Optimization of Packet Length for MIMO systems

Optimization of Packet Length for MIMO systems
Ghassan Alnwaimi1
, Hatem Boujemaa2
, Kamran Arshad3
1
King Abdulaziz University, Kingdom of Saudi Arabia
2
Sup’Com, COSIM Laboratory, Tunisia
3
Ajman University, United Arab Emirates
1 2
3
Abstract
In this article, a method to enhance the throughput for Multiple Input Multiple Output
(MIMO) systems by optimizing packet length is proposed. Two adaptation algorithms are
proposed. In the first algorithm, we use the Average Signal to Noise Ratio (ASNR) to choose
the optimal packet length and Modulation and Coding Scheme (MCS) in order to maximize
the throughput. In the second algorithm, packet length and MCS are adapted with respect
to the Instantaneous received SNR (ISNR). This article concludes that the variable packet
length gives up to 1.8 dB gain with respect to the Fixed Packet Length (FPL).
Index Terms : MIMO systems, Optimal packet length, Rayleigh fading channels.
1 Introduction
Generally, MIMO systems offer larger throughput than Single Input Single Output (SISO) sys-
tems [1-5]. Multiple antennas are used to benefit from transmit/receive diversity and providing
enhanced system performance. For instance, in wireless systems, Space Time Coding (STC) is
used at the transmitter benefiting from transmit diversity. Accordingly, at the receiver, Space
Time Decoding (STD) and Maximum Ratio Combining (MRC) can be used to benefit from
spatial diversity and providing higher data rates [1-5].
Channel estimation for MIMO systems has been investigated in the literature e.g. [1]. Power
allocation was studied in [2-3] to improve the overall system performance. Scheduling for MIMO
systems was considered in [4] and full duplex MIMO was studied in [5-6]. Antenna selection
allows to benefit from spatial diversity [7-8]. The antenna with the highest SNR is selected in
Selection Combining (SC) to provide higher capacity gains. The theoretical capacity of MIMO
systems was derived in [9-10]. A combination of spatial and cooperative diversity was proposed
in [11-14]. Relay nodes are used to amplify or decode the source signal. The destination combines
signals from the source and other relays. Multihop relaying for MIMO systems was suggested
in [13]. The security aspect of MIMO systems was analyzed in [15] in the presence of an eaves-
dropper. MIMO systems can also be deployed for Underwater Visible Light Communications
(UVLC) [16].
In the existing literature [1-16], the packet length is kept fixed. In this paper, we propose
methods to enhance the system throughput of MIMO systems by optimizing packet length. The
major contributions of the article are as follows :
1
International Journal of Computer Networks & Communications (IJCNC) Vol.11, No.1, January 2019

• An algorithm is proposed in this article to compute the Optimal Packet Length (OPL)
that maximizes the average throughput of the MIMO system.
• An algorithm is proposed to calculate the OPL that maximizes the instantaneous through-
put of MIMO system.
• The two adaptation algorithms are also compared in this article. The ﬁrst one uses the
ASNR to optimize packet length and the MCS. The second algorithm uses ISNR to opti-
mize packet length as well as the MCS.
The remainder of this article is organized as follows; Preliminary results on Packet Error
Probability (PEP) are given in section 2. Section 3 proposes an algorithm to compute the OPL
using average SNR. Section 4 derives the OPL using the instantaneous SNR. Numerical results
are provided in section 5 whereas section 6 concludes the paper.
2 Preliminary Results
In this section, we derive the expression of the average Packet Error Probability (PEP) for single
carrier systems. The PEP can be upper bounded by [17]
PEP ≤
∫ w0
0
fΓ(γ)dγ (1)
where fΓ(γ) is the Probability Density Function (PDF) of instantaneous SNR Γ and w0 is a
waterfall threshold.
Equation (1) shows that the PEP for γ ≤ w0, can be approximated to 1. However, the PEP
for γ > w0 can be approximated to 0 [17].
Hence,
PEP ≤ FΓ(w0), (2)
where FΓ(x) is the Cumulative Distribution Function (CDF) of the received SNR. We denote
Γ = Eb
N0
as the average SNR, where Eb is the transmitted energy per bit, N0 is the noise Power
Spectral Density (PSD) and w0 is a waterfall threshold written as [17],
w0 =
∫ +∞
0
g(γ)dγ, (3)
where g(γ) is the PEP for a given instantaneous SNR, γ = Γ|h|2 and h is the channel coeﬃcient.
2.1 Packet Error Probability (PEP) without error correction
For uncoded transmission and QAM modulation with size M, we have
g(γ) = 1 − (1 − Pes(γ))
N+nd
log2(M) , (4)
where N is packet length in bits per, nd is the number of CRC (Cyclic Redundancy Check)
bits per packet and Pes is the Symbol Error Probability (SEP) given as, [18]
Pes(γ) ≃ 2(1 −
1
√
M
)erfc
(√
log2(M)3γ
(M − 1)2
)
. (5)
erfc(x) ≤ exp(−x2
) (6)
2

