Comparison of the Level Crossing Rate and Average Fade Duration of
Rayleigh, Rice, and Nakagami Fading Models with Mobile Channel Data
Ali Abdi1, Kyle Wills2, H. Allen Barger3, Mohamed-Slim Alouini1, and Mostafa Kaveh1
1
Dept. of Elec. and Comp. Eng., University of Minnesota, Minneapolis, Minnesota 55455, USA
2
Raytheon Systems, Dallas, Texas 75243, USA
3
Dept. of Elec. Eng., University of Texas at Tyler, Tyler, Texas 75799, USA
Emails: {abdi,alouini,kaveh}@ece.umn.edu, [email protected], [email protected]
Abstract
Level crossing rate (LCR) and average fade duration (AFD)
of the signal envelope are two important second-order channel
statistics, which convey useful information about the dynamic
temporal behavior of multipath fading channels. In this paper
and for a general non-isotropic scattering scenario, in which
the mobile receives signal only from particular directions with
different probabilities, we derive expressions for the LCR and
AFD of Rayleigh, Rice, and Nakagami fading models, including
the effect of non-uniform signal angle-of-arrival distribution.
The merits and limitations of all the above models in describing
the first- and second-order statistics of multipath fading
channels are explored through an extensive comparison of
theoretical results with narrowband measurements taken in
urban and suburban areas at 910.25 MHz.
1. Introduction
In mobile channels with multipath fading (fast fading),
performance of communications systems degrade significantly
due to the signal envelope fluctuations and it is vital to
characterize those random variations in terms of the fading
statistics. The level crossing rate (LCR), how often the envelope
crosses a certain threshold, and the average fade duration (AFD),
how long the envelope stays below a given threshold, are two
important second-order statistics of multipath fading channels
[1]. Since the impact of long-time but rare envelope fades is
totally different from the influence of short-time but frequent
envelope fades [2], the LCR and AFD, which characterize some
aspects of the dynamic temporal behavior of envelope
fluctuations, can help us better understand/combat the disturbing
effects of signal fades. For example, the AFD determines the
average length of error bursts in fading channels. Hence, in
fading channels with relatively large AFD, long data blocks are
more likely to be significantly affected by the channel fades than
short blocks. This should be kept in mind in choosing the frame
length for coded packetized systems, designing interleaved or
non-interleaved concatenated coding methods [3], optimizing
the interleaver size, choosing the buffer depth for adaptive
modulation schemes [4], throughput (efficiency) estimation of
communication protocols [5], etc.. For all of these applications
we need empirically-verified formulas for the LCR and AFD of
common multipath fading models.
Among the available models for multipath fading, Rice and
Nakagami models [6] have a special place. These two
distributions both include the Rayleigh distribution (the basic
model of multipath fading) as a special case. Moreover, for
calculating important system performance measures, such as bit
error rates with single- and multi-channel (diversity) reception in
fading channels, they usually result in closed-form expressions.
However, in contrast with the Rayleigh model, most often the
comparison of these two models with measured data have been
limited to the first-order statistics and little effort has been made,
thus, to compare the empirical and theoretical second-order
statistics. In other words, the probability density function (PDF)
or the cumulative distribution function (CDF) of Rice and
Nakagami models have been fitted extensively to different types
of mobile fading channels, while the merits and limitations of
these two models in describing such important dynamic
characteristics of mobile channels as LCR and AFD have not
been explored in terms of measured data. The main purpose of
this paper is to empirically investigate the ability of Rayleigh,
Rice, and Nakagami fading models in characterizing the LCR
and AFD of mobile channel data.
Furthermore, since both LCR and AFD depend on the fading
correlation (or equivalently, fading spectrum), we also develop
empirically-verified new expressions for the LCR and AFD of
Rayleigh model when the fading correlation is significantly far
from the Clarke’s classic model of isotropic scattering [6].
Appropriate expressions for the LCR and AFD of Rice and
Nakagami models, assuming non-isotropic scattering in which
the mobile receives signal only from particular directions, are
derived as well.
