DOI: 10.1002/ett.2719 RESEARCH ARTICLE Exact outage analysis of underlay cooperative cognitive networks with maximum transmit-and-interference power constraints and erroneous channel inf
Trang 1Trans Emerging Tel Tech (2013) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/ett.2719
RESEARCH ARTICLE
Exact outage analysis of underlay cooperative
cognitive networks with maximum
transmit-and-interference power constraints and
erroneous channel information
Khuong Ho-Van*
HoChiMinh City University of Technology, 268 Ly Thuong Kiet street, District 10, HoChiMinh City, Vietnam
ABSTRACT
This paper presents exact analysis of interference probability and outage probability of underlay cooperative cognitive net-works under quite general conditions such as imperfect channel information, maximum transmit-and-interference power constraints, correlation among received signal-to-noise ratios and non identically distributed fading channels In addition, asymptotic outage analysis at either large maximum transmit power or large maximum interference power is proposed
to have useful insights into performance limits of underlay cooperative cognitive networks Analytical results reveal that underlay cooperative cognitive networks dramatically suffer outage saturation phenomenon, and saturation degree signifi-cantly depends on channel estimation quality Moreover, the performance of both licenced network in terms of interference probability and unlicenced network in terms of outage probability is considerably deteriorated by channel estimation error Furthermore, optimum relay position is dependent of several factors such as maximum transmit power, maximum interference power, licensee position and channel estimation quality Copyright © 2013 John Wiley & Sons, Ltd
*Correspondence
Khuong Ho-Van, HoChiMinh City University of Technology, 268 Ly Thuong Kiet street, District 10, HoChiMinh City, Vietnam E-mail: khuong.hovan@yahoo.ca
Received 28 June 2013; Revised 11 August 2013; Accepted 4 September 2013
1 INTRODUCTION
Underlay decode-and-forward (DF*) cooperative†
cogni-tive network is a feasible-and-efficient solution to
prob-lems of spectrum scarcity, spectrum under-utilisation
and coverage extension [1–4] For system design
opti-mization such as power allocation optiopti-mization, channel
information must be available However, it is almost
impossibly expected to have full knowledge of channel
information from existing channel estimation algorithms
As such, the investigation on the impact of imperfect
channel information on the system performance is
nec-essary On the other hand, outage probability is a useful
metric in providing insights into the information-theoretic
* Popularly, the relay in most cooperative relaying schemes operates in
either DF or amplify-and-forward type [33, 34].
† Multi-hop communication (e.g [2], [6]) differs from cooperative
relaying (e.g [7, 8]) in that the former bypasses the direct
chan-nel between the source and the destination while for enhanced space
diversity, the latter does.
performance limit [5] As a result, a general outage analysis framework for underlay DF cooperative cogni-tive networks, accounting for multiple practical operation conditions such as imperfect channel information on all channels, two maximum transmit-and-interference power constraints, non-identically distributed (i.d.) fading chan-nels and the correlation among received signal-to-noise ratios (SNRs) is urgent and essential Nevertheless, current publications have not considered all these conditions con-currently For instances, the authors in [2–4, 6–8], [9–32] partially investigated them
More specifically, the authors in [3, 7, 8], [16–20],
[22–25] analysed the outage performance of underlay DF
cooperative cognitive networks‡in different aspects under
the assumption of perfect channel information: (i) about
‡ This paper studies underlay DF cooperative cognitive networks, and hence, the works on other modes (e.g the interweave mode in [31]), error probability analysis (e.g [4], [27–30]), the amplify-and-forward type (e.g [14, 32]), and multi-hop communications (e.g [2, 6], [9–15], [21]) are not necessarily surveyed.
Trang 2power constraints, the authors in [3, 7, 17, 20], [22–25]
solved a more general problem than [8,16,18,19] by
study-ing both constraints (interference power constraint and
maximum transmit power constraint) while [8, 16, 18, 19]
aim to deal with only the interference power constraint; (ii)
about the correlation among received SNRs, the authors in
[3, 7], [16–20], [22–25] investigated this problem, hence
obtaining more precise results than [8] which assumes
independent SNRs; (iii) about the assumption on i.d
fad-ing channels, only [3, 8], [16–20], [22, 23, 25] relaxes it
Additionally, under consideration of channel estimation
error, the authors in [13, 14, 26] proposed outage
proba-bility expressions for different scenarios, but among them,
only [26] is really related to our work on underlay DF
cooperative cognitive networks because both [13, 14]
stud-ied two-hop communications The drawbacks of [26] are
assumptions of the perfect channel knowledge among the
unlicenced network but the imperfect channel knowledge
between the licenced network and the unlicenced network,
and only the interference power constraint.
