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Figure 3.1 The shape of an N 5 4 density. in Java Writer pdf417 2d barcode in Java Figure 3.1 The shape of an N 5 4 density.

Figure 3.1 The shape of an N 5 4 density. using barcode writer for j2se control to generate, create pdf417 2d barcode image in j2se applications. Recommended GS1 barcodes for mobile apps p (u). 3.1 Gaussian basics Figure 3.2 The and Q function jvm pdf417 s are obtained by integrating the N 0 1 density over appropriate intervals..

p(u). x (x ) Q(x ). Conversion of a Gaussian rand om variable into standard form If X N m v2 , then X m /v N 0 1 . We set aside special notation for the cumulative distribution function (CDF) x and complementary cumulative distribution function (CCDF) Q x of a standard Gaussian random variable. By virtue of the standard form conversion, we can easily express probabilities involving any Gaussian random variable in terms of the or Q functions.

The definitions of these functions are illustrated in Figure 3.2, and the corresponding formulas are specified below..

x =P N 0 1 x = Q x =P N 0 1 >x = 1 2 1 2. exp exp t2 2 t2 2 (3.2). (3.3). See Figure 3.3 for a plot of tomcat PDF417 these functions. By definition, x + Q x = 1.

Furthermore, by the symmetry of the Gaussian density around zero, Q x = x . Combining these observations, we note that Q x = 1 Q x , so that. Figure 3.3 The functions. and Q 1 Q(x ) (x ). Demodulation it suffices to consider only PDF-417 2d barcode for Java positive arguments for the Q function in order to compute probabilities of interest. Example 3.1.

1 X is a Gaussian random variable with mean m = 3 and variance v2 = 4. Find expressions in terms of the Q function with positive arguments for the following probabilities: P X > 5 , P X < 1 , P 1 < X < 4 , P X2 + X > 2 . Solution We solve this problem by normalizing X to a standard Gaussian random variable X mv = X + 3/2: P X>5 =P X +3 5+3 > =4 =Q 4 2 2 1 = 1 Q 1.

P X < 1 = P X + 3 1 + 3 < =1 = 2 2 P 1<X<4 =P 2= 1+3 X +3 4+3 < < =35 = 2 2 2 35 . = Q 2 Q 3 5 Computation of t Java pdf417 2d barcode he last probability needs a little more work to characterize the event of interest in terms of simpler events: P X2 + X > 2 = P X2 + X 2 > 0 = P X + 2 X 1 > 0 The factorization shows that X 2 + X > 2 if and only if X + 2 > 0 and X 1 > 0, or X + 2 < 0 and X 1 < 0. This simplifies to the disjoint union (i.e.

, or ) of the mutually exclusive events X > 1 and X < 2. We therefore obtain 1+3 2 + 3 P X 2 + X > 2 = P X > 1 + P X < 2 = Q + 2 2 = Q2 + 1 2 = Q 2 +1 Q 1 2. The Q function is ubiquitous in communication systems design, hence it is worth exploring its properties in some detail. The following bounds on the Q function are derived in Problem 3.3.

Bounds on Q x for large arguments e x /2 e x /2 Q x x 0 (3.4) x 2 x 2 These bounds are tight (the upper and lower bounds converge) for large values of x. 1 1 x2.

3.1 Gaussian basics Upper bound on Q(x) useful fo r small arguments and for analysis 1 2 Q x e x /2 2 x 0 (3.5). This bound is tight for small jar barcode pdf417 x, and gives the correct exponent of decay for large x. It is also useful for simplifying expressions involving a large number of Q functions, as we see when we derive transfer function bounds for the performance of optimal channel equalization and decoding in s 5 and 7, respectively. Figure 3.

4 plots Q x and its bounds for positive x. A logarithmic scale is used for the values of the function to demonstrate the rapid decay with x. The bounds (3.

4) are seen to be tight even at moderate values of x (say x 2). Notation for asymptotic equivalence Since we are often concerned with exponential rates of decay (e.g.

, as SNR gets large), it is useful to introduce the notation P = Q (as we take some limit), which means that log P/ log Q 1. An analogous notation p q denotes, on the other hand, that p/q 1. Thus, P = Q and log P log Q are two equivalent ways of expressing the same relationship.

Asymptotics of Q(x) for large arguments For large x > 0, the exponential decay of the Q function dominates. We denote this by Q x = e x. 2 /2. (3.6).
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