### I. Introduction

### II. Basic Description

### 1. Alert–Confirm Detection

*T*

*is the time of the range bin,*

_{res}*N*

*is the number of range bins,*

_{rng}*N*

*is the number of coherent integrations,*

_{cn}*N*is the number of non-coherent integrations, and

*T*

*is the coherent dwell time.*

_{coh}*T*

*is fixed and that the dwell time*

_{coh}*T*

*is dependent on*

_{d}*N*. The probability density function (PDF) of

*y*is

*I*

_{N}_{−1}is the modified Bessel function of order

*N*–1, and

*R*

*= 2 ×*

_{p}*SNR*[14]. The SNR is normally calculated by the radar range equation, which is

*P̄*= average transmitted power*G*= antenna gain*T*_{d}*A*_{e}*σ*= radar cross section (RCS)*A*= amplitude related to*σ*(*σ*=*A*^{2}/2)*k*= Boltzmann constant*T*_{eff}*L*= Loss factor

*A*is described by a Rayleigh distribution expressed as follows:

*σ̄*is the average RCS.

*i*= 1 for alert detection and

*i*= 2 for the confirm detection. The false alarm probability per cell can be expressed as

*m*resolution cells is

*T*

_{beam}*,*for the alert–confirm detection is limited by target movement,

*T*

_{d}_{1}is the dwell time for alert detection,

*T*

_{d}_{2}is the dwell time for confirm detection,

*T*

*is the time between alert and confirm detections including the processing time, and*

_{in}*T*

*is the time that the target dwells in a beam direction. By substituting Eq. (1),*

_{stay}*N*

_{1}and

*N*

_{2}are the numbers of non-coherent integrations in the alert and confirm detections, respectively.

*ν*is the mean new target appearance rate within the search volume,

*N*

*is the number of search beams in a frame time,*

_{beam}*T*

*is the time for tracking the beams for multiple targets, and*

_{track}*η̄*is the average number of unconfirmed search alarms per new target [16]. The second term,

*mP*

_{f}_{1}×

*N*

*, represents the number of confirm beams from false alarms, and the third term,*

_{beam}*T*

*×*

_{s}*ν*(1 +

*η̄*), is the number of confirm beams from the true target detection.

*T*

*can include any time required for tasks such as calibration, among others.*

_{track}*T*

*is expressed as a ratio of the frame time, i.e.,*

_{track}*ω*is the solid angle of the radar beam, and Ω is the solid angle of the search frame. As

*η̄*is dependent on

*T*,

*T*cannot be expressed in the closed form, and iteration is required to calculate

*T*and

*η̄*.

### 2. Cumulative Detection Probability

*R*can be expressed as [9]

*P*

*(*

_{d}*R*′) is the per-scan probability at

*R*′ for a given false alarm probability

*P*

*; Δ =*

_{f}*V*

*×*

_{c}*T*is the distance a target radially travels during a search frame, where

*V*

*is the target radial velocity, and*

_{c}*l*is the number of cumulations that equals the number of frames by the time the target reaches a given range from the initial detection range. The initial detection range is set large enough that the detection probability can be ignored.

*P*

*= 0.85 as a function of the number of non-coherent integrations,*

_{c}*N*in Eq. (1). The scan area was defined as shown in Fig. 2 using a common 4 bar, ±60° scan raster. The bars were spaced 2.8° apart, and the number of beams in a bar was 30. The other parameters used in the simulation are listed in Table 1 [17].

*R*increases to its maximum as

*N*increases from zero. In this interval, the effect of the increased per-scan detection probability exceeds that of the decreased scan rate. However, once

*R*reaches its maximum, the effect of the scan rate is dominant, and the range decreases. The maximum range

*R*

_{0}was 38.07 (NM) at

*N*= 9. This result was verified by comparing with that in [9].