To the best of our knowledge, the exact outage analysis
of underlay DF cooperative cognitive networks under all
aforementioned operation conditions simultaneously has
been left open In this paper, we will solve this problem
Towards this end, we derive the exact closed-form
out-age probability expression Because of the channel
esti-mation error, the derivation is significantly complicated
even though some initial derivation steps can be
bor-rowed from those in the case of perfect channel
estima-tion (e.g [17]) Of course, the derived expression includes
some previous works such as [7, 16, 17, 26] as special
cases The most advantage of the derived expression is to
facilitate in investigating the performance behaviour in
dif-ferent system parameters without time-consuming
simula-tions as well as in optimising system design Likewise, we
perform asymptotic analysis to have more insights into
system performance limits such as the performance
satu-ration of underlay cooperative cognitive networks at large
maximum interference power or large maximum
trans-mit power Also, the asymptotic analysis can be used to
deduce the outage performance of traditional cooperative
networks [33] and underlay cooperative cognitive networks
with only interference power constraint [16] Moreover,
we derive an exact closed-form interference probability
expression, where the interference probability is defined
as the probability that the interference power constraint is
invalid, to further investigate the effect of channel
estima-tion error on licenced networks All derived expressions
are validated through extensive comparisons with
com-puter simulation results Notably, analytical results show
that underlay cooperative cognitive networks considerably
suffer the outage saturation phenomenon, and the
satu-ration level is drastically dependent of channel
estima-tion error Addiestima-tionally, the performance of both licenced
network in terms of interference probability and
unli-cenced network in terms of outage probability is
signif-icantly degraded by channel estimation quality
Further-more, the derived expressions can be applied to search
optimum relay positions, which are shown to be dependent
of several factors such as maximum transmit power, max-imum interference power, licensee position and channel estimation error
We start with the system model in Section 2 Then,
we provide the elaborate derivation of exact and asymp-totic outage probability expressions in Sections 3 and 4, respectively Next, the interference probability analysis is discussed in Section 5 Simulations used to validate the proposed expressions are presented in Section 6 Also, this section demonstrates numerous analytical and simu-lation results to have insights into the system performance Finally, conclusions close this paper in Section 7, and some complex derivations are deferred to appendices
2 SYSTEM MODEL
We study a typical underlay DF cooperative cognitive net-work in Figure 1 (e.g [18]) In the unlicenced netnet-work, information transmission from the source to the destination
is assisted by the relay in two stages In the first stage, both the relay and the destination receive the signal from the source Upon successfully decoding the source signal, the relay will re-encode the decoded information before for-warding it to the destination in the next stage Then, the source information is restored at the destination by maxi-mum ratio combining both signals from the source and the relay In case of unsuccessfully decoding the source signal, the relay stands by in the second stage Then, the desti-nation restores the source information based on only the signal received from the source in the first stage
We denote qmnas the channel coefficient between the transmitter m and the receiver n with m 2 fS ; Rg and
n 2 fR; D; Lg Under the assumption of independent frequency-flat Rayleigh fading, qmn CN 0; mn D
dmn/ where q CN k; l/ stands for a circular symmetric
S
D
R
L
Stage 1
Stage 2
source
destination
relay
licensed receiver
unlicensed network
licensed network
RL
q
SR
q
SD
q
SL
q
RD
q
Tx
licensed transmitter
Figure 1 System model.
