Abstract Channel information (CI) plays an important role in system design optimization.It is obtained through channel estimation process and hence, inevitably imperfect at a certaindegree. Numerical evaluation of the outage performance of cooperative underlay cognitivenetworks under the presence of channel estimation error has been left open. This paper healsthis literature gap with the proposal of a precise outage probability expression. Various resultsshow the excellent match between analysis and simulation, and the advantage of the derivedexpression in studying the effect of CI imperfection on system performance.
We are beholding a conflict: spectrum resources for new wireless applications are scarcewhile licensed spectrum utilization efficiency is low [1]. Cognitive radio technology thatpermits unlicensed users to operate in the frequency band allotted to licensed users1 emergesas a feasible and efficient solution to this conflict [2]. When operating in the underlay mode,the former must adjust their power to guarantee the resulting interference power at the latterbelow an acceptable level [3]. The negative effect of this power adjustment is the reducedcoverage of unlicensed users. Cooperative communications2 with the capability of increas-ing system capacity, extending transmission range, etc. [4] can solve the above drawbackof underlay cognitive networks, hence forming new networks called cooperative under-
1 In this paper, unlicensed/licensed and secondary/primary are interchangeable.2 Difference between multi-hop communication (e.g., [3,5]) and cooperative one (e.g., [6–9]) is that the formerdoes not consider the direct communication link between the source and the destination while for improvedspatial diversity, the latter does.
K. Ho-Van (B)268 Ly Thuong Kiet street, District 10, HoChiMinh City, Vietname-mail: [email protected]
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K. Ho-Van
lay cognitive networks. In most cooperative communications protocols, the helping node(namely relay) works in the decode-and-forward (DF) or the amplify-and-forward (AF)manner.
The outage probability analysis of DF-type cooperative underlay cognitive networks hasbeen studied in [6–9] under different power constraints. Specifically, both [8] and [9] con-sider only the interference power constraint while [6] and [7] generalizes the work in [8] and[9] with the consideration of both constraints (interference power constraint and maximumtransmit power constraint). Additionally, [6–8] investigate the correlation among receivedsignal-to-noise ratios (SNRs), hence providing more accurate results than [9] in which inde-pendent SNRs are assumed. However, the shortage of all above mentioned works lies in theassumption of channel information perfection.
System design optimization such as beam-former design requires channel information.Notwithstanding, perfect channel information is inevitably unavailable from current channelestimators. Therefore, the study on the effect of channel estimation error on the outage per-formance is essential. For AF-type dual-hop underlay cognitive networks, [10] proposed anoutage probability expression. To the best of our knowledge, the numerical evaluation of theoutage probability of DF-type cooperative underlay cognitive networks with imperfect chan-nel information is still left open. This paper fills in this gap. Firstly, we explain the consideredsystem model in Sect. 2. Then, we derive the precise outage probability expression, takingthe channel information imperfection into account in Sect. 3. Its accuracy and advantage inquickly investigating the impact of channel estimation error on the outage performance arepresented in Sect. 4. Finally, we close the paper in Sect. 5. For convenience of presentation,Table 1 lists all notations used throughout this paper.
Table 1 List of notations Notation Meaning
mab Channel coefficient between transmitter a and receiver bβab Fading power of the channel between transmitter a and
receiver bcab Distance between transmitter a and receiver bτ Path loss exponentxab Signal received at receiver b from transmitter ada Symbol transmitted by transmitter aεab Noise at receiver bρ Noise varianceζa Transmit power of the secondary transmitter a with perfect
CI�{·} Statistical expectationδ Maximum interference power that the primary user can
a Transmit power of the secondary transmitter a withimperfect CI
γ Power control factorμab Signal-to-noise ratio at receiver b from transmitter aR Required transmission rate of secondary networkOP Outage probabilityPr{w} Probability of the event w
L p Number of pilot symbols used for channel estimationζa,p Pilot power
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Performance Analysis of Cooperative Underlay Cognitive Networks
Fig. 1 System model
1 3
2
0
Stage 1
Stage 2
source destination
relay
primary user
secondary network
primary network
13m
12m23m
10m20m
2 System Model
Consider a typical cooperative underlay cognitive network in Fig. 1. We denote chan-nel coefficients3 between the source and the relay, between the source and the destination,between the source and the primary user, between the relay and the destination, betweenthe relay and the primary user as m12, m13, m10, m23, m20, respectively. Also, we assumeindependent frequency-flat Rayleigh fading and hence, mab ∼ CN (
0, βab = c−τab
)
wherecab is the distance between two users and τ is the path loss exponent [11], a ∈ {1, 2} andb ∈ {2, 3, 0}. Meanwhile, m ∼ CN (x, y) denotes a circular symmetric complex Gaussianrandom variable with mean x and variance y.
