## 2 SFN DVB-T Description
### Ofdm  
OFDM is a digital multi-carrier modulation method. It divides the bandwidth into a large number of closely-spaced sub-bands. On each sub-band, a Quadrature Amplitude Modulation (QAM) or Quadrature Phase-Shift Keying (QPSK) is used, and all the sub-band signals are summed together. The symbol rate is the same for each subcarrier and is equal to the sub-band bandwidth. This way, the subcarriers are orthogonal, and the spectra can overlap without causing Inter-Carrier Interference (ICI) when the receiver is well synchronized. To avoid Inter-Symbol Interference (ISI) in multipath fading channels, a guard interval is inserted prior to the OFDM symbol. This interval is used to transmit an exact replica at the end of the OFDM symbol, referred to as CP. This gives the OFDM system an excellent multipath resistance: the receiver can easily avoid the ISI if the multipath time-spreading is shorter than the guard interval.  
The Fast Fourier Transform (FFT) algorithm can be used to implement the OFDM modulation, and the good efficiency of this algorithm allows a large number of subcarriers in operation. In Figure 1, the block diagram of an OFDM transmission system is presented. The output of the parallel to the serial block can be expressed as:  
\[\begin{split}\tilde{s}_{n}^{k}=\mathrm{IFFT}(c_{p}^{k})[n]& =\frac{1}{N_{FFT}}\sum_{p=0}^{N_{FFT}-1}c_{p}^{k}\exp\left(j2\pi \frac{pn}{N_{FFT}}\right)\\ &\mathrm{with}\ 0\leq n\leq N_{FFT}-1\end{split} \tag{1}\]  
where:  
* \(k\) is the OFDM symbol index,
* \(n\) is the sample index,
* \(p\) is the subcarrier index,
* \(N_{FFT}\) is the FFT size,
* \(\tilde{s}_{n}^{k}\) is the \(n\)-th sample of the \(k\)-th OFDM symbol and
* \(c_{p}^{k}\) is the data (constellation complex value) on the \(p\)-th subcarrier of the \(k\)-th OFDM symbol.  
Normally, in OFDM modulation, the number of subcarriers, \(N_{FFT}\), is chosen as a power of two in order to efficiently use the FFT algorithm. Before the samples are transmitted, the CP is inserted at the beginning of each OFDM symbol. Therefore, the emitted signal with the CP is:  
\[\begin{split} s_{n}^{k}&=\tilde{s}_{mod(n+N_{FFT}- N_{CP},N_{FFT})}^{k}\\ \mathrm{with}&\ 0\leq n\leq N_{FFT}+N_{CP}-1\end{split} \tag{2}\]  
where:  
* \(s_{n}^{k}\) is the \(n\)-th emitted sample of the \(k\)-th OFDM symbol and
* \(N_{CP}\) is the length of the cyclic prefix guard interval.  
On the receiver side, the CP has to be removed. To do this, a coarsely-timed synchronization is needed to find the starting instant of the OFDM symbol. The synchronization can be achieved, for example, by using the Van de Beek algorithm [PERSON] (1984). This algorithm is based on the fact that the guard interval is a replica of the end of the OFDM symbol. The correlation of two groups of \(N_{cp}\) samples spaced by \(N_{FFT}\) is calculated, and the absolute maximum of the correlation is used to estimate the beginning of the OFDM symbol. The synchronization can be achieved more precisely by adopting a delay estimation method. If the starting point of the FFT window is slightly wrong, but still within the CP, the receiver still has all wanted samples. These samples are circularly shifted, comparing the samples originally emitted. This cyclic permutation only affects the phase of the received symbol output by the FFT. This additional phase rotation is typically compensated for by the frequency equalizer, since it is summed to the phase rotation introduced by the channel.