Using (5) and (6), we have
Pes ≃ a1e−c1γ
(7)
where,
a1 = 2
(
1 −
1
√
M
)
, (8)
c1 =
3 log2(M)
2(M − 1)
(9)
2.2 PEP with error correction
If convolutional encoding is used, the PEP is written as
g(γ) = 1 − (1 − PE(γ))
N+nd
log2(M) , (10)
where
PE(γ) ≤
+∞∑
d=df
adPd(γ) (11)
df and ad are distance spectral of the convolutional encoder,
Pd(γ) ≃ 2
(
1 −
1
√
M
)
erfc
(√
3Rcdγ log2(M)
2(M − 1)
)
. (12)
where Rc is channel encoder rate.
Using (6), we have
PE(γ) ≃ a2e−c2γ
(13)
where
a2 = adf
2
(
1 −
1
√
M
)
, (14)
c2 =
3Rcdf log2(M)
2(M − 1)
. (15)
Hence, we can write g(γ) as follow,
g(γ) ≃ 1 − (1 − aie−ciγ
)
N+nd
log2(M) , (16)
where i = 1 for uncoded communications and i = 2 for coded communications.
3

2.3 Waterfall Threshold
Using (3) and (16), one can approximate waterfall threshold as follows [19],
w0 ≃ k1 ln
(
N + nd
log2(M)
)
+ k2 (17)
where,
k1 =
1
ci
, (18)
k2 =
E + ln(ai)
ci
, (19)
and E ≃ 0.577 is the Euler constant.
3 OPL using ASNR
In this section, we derive OPL for MIMO systems. Consider a system with one transmitting and
two receiving antennas. It is assumed that Space Time Coding (STC) is used at the transmitter
and Space Time Decoding (STD) is used at the receiver. The PEP can be written as follows
(see Appendix A for Proof),
PEP(2) = 1 − e−
w0
Γ
[
1 +
w0
Γ
]
(20)
where PEP(2) refers to the case of two receiving antennas.
Similarly, when there is one transmitting and three receiving antennas, we show in Appendix
B that the PEP can be written as follows,
PEP(3) = 1 − e−
w0
Γ
[
1 +
w0
Γ
+
1
2
(
w0
Γ
)2
]
(21)
When there are one transmitting and n receiving antennas, we show in appendix C that the
PEP is equal to
PEP(n) = 1 − e−
w0
Γ
n−1∑
i=0
1
i!
(
w0
Γ
)i
(22)
Hence, for the case of nt transmitting and nr receiving antennas, the SNR per bit can be
expressed as follows,
γ =
Eb
ntN0
nt∑
i=1
nr∑
j=1
γj,i (23)
where γj,i is the instantaneous SNR between the i-th transmitting antenna and j-th receiving
antenna.
Using the results of Appendix C, the PEP when there are nt transmitting and nr receiving
antennas can be written as follows,
PEP(ntnr) = 1 − e−
w0
Γ
ntnr−1∑
i=0
1
i!
(
w0
Γ
)i
= 1 − Γ
(
w0
Γ
, ntnr
)
(24)
4

