entropy
Article
On the Definition of Diversity Order Based on Renyi
Entropy for Frequency Selective Fading Channels
Seungyeob Chae and Minjoong Rim *
Department of Information and Communication Engineering, Dongguk University-Seoul, 30 Pildong-ro 1 Gil,
Jung-gu, Seoul 100-715, Korea; [email protected]
* Correspondence: [email protected]; Tel.: +82-2-2260-3595
Academic Editors: Luis Javier Garcia Villalba, Anura P. Jayasumana and Jun Bi
Received: 23 November 2016; Accepted: 18 April 2017; Published: 20 April 2017
Abstract: Outage probabilities are important measures of the performance of wireless communication
systems, but to obtain outage probabilities it is necessary to first determine detailed system
parameters, followed by complicated calculations. When there are multiple candidates of diversity
techniques applicable for a system, the diversity order can be used to roughly but quickly compare
the techniques for a wide range of operating environments. For a system transmitting over frequency
selective fading channels, the diversity order can be defined as the number of multi-paths if
multi-paths have all equal energy. However, diversity order may not be adequately defined when
the energy values are different. In order to obtain a rough value of diversity order, one may use the
number of multi-paths or the reciprocal value of the multi-path energy variance. Such definitions are
not very useful for evaluating the performance of diversity techniques since the former is meaningful
only when the target outage probability is extremely small, while the latter is reasonable when the
target outage probability is very large. In this paper, we propose a new definition of diversity order
for frequency selective fading channels. The proposed scheme is based on Renyi entropy, which
is widely used in biology and many other fields. We provide various simulation results to show
that the diversity order using the proposed definition is tightly correlated with the corresponding
outage probability, and thus the proposed scheme can be used for quickly selecting the best diversity
technique among multiple candidates.
Keywords: diversity; diversity order; Renyi entropy; frequency selective fading; outage probability
1. Introduction
The outage probability is a critical measurement for evaluating the performance of wireless
communication systems over fading channels. In order to obtain an outage probability, however,
it is necessary to determine detailed system parameters, followed by complicated high-precision
calculations. On the other hand, a diversity order can be used to roughly but quickly compare multiple
diversity techniques for a wide range of system parameters and operating environments [1–4].
5G cellular communication systems consider various usage scenarios such as enhanced
mobile broadband, massive machine type communications, and ultra-reliable and low latency
communications [5]. Especially in order to achieve reliable communications over fading channels,
various diversity techniques need to be considered including antenna, time, frequency, polarity, angle,
and site diversity techniques [6–8], and the best scheme or the best combination of techniques should
be selected within a short time to satisfy the latency requirement.
If the average energy values of multiple paths are all equal in a frequency selective fading channel,
the diversity order can be defined as the number of multi-paths and the outage performance can be
reduced as the diversity order becomes larger. However, if the average energy values of multiple
paths are different, there is no adequate definition to represent the diversity order. In order to roughly
Entropy 2017, 19, 179; doi:10.3390/e19040179
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represent the diversity achievable in a frequency selective fading channel, one may use the number
of multi-paths regardless of the average energy values of the multi-paths, or some others use the
reciprocal value of the multi-path energy variance [9–12]. Diversity order values produced by these
schemes, however, are not tightly correlated with the corresponding outage probabilities, and thus
they are not very suitable for evaluating the performance of diversity techniques. In this paper, we
propose a new definition of a diversity order for a frequency selective fading channel. The proposed
scheme is based on Renyi entropy, which is widely used in biology and many other fields. We provide
various simulation results showing that diversity orders using the proposed definition are very useful
for comparing diversity techniques. The diversity order obtained from the proposed definition is
tightly correlated with the corresponding outage probability, and the proposed scheme can be used for
quickly selecting the best diversity technique among multiple candidates.
This paper is organized as follow: Section 2 describes the relationships between the outage
probabilities and the diversity orders in frequency selective fading channel environments. Section 3
proposes a new definition of a diversity order based on Renyi entropy for frequency selective fading
channels and Section 4 shows the effectiveness of the proposed definition of diversity order through
various simulation results. Section 5 provides an application example of the proposed scheme and
finally, conclusions are drawn in Section 6.
