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We outline the basic steps in the proof due to Stam [505] and Blachman [61]. A different proof is given in Section 17.8. Stam s proof of the entropy power inequality is based on a perturbation argument. Let n = 1. Let Xt = X + f (t)Z1 , Yt = Y + g(t)Z2 , where Z1 and Z2 are independent N(0, 1) random variables. Then the entropy power inequality for n = 1 reduces to showing that s(0) 1, where we de ne 22h(Xt ) + 22h(Yt ) . (17.87) 22h(Xt +Yt ) If f (t) and g(t) as t , it is easy to show that s( ) = 1. If, in addition, s (t) 0 for t 0, this implies that s(0) 1. The proof of the fact that s (t) 0 involves a clever choice of the functions f (t) and g(t), an application of Theorem 17.7.2 and the use of a convolution inequality for Fisher information, s(t) = 1 1 1 + . J (X + Y ) J (X) J (Y ) (17.88)

Toward this end, let f : X n {0, 1} (where {0, 1} = { , 0, 1, 00, 01, } is the set of all nite-length binary sequences) be a mapping f (X1 , X2 , , Xn ) = (Z1 , Z2 , , ZK ), where Zi Bernoulli ( 1 ), and K may depend on (X1 , , Xn ) 2 In order that the sequence Z1 , Z2 , appear to be fair coin ips, the map f from bent coin ips to fair ips must have the property that all 2k sequences (Z1 , Z2 , , Zk ) of a given length k have equal probability (possibly 0), for k = 1, 2, ..

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For example, for n = 2, the map f (01) = 0, f (10) = 1, f (00) = f (11) = (the null string) has the property that Pr{Z1 = 1|K = 1} = Pr{Z1 = 0|K = 1} = 1 Give reasons for the following inequalities: 2 nH (p) = H (X1 , , Xn ).

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Bandwidth is also determined by the regulating authority (e.g., the FCC in the United States) for a particular service/ application. Through bit packing techniques, described in 9, the information-carrying capacity of a unit of bandwidth is considerably greater than 1 bit per Hz of bandwidth. On line-of-sight microwave systems, 5, 6, 7, and 8 bits per Hz of bandwidth are fairly common. 9 provides a more detailed discussion of radio systems.

we will provide an overview of the J2EE facilities provided. These facilities are part of the J2EE speci cation and therefore we would expect to nd them in any J2EE platform implementation alternative. There might also be nonstandard facilities provided by a particular implementation, or nonstandard extensions to the standard facilities considered here. Similarly, we also have the ability to include other software packages that are not de ned in the J2EE-SPEC including packages that are not part of any J2EE platform implementation alternative. For example, we can always select additional middleware packages (our J2EE-SKIT is a simplistic example of this). However, with the exception of the J2EE-SKIT, we will not discuss these other nonstandard facilities further.

Note. All items were coded. 1 = never or rarely during the past week; 2 = sometimes, a lot, most, or all of the time during the past week.

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