2. Theoretical results
In a frequency non-selective multipath fading channel, let
ℜ(t ) = R(t ) exp[− j Ψ (t )] denote the stationary narrowband
lowpass complex envelope at the mobile receiver, where
j 2 = −1 , and R(t ) and Ψ (t ) are the envelope and phase
processes, respectively (needless to say, ℜ(t ) is a complex
process while R(t ) and Ψ (t ) are real). For the case in which
R(t ) is either a Rayleigh or Rice process, ℜ(t ) is a complex
Gaussian process, while Ψ (t ) has to be a uniform process [6].
When R(t ) is a Nakagami process, there is no presumption
about the statistics of Ψ (t ) and consequently, ℜ(t ) . So, in
order to have a full statistical characterization of ℜ(t ) , one has
to assign a model to Ψ (t ) [7] (for other envelope processes
such as Weibull, see [8]). This comes from the fact that the
Nakagami model was merely developed for the envelope [9],
while Rayleigh and Rice models implicitly convey information
about both the envelope and phase. For Rayleigh, Rice, and
Nakagami models, the univariate PDFs of R(t ) are given by [6]:
⎛ −r2 ⎞
2r
⎟⎟ ,
f Rayleigh ( r ) =
exp⎜⎜
(1)
Ω
⎝ Ω ⎠
f Rice (r ) =
⎛
2( K + 1)r
( K + 1)r 2 ⎞ ⎛⎜ K ( K + 1)
⎟I0 2
exp⎜⎜ − K −
⎟ ⎜
Ω
Ω
Ω
⎝
⎠ ⎝
⎞
r ⎟ ,(2)
⎟
⎠
⎞
⎟⎟ ,
(3)
⎠
where Ω = E[ R 2 ] is the average power with E as the
expectation, K ≥ 0 is the Rice parameter, I n (.) is the nth-order
modified Bessel function of the first kind, m ≥ 1 2 is the
Nakagami parameter, and Γ(.) denotes the gamma function.
The CDFs of these models can be written in terms of the
normalized envelope level ρ = r Ω as:
(4)
FRayleigh ( ρ ) = 1 − exp(− ρ 2 ) ,
f Nakagami (r ) =
⎛ − mr 2
2m m
r 2 m −1 exp⎜⎜
m
Γ( m) Ω
⎝ Ω
FRice ( ρ ) = 1 − Q⎛⎜ 2 K , 2( K + 1) ρ 2 ⎞⎟ ,
⎝
⎠
FNakagami ( ρ ) = Γ(m, mρ 2 ) Γ(m) ,
(5)
(6)
where Q (.,.) and Γ(.,.) are the Marcum-Q and incomplete
gamma functions, respectively, defined by:
∞
⎛ x2 + a2 ⎞
⎟ I 0 (ax) dx ,
(7)
Q (a, b) = ∫ x exp⎜⎜ −
b
2 ⎟⎠
⎝
b
Γ(a, b) = ∫ x a −1 exp(− x ) dx .
0
(8)
For K = 0 and m = 1 , all the equations of Rice and Nakagami
models simplify to those of the Rayleigh model. This could be
easily verified using the relations Q (0, b) = exp(− b 2 2) and
Γ(1, b) = 1 − exp(−b) .
Before going through the details of the LCR and AFD
formulas, we define the appropriate correlation functions for all
the three models and also introduce the notion of spectral
moments. For Rayleigh and Rice models, consider the
autocovariance of the complex envelope ℜ(t ) , defined as [6]
2 Bℜ (τ ) = E[ℜ∗ (t )ℜ(t + τ )]− | E[ℜ(t )] |2 , where ∗ is the
complex conjugate operator. The nth spectral moment, b n ,
n = 0, 1, 2, ... , is defined to be:
bn =
d n Bℜ (τ )
.
