# OFDM coarse freq estimation

I am trying to read OFDM synchronization paper by Schmidl and Cox. The math is slightly overwhelming.

What is the formula that they use in implementation that distinguishes between coarse estimation and fine estimation?

Secondly once fine frequency offset is done, how do you determine that you need to do coarse frequency synchronization or not? What is it that you will check to determine it? If anybody could explain with a set of equations that would be very helpful.

• I don't have time to read it right now, but this appears to be the paper that the OP is talking about. Apr 19, 2013 at 12:45
• Yes Jason, that's the paper that is being used in gnuradio and all other ofdm sync implementations. Apr 19, 2013 at 13:11
• You might try clarifying your question. It's not clear what you're asking, other than saying that you don't understand the paper. That's a little vague. Apr 19, 2013 at 13:40
• What makes you think that there's a formula or method that is able to determine whether the second training symbol is required? As I understand it, this has to be known in advance as they say "if $|\hat\phi|$ can be guaranteed to be less than $\pi$ then [..] the second training symbol would not be needed."
– Deve
Apr 19, 2013 at 13:48
• Can you give us some reference? To me it seems useless to apply fine sync before coarse sync.
– Deve
Apr 20, 2013 at 11:41

## 3 Answers

I think I understand your question better now. In the Schmidl & Cox paper they do frequency synchronization in two steps by splitting the frequency offset $\Delta f$ into two parts: $$\Delta f = \Delta f_1 + \frac{2z}{T}\text{,} \quad z \in \mathbb{Z}\text{, }\Delta f_1 < \frac{2}{T}$$ These two parts are estimated and corrected in two steps:

1. Correct any frequency offset $\Delta f_1$ with the help of the first training symbol, where $T$ is the OFDM symbol duration without guard interval.
2. Correct the residual frequency offset $2z/T$ with the help of the second training symbol. If $\Delta f < 2/T$ (i.e. $z=0$) the result of the maximization of $B(g)$ should be $\hat g = 0$ and no further correction is necessary in this step.

I think you were referring to step 1 as "fine sync" and to step 2 as "coarse sync". If you're aksing how it can be determined whether step 2 is necessary then my answer is: you can't unless you can guarantee that $\Delta f < 2/T$. In that case it would never be necessary and the second training symbol would not be needed.

The key part of the paper seems to be section IV. There, they estimate the frequency offset by:

where $P(d)$ is given by:

And then $\widehat{\Delta f}$ is given by

if $|\hat{\phi}| < \pi$, or a correction is added:

otherwise.

This paper is specific to LTE, but it describes the synchronization of the OFDM physical layer really well: "A Closed Concept for Synchronization and Cell Search in 3GPP LTE Systems"

I highly recommend it and based an academic paper from it by verifying the results.

• Thanks. I checked the paper, i understood time synchronisation well. But still have some doubts on coarse synchronsiation as to why we need to take an FFt again on an input FFT signal to obtain the number of integer offsets of frequency. Apr 26, 2013 at 6:18
• I'm unsure what you mean, but after you get the sync. symbols (by taking an FFT of input time samples), you just need to complex correlate those symbols against what's expected with integer offsets of ... -2,-1,0,1,2 ... and keep track of whichever is highest. Apr 26, 2013 at 13:18