2. Conventional Definitions of Diversity Order
The average power information of multi-paths in a frequency selective fading channel is called
the power delay profile (PDP) and a PDP with L multi-paths can be presented as:
p = Ptotal [ p0 , p1 , · · · , p L−1 ]
(1)
where Ptotal is the total energy of the multi-paths, and the vector [ p0 , p1 , · · · , p L−1 ] is assumed to satisfy
the following equation:
L −1
∑
pl = 1
(2)
l =0
2.1. Outage Probability
The outage probability for a communication system is the probability that the channel capacity
will fail to satisfy its target data transmission rate. When instantaneous channel information is not
available but the PDP is given, the outage probability can be calculated as:
L −1
2
pout = Pr W log2 1 + Ptotal
h
p
<
R
|
|
l
l
σ2 ∑
l =0
L −1
2
= Pr W log2 1 + SNR ∑ |hl | pl < R
l =0
L −1
2
= Pr ∑ |hl | pl < ξ
(3)
l =0
where hl is an independent complex Gaussian random variable with zero mean and unit variance, R is
the target data transmission rate, W is
σ2 is the noise variance, SNR is the signal-to-noise
the bandwidth;
ratio defined as Ptotal /σ2 , and ξ ≡ 2R/W − 1 /SNR. The outage probability in Equation (3) can be
calculated with the equations described in [12,13].
2.2. Equal Energy Values of Multi-Paths
If the average energy values of multi-paths in a frequency selective fading channel are all equal,
the diversity order can be precisely defined. There are L paths with the same energy, the outage
probability can be written as Equation (4) [12] and thus the diversity order can be written as L:
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L −1
∑
pout = 1 −
l =0
!
Lξ
e− Lξ
Γ ( l + 1)
(4)
2.3. Different Energy Values of Multi-Paths
If the average energy values of multi-paths are different, there is no adequate definition of the
diversity order. One scheme uses the number of multi-paths as the diversity order as shown in
Equation (5) assuming that the operating SNR is very large, in other words, the outage probability is
very small [12,14]:
D0 = L
(5)
In Equation (5), the effect of different average energy values of multi-paths are not taken
into account and thus it may not be tightly correlated with the outage probability except when
the operating SNR is very high and thus all energy values of multi-paths are meaningfully large.
Another scheme, called effective frequency diversity, uses the reciprocal value of the multi-path energy
variance [10,14–16], written as follows:
L −1
∑
D2 =
! −1
p2l
(6)
l =0
In Equation (6), if the average energy values of multi-paths are all equal, in other words, pl = 1/L
with L multi-paths, the diversity order becomes L, which coincides with the result in Section 2.2.
The scheme described in Equation (6) mainly takes into account paths with large average energy
values and undervalues the effect of small-energy paths. Therefore, this diversity order is not tightly
correlated with the outage probability when the operating SNR is large, where a path with a relatively
small energy value can contribute to the reduction of outage probabilities.
3. New Definition of Diversity Order
In this paper, we propose a new definition of a diversity order, which is a superset of the two
conventional definitions described in Section 2 and applicable to a wide range of target outage
probabilities. In order to define a diversity order, we borrow the definition of Renyi entropy, which
is widely used to define a diversity order in biology and many other fields [17–22]. Renyi entropy is
written as:
!
n
1
α
Hα =
log ∑ ρi (0 ≤ α, α 6= 1)
(7)
1−α
i =1
where ρi is a probability of an event. The diversity order can be represented as two to the power of
Renyi Entropy:
! 1
Dα = 2 Hα =
L −1
1− α
∑ ραl
(0 ≤ α ≤ 2, α 6= 1)
(8)
l =0
Although PDP values are not probabilities but energy values, we can use a similar definition as
Renyi entropy for a frequency selective fading channel, written as:
1
Hα =
log
1−α
!
n
∑
piα
(0 ≤ α, α 6= 1)
(9)
(0 ≤ α ≤ 2, α 6= 1)
(10)
i =1
and the diversity order can be expressed as follows:
Dα = 2
Hα
!