Trang 3complex Gaussian random variable with mean k and
vari-ance l; dmnand parameters denote the distance between
two users and the involved path-loss exponent, respectively
[35] We investigate non-i.d fading channels, and hence,
all mnare not necessarily identical as assumed in [7, 24]
The equivalent complex baseband model for any
chan-nel between the unlicenced transmitter m 2 fS ; Rg and the
unlicenced receiver n 2 fR; Dg can be expressed as
where ymn is the received signal, mn CN 0; N0/ is
the noise§at the receiver n and xmis the transmitted
sym-bol with the symsym-bol energyPm (i.e Efjxmj2g DPmin
which Efg represents the expectation) It should be noted
that more practically, mnat the unlicenced receiver
con-sists of two terms: (i) the actual noise at the receiver and
(ii) the interference from the licenced transmitters, [9–11],
[19, 20, 23] According to the central limit theorem [36],
the interference term becomes Gaussian distributed as long
as the number of interfering licensees is large enough In
underlay cognitive radio networks, unlicenced users
oper-ate in the opportunistic manner, and hence, they may be
interfered by a large number of licensees Consequently,
the assumption of Gaussian-distributed interference from
licensees can be valid in several practical scenarios and
is widely accepted and explored in most recent research
works [2–4, 6–8], [13–18], [21, 22], [24–31] Under this
assumption, N0 is considered as the total variance of
the noise at the unlicensee and the interference from the
licenced transmitters¶ Furthermore, in the underlay mode
(e.g [2], [7], [17]), the selection ofPmmust strictly meet
both interference power constraint, Pm IT
jq mL j2, and maximum transmit power constraint, Pm Pt, with
IT being the maximum interference power tolerated by
the licensee andPt being the maximum transmit power
designed for the unlicensees For the maximum coverage,
we setPmD min
IT
jqmLj2;Pt
As analysed thoroughly
in [2–4, 6–8], [9–31],IT stands implicitly for the
interfer-ence limit from unlicensees and excludes interferinterfer-ence
gen-erated by licensees Equivalently, the licenced networks are
implicitly assumed to operate reliably for interference
lev-els caused by unlicensees up toIT, regardless of the
inter-ference already existing in these networks In other words,
licensee-to-licensee interference has not been necessarily
considered when formulating Pm Also, in order to set
the transmit power of unlicensees, the channel coefficient
qmLmust be available at the unlicensees Although
obtain-ing channel state information (CSI) at a certain degree of
accuracy is almost feasible and elaborately discussed in
most works on the cognitive radio technology, for example
§ Without loss of generality, the noise at unlicenced receivers is
nor-malised to have the same variance.
¶ The case of non-Gaussian interference from licenced transmitters is
left to future work.
[2, 23], the channel estimation error is inevitably unavoid-able Therefore, the investigation of its effect on the sys-tem performance is essential The channel estimation error model widely accepted in most literature can be expressed
as [37–40]
where bqmn is the estimate of the m n channel and „mn CN 0; ˇmn/ stands for channel estima-tion error The value of ˇmn reveals the reliability of the channel estimator For instance, given the linear-minimum-mean-square-error estimator in [37], the vari-ance of the channel estimation error is expressed as
NpPm;pi lotmn=N0C 1
where Np is the number of pilot symbols and Pm;pi lot is the pilot power Furthermore, for the channel information imper-fection model in Equation (2), qmn andbqmnare jointly ergodic and stationary Gaussian processes, andbqmn
CN0;˛1
mn D mn ˇmn
Notably, without full knowledge of channel informa-tion, the transmitter m must adjust its power according to [41, Equation (2)], namely,
O
j OqmLj2;Pt
!
(3)
which results in the following interference power at the licensee as,
