### Abstract

We consider a problem in which exactly one of n+1 distinct signals {S_{0}, ..., S_{n}}, say S_{0}, may be present in a noisy environment and the presence or absence of S_{0} needs to be determined. The performance is given by the false alarm probabilities P_{f(i)}, i = 1, ..., n, where P_{f(i)} is the probability that S_{i} is declared as S_{0}, and the detection probability P_{d} which is the probability that S_{0} is correctly recognized. It is required that P_{d} be maximized while P_{f(i)}≤c_{i}, i = 1, ..., n, where c_{1}, ..., c_{n} are prescribed non-negative constants. The solution is an extended Neyman-Pearson test in which p_{0}(x) is tested against (w_{1}p_{1}(x)+...+w_{n}p_{n}(x)) where p_{i}(x) is the probability density function of observation X when S_{i} occurs. We propose two methods to obtain w_{1}, ..., w_{n} for any given c_{1}, ..., c_{n}. For certain simple cases, analytical or graphical solution is used. For the general case, we propose a search algorithm on w_{1}, ..., w_{n} which provides a sequence of extended Neyman-Pearson tests that converge to the optimal solution. Numerical examples are given to illustrate these methods.

Original language | English (US) |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Publisher | Society of Photo-Optical Instrumentation Engineers |

Pages | 378-389 |

Number of pages | 12 |

Volume | 3720 |

State | Published - 1999 |

Event | Proceedings of the 1999 Signal Processing, Sensor Fusion, and Target Recognition VIII - Orlando, FL, USA Duration: Apr 5 1999 → Apr 7 1999 |

### Other

Other | Proceedings of the 1999 Signal Processing, Sensor Fusion, and Target Recognition VIII |
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City | Orlando, FL, USA |

Period | 4/5/99 → 4/7/99 |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(Vol. 3720, pp. 378-389). Society of Photo-Optical Instrumentation Engineers.