The received signal, xab, over any channel between the transmitter a and the receiver b ismodeled as
xab = mabda + εab, (1)
where εab ∼ CN (0, ρ) is the noise at the receiver b, and da denotes the symbol sent fromthe transmitter a.
For the underlay mode [8,9], the transmit power of the secondary transmitter a, ζa (i.e.,�{|da |2} = ζa in which �{·} stands for the expectation), must be controlled such that theinterference power at the primary user is below the maximum interference power, δ, at whichthe performance of the primary user is still acceptable. More specifically, this interferencepower constraint leads to the setting of ζa = δ/|ma0|2.
Channel estimation error, ab ∼ CN (0, αab), with αab reflecting the accuracy of thechannel estimator is popularly modeled in [12–15] as
mab = m̂ab + ab, (2)
where m̂ab ∼ CN(
0, 1ηab
= βab − αab
)
is the estimate of the a −b channel coefficient; mab
and m̂ab are jointly ergodic and stationary Gaussian processes.
3 Channel coefficient reflects the severe effect of fading channel on transmitted signals.
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K. Ho-Van
3 Outage Probability Analysis
Under investigation of imperfect channel information, the secondary transmitter a must adjustits transmit power as ζ ′
a = δ/|m̂a0|2. Then, the actual interference power becomes ζ ′a |ma0|2 =
δ|ma0|2/|m̂a0|2. Obviously, the interference power constraint is violated if |m̂a0|2 < |ma0|2.It is proved that this possibility happens with the relatively high probability of 0.75 in [10].Consequently, the inaccurate channel estimation significantly affects the performance ofprimary networks. To decrease this probability, it is mandatory to reduce the transmit powerby a factor of γ with 0 ≤ γ ≤ 1 as in [10], leading to the actual transmit power of thetransmitter a as γ ζ ′
a .Under the presence of the channel estimation error, (1) is rewritten as
xab = m̂abda + (abda + εab). (3)
Then, the received SNR accounting for channel estimation imperfection and the power reduc-tion is given as
μab =∣
∣m̂ab∣
∣
2�
{|da |2}
�{|abda + εab|2
} = γ ζ ′a
∣
∣m̂ab∣
∣
2
γ ζ ′aαab + ρ
= gab
rab, (4)
where
gab = ∣
∣m̂ab∣
∣
2, (5)
rab = αab + ga0/ν, (6)
ν = γ δ
ρ. (7)
Since m̂ab ∼ CN(
0, 1ηab
)
, we have results as follows:
– Result 1 (R1): The estimated channel gain gab = |m̂ab|2 is an exponentially distributedrandom variable with the probability density function (pdf) fgab (x) = ηabe−ηab x forx ≥ 0.
– Result 2 (R2): Pr {gab < u} = ∫ u0 ηabe−ηab x dx = 1 − e−ηabu , where Pr{w} denotes the
probability of the event w.– Result 3 (R3): Pr {gab > u} = ∫ ∞
u ηabe−ηab x dx = e−ηabu .