QmLD OPmjqmLj2D min IT
j OqmLj2;Pt
!
jqmLj2 (4)
Becausej OqmLj ¤ jqmLj, the interference power accord-ing to Equation (4) can not be guaranteed belowIT at all times, which can be proven particularly detrimental
to the operation of licensees As such, it is undoubtedly important to determine the percentage of the interference power constraint that is violated under imperfect channel information Section 5 will discuss this issue
3 EXACT OUTAGE PROBABILITY ANALYSIS
This section derives the exact closed-form outage probabil-ity expression for underlay DF cooperative cognitive net-works As mentioned in Section 1, the channel estimation error makes the derivation significantly complicated For-tunately, all involved integrals are expressed in the closed form, highlighting our contributions even though some ini-tial derivation steps can be borrowed from those in the case
of perfect channel estimation (e.g [17]) To this effect, by plugging Equation (2) in Equation (1), one obtains
ymnD OqmnxmC „mnxmC mn/ (5) which results in the received SNR at the unlicensee n as
Trang 4mnD Enj Oqmnxmj2o
Enj„mnxmC mnj2o
D min IT=zmL;Pt/ zmn
min IT=zmL;Pt/ ˇmnCN0
(6) where
Becausebqmn CN0;˛1
mn
, zmnis exponentially distributed with the probability density function (pdf):
fz mn.x/ D ˛mne˛mn x
(8) where x 0
The Shannon information theory states that given the
received SNR and the transmission rate of the
unli-cenced network R, the outage event happens if the
inequal-ity R 12log2.1 C / or v with v D 22R 1
holds Here, the factor of 12 before the logarithm is due
to the two-stage nature of the cooperative relaying scheme
(Figure 1) According to the principle of this scheme, there
are two events causing the outage at the destination The
first event corresponds to the case that both relay and
des-tination are in outage (i.e SR < v and SD < v) The
second event happens when the relay is not in outage (i.e
SR v) while the destination is (i.e SDC RD< v)
Therefore, the outage probability of underlay cooperative cognitive networks is addressed as
PoD Pr fSD< v; SR< vg
G1
C Pr fSDC RD< v; SR vg
G2
(9)
where PrfX g denotes the probability of the event X The next two subsections will present the derivation of
G1andG2in details
3.1 Derivation ofG1
Plugging SD and SR, both obeying the form in Equation (6), inG1, one obtains Equation (13) By substi-tuting fzS n.x/ with n 2 fR; D; Lg and performing some basic algebraic manipulations, one achieves the closed form of Equation (13) as Equation (14) where
EmnD ˛mn
ˇmnCN0
Pt
(10)
KmnD e
˛mnIT
GmnD˛mLIT
˛mnN0
(12)
G1D Pr
8
<
:
minI
T
z SL;Pt
zSD minI
T
zSL;Pt
ˇSDCN0
< v;
minI
T
z SL;Pt
zSR minI
T
zSL;Pt
ˇSRCN0
< v
9
=
; D
1 Z 0
0
@Z v
min
IT
x ;Pt
ˇSDCN0
= min
IT
x ;Pt
0
fzSD.y/ dy
1 A
0
@Z v
min IT
x ;Pt
ˇ SR CN0
= min IT
x ;Pt
0
fzSR.z/ dz
1
A fzSL.x/ dx
(13)
G1 D
1
Z
0
0 B
B 1 e
˛SD v
min
IT
x ;Pt
ˇSD CN0
min
IT
x ;Pt
1 C C
0 B
B 1 e
˛SRv
min
IT
x ;Pt
ˇSRCN0
min
IT
x ;Pt
1 C
C ˛ SL e ˛SLx dx
D
1
Z
1 e ˛SDv.ˇSDCN0 x=IT /
1 e ˛SRv.ˇSRCN0 x=IT /
˛ SL e ˛SLx dx
C
0
1 e ˛SDv.ˇSDCN0 =Pt /
1 e ˛SRv.ˇSRCN0 =Pt /
˛ SL e ˛SLx dx
D K SL 1 GSRe
ESRv
G SR C v
G SD e ESDv
G SD C v C
G SR G SD e .ESDCESR/v
G SR G SD C G SR C G SD / v
!
C 1 K SL /
1 e E SR v
1 e E SD v
(14)
Trang 53.2 Derivation ofG2
Inserting SR, SDand RD, all having the general form
in Equation (6), in G2, we reduceG2 to Equation (15)
where fRD.x/ is the pdf of RD Its closed form is given
in the following lemma:
G2D Pr
8
<
:
minI
T
zSL;Pt
zSD min
IT
zSL;Pt
ˇSDCN0
C RD< v;
minI
T
zSL;Pt
zSR min
IT
zSL;Pt
ˇSRCN0
v
9
=
;
D
v Z
0
2 6 6 4
1 R 0
R.vy/ min
IT
x ;Pt
ˇSDCN0
= min IT
x ;Pt
!
v min IT
x ;Pt
ˇ SR CN0
= min IT
x ;Pt
fzSR.w/ dw
!
fzSL.x/ dx
3 7 7
Lemma 1. Given theRDterm in Equation (6), its pdf
can be expressed by Equation (16) withERD,KRL and
GRDdefined in Equations (10), (11) and (12),
correspond-ingly.
f RD.x/ D
.1 KRL/ ERDCGRDKRLERD
GRDC x
.GRDC x/2
eERD x
(16)
Proof Please see the Appendix 1.
