DSpace at VNU: Impact of Imperfect Channel Information on the Performance of Underlay Cognitive DF Multi-hop Systems

Wireless Pers Commun
DOI 10.1007/s11277-013-1301-y

Impact of Imperfect Channel Information
on the Performance of Underlay Cognitive DF
Multi-hop Systems
Khuong Ho-Van

© Springer Science+Business Media New York 2013

Abstract This paper presents an analysis framework for performance evaluation of underlay
cognitive decode-and-forward (DF) multi-hop systems over Rayleigh fading channel under
imperfect channel information. Specifically, we derive the exact closed-form bit error rate
(BER) and interference probability (i.e., the probability that the interference power constraint
is invalid) expressions. The derived expressions are well supported by simulations and serve
as useful tools for fast system performance evaluation under different aspects. To reduce the
interference probability, we consider the back-off power control mechanism. Various results
demonstrate the effect of channel information imperfection on the system performance and
the trade-off between the interference probability and BER. Also, the performance of underlay
cognitive DF multi-hop systems depends both network topology and the number of hops.
Keywords Imperfect channel information · Decode-and-forward · Cognitive radio ·
Underlay · Multi-hop communication · Fading channel

1 Introduction
The Federal Communications Commission (FCC) pointed out in a survey of spectrum utilization that the currently licensed spectrum is significantly under-utilized [1]. On the other
hand, the spectrum resources for many emerging wireless applications such as video calling,
online high-definition video streaming, high-speed Internet access through mobile devices,
etc. are very scarce. To improve the spectrum utilization, the cognitive radio technology is
proposed [2]. In cognitive radio, secondary users-SUs (or unlicensed users) are generally
allowed to use the licensed band primarily allotted to primary users-PUs (or licensed users)
unless their operation does not interfere with the normal communication of PUs in three

modes: underlay, overlay, and interweave [3]. In the underlay mode, SUs are allowed to use

K. Ho-Van (B)
Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam
e-mail:

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K. Ho-Van

the spectrum when the interference caused by SUs on PUs is within the range tolerated by
PUs. This mode is more preferable than the others for its low implementation complexity [4].
Due to the interference power constraint imposed on SUs operating in the underlay
mode, their transmit power is limited and as such, their transmission range is significantly
reduced. To overcome this shortage, SUs can apply relaying techniques, which take advantage of shorter range communication for lower path loss. Among various relaying techniques,
decode-and-forward (DF) and amplify-and-forward (AF) have been extensively investigated [5]. In DF, each relay decodes information from the source, re-encodes it, and then
forwards it to the destination. In AF, each relay simply amplifies the received signal and
forwards it to the destination. Due to its capability of regenerating noise-free relayed signals,
DF is selected in this paper.
This paper investigates underlay cognitive DF multi-hop networks with arbitrary number
of hops. Most performance analyzing works dedicated to these networks in terms of outage
probability (e.g., [3,6–11]) and BER (e.g., [12–14]1 ) assume perfect channel estimation and
two-hop communication. It is well-known that channel state information (CSI) is essential
for coherent detection. Nevertheless, existing channel estimators are unable to provide the
perfect CSI. As such, the impact of imperfect CSI on the system performance should be
considered.
In [15], interference probability and BER analysis for cognitive single-hop networks is
presented in the assumption of the imperfect CSI only for SU-PU links. In [16], exact interference probability and outage probability expressions for cognitive AF dual-hop networks. To
the best of our knowledge, the exact interference probability and BER analysis for cognitive

DF N -hop networks with N being the arbitrary integer and imperfect CSI on all channels is
still open. This paper fills in this gap with the proposal of new closed-form exact interference
probability and BER expressions. All derived expressions are validated by simulations and
are useful in evaluating the system performance without time-consuming simulations.
The rest of this paper is organized as follows. The next section presents the system model
and the CSI imperfection model. The performance analysis in terms of interference probability and BER is discussed in Sect. 3. Simulated and analytical results are presented in Sect. 4
for derivation validity and performance evaluation. Finally, the paper is concluded in Sect. 5.

2 System Model
The underlay cognitive DF multi-hop network model under consideration is depicted in
Fig. 1, where N − 1 SRs numbered from 1 to N − 1 assist the transmission of SS 0 to SD
N , and SS and SRs use the same spectrum as a primary user P. The direct communication
between SS and SD is bypassed, which may be reasonable in scenarios where SS and SD
are too far apart or their communication link is blocked due to severe shadowing and fading.
We assume that the channel between any pair of transmitter and receiver experiences independent block frequency-flat Rayleigh fading (i.e., frequency-flat fading is invariant during
one phase but independently changed from one to another). Therefore, the channel coefficient between the transmitter t ∈ {0, 1, . . . , N − 1} and the receiver r ∈ {1, 2, . . . , N , P} is
h tr ∼ CN 0, ηtr = dtr−α ,2 where dtr is the distance between the two terminals and α is the
path-loss exponent [17].