Recall that the outage criterion according to the Shannon information theory is R ≥12 log2 (1 + μ) or μ ≤ z with z = 22R − 1 where μ is the received SNR, R is the requiredrate of the secondary network, the factor of 1
2 prior to the logarithm is due to the two-stage cooperative communications protocol (see Fig. 1). In this protocol, the relay and thedestination listen to the source in the first stage. If the relay accurately restores the sourceinformation (i.e., μ12 ≥ z), it will transmit the processed signal to the destination in thesecond stage. Then, the destination recovers the source information with maximum-ratio-combining both signals, one from the source and the other from the relay. As such, thedestination is in outage if μ13 + μ23 < z. Otherwise (i.e., μ12 ≤ z), the relay is idle in thesecond stage, and therefore, the destination restores the source information only based onreceived signal from the source. In such a case, an outage event occurs if μ13 < z. Briefly,the outage probability can be expressed in the compact form as
OP = Pr {μ13 < z, μ12 < z}︸ ︷︷ ︸
I1
+ Pr {μ13 + μ23 < z, μ12 > z}︸ ︷︷ ︸
I2
. (8)
The elaborate derivation of I1 and I2 is presented in the next two subsections.
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Performance Analysis of Cooperative Underlay Cognitive Networks
3.1 Derivation of I1
By plugging μ13 and μ12, both having the general form in (4), into I1 we obtain
I1 = Pr
{
g13
α13 + g10/ν< z,
g12
α12 + g10/ν< z
}
=∞
∫
0
Pr {g13 < z (α13 + x/ν)} Pr {g12 < z (α12 + x/ν)} fg10 (x) dx . (9)
Applying the results R1, R2, R3, we get the exact closed form of (9) as
I1 =∞
∫
0
(
1 − e−η13z(α13+x/ν)) (
1 − e−η12z(α12+x/ν))
η10e−η10x dx
= 1 − η10
(
e−η13α13z
η10 + η13z/ν+ e−η12α12z
η10 + η12z/ν− e−(η13α13+η12α12)z
η10 + (η13 + η12) z/ν
)
. (10)
As a double check, in the case of the perfect channel estimation (i.e., α13 = α12 = 0), itis seen that (10) becomes [8, eq. (5)].
3.2 Derivation of I2
Again inserting μ13, μ12, and μ23, all following (4), into I2 results in
I2 = Pr
{
g13
α13 + g10/ν+ μ23 < z,
g12
α12 + g10/ν> z
}
= Pr {g13 < (z − μ23) (α13 + g10/ν) , g12 > z (α12 + g10/ν)}
=z
∫
0
⎡
⎣
∞∫
0
Pr{
g13 < (z−y)(
α13+ x
ν
)}
Pr{
g12 > z(
α12+ x
ν
)}
fg10 (x) dx
⎤
⎦ fμ23 (y) dy,
(11)
where fμ23 (y) is the pdf of μ23. The closed form of fμ23 (x) is given as
fμ23 (x) =∞
∫
0
y fg23 (yx) fr23 (y) dy = κ23νeη20να23
(x + κ23ν)2 , (12)
where we define the constant
κab = ηa0/ηab (13)
with a = {1, 2} and b = {2, 3}.The proof for (12) is as follows. Since fg20 (x) = η20e−η20x , the pdf of r23 in (6) is
fr23 (x) = η20νe−η20ν(x−α23). As a result, with the help of [16, eq. (6–60)], the pdf ofμ23 = g23/r23 in (4) follows (12), completing the proof.
123
K. Ho-Van
Using the results R1, R2, R3 for the square bracket in (11), we reduce (11) to
I2 =z
∫
0
⎡
⎣
∞∫
0
(
1 − e−η13(z−y)(α13+x/ν))
e−η12z(α12+x/ν)η10e−η10x dx
⎤
⎦ fμ23 (y) dy
= η10
eη12α12z
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1
η10 + η12zν
z∫
0
fμ23 (y) dy
︸ ︷︷ ︸
I21
−z
∫
0
e−η13(z−y)α13 fμ23 (y)
η10 + η12zν
+ η13(z−y)ν
dy
︸ ︷︷ ︸
I22
⎞
⎟
⎟
⎟
⎟
⎟
⎠
. (14)
The problem now is to solve I21 and I22 in the closed form. With fμ23 (x) given in (12),I21 has the closed form as
I21 = zeη20να23
z + κ23ν. (15)
To simplify notations in computing I22, we define the constants
A = 1
(κ13ν + (κ13/κ12 + 1) z + κ23ν)2 , (16)
B = ν A
η13, (17)
C = ν√
A
η13. (18)
Also, we will present the results in terms of the exponential integral function Ei(x), whichis defined in [17] as Ei(x) = − ∫ ∞
−xe−t
t dt . Ei(x) is a built-in function in most computationsoftwares (e.g., Matlab).