where
Γ(x, n) =
1
(n − 1)!
∫ +∞
x
tn−1
e−t
dt. (25)
The average SNR (ASNR) per antenna is
Γ =
Eb
ntN0
(26)
The average number of attempts of Hybrid Automatic Repeat reQuest (HARQ) protocol is
equal to
Tr =
+∞∑
i=1
PEPi−1
(1 − PEP) =
1
1 − PEP
(27)
Therefore, the system throughput is expressed as follows,
Thr =
N log2(M)
(N + nd)Tr
=
N log2(M)
(N + nd)
(1 − PEP)
=
N log2(M)
(N + nd)
Γ
(
w0
Γ
, ntnr
)
(28)
The OPL can be obtained using the Gradient algorithm:
Ni+1 = Ni + µ
∂Thr(N = Ni)
∂N
(29)
where
∂Thr
∂N
=
log2(M)nd
(N + nd)2
Γ
(
w0
Γ
, ntnr
)
−
log2(M)N
(N + nd)(ntnr − 1)!
(
w0
Γ
)ntnr−1
e−
w0
Γ
k1
(N + nd)Γ
(30)
The principle of the OPL using average SNR is shown in Fig. 1.
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8*:"&(;&1<1#1=(
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Figure 1: OPL using Average SNR.
4 OPL using ISNR
4.1 Adaptive Packet Length without Channel Coding
The instantaneous throughput can be expressed as
5

Thr(γ) =
N log2(M)
N + nd
(1 − Pbloc(γ))
=
N log2(M)
N + nd
(1 − PesM−QAM (γ))
N+nd
log2(M) , (31)
where Pbloc(γ) is the packet error probability, γ is the instantaneous SNR at the receiver
expressed as
γ =
Eb
ntN0
nt∑
i=1
nr∑
j=1
γj,i (32)
PesM−QAM is the symbol error probability of M-QAM modulation:
1 − PesM−QAM (γ) = (1 − PesASK)2
, (33)
where
PesASK(γ) =
(
1 −
1
√
M
)
erfc
(√
γ
3 log2(M)
2(M − 1)
)
. (34)
By maximizing the throughput (31), the OPL is given by
N(γ) =
nd
2
[√
1 −
2 log2(M)
nd ln(1 − PesASK(γ))
− 1
]
(35)
4.2 Adaptive Packet Length with Channel Coding
In this section, a convolutional encoder is employed and similar methodology can be used to
derive OPL for other types of channel coding. The throughput can be written as,
ThrQPSK,Rc(γ) =
Rc2N
(N + nd)
(1 − Pbloc(γ))
=
Rc2N
N + nd
(1 − PE(γ))
N+nd
2 (36)
where
PE(γ) < min
(
1,
+∞∑
d=df
adQ
(√
2dRcγ
) )
(37)
When convolutional coding is employed, the OPL can be written as follows
N(γ) =
nd
2
[√
1 −
8
nd ln(1 − PE(γ))
− 1
]
(38)
The expression of OPL is obtained by maximizing the throughput given in (36).
6

4.3 OPL and optimal MCS
From Fig. 2, it can be deduced easily that different threshold should be used to select different
MCS:
• Case 1 : for ( γ ≤ F1 = 6.3dB ) → use a coded QPSK modulation with rate Rc = 1/2..
• Case 2 : for ( F1 < γ ≤ F2 = 10dB ) → use uncoded QPSK.
• Case 3 : for ( F2 < γ ≤ F3 = 15.2dB ) → use uncoded 16-QAM.
• Case 4 : for ( F3 < γ ) → use uncoded 64 QAM.
Packet length is given in (38) or (35).
Thresholds Fi are defined in Figure 2.
−5 0 5 10 15 20 25 30
0
1
2
3
4
5
6
γ : Instantaneous SNR
InstantaneousThroughputbit/s/Hz
Optimal Packet Length
MCS1 : QPSK, R
c
=0.5
MCS2 : QPSK, R
c
=1
MCS3 : 16QAM, Rc
=1
MCS4 : 64QAM, Rc
=1
F1
F
2
F3
Figure 2: Thresholds to select MCS using instantaneous SNR : OPL.
When the MCS is adapted for N = 590 and nd = 10, Fig. 3 shows that we have to use other
thresholds : Z1 = 5.4 dB,Z2 = 9 dB and Z3 = 13.8 dB.
7

