# Statistical Method Based Waveform Optimization in Collocated MIMO Radar Systems
## 5 Simulation Results and Discussions
### Minimization of \(P_{out}(\gamma)\)  
In order to test the effectiveness of Algorithm 1 given in Section 4, we optimize transmit and receive waveform. The network configurations are kept as in Section 5.1. The results are given in the following three ways:  
1. Using Figure 4, we reflect on the performance while selecting a specific \(\mathbf{f}\). Initially, we select \(\mathbf{f}\) using random complex Gaussian vector and normalize it to have power 1. Next, we employ a match filter bound and achieve local solutions of \(\mathbf{f}\) and also link it to obtain \(\mathbf{h}\). This solutions shows consistent improvement across the gamma range and it reflects on the effectiveness of such a solution over a random selection of \(\mathbf{f}\) and \(\mathbf{h}\). We set the local solution of \(\mathbf{f}\), referred in Figure 4 as, 'Init. \(\mathbf{f}\) using MFB', as an initialization vector and perform CMM and UMM optimization using Algorithm 1. While the improvement is for both methods, the later, i.e., unconstrained approach finds excellent solutions across the \(\gamma\) range. Hence, in the forthcoming analysis, only UMM is considered while CMM shall have similar results;
2. Using Figure 5, we indicate the convergence of Algorithm 1 at three different values of \(\gamma\), i.e., at 1, 2, and 3 respectively. The plot shows that the convergence is achieved at the 39 th, 37 th, and 34 th iteration, having terminal values of 0.590396, 0.761357, and 0.833775, for \(\gamma=1\), 2, and 3 respectively. Hence, a faster convergence at higher values of threshold is observed. Note that the convergence is based on the objective function (outage probability), which is non-decreasing in a viable direction to within the the optimality tolerance threshold set as \(10^{(-6)}\);
3. Using Figure 6, we demonstrate the time complexity for three sub-routines of Algorithm 1, namely, interior-point, sequential quadratic programming, and active-set approach. It is observed that the convergence of each of the three schemes is exactly the same and hence they find identical solutions; however, the time taken for each approach is different, hence the need for this analysis. The processor considered for this task is Intel(R) Core(TM) m7-6Y75 CPU 1.20 GHz 1.50 GHz with 16 GB RAM. Figure 6a is  
Figure 3: Comparison of outage probability with respect to the predefined threshold \(\gamma\) for different filter order of transceiver units.  
the scatter plot of the three sub-routines and it is observed that the 'active-set' method is finding solutions faster compared to its counterparts. The marked difference is especially notable at high threshold values where 'active-set' method is consuming just 0.05 s as compared with 0.6 s required for other methods. Figure 6b shows the box plot for the three sub-routines indicating the mean value and standard deviation with respect to time in seconds. Again, the 'active-set' method fairs better.  
Figure 4: Comparison of outage probability with respect to the predefined threshold \(\gamma\) for multiple design criterion of transmit waveform \(\mathbf{f}\).  
Figure 5: Convergence minimization of outage probability with respect to the number of iterations for different values of \(\gamma\).