Resolution 



Large range bins for stage A can be produced by transmitting pulses of 

 long duration. When only one range bin is used in stage A, the minimum range 

 of the bin must be at least half of R^^^, because of propagation time. The two- 

 stage mode would then be an acquisition mode, and a single-stage high-resolution 

 mode could be included for close targets; in fact, for any application in which each 

 stage-A pulse is very long, detection for the resulting blind range must be provided 

 for in some other way. If each stage-A pulse is a very long uncoded waveform, the 

 width of the signal frequency spectrum might be small compared with the doppler 

 frequency shift. Then, to cover the range of expected doppler frequency, it would 

 be necessary to use in stage A either a bank of doppler filters or a receiver band- 

 width considerably larger than the width of the signal frequency spectrum. The 

 analysis of a two-stage system with doppler filters used in this manner is more 

 complicated and will not be considered here. If the doppler shift is provided for 

 by making the receiver bandwidth large or if sampling long pulses results in a 

 loss in SNR, then the decrease in the stage-A SNR must be taken into account. 

 As the compai-isons presented in this paper are in terms of a per-pulse SNR 

 common to both stages, power saving can be measured only when assuming that 

 both the transmitted energy and the resulting SNR are the same for stage-A 

 pulses as for stage-B pulses. Otherwise, one must compute for each case the 

 average amount of transmitted energy required per beam position. 



Even though the resolution cells will be referred to as range bins, the 

 results hold for any type of resolution such that (1) each of the M^ cells is sub- 

 divided (in range andy'or doppler frequency or other) into Mg cells, and (2) a 

 stage-B cell is tested if and only if the one stage-A cell containing it alarms. 



Accepting the fact that the resolution-varying property of two-stage 

 detection contributes to its efficiency, the logical conclusion is that the range 

 of surveillance (in a clutter-free environment) should be covered with as few 

 stage-A range bins as possible under the limitations on minimum range and 

 doppler shift mentioned above. This is supported by the P^-versus-S curves 

 given in figure 3 for fixed-sample stages, a Rice distribution, and N= 1.2. 

 Since the optimum value of Ng decreases with increase in M^, each curve was 

 computed using the Ng which was optimum for Pq about 0.5; these values of /Vg 

 were 14, 10, 7, 5, and 4, respectively, for M^ = 1, 4, 16, 64, and 256. Note that 

 quadrupling the number of stage-A range bins causes an increase of roughly 0. 6 

 to 1.0 dB/pulse in the SNR required for a single-scan detection probability of 

 0.5. For a fair comparison, the M^ bins would cover the same search zone in 

 each of the cases; then the product M^ Mg must be the same for each case in 

 order that the FAR and final resolution be the same. (FAR = 3 PRF x M^Mg x lO""-) 

 The correspondence of range bins (bin centers) to values of S is determined by 

 whatever value is assigned to S^. The P^-versus-S curves are valid for any S^ 

 (in the range of the graph) as long as the threshold assigned to each range-bin 

 test is the optimum one for the value of S corresponding to that range (for that S^) 

 and the target is indeed of the design cross section. For any other situation, 

 where S is the actual rather than design value of the SNR, the curves give only 

 upper bounds to Pq, generally close bounds, though, since (as was suggested by 

 fig. 2) the thresholds would still be at least nearly optimum. 



15 