−5 0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
4
γ : Instantaneous SNR
InstantaneousThroughputbit/s/Hz
Fixed Packet Length N=590
MCS1 : QPSK, Rc
=0.5
MCS2 : QPSK, R
c
=1
MCS3 : 16QAM, R
c
=1
MCS4 : 64QAM, Rc
=1
Z
1
Z
2
Z3
Figure 3: Thresholds to select MCS using instantaneous SNR : FPL.
Fig. 4 shows the principle of the proposed packet length and MCS adaptation using ISNR.
The ISNR is used to select the optimal MCS and OPL as shown in (35) and (38).
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Figure 4: OPL and Adaptive MCS using Instantaneous SNR.
For a given average SNR, Γ = Eb
ntN0
, the average throughput is written as
8

Thr(Γ) =
∫ F1
0
ThrQPSK,Rc(γ)p(γ)dγ
+
∫ F2
F1
ThrQPSK(γ)p(γ)dγ
+
∫ F3
F2
Thr16−QAM (γ)p(γ)dγ
+
∫ +∞
F3
Thr64−QAM (γ)p(γ)dγ, (39)
where p(γ) is the PDF of the instantaneous SNR γ. For Rayleigh fading channel, the SNR
in (23) follows a central chi-square distribution with 2ntnr degrees of freedom given by
p(γ) =
1
σntnr 2ntnr/2(ntnr/2 − 1)!
γntnr/2−1
e− γ
2σ2 , γ ≥ 0, (40)
where σ2 = Eb
2N0nt
.
Equation (39) gives the throughput for OPL when MCS is also adapted to ISNR. The
expressions of throughput given in (31) and (35) must be used with packet length N(γ) as given
in (35) and (38) that depends on instantaneous SNR γ. When MCS is adapted for a FPL
N(γ) = N, we have to use (39) with diﬀerent thresholds Z1, Z2 and Z3.
5 Theoretical and Simulation Results
Simulation results were obtained using MATLAB as the simulation environment.
Simulation results were performed by measuring the Packet Error Rate (PER) to deduce the
throughput. The packet error rate is the number of erroneous packets/number of transmitted
packets. We made simulation until 1000 packets are erroneously received.
Fig. 5 shows the average throughput of MIMO systems with respect to packet length for
QPSK modulation and for average SNR per bit of Eb/N0 = 6 dB. We studied SISO systems
nt = nr = 1 as well as MIMO for (nt, nr) = (1, 2), (nt, nr) = (2, 2), (nt, nr) = (2, 3), (nt, nr) =
(3, 3). Space Time Coding (STC) is used at the transmitter and Space Time Decoding (STD)
is used at the receiver in our simulation. It is concluded in this article that the packet length
optimization allows increasing data rates for MIMO systems. Furthermore, the packet length is
also dependent on the number of transmitting and receiving antenna; packet length should be
increased as the number of transmitting and receiving antennas increase.
9

0 100 200 300 400 500 600 700 800 900 1000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Packet Length N
AverageThroughputbit/s/Hz
E
b
/N
0
=6 dB, QPSK
SISO
MIMO n
t
=1, n
r
=2
MIMO n
t
=2, n
r
=2
MIMO n
t
=2, n
r
=3
MIMO n
t
=3, n
r
=3
Figure 5: Throughput of MIMO systems with respect to packet length.
Fig. 6 shows the OPL for QPSK modulation and diﬀerent MIMO systems. We notice that
packet length increases with Average SNR (ASNR). Packet length must be increased as the
number of transmitting and receiving antennas increase.
10

0 1 2 3 4 5 6 7 8 9 10
0
500
1000
1500
2000
2500
3000
3500
4000
Average SNR : E
b
/N
0
(dB)
Optimalpacketlength
SISO
MIMO nt
=1, nr
=2
MIMO nt
=2, nr
=2
MIMO nt
=2, nr
=3
Figure 6: OPL for MIMO systems.
Fig. 7 shows the throughput of MIMO systems using OPL for nt = 2 transmitting and
nr = 2 receiving antennas. Diﬀerent Modulation and Coding Schemes (MCS) are studied :
QPSK with convolutional channel coding and rate Rc = 0.5, generator polynomials (133,171),
memory 6 and free distance 10. uncoded QPSK, 16 QAM and 64 QAM. From Fig. 7, we can
note that the following strategy should be employed :
• Case 1 : for ( Γ ≤ S1 = 2.3dB ) → We should use a coded QPSK modulation with rate
Rc = 0.5.
• Case 2 : for ( S1 < Γ ≤ S2 = 6.8dB ) → We should use an uncoded QPSK modulation.
• Case 3 : for ( S2 < Γ ≤ S3 = 12.6dB ) → We should use an uncoded 16 QAM modulation.
• Case 4 : for ( S3 < Γ ) → We should use an uncoded 64 QAM modulation.
Thresholds Si are deﬁned in Fig. 7.
11

