Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
5.5 Small-Scale Fading with a Dominant Component
Fading statistics change when a dominant MPC – e.g., an LOS component or a dominant specu- lar component – is present. We can gain some insights by repeating the computer experiment of Section 5.4.1, but now adding an additional wave with the (dominant) amplitude 5:
|a9 φ9 ϕ9
E9(x, y)=5.0 exp[−j k0(xcos(0◦)+ysin(0◦))] exp(j0◦) 5.0 0◦ 0◦ Figure 5.17 shows the real part of E, and the contribution from the dominant component is visible; while Figure 5.18 shows the absolute value. The histogram of the absolute value of the field strength is shown in Figure 5.19. It is clear that the probability of deep fades is much smaller than in the Rayleigh-fading case.
Re{E}
x y
−10 +10
Figure 5.17 Re(E) – i.e., the instantaneous value att=0, in the presence of a dominant MPC.
5.5.2 Derivation of the Amplitude and Phase Distribution
The pdf of the amplitude can be computed in a way that is similar to our derivation of the Rayleigh distribution (Appendix 5.B – seewww.wiley.com/go/molisch). Without restriction of generality, we assume that the LOS component has zero phase, so that it is purely real. The real part thus has a non- zero-mean Gaussian distribution, while the imaginary part has a zero-mean Gaussian distribution.
Performing the variable transformation as in Appendix 5.B, we get the joint pdf of amplituderand phaseψ [Rice 1947]:
pdfr,ψ(r, ψ )= r 2π σ2exp
−r2+A2−2rAcos(ψ ) 2σ2
(5.26) whereAis the amplitude of the dominant component. In contrast to the Rayleigh case, this distri- bution is not separable. Rather, we have to integrate over the phases to get the amplitude pdf, and vice versa.
|E|
x y
10 0
Figure 5.18 Magnitude of the electric field strength,|E|, in an example area, in the presence of a dominant MPC.
0.25
00 5 10 12
pdf(|E| )
|E|
Figure 5.19 Histogram of the amplitudes in the presence of a dominant MPC.
The pdf of the amplitude is given by theRice distribution (solid line in Figure 5.19):
pdfr(r)= r
σ2ãexp −r2+A2 2σ2
ãI0 rA
σ2
0≤r <∞ (5.27) I0(x)is the modified Bessel function of the first kind, zero order [Abramowitz and Stegun 1965].
The mean square value of a Rice-distributed random variabler is given by:
r2=2σ2+A2 (5.28)
The ratio of the power in the LOS component to the power in the diffuse component,A2/(2σ2), is called the Rice factorKr.
Figure 5.20 shows the Rice distribution for three different values of the Rice factor. The stronger the LOS component, the rarer the occurrence of deep fades. For Kr→0, the Rice distribution
Statistical Description of the Wireless Channel 85
Rice factor Kr= A2/2s2
Kr= 10 pdf(r)
Kr= 1
0 A r
Kr= 0
s
Figure 5.20 Rice distribution for three different values ofKr– i.e., the ratio between the power of the LOS component and the diffuse components.
becomes a Rayleigh distribution, while for large Kr it approximates a Gaussian distribution with mean valueA.
Example 5.2 Compute the fading margin for a Rice distribution withKr=0.3,3 and 20 dB so that the outage probability is less than 5%.
Recall that the outage probability can be expressed in terms of thecdf of the Rician envelope:
Pout=cdf(rmin) (5.29)
For the Rician distribution the cdf is given as:
cdf(rmin)= rmin
0
r
σ2 ãexp −r2+A2 2σ2
ãI0 rA
σ2
dr 0≤r <∞
=1−QM
A σ,rmin
σ
(5.30) whereQM(a, b)is Marcum’sQ-function (see also Chapter 12) given by:
QM(a, b)=e−(a2+b2)/2 ∞ n=0
a b
n
In(ab) (5.31)
In(ã)is the modified Bessel function of the first kind, ordern. The fading margin is given by:
r2
rmin2 = 2σ2(1+Kr)
rmin2 (5.32)
The Rice power cdf is plotted in Figure 5.21. The required fading margins at different Kr can be found from that figure: they are 11.5, 9.7, and 1.1 dB, for Rice factors of 0.3, 3, and 20 dB, respectively.