L −1
=
∑
l =0
pαl
1
1− α
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If the average energy values of multi-paths are all equal, in other words, pl = 1/L with L paths,
the diversity order in Equation (10) becomes L, which also coincides with the result in Section 2.2.
Note that in the case of α = 0, Equation (10) is the same to Equation (5), and in the case of α = 2, it
becomes Equation (6). In Equation (10), α is a value which needs to be determined according to the
target outage probability. If a target outage probability is very small, a small α can be used. On the
other hand, if it is very large, a large α will be preferred. Equation (10) includes the definitions of
Equation (5) (with α = 0) and Equation (6) (with α = 2), and can be used in a wide range of target
outage probabilities assuming that a proper value of α can be chosen.
In order to make diversity orders in Equation (10) highly correlated with the corresponding
outage probabilities, it is important to choose a proper value of α. We define mean square error (MSE)
as the expected value of the square of the difference between a linear interpolation function of an
outage probability and a diversity order obtained from Equation (10), written as:
n
o
MSE = E ( f ( Pout (pk )) − Dα (pk ))2
(11)
where f is a linear interpolation function created from the cases in which the diversity order is precisely
defined, in other words, all paths have equal energy values. To express the equation more clearly,
the inverse function of f , denoted as g, is defined as:
e
g D
e
= f −1 D
(
= log10
)
e [ De ]+1
Pout p
e
D
e [ De ]
Pout p
h i
e + Pout p
e [ De ]
− D
(12)
e L is the PDP with L equal-energy multi-paths. A proper α(0 ≤ α ≤ 2, α 6= 1) can be chosen as
where p
the value that minimizes the MSE in Equation (11).
4. Simulation Results
In order to observe the correlation of diversity orders and outage probabilities, PDPs are randomly
generated and the corresponding diversity orders and outage probabilities are computed. The ratio of
the target data transmission rate and bandwidth (R/W) is assumed to be 2. Figure 1 shows outage
probabilities versus diversity orders calculated by Equation (6) when PDPs with 5 multi-paths are
generated with random energy values. In the figure, “+” represents the points where the diversity
order is clearly defined, in other words, all paths have equal energy values. It can be seen that outage
probabilities are widely spread, meaning that diversity orders obtained by Equation (6) is not tightly
correlated with the outage probabilities. Increasing the diversity order using Equation (6) may not
mean that the outage probability can be reduced. If we use the diversity order definition in Equation (5),
all cases in Figure 1 correspond to the diversity order of 5 and the diversity order is not correlated with
the outage probabilities.
Figure 2 shows the results generated by Equation (10) with α = 0.57. For each value of a diversity
order, the outage probability distribution is far narrower than that in Figure 1 and thus we can say
that Equation (10) can provide diversity orders tightly correlated with the corresponding outage
probabilities compared to the conventional schemes. The diversity order using Equation (10) with a
proper α allows quick performance evaluation of diversity techniques.
A proper value of α according to each target operating SNR can be found by evaluating
Equation (11) and selecting the value resulting in the minimum MSE. Figures 3–5 show the MSEs
in Equation (11) according to α with different SNR values, and each figure includes three curves
representing 3, 5, and 7 multi-paths, respectively. Note that the optimal α is highly related to the SNR,
but independent of the number of multi-paths. If the operating SNR or target outage probability is
given, a proper value of α can be determined regardless of the number of multi-paths.
order is clearly defined, in other words, all paths have equal energy values. It can be seen that outage
probabilities are widely spread, meaning that diversity orders obtained by Equation (6) is not tightly
correlated with the outage probabilities. Increasing the diversity order using Equation (6) may5 not
Entropy 2017, 19, 179
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mean that the outage probability can be reduced. If we use the diversity order definition in Equation
(5), all cases in
Figure 1 correspond to the diversity order of 5 and the diversity order is not correlated
Entropy Figure
2017, 19, 2
179shows the results generated by Equation (10) with α = 0.57. For each value5 of
of 9a
with
the
outage
probabilities.
diversity order, the outage probability distribution is far narrower than that in Figure 1 and thus we
can say that Equation (10) can provide diversity orders tightly correlated with the corresponding
10
outage probabilities compared
to the conventional schemes. The diversity order using Equation (10)
with a proper α allows quick performance evaluation of diversity techniques.
