## 2 Signal Modeling and CFR Method
### CFR Algorithm to Single-Band Signals with PAPR Target  
The peak windowing for CFR lowers the PAPR, attenuating the waveform for a short time interval surrounding a peak. Once a peak is detected in the waveform, the peak-centered window function attenuates the waveform at the peak and its neighboring samples. The window function will typically have a Gaussian shape that is time domain truncated. Other suitable functions include the Hann, Hamming, and Kaiser windows. The peak with waveform window \(x_{w}\) is:  
\[x_{w}=\left(1-c^{\prime}(n)\right)x(n), \tag{7}\]  
where \(x(n)\) is the input signal, and \(c^{\prime}(n)\) is the clipping algorithm defined by the following equation:  
\[c^{\prime}(n)=\sum_{m}\gamma_{m}\sum_{\tau}g(\tau),\delta_{peaks}(n-\lambda_{ m}-\tau) \tag{8}\]  
where \(\lambda_{m}\) is the location of peak \(m\), \(\delta_{peaks}\) is the location in time of the detected peak, \(g(\tau)\) is a Gaussian window and \(\gamma_{m}\) can be expressed as:  
\[\gamma_{m}=\max\left(1-\frac{L}{|x|},0\right). \tag{9}\]where \(L\) is the desired level of clipping. Algorithm 1 shows the pseudocode of the CFR-based technique to evaluate the PAPR performance. It should be noted that the purpose of developing a CFR system is to reduce PAPR levels of at least 3 dB in order to compare against DPD methods and alternative spectral improvements works lately reported. The CFR algorithm implementation is used for a 2.40 GHz single carrier frequency for 64-QAM digital modulation multiplexed by OFDM. In Algorithm 1, it can be observed that the inputs are described as \(x(n)\) in which a signal \(cp(n)\) is obtained based on a reference power. This signal is filtered to compensate for the out of band components and is obtains an \(x_{c}(n)\) and obtain a clipped signal \(cp(n)\), so that based on Equation (5) a comparable signal is obtained with hardware-based linearization methods. ```
Input:\(n\geq 0\) Output:\(y=x^{n}\) \(y\gets 1\); \(X\gets x_{i}\) for\(N\
eq 0\)do if\(N\) is eventhen \(x_{c}(n)=x(n)-filter[cp(n)]\); \(N\leftarrow\frac{N}{2}\) end if N is odd \(y\gets y\times X\); \(N\gets N-1\); \(cp_{i}\gets N-1\); \(x_{c}(n)=x_{c}(n)\times\odot(cp_{i})\)\(cp_{i}(n)\times x(cp_{i})\) / \(|x|\times(cp_{i})\);  Set Noise Floor \(\gets 2^{-12}(1,\text{length}(1:(N/2-k)))\);  Calculate PAPR \(=\ \sum_{n}^{N-1}|x(n)|^{2}\) ; end
```  
**Algorithm 1**CFR algorithm with PAPR target. Table 1 shows the relationship obtained with each CFR test performed. It can be seen that the EVM obtained based on the clipping process varies in each stage, given the deterioration of the vector to be analyzed. The variation due to the improvement of the EVM can be related to corrections of the I/Q umbalance, phase gain, DC offset, among others. Figure 3 shows the actual LTE 10-MHz signal when comparing it with the resulted clipping signal both estimated using a frame with \(N=76\),800 complex-valued data samples while peak integrity is guaranteed. In Figure 4, the PAPR achieved in the transmitter Tx due to the clipping and filtering stages is shown on the x-axis. An example of an algorithm with lower computational complexity analysis related to CFR and DPD modeling with spline-interpolation coefficients LUT, which aims to preserve a learning rule based on parameters is reported in [24]. Equation (6) shows the \(P(t)_{peak}\) represents the peak measured at the fundamental for the LTE implementations in this case of \(-9.18\) dBm for LTE 5-MHz and \(-14.04\) dBm for LTE 10-MHz, in addition to the values \(P(t)_{rms}\), for each transmitted symbol of 64-QAM digital modulation under OFDM multiplexing.