j n dτ n τ = 0
(9)
It is shown in [10] that with an omnidirectional antenna at the
mobile and upon the application of the von Mises PDF for
Φ (t ) , the angle of arrival (AOA) in the horizontal plane:
exp[κ cos(ϕ − μ )]
(10)
, ϕ ∈ [ −π , π ) ,
f Φ (ϕ ) =
2πI 0 (κ )
with μ ∈ [−π , π ) as the mean direction of the AOA and κ ≥ 0
as the width control parameter of the AOA, the autocovariance
of the complex envelope is given by:
2
I 0 ⎛⎜ κ 2 − 4π 2 f d τ 2 + j 4π κ cos( μ ) f d τ ⎞⎟
⎝
⎠
,
(11)
Bℜ (τ ) = b0
I 0 (κ )
in which f d is the maximum Doppler frequency. According to
the empirical results reported in [10], this model has shown very
good fit to measurements. Note that for κ = 0 , the von Mises
PDF reduces to the uniform PDF, i.e. f Φ (ϕ ) = 1 (2π ) , which
represents Clarke’s model of isotropic scattering. This in turn
yields Bℜ (τ ) = b0 J 0 (2π f d τ ) , the well-known autocovariance
of the complex envelope assuming isotropic scattering [6].
Upon the application of the von Mises PDF for the AOA,
the spectral moments b1 and b 2 for Rayleigh and Rice models
can be derived by successive differentiation of Eq. (11):
2 π f d cos(μ ) I 1 (κ )
,
(12)
b1 = b0
I 0 (κ )
b 2 = b0
2 π 2 f d2 [ I 0 (κ ) + cos(2 μ ) I 2 (κ )]
.
I 0 (κ )
(13)
For the special case of isotropic scattering ( κ = 0 ), the above
results simplify to b1 = 0 and b2 = b0 2 π 2 f d2 [6].
For the Rayleigh model, the autocovariance of R 2 (t ) ,
BR 2 (τ ) = E[ R 2 (t ) R 2 (t + τ )] − {E[ R 2 (t )]}2 , can be expressed in
terms of Bℜ (τ ) as BR 2 (τ ) =| Bℜ (τ ) |2 [6] (the result for the Rice
model is given in [11]). If we model the AOA by the von Mises
PDF, then according to (11) we obtain:
2
B R 2 (τ ) = b02
2
I 0 ⎛⎜ κ 2 − 4π 2 f d τ 2 + j 4π κ cos(μ ) f d τ ⎞⎟
⎝
⎠
. (14)
I 02 (κ )
In
the
case
of
isotropic
scattering
we
have
BR 2 (τ ) = b02 J 02 (2π f dτ ) . Interestingly, it has been observed in
[10] that Eq. (14) shows very good fit even to those signal
records whose envelope CDFs are far from Rayleigh. This
means that Eq. (14) can be used as a flexible parametric model
for the autocovariance of the envelope-squared, regardless of the
distribution of the envelope. We specifically employ this result
by assuming that the autocovariance of the envelope-squared in
the Nakagami model follows (14). We will see in the sequel that
such an assumption yields a general expression for the LCR of
the envelope in non-isotropic scattering Nakagami fading
channels, which is also consistent with the simpler result derived
for isotropic scattering and integer values for m [12].
For a fading signal, the LCR, N, is by definition the average
number of times per second that the signal envelope, R (t ) ,
crosses a specific level with positive slope. Consider the general
scenario of non-isotropic scattering, where the mobile receives
signals only from specific directions with different probabilities.
This is in contrast with Clarke’s isotropic scattering model in
which the signals impinge the mobile from all directions with
equal probability. The LCR expressions for Rayleigh [13], Rice
[14], and Nakagami [15] models are then:
b2 b12 ρ exp(− ρ 2 )
,
−
b0 b02
π
N Rayleigh ( ρ ) =
N Rice ( ρ ) =
π 2
b2 b12 2 K + 1
ρ exp − K − ( K + 1) ρ 2
−
b0 b02 π 3 2
(
(
× ∫ cosh 2 K ( K + 1) ρ cosθ
0
(15)
) [exp(− χ
]
sin 2 θ
2
)
⎛bb
⎞
+ π χ sin(θ ) erf ( χ sin θ ) dθ , χ = K ⎜⎜ 0 2 2 − 1⎟⎟
b
⎝ 1
⎠
B R′′ 2 (0)
N Nakagami ( ρ ) =
(mρ )
m − (1 2 )
,
exp(− mρ )
2
2π Γ(m)
B R 2 (0 )
(16)
,
(17)
where cosh(.) is hyperbolic cosine, erf(.) is the error function:
z
erf ( z ) = 2π −1 2 ∫ exp(−u 2 ) du ,
(18)
0
and prime denotes differentiation with respect to τ . The
accuracy of the rather complicated formula in (16) is verified via
numerical comparison with the Rice LCR formula derived in
[16] using the characteristic function-based approach. The above
three expressions hold for any arbitrary combination of AOAs
from different directions. However, if we consider a single von
Mises-distributed AOA, which indicates that the mobile unit
receives signal only from the specific direction μ such that the
width of the AOA is κ , 360 (π κ ) in degree [10], then the
spectral moments in (15) and (16) for Rayleigh and Rice models
can be replaced by (12) and (13); while for the Nakagami model,
(14) should be inserted into (17). Note that after these
substitutions, all the LCR formulas in (15)-(17) become
independent of b0 . For the Rayleigh model, after some
manipulations, we obtain:
N Rayleigh ( ρ ) = 2π f d ρ exp(− ρ 2 )
×
I 02 (κ ) − I 12 (κ ) + cos(2 μ )[ I 0 (κ ) I 2 (κ ) − I 12 (κ )]
I 0 (κ )
.