-1
-2
10
-1
10
-3
Outage
10
-2
10
-4
10
-3
Outage
10
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-5
10
-4
10
Figure 2 shows the results
generated by Equation (10) with α = 0.57. For each value of a
10
diversity order, the outage probability
distribution
than that
in Figure 1 and thus we
1
1.5
2
2.5
3is far
3.5narrower
4
4.5
5
Diversity Order
can say that Equation (10)10can provide diversity
orders tightly correlated with the corresponding
outage
probabilitiesprobabilities
compared to the conventional
schemes. The
diversity order
using Equation
(10)
Figure
Figure1.1.Outage
Outage probabilitiesversus
versusdiversity
diversityorders
orderscalculated
calculatedusing
usingEquation
Equation(6)
(6)(or
(orEquation
Equation(10)
(10)
withwith
a proper
allows
quick
performance evaluation of diversity techniques.
α
10
= 17.8
dB.
withα =
), SNR
= 17.8
dB.
α =2),2 SNR
1
1.5
2
2.5
3
3.5
4
4.5
5
-6
-5
-6
Diversity Order
-1
10
Figure 2. Outage probabilities versus diversity orders calculated using Equation 10 with α = 0.57 ,
-2
10
SNR = 17.8 dB.
A proper value of α10 according to each target operating SNR can be found by evaluating
Equation (11) and selecting the value resulting in the minimum MSE. Figures 3–5 show the MSEs in
Equation (11) according to α with different SNR values, and each figure includes three curves
10
representing 3, 5, and 7 multi-paths, respectively. Note that the optimal α is highly related to the
SNR, but independent of the number of multi-paths. If the operating SNR or target outage probability
is given, a proper value of 10α can be determined regardless of the number of multi-paths.
Figure 6 shows the optimal α values according to SNR. A large value of α is required for low
operating SNRs, where the10outage probability is relatively large and paths with low average energy
1
1.5
2
2.5
3
3.5
4
4.5
5
Diversity Order
values make little contribution to the outage probability.
On the other hand, a small value of α is
preferred for high operating SNRs, where the average channel energy is sufficiently strong and even
Figure 2. Outage probabilities versus diversity orders calculated using Equation 10 with α = 0.57,
a path
with 2.a relatively
low average
energy
valueorders
can help
the reduction
of the outage
Figure
Outage probabilities
versus
diversity
calculated
using Equation
10 with probability.
α = 0.57 ,
Outage
-3
-4
-5
-6
SNR = 17.8 dB.
SNR = 17.8 dB.
SNR = 7.8dB
2
A proper value of α10 according to each target operating SNR can be found by evaluating
Equation (11) and selecting the value resulting in the minimum MSE. Figures 3–5 show the MSEs in
10
Equation (11) according to α with different SNR values, and each figure includes three curves
representing 3, 5, and 7 multi-paths,
respectively. Note that the optimal α is highly related to the
10
SNR, but independent of the number of multi-paths. If the operating SNR or target outage probability
is given, a proper value of 10α can be determined regardless of the number of multi-paths.
Figure 6 shows the optimal α values according to SNR. A large value of α is required for low
operating SNRs, where the10outage probability is relatively large and paths with low average energy
values make little contribution to the outage probability. On the other hand, a small value of α is
10 SNRs, Path
preferred for high operating
where
Lengththe
= 3 average channel energy is sufficiently strong and even
Length = 5
a path with a relatively low averagePath
energy
value can help the reduction of the outage probability.
Path Length = 7
1
Variance
0
-1
-2
-3
-4
10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
α
2
SNR = 7.8dB
10
Figure 3. Mean square errors according to α (SNR = 7.8 dB).
Figure 3. Mean square errors according to α (SNR = 7.8 dB).
