# Low-Complexity Synchronization Scheme with Low-Resolution ADCs
## 2 OFDM Equivalent Signal Model
### Effect of Synchronization Bias on OFDM System  
The advantage of the OFDM can be useful only when the orthogonality is maintained. In case the orthogonality is not sufficiently warranted by any means, its performance may be degraded due to inter-symbol interference (ISI) and inter-channel interference (ICI) [22]. Since the orthogonality can be destroyed by carrier frequency offset (CFO), the accuracy of frequency offset estimation is usually required to reach 1-2% of the subcarrier spacing [23]. Thus, we discuss the impact of CFO on OFDM systems in this subsection. The OFDM-UWB baseband transmission model is shown in Figure 1.  
In digital communications, there is often a mismatch between the local oscillator of the transmitter and the receiver. This translates to a carrier frequency error when downconverting at the receiver, which effectively rotates the signal constellation from sample to sample [24]. Therefore, the CFO can be seen as composed of two parts, the fractional carrier frequency offset (FFO) and integer carrier frequency offset (IFO) [25]. Because IFO only causes cyclic shift between subcarriers [26], we focus on the impact of FFO. Let us define the normalized CFO, \(\varepsilon\), as a ratio of the CFO to subcarrier spacing \(\Delta f\), shown as  
\[\varepsilon=\frac{f_{\text{offset}}}{\Delta f}=\varepsilon_{i}+\varepsilon_{f}, \tag{5}\]  
where \(f_{\text{offset}}\) represents the actual frequency offset, \(\varepsilon_{i}\) is the fractional part of \(\varepsilon\), \(\varepsilon_{f}\) is the integer part of \(\varepsilon\), and \(\Delta f\) is the subcarrier spacing. Furthermore, we assume that only a CFO of \(\varepsilon\) exists between transmitter and receiver, without any phase noise [27]. Thus, the time-domain received signal of an OFDM system can be expressed as  
\[y_{l}[n]=\frac{1}{N}\sum\limits_{k=0}^{N-1}H[k]X_{l}[k]\exp\left(\frac{j2\pi n (k+\varepsilon)}{N}\right)+z_{l}[n], \tag{6}\]  
where \(H[k]\) is channel frequency response at the \(k\)th subcarrier, \(X_{l}[k]\) is the frequency-domain transmitted symbol, and \(z_{l}[n]\) is a noise term. Taking the FFT of \(\{y_{l}[n]\}\) in Equation (6), the frequency-domain received signal with an FFO of \(\varepsilon_{f}\) can be expressed as  
\[\gamma_{l}[k]=\frac{\sin\!\left(\pi\varepsilon_{f}\right)}{N\sin\!\left(\pi \varepsilon_{f}/N\right)}\exp\!\left[j\pi\varepsilon_{f}(N-1)/N\right]H_{l}[ k]X_{l}[k]+I_{l}[k]+Z_{l}[k], \tag{7}\]  
where \(H_{l}[k]\) is channel frequency response at the \(k\)th subcarrier of the \(l\)th symbol, \(Z_{l}[k]\) is white Gaussian noise, and \(I_{l}[k]\) is the ICI which can be expressed by Equation (8).  
Figure 1: Orthogonal frequency division multiplexing ultra-wideband (OFDM-UWB) baseband transmission model.  
It can be seen from Equation (7) that the first term represents the amplitude and phase distortion of the \(k\)th subcarrier frequency component due to FFO. Meanwhile, \(I_{l}[k]\) in Equation (8) represents the ICI from other subcarriers into \(k\)th subcarrier frequency component, which implies that the orthogonality among subcarrier frequency components is not maintained any longer due to the FFO. In general, the performance degradation caused by FFO is more serious than that of IFO. However, unless the cyclic shift is compensated, it incurs a significant degradation in BER performance.