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Detection Chia-Hsin Cheng

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 2 Outlines Detection Theory Simple Binary Hypothesis Tests Bayes Criterion The MAP Criterion The ML Criterion Neyman-Pearson Criterion M Hypotheses Composite Hypothesis GLRT (Generalized LRT) The General Gaussian Problem Course Information

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 3 Detection Theory Example: Know Signal in Noise Problem We are faced with the problem of decision which of two possible signals was transmitted. Detection problem: observes r(t) and guess whether s 1 (t) or s 2 (t) was sent. Decision rule decision

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 4

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CCU Wireless Access Tech. Lab. 5 Classical Detection Theory The source generates outputs of two choices (hypotheses), H 0 and H 1.We do not know which hypothesis is true. The transition mechanism can be viewed as a device that knows which hypothesis is true.(i.e., channel model, likelihood function) Based on this knowledge, it generates a point in the observation space according to some probability law.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 6 Classical Detection Theory Example: symbol rate sampling Example: over the symbol rate sampling We confine our discussion to problems in which the observation space is finite-dimensional. The observations consist of a set of N numbers and can be represented as a point in a N-dimensional space.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 7 After observing the outcome in the observation space, we shall guess which hypothesis is true, and to accomplish this, we develop a decision rule that assign each point in the observations space to one of the hypotheses.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 8 Simple Binary Hypothesis Tests We assume that the observation space corresponds to a set of N observations:, or in a vector r, The probabilistic transition mechanism generates points in accord with the two known conditional probability densities and. The objective is to use this information to develop a suitable decision rule. Example:

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 9 Decision Criteria In the binary hypothesis problem, we know that either H 0 or H 1 is true. Each time the experiment is conducted, one of things can happen:

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 10 Bayes Criterion Two assumptions 1. A Priori probabilities are known 2. Costs are assigned C ij : We should like to design our decision rule so that on the average the cost will be as small as possible. Average cost = risk

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 11 Bayes Criterion (cont.) Because the decision rule must say either H 1 or H 0,we can view it as a rule for dividing the total observation space Z into two parts Z 0 and Z 1. When an observation falls in Z 0, we say H 0, and whenever and observation falls in Z 1,we say H 1. Optimal Bayes test: design Z 0 and Z 1 to minimize

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 12 Bayes Criterion (cont.) The risk function of (1) can be written in terms of the transition probabilities and the decision regions: We shall assume throughout our work that the cost of a wrong decision is higher than the cost of a correct decision, i.e., (3) (2)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 13 Bayes Criterion (cont.) To find the Bayes test, we must choose the decision regions Z 0 and Z 1 in such a manner that the risk will be minimized. Because we require that a decision be made, this means that we must assign each point R in the observation space Z to Z 0 or Z 1. Thus Rewriting (2), we have Observing that Substituting into (5) (4) (5) (6)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 14 Bayes Criterion (cont.) Then, we have The first two terms of (7) represent the fixed cost. The assumptions in (3) imply that the two terms inside the brackets are positive, the second term is larger than the first should be included in Z 0 because they contribute a negative amount to the integral. The decision regions are defined by the statement: (7) (8) fixed cost >0 H1H1

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 15 Bayes Criterion (cont.) The quantity on the right of (9) is the threshold of the test and is denoted by : (10) (9) Likelihood ratio:

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 16 Bayes Criterion (cont.) The Bayes criterion leads us to a likelihood ratio test (LRT) Because the natural logarithm is a monotonic function, and both sides of (11a) are positive, an equivalent test is (log LRT) (11a) (11b)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 17 The MAP Criterion A priori (before we observe R = r): P 0 and P 1 A posteriori (after we have observed R = r): When C 10 =C 01 =1, C 00 =C 11 =0 Form (9) and dividing by P r (R) MAP(maximum a posteriori probability) Criterion:

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 18 The ML Criterion The possible likelihoods of r: and When P 0 = P 1 =1/2, C 10 =C 01 =1,and C 00 =C 11 =0 ML(maximum likelihood) Criterion: MAP Criterion = ML Criterion (when all P i are the same) Form (9)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 19 Bayes Criterion Example Example: We assume that under H 1 the source output is a constant voltage m and that under H 0 the source output is zero. Before observation that voltage is corrupted by an Gaussian noise. because the noise samples are Gaussian.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 20 Bayes Criterion Example (cont.)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 21 Bayes Criterion Example (cont.) The likelihood ratio test is Thus, the log LRT is or, equivalently

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 22 Bayes Criterion Example (cont.) If C 00 = C 11 = 0 and C 01 = C 10 = 1, the risk function of (5) reduces to the probability of error i.e., the Bayes test is minimizing the total probability of error. When the decision regions are chosen, the values of the integrals in (5) are determined. We denote the probabilities of false alarm, detection, and miss, respectively, as

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 23 Bayes Criterion Example (cont.) For any choice of decision regions, the risk function can be written from (5) as Because Then

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 24 Neyman-Pearson Criterion In many physical situations, it is difficult to assign realistic costs or a priori probabilities. A simple procedure to bypass this difficulty is to work with the conditional probabilities P F and P D. Min. F >0<0 For >0,

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 25 Neyman-Pearson Criterion (cont.) For any positive value of an LRT will minimize F. (A negative value of gives an LRT with the inequalities reversed) Thus F is minimized by the likelihood radio test To satisfy the constraint we choose so that, i.e., Observe that decreasing is equivalent to increasing Z 1 ; thus P D increase as decreases. (12) Solving (12) for gives the threshold.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 26 Neyman-Pearson Criterion (cont.) 21 ???

