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So far, we have focussed on clustering as the main in- dicator of success in locating the true global maximum. Does this imply that in the trials in which there is no clustering, PSO fails to locate the global maximum? To address this, we carried out the following test. First, we retain the maximum among the N$_{rep}$ fitness values from each trial. For each combination of N$_{p}$ and N$_{t}$ , we divide the set of maximum fitness values into two disjoint sub- sets: one in which all parameters, the two chirp-times and the fitness, were clustered and the other in which at least one parameter did not show clustering. For the former set, clustering of all three parameters is a strong indica- tor of successful termination near the global maximum. A two-sample Kolmogorov-Smirnov (KS) test [19] is car- ried out to see if the two subsets were drawn from the same parent distribution. The results are summarized in Table III. 

As can be seen from the table, in all cases the test supports the hypothesis that the maximum fitness value is drawn from the same distribution irrespective of the clustering of the parameters. That this is a non-trivial result is further supported by the fact that if the same test is done with fitness values other than the maximum one, the null hypothesis is rejected strongly. Table III shows the results from the same test but using the set of minimum fitness values. In this case, it is seen that the values are drawn from different distributions, at least for N$_{p}$ = 81. Thus, we conclude that even in the absence of 

## C. Trials with no clustering 

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clustering, the PSO run that yields the maximum fitness value terminates, with high probability, in the vicinity of the global maximum for the N$_{p}$ = 81 and N$_{t}$ = 80 combination. 

We have traced the lack of clustering to the presence of distant peaks in the fitness function that are similar in value. The probability of this happening in the presence of a sufficiently strong signal is very small, but this need not be so for noise-only data. 

We have demonstrated a systematic tuning procedure for the design variables of PSO. It is important to note that no prior information about any special features of the fitness function was used. Hence, the procedure would stay the same if the testbed were changed. 

A larger number of trials or a finer spacing of grid points in N$_{p}$ and N$_{t}$ will probably lead to a different end result. Instead of N$_{p}$ = 81, for example, N$_{p}$ = 121 may turn out to be the right choice. However, the main goal in this paper is to test the viability of PSO and, for this purpose, a coarse tuning such as the one presented here is adequate. Besides, a significant investment in refining the results of the tuning procedure would be rendered ob- solete with future improvements in the implementation of PSO. Until a version of PSO is developed that is hard to improve upon, a strong focus on the results from tun- ing, as opposed to improvements to the tuning procedure itself, is not of much use. 

For N$_{p}$ = 121, Table III shows that the minimum fit- ness values obtained with and without clustering are also mutually consistent. This observation suggests an alter- native approach to tuning where one of the measures used for picking the optimal combination is this type of con- sistency. We leave this for future work to address. 

## D. Comments 

