# OFDM symbol synchronization

In an OFDM receiver, how could we find the start of the symbol for FFT operation. I know the CP-based method to find the start of symbol. However, what if the sampling is off by some duration. We will not get the same digital signal at the transmitter. How serious is that issue? Is it simply addressed in the channel estimation in the frequency domain?

• First use correlation between two duration of signal separated by FFT window to find the start of OFDM symbols. This works mediocrely and you will not get the start of OFDM symbol. Nonetheless, you will probably fall within the CP so that Inter Symbol Interference (ISI) can be avoided. Next, both data and pilots suffers the offset to the real symbol beginning, hence the estimated channel by using pilots include the effect of the offset and, therefore, will be compensated by simple channel equalization. Apr 21 '20 at 15:21

Asuume your FFT/IFFT size is $$N$$, and CP length is $$N/4$$, and channel taps is $$L$$. The condition of avoiding ISI is $$L \le N/4$$. So if you start your FFT processing from $$n=L$$ to $$n=N/4$$, you can avoid ISI as well as meet the circular convolution condition. Fine symbol timing offsets result in phase shift $$e^{-j2\pi \frac{\Delta t}{T} k}$$ after FFT if you take symbols from good region for FFT . This means rotation of data after FFT, which linearly increases for each sub carrier $$k$$, which you can estimate and correct.

If you take symbols for FFT outside good region, you will end up getting ICI(Inter carrier Interference for $$n \gt N/4$$) or ISI from previous OFDM symbol $$n \lt L$$. This results in severe degradation even at high SNR.

• So basically, in time domain, we try to fix the starting point within the good region. It is in the frequency domain we compensate for the fine misalignment? I was baffled by the completely rotated constellation. I thought it is not possible to recover. Because it is not a small constant rotation. Apr 21 '20 at 19:47
• Yes, if we know the power delay profile of channel we can start anywhere in good region and take N samples for FFT. It is possible to find out the fine timing offset by using pilot symbols phase because of linear relationship with subcarrier position index k. Apr 22 '20 at 1:27

CP based Symbol Boundary detection algorithm which is fundamentally based on correlation will still work. You just need to do a timing recovery before FFT operation.

A small offset in sampling times will cause a small phase rotation in pilots and these small rotations are used to compute the timing offset.

Once you have a good timing offset estimate, you do timing recovery on the received samples of OFDM symbol. Timing recovery is nothing but doing some low cost polynomial interpolation to get the actual values of the samples at correct sampling times.

Or in OFDM once you have these timing offsets as phase rotation, derotate rest of the data symbols after FFT before slicing to get the complex QAM symbols.