# Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions
## 3 DFT-Based LS Channel Estimation Methods
### Proposed Modification  
Implementation simplicity is a big advantage of noise energy controlled method presented in Section 3.2.1. However, in the case of compressible channel impulse response, the individual decisions about assigning a given coefficient of \(\hat{\mathbf{h}}_{C}\) to the vector \(\hat{\mathbf{h}}_{N}\) (see Equation (10)) can be a shortcoming of the solution. Two unfavorable situations can be distinguished. In the first one, a coefficient is rejected only because its amplitude unexpectedly took a value below the reference threshold due to noise interference. In the second, the opposite one, when a large value of a wrongly assigned coefficient is caused by the influence of noise. To minimize the risk of the above incorrect interpretations, a modification with the use of an additional moving average filter of length \(U\) is proposed in this article. The energy values of \(U\)-element groups of the subsequent coefficients of \(\hat{\mathbf{h}}_{C}\), obtained at the filter output using circular convolution, is tested against a new threshold level \(\beta\). The coefficients of \(\mathbf{h}_{C}\) that belong to the groups with total energy larger than the threshold \(\beta\) are assigned to a new impulse response vector \(\mathbf{h}_{G}\). Thus, not a single but a group of coefficients is selected to \(\mathbf{h}_{G}\) after every positive verification against the threshold. Referring to the previous notations, the impulse response can be defined as follows:  
\[\hat{\mathbf{h}}_{G}=\mathbf{I}_{N\{G\}}\hat{\mathbf{h}}_{C}, \tag{19}\]where  
\[\mathbf{I}_{N\{G\}}(n,n)=\bigg{\{} \begin{array}{ll}1&\text{if}\quad n\in\mathbb{ G}\\ 0&\text{otherwise}\end{array} , \tag{20}\]  
and  
\[\mathbb{G}=\bigg{\{}n|n\in\mathbb{M}\wedge\sum_{i=n}^{(n+U-1)}\text{mod }N|\hat{\mathbf{h}}_{\mathbb{C}}(i)|^{2}\geq\beta\bigg{\}}. \tag{21}\]  
Next important issue is the determination of the threshold value \(\beta\) and the length \(U\). The former can be defined in relation to the value in (14). Because \(U\) impulse response coefficients are analyzed by the filter at the same time, a direct and straightforward proposition of the threshold level is:  
\[\beta=2U\sigma_{W}^{2}. \tag{22}\]  
This takes into account the cumulative effect of potential noise samples in the window of analysis. If the energy of \(U\) adjacent impulse response coefficients is less then (22), they all are considered as noise. Otherwise, it is possible that among them can be real channel coefficients so they are all appended to the reconstructed impulse response. The choice of the length \(U\) requires a little bit more investigation. The research in [16] reveals that the best estimate of compressible signal in terms of MSE is its sparse approximation. This means that only large-amplitude components of the signal should be recovered, and their number is inversely related to noise power. Finally, the low amplitude ones are set to zero because if they were reconstructed, it would reduce the overall quality of the estimation. The above explanation implies that the window length \(U\) should be smaller for low SNR than in the opposite situation. A trade-off solution for \(U\) can be found using the relation between two error components--estimation error \(\epsilon_{E}\) and approximation error \(\epsilon_{A}\)--that are responsible for the total MSE value of the channel impulse response \(\mathbf{h}_{U}\) estimate, i.e. :  
\[\text{MSE}=\epsilon_{E}+\epsilon_{A}. \tag{23}\]  
Consider a hypothetical compressible impulse response \(\mathbf{h}_{\mathbb{C}}\) of length limited to \(M\) (please note that the remaining N-M coefficients are considered 0; therefore, they have no effect on further processing). Only \(U\) of its largest coefficients are recovered in a sparse counterpart \(\mathbf{h}_{U}\). The estimation error \(\epsilon_{E}\) is defined as the variance of the LS estimation of \(\hat{\mathbf{h}}_{U}\).