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 27 Q-function Gaussian (normal) distribution erfc-function Q-function The pdf of a Gaussian or normal distribution: The cumulative distribution function (CDF) of a N(0,1) The complementary CDF of a N(0,1)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 28 Q-function (cont.)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 29 Q-function (cont.)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 30 Neyman-Pearson Criterion (cont.)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 31

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 32 Receiver operating characteristic(ROC)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 33 Summary Using either a Bayes criterion or a Neyman-Pearson criterion, we find that the optimum test is a likelihood ratio test. Thus, regardless of the dimensionality of the observation space, the test consists of comparing a scalar variable with a threshold. In many cases, construction of the LRT can be simplified if we can identifies a sufficient statistic. A complete description of the LRT performance was obtained by plotting the conditional probabilities P D and P F as the threshold was varied.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 34 M Hypotheses In the simple M-ary test, there are M source outputs, each of which corresponds to one of M hypotheses. As before, we are forced to make a decision. The Bayes criterion assigns a cost to each of the alternatives, assumes a set of a priori probabilities, and minimizes the risk. The cost C ij denotes that the i-th hypothesis is chosen and the j-th hypothesis is true.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 35 M Hypotheses (cont.) The risk function for the M-ary hypothesis problem is To find the optimum test, we vary the Z i to minimize R. We consider the case of M=3 below.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 36 Noting that Z 0 =Z-Z 1 -Z 2, because the regions are disjoint, we obtain

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 37

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 38 Bayes criterion We see that the decision rules correspond to three lines in the 1, 2 plane. It is easy to verify that these three lines intersect at a common point. The optimum Bayes test becomes (I) (II) (III) Choose H 1 (I0>I1)(I0>I1) (I0<I1)(I0<I1) (I0>I2)(I0>I2) (I0<I2)(I0<I2) (I1<I2)(I1<I2) (I1>I2)(I1>I2)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 39 (I) (II) (III)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 40 1

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 41

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 42 When, MAP ML

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 43 Some Points of M-ary Detection The minimum dimension of the decision space is no more than M-1. The boundaries of the decision regions are hyperplanes in the( 1, …, M-1 ). A particular test of importance is the minimum total probability of error test. Here we compute the a posteriori probability of each hypothesis Pr(H i |R) and choose the largest.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 44 Composite Hypothesis Example:

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 45 Composite Hypothesis (cont.) If is a random variable with a know pdf and the probability density of on the two hypotheses as the likelihood ratio is Above ex: Let = M Reduce the problem to a simple hypothesis-testing problem ( knowing a pdf of ) (14)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 46 Composite Hypothesis (cont.) Example (continued) We assume that the probability density governing m on H 1 is Then, Integrating and taking the logarithm of both sides, we obtain

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 47 GLRT (Generalized LRT) Using ML (maximum likelihood) estimate the value of under the two hypotheses (H 0, H 1 ), the result is called a generalized likelihood ratio test: where 1, ranges over all in H 1 and 0, ranges over all in H 0. In other words, we make a ML estimate of 1, assuming that H 1 is true. We then evaluate for and use this value in the numerator. A similar procedure gives the denominator.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 48 The General Gaussian Problem Definition: A set of random variables are defined as jointly Gaussian if all their linear combinations are Gaussian random variables. Definition: A vector r is a Gaussian random vector when its components are jointly Gaussian random variables. Definition: A Gaussian random vector r is characterized by its mean m and covariance matrix, i.e.,

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 49 (16) (15) Consider the following binary hypothesis problem

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 50 (17) Equal covariance matrices:

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 51 we defined d as the distance between the means on the two hypothesis when the variance was normalized to equal one. The performance of this binary detection problem depends on d (18)

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 52 PDPD d

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 53 Case1. Independent Components with Equal Variance. Substituting to (18) We see that d corresponds to the distance between the two mean-value vectors divided by the standard deviation of R i.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 54 Case 2. Independent Components with Unequal Variance. Case 3. A general case. PLS refer to textbook pp.101-107.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 55 Course Information Instructor: Chia-Hsin Cheng Room: 525 Tel:05-2720411 ext23240 E-mail: vincent@wireless.ee.ccu.edu.tw Text book: H.L. Van Trees, Detection, Estimation and Modulation Theory, Wiley, 2001, pt. I, Chap1~ chap2. Reference books: S.M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Prentice Hall, 1998, pt. II. H.V. Poor, An Introduction to Signal Detection and Estimation, 2nd ed., Springer-Verlag, 1994.

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Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 56 Ultra wideband UWB First Reading [1]Moe Z. Win & Robert A. Scholtz, Impulse Radio: How it works, IEEE Communication Letters,February 1998. [2] R. A. Scholtz, Multiple access with time-hopping impulse modulation,in Proc. MILCOM, Oct. 1993. [3] Durisi, G. Romano, On the validity of Gaussian Approximation to Characterize the Multiuser Capacity of UWB TH PPM, IEEE Conference on Ultra Wideband Systems and Technologies. Digest of Papers, Baltimore, USA,pp.157 - 161, 2002. [4]PlusON Technology Overview, http://www.timedomain.com,July 2000. [5] K. Mandke et al., The Evolution of Ultra Wide Band Radio for Wireless Personal Area Networks, High Frequency Electronics, September 2003, pp. 22-32.

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