# Robust Sampling Frequency Offset Estimation for OFDM over Frequency Selective Fading Channels
## 2 Signal Model  
We assume a baseband OFDM system with \(N\) subcarriers and \(N_{g}\) cyclic prefix (CP) samples. A transmission signal is modulated by using the inverse fast Fourier transform (IFFT) on the transmitter side and the last \(N_{g}\) IFFT samples are appended at the beginning to form the guard interval (GI). OFDM is highly sensitive to frequency offset and timing mismatch, so the synchronization problem between the transmitter and the receiver is of paramount importance to OFDM. Synchronization algorithms can, in general, be divided into pre-FFT and post-FFT classes. The goal of pre-FFT processing is to obtain the initial timing and fractional CFO based on GI correlation. In the post-FFT stage, residual CFO, SFO, and integer CFO are estimated by using the periodic nature of pilots. Since the focus here is the estimation of SFO, it is assumed that the symbol timing mismatch has been perfectly recovered at the receiver and coarse CFO acquisition has been successfully applied at the beginning of the reception of OFDM frame. For this purpose, many schemes are currently available in the literature [17; 18; 19]. During the data section, therefore, relatively small SFO and residual CFO will be present.  
At the receiver, the \(l\)-th received OFDM symbol at the \(k\)-th subcarrier after FFT demodulation is given in [20; 21]  
\[R_{l}(k)=\alpha(\phi(k))H_{l}(k)X_{l}(k)e^{j2\pi\phi(k)(IN_{u}+N_{g})/N}e^{j\pi( N-1)\phi(k)/N}+I_{l}(k)+Z_{l}(k),\ 0\leq k<N \tag{1}\]  
where \(\alpha(\phi(k))\) is the attenuation of the subcarriers magnitudes due to the frequency offset, \(N_{u}=N+N_{g}\), \(H_{l}(k)\) is the zero-mean channel's frequency response with variance \(\sigma_{H}^{2}\), \(Z_{l}(k)\) is a zero-mean complex Gaussian noise with variance \(\sigma_{Z}^{2}\) during the \(l\)-th symbol period, and \(I_{l}(k)\) is the inter-carrier interference (ICI) term introduced by frequency offsets. In (1), \(\alpha(\phi(k))\) and \(I_{l}(k)\) are given by  
\[\alpha(\phi(k))=\frac{\sin(\pi(\phi(k))}{N\sin(\pi\phi(k)/N)} \tag{2}\]  
and  
\[I_{l}(k)=\sum_{n=0,n\
eq k}^{N-1}H_{l}(n)X_{l}(n)e^{j2\pi\phi(n)(IN_{u}+N_{g} )/N}\frac{\sin(\pi(\phi(n)+n-k))}{N\sin(\pi(\phi(n)+n-k)/N)}e^{j\pi(N-1)(\phi (n)+n-k)N} \tag{3}\]  
where \(\phi(k)\) is formed with CFO \(\Delta_{f}\) and SFO \(\Delta_{s}\). The former is the same for all subcarriers, while the latter occurs linearly according to the subcarrier index \(k\), i.e., \(\phi(k)=\Delta_{f}+k\Delta_{s}\). Here, \(\Delta_{f}\) and \(\Delta_{s}\) denote the normalized CFO and SFO by the sampling rate, respectively. Since CFO and SFO are small during the data section of the frame, we approximate \(\alpha(\phi(k))\approx 1\) and the ICI term is negligible compared to the additive noise \(Z_{l}(k)\) for medium SNR conditions [21]. In this paper, however, its impact on the system performance will be addressed. From [22], \(I_{l}(k)\) may be handled as Gaussian noise with zero mean and variance \(\sigma_{l}^{2}\).  
In the DRM standard, three FRCs are embedded for frequency tracking in subcarriers \(\{k_{1},k_{2},k_{3}\}\), which are fixed for all DRM modes in Hz and are transmitted at a boosted power level of \(E_{X}=|X_{l}(k)|^{2}=2\). Denote \(k_{m}\) the subcarrier index of the \(m\)-th FRC, where \(m\in\{1,2,3\}\). For subsequent symbols, the \(l\)-th pilot symbol \(X_{l}(k_{m})\) is derived from the first pilot symbol in the frame denoted by \(X_{l_{1}}(k_{m})\) as follows \(X_{l}(k_{m})=d_{l}(k_{m})X_{l_{1}}(k_{m})\), where \(d_{l}(k_{m})\in\{1,-1\}\). In [2], the phases of FRCs are chosen to ensure the tones are continuous: (1) \(d_{l}(k_{m})=1\) for all \(l\)'s and \(m\)'s (modes A, B, C, and \(k_{3}\) only of mode D), and for even values of \(l\) (\(k_{1}\) and \(k_{2}\) of mode D) (2) \(d_{l}(k_{m})=-1\) for odd values of \(l\) (\(k_{1}\) and \(k_{2}\) of mode D). The GRCs are used mainly to get a proper estimate of the channel transfer function, which are spread equally in time and frequency directions. The GRCs are also used to track the frequency errors.