modulation - OFDM training symbol format

For OFDM synchronisation, we are using two training symbols according to the Schmidl and Cox method of frequency synchronization and timing synchronization. The paper says:

The first OFDM training symbol has only even numbered subcarriers applying a PN sequence.The result is two identical half symbols in time domain each consisting of Nc/2 samples each.

The second training symbol consists of even numbered subcarriers that are differentially modulated with even numbered subcarriers of first training symbol using a PN sequence. The odd numbered subcarriers of second training symbol can be used for data, pilot or reference symbols.

I have two questions on this.

1. 
   

   In OFDM, every subcarrier corresponds to one symbol which is a complex value. So what does it mean by saying even numbered subcarriers are modulated in a symbol using a PN sequence? How does that ensure that it results in two identical halves of a time domain symbol.

   
   


2. 
   
   

   what is differential modulation using a PN sequence. How is it achieved?

   
   

Answer

To answer your first question, what they mean is that the first training symbol only encodes data on the even-numbered subcarriers. The other subcarriers are set to zero. That is, the frequency-domain,

$$ X[k] = \begin{cases} s_k, &k \text{ mod } 2 = 0 \\ 0, &\text{otherwise} \end{cases} $$

The symbols to encode on the even-numbered subcarriers $s_k$ are chosen from a pseudorandom noise (PN) sequence. They assert that this results in a time-domain symbol that has two identical half-symbols in it (i.e. the first half of the symbol period is equal to the second half). Why is that?

Recall one of the properties of the discrete-time Fourier transform: if you zero-stuff a signal in the time-domain (by inserting zeros between each pair of samples), then in the frequency domain, the spectrum is repeated periodically. You might have seen this before in a description of discrete-time interpolation: insert zeros, then add a lowpass filter to eliminate the spectrum duplicates.

The property is dual in the sense that it works the other way also. When assigning symbols to subcarriers, you're working in the frequency domain. An inverse DFT is used to generate the time-domain signal for transmission. If you set all of the odd-numbered subcarriers to zero, then you're generating a frequency-domain signal that is zero-stuffed by a factor of two. When you inverse transform that to the time domain, you get the same effect: the resulting symbol consists of two periodic half-symbol waveforms.

I haven't read the paper in detail, but I'm assuming their synchronization technique takes this redundancy into account in some way.