## 3 PAPR Reduction with Clipping and Modified Iterative Detection Scheme
### Iterative Soft Compensation  
Here, it is assumed that the signal vector before clipping is \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{N}\}\). According to the existing literature [3; 18; 19; 24], the nonlinear distortion introduced by \(c(\mathbf{x})\) can be separated into two parts as  
\[c(\mathbf{x})=\alpha\mathbf{x}+\mathbf{d}, \tag{11}\]  
where \(\mathbf{d}\) is the clipping noise vector, usually considered to be Gaussian noise uncorrelated with the signal. And, \(\alpha\) is an accelerating factor which is calculated as  
\[\alpha=\frac{\mathrm{E}\big{[}\mathbf{x}^{H}c(\mathbf{x})\big{]}}{\mathrm{E} \Big{[}\big{\|}\mathbf{x}\big{\|}^{2}\Big{]}}. \tag{12}\]  
Then, the signal at the receiving end can be expressed as  
\[\mathbf{y}=\alpha\mathbf{x}+\mathbf{d}+\mathbf{n}, \tag{13}\]  
where \(\mathbf{n}\) is the additive Gaussian noise of the channel. It is then necessary to consider the probability distribution model of \(\mathbf{d}\). In general, \(\mathbf{d}\) is usually considered a zero-mean Gaussian noise, then at this point, \(\mathbf{d}\) can be combined with \(\mathbf{n}\) into a total Gaussian noise \(\mathbf{w}_{1}=\mathbf{n}+\mathbf{d}\) with the total noise variance \(\sigma_{w_{1}}^{2}=\sigma_{x}^{2}+\sigma_{d}^{2}\). Therefore, only a correction forthe estimated noise power at the receiver side is required under this assumption, while the variance of \(\mathbf{d}\) can be expressed as  
\[\sigma_{d}{}^{2}=(1-\alpha^{2})\sigma_{x}{}^{2}, \tag{14}\]  
where the transmit power \(\sigma_{x}{}^{2}\) is usually normalized to 1. Meanwhile, another way of estimating \(\mathbf{d}\) is to model it as a Gaussian variable with a non-zero mean. Thus, the parameters \(\mathbf{d}\) are constantly updated through iterative testing to obtain a more accurate noise estimate to improve the decoder performance. So the signal to be demodulated at the receiver side can be written as  
\[\mathbf{y}-\mathbf{\bar{d}}=\alpha\mathbf{x}+(\mathbf{d}-\mathbf{\bar{d}})+ \mathbf{n}=\alpha\mathbf{x}+\mathbf{w_{2}}, \tag{15}\]  
where \(\mathbf{w_{2}}=(\mathbf{d}-\mathbf{\bar{d}})+\mathbf{n}\) and \(\mathbf{\bar{d}}\) is the mean value of \(\mathbf{d}\). In the previous section, we assumed that the signal time domain is Gaussian distributed, then \(\mathbf{x}\) can be completely described by the mean \(E[\mathbf{x}]\) and variance \(V[\mathbf{x}]\). Thus, the mean value of \(\mathbf{d}\) can be calculated by the following equation  
\[\bar{d}[n] = \mathrm{E}[c(x[n])-\alpha x[n]] \tag{16}\] \[= \int\frac{c(x)-\alpha x}{\pi\mathrm{V}[x[n]]}e^{-\frac{|x-E[x[n] ]|^{2}}{V[x[n]]}}dx.\]  
Correspondingly, the variance of \(\mathbf{d}\) can be expressed as  
\[\mathrm{V}[d[n]] = \mathrm{E}\Big{[}|d[n]-\bar{d}[n]|^{2}\Big{]} \tag{17}\] \[= \int\frac{|c(x)-\alpha x-d[n]|^{2}}{\pi\mathrm{V}[x[n]]}e^{-\frac {|x-E[x[n]]|^{2}}{V[x[n]]}}dx.\]  
Then, the total noise variance can be expressed as  
\[\sigma_{w_{2}}^{2} = \sigma_{n}^{2}+\sigma_{d}^{2} \tag{18}\] \[\alpha=\frac{\mathrm{E}\big{[}\mathbf{x}^{H}c(\mathbf{x})\big{]}}{\mathrm{E} \Big{[}\big{\|}\mathbf{x}\big{\|}^{2}\Big{]}}. \tag{12}\]0  
and this corrected noise variance is used in the log-likelihood ratio calculation of the decoder. In summary, the estimation algorithm for clipping noise can be expressed as follows. 1. The first round of SIC detection is performed by initializing \(\mathbf{\bar{d}}\) to 0. After obtaining the updated \(E\{\hat{x}_{k}[l]\}\) and \(\mathrm{Var}\{\hat{x}_{k}[l]\}\) for each layer, they are superimposed in the same way as in the modulation process. If \(\tau\
eq 1\), then the inter-symbol interference coefficient \(g_{n}\) must also be taken into calculation. Thus, the \(E[\mathbf{x}]\) and \(V[\mathbf{x}]\) of the overall signal can be estimated. 2. Calculate \(\mathbf{\bar{d}}\) via (16) and let \(\mathbf{\hat{y}}=\mathbf{y}-\mathbf{\bar{d}}\). Calculate \(\sigma_{d}^{2}\) via \(\mathbf{\bar{d}}\) and (17) and modify the total noise variance \(\sigma_{w_{2}}^{2}\) for the next iteration.