# 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?

• Do you have a reference for this scheme that you describe? Some context might help decipher exactly what it means. – Jason R May 6 '13 at 12:42
• Regarding question #2, look up "differential encoding". – Jim Clay May 6 '13 at 12:46
• @JasonR The reference is this home.mit.bme.hu/~kollar/papers/Schmidl2.pdf. The wording is in second page. – Karan Talasila May 7 '13 at 4:40
• what does he really mean by even and odd frequencies. Does he mean for example if N=64,he sends a PN modulated sequence on even subcarrier frequencies of the entire 64 subcarrier OFDM symbol or does he send 2 training symbols in 2 subcarriers in an ofdm symbol. I am unable to get that distinction. Is there anything fundamental that i am missing. – Karan Talasila May 9 '13 at 6:33

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.

• I got it now.Very well explained answer. I have one question. In OFDM transmission say FFT Length=512.suppose say you are sending one training symbol, does that mean you are assigning 512 subcarriers to correspond to one symbol or does one training symbol mean one subcarrier in the 512 subcarrier sequence where the rest are data or zero padding. Let me know if i have to edit the question to make it clear. – Karan Talasila May 23 '13 at 13:56
• In the nomenclature of OFDM, a "symbol" usually refers to one set of all of the subcarriers. So, an OFDM training symbol in your case could consist of up to 512 individually-modulated symbols. – Jason R May 23 '13 at 13:57
• ok. That clears my doubt. why I got this doubt is when i am implementing ofdm in GNU Radio, I see that one symbol sent is giving me a constellation diagram of a lot of points. So I suppose that constellation has 512 points for N=512 where each point corresponds to individually modulated symbols.Am I right? – Karan Talasila May 23 '13 at 14:01
• No. Each of the subcarriers is modulated independently, so you don't visualize the entire OFDM symbol on a constellation. Each subcarrier is modulated with something like BPSK, QPSK, or QAM, all of which have their own constellation formats. – Jason R May 23 '13 at 15:21
• So does that mean for example, to send one training symbol, assuming N=512 you send 512 subcarriers each of which is modulated using a modulation technique of your choice. In my example I am only using bpsk as modulation technique, so in that can i correspond that the no of points on the constellation diagram obtained are equal to N for each symbol. – Karan Talasila May 23 '13 at 15:30