(19)
In Clrake’s isotropic scattering model, based on b1 = 0 and
b2 = b0 2 π 2 f d2 , Eqs. (15)-(17) simplify to the results given in
[6] for the Rayleigh and Rice models, and derived in [12] for the
Nakagami model, respectively:
N Rayleigh ( ρ ) = 2π f d ρ exp(− ρ 2 ) ,
(
N Rice ( ρ ) = 2π ( K + 1) f d ρ exp − K − ( K + 1) ρ
(
)
× I 0 2 K ( K + 1) ρ ,
(20)
2
)
(21)
m m − (1 2 ) 2 m −1
exp(−mρ 2 ) .
ρ
Γ( m )
(22)
For a fading signal, the AFD, T, is by definition the average
time over which the signal envelope, R (t ) , remains below a
certain level. For any fading model, the corresponding AFD is:
F
(ρ)
TModel ( ρ ) = Model
,
(23)
N Model ( ρ )
with FModel ( ρ ) given in (4)-(6), and N Model ( ρ ) presented in
(15)-(17), (19), and (20)-(22). Below we list the equations that
we need in the next section for comparison with measured data.
For the Rayleigh case with non-isotropic scattering (where the
AOA is modeled by a single von Mises PDF), AFD can be
obtained via dividing (4) by (19):
exp( ρ 2 ) − 1
T Rayleigh ( ρ ) =
2π f d ρ
×
−1 2
12
2
)
N Nakagami ( ρ ) = 2π f d
I 0 (κ )
I (κ ) − I (κ ) + cos(2 μ )[ I 0 (κ ) I 2 (κ ) − I 12 (κ )]
2
0
2
1
.
(24)
On the other hand, in a propagation environment with isotropic
scattering, the AFD for Rayleigh, Rice, and Nakagami models
can be simply obtained after dividing (4)-(6) by (20)-(22),
respectively:
exp( ρ 2 ) − 1
TRayleigh ( ρ ) =
,
(25)
2π f d ρ
(
)
⎡1 − Q⎛ 2 K , 2( K + 1) ρ 2 ⎞⎤ exp K + ( K + 1) ρ 2
⎜
⎟⎥
⎢
⎝
⎠⎦
,(26)
TRice ( ρ ) = ⎣
2π ( K + 1) f d ρ I 0 (2 K ( K + 1) ρ )
T Nakagami ( ρ ) =
Γ(m, mρ 2 ) exp(mρ 2 )
2π m 2 m −1 f d ρ 2 m −1
.
(27)
3. Empirical results
In this section, we compare the theoretical results of the
previous section with our measured data. Before that, we review
the empirical investigations on the LCR and AFD of fading
signals, reported in the literature so far. The LCR and AFD of
the Rayleigh model with isotropic scattering, i.e. (20) and (25),
are compared with measured data taken in an urban location at
836 MHz [17] [18], in a suburb at 836 MHz and 11.2 GHz [19],
and in rural areas at 900 MHz [20]. Moreover, the LCR of the
Rayleigh model with isotropic scattering, i.e. (20), seems to have
been compared with the experimental data taken in urban and
suburban areas at 11.2 GHz [21]. Except for [20], fairly good
agreement between the Rayleigh model (with isotropic
scattering) and empirical results are observed in [17] - [19] [21].