Then, by substituting (12) in I22, we can decompose I22 as
where I221, I222, and I223 are completely solved in the exact closed form as
I221 =z
∫
0
eη13α13 y
η10 + (η12 + η13) z/ν − η13 y/νdy
= ν
η13
Ei (−α13 (η10ν + (η12 + η13) z)) − Ei (−α13 (η10ν + η12z))
e−α13(η10ν+(η12+η13)z), (20)
I222 =z
∫
0
eη13α13 y
y + κ23νdy
=z+κ23ν∫
κ23ν
eη13α13(x−κ23ν)
xdx
= Ei (η13α13 [z + κ23ν]) − Ei (η13α13κ23ν)
eη13α13κ23ν, (21)
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Performance Analysis of Cooperative Underlay Cognitive Networks
I223 =z
∫
0
eη13α13 y
(y + κ23ν)2 dy
= 1
κ23ν− eη13α13z
z + κ23ν+ η13α13
z∫
0
eη13α13 y
y + κ23νdy
= 1
κ23ν− eη13α13z
z + κ23ν+ η13α13I222. (22)
4 Numerical Results
This section illustrates the validity of the derived outage probability expression in (8) andits behaviour. Without any loss of generality, we randomly choose the coordinates for theprimary user, the source, the relay, and the destination as (0.7, 0.5), (0, 0), (0.6, 0.2), and(1, 0), correspondingly. We assume the power reduction is not applied for the case of perfectchannel information (i.e., γ = 1).
To evaluate the outage performance in the presence of channel estimation error, we con-sider the Linear-Minimum-Mean-Square-Error estimator. It is shown in [12] that this estima-tor results in the variance of channel estimation error as αab = 1/
(
L pζa,pβab/ρ + 1)
whereL p is the number of pilot symbols and ζa,p is the pilot power. The selection of ζa,p mustguarantee the average interference power at the primary user below δ (i.e., ζa,pβa0 ≤ δ).Therefore, for maximum transmission range, we set ζa,p = δ/βa0.4
Figure 2 demonstrates simulated5 and numerical results with parameters: perfect CI (i.e.,L p −→ ∞) and imperfect CI with different number of pilot symbols L p = {1, 5}6); therequired rate of R = 1 bps/Hz; the power control factor γ ∈ {0.5, 1}; the path loss exponentτ = 3. The excellent agreement between analysis and simulation is observed. This affirmsthe accuracy of (8). In addition, the results in Fig. 2 are justified in the sense that the outageprobability is proportional to channel information imperfection degree. For example, theSNR gain of 1 dB for γ = 1 is achievable for increasing the number of pilot symbols from1 to 5; and no significant gain is observed for further increasing L p since the performancefor L p = 5 almost approaches that for the case of the perfect CI. Moreover, Fig. 2 showsthe performance improvement with the increase in the maximum interference power δ. Thisis because underlay cognitive networks set the transmit power of secondary users propor-tionally to δ. As such, the larger δ, the higher the transmit power, eventually reducing theoutage probability. Furthermore, as predicted in Sect. 3 the power reduction (i.e., decreas-ing γ ) deteriorates the performance of cooperative underlay cognitive networks in exchangefor lower interference to primary networks. Therefore, the performance trade-off betweenprimary network and secondary network is of concern in system design.
4 Channel estimation is not the main objective of this paper. As such, this setting of ζa,p is only a solutionto illustrate the impact of channel estimation error on the performance of cooperative underlay cognitivenetworks.5 Simulated results are achieved with 107 channel realizations.6 It is noted that the quality of the channel estimator depends on the number of pilot symbols. The highernumber of pilot symbols, the better quality of the channel estimator and the larger transmission bandwidthloss.