1
10
0
Variance
10
-1
10
-2
10
-3
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Entropy 2017, 19, 179
SNR = 10.8dB
SNR = 10.8dB
2
10
2
10
1
10
1
10
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Variance
Variance
0
10
0
10
SNR = 10.8dB
2
10
-1
10
-1
10
1
10
-2
Variance
10
-2
10
0
10
-3
10
-30
10
-1 0
10
Path Length = 3
Path Length
Length == 53
Path
Path Length
Length == 75
Path
Path Length = 7
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1
1.2
1.2
α1
α
1.4
1.4
1.6
1.6
1.8
1.8
2
2
Figure 4. Mean square errors according to α (SNR = 10.8 dB).
Figure
4.10 Mean
square
to α
α (SNR
10.8dB).
dB).
Figure
4. Mean
squareerrors
errors according
according to
(SNR =
= 10.8
-2
Path Length = 3
Path Length = 5
Path Length = 7
1
10-3
10
1
10 0
0.2
0.4
0.6
SNR = 17.8dB
SNR = 17.8dB
0.8
1
1.2
1.4
1.6
1.8
2
α
0
Variance
Variance
Figure10
4.
Mean square errors according to α (SNR = 10.8 dB).
0
10
SNR = 17.8dB
-1
101
10
-1
10
-2
Variance
100
10
-2
10
-3
10-1
10
-30
10
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1
1.2
1.2
α1
α
1.4
1.4
Path Length = 3
Path Length
Length == 53
Path
Path Length
Length == 75
Path
Path Length = 7
1.6
1.8
2
1.6
1.8
2
Figure
5.10 Mean
to αα (SNR
17.8dB).
dB).
Figure
5. Meansquare
squareerrors
errors according
according to
(SNR =
= 17.8
Figure 5. Mean square errors according to α (SNR = 17.8 dB).
-2
Path Length = 3
Path Length = 5
Path Length = 7
Figure 6 shows the optimal α values according to SNR. A large value of α is required for low
10 2
0.2
0.4
0.6
0.8 relatively
1
1.2
1.4
1.6 and
1.8 paths
2
20
operating SNRs, where the outage
probability
is
large
with low average energy
1.8
α
1.8
values make little contribution to the outage probability. On the other hand, a small value of α is
1.6
1.6
preferred for high operating
SNRs,
where
theerrors
average
channel
sufficiently
strong and even a
Figure
5. Mean
square
according
to αenergy
(SNR = is
17.8
dB).
1.4
1.4
path with a relatively low average
energy value can help the reduction of the outage probability.
-3
α
αα
1.2
1.2
21
1
0.8
1.8
0.8
0.6
1.6
0.6
0.4
1.4
0.4
0.2
1.2
0.2
10
5
0
5
0.8
10
10
15
15
20
20
25
30
25
30
SNR(dB)
SNR(dB)
35
35
40
40
45
45
50
50
0.6
Figure 6. Optimal α according to operating SNR.
6. Optimal α according to operating SNR.
0.4
Figure
0.2
0
5
10
15
20
25
30
35
40
45
50
SNR(dB)
Figure
Optimal ααaccording
accordingtotooperating
operating SNR.
Figure
6.6.Optimal
SNR.
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5. Application of Diversity Order
5. Application of Diversity Order
The proposed definition of a diversity order can be applied for quick evaluation of system
The
proposed
definition
of a diversity
ordertechniques
can be applied
forthe
quick
evaluation of
ofdetailed
system
performance
with multiple
candidates
of diversity
without
determination
performance
with
multiple
candidates
of
diversity
techniques
without
the
determination
of
detailed
system parameters or complicated calculation of exact outage probabilities. For example, the
parameters
or
complicated
calculation
of exact
outage
probabilities.
For example,
the
system
parameters
complicated
of
exact
outage
probabilities.
example,
the technique
proposed
proposed
definitionorcan
be used tocalculation
determine
the
optimal
delay
value
for aFor
delay
diversity
proposed
definition
can
be
used
to
determine
the
optimal
delay
value
for
a
delay
diversity
technique
definition
can
be
used
to
determine
the
optimal
delay
value
for
a
delay
diversity
technique
in
a
in a multiple input single output (MISO) system. Consider a MISO system with two transmission
in
a
multiple
input
single
output
(MISO)
system.