Likewise, after substituting fzS n.y/ into Equation (15)
with n 2 fR; D; Lg and performing some basic
manipula-tions, one reducesG2to
G2D
1 KSLCGSRKSL
GSRC v
eESR vFRD.v/
GSRGSDKSLe.ESD CESR/v
v
Z
0
eESD yf RD.y/
GSRGSDC GSRC GSD/ v GSRydy
G21
1 KSL/ e.ESD CESR/v
v Z 0
eESD yfRD.y/dy
G22
(17) where the FRD.x/ function is expressed by Equation (49)
in the Appendix 1
Evidently, the derivation of Equation (17) is subject to
analytic solutions for the integrals G21 and G22 Before
doing so, the derivation of some essential analytic results
is necessary
Lemma 2. The following integrals can be expressed
in closed-form as Equations (19), (20), (21), (22) and (23) where ln(x) is the natural logarithm of x,
Ei x/ D
1 s
x
e t
t dt is the exponential integral function
in [42, Equation (8.211.1)] and
‡ D GSRGSDC GSRC GSD/ v C GSRGRD (18)
f a; b; d ; k/ D
k Z 0
e ax
b C dxdx D
(
e ab=d
d
˚
E i a.bCd k/
d
E i ab
d ; a ¤ 0 1
d ln bCd k b
(19)
G211 D
v Z 0
e ESDERD/ y
G SR G SD C G SR C G SD / v G SR ydy
D f E SD E RD ; G SR G SD C G SR C G SD / v; G SR ; v/
(20)
Q
G211D v Z 0
e ESDERD/ y
G RD C y dy D f ESD E RD ; G RD ; 1; v/
(21)
G212 D
v Z 0
e E SD E RD / y
fG SR G SD C G SR C G SD / v G SR yg G RD C y/dy
DGSRG211 C QG211
(22)
G213 D
v Z 0
e E SD E RD / y
fG SR G SD C G SR C G SD / v G SR yg G RD C y/ 2 dy
D 1
GSRG211 2
G SR
C ESD E RD
Q
G211C 1
G RD
e.
E SD E RD / v
G RD C v
)
(23)
Proof Please see the Appendix 2.
Trang 6Likewise, by substituting Equation (16) into theG21
term in Equation (17), one obtains
G21D 1 KRL/ ERDG211C GRDKRLERDG212
whereG211;G212;G213are expressed in Equations (20),
(22) and (23), respectively
Similarly, by substituting Equation (16) into theG22
term in Equation (17), we expressG22 as Equation (25)
Importantly, the first integral ofG22 is straightforwardly
computed as Equation (26) while its third integral is
already expressed by Equation (51) in the Appendix 2 By
substituting the analytic solutions of these integrals into
Equation (25), one achieves the exact closed form ofG22
as Equation (27)
G22D 1 KRL/ ERD
v Z 0
e.ESD ERD/xdx
C GRDKRLERD
v Z 0
e.ESD ERD/x
Q
G211
C GRDKRL
v Z 0
e.ESD ERD/xdx GRDC x/2
(25)
v Z 0
e.ESD ERD/xdx D
(
e ESD ERD / v 1
(26)
G22D
8
<
:
.1 KRL/ ERDv C KRL v
ERD
e ESD ERD / v 1
.1K RL / 1 E SD E RD /C GRDKRL
1
GRDe.ESD ERDG /v
RD Cv
C GRDKRLESDGQ211 ; ESD¤ ERD
(27)
3.3 Special case of perfect
channel information
Our derived outage probability expression takes into
account quite practical conditions such as channel
informa-tion imperfecinforma-tion on all channels, maximum
transmit-and-interference power constraints, non-i.d fading channels,
and the correlation among received SNRs Therefore, it can
include some previous works such as [7, 16, 17, 26] as
spe-cial cases For example, in the case of the perfect channel
information (i.e ˇmnD 0), it is easily seen that Equations
(14) and (17) become [17, Equation (6)] and [17, Equation
(7)], respectively
4 ASYMPTOTIC OUTAGE PROBABILITY ANALYSIS
The asymptotic analysis for the outage probability is derived by considering two extreme scenarios as follows: (i) the large maximum transmit power and (ii) the large maximum interference power Based on this
For the specific scenario of large maximum transmit power, it follows thatPt ! 1 To this effect and recalling that Emn ! ˛mnˇmn and Kmn ! 1, Equations (14) and (17) become
G1D 1 GSRe
ESRv
GSDeESD v
GSDC v
C GSRGSDe
.ESDCESR/v
GSRGSDC GSRC GSD/ v (28)
G2DGSRe
ESRv
GRDeERD v
v C GRD
!