123
K. Ho-Van
0 5 10 15 2010
−3
10−2
10−1
100
δ/ρ (dB)
OP
Perfect CI (Analysis)Perfect CI (Simulation)Imperfect CI (Analysis) − L
p=1
Imperfect CI (Simulation) − Lp=1
Imperfect CI (Analysis) − Lp=5
Imperfect CI (Simulation) − Lp=5
γ=0.5
γ=1
Fig. 2 Outage probability versus δ/ρ
2 2.5 3 3.5 4 4.5 510
−4
10−3
10−2
τ
OP
Perfect CIImperfect CI − L
p=1
Imperfect CI − Lp=5
γ=0.5
γ=1
Fig. 3 Outage probability versus path loss exponent τ
Due to the perfect match between analytical and simulated results, for time savings all thefollowing figures plot only analytical ones. First of all, the effect of the path loss on the outageperformance is illustrated in Fig. 3 with δ/ρ = 25 dB and R = 1 bps/Hz. We consider samplevalues of the path loss exponent τ ∈ [2, 5]. This figure shows the performance degradationwith respect to the increase in τ . This is reasonable since the transmitter-receiver distanceunder investigation is less than 1, the larger τ results in the larger path loss and the higheroutage.
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Performance Analysis of Cooperative Underlay Cognitive Networks
1 1.5 2 2.5 3 3.5 4 4.5 510
−5
10−4
10−3
10−2
10−1
100
R (bps/Hz)
OP
Perfect CIImperfect CI − L
p=1
Imperfect CI − Lp=5
γ=0.5
γ=1
Fig. 4 Outage probability versus required transmission rate R
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
−5
10−4
10−3
10−2
γ
OP
Perfect CIImperfect CI − L
p=1
Imperfect CI − Lp=5
Fig. 5 Outage probability versus power reduction factor γ
Secondly, the performance trend versus the required transmission rate is investigated inFig. 4 with δ/ρ = 30 dB and τ = 3. As expected, the outage probability significantlyincreases with respect to R. Therefore, the trade-off between the outage performance and thebandwidth efficiency should always be considered in system design.
Finally, we evaluate the outage performance versus the power reduction factor γ forδ/ρ = 30 dB, R = 1 bps/Hz, and τ = 3. Results are displayed in Fig. 5. It is seen that theperformance with imperfect CI is dramatically deteriorated with respect to the decrease inγ and significantly worse than that with perfect CI for small values of γ . Additionally as
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K. Ho-Van
expected, without the power reduction (γ = 1) the performance gap between CI imperfectionand perfection is minor, especially for good estimators (i.e., large L p).
5 Conclusions
This paper illustrates the effect of the channel estimation error on the outage performance ofDF-type cooperative underlay cognitive networks. For time-savings in evaluating the systemperformance in different operation parameters, we proposed the accurate outage probabilityformula and confirmed its validity. Analysis reveals that the imperfect channel informationdeteriorates the performance of both secondary network (e.g., increase the outage probability)and primary network (e.g, increase the interference power), and its impact can be alleviated byimproving the quality of the channel estimator. Additionally, the power reduction mechanismis useful to protect primary networks from channel information imperfection at the expenseof the performance degradation of secondary networks.
Acknowledgments This research is funded by Vietnam National Foundation for Science and TechnologyDevelopment (NAFOSTED) under Grant number 102.04-2012.39.
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Performance Analysis of Cooperative Underlay Cognitive Networks
Khuong Ho-Van received the B.E. (with the first-rank honor) andthe M.S. degrees in Electronics and Telecommunications Engineeringfrom HoChiMinh City University of Technology, Vietnam, in 2001and 2003, respectively, and the Ph.D. degree in Electrical Engineeringfrom University of Ulsan, Korea in 2006. During 2007–2011, he joinedMcGill University, Canada as a postdoctoral fellow. Currently, he is anassistant professor at HoChiMinh City University of Technology. Hismajor research interests are modulation and coding techniques, diver-sity technique, digital signal processing, and cognitive radio.
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