1 Khuong and Bao [14] only derives the approximate closed-form BER expression.
2 h ∼ CN (m, v) denotes a m-mean circular symmetric complex Gaussian random variable with variance v.

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Impact of Imperfect Channel Information
Fig. 1 System model

P

h1P

h0 P
1

Re(1)

h01
0

Re(2)

h12

h2 P

h (N-1)P

2
N-1

Re(N)

h(N-1)N

N

A N -hop communication time interval consists of N phases. In the first phase, SS
0 transmits a modulated symbol x0 with the symbol energy, B0 (i.e., E{|x0 |2 } = B0
where E{·} denotes the expectation). SR 1 demodulates the received signal from SS 0 and

re-modulates the demodulated symbol as x1 with the symbol energy, B1 , before forwarding to
SR 2 in the second phase. The process continues until the signal reaches SD N . The received
signal through the hop r can be expressed as
ytr = h tr xt + n tr ,

(1)

where ytr denotes a signal received at the node r from the node t = r −1 and n tr ∼ CN (0, N0 )
is additive white Gaussian noise at the node r .
In the underlay cognitive relay systems [10,18], the SU t’s transmit power is limited such
that the interference imposed on PU is under control. Without CSI errors, this interference
constraint can be addressed as Bt ≤ IT /|h t P |2 where IT is the maximum interference level
that PU still operates reliably. For the maximum transmission range, Bt = IT /|h t P |2 is set.
Following [19–22], we choose the CSI imperfection model as
h tr = h tr + εtr ,

(2)

where h tr is the estimate of the t − r channel and εtr is the CSI error.
We assume that h tr and h tr are jointly ergodic and stationary Gaussian processes. Therefore, εtr ∼ CN (0, σtr ) and h tr ∼ CN 0, λ1tr = ηtr − σtr . σtr represents the quality of the
channel estimator. For example [19], for the linear-minimum-mean-square-error (LMMSE)
2
= 1/ L p γ¯tr,training + 1 where L p is the number
estimator, σtr = E |h tr |2 − E h tr
of pilot symbols, γ¯tr,training = E γtr,training = Bt,training ηtr /N0 is the average SNR of
pilot symbols for the t − r channel, and Bt,training is the pilot power.

3 Performance Analysis
Due to CSI errors, the transmit power of the node t is modifed as Bt = IT /|h t P |2 . Then, there
are two possibilities: |h t P |2 ≤ |h t P |2 and |h t P |2 > |h t P |2 . Setting the transmit power as

Bt = IT /|h t P |2 meets the interference power constraint for |h t P |2 ≤ |h t P |2 (since this case
results in the interference power as Bt |h t P |2 = IT |h t P |2 /|h t P |2 ≤ IT ) but not for |h t P |2 >
|h t P |2 (since this case results in the interference power as Bt |h t P |2 =IT |h t P |2 /|h t P |2 >IT ).
Since E

ht P

2

≤ E |h t P |2

where the equality holds for no CSI errors, on the

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K. Ho-Van

average such transmit power setting may not meet the interference power constraint (i.e.,
the interference at P is greater than IT ). Therefore, the primary system performance may be
severely degraded if the channel estimator is not efficient. Consequently, in order to propose
solutions to interference reduction on primary systems, statistics of interference at the PU
receiver should be analyzed. The most important statistics is the probability that the interference exceeds IT , namely the interference probability PI as used in [16]. It is noted that
PI is derived for underlay cognitive AF dual-hop networks [16] and for underlay cognitive
single-hop networks [15] with the CSI imperfection model slightly different from mine.3
Obviously, by backing-off the transmit power of SUs, PI can be reduced. This mechanism
is applied in [15,16] at the expense of performance degradation of SUs due to lower transmit
power. Specifically, the transmit power of the SU t is just a fraction of Bt . Therefore, the
transmit power of the SU t taking into account both the imperfect CSI and the back-off power
control (BPC) is B˜ t = ρ Bt , where 0 ≤ ρ ≤ 1 is the back-off power control coefficient.