0 5 10 15 20 25 30
0
1
2
3
4
5
6
Average SNR : Eb
/N0
(dB)
MIMO nt
=nr
=2, optimal packet length
S1
S2
S3
QPSK Rc
=1/2
QPSK Rc
=1
16 QAM Rc
=1
64 QAM Rc
=1
Figure 7: Throughput of MIMO systems for OPL and different modulations.
Fig. 8 shows the throughput of MIMO systems using FPL N=590 for nt = 2 transmitting
and nr = 2 receiving antennas. Different Modulation and Coding Schemes (MCS) are studied :
QPSK with convolutional channel coding and rate Rc = 0.5, generator polynomials (133,171),
memory 6 and free distance 10. uncoded QPSK, 16 QAM and 64 QAM. From Fig. 8, it can be
noted that the following strategy should be employed :
• Case 1 : for ( Γ ≤ D1 = 4.7dB ) → We should use a coded QPSK modulation with rate
Rc = 0.5.
• Case 2 : for ( D1 < Γ ≤ D2 = 8.9dB ) → We should use an uncoded QPSK modulation.
• Case 3 : for ( D2 < Γ ≤ D3 = 14.3dB ) → We should use an uncoded 16 QAM modulation.
• Case 4 : for ( D3 < Γ ) → We should use an uncoded 64 QAM modulation.
Thresholds Di are defined in Fig. 8.
12

0 5 10 15 20 25 30
0
1
2
3
4
5
6
Average SNR : Eb
/N0
(dB)
MIMO nt
=nr
=2, Fixed Packet Length N=590
D1
D2
D3
QPSK Rc
=1/2
QPSK R
c
=1
16 QAM R
c
=1
64 QAM Rc
=1
Figure 8: Throughput of MIMO systems for FPL and different modulations.
Fig. 9 shows the average throughput of MIMO systems with two transmitting and two
receiving antennas. In Fig. 9, we compare the two approaches: the first one consists to optimize
the MCS using average SNR with FPL, N = 590. The second approach is to optimize the MCS
and packet length using the average SNR. The proposed strategy offers 2 dB gains with respect
to FPL for a throughput of 4 bit/s/Hz.
13

0 5 10 15 20 25 30
0
1
2
3
4
5
6
Average SNR : E
b
/N
0
(dB)
Fixed Packet Length N=590 : AMC using ASNR
Optimal packet length : AMC using ASNR
Figure 9: Throughput of MIMO systems with AMC using ASNR : ﬁxed versus OPL.
Fig. 10 shows the throughput for two transmitting and two receiving antennas nt = nr = 2:
We studied:
- Optimal packet length (OPL) and MCS using ASNR
- OPL and MCS using ISNR
- Optimal MCS using ASNR with Fixed Packet Length (FPL).
- Optimal MCS using ISNR with FPL.
First, we noticed that the best strategy is to use the OPL and MCS using ISNR as explained
in section 4. The proposed protocol oﬀers 1.8 dB gain with respect to optimal MCS and FPL
as considered in [1-16].
14

