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Since the mid-1980s, the generation of functional bases has become an advanced science in its own right in the context of wavelet theory We consider wavelet representations in Section 38 This section describes a startling approach to discrete representation developed by Shannon [219] in 1949 Under the Whittaker Shannon sampling theorem, a continuous function is represented on a basis such that the representation coef cients are equal to periodically spaced values of the function itself The core idea of the sampling theorem is that a signal may be fully represented by capturing a discrete sequence of signal values Detailed consideration of the relationship between sampling in measurement, analysis, and display is critical to understanding optical sensors It is not possible in practice to actually measure the eld at speci c points in space or time (Real detectors integrate eld values over nite space time windows.



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where (7.95) follows from the fact that the entropy of a collection of random variables is less than the sum of their individual entropies, and (7.97) follows from the de nition of capacity. Thus, we have proved that using the channel many times does not increase the information capacity in bits per transmission. We are now in a position to prove the converse to the channel coding theorem. Proof: Converse to Theorem 7.7.1 (Channel coding theorem). We have to show that any sequence of (2nR , n) codes with (n) 0 must have R C. If the maximal probability of error tends to zero, the average probability of error for the sequence of codes also goes to zero [i.e., (n) 0 implies Pe(n) 0, where Pe(n) is de ned in (7.32)]. For a xed encoding rule X n ( ) and a xed decoding rule W = g(Y n ), we have W X n (W ) Y n W . For each n, let W be drawn according to a uniform distribution over {1, 2, . . . , 2nR }. Since W has a uniform distribution, Pr(W = W ) = (n) 1 Pe = 2nR i i . Hence,

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) In 7 we examine the tension between measurement sampling and analytic sampling in more detail and consider the process of measurement itself as an inner product with the eld For the present purpose of exploring the mathematical tools needed to analyze the eld, however, it is important to understand the classical sampling theorem..

Using the relation (10.195), we evaluate the score function at X(z) = 0 log p(y; W, r) |X(z)=0 = i (yi (k))yj (k p), Xp,ij (10.221)

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FIGURE 12.28 We re using four random files from the Displacement maps folder (typically a subfolder of the Plug-ins folder). Note that the red (left column) and green (middle column) channels often differ, thus conveying unique horizontal and vertical displacement information. The final column shows the full-color dmap, with the blue channel set to 50 percent gray. Schnable Effect

Behavior(s) (Continued) adherence to sport injury rehabilitation, 411 413 coping, 413 indeterminate, 224 227 responses, rehabilitation, 411 413 Behavioral changes over the life span, 654 655 Behavioral features, eating disorders, 354 Behavioral handicaps, 145 Behavioral modification, adherence, 520 Behavioral outcomes, motivation, 67, 71 72 Behavioral Regulation in Exercise Questionnaire (BREQ), 62, 70 Behavioral theories, choking, 434 435 Behavior change interventions: adherence, 519 521 cancer survivors, 590 591 Belief(s)/values: about causes of success, goals orientations and, 7 parental, 686 695 Belief-attitude approaches, theoretical frameworks, 538 543 implementation intentions, 542 543 theory of planned behavior (TPB), 522, 540 542, 590, 591 theory of reasoned action (TRA), 522, 539 540 Bem Sex Role Inventory (BSRI ), 827, 828 Best information criterion (BIC), 748 Big-fish-little-pond effect (BFLPE), 195 Bilder-Angst-Test, 741 Bioecological model of human development, 685 Biomedical status, adherence and, 513 Biopsychosocial model, 404 406 Birth order, 704 705 Birthplace effect, 194 195 Bland and Altman s method, 759 760 Bodily experiences, idiosyncratic, 36 Body Awareness Scale, 41 Body dissatisfaction, eating disorders and, 366 367 Body Dysmorphic Disorder, 356 Body image, gender and physical activity, 829 830 Bootstrapping, 749 Bottlenecks, information-processing, 247 Boundaries, professional, 323 324 Brain-derived neurotrophic factor (BDNF) hypothesis, 473, 479 Brain processes.

(b) Introduce the eigenvalue decomposition of R,, say R, = UAU*,where U is a unitary matrix and A is a diagonal matrix with positive entries {A,, Az, . . ,A M } . Let vl(k) denote the k-th

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