3.1 Interference Probability
There are N secondary transmitters t ∈ {0, 1, . . . , N − 1} in the considered multi-hop
networks and thus, an interference event occurs if and only if the current transmitter n causes
the interference to exceed IT while the previous ones m ∈ {0, 1, . . . , n −1} do not. According
to the total probability law the interference probability is expressed as
N −1

PI =

n−1

(1 − Pm ),

Pn
n=0

(3)

m=0

where Pt = Pr B˜ t |h t P |2 > IT = Pr ρ|h t P |2 > hˆ t P

2

with t ∈ {n, m} is the probability

that the SU t causes the interference to exceed IT .
Let
2
σt P

2
θ =
σt P

τ =

=

ρηt P
ηt P − σ t P
ρηt P
1−
ηt P − σ t P
1+

τ2 −

16ρ
σt2P

(4)
(5)
(6)

Then
Pt =

1
θ
1−

2

,

(7)

The proof of (7) is given in the “Appendix”. Plugging (7) in (3) results in the closed-form
expression of PI .
3 The CSI imperfection model in [15] and [16] is hˆ = ρ h +
tr
tr tr

coefficient between hˆ tr and h tr .

123

2 ε where ρ is the correlation
1 − ρtr
tr
tr


Impact of Imperfect Channel Information

3.2 BER Derivation
Using the CSI imperfection model in (2), we rewrite (1) as
hˆ tr xt

ytr =

+ εtr xt + n tr .

desired signal

(8)

effective noise

According to (8), the effective SNR of the t − r channel taking CSI errors and the BPC
into account is expressed as
B˜ t h tr

2

γtr =

h tr E |xt |2
E |εtr xt + n tr |2
2

=

2

B˜ t σtr + N0

h tr

=

2
2

σtr + h t P /μ

=

z tr
,
dtr

(9)

2

where z tr = h tr , dtr = σtr + h t P /μ, and μ = ρ IT /N0 .
The average BER at the node r for square M-QAM with M = 2q (q even) and rectangular
M-QAM with M = 2q (q odd) modulation schemes,4 respectively, as
Re (r ) =

∞
0 {ψ (I,√u, M; γ ) + ψ
∞
2 0 ψ
M, g, M; γ

(J, u, M; γ )} f γtr (γ ) dγ ,

k odd

f γtr (γ ) dγ ,

k even

,

(10)

where g = 3/(M − 1), u = 6/(I 2 + J 2 − 2), I = 2(q−1)/2 , J = 2(q+1)/2 , and
2
ψ (s, v, M; γ ) =
slog2 M
Δ

log2 s

1−2−k s−1

k=1

i=0

(−1)

i2k−1
s

Q

2k−1 −

(2i + 1)2 vγ
i2k−1
s

+

1
2

−1

.

(11)

The expression in (11) is cited from [23].
Next, we derive f γtr (γ ) to have explicit expression for (10). Since h tr ∼ CN 0, λ1tr
and h t P ∼ CN 0, λ1t P , the pdf’s of z tr and dtr are f z tr (x) = λtr e−λtr x and f dtr (x) =

λt P μe−λt P μ(x−σtr ) , respectively. As a result, the pdf of γtr = z tr /dtr in (9) is given as
[24, eq. (6–60)]
∞

f γtr (x) =

y f z tr (yx) f dtr (y) dy
0

=

κtr μeλt P μσtr
(x + κtr μ)2

,

(12)

where κtr = λt P /λtr .
Inserting (12) into (10) results in
Re (r ) =

θ (I, u, Wtr ) + θ (J, u, Wtr ) ,
√
2θ
M, g, Wtr ,

k odd

(13)

k even

where Wtr = {M, κtr , μ, λt P , σtr } is a set of parameters.
4 The average BER of other modulation schemes such as M-PSK can be derived in the same approach.

123


K. Ho-Van

In (13), we define
2
θ (s, v, Wtr ) =
slog2 M
Δ

log2 s

1−2−k s−1

k=1

i=0

i2k−1
s

(−1)

κtr μeλt P μσtr ζ (2i + 1)2 v, κt P μ
2k−1 −

i2k−1
s

+

1
2

−1

.
(14)

Here we define
∞

√

Q

ζ (β, a) =

βx

(x + a)2

0

d x.

(15)

Applying the integration by parts, we obtain the closed-form of ζ (β, a) as
ζ (β, a) =

√
β

1
− √
2a
2 2π

∞

0

√ βa
βe 2
1
=
− √
2a
2 2π
=
where er f (x) =

√2
π

x

1
−
2a

βx

e− 2
√ dx
(x + a) x

∞

a
βa
2

βπ e
2a 2

βy

e− 2
dy
√
y y−a
1 − er f

βa
2

,

(16)

e−t dt is the error function [27] and the closed-form expression of
2

0

the integral in the second equality is borrowed from [27, eq. (3.363.2)].
Given the set of the average BERs of all hops {Re (1), . . . , Re (N )}, the exact closed-form
average BER of the underlay cognitive DF multi-hop networks is expressed as [25]
⎤
⎡
N

Re =
n=1

⎣ Re (n)

N

(1 − 2Re ( j))⎦ .

