## 4 Low Complexity Non-Linear Detection
### Two Approaches for Non-Linear Data Detection
#### 4.2.2 Near ML Detection
``` Input:\(\mathbf{R}\), \(\mathbf{\hat{v}}\), \(\mathbf{P}\), \(N_{c}\), \(\Omega\), \(N_{D}\) Output:\(\mathbf{\hat{\lambda}}\) estimated symbol \(\triangleright\) The bold uppercase variables represent matrices \(\triangleright\) The bold lowercase variables represent vectors \(\triangleright\) All the variables are considered to be global, visible in any part of the function \(\triangleright\) The use of subindexes in the variables indicates that a vector or row of a matrix is being referenced {Main program: Recursive near ML detector} \(D_{min}\leftarrow\infty\) \(skip\gets false\) level \(\leftarrow\left\{\quad\right\}\), void vector \(\chi\leftarrow\left\{\quad\right\}\), void vector for\(\ell=N_{D}:-1:1\)do \(\mathbf{d}_{



the demapping of the OFDM symbols. The second stage consists of estimating the channel with the use of the pilot symbols. The channel estimator block has the channel matrices \(\mathbf{G}\) and \(\mathbf{K}\) as output, necessary to carry out the ICI mitigation and data estimation, respectively. In the third stage, the \(D_{T}\) truncation is carried out once the \(B\) bands of the matrix \(\mathbf{K}\) are selected. Afterward, the QR decomposition of the truncated matrix \(\mathbf{K}\) under the ZF or MMSE criterion is carried out. Next, the vector \(\mathbf{\tilde{v}}\) is calculated by multiplying \(\mathbf{v}\) and \(\mathbf{Q}^{H}\). The following step is to carry out either OSIC or Near-ML non-linear detection. At the final stage, bit error correction is performed on the estimated vector \(\hat{\boldsymbol{\chi}}\). The estimated vector enters a feedback loop where ICI mitigation is performed for the data subcarriers. This process reduces the channel estimation error and improves the detection of the data. The process is repeated until the number of iterations configured in the receiver is executed. ## 6 Computational Complexity

The computation complexity is first presented for the QR decomposition; it is given in terms of \(\mathcal{O}\) and verified by counting the number of complex operations, where Givens rotations are implemented using conventional arithmetic. The parameters considered for the system are:

* An OFDM system with \(N_{D}=48\) data subcarriers;
* \(4-QAM\) modulation scheme with unitary average power per symbol;
* \(N_{c}=4\) is the maximum number of surviving branches per level in the Near ML V2V algorithm;
* The dimensions of \(\mathbf{X}_{N_{D}}\) is \(48\times 49\) for the ZF criterion and \(96\times 49\) for the MMSE criterion;
* The algorithms were performed for 2000 channel realizations, and the complex operations employed were counted independently for each algorithm and finally averaged;
* Based on simulations results, it was determined the number \(B=27\) bands for truncating matrix \(\underline{\mathbf{K}}\). As can be observed in Table 1, the complexity obtained in this proposal, in terms of complex operations, is in quadratic order with respect to the \(B\) number of bands and in linear order with respect to the \(N_{D}\) number of data subcarriers. Systems with similar orders of complexity can be found in [8; 14], but the work proposed in this paper provides a significant reduction in the complexity required for the conventional MMSE linear criterion, which is reported in the literature in \(\mathcal{O}(N_{D}^{3})\) order. A system that uses the real model and LDL decomposition with a complexity similar to \(\mathcal{O}(B^{2}N_{D})\) was proposed in [8] for an OFDM system in doubly dispersive channels. However, that work does not achieve to improving the performance in terms of BER when compared to the conventional MMSE detection. In the case of the OSIC and Near ML detector complexity, the necessary complex operations were assessed for the detection of \(N_{D}\) symbols transmitted by the system. The number of required complex operations was counted depending on the signal-to-noise ratio, both for the ZF criterion and for the MMSE. The results of these simulations are presented in Figure 8.

Figure 4.0 Model of the LP-OFDM transmitter.