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vector corresponding to the pair of complex symbols generated for a particular group of 2b input bits, each chosen from the abovementioned constellation, to be transmitted in the nth transmission interval,8 nTs t (n + 2) Ts , from the two transmit antennas where the T superscript denotes the transpose operation The communication system is such that for each group of, say, b bits pair of com(n) (n) plex symbols x1 , x2 are chosen from a signal constellation For example, xn could consist of a pair of independently chosen M -PSK symbols or alternatively a pair of such symbols chosen from a four-dimensional 2 M-PSK constellation as might be the case if the input data were rst trellis-encoded [17].



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In Alamouti s scheme [26] this joint information is sent twice per transmission interval (hence the information symbol rate remains unchanged) each time with a different assignment to the two transmit antennas Thus, an Alamouti symbol consists of two information symbols, and in our notation, a transmission interval corresponds to the duration of an Alamouti symbol Speci cally, antenna 1 (n) (n) and antenna 2 rst transmit x1 and x2 , respectively, in the interval nTs t (n) (n) (n + 1) Ts ; then they transmit ( x2 ) and (x1 ) in the interval (n + 1) Ts t (n + 2) Ts a simple 2 2 space-time block code Thus, for a block of N successive information vectors x1 , x2 , , xN , the transmitted sequences from each antenna would be a block of 2N complex symbols as follows.

(4.28)





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As already mentioned, the coaxial cable was almost universal around 1990, At that time the standard speed for Ethernet networks was 10 Mbps, and it looked as if that would be

Antenna 1 would (1) (1) (2) (2) (N) (N) transmit x1 , ( x2 ) , x1 , ( x2 ) , ., x1 , ( x2 ) , whereas antenna 2 would (1) (1) (2) (2) (N) (N) transmit x2 , (x1 ) , x2 , (x1 ) , ., x2 , (x1 ) The channel during the nth transmission interval is described by the Lr Lt (n) (n) matrix Cn = [cj i ] where cj i denotes the complex channel gain (fading) between (n) the i th transmit antenna and the j th receive antenna and the cj i terms are assumed to be independent of one another For example, for a Rayleigh fading channel, (n) cj i would be a complex Gaussian random variable (RV) with average fading power.

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The most time consuming step in the calculation of the fluxes is the evaluation of the matrix [B]. For problems involving only a few (three or four) components the use of Sylvester's theorem is recommended. Unfortunately, Sylvester's expansion formula demands an increasing percentage of computer time as the number of components and, therefore, the size of the matrices involved increases. An alternative to the use of Sylvester's formula is to evaluate a matrix function from its power series definition. Thus, for example, the exponential matrix in Eq. A.6.1 could be evaluated directly from Eq. A.5.4. The number of terms needed to evaluate exp[^>] in this way depends on the magnitude of the dominant eigenvalue &d of [<1>] (the eigenvalue with the largest absolute value). Convergence will be rapidly obtained for small <t>d. However, for large &d we may need a great many terms of the series to evaluate the exponential. Moler and Van Loan (1978) discuss at some length "Nineteen dubious ways to compute the exponential of a matrix." They consider three or four methods as candidates for the

H.6.

equipment (used with narrowband ISDN services).

speaking, photo-oxidative degradation refers to reactions initiated by UV radiation because few commodity polymers absorb visible light when they are rst manufactured (unless they are speci cally designed to degrade in visible light or unless they contain impurities). The photo-oxidative degradation mechanism has been extensively investigated1 6,9 for polyole ns and selected other polymers, and there is general agreement that the initial phase in the degradation is an auto-oxidation cycle, shown in its most general form in scheme 1. Of course, some polymers degrade by photoreactions that occur in addition to oxidative degradation. PVC, for example, reacts photochemically to eliminate HCl and form C C bonds.10,11 Even in these cases, however, oxidative degradation of the polymer still occurs.

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