# Mdpi
## 3 Proposed IN Mitigation Scheme
### Iterative IN Reconstruction  
Figure 4 details the proposed iterative IN reconstruction block. After re-mapping and pilot-insertion, we can estimate the input of the IDWT block on the transmitter side. Thus, if an estimate of the CIR \(\hat{h}[n]\) is available, then it is possible for the IN to be re-generated as follows:  
\[\hat{d}[n]=\hat{r}[n]-\hat{s}[n]\otimes\hat{h}[n]. \tag{17}\]  
However, \(\hat{s}[n]\) and \(\hat{h}[n]\) are not exactly the same with \(s[n]\) and \(h[n]\), respectively. Thus, for the samples in \(\hat{d}[n]\), only the samples with large amplitude are reliable estimation for the IN. This leads us to use a peak detector to enhance the reliability of the estimation process, such that only significant estimates of \(\hat{d}[n]\) are retained. The operation involving the peak detector can be expressed as follows:  
\[\hat{a}[n]=\begin{cases}\hat{d}[n],&\left|\hat{d}[n]\right|^{2}>T_{2}[n]\\ 0,&\text{otherwise}\end{cases}, \tag{18}\]  
where \(T_{2}[n]=4.12\cdot\mathbb{E}\left(\left|\hat{d}[n]\right|\right)\) is an adaptive threshold for the peak detector [22]. Channel estimation can be performed using the following recursions:  
\[\hat{\mathbf{h}}[n]=\hat{\mathbf{h}}[n-1]+\mu_{h}[n]\frac{\mathbf{x}_{h}[n] \epsilon_{h}[n]}{\mathbf{x}_{h}^{T}[n]\mathbf{x}_{h}[n]+\delta_{h}}, \tag{19}\]  
where \(\hat{\mathbf{h}}[n]=\left[\hat{h}_{0}[n],\hat{h}_{1}[n],\ldots,\hat{h}_{N_{h}- 1}[n]\right]^{T}\) denotes the weight vector with length \(N_{h}\) at time index \(n\); \(\mathbf{x}_{h}[n]=\left[\hat{s}[n],\hat{s}[n-1],\ldots,\hat{s}[n-N_{h}+1] \right]^{T}\) denotes the input regressor vector; \(\delta_{h}\) is a small positive constant used to prevent numerical errors during the weight updating process; and \(e_{h}[n]\) is an error signal, which can be expressed as follows:  
\[e_{h}[n] =\tilde{r}[n]-y_{h}[n] \tag{20}\] \[=\tilde{r}[n]-\hat{\mathbf{h}}^{T}[n-1]\mathbf{x}_{h}[n].\]  
\(\mu_{h}[n]\) is the variable step-size, which can be calculated as follows [24]:  
\[\mu_{h}[n]=\left|1-\frac{\sqrt{\left|\hat{\sigma}_{d,h}^{2}[n]-\hat{\sigma}_{ y,h}^{2}[n]\right|}}{\hat{\sigma}_{e,h}[n]+\zeta_{h}}\right|, \tag{21}\]  
Figure 3: Details in an adaptive DWT block with \(M\) subchannels.  
where \(\zeta_{h}\) is a small positive number; \(\hat{\sigma}_{d,h}^{2}[n]\) is the estimated variance of the desired signal, which can be calculated using the following recursion:  
\[\hat{\sigma}_{d,h}^{2}[n]=\lambda_{h}\hat{\sigma}_{d,h}^{2}[n-1]+(1- \lambda_{h})\hat{d}^{2}[n], \tag{22}\]  
\(\hat{\sigma}_{y,h}^{2}[n]\) is the estimated variance of the output signal of the channel estimator which can be calculated as follows:  
\[\hat{\sigma}_{y,h}^{2}[n]=\lambda_{h}\hat{\sigma}_{y,h}^{2}[n-1]+(1- \lambda_{h})y_{h}^{2}[n], \tag{23}\]  
and \(\hat{\sigma}_{e,h}^{2}[n]\) is the estimated variance of the error signal of the channel estimator which can be calculated as follows:  
\[\hat{\sigma}_{e,h}^{2}[n]=\lambda_{h}\hat{\sigma}_{e,h}^{2}[n-1]+(1- \lambda_{h})e_{h}^{2}[n], \tag{24}\]  
where \(\lambda_{h}\in(0,1)\) is the smoothing factor.  
When the estimated IN \(\hat{a}[n]\) becomes available, we are able to obtain a cleaner received \(\hat{r}[n]\) by subtracting \(\hat{a}[n]\) from \(\hat{r}[n]\), such that in the following iteration, \(\hat{r}[n]\) becomes less contaminated. As a result, the mis-adjustments imposed by the equalizers \(\mathbf{w}_{k}[n]\) and channel estimator \(\hat{\mathbf{h}}[n]\) are iteratively reduced. In cases where more iterations are unnecessary, the output of the de-mapper \(\hat{X}_{k}[n]\) passes through a P/S block to produce the estimation of the information \(\hat{x}[n]\) that has been transmitted.