Principles of Signal Detection and Parameter Estimation by Bernard C. Levy

By Bernard C. Levy

This new textbook is for modern sign detection and parameter estimation classes provided on the complex undergraduate and graduate degrees. It provides a unified therapy of detection difficulties bobbing up in radar/sonar sign processing and sleek electronic conversation structures. the fabric is entire in scope and addresses sign processing and conversation functions with an emphasis on basic rules. as well as average subject matters often lined in one of these direction, the writer contains fresh advances, corresponding to the asymptotic functionality of detectors, sequential detection, generalized chance ratio exams (GLRTs), strong detection, the detection of Gaussian signs in noise, the expectancy maximization set of rules, and the detection of Markov chain signs. quite a few examples and certain derivations in addition to homework difficulties following each one bankruptcy are included.

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Example text

5. ROC for two zero-mean Gaussian vectors with unequal variances σ12 and σ02 . The ROC is plotted for the values r = 1, 2, 4 and 8 of the variance ratio r = σ12 /σ02 . 3. 40) the optimum test is given by H1 η Y H0 with η= λ1 − λ0 ln(τ ) + . ln(λ1 /λ0 ) ln(λ1 /λ0 ) Note that Y takes integer values whereas the threshold η is real. Let n = η denote the smallest nonnegative integer greater than or equal to η. Then the probabilities of detection and of false alarm can be expressed as +∞ PD (n) = λm 1 exp(−λ1 ) m!

Binary non-symmetric channel model. In other words, the probability that bit B = j is flipped is qj . , 0 < qj < 1. Since we seek to minimize the probability of error, the cost function is given by Cij = 1 − δij for i, j = 0 or 1. Then depending on whether Y = 0 or 1, the optimum receiver examines the likelihood ratio values L(0) = 1 − q1 q1 , L(1) = . 1 − q0 q0 To simplify our analysis, we assume q0 + q1 < 1, which ensures L(0) < L(1). Note that this is a reasonable assumption since in most communication systems the bit flipping probability is small.

1. 35), the optimum test can be expressed as S= with 1 N N H1 Yk k=1 η H0 σ 2 ln(τ ) E 1/2 + , 2 N E 1/2 η= where S admits N (0, σ 2 /N ) and N (E 1/2 , σ 2 /N ) distributions under H0 and H1 , respectively. 67) s2 ds 2σ 2 /N 1 exp(−z 2 /2)dz . 69) and the distance d = (N E)1/2 /σ between the mean vectors m and 0 of Y under H1 and H0 , measured in units of the standard deviation σ. Noting that d ln(τ ) (η − E 1/2 )N 1/2 =− + , σ 2 d and using the symmetry property Q(−x) = 1 − Q(x), we find d ln(τ ) ) PD (τ ) = 1 − Q( − 2 d d ln(τ ) PF (τ ) = Q( + ).

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