(17)

j=n+1

4 Illustrative Results
For illustration purpose, we randomly select user coordinates as shown in Fig. 2: P at
(0.7, 0.5), SS 0 at (0, 0), SR 1 at (0.6, 0.2), SR 2 at (0.8, 0.3), SD 3 at (1, 0). SS 0, SD
3, and P are always fixed and thus, for 2-hop case only SR 1 is considered. Also, the number
on the line is the distance between two corresponding terminals. The network topology in
Fig. 2 is applied to all following results.
We consider the path-loss exponent of α = 3 and the CSI error variance of σtr =

1/ L p Bt,training ηtr /N0 + 1 [19]. The value of Bt,training is selected such that the average received power at P does not exceed IT (i.e., Bt,training ηtr ≤ IT ).5 As a result, for
illustration purpose we select Bt,training = IT /ηt P .
5 The study of channel estimators is outside the scope of this paper. Therefore, the selection of B
t,training

in this paper is just an example to demonstrate the effect of CSI imperfection on BER of underlay cognitive
relay systems.

123


Impact of Imperfect Channel Information

Fig. 2 Network topology
0

10

Simulation: 2−hop
Analysis: 2−hop
Simulation: 3−hop
Analysis: 3−hop

−1

PI

10

Lp=1

−2

10

Lp=3

−3

10

−4

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ

Fig. 3 Interference probability versus back-off power control coefficient (I T /N0 = 20 dB)

Figure 3 plots the interference probability versus the back-off power control coefficient
for IT /N0 = 20 dB, N = {2, 3}, and L p = {1, 3}. Both simulation and analysis are in
perfect agreement, validating the accuracy of (3). Additionally as expected in the analysis of
Sect. 3.1, PI decreases with the decrease of ρ. Nevertheless, this reduction of PI degrades
the BER performance of secondary networks as seen in the following results. Moreover,
these results are reasonable in the sense that PI is proportional to the number of hops. This
is because the higher number of hops, the more transmitters can cause interference. Finally,
the smaller the CSI error (i.e., the larger the L p ), the smaller the PI .
Figures 4 and 5 compares simulated and numerical results for two typical modulation
levels (2-QAM for odd q and 4-QAM for even q), N = {2, 3}, different degrees of CSI
availability (perfect CSI and imperfect CSI with L p = 1), with/without BPC. For the case
of perfect CSI, no BPC is assumed (i.e., ρ = 1). It is seen that analytical results are well
matched with simulated ones. Additionally, the BER performance is improved with respect
to the increase in IT . This is obvious since IT imposes a constraint on the transmit power and

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K. Ho-Van

3 hops

−1

10

BER

2 hops

Perfect CSI (Analysis)
Perfect CSI (Simulation)
Imperfect CSI: ρ=1 (Analysis)
Imperfect CSI: ρ=1 (Simulation)
Imperfect CSI: ρ=0.7 (Analysis)
Imperfect CSI: ρ=0.7 (Simulation)

−2

10

0

5

10

15

20

I /N (dB)

T

0

Fig. 4 BER versus I T /N0 (2-QAM)

3 hops
2 hops

−1

BER

10

Perfect CSI (Analysis)
Perfect CSI (Simulation)
Imperfect CSI: ρ=1 (Analysis)
Imperfect CSI: ρ=1 (Simulation)
Imperfect CSI: ρ=0.7 (Analysis)
Imperfect CSI: ρ=0.7 (Simulation)

−2

10

0

5

10

15

20

I /N (dB)
T

0

Fig. 5 BER versus I T /N0 (4-QAM)

the higher IT , the higher the transmit power, eventually enhancing communication reliability.
Moreover, the BER performance is deteriorated with the decrease of ρ and the lack of CSI. It
is recalled that PI is proportional to ρ. Therefore, the trade-off between BER and PI should
be noted in system design.
Figure 6 investigates the impact of the quality of the channel estimator on BER without
the BPC (i.e., ρ = 1). The quality of the channel estimator can be enhanced by increasing
the number of pilot symbols L p at the cost of the bandwidth loss due to increased overhead. The results are reasonable since the BER performance is improved with the increased
L p . Furthermore, for the selected channel estimator model, the performance is saturated
at L p = 4.

