## 4 Low Complexity Non-Linear Detection
### Two Approaches for Non-Linear Data Detection
#### 4.2.2 Near ML Detection
Figure 4.1 Schematic structure of the 802.11p link-level receiver proposed with no-linear detection and iterative ICI cancellation. Figure 8 demonstrates the complexity in terms of complex operations required for both proposed detectors. For the case of the OSIC detector the complexity, using the MMSE criterion in the calculation of the QR decomposition, significantly reduces the complexity of the data detection task. For the Near ML detector, none of the two mentioned criteria for QR decomposition is affecting the complexity of finding the optimal vector. The complexity in the OSIC detector is constant and it does not depend on the receiver's SNR. This approach presents the lowest complexity but its performance is not the best, as shown in Figure 4.2 In the case of the Near ML detector, Figure 8 shows that its complexity tends towards a low constant value starting at an SNR of 15 dB, where it achieves a very similar value to the complexity obtained by the OSIC detector. On the other hand, its performance, in terms of BER, is quite close to the ML detector, as will be discussed in detail in the following section. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**QR** & \
\begin{tabular}{c} **Number of** \\ **Rotations with** \\ **Banded Matrix** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Rotations with** \\ **Full Matrix** \\ \end{tabular} & **Complexity** \\ \hline ZF & 37,900 & 114,020 & \(\mathcal{O}(B^{2}N_{D})\) \\ \hline MMSE & 59,650 & 172,780 & \(\mathcal{O}(2B^{2}N_{D})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational complexity required for sorted-QR decomposition. Figure 8: Computational complexity of OSIC and Near ML detectors. Table 2 summarizes the complexity required by the most representative detection algorithms used in doubly selective channels (DSC). The main objective is to provide the reader with an overview of the computational complexity of the compared approaches. Table 2 shows the closeness in computational cost of Near ML detection compared to Sorted OSIC detection. The LMMSE detection requires approximately three times the computational cost of the OSIC and Near ML detection. The proposed algorithms achieve a cost reduction in \(10^{25}\) order compared with the Full-ML detection. ## 7 Simulations Results

The numerical results presented below were obtained from a simulator implemented in Matlab-Simulink, replicating compatible simulation environments with the 802.11p link [11]. The blocks that describe the internal signal processing algorithms considered for the transmitter and the iterative receiver were discussed in Section 5. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Complexity**} & \multicolumn{2}{c}{**Normalized Cost**} \\ \cline{3-4}  & & \(N_{D}=48\) & \(N_{D}=512\) \\ \hline ML & \(\mathcal{O}(\Omega^{N_{D}}N_{D})\) & 5.43 \(\times 10^{25}\) & Inf \\ \hline LMMSE & \(\mathcal{O}(N_{D}^{3})\) & 1.5802 & 179.7970 \\ \hline Near ML Proposed & \(\mathcal{O}((2B^{2}+N_{c}^{2})N_{D})\) & 1.0110 & 1.0110 \\ \hline OSIC Proposed & \(\mathcal{O}(2B^{2}N_{D})\) & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Computational complexity in terms of complex products per OFDM symbol required for signal detection, considering one iteration with \(N_{c}=4\), \(B=27,\Omega=4\), and \(\text{SNR}=15\) dB. Figure 9: BER vs. SNR comparison without convolutional encoder of the LMMSE with channel knowledge vs. the proposed OSIC and Near ML detections using QR decomposition with ZF and MMSE as criteria. The channel model uses a Jake’s Doppler profile for each channel tap. The power delay profile is exponentially decaying with RMS delay spread of 0.4 μs and frequency Doppler spread of 1 kHz modeling a Rayleigh fading NLOS scenario. The standard 802.11p uses a bandwidth of \(BW=10\) Mhz, a cyclic prefix containing CP = 16 samples, due to which the system can absorb a maximum tolerable delay of 1.6 us in the duration of the CIR from the V2V channel.