# Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions
## 1 Introduction
Hence, the impulse response obtained as IFFT (Inverse FFT) of the LS result is divided into two time-continuous parts. The first one of the length of the guard period is the channel part and consists of the highest energy impulse response coefficients. The second, a noise part, is formed with the remaining coefficients that can be further used for noise variance estimation. Finally, only the channel part is used to recalculate the channel frequency characteristic. In FFT processing, the coefficients considered as noise are replaced with zero values thus contributing to the smoothing/averaging effect in the frequency domain. All channel refinement is performed in the time domain; therefore, the double use of the Fourier transform is the inherited feature of the methods known as DFT-based. A detailed inspection of the channel part reveals that the number of significant impulse response coefficients, whose reconstruction is necessary, may be even smaller. Numerous studies indicate that it can be a typical situation especially in a wireless environment where only a few signal propagation paths transport most of the energy between the transmitter and receiver [10; 11]. These types of channels have a sparse or, even more frequently, a compressible impulse response (see a comparison in Figure 1). The channel compressibility means that the impulse response coefficients form clusters around the dominant ones with rapidly decreasing amplitudes within the cluster. The compressible CIR can be successfully reconstructed supplementing the DFT-based estimation idea with iterative greedy algorithms belonging to the broad class of compressed sensing (CS) methods [12]. For example, a simple modification of the classical OMP method that forms groups of the reconstructed coefficients was proposed in [13]. Unfortunately, its quite good reconstruction efficiency comes at the cost of high computational complexity due to the inherent iterative procedure of the greedy algorithms. This paper presents a solution that reduces the computational complexity and retains the group-based reconstruction principle. The proposition is a modification of the estimation procedures described in [14; 15], where the restored CIR coefficients have been individually selected by comparing their energy to the power of noise. However, the channel compressibility nature has not been exploited there. The authors' method is mathematically explained using a relation between estimation and approximation errors--the idea with a detailed description presented in [16]. A simulation comparative analysis with other popular DFT-based methods shows quality improvement of the uncoded transmission in terms of Bit Error Rate (BER) and Mean Square Error (MSE) measures in low and middle ranges of signal-to-noise ratio (SNR). On the other hand, a slightly worse performance for high SNR is not a serious obstacle to use this proposition because error correction codes commonly used in practical systems can compensate this loss. Especially implementation efficient LDPC (Low Density Parity Check) codes including nonbinary ones [17; 18] and polar codes [19] are designated for use in modern communications systems. The original contributions of the paper are as follows:  
* to complement the DFT-based scheme with an additional moving average filter in order to perform the final CIR coefficients selection in groups (Section 3.3);
* to determine the moving average filter order using the relationship between approximation and estimation parts of the MSE error (Section 3.3);
* to perform a comparison analysis between the proposed method and the related ones using simulation tests in OFDM system model (Section 4)  
In order to outline a relation between the proposed and the most closely related solutions (discussed here for comparison purposes), Table 1 highlights their main distinctive features.