Consider
a
MISO
system
with
two
transmission
multiple
input
single
output
(MISO)
system.
Consider
a
MISO
system
with
two
transmission
antennas
antennas and one reception antenna, shown in Figure 7. Suppose that the PDP of each antenna is
antennas
and one antenna,
receptionshown
antenna,
shown7.inSuppose
Figure that
7. Suppose
the PDP
of each
antenna
and
one
in Figure
the PDPthat
of each
antenna
is given
as: is
given
as:reception
given as:
= p total
psinglep=
· · , ,ppL −L1−] ,1 ],
(13)
[ p0[ p, 0p, 1p,1 ·, 
singlep total
p single = p total [ p 0 , p1 ,  , p L −1 ] ,
(13)
and the
thePDPs
PDPsofofthe
thetwo
two
antennas
independent.
When
the delay
diversity
technique
with
the
and
antennas
areare
independent.
When
the delay
diversity
technique
with the
delay
and
the
PDPs
of
the
two
antennas
are
independent.
When
the
delay
diversity
technique
with
the
delay
of K symbols
is applied,
the resulting
PDP
at the can
receiver
can be represented
as follows:
of
K symbols
is applied,
the resulting
PDP at the
receiver
be represented
as follows:
delay of K symbols is applied, the resulting PDP at the receiver can be represented as follows:
1
1
p MISO
,  , 0, p single ]
1 = 12 p total p single +112 p total [0

p MISO =
]
+ ptotal
·K· 
p MISOp=totalpptotal
p total [[0,
0, 
,·0, p0, p ]
sinpgle
single +
 }singlesingle
{z
|
2 2
22
(14)
(14)
(14)
K
K
1
12 Ptotal {P0 , P1 ,, PL −1 ,0,0, }
Ptotal {P0 , P1 ,, PL −1 ,0,0, }
2
1
12 Ptotal {d 0 , d1 ,, d K , P0 , P1 ,, PL −1 ,0,0, }
Ptotal {d 0 , d1 ,, d K , P0 , P1 ,, PL −1 ,0,0, }
2
Figure 7. Power delay profile for each antenna with delay diversity.
Figure
Figure 7.
7. Power
Power delay
delay profile
profile for
for each
each antenna
antenna with
with delay
delay diversity.
diversity.
Without calculating the exact outage probabilities, the optimal delay value K opt (0 ≤ K opt ≤ K max
Without calculating the exact outage probabilities, the optimal delay value K opt (0 ≤ K opt ≤ K max
calculating
exact outage
thefor
optimal
value
Kopt (0the
≤ maximum:
Kopt ≤ Kmax )
K and
) canWithout
be quickly
found bythe
evaluating
the probabilities,
diversity order
each delay
selecting
K and
) canbebequickly
quicklyfound
foundbybyevaluating
evaluatingthe
thediversity
diversityorder
orderfor
foreach
eachK and
selecting
the maximum:
can
selecting
the maximum:
K opt = argmax Dα (p MISO (K ))
(15)
0 ≤ K ≤ K max D (p
K opt = argmax
α
MISO (K ))
(15)
0 ≤ K ≤ K maxD
Kopt = argmax
p
K
(15)
α ( MISO ( ))
0≤K ≤Kmax
In the simulation, it was assumed that a single-antenna channel has a cluster exponential
In
the simulation,
simulation,
was assumed
assumed
that aa single-antenna
single-antenna
channel
has
cluster
exponential
In
the
itFigure
was
that
aa cluster
K max is 5.has
Figures
9 andexponential
10 illustrate
distribution,
as shown init
8 and the maximum
delay value channel
K
is
5.
