10 FIG. 4: Scatterplot of maximum fitness value found by PSO (Y axis) against the value at the known signal location, τ 0 = 10 . 0 sec, τ 1 . 5 = 0 . 75 sec, for all signal SNR values. In one trial, the maximum fitness value (near 6.0 on the Y axis) dips below the line of equality (dashed). 4 5 6 7 8 9 10 11 4 5 6 7 8 9 10 11 Fitness at signal location Max. Fitness found by PSO C. Signal detection and parameter estimation In order to cast the results obtained so far in terms of signal detection and parameter estimation performance, we choose the maximum fitness value found over the N$_{rep}$ runs as the detection statistic and the location corre- sponding to the maximum fitness value as the estimator for the chirp time parameters. 1. Detection It was discussed in Sec IV C that, after tuning PSO, the distribution of the detection statistic in trials with and without clustering remains the same. As the simul- taneous clustering of the two chirp-times and the fitness values indicates termination in the vicinity of the global maximum, it follows that the probability distribution of the PSO detection statistic is about the same as that of the global maximum. Strictly speaking, the PSO detec- tion statistic will always have a value less than the global maximum but, given our termination criterion, the rel- ative difference between the two is less than 3%. Thus, the false alarm probabilities, for a given detection thresh- old, corresponding to the PSO detection statistic and the true global maximum are also nearly the same, with the former being slightly smaller. In the presence of signals with an SNR of 8 or higher, which is the typical value sought in a real detection, it was shown that almost all trials exhibit clustering and that the detection statistic value was always higher than the fitness at the true signal location. Hence, the distri- bution of the detection statistic in the presence of a sig- nal also closely follows that of the true global maximum. Taken together, the figures of merit show that PSO almost always terminates near the true global maximum when a sufficiently strong signal is present. When the signal is weak, we recover the performance ensured by the tuning procedure for the pure noise case. As far as the second figure of merit is concerned, we find that it is unity for all combinations of signal SNR and locations except for one, namely, SNR = 8 . 0, τ$_{0}$ = 10 . 0 sec and τ$_{1}$$_{.}$$_{5}$ = 0 . 75 sec, for which it was 0.98. Fig. 4 shows the scatterplot between the maximum fit- ness found by PSO and the value at this signal location for all signal SNR values. It is seen that in one trial the maximum fitness fell below the value at the signal loca- tion. However, the two values are so close that the figure of merit should be considered to be practically unity for this case too. Table IV reports the first figure of merit for each com- bination of signal SNR and location. As expected, for the case of strong signals (SNR ≥ 7) the probability of clus- tering is always, and often significantly, higher than 90%. For the weak signal SNR of 6 . 0, the probability of clus- tering has an average value of 91% which is statistically consistent with the pure noise case of 90%. ness in the cluster is larger than the value at the signal location. Our second figure of merit, therefore, is the fraction of trials in which this occurs. Ideally, this figure of merit should be unity. ( τ 0 , τ 1 . 5 ) = (5 . 0 , 0 . 6) (10 . 0 , 0 . 75) (16 . 0 , 0 . 762) (20 . 0 , 0 . 9) SNR=9.0 ( P$_{λ}$ ) 98 ( P$_{τ}$$_{0}$ )94 ( P$_{τ}$$_{1}$$_{.}$$_{5}$ )94 94 92 94 94 96 94 90 98 94 8.0 96 96 92 98 96 96 98 92 92 94 92 96 7.0 96 96 90 92 90 88 98 94 98 92 88 90 6.0 92 94 84 82 86 86 88 86 84 94 96 96 TABLE IV: Probability of clustering for simulations with sig- nal present in the data. For each combination of signal SNR and location, the fraction of trials (in %) P$_{λ}$ , P$_{τ}$$_{0}$ and P$_{τ}$$_{1}$$_{.}$$_{5}$ for which the fitness, τ 0 and τ 1 . 5 values respectively were found to be clustered are listed. The probability of clustering, shown in bold, is the maximum over P$_{λ}$ , P$_{τ}$$_{0}$ and P$_{τ}$$_{1}$$_{.}$$_{5}$ . The number of trials for each combination is 50.