11 5 10 15 20 25 30 35 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 $_{0}$ (sec) $_{τ}$1.5  (sec) ( $_{0}$=5.0, $_{1.5}$=0.6) (10.0,0.75)  + (16.2,0.762) (20.0,0.9) FIG. 5: Estimation of parameter values for a signal SNR of 8.0. The true locations of the signals are indicated by the ⋆ marker and each of the markers, • , +, ∗ and × , indicates an estimated location corresponding to one of the true locations. The association between the markers and the true signal loca- tions is indicated in the figure. For each true signal location, the simulation consisted of 50 trials. Strictly speaking, as with the false alarm probability, the detection probability will be slightly smaller for the PSO detection statistic, for a given threshold, as compared to that for the true global maximum. The above line of reasoning suggests that the Receiver Operating Characteristics (ROC) of the PSO detection statistic should nearly be the same as that of the true global maximum. The only way to rigorously verify this is to carry out simulations with a large number of tri- als in which both PSO and grid-based searches are per- formed. This is a computationally expensive task which we plan to undertake in the future. However, it is im- portant to note that such a comparison may not be pos- sible for searches that are too expensive for a grid-based search. TABLE VI: Computational cost of PSO on data containing a signal. For each combination of signal SNR and location, the mean number of fitness function evaluations, over 50 trials, is listed along with the maximum (superscript) and minimum (subscript). All numbers are in units of 10 4 and rounded off to a single digit of precision. The number of fitness function evaluations for each combination of signal SNR and location are shown in Table. VI. It is seen that for a signal SNR of 9.0, the maximum number of evaluations is about the same as the mean in the pure noise case (c.f., Table I). This reduction is consistent with the fact that a strong signal makes it easier for the swarm to find the global maximum. D. Computational cost A comparison of Table V with existing results [20] shows that the estimation errors due to PSO are con- sistent with grid-based searches. Since the search region in the current testbed includes unphysical chirp time parameters, the global maximum and, hence, the estimated chirp times fall there in some trials. Fig. 5 shows an example where the estimates from all the trials are shown for a signal SNR of 8. In an improved implementation of PSO, blocking the unphysi- cal region should improve parameter estimation accuracy significantly. As an indicator of this, we also show in Ta- ble V, the rmse obtained by dropping the trials where the estimate fell in the unphysical region. It is seen that the errors are reduced significantly, especially for the lower signal SNR value for which there is more scatter into the the unphysical region. where ̂ θ is the estimator of θ . The rmse includes the effects of both estimator variance and bias. rmse( θ ) = [ E [ ( ̂ θ − θ ) 2 ]] 1 / 2 , (17) Table V summarizes the errors in the estimation of the chirp time parameters for signal SNR values ≥ 8 at the different signal locations used in the simulations. Each entry in the table is an estimate of the root mean-square error (rmse) defined as, 2. Estimation ( τ 0 , τ 1 . 5 ) = (5 . 0 , 0 . 6) (10 . 0 , 0 . 75) (16 . 2 , 0 . 762) (20 . 0 , 0 . 9) SNR=9.0 2(13) 2(12) 1(11) 0 . 5(6) 0 . 3(6 . 0) 0 . 2(4) 1(11) 0 . 2(4) 8.0 44(13) 10 . 5(10) 32(16) 1(10) 12(16) 0 . 3(5) 11(17) 12(10) ( τ 0 , τ 1 . 5 ) = (5 . 0 , 0 . 6) (10 . 0 , 0 . 75) (16 . 0 , 0 . 762) (20 . 0 , 0 . 9) SNR=9.0 4 . 4 5 . 2 3 . 8 4 . 7 5 . 7 4 . 1 4 . 8 5 . 6 4 . 2 4 . 7 5 . 8 4 . 2 8.0 4 . 5 6 . 0 3 8 4 . 7 6 . 5 3 7 4 . 8 5 . 6 4 1 4 . 8 5 . 9 4 2 7.0 4 . 7 6 . 4 3 . 6 4 . 8 6 . 3 3 . 9 4 . 9 6 . 2 3 . 7 4 . 8 5 . 9 3 . 8 6.0 4 . 5 6 . 1 3 . 3 4 . 8 7 . 0 2 . 9 4 . 7 5 . 9 3 . 2 4 . 7 6 . 0 3 . 6 TABLE V: Signal parameter estimation errors with PSO. Each entry in the table is of the form a ( b ), where a and b are the estimated root mean square errors (rmse) for τ 0 and τ 1 . 5 respectively (expressed as a percentage of the true parameter value). In each row, the top and bottom pairs of numbers refer to mse obtained without and with the physical boundary cut respectively. All the numbers have been rounded off, given the expected precision from the 50 trials used per combination of signal SNR and location.