Figures
9
and
10 9illustrate
distribution, as
shown
in
Figure
8
and
the
maximum
delay
value
as
shown
in
Figure
8
and
the
maximum
delay
value
and K10.
max Kmax is 5. Figures
the diversity orders and the outage probabilities, respectively, according to the delay value
illustrate
the
diversity
orders
and
the
outage
probabilities,
respectively,
according
to
the
delay
value
KK..
the
diversity
orders
and
the
outage
probabilities,
respectively,
according
to
the
delay
value
Comparing Figures 9 and 10, we can see that maximizing the diversity order can achieve minimizing
Comparing
Figures
9
and
10,
we
can
see
that
maximizing
the
diversity
order
can
achieve
minimizing
the outage probability. Therefore, using the new definition of a diversity order, quick evaluation of a
the
probability.
Therefore,
using
definition
of a diversity
quickcalculation.
evaluation of a
outage
probability.
order,
large
number
of diversity
techniques
canthe
benew
performed
without
complicated
outage
large number of diversity techniques can be
be performed
performed without
without complicated
complicated outage
outage calculation.
calculation.
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
2
4
6
8
10
12
14
16
0
2
4
6
8
10
12
14
16
Figure 8. PDP for a single transmission antenna.
Figure 8. PDP for a single transmission antenna.
Figure 8. PDP for a single transmission antenna.
Entropy 2017, 19, 179
Entropy2017,
2017,19,
19,179
179
Entropy
8 of 9
of99
88of
12
12
11
11
Diversity
DiversityOrder
Order
10
10
9
9
8
8
7
7
6
6
5
5
4
4 0
0
alpha = 0.55
alpha = 0.55
alpha = 0.8
alpha = 0.8
alpha = 1.2
alpha = 1.2
alpha = 2
alpha = 2
Max Diversity Order
Max Diversity Order
0.5
1
1.5
2
0.5
1
1.5
2
2.5
2.5
Delay
Delay
3
3
3.5
3.5
4
4
4.5
4.5
5
5
Figure9.9.Diversity
Diversityorders
ordersaccording
accordingto
todelay.
delay.
Figure
Figure
9. Diversity orders
according to
delay.
0
100
10
-2
10-2
10
Outage
OutageProbability
Probability
-4
10-4
10
-6
10-6
10
-8
10-8
10
-10
10-10
10
-12
10-12
10
-14
10-14
10 0
0
alpha = 0.55
alpha = 0.55
alpha = 0.8
alpha = 0.8
alpha = 1.2
alpha = 1.2
alpha = 2
alpha = 2
Min Outage
Min Outage
0.5
1
1.5
0.5
1
1.5
2
2
2.5
2.5
Delay
Delay
3
3
3.5
3.5
4
4
4.5
4.5
5
5
Figure 10. Outage probabilities according to delay.
Figure10.
10.Outage
Outageprobabilities
probabilitiesaccording
accordingto
todelay.
delay.
Figure
6. Conclusions
6.Conclusions
Conclusions
6.
Diversity
orders
can
bebe
used
toto
roughly
but
candidates
ofdiversity
diversity
Diversity
orders
can
be
used
to
roughly
butquickly
quicklycompare
comparemultiple
multiple candidates
candidates of
of
diversity
Diversity
orders
can
used
roughly
but
quickly
compare
multiple
techniques
in
a
wide
range
of
operating
environments.
For
a
system
transmitting
over
a
frequency
techniques
in
a
wide
range
of
operating
environments.
For
a
system
transmitting
over
a
frequency
techniques in a wide range of operating environments. For a system transmitting over a frequency
selective
fading
channel,
however,
the average
average
energy
values
of multi-paths
multi-paths
are different,
different,
the
selective
fading
channel,
however,
if theififaverage
energyenergy
valuesvalues
of multi-paths
are different,
the diversity
selective
fading
channel,
however,
the
of
are
the
diversity
order
has
not been
been adequately
adequately
defined.
In this
this
paper,
we proposed
proposed
new definition
definition
of
order
has not
beenhas
adequately
defined.
Indefined.
this paper,
wepaper,
proposed
a new definition
of diversity
diversity
order
not
In
we
aa new
of
diversity
order
for
a
frequency
selective
fading
channel.