0 5 10 15 20 25 30
0
1
2
3
4
5
6
Average SNR : E
b
/N
0
(dB)
MIMO nt
=nr
=2
Optimal packet length : AMC using ASNR
Adaptive packet Length and Adaptive MCS using ISNR
Fixed packet length and adaptive MCS using INSR
Figure 10: Throughput of MIMO systems with AMC using ASNR or ISNR : ﬁxed versus OPL.
6 Conclusion
In this article, we have proposed strategies to enhance the throughput for MIMO systems by
optimizing packet length. Two adaptation algorithms are proposed and evaluated in this article.
In the ﬁrst one, for each received ASNR, the OPL and optimal Modulation and Coding Scheme
(MCS) can be selected. Therefore, packet length adaptation is performed with respect to ASNR.
In the second approach, the adaptation is made with respect to ISNR. The proposed OPL allows
1.8 dB gains with respect to FPL.
Appendix A :
When there are two receiving antennas, the PEP can be upper bounded by
PEP(2) ≤
∫ ∫
x1+x2≤w0
fΓ(x1)fΓ(x2)dx1dx2, (41)
where fΓ(x) is the PDF of SNR Γ.
For Rayleigh fading channels, the SNR follows an exponential distribution so that we can
write
PEP(2) ≤
∫ w0
0
e−
x1
Γ
Γ
∫ w0−x1
0
e−
x2
Γ
Γ
dx2dx1 = 1 − e−
w0
Γ
[
1 +
w0
Γ
]
(42)
Appendix B :
When there are three receiving antennas, the PEP can be upper bounded by
PEP(3) ≤
∫ ∫ ∫
x1+x2+x3≤w0
fΓ(x1)fΓ(x2)fΓ(x3)dx1dx2dx3, (43)
15

For Rayleigh fading channels, we can write
PEP(3) ≤
∫ w0
0
e−
x1
Γ
Γ
∫ w0−x1
0
e−
x2
Γ
Γ
∫ w0−x1−x2
0
e−
x3
Γ
Γ
dx3dx2dx1 (44)
We deduce that
PEP(3) ≤
∫ w0
0
e−
x1
Γ
Γ
∫ w0−x1
0
e−
x2
Γ
Γ
[
1 − e
x1+x2−w0
Γ
]
dx2dx1 (45)
Therefore, we have
PEP(3) ≤
∫ w0
0
e−
x1
Γ
Γ
[
1 − e−
x1−w0
Γ +
x1 − w0
Γ
e
x1−w0
Γ
]
dx1 (46)
Finally, we have
PEP(3) ≤ 1 − e−
w0
Γ
[
1 +
w0
Γ
+
w2
0
2Γ
2
]
(47)
Appendix C:
When there are n receiving antennas, the PEP can be written as
PEP(n) ≤
∫ ∫
...
∫
x1+...+xn≤w0
fΓ(x1)fΓ(x2)...fΓ(xn)dx1dx2...dxn, (48)
We deduce that
PEP(n) ≤
∫ w0
0
e−
x1
Γ
Γ
∫ w0−x1
0
e−
x2
Γ
Γ
...
∫ w0−
∑n−1
i=1 xi
0
e−xn
Γ
Γ
dxndxn−1...dx1 (49)
and
PEP(n) ≤
∫ w0
0
e−
x1
Γ
Γ
∫ w0−x1
0
e−
x2
Γ
Γ
...
∫ w0−
∑n−2
i=1 xi
0
[
1 − e
∑n−1
i=1
xi−w0
Γ
]
e−
xn−1
Γ
Γ
dxn−1...dx1 (50)
We compute the last integral to obtain
PEP(n) ≤
∫ w0
0
e−
x1
Γ
Γ
∫ w0−x1
0
e−
x2
Γ
Γ
... (51)
∫ w0−
∑n−3
i=1 xi
0
[
1 − e
∑n−2
i=1
xi−w0
Γ (1 −
∑n−2
i=1 xi − w0
Γ
)
]
e−
xn−2
Γ
Γ
dxn−2...dx1
The ﬁrst two terms of (51) are computed similarly to (50) to have
PEP(n) ≤
∫ w0
0
e−
x1
Γ
Γ
∫ w0−x1
0
e−
x2
Γ
Γ
... (52)
∫ w0−
∑n−4
i=1 xi
0
[
1 − e
∑n−3
i=1
xi−w0
Γ (1 −
∑n−3
i=1 xi − w0
Γ
+
(
∑n−3
i=1 xi − w0)2
2Γ
2 )
]
e−
xn−3
Γ
Γ
dxn−3......dx1
16

The last integral is as follows
PEP(n) ≤
∫ w0
0
e−
x1
Γ
Γ
[
1 − e
x1−w0
Γ (1 −
x1 − w0
Γ
+
(x1 − w0)2
2Γ
2 + ... + (−1)n (x1 − w0)n
n!Γ
n )
]
dx1
(53)
We finally obtain
PEP(n) ≤ 1 − e
−w0
Γ
n∑
i=0
1
i!
(
w0
Γ
)i
(54)
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