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Here, a is the amplitude of the LOS, and z is the multipath components of the total signal; 2b0 E A2 is the average power of the multipath component; O E Z 2 is the average power of the LOS component. The Nakagami parameter m with variance Var Z 2 can be presented as m E Z 2 2 Var Z 2 14:35
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and G m is the gamma function. The major advantage of this model is that it leads to a closed-form and mathematically tractable expressions for the fundamental channel statistics such as the envelope probability density function, moment generating function of the instantaneous power (the so-called rst-order statistics), the level crossing rate (LCR), and the average fade duration (ADF) (the so-called second-order statistics) [13]. Below, we will discuss the rst-order statistics only, on the basis of derivation of the PDF of the signal envelope in LSC channel. The model is very convenient for analytical and numerical performance predictions of complicated narrowband and wideband land mobile satellite systems, with different types of decoded/encoded modulations, with or without diversity. Comparisons of the rst- and second-order statistics of the proposed model with different sets of published statistics demonstrate the exibility of the new model in characterizing a variety of channel conditions and propagation mechanisms over satellite links. At the same time, the proposed model provides a similar t to the experimental data as the well-accepted Loo s model, which is characterized by signi cantly less computational operations. The model mainly focuses on narrowband models, which are the basic building blocks of wideband models. The available statistical models for narrowband LMS channels can be placed into two categories: single and mixture models. In a single model, the channel is characterized by a single statistical distribution, whereas a mixture model refers to a combination (weighted summation) of several statistical distributions. Single models are valid for stationary conditions, where the channel statistics remain approximately constant over the time period of interest in a small area. On the contrary, the mixture models are developed for nonstationary channels, where the signal statistics vary signi cantly over the observation interval in large areas. The random uctuations of the signal envelope in a narrowband LMS channel can be attributed to two types of fading: multipath (fast) fading and shadow (slow) fading. LOS shadow fading (called at, see 1) comes from the complete or partial blockage of the LOS by buildings, trees, hills, mountains, and so forth, which in turn makes the amplitude of the LOS component a random variable. In comparison, multiplicative fading (see de nition in 1) refers to the random variations of the total power of the multipath components, both the LOS and scatter components.
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As for the rst-order statistic of the model, the shadowed Abdi s PDF for the signal envelope in an LMS channel can be written as [13] pR r  2b0 m 2b0 m O m ! & ' r r2 Or 2 ; r ! 0 14:36 exp 1 F1 m; 1; 2b0 2b0 2b0 m O b0
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where 1 F1 is the con uent hypergeometric function. As was shown in Reference [13], the Loo s signal envelope distribution and the above distribution (14.36) closely match each other. This is particularly useful when we wish to apply the new model with unknown parameters to a set of data collected previously, but the measured data is not available for parameter estimation, or we may not want to go through the time-consuming procedures of parameter estimation. Now we want to show the equivalency of the proposed pure stochastic models based on the well-known statistical distributions and their combinations. Let us compare the model proposed by Abdi [13] with that proposed by Loo [7,9]. In Figure 14.12, we plotted the envelope complementary CDF CCDF 1 CDF for Loo s PDF and Abdi s PDF, together with the measured data (points) according to Reference [13].
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FIGURE 14.12. CCDF of the signal envelope in a land mobile satellite channel in Canada, under different shadowing conditions: measured data [13], Loo s model [7 9], and the Abdi s model [13]. (Reprinted with permission # 2003 IEEE)
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