f/f in Software Compose Code 39 in Software f/f

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0 f/f generate, create ansi/aim code 39 none in software projects .NET Framework 2.0 Figure 3.6: In-Phase and Q Software barcode 3/9 uadrature PSD: SrI (f ) = SrQ (f ). f ( Dcos( )=f ). S r (f). S r (f). f D f D( ) f Figure 3.7: Cosine and PSD Approximation by Straight Line Segments and shadowing are constant we obtain Figure 3.9, where we show dB uctuation in received power versus linear distance d = vt (not log distance).

In this gure the average received power P r is normalized to 0 dBm. A mobile receiver traveling at xed velocity v would experience the received power variations over time illustrated in this gure..

3.2.2 Envelope and Power Distributions For any two Gaussian rando m variables X and Y , both with mean zero and equal variance 2 , it can be shown that Z = X 2 + Y 2 is Rayleigh-distributed and Z 2 is exponentially distributed. We saw above that for n (t) uniformly distributed, rI and rQ are both zero-mean Gaussian random variables. If we assume a variance of 2 for both in-phase and quadrature components then the signal envelope z(t) = .

r(t). = is Rayleigh-distributed with distribution pZ (z) = 2z z exp[ z 2 /Pr ] = 2 exp[ z 2 /(2 2 )], x 0, Pr 69 (3.32). 2 2 rI (t) + rQ (t). (3.31). Shadowing K (dB) Narrowband Fading (dB). Path Loss log (d/d 0 ). Figure 3.8: Combined Path Loss, Shadowing, and Narrowband Fading. 0 dBm -30 dB Figure 3.9: Narrowband Fading. 2 where Pr = n E[ n ] = 2 Code 3 of 9 for None 2 is the average received signal power of the signal, i.e. the received power based on path loss and shadowing alone.

We obtain the power distribution by making the change of variables z 2 (t) = . r(t). 2 in (3.32) to obtain pZ 2 (x) =. 1 x/Pr 1 2 e = 2 e x/(2 ) , x 0. Pr 2 (3.33). Thus, the received signal Software Code-39 power is exponentially distributed with mean 2 2 . The complex lowpass equivalent signal for r(t) is given by rLP (t) = rI (t) + jrQ (t) which has phase = arctan(rQ (t)/rI (t)). For rI (t) and rQ (t) uncorrelated Gaussian random variables we can show that is uniformly distributed and independent of .

rLP . So r(t) has a Rayleigh-d Software barcode 3 of 9 istributed amplitude and uniform phase, and the two are mutually independent. Example 3.

2: Consider a channel with Rayleigh fading and average received power P r = 20 dBm. Find the probability that the received power is below 10 dBm. Solution.

We have Pr = 20 dBm =100 mW. We want to nd the probability that Z 2 < 10 dBm =10 mW. Thus p(Z 2 < 10) =.

0 10. 1 x/100 e dx = .095. 100 If the channel has a xed LOS component then r I (t) and rQ (t) are not zero-mean. In this case the received signal equals the superposition of a complex Gaussian component and a LOS component. The signal envelope in this case can be shown to have a Rician distribution [9], given by pZ (z) = z (z 2 + s2 ) zs exp , z 0, I0 2 2 2 2 (3.

34). 2 2 where 2 2 = n,n=0 E[ Software 3 of 9 n ] is the average power in the non-LOS multipath components and s 2 = 0 is the power in the LOS component. The function I0 is the modi ed Bessel function of 0th order. The average received power in the Rician fading is given by .

Pr = z 2 pZ (z)dx = s2 + 2 2 . (3.35). The Rician distribution is Software Code 3/9 often described in terms of a fading parameter K, de ned by K= s2 . 2 2 (3.36).

Thus, K is the ratio of th e power in the LOS component to the power in the other (non-LOS) multipath components. For K = 0 we have Rayleigh fading, and for K = we have no fading, i.e.

a channel with no multipath and only a LOS component. The fading parameter K is therefore a measure of the severity of the fading: a small K implies severe fading, a large K implies more mild fading. Making the substitution s 2 = KP/(K + 1) and 2 2 = P/(K + 1) we can write the Rician distribution in terms of K and P r as 2 2z(K + 1) (K + 1)z K(K + 1) exp K I0 2z , z 0.

(3.37) pZ (z) = Pr Pr Pr Both the Rayleigh and Rician distributions can be obtained by using mathematics to capture the underlying physical properties of the channel models [1, 9]. However, some experimental data does not t well into either of 71.

these distributions. Thus, ANSI/AIM Code 39 for None a more general fading distribution was developed whose parameters can be adjusted to t a variety of empirical measurements. This distribution is called the Nakagami fading distribution, and is given by 2mm z 2m 1 mz 2 exp , m .

5, (3.38) pZ (z) = m (m)Pr Pr where Pr is the average received power and ( ) is the Gamma function. The Nakagami distribution is parameterized by Pr and the fading parameter m.

For m = 1 the distribution in (3.38) reduces to Rayleigh fading. For m = (K + 1)2 /(2K + 1) the distribution in (3.

38) is approximately Rician fading with parameter K. For m = there is no fading: Pr is a constant. Thus, the Nakagami distribution can model Rayleigh and Rician distributions, as well as more general ones.

Note that some empirical measurements support values of the m parameter less than one, in which case the Nakagami fading causes more severe performance degradation than Rayleigh fading. The power distribution for Nakagami fading, obtained by a change of variables, is given by pZ 2 (x) = m Pr. xm 1 exp (m).
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