GRDERDG212C GRDG213
GSR1GSD1e.E SR CE SD /v (29) Notably, according to IT, the value of Po D
G1 C G2 remains constant This clearly indicates that the system performance saturates at large val-ues of Pt This performance saturation level is
dependent of Emnor equivalently, the channel esti-mation reliability
For the specific scenario of large maximum inter-ference power, it follows that IT ! 1 To this effect and recalling that Kmn! 0 and Gmn! 1, Equations (14) and (17) reduce to Equations (30) and (31), correspondingly As such, when conditioned on
Pt, the value of PoDG1CG2remains constant and therefore, the network suffers the error floor for large
IT Again, this error floor depends on Emn or the channel estimation error
Trang 71 eESR v
1 eESD v
(30)
G2D
8 ˆ ˆ
eESR v
1 eESD v ESDveESD v
; ESDD ERD
eESR v
1 eERD vERD.e ERD veESD v/
E SD E RD
; ESD¤ ERD
(31)
Briefly, the asymptotic analysis is important and
pro-vides the following useful insights for the performance of
underlay cooperative cognitive networks:
Underlay cooperative cognitive networks suffer the
performance saturation at either the large maximum
transmit power or the large maximum interference
power
The asymptotic outage performance corresponds to
the outage probability of the previous works (e.g
[16, 33]):
– Underlay cooperative cognitive networks with only
interference power constraint
This paper takes into account both
maxi-mum transmit power constraint and interference
power constraint in investigating the outage
perfor-mance of underlay cooperative cognitive networks
Some other works such as [16] just consider the
interference power constraint Based on our
asymp-totic analysis, it is obvious that the outage
probability expression for underlay cooperative
cognitive networks under only interference power
constraint is our asymptotic performance asPt!
1, and hence, the outage probability for this case
is computed by using Equations (28) and (29)
– Traditional cooperative systems with perfect CSI
It is straightforwardly seen that traditional
coop-erative systems with perfect CSI such as [33], are
underlay cooperative cognitive networks without
the interference power constraint and the channel
estimation error As such, the outage probability
for these systems can be computed with the help of
Equations (30) and (31), and by setting ˇmnD 0
5 INTERFERENCE
PROBABILITY ANALYSIS
As discussed previously, erroneous channel information
can cause the interference power at licensees to exceedIT
The probability that this event happens in underlay
coop-erative cognitive networks under consideration is defined
as the interference probability, PI According to the
two-stage nature of the cooperative relaying scheme (Figure 1),
there are two cases causing this event:
Case 1: In the first stage, the source adjusts the
trans-mit powerPS incorrectly and causes an interference
higher thanIT
Case 2: In the first stage, the source does not
inter-fere with the licensee Nevertheless, in the second
stage, the relay is active, adjusts its transmit power
PR wrongly and causes an interference higher than
IT
Therefore, based on the total probability law, PIis given by
PID Pr fQSL>ITg
C Pr fQSLIT;QRL>IT; SR vg
D Pr fQSL>ITg C Pr fQRL>ITg
Pr fQSLIT; SR vg
W
(32)
It should be emphasised here that Equation (32) is novel and completely different from [14, Equation (3)] Also,
QSLand SRare correlated because both containj OqSLj2 Therefore, the derivation ofW is cumbersome (please see
the Appendix 4 for the detailed derivation ofW) but does
not provide more insights into the performance limit Con-sequently, the present section assumes thatQSLand SR are independent Nevertheless, as shown in Figure 2, such assumption negligibly affects the analytic accuracy To this end, PIis rewritten as
PID Pr fQSL>ITg C 1 Pr fQSL>ITg/
Pr fQRL>ITg Pr fSR vg (33)
The PrfSR vg term in Equation (33) is straightfor-wardly deduced as
D
1 KSLv
v C GSR
eESR v
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Correlation coefficient ρ
Analysis Simulation
Pt→ ∞
Figure 2 Interference probability versus .
Trang 8where FSR.x/ is the cumulative distribution function of
SR The derivation of FSR.x/ is same as that of
Equa-tion (49) in the Appendix 1 As a result, it is obvious that
the derivation of Equation (33) is subject to the analytical
evaluation of PrfQmL>ITg with m 2 fS ; Rg
Actu-ally, PrfQmL>ITg was established for only interference
power constraint in [14, 26] while for both maximum
trans-mit power constraint and interference power constraint in
[41] The same channel estimation error model, Oqmn D
qmnCp
1 2„mnwhere the same for different links
is used to characterise the CSI imperfection, is employed in
[14, 26, 41] which differs from our model in Equation (2)
Therefore, our result on PrfQmL>ITg is not totally new
(We do not consider the derivation of PrfQmL>ITg as
our contribution because some derivation steps can imitate
[14, 26, 41] However, the proposal of Equations (32) and
(33) is surely our contribution.) but really essential to
pro-vide the consistent and complete study on the performance
of underlay cooperative cognitive networks under the
pop-ular channel estimation error model, because it is proved
that the channel estimation error impacts not only the
unli-cenced network but also the liunli-cenced network Likewise,
it can complement [14, 26, 41] with the different channel
estimation error model in order to diversify the literature
Before deriving each term PrfQmL>ITg with m D
fS ; Rg in Equation (33), it is essential to establish some
preliminaries Subsequently, we let WmL D pXmL D
jqmLj and OWmLDpzmLD j OqmLj Then, the joint pdf
of XmLand zmLis addressed in the following lemma:
Lemma 3. Given the Rayleigh distributions ofWmLD
jqmLj and OWmLD j OqmLj, the joint pdf of XmLandzmL
is expressed as follows:
fXmL;zmL.x; y/ De
mLxCmLyO
mL OmL 1mL /
mLOmL.1 mL/
I0
2pmLxy p
mLOmL.1 mL/
!