The unsatisfactory predictions of the Rayleigh model (with
isotropic scattering) in [20] comes from the expected fact that in
rural areas, Rice could be a better model due to the presence of a
rather clear line-of-sight, as has been confirmed in [20]
empirically via statistical goodness-of-fit tests. So, it is
anticipated that (21) and (26) may show better fit to the
empirical LCR and AFD curves in [20]. Finally, in [22], the
LCR of Rice and Nakagami models with isotropic scattering, i.e.
(21) and (22), are compared with the measurements conducted
in an urban location at 870.9 MHz. It has been observed that for
some cases the Rice model fits reasonably, while for some other
situations Nakagami model shows a good fit.
Our data was collected in twelve different locations in urban
and suburban areas in Texas, all at 910.25 MHz. Each data set
represents the signal envelope (and phase) over a traveled
distance of 47 m, or 7 s of time. Both antennas at the fixed
transmitter and mobile receiver had omnidirectional patterns.
More details about the data can be found in [10] [23] [24].
All the records were first preprocessed by removing the
shadow fading (slow fading) component from the envelope,
using a standard local sliding window technique [24]. Before
going through the details of parameter estimation, let us
introduce the useful parameter γ , the amount of fading, defined
by γ = V [ R 2 ] ( E[ R 2 ])2 [25], where V is the variance. It is
straightforward to verify that the Rice K parameter [26] and the
Nakagami m parameter [9] could be conveniently expressed in
terms of the amount of fading as:
K=
1−γ
1− 1− γ
, m=
1
.
γ
(28)
If in (28) we replace any theoretical moment E[ R k ] by its
n
sample estimate n −1 ∑ i =1 Rik , where n ≈ 250,000 is the number
of envelope samples for each record, then the formulas in (28)
serve as simple and easy-to-use moment-based estimators [27]
for K and m ( Ω can be estimated using a simple moment-based
estimator as well). Based on the Monte Carlo simulations in
[28], the m-estimator in (28) is a good estimator with reasonable
finite sample bias-variance-efficiency characteristics (see [29]
for some theoretical results). However, we are unaware of the
finite sample bias-variance-efficiency properties of the Kestimator in (28). This issue has to be studied at least via Monte
Carlo simulations, in a manner similar to [28]. Anyway, we use
the moment-based estimators in (28) for the sake of simplicity,
similar to [22].
In this paper we consider the fast fading components of six
typical envelope records. Records #0011, #0012, #0014, #0015,
and #0018 are taken in suburban areas, while record #0019 is
collected in an urban location. Fig. 1 shows the empirical CDFs,
together with the estimated CDFs of Rayleigh, Rice, and
Nakagami models. The estimated parameters on the plots are
calculated using the moment-based estimators, discussed in the
previous paragraph. The ranges of the estimated parameters in
all of our twelve records are 1.19 ≤ Ω ≤ 1.42 , 0.53 ≤ K ≤ 4.51 ,
and 0.65 ≤ m ≤ 3.03 . For records #0011, #0012, and #0019, the
Rayleigh model fits very well, while for the rest it deviates
significantly. In general, Rice and Nakagami models show better
CDF fit than Rayleigh, although, for example, for record #0014,
the fit is not so satisfactory. In all cases, the estimated Rice and
Nakagami CDFs are very close to each other. This is due to the
implicit relationship between K and m depicted in (28), which
makes the two CDFs almost indistinguishable [6]. Note that for
records #0011, #0012, and #0019, the Rice CDF is not included,
as the estimated K were complex numbers since the estimated
γ s were larger than one.
With regard to theoretical LCR and AFD formulas, we
consider the Rayleigh model with non-isotropic scattering, Eqs.