### III. Optimization for the Alert-Confirm Detection

### 1. Dwell Time Optimization

*P*

*= 10*

_{fa}^{−6}and that the maximum allowable number

*N*

*= 50. Optimization aimed to find the parameters for*

_{c}*P*

_{f}_{1},

*P*

_{f}_{2},

*N*

_{1}, and

*N*

_{2}that maximize the detection range for a given cumulative detection probability (

*P*

*= 0.85). The overall optimization is summarized in Fig. 4.*

_{c}*R*as a function of

*N*

_{1}when

*m*= 32 and

*α*= 0.

*N*

_{1}increases the per-scan detection probability but decreases the scan rate. Therefore, as

*N*

_{1}increases,

*R*increases to its maximum point and then decreases monotonically. The change in

*R*around the optimum is small. Fig. 6 shows the contour plots of

*R*for

*N*

_{2}and

*P*

_{f}_{1}at

*N*

_{1}= 1 and at

*N*

_{1}= 5. The change in

*R*due to

*N*

_{2}is also small when

*N*

_{1}approaches its optimum (

*N*

_{1}= 5).

*ν*. If

*ν*= 0,

*R*/

*R*

_{0}= 1.163 and 1.043 for QC and LC, respectively, as shown in Fig. 7. The increment is much larger for QC than for LC. In other words, QC, which keeps the target RCS constant between the alert and the confirm detection, has a significantly longer detection range than LC. Late confirmation even extends the detection range compared with the uniform scanning detection. The maximum range decreases as

*ν*increases because more confirm detections are triggered. For LC, if

*ν*is greater than 0.7, then the range drops below

*R*

_{0}. Fig. 8 shows the optimal values for

*N*

_{1},

*N*

_{2}, and

*T*.

*N*

_{2}is larger than

*N*

_{1}and the frame time is shorter than that of the uniform scanning detection. At

*ν*= 0,

*T*= 7 and 4.56 seconds for QC and LC, respectively, which are significantly less than

*T*= 8.6 seconds for the uniform scanning. The total dwell time per beam position for QC is shorter than that for LC, e.g.,

*N*

_{1}+

*N*

_{2}= 24 and 30, respectively, at

*ν*= 0.6. If the RCS fluctuation time can be estimated in advance, the constraint Eq. (11) can be used for the QC condition. In this simulation, the constraint Eq. (10) appears only where

*ν*= 0.2 – 0.3 in LC. Despite the shorter dwell time for QC, its frame time is longer than that of LC because its confirm detections by false alarms are greater than that of LC.

*P*

_{f}_{1}= 10

^{−3}for QC when

*ν*is less than 0.5. Therefore,

*P*

*= 3.125 × 10*

_{f2}^{−5}from (8). The false alarm probability for confirm detection is lesser than that for alert detection. That is, the threshold level for confirm detection is higher than that for alert detection.

*ν*is smaller than 0.3, the detection range is maximized when

*N*

_{1}is 5,

*N*

_{2}is 28, and

*P*

_{f}_{1}is 10

^{−3}with quick confirmation.

*m*increases. The rate of the maximum range decrease is similar for all values of

*ν*. Fig. 12 illustrates the range reduction as a function of the tracking load

*α*, where the dashed line shows the range reduction for the uniform scanning detection. At

*α*= 0.5, the maximum range decreases by 1 dB. For QC and LC, the reduction is 0.66 dB and 0.58 dB, respectively, which is less than that for the uniform scanning detection. LC shows the smallest reduction because it has the shortest frame time among the three cases.

### 2. Monte Carlo Simulation

*ν*was 0.3. The reflective signals from the targets were synthesized according to the RCS distribution as in Eq. (4), and the Gaussian random noise was added by SNR in Eq. (3). The detection range was defined as the range at which 85% of the targets were detected. The detection was decided by the false alarm probability or equivalently by the threshold level in Eq. (7).

*ν*= 0.3,

*N*

_{1}= 5, and

*P*

_{fa}_{1}= 10

^{−3}. This model is useful to simulate complex scenarios or system diversity, including beam broadening at off-boresight angles, in which numerical calculation are difficult. Modifying the confirmation strategy (e.g., more than 1 out of 3 confirmation detections) is also possible to improve the detection range. This issue will be investigated in future research.