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