# Performance of a practical two-step detector for non-fluctuating targets

## Summaryย (2 min read)

### Introduction

- Individual sensors are capable of producing detection statistics at a high data rate.
- The two-step detection scheme arises when an aggregate data rate restriction is imposed on a distributed sensor system.
- After being shared, the detection statistics are passed to Stage 2 (second-stage detection) of the two-step detection scheme where they are integrated and compared to a final threshold corresponding to the desired overall probability of false alarm.
- Thus an individual platformโs data rate is directly proportional to the first-stage probability of false alarm (PFA).

### II. TWO-STEP DETECTION RULE

- In [4] the Neyman-Pearson (NP) two-step detection rule (2SD) for the Swerling 2 (SW2) [5] fluctuating target model was derived and was shown to be clairvoyant [7] because of its dependence on a priori knowledge of the unknown target SNR.
- [4] presented empirical evidence that the performance of the detector is only weakly dependent on this a priori knowledge.
- Thus, by assuming a reasonable target SNR, the detector becomes a practical one.
- Additionally, the detection rule is linear, making threshold selection and performance analysis straightforward.
- In this section the authors complement the SW2 results in [4] by presenting the NP two-step detection rule for the Swerling 0 (SW0) [5] target model.

### A. General NP Two-Step Detection Rule

- In [4] the general NP two-step detection rule was derived based on the following assumptions: i) Noise at each sensor is independent and identically distributed (IID). ii).
- The target SNR measured by each sensor is also IID. iii).
- The detection cells for each platform align exactly (i.e. there are no registration errors).

### B. Swerling 0 NP Two-Step Detection Rule

- The authors now make several important observations: i).
- The second-stage detection statistic is a non-linear combination the shared first-stage detection statistics. iii).

### C. A Practical Two-Step Detection Rule

- Aside from being clairvoyant, the Swerling 0 NP two-step detector is also non-linear.
- This not only makes its real-world implementation impractical from a computational and system engineering stand-point, it also makes threshold selection and performance analysis difficult.
- The major caveat of using the detection rule above is that observation (iii) above is no longer valid, making the process of optimal threshold selection non-obvious.
- In order to circumvent this issue the authors will select the second-stage thresholds using results from [4] by assuming the target is SW2.

### B. Second-Stage Probability of Detection

- For the SW2 target model the authors were able to derive a closed form for (8) [4].
- In their analyses the approximation proved comparable to Monte Carlo simulations of the true distribution.

### IV. RESULTS

- Where the second-stage detection thresholds were selected based on the results in [4] by assuming the target is SW2.the authors.
- In part A the authors present plots illustrating how ๐๐2 varies with ๐๐๐1, and in part B they present a series of plots illustrating how ๐๐2 varies with target SNR.
- In their analysis the authors consider the Cooperative Networked Radar (CNR) [3] implementation of statistical MIMO radar.
- In this scenario each platform transmits a single pulse and noncoherently integrates its own pulse plus those transmitted by the other participating platforms prior to first-stage thresholding.

### A. ๐๐2 vs. ๐๐๐1

- In the following the authors present two plots (Fig. 2 and 3).
- The first illustrates how ๐๐2 varies with ๐๐๐1 using the practical detector (7) under the SW0 target model.
- The second plot shows the equivalent SNR loss due to first-stage thresholding .
- The SNR loss is defined as the additional SNR needed to achieve the single-stage detection performance using the two-step detector.

### B. ๐๐2 vs. Target SNR

- The focus of each plot is the comparison of a practical detector that can be easily implemented, to the optimal, yet unrealizable one, for the SW0 case.
- Hence, each plot displays two series: one series shows the performance of the practical 2SD (7) where the second-stage thresholds are fixed and were selected based on an assumed SNR yielding a PD of 0.5 for the single-stage detector (2.3 dB and -1.4 dB per pulse per platform for ๐ = 4 and ๐ = 8 respectively).
- The second series shows the performance of the Swerling 0 NP two-step detector (4) in which the second-stage thresholds were selected assuming prior knowledge of the target SNR.
- For reference, these plots also include an upper curve for the unrestricted case and a lower curve for the single platform case.

### V. CONCLUSION

- The authors have examined the two-step detection scheme that arises when practical data-rate limits are imposed on the Cooperative Networked Radar implementation of statistical MIMO radar.
- The authors results echo those in [4] and indicate that the data rates in a distributed detection system can be reduced by several orders of magnitude using the two-step detection scheme without fully compromising the detection capability of the distributed sensor system.
- Additionally, the results presented in Subsection B indicate that the performance of the linear detector is very close to optimal.
- Thus, the authors should expect the linear detector (with thresholds selected by assuming the target is SW2), to perform close to optimal over a range of fluctuating target models.

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### Additional excerpts

...When individual sensor platforms are aggregated to improve sensitivity, such as in statistical MIMO radar [1], [2], [3], the aggregate data rate can saturate both the communication and computational capacity of the system....

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### "Performance of a practical two-step..." refers methods in this paper

...In this section we complement the SW2 results in [4] by presenting the NP two-step detection rule for the Swerling 0 (SW0) [5] target model....

[...]

...In [4] the Neyman-Pearson (NP) two-step detection rule (2SD) for the Swerling 2 (SW2) [5] fluctuating target model was derived and was shown to be clairvoyant [7] because of its dependence on a priori knowledge of the unknown target SNR....

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##### Frequently Asked Questions (2)

###### Q2. What are the future works mentioned in the paper "Performance of a practical two-step detector for non-fluctuating targets" ?

In future work the authors plan to compliment the SNR sensitivity results in [ 4 ] by investigating the loss in detection capability from using the linear detection scheme over the true optimal NP detection rule for the Swerling 0 target model.ย