The presence of a dominant component also changes the phase distribution. This becomes intuitively clear by recalling that for a very strong dominant component, the phase of the total signal must be very close to the phase of the dominant component – in other words, the phase
−30
−3
−2.5
−1.5
−1
−0.5 0
−2
r2min / E[r2] (dB) Pout = 5%
Kr= 0.3 dB Kr= 3 dB Kr= 20 dB
log10 Pr {envelope < abscissa}
−25 −20 −15 −11.5−9.7 −5 −1.1 0 5
Figure 5.21 The Rice powercdf,σ=1.
distribution converges to a delta function. For the general case (remember that we define the phase of the LOS component asψ=0), the pdf of the phase can be computed from the joint pdf of r andψand becomes [Lustmann and Porrat 2010]:
pdf(ψ )= 1+√
π KreKrcos2(ψ)cos(ψ )1+erf√
Krcos(ψ )
2π eKr (5.33)
whereerf(x) is the error function [Abramowitz and Stegun 1965]:
erf(x)=(2/√ π )
x
0 exp(−t2) dt
Figure 5.22 shows the phase distribution forσ =1 and different values ofA.
The pdf of the power is given as pdfP(P )=1+Kr
exp
−Kr−(Kr+1)P
I0
2
Kr(Kr+1)P
forP≥0 (5.34) From a historical perspective, it is interesting to note that all the work about Rice distributions was performed without the slightest regard for wireless channels. The classical paper of Rice [Rice 1947] considered the problem of a sinusoidal wave in additive white Gaussian noise. However,
Statistical Description of the Wireless Channel 87
4
3
2
1
0 −0.3 −0.1 0.1
Phase ψ/(2p)
pdf(y)
0.3 0.5
A = 0
−0.5
A = 1 A = 3
A = 10
Figure 5.22 Pdf of the phase of a non-zero-mean complex Gaussian distribution, withσ =1, A=0, 1, 3, 10.
from a mathematical point of view, this is just the problem of a deterministic phasor (giving rise to a non-zero-mean) added to a zero-mean complex Gaussian distribution – exactly the same problem as in the field strength computation. Existing results thus just had to be reinterpreted by wireless engineers. This fact is so interesting because there are probably other wireless problems that can be solved by such “reinterpretation” methods.
5.5.3 Nakagami Distribution
Another probability distribution for field strength that is in widespread use is the Nakagami m- distribution. The pdf is given as:
pdfr(r)= 2 (m)
m
m
r2m−1exp
−m r2
(5.35) for r≥0 and m≥1/2;(m)is Euler’s Gamma function [Abramowitz and Stegun 1965]. The parameteris the mean square value=r2, and the parametermis
m= 2
(r2−)2
(5.36) It is straightforward to extract these parameters from measured values. If the amplitude is Nakagami- fading, then the power follows a Gamma distribution:
pdfP(P )= m (m)
mP
m−1
exp
−mP
(5.37)
Nakagami and Rice distribution have a quite similar shape, and one can be used to approximate the other. Form >1 them-factor can be computed fromKr by Stueber [1996]:
m= (Kr+1)2
(2Kr+1) (5.38)
while
Kr=
√m2−m m−√
m2−m (5.39)
While Nakagami and Rice pdfs show good “general” agreement, they have different slopes close tor=0. This in turn has an important impact on the achievable diversity order (see Chapter 13).
The main difference between the two pdfs is that the Rice distribution gives theexactdistribution of the amplitude of a non-zero-mean complex Gaussian distribution – this implies the presence of one dominant component, and a large number of non-dominant components. The Nakagami distribution describesin an approximate way the amplitude distribution of a vector process where the central limit theorem is not necessarily valid (e.g., ultrawideband channels – see Chapter 7).