(35)
whereI0.:/ is the modified Bessel function of the first kind
andzerot horder [42, Equation (8.431.1)] while
mLD En
WmL2 o
OmLD En
O
WmL2 o
Proof Please see the Appendix 3.
The result in Lemma 3 is important to derive
PrfQmL> zg, which is expressed as
PrfQmL> zg D Pr
min IT
zmL
;Pt
XmL> z
D Pr
XmL> z1;XmL
zmL
> z2
D
1 Z
z1
x=zZ 2
0
fXmL;zmL.x; y/ dydx (39)
where z1 D z=Pt, z2 D z=IT To this effect, by sub-stituting Equation (35) in Equation (39) and applying the variable transformation t Dp
y, it follows that
mLOmL.1 mL/
1 Z
z1
p x=z 2
Z 0
t I0
2pmLxt p
mL OmL.1mL/
e
x
mL 1mL /e
t 2 O
mL 1mL /
Importantly, the inner integral can be expressed in terms of the first-order Marcum Q-function in [41, Equation (30)]
.1 mL / O mL and performing some basic manipulations, one obtains
PrfQmL> zg D emLz1 1
mL
1 Z
z 1
emLx
Q
s 2mLx 1 mL/ mL; (41) s
2x 1 mL/ OmLz2
! dx
We simplify the aforementioned integral by applying the variable transform t Dp
x and using [45, Equation (55)]
To this end, one obtains Equation (44) where
smLD2˛mL.˛mLmLC 1/
rmLD 2˛mL
s
˛mLmLC 3
Trang 9PrfQmL>ITg D e
IT
mLPt emLPt IT Q
0
@
s
2IT ˛mLmL 1/ mLPt
;
s 2˛2mLmLIT ˛mLmL 1/Pt
1 A
0
@
s smL rmL/IT
s smLC rmL/IT
2Pt
1 A
2
1 2˛mL
rmL
esmLIT2Pt I0
2˛mLIT ˛mLmL 1/Pt
(44)
By making the necessary change of variables in
Equation (44) and substituting the results together with
Equation (34) in Equation (33), one obtains the exact
closed form of the interference probability Furthermore,
under consideration of only interference power constraint
(e.g [14, 26]), we can reduce Equations (34) and (44) to
lim
Pt !1PrfSR vg D˛SLITe˛SR ˇSRv
˛SRN0v C ˛SLIT
(45)
lim
Pt !1PrfQmL>ITg D1
s
˛mLmL 1
˛mLmLC 3
!
(46)
It should be noted here that although the derivation
of Equation (44) is not completely novel, it
demon-strates that different channel estimation error models
lead to different results As a simple example, the same
result of lim
Pt !1PrfQmL>ITg D 0:5 is shown in
[14, 26, 41], but our result shows the dependence of
lim
Pt !1PrfQmL>ITg on the channel estimation quality
More results will illustrate this observation in details in
Section 6
6 NUMERICAL AND
SIMULATION RESULTS
This section provides various results to verify the
accu-racy of the derived expressions, show the performance
behaviour of underlay cooperative cognitive networks in
key parameters and optimise relay position It is recalled
that the mnquantity in Equation (38) represents the
cor-relation between jqmnj2and jbqmnj2, and hence,
character-ising the quality of the channel estimator Because channel
estimation is not the main focus of this study, we assume all
channel estimators of the same quality, that is, mnD for
all m and n Moreover, the path-loss exponent D 3 and
the required transmission rate R D 1 bps/Hz are assumed
to limit case-studies Furthermore, Figures 2–4 investigate
the same network topology where the coordinates of the
source, the relay, the destination and the licensee are
arbi-trarily selected as (0, 0), (0.4, 0.3), (1, 0) and (0.6, 0.5),
correspondingly, while Figures 5–7 consider a linear
net-work topology in which the relay lies on the straight line
connecting the source and the destination
0 5 10 15 20 25 30 35 40
10−3
10−2
10−1
100
IT/N0 (dB)
Po
Analysis Simulation
IT→ ∞
ρ=0.1
ρ=0.9
perfect CSI
Figure 3 Outage probability versusIT=N0.