(19) and (24), respectively, and Rayleigh, Rice and Nakagami
models with isotropic scattering, i.e. (20)-(22) for LCR and (25)(27) for AFD. Only the normalized LCR and AFD expressions,
i.e. N Model ( ρ ) f d and f d TModel ( ρ ) are plotted in Figs. 2-3 in
terms of 20 log 10 ( ρ ) (in our experiments f d = 20 Hz ). The
estimated values of κ and μ in (10) for #0011, #0012, #0014,
#0015, #0018 and #0019 are given in [10] as
o
o
o
o
o
o
(κ , μ ) = {(2.4,19.8 ),(3,36 ),(2.4,0 ),(2.4,0 ),(3.3,0 ),(0.6,0 )}.
The empirical LCR curves for the six records are shown in
Fig. 2, together with the theoretical curves. Except for #0014
and #0019, the simple Rayleigh model with isotropic scattering,
Eq. (20), shows reasonable fit in terms of LCR, even for records
such as #0015 and #0018 where the Rayleigh CDF fit is bad.
Note that the excellent match between the Rayleigh and
empirical CDFs for #0019 does not guarantee a good match for
the corresponding LCR curves. It can also be observed that for
all cases, the LCR curves of the non-isotropic scattering
Rayleigh model, Eq. (19), and the isotropic ones, Eq. (20), are
almost alike in the scale shown, although the difference between
the associated fading correlation functions are very significant
[10]. This may be attributed to the fact that LCR depends on the
slopes of the fading correlation function only at τ = 0 and not
on the entire range of τ (see (9)). Except for #0011, #0014, and
#0019, both Rice and Nakagami LCR formulas, (21) and (22),
fits reasonably to the data. Note the significant difference
between Rice and Nakagami LCR curves over the low envelopelevel region (say, less than –20 dB) for #0015, in contrast with
the close CDFs in Fig. 1. The same behavior is also reported in
[22], confirming the obvious fact that close match for the firstorder statistics may yield dissimilar second-order statistics. As
the last comment, observe that the Nakagami LCR formula is
inferior to both Rayleigh formulas for #0011 in spite of rather
similar CDF match to data in Fig. 1, while all the LCR equations
in (19)-(22) fail to follow the empirical LCR curve for #0019.
The empirical AFD curves for the six records are shown in
Fig. 3, together with the theoretical curves. Except for #0019, all
the Eqs. (24)-(27) are close to each other and match closely to
the data in the scale shown. Even the simple Rayleigh model
with isotropic scattering, Eq. (25), shows reasonable fit in terms
of AFD, although the Rayleigh CDF fit to the empirical CDF is
not so good for #0014, #0015, and #0018. Note that the
excellent match between the Rayleigh and empirical CDFs for
#0019 does not guarantee a good match for the corresponding
AFD curves (as was the case for the LCR curves of this
particular record). Interestingly, the AFD predictions of Rice
and Nakagami models diverge significantly from those of both
Rayleigh models, as well as the measured data, for high values
of the envelope level (say, more than 5 dB). Heuristically, this
does not comply with the better Rice and Nakagami CDF match
to the data, in comparison with the Rayleigh CDF. Such an
observation again confirms fairly uncorrelated behavior of firstand second-order fading statistics.
4. Conclusion
In this paper we have derived expressions for the LCR and
AFD of Rayleigh, Rice, and Nakagami models of multipath
fading, assuming a general non-isotropic scattering scenario in
which multipath signals impinge the mobile from different
directions. The impact of the distribution of the angles-of-arrival
at the mobile (which is assumed to be uniform in Clarke’s
classic model of isotropic scattering) is modeled via the
application of von Mises distribution. Visual comparison of the
theoretical results with measured data revealed that all the above
models (even the simple Rayleigh model with isotropic
scattering) show reasonable fit in terms of LCR and AFD for
most of the records, independent of the CDF fits. This implies
that the goodness of fit for second-order statistics (LCR and
AFD) do not appear to be dependent on the accuracy of fit for
first-order statistics (CDF). Extension of the results of this paper
to generalized (wideband) fading channels are discussed in [30].
5. Acknowledgement
The work of the first, the fourth, and the fifth authors was
supported in part by NSF, under the Wireless Initiative Program,
Grant #9979443. The work of the second author was supported
in part by the NSF Summer 1999 Research Experience for
Undergraduates (REU) program at the University of Minnesota.
