# A Frequency-Domain Estimation Scheme for Frequency Offset with Large Range in OFDM Systems
## 2 System Description
### The OFDM System Model  
Figure 1 shows the baseband transmitter and receiver structure of an OFDM system with the pilot-based frequency offset estimation. At the transmitter, the input bits are firstly modulated by constellation mapping to obtain data symbols. Then, the data symbols or pilot symbols are fed into a serial to parallel (S/P) converter to obtain a complex vector \(\mathbf{X}=(\mathbf{X}(0),\mathbf{X}(1),\ldots,\mathbf{X}(N-1))\), where \(N\) is the number of subcarriers in the OFDM system. The pilot symbols are known at the transmitter and receiver and typically used in phase/frequency offset estimation or channel estimation at the receiver to assist in data demodulation. It is noted that the complex vector \(\mathbf{X}\) could contain both data symbols and pilot symbols, and \(\mathbf{X}\) is a pilot OFDM symbol if \(\mathbf{X}\) contains pilot symbols; otherwise, \(\mathbf{X}\) is a data OFDM symbol. Finally, the vector \(\mathbf{X}\) is transformed by the inverse fast Fourier transform (IFFT) method to obtain a time-domain OFDM signal \(\mathbf{x}=(\mathbf{x}(0),\mathbf{x}(1),\ldots,\mathbf{x}(N-1))\). At the receiver, the received time-domain OFDM signal \(\mathbf{y}=(\mathbf{y}(0),\mathbf{y}(1),\ldots,\mathbf{y}(N-1))\), is firstly transformed by the fast Fourier transform (FFT) method to obtain the received \begin{table}
\begin{tabular}{l l l} \hline
**Notation** & **Dimension** & **Definition** \\ \hline \(M\) & \(1\times 1\) & The number of non-zero elements in the simplified compensation coefficients \(\mathbf{Z}_{l}\). \\ \(N\) & \(1\times 1\) & The number of FFT points for an OFDM symbol. \\ \(N_{s}\) & \(1\times 1\) & The number of elements in the pilot-subcarrier index set \(\mathcal{S}\). \\ \(N_{rb}\) & \(1\times 1\) & The number of configured RBs in the NR PUSCH. \\ \(\mathcal{S}\) & \(1\times N_{s}\) & The pilot-subcarrier index set. \\ \(\mathbf{x}\) & \(1\times N\) & The transmitted time-domain vector. \\ \(\mathbf{X}\) & \(1\times N\) & The transmitted frequency-domain vector. \\ \(\mathbf{X}^{(0)}(k)\) & \(1\times 1\) & The transmitted symbol on the \(k\)-th subcarrier of pilot OFDM symbol 0, and \(k\in\mathcal{S}\). \\ \(\mathbf{X}^{(1)}(k)\) & \(1\times 1\) & The transmitted symbol on the \(k\)-th subcarrier of pilot OFDM symbol 1, and \(k\in\mathcal{S}\). \\ \(\mathbf{y}\) & \(1\times N\) & The received time-domain vector. \\ \(\mathbf{Y}\) & \(1\times N\) & The received frequency-domain vector. \\ \(\mathbf{Y}^{(0)}(k)\) & \(1\times 1\) & The received symbol on the \(k\)-th subcarrier of pilot OFDM symbol 0, and \(k\in\mathcal{S}\). \\ \(\mathbf{Y}^{(1)}(k)\) & \(1\times 1\) & The compensated pilot sequence with the estimated candidate normalized frequency offset \(\hat{e}(l)\). \\ \(\bar{\mathbf{Y}}_{l}\) & \(1\times N_{s}\) & The time-domain compensation coefficient vector, calculated with \(\hat{e}(l)\). \\ \(\mathbf{z}_{l}\) & \(1\times N\) & The frequency-domain compensation coefficient vector, calculated with \(\hat{e}(l)\). \\ \(\bar{\mathbf{Z}}_{l}\) & \(1\times N\) & The simplified frequency-domain compensation coefficient vector \\ \(\text{mod}(x,N)\) & – & \(\text{mod}(x,N)\) denotes the integer \(x\) modulo \(N\). \\ \(\text{angle}(x)\) & – & \(\text{angle}(x)\) denotes the phase of the complex \(x\). \\ \(\left\lceil x\right\rceil\) & – & \(\left\lceil x\right\rceil\) denotes the upper integer of \(x\). \\ \(\left\lfloor x\right\rfloor\) & – & \(\left\lfloor x\right\rfloor\) denotes the lower integer of \(x\). \\ \(x^{*}\) & – & \(x^{*}\) denotes the conjugation of the complex \(x\). \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._frequency-domain vector \(\mathbf{Y}=(\mathbf{Y}(0),\mathbf{Y}(1),\ldots,\mathbf{Y}(N-1))\). The received pilot symbols in \(\mathbf{Y}\) are then processed in a frequency offset estimation module to estimate the frequency offset, which is later used to compensate the received data and pilot symbols. The compensated data symbols are finally equalized by the estimated channel, and demodulated and decoded to obtain the information bits.