The
proposed
scheme
is
based
on
Renyi
order
for
a
frequency
selective
fading
channel.
The
proposed
scheme
is
based
on
Renyi
entropy
diversity order for a frequency selective fading channel. The proposed scheme is based on Renyi
entropy
andititproduces
produces
thediversity
diversity
ordertightly
tightly
correlated
with
the
outage
probability.Hence,
Hence,the
the
andentropy
it produces
the
diversity
order tightly
correlated
withwith
the the
outage
probability.
Hence,
the
and
the
order
correlated
outage
probability.
proposed
definition
of
a
diversity
order
can
be
used
to
quickly
evaluate
various
frequency
diversity
proposed
definition
of of
a diversity
order
variousfrequency
frequencydiversity
diversity
proposed
definition
a diversity
ordercan
canbe
beused
usedto
toquickly
quickly evaluate
evaluate various
techniques
without
calculating
the
exact
outageprobabilities.
probabilities.For
Forexample,
example,as
as
described
in
Section
5,5,
techniques
without
calculating
the
exact
outage
probabilities.
5,
techniques
without
calculating
the
exact
outage
example,
asdescribed
describedin
inSection
Section
the optimal
optimal
delay
value
of aa delay
delay
diversity
technique
can
be quickly
quickly
found
by computing
computing
the
delay
of
diversity
technique
be
found
by
the
the the
optimal
delay
valuevalue
of a delay
diversity
technique
can becan
quickly
found by
computing
the diversity
diversity
orders
according
to
delay
values.
In
the
future,
more
theoretical
analysis
will
be
performed
diversity
orderstoaccording
to delay
In the
future,
more theoretical
be performed
orders
according
delay values.
In values.
the future,
more
theoretical
analysisanalysis
will be will
performed
for the
forthe
therelationships
relationshipsbetween
between diversityorders
ordersand
andoutage
outageprobabilities.
probabilities.
for
relationships
between diversity diversity
orders and outage
probabilities.
Acknowledgments: This
This work
work was
was supported
supported by
by the
the National
National Research
Research Foundation
Foundation of
of Korea (NRF)
(NRF) grant
grant
Acknowledgments:
Acknowledgments:
This work
was supported
by the National
Research Foundation
of KoreaKorea
(NRF) grant funded
funded
by
the
Korea
government
(MSIP)
(No.
2016R1A2B1008953).
This
work
was
also
supported
by
the
by the
Korea
(MSIP) (No.
2016R1A2B1008953).
This work
was
alsowas
supported
by the research
funded
bygovernment
the Korea government
(MSIP)
(No. 2016R1A2B1008953).
This
work
also supported
by the
research
programof
ofDongguk
DonggukUniversity,
University,2016.
2016.
program
of Dongguk
University,
2016.
research
program
Author
Contributions:
SeungyeobChae
Chaeand
andMinjoong
Minjoong
Rim
conceived
andand
designed
thestudy.
study.
Seungyeob
Chae
Author
Contributions:
Seungyeob
MinjoongRim
Rim
conceived
designed
the study.
Seungyeob
Author
Contributions:
Seungyeob
Chae
and
conceived
and
designed
the
Seungyeob
Chae
Chae
and
Minjoong
Rim
conceived
and
designed
the
simulations.
Seungyeob
Chae
performed
the
simulations;
and
Minjoong
Rim
conceived
and
designed
the
simulations.
Seungyeob
Chae
performed
the
simulations;
and Minjoong Rim conceived and designed the simulations. Seungyeob Chae performed the simulations;
Minjoong
RimRim
analyzed
thethe
data.
Seungyeob
authors
have
read and
and approved
approvedthe
the
Minjoong
analyzed
data.
SeungyeobChae
Chaewrote
wrotethe
thepaper.
paper.All
All authors
authors have
have read
read
Minjoong
Rim analyzed
the data.
Seungyeob
Chae
wrote
the
paper.
All
and approved
the
finalfinal
manuscript.
manuscript.
final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Entropy 2017, 19, 179
9 of 9
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