# Data-Aided Frequency Offset Estimation for CE-OFDM Broadband Satellite Systems
## 3 CE-OFDM Transceiver Structure and the Impact of CFO  
The difference between the CE-OFDM system and the OFDM system is that the CE-OFDM system adjusts the symbol structure before the inverse discrete Fourier transform (IDFT) through the conjugate symmetric mapping module so that the result of the IDFT is a real-valued sequence, and then the real-valued sequence is modulated by the phase modulation module. The detailed description of the CE-OFDM transceiver structure can be found in [11], and we give a brief review here.  
Figure 11 shows the CE-OFDM transceiver structure. Modulation symbols are sent to the conjugate symmetry mapping module, and the output is conjugated symmetric sequence as:  
\[S=[0,S(1),\ldots,S(N_{sym}),0_{1\times N_{L}},0,S^{*}(N_{sym}),\ldots,S^{*}(1)], \tag{1}\]  
where \(S(k)\big{|}_{k=1}^{N_{sym}}\) is the input modulation symbols in the frequency domain, \(N_{sym}\) is the length of the useful symbols, and \(N_{L}\) is the length of the row vector comprised of zeros. Therefore, the spectrum efficiency of the CE-OFDM signal is reduced to half that of the regular OFDM signal. The CE-OFDM signal can be expressed as \(x(n)\big{|}_{n=-N_{QP}}^{N-1}\), where \(x(n)=Ae^{j2n\text{fs}(n)},x(n)=x(N+n)\), \(n=-N_{QP},\ldots,-2,-1\). \(h\) is the modulation index, \(s(n)\big{|}_{n=0}^{N-1}\) is the IDFT of \(S(k)\big{|}_{k=0}^{N-1}\), \(N\) is the IDFT point, and \(N_{QP}\) is the length of a cyclic prefix (CP).  
\(x(n)\) is the time domain signal finally transmitted by the CE-OFDM transmitter. It can be seen that the \(x(n)\) is obtained by applying a nonlinear phase modulation to the \(s(n)\). Since \(s(n)\) is the IDFT of the \(S(k)\), and \(S(k)\) has a conjugate symmetry structure, \(s(n)\) is a real-valued OFDM signal. Thus, the envelope of the CE-OFDM signal \(x(n)\) is a fixed value and the PAPR of the CE-OFDM signal \(x(n)\) is 0 dB.  
As the signal passes through the flat fading channel with CFO, the received signal is  
\[\begin{array}{ll}y(n)&=\ px(n)e^{\frac{(2 mAf}{N}}e^{j\phi(n)}+w(n)\\ &=\ \overline{A}e^{j[2n\text{fs}(n)+\phi+\frac{2 mAf}{N}+\phi(n)]}+w(n)\end{array} \tag{2}\]  
Figure 2: Constant-envelope orthogonal frequency-division multiplexing (CE-OFDM) transceiver structure.  
where \(\rho=\alpha e^{j\theta}\) represents the channel fading factor, \(\overline{A}\) is the signal amplitude with channel attenuation, \(\Delta f\) is the normalized Doppler frequency offset, \(\phi(n)\) is the phase noise, and \(w(n)\) is the additive noise (\(w(n)=\left|\!\!\left|w(n)\right|\!\right|e^{jn_{\phi}(n)}\)). We assume that the timing offset is perfectly compensated at the receiver. Therefore, there is only frequency offset and phase noise in Equation (2).  
The phase information obtained after the phase demodulator is  
\[y_{\theta}(n)=2\pi hs(n)+\frac{2\pi n\Delta f}{N}+\theta+\phi(n)+w_{\theta}(n) \tag{3}\]  
where \(w_{\theta}(n)\) is the noise component after the phase demodulator. From Equation (3), we can see that the Doppler frequency offset component \(\Delta f\) and the phase noise component \(\phi(n)\) are added to the \(s(n)\), and this will cause interference after the DFT module, causing the symbol \(\left.S(k)\right|_{k=1}^{N_{sym}}\) to be demodulated incorrectly.  
The phase diagram of the received signal is given in Figure 11. It was proven in [11] that any fixed phase offset \(\theta\) does not affect the BER performance. Therefore, only the CPO components need to be estimated and compensated. In this paper, we assume that the phase noise is ideally compensated at the receiver, and we concentrate on the estimation of \(\Delta f\). In the following text, \(\phi(n)\) and \(w_{\theta}(n)\) are combined with \(\overline{w}_{\theta}(n)\) for simplicity.