0 5 10 15 20 25
10−1
100
Pt/N0 (dB)
Po
Analysis Simulation
Pt→ ∞ ρ=0.1
ρ=0.9
perfect CSI
Figure 4 Outage probability versusPt=N0.
Figure 2 shows the interference probability with respect
to the correlation coefficient for IT=N0 D 10 dB and
Pt=N0 D 20 dB It is seen that the analysis and the simulation|| are in good agreement, confirming the accuracy of the derived expression in Equation (33) In addition, the asymptotic interference probability using
|| 10 8 channel realisations are produced to have simulated results.
Trang 100 0.2 0.4 0.6 0.8 1
10−2
10−1
Source−Relay Distance
Po
ρ=0.97 ρ=1 (perfect CSI)
Figure 5 Outage probability versus source-relay distance.
0 5 10 15 20 25
0.35
0.4
0.45
0.5
0.55
Pt/N0 (dB)
dopt
ρ=0.9 & licensee at (0.2, 0.7) ρ=1 & licensee at (0.2, 0.7) ρ=0.9 & licensee at (0.4, 0.3) ρ=1 & licensee at (0.4, 0.3)
Figure 6.d opt versusPt=N0.
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
IT/N0 (dB)
dopt
ρ=0.9 & licensee at (0.2, 0.7) ρ=1 & licensee at (0.2, 0.7) ρ=0.9 & licensee at (0.4, 0.3) ρ=1 & licensee at (0.4, 0.3)
Figure 7.d opt versusIT=N0.
Equations (45) and (46) as Pt ! 1 depends on the
channel estimation quality, consistent with the remark in
Section 5 Notably, these illustrative results demonstrate
that PI slightly increases with respect to the better
chan-nel estimation quality for large values of (e.g > 0:6)
This comes from the fact that SRsignificantly increases with larger , and hence, the relay has more chances to participate in the cooperative relaying, eventually increas-ing the interference probability However, at** D 1, this probability is zero, as expected Moreover, over the wide range of the correlation coefficient 0:01 0:99, the interference probability is very high, for example, PI > 0:7, detrimentally inducing the operation of the licensees Therefore, underlay cooperative cognitive networks oper-ating in practical conditions (e.g with channel estimation error) should take into account some solutions to limit the interference level so as to protect licenced networks One of them is the back-off power mechanism in [14, 41] Incorporating this mechanism into our derived expression
is straightforward and hence, omitted in this paper Figure 3 illustrates analytical and simulated results with different channel estimation error levels D f0:1; 0:9; 1g andPt=N0D 15 dB The results show the perfect match between exact analysis and simulation, verifying the valid-ity of the derived expression Additionally, the asymptotic performance well agrees with the simulation at large values
ofIT, for example,IT=N0> 35 dB It should be noted that the realistic values ofIT=N0may be much less than
40 dB, and hence, this figure shows largeIT=N0just to validate the performance saturation of underlay coopera-tive cognicoopera-tive networks Moreover, as expected, the outage performance is significantly enhanced with respect to bet-ter estimated channel information (i.e decrease in chan-nel information error or equivalently, increase in ) As such, the effect of the channel estimation error on the performance of underlay cooperative cognitive networks
is considerable and must be accounted in system design process Furthermore, the results in Figure 3 are reason-able in the sense that the outage probability is inversely proportional to the maximum interference power IT for low-to-moderate values ofIT However, at largeIT (e.g larger than 35 dB), the outage saturation phenomenon occurs This performance saturation level is dramatically dependent of the channel estimation error, as discussed in Section 4 This phenomenon can be explained as follows The transmit power of unlicensees is controlled by the min-imum of the maxmin-imum interference power,IT, and the maximum transmit power,Pt Therefore, it is completely determined byPtwhenIT is larger than a threshold (e.g about 35 dB in Figure 3), making the outage probability unchanged for any increase inIT
Figure 4 reveals the effect of Pt on the outage per-formance for IT=N0 D 15 dB It is observed that the exact analytical results perfectly support the simulated ones while the asymptotic performance approaches the exact ones at large values ofPt (e.g.Pt=N0 > 20 dB), again affirming the accuracy of the derived expressions Moreover, the results illustrate the outage saturation phe-nomenon at large values of Pt, and the error floor level
is drastically dependent of the channel estimation error, as
** D 1 corresponds to the case of perfect CSI.