# Low Complexity Equalization of Orthogonal Chirp Division Multiplexing in Doubly-Selective Channels
## 4 Numerical Simulations
### Performance Comparison of Proposed Equalization Algorithms  
In this section, we compare the performance of the proposed low complexity equalization algorithms for UP-OCDM over doubly selective channels with varying normalized Doppler spread set as 0.1 and 0.3. We simulate the band MMSE block equalizer and the iterative LSQR equalizers with varying parameters. For the band MMSE block equalizers, we set the bandwidth \(Q\) as 2, 4, and 6, while for the iterative LSQR equalizers, we set the iteration number \(i\) as 1, 4, and 8, respectively. To make a fair comparison, we group the equalizers with various \(Q\) and \(i\) into three pairs and confirm the computational complexity of the two low complexity equalizers in each pair to be similar. We set the parameters of the low complexity equalizers in each pair as \(Q=2/i=1\), \(Q=4/i=4\), and \(Q=6/i=8\), respectively. In this paper, we denote the performance of equalizers in the same pair with curves that plotted using the same color.  
The simulation results are shown in Figure 4(a),b. The blue curves denote the pair with the lowest complexities, or the \(Q=2/i=1\) pair. The red curves denote the pair with larger complexities, that is, the \(Q=4/i=4\) pair. Moreover, the green curves represent the pair with \(Q=6\) and \(i=8\), which leads to the largest computational complexities among the three pairs. Besides, the MMSE block equalizers, whose complexities are \(\mathcal{O}(N^{3})\), are plotted with black curves. From Figure 4(a), it can be seen that when the Doppler spread is relatively low, the performance of the band MMSE equalizer can outperform the iterative equalizer with similar computational complexity. Whereas, when it comes to the fast-moving scenario, as shown in Figure 4(b), the performance of the band MMSE equalizer will suffer huge degradation, and the performance of the band MMSE equalizers becomes worse than the iterative LSQR equalizers in the second and the third pairs (e.g., the red and the green curves). Moreover, it can be seen from the curves with different colors (e.g., different computational complexity) in both Figure 4(a),b that, the performance of the LSQR iterative equalizer can obtain a huge improvement with the increase of computational complexity and might be comparable to the MMSE block equalizer. In other words, the improvement brought by extra computational complexity for the band MMSE equalizer is not so significant as the iterative LSQR equalizers. Moreover, it should be noted that the complexity of the iterative LSQR equalizer is still much lower than the MMSE block equalizer when a relatively large \(i\) (e.g., \(i=8\) in this section) is used. Finally, we can conclude that, when the Doppler spread is small, or the capacity of computational complexity in the system is severely limited, it might  
Figure 4: BER performance of the UP-OCDM and OFDM systems with iterative equalization algorithms under different iteration number \(i\) with the normalized Doppler spread of 0.1 (a) and 0.3 (b).  
be wise to choose the band MMSE equalizer. On the other hand, when the Doppler spread is large, or the system can bear a relatively higher computational cost, the iterative LSQR equalizer is preferred then.