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In this problem, we consider target detection in a radar system. Upon observing a realization x of the random variable X which represents the return signal in noise, a decision has to be made as to whether the target is present or absent. The target amplitude is known and equal to A > 0. The observation is made in zero-mean Gaussian noise W (real-valued), whose variance is known and equal to o?. If the target is present (hypothesis H1), X = A + W. If the target is absent (hypothesis Ho), X = W. The likelihood ratio test for this problem is stated as follows: If Si(x)/ fo(x) > n decide in favor of H1: otherwise, decide in favor of Ho. If n 1, determine the probability of correct detection (target declared present when indeed present) and the probability of false alarm (target declared present when in fact absent). Express these probabilities in terms of the ratio a = Alo. If a = 4, which of the following represents the correct probabilities of false alarm and correct detection, respectively? Answers are rounded to the closest whole percentage value.

In this problem, we consider target detection in a radar system. Upon observing a realization x of the random variable X which represents the return signal in noise, a decision has to be made as to whether the target is present or absent. The target amplitude is known and equal to A > 0. The observation is made in zero-mean Gaussian noise W (real-valued), whose variance is known and equal to o?. If the target is present (hypothesis H1), X = A + W. If the target is absent (hypothesis Ho), X = W. The likelihood ratio test for this problem is stated as follows: If Si(x)/ fo(x) > n decide in favor of H1: otherwise, decide in favor of Ho. If n 1, determine the probability of correct detection (target declared present when indeed present) and the probability of false alarm (target declared present when in fact absent). Express these probabilities in terms of the ratio a = Alo. If a = 4, which of the following represents the correct probabilities of false alarm and correct detection, respectively? Answers are rounded to the closest whole percentage value.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 21E
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This is a multi-part question 问题2 In this problem, we consider target detection in a radar system. Upon observing a realization x of the random variable X which represents the return signal in noise, a decision has to be made as to whether the target is present or absent. The target amplitude is known and equal to A > 0. The observation is made in zero-mean Gaussian noise W (real-valued), whose variance is known and equal to o?. If the target is present (hypothesis H1), X = A + W. If the target is absent (hypothesis Ho), X = W. The likelihood ratio test for this problem is stated as follows: If fi(x)/fo(x) > n = decide in favor of H; otherwise, decide in favor of Ho. If n = 1, determine the probability of correct detection (target declared present when indeed present) and the probability of false alarm (target declared present when in fact absent). Express these probabilities in terms of the ratio a = Alo. If a = 4, which of the following represents the correct probabilities of false alarm and correct detection, respectively? Answers are rounded to the closest whole percentage value. 2% and 98% None of the answers are correct. 5% and 95% 5% and 98% 问题3 In the previous problem, if instead of 7 = 1 the threshold is set so as to ensure probability of false alarm of 1%, what will the probability of correct detection be? Mark the answer closest to the true value. O 89% 95% 99%
Transcribed Image Text:This is a multi-part question 问题2 In this problem, we consider target detection in a radar system. Upon observing a realization x of the random variable X which represents the return signal in noise, a decision has to be made as to whether the target is present or absent. The target amplitude is known and equal to A > 0. The observation is made in zero-mean Gaussian noise W (real-valued), whose variance is known and equal to o?. If the target is present (hypothesis H1), X = A + W. If the target is absent (hypothesis Ho), X = W. The likelihood ratio test for this problem is stated as follows: If fi(x)/fo(x) > n = decide in favor of H; otherwise, decide in favor of Ho. If n = 1, determine the probability of correct detection (target declared present when indeed present) and the probability of false alarm (target declared present when in fact absent). Express these probabilities in terms of the ratio a = Alo. If a = 4, which of the following represents the correct probabilities of false alarm and correct detection, respectively? Answers are rounded to the closest whole percentage value. 2% and 98% None of the answers are correct. 5% and 95% 5% and 98% 问题3 In the previous problem, if instead of 7 = 1 the threshold is set so as to ensure probability of false alarm of 1%, what will the probability of correct detection be? Mark the answer closest to the true value. O 89% 95% 99%
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