By Harry L. Van Trees

Paperback reprint of 1 of the main revered classics within the heritage of engineering publicationTogether with the reprint of half I and the hot half IV, this may be the main whole remedy of the topic availableProvides a highly-readable dialogue of sign Processing and NoiseFeatures a number of difficulties and illustrations to assist advertise figuring out of the topicsContents are hugely acceptable to present structures

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**Extra info for Detection, estimation, and modulation theory**

**Example text**

In this case, we have the property n(a ∪ b) = n(a) + n(b). As a third definition, the complementary event of x is the event that occurs if x does not occur. It is noted x¯ or sometimes ¬x. In Fig. 4, the complementary event of a ∪ b, which is noted a ∪ b, is represented by the shaded surface spreading inside the space S and outside the surface of a ∪ b. 3 Combined, joint, and conditional probabilities where p(a ∪ b) ≡ p(a + b) is the combined probability of the two events a, b and p(a ∩ b) ≡ p(a, b) is the corresponding joint probability (one must be careful to read the + sign in p(a + b) as meaning “or” and not “and”).

The mean and variance of the binomial distribution is N = nq and σ 2 = N (1 − q), respectively. 4 shows plots of the binomial PDF for n = 10 and different values of q. 5, in which case the peak value is exactly centered about N . A physical illustration of the Bernoulli distribution is the passing, one at a time, of a stream of individual particles (like light quanta or photons) through a piece of absorbing material. These particles may either successfully pass through the material (success, probability q), or absorbed by it (failure, probability 1 − q).

It can also be shown that, for large mean numbers N , the Poisson (or binomial) PDF asymptotically converges towards the Gaussian or normal distribution, which is described in the next section. 3 Continuous distributions At the beginning of this chapter, I discussed the possibility that the events form a continuous and infinite suite of real numbers, which, in the physical world, represent the unbounded set of measurements of a physical quantity. The likelihood of the measurement then corresponds mathematically to a continuous probability distribution.