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0.4
0.5
1
1.5
2
Normalized envelope level
Fast fading component of record #0014
1
Ω = 1.24
K = 0.75
m = 1.23
0.8
0.6
0.4
Empirical
Rayleigh
Rice
Nakagami
0.2
0
0.5
1
1.5
2
Normalized envelope level
Fast fading component of record #0018
1
Ω = 1.29
K = 1.29
m = 1.47
0.6
0.4
Empirical
Rayleigh
Rice
Nakagami
0.2
0
0
0.5
1
1.5
2
Normalized envelope level
Ω = 1.42
m = 0.98
0.8
0.6
0.4
Empirical
Rayleigh
Nakagami
0.2
1
2
Normalized envelope level
Fast fading component of record #0015
1
0
Ω = 1.19
K = 4.51
m = 3.03
0.8
0.6
0.4
Empirical
Rayleigh
Rice
Nakagami
0.2
0
0
0.8
Fast fading component of record #0012
1
0
0
Cumulative distribution function
Cumulative distribution function
Empirical
Rayleigh
Nakagami
0.2
Cumulative distribution function
0.6
0
Cumulative distribution function
Ω = 1.28
m = 0.65
0.8
Cumulative distribution function
Cumulative distribution function
Fast fading component of record #0011
1
0
0.5
1
1.5
2
Normalized envelope level
Fast fading component of record #0019
1
Ω = 1.40
m = 0.95
0.8
0.6
0.4
Empirical
Rayleigh
Nakagami
0.2
0
0
0.5
1
1.5
2
Normalized envelope level
Figure 1. Empirical envelope CDFs, together with the CDFs of Rayleigh, Rice, and Nakagami fading models.
−5
10
−10
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0014
0
10
−5
10
−10
10
Empirical
Rayleigh
Non−iso. Ray.
Rice
Nakagami
−40
5
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0018
10
0
10
−5
10
−10
10
−15
10
−40
Empirical
Rayleigh
Non−iso. Ray.
Rice
Nakagami
−20
0
20
Normalized envelope level (dB)
0
10
−5
10
−10
Empirical
Rayleigh
Non−iso. Ray.
Nakagami
−40
10
Normalized level crossing rate
10
Fast fading component of record #0012
10
10
−40
5
Normalized level crossing rate
Empirical
Rayleigh
Non−iso. Ray.
Nakagami
5
Normalized level crossing rate
0
10
10
Normalized level crossing rate
Fast fading component of record #0011
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0015
10
0
10
−10
10
−20
10
−30
10
Empirical
Rayleigh
Non−iso. Ray.
Rice
Nakagami
−40
2
Normalized level crossing rate
Normalized level crossing rate
5
10
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0019
10
0
10
−2
10
−4
10
−6
10
−40
Empirical
Rayleigh
Non−iso. Ray.
Nakagami
−20
0
20
Normalized envelope level (dB)
Figure 2. Empirical LCRs, together with the LCRs of Rayleigh, Rice, and Nakagami fading models.
Empirical
Rayleigh
Non−iso. Ray.
Nakagami
5
10
0
10
−5
10
−40
Fast fading component of record #0012
Normalized average fade duration
Normalized average fade duration
Fast fading component of record #0011
10
10
Empirical
Rayleigh
Non−iso. Ray.
Rice
Nakagami
5
10
0
10
−5
10
−40
Normalized average fade duration
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0018
15
10
10
10
Empirical
Rayleigh
Non−iso. Ray.
Rice
Nakagami
5
10
0
10
−5
10
−40
−20
0
20
Normalized envelope level (dB)
Normalized average fade duration
10
10
Empirical
Rayleigh
Non−iso. Ray.
Nakagami
5
10
0
10
−5
10
−40
30
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0015
10
Empirical
Rayleigh
Non−iso. Ray.
Rice
Nakagami
20
10
10
10
0
10
−10
10
Normalized average fade duration
Normalized average fade duration
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0014
10
10
−40
−20
0
20
Normalized envelope level (dB)
Fast fading component of record #0019
10
10
5
10
Empirical
Rayleigh
Non−iso. Ray.
Nakagami
0
10
−5
10
−40
−20
0
20
Normalized envelope level (dB)
Figure 3. Empirical AFDs, together with the AFDs of Rayleigh, Rice, and Nakagami fading models.