123


Impact of Imperfect Channel Information

BER

2−QAM & 2 hops
4−QAM & 2 hops
2−QAM & 3 hops
4−QAM & 3 hops

−1

10

1

2

3

4

5

6

7

8

9

10

Lp

Fig. 6 BER versus L p (I T /N0 = 10, ρ = 1 dB)

Given the specific network topology in Fig. 2, results in Figs. 4, 5, and 6 illustrate that
3-hop communication is worst than 2-hop communication for any set {ρ, L p , α, IT , M}.
This means that in underlay cognitive DF multi-hop networks the advantage of the 3-hop
communication over 2-hop communication in terms of the path loss reduction [e.g., the
distance from the last relay to the destination in the 3-hop case (i.e., SR 2) is smaller than that
in the 2-hop case (i.e., SR 1)] can not sometimes turn into the performance improvement.
This is because the last relay in the 3-hop case is closer to the primary user than in the
2-hop case, causing higher interference. Thus, the last relay in the 3-hop case should utilize
lower transmit power than in the 2-hop case for reducing the interference level to the primary
user, leading to higher performance degradation. These results recommend that the relay
selection in underlay cognitive DF multi-hop networks is crucial in enhancing the network
performance. A good relay not only provides reliable communication to the destination but
also causes less interference to the primary user. The problem of the relay selection is outside
the scope of this paper.

5 Conclusion
This paper investigates the interference probability and the BER of underlay cognitive DF
multi-hop systems over Rayleigh fading channel in consideration of imperfect CSI. We
quickly obtain results owing to the derived expressions of PI and BER. Simulated results
are well matched with numerical ones. Various results demonstrate that the imperfect CSI
significantly affects the BER performance of underlay cognitive DF multi-hop networks and
the interference power constraint imposes the trade-off between PI and BER. Additionally,
the BER performance is dependent on both the number of hops and the network topology.
Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 102.04-2012.39.

123

K. Ho-Van

Appendix
This appendix derives Pt in (7). Let x = hˆ t P and y = |h t P |. Then the joint pdf of x and y
is expressed as [26]
−

η y x 2 +ηx y 2

4x ye ηx η y (1−ρx y )
I0
f x,y (x, y) =
ηx η y 1 − ρx y

√

√
2 ρx y x y
ηx η y 1 − ρx y

,

(18)

where ηx = E x 2 = ηt P − σt P , η y = E y 2 = ηt P , and ρx y is the power correlation
coefficient.
var x 2 var y 2 , we finally obtain
Using (2) and the definition of ρx y = cov x 2 , y 2
ρx y = 1 −

σt P
.
ηt P

(19)

The joint pdf of z = x 2 and w = y 2 can be achieved from that of x and y after the variable
transformation. After some simplifications, we get the joint pdf of z and w as
−

ηt P z+(ηt P −σt P )w

e (ηt P −σt P )σt P
f z,w (z, w) =
I0
(ηt P − σt P ) σt P

√
2 zw
,
σt P

(20)

where z, w > 0 and I0 () is the zeroth-order modified Bessel function of the first kind [27,
eq. (8.431.1)].
We express Pt as
∞ ρy

Pt = Pr {ρw > z} =

f z,w (x, y) d xd y =
0

0

∞ ρy − ηt P x+(ηt P −σt P ) y
e (ηt P −σt P )σt P
0

0

(ηt P − σt P ) σt P

I0

√
2 xy
σt P

d xd y.
(21)

After changing the variable of t =
Pt = 1 −

1
ηt P

∞

e
0

− ηy

tr

Q

√

x and applying [28, eq. (10)], we simplify (21) as

2 (ηt P − σt P ) √
y,
ηt P σ t P

2ρηt P
√
y dy,
(ηt P − σt P ) σt P

(22)

where Q(a, b) is the first-order Marcum Q-function [28, eq. (1)].
√
Finally, we reduce (22) to (7) after changing the variable of t = y and applying
[28, eq. (55)].

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K. Ho-Van

Author Biography
Khuong Ho-Van received the B.E. (with the first-rank honor) and
the M.S. degrees in Electronics and Telecommunications Engineering
from HoChiMinh City University of Technology, Vietnam, in 2001
and 2003, respectively, and the Ph.D. degree in Electrical Engineering from University of Ulsan, Korea in 2006. From April 2001 to
September 2004, he was a lecturer at Telecommunications Department, HoChiMinh City University of Technology. During 2007–2011,
he joined McGill University, Canada as a postdoctoral fellow. Currently, he is an assistant professor at HoChiMinh City University of
Technology. His major research interests are modulation and coding techniques, MIMO system, digital signal processing, cooperative
communications.

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