exchange so-called frames, which consist of several fields, which in turn accomplish different, specific tasks. Typically, the first field of a frame is a so-called preamble, which has the mere purpose of

• detecting arriving frames,
• synchronizing the receiver with the transmitter,
• performing automatic gain correction (AGC) at the receiver (required in wireless communication systems).

The preamble typically consists of a Barker sequence, which is a binary code with minimal off-peak autocorrelation. This code doesn't even necessarily have to be OFDM-modulated, but it may be BPSK-modulated on a single carrier within the available frequency band. The receiver applies a matched filter to the incoming stream of samples. If the matched filter's output exceeds a specific threshold, it is very likely that it has detected an incoming preamble. As the Barker code's off-peak autocorrelation coefficients are minimal, the peak of the matched filter's output provides the required information to align the subsequent fields of the frame with the receiver's FFT.

# Training Sequence

After the preamble, the next field of a frame is typically some sort of an OFDM training sequence. The main purpose of training sequences is to estimate channel coefficients of individual subcarriers, not synchronization. Some protocols distinguish also between long and short training sequences, whereas a long training sequence can be found directly after the preamble and short training sequences are spread in the rest of the frame. Generally, the receiver knows in advance

• the positions of training sequences in the frame and
• the values of the pilot symbols contained in the training sequences.

As the channel coefficients may change over time due to mobility of nodes and obstacles in the environment, they have to be re-estimated within the so-called coherence time, which is accomplished by short training sequences (i.e., pilot symbols) between payload OFDM symbols. The coherence time can be approximated as the inverse of the maximum Doppler spread. Also, in some protocols, training sequences are transmitted only on a few, equally-spaced subcarriers, while all other subcarriers in between continue payload transmissions. This works since the channel coefficients of neighboring subcarriers are correlated to each other. The coherence bandwidth of a fading channel can be estimated as the inverse of the channel delay spread.

Also note that in practical systems, the pilot symbols may also be used for other purposes, such as to estimate the SNR of individual subcarriers or to perform estimation of the carrier frequency offset (see below).

# Cyclic prefix

The main purpose of the cyclic prefix inserted between successive OFDM symbols is mitigation of ISI (Inter-Symbol-Interference) and ICI (Inter-Carrier-Interference), not synchronization or determining symbol starts or ends.

## Mitigation of ISI

Due to multipath propagation, multiple copies of the transmitted waveform arrive at the receiver at different time instants. Hence, if there was no guard space between successive OFDM symbols, a transmitted OFDM symbol may overlap with its subsequent OFDM symbol at the receiver, causing ISI. Inserting a guard space between successive OFDM symbols in the time domain mitigates this effect. If the guard space is larger than the maximum channel delay spread, all of the multi-path copies arrive within the guard space, keeping the subsequent OFDM symbol unaffected. Note that the guard space may also contain zeros to mitigate the effect of ISI. In fact, no cyclic prefix is required in the guard space in any digital communication technique to mitigate the effect of ISI.

## Mitigation of ICI

In OFDM, guard spaces are filled with a cyclic prefix to maintain orthogonality between subcarriers on condition that multiple delayed copies arrive at the receiver due to multi-path propagation. If the guard space was actually filled with zeros at the transmitter, the multiple copies arriving at the receiver would be non-orthogonal (i.e., somehow correlated) to each other, causing ICI.

# Carrier Frequency Offset (CFO) and Phase Noise

In practical systems, the transmitter's and the receiver's carrier frequency oscillators typically have a slight offset in frequency, which causes a phase drift over time. In addition, the power spectral density of a practical oscillator is not an ideal delta function, resulting in phase noise. Phase noise causes the CFO to continuously change, resulting in a change of the phase drift's speed and direction. There are various techniques to resynchronize the receiver to the received signal, i.e., to track the phase of the incoming signal. These techniques may additionally exploit the presence of pilot symbols in the signal, and/or apply blind estimation and correlation techniques.

I also maintain an open-source OFDM framework for software defined radios, which covers the techniques described above in Matlab code.

• I'm unsure about the terminology. Would "OFDM symbol" be a synonym for "field"? – sellibitze Nov 5 '13 at 15:33
• Also, I'm not always sure about what you mean by "synchronization" because there are so many kinds of synchronizations (firequency, symbol, frame). – sellibitze Nov 5 '13 at 15:35
• I'm unsure about what you mean by "field". By the term "OFDM symbol", I mean the sequence of samples you get by calculating the IDFT of an array of complex values (which are the symbols on the subcarriers). Synchronization is all about getting such a sequence of samples correctly aligned at the receiver. – Robin Klose Nov 6 '13 at 17:07
• I'm clear on the meaning of "OFDM symbol". But you used the word "field" in the second sentence of your answer ("...so-called frames, which consist of several fields..."). – sellibitze Nov 12 '13 at 15:23
• I see. By "field" I meant a portion of a frame which accomplishes a specific task. So no, "OFDM symbol" would not be a synonym for "field". But a field may contain several OFDM symbols if that field contains payload data or pilot symbols. – Robin Klose Nov 12 '13 at 22:29

To roughly summarize Deve & Jim Clay's excellent responses:

Symbol Synchronization consists of two different tasks -- rough symbol synchronization, where the symbol boundaries are approximated, and fine symbol synchronization, where the rough synchronization is slightly adjusted. Often the fine synchronization is less computationally intensive and so can be done more frequently to adjust for changes in the channel.

Pilot symbols, which are special predefined symbols that are known to the transmitter and receiver can be used to do rough synchronization by searching for the symbol in the time-domain ("auto-correlation")

The phase of a sub-carrier should change in a predictable way from one window to the next. For example, in BPSK, the phase should be 0 or pi radians away from its expected value from one window to the next. By trying different window positions, and testing multiple sub-carriers (for better noise immunity) rough symbol synchronization can be achieved. This is an "exotic" method.

Cyclic prefixes, which are a continuation of the symbol that is prefixed to the beginning, can be used for fine correlation through auto-correlation.

Pilot tones are specific sub-carriers that are chosen ahead of time. They carry a specific repeating pattern. They are used for channel estimation and additionally can be used for fine synchronization.

• Some additions: 1) Pilot symbol based synchronization can also yield sufficiently exact sync but can't be updated very frequently due to the overhead. This might be ok for slow varying channels but fast varying channels require some additional technique for updating the delay time more often. That said, pilot symbol sync does not necessarily use a subsequent fine sync method whereas the fine sync methods need some initial rough estimate for delay time. 2) The "exotic" method that I numbered 3 shifts the incoming signal on a per-sample basis. I would consider it a "fine sync" method. – Deve Feb 9 '13 at 8:13