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

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

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 uti-lization 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: khuong.hovan@yahoo.ca

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 advan-tage of shorter range communication for lower path loss Among various relaying techniques, decode-and-forward (DF) and amplify-and-forward (AF) have been extensively investi-gated [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

interfer-ence 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 probabil-ity 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 inde-pendent 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

coeffi-cient between the transmitter t ∈ {0, 1, , N − 1} and the receiver r ∈ {1, 2, , N, P} is

h tr∼CN

0, η tr = d tr −α,2where d tris the distance between the two terminals andα is the

path-loss exponent [17]

1 Khuong and Bao [ 14 ] only derives the approximate closed-form BER expression.

2h ∼ CN (m, v) denotes a m-mean circular symmetric complex Gaussian random variable with variance v.

Fig 1 System model

N-1

P

P

h0 h1P

P

h2

01

h

12

h

h (N- 1 ) P

h (N- 1 ) N

R e(1)

R e(2)

R e (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 x1with 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

where y tr 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 B t ≤ I T /|h t P|2where I T is the maximum interference level

that PU still operates reliably For the maximum transmission range, B t = I T /|h t P|2is set Following [19–22], we choose the CSI imperfection model as

where h tr is the estimate of the t − r channel and ε tr is the CSI error

We assume that h trand h trare jointly ergodic and stationary Gaussian processes There-fore,ε tr∼CN (0, σ tr ) and  h tr ∼CN

0, 1

λ tr = η tr − σ tr

 σ trrepresents the quality of the channel estimator For example [19], for the linear-minimum-mean-square-error (LMMSE) estimator,σ tr = E|h tr|2

= 1/L p ¯γ tr ,training+ 1where L pis the number

of pilot symbols, ¯γ tr ,training = Eγ tr ,training

= B t ,training η tr /N0is the average SNR of

pilot symbols for the t − r channel, and B t ,trainingis the pilot power

3 Performance Analysis

Due to CSI errors, the transmit power of the node t is modifed as B

t = I T /| 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

B

t = I T /| h t P|2meets the interference power constraint for|h t P|2≤ |h t P|2(since this case

results in the interference power as B

t |h t P|2 = I T |h t P|2/| h t P|2≤ I T) but not for|h t P|2>

|h t P|2(since this case results in the interference power as B

t |h t P|2=I T |h t P|2/| h t P|2>I T)

Since E h t P 2

≤ E|h t P|2

where the equality holds for no CSI errors, on the

average such transmit power setting may not meet the interference power constraint (i.e.,

the interference at P is greater than I T) 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

inter-ference exceeds I T , namely the interference probability P I as used in [16] It is noted that

P I 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, P Ican 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 B

t 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 B

t, 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 I T while the previous ones m ∈ {0, 1, , n−1} do not According

to the total probability law the interference probability is expressed as

P I =

N−1

n=0

P n

n−1

m=0

where P t= Pr ˜B t |h t P|2> I T

= Pr



ρ|h t P|2 > t P

2

with t ∈ {n, m} is the probability that the SU t causes the interference to exceed I T

Let

σ t P



η t P − σ t P



(4)

σ t P



η t P − σ t P



(5)

=



τ2−16ρ

σ2

t P

(6)

Then

P t= 1 2



1−θ



The proof of (7) is given in the “Appendix” Plugging (7) in (3) results in the closed-form

expression of P I

3 The CSI imperfection model in [ 15 ] and [ 16 ] is ˆh tr = ρ tr h tr+1− ρ2

tr ε trwhereρ tr is the correlation coefficient between ˆh tr and h tr.

3.2 BER Derivation

Using the CSI imperfection model in (2), we rewrite (1) as

y tr =   ˆh tr x t

desired signal

+ εtr x t+ n tr

effective noise

According to (8), the effective SNR of the t − r channel taking CSI errors and the BPC

into account is expressed as

γ tr= h tr

2

E

|x t|2

E

|ε tr x t + n tr|2 = ˜B t h tr

2

˜B t σ tr + N0 =

h tr 2

σ tr+ h t P2/μ =

z tr

d tr , (9)

where z tr = h tr 2, d tr = σ tr+ h t P 2/μ, and μ = ρ I T /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,4respectively, as

R e (r) =

 ∞

0 {ψ (I, u, M; γ ) + ψ (J, u, M; γ )} f γ tr (γ ) dγ , k odd

2∞

0 ψ√

M , g, M; γf γ tr (γ ) dγ , k even , (10)

where g = 3/(M − 1), u = 6/(I2+ J2− 2), I = 2 (q−1)/2 , J = 2 (q+1)/2, and

ψ (s, v, M; γ )=Δ 2

slog2M

log2s

k=1



1 −2−k

s−1

i=0

(−1)



i 2k−1 s



Q



(2i + 1)2vγ





2k−1−i 2 k−1

2

The expression in (11) is cited from [23]

Next, we derive f γ tr (γ ) to have explicit expression for (10) Since h tr ∼ CN

0, 1

λ tr

 and h t P ∼ CN

0, 1

λ t P



, the pdf’s of z tr and d tr are f z tr (x) = λ tr e −λ tr x and f d tr (x) =

λ t P μe −λ t P μ(x−σ tr ), respectively As a result, the pdf ofγ tr = z tr /d tr in (9) is given as [24, eq (6–60)]

f γ tr (x) =

∞

0

y f z tr (yx) f d tr (y) dy

= κ tr μe λ t P μσ tr

whereκ tr = λ t P /λ tr

Inserting (12) into (10) results in

R e (r) =



θ (I, u, W tr ) + θ (J, u, W tr ) , k odd

2θ√

M , g, W tr



where W tr = {M, κ tr , μ, λ t P , σ tr} is a set of parameters

4The average BER of other modulation schemes such as M-PSK can be derived in the same approach.

In (13), we define

θ (s, v, W tr )=Δ 2

slog2M

log2s

k=1

 1−2−k

s−1

i=0

(−1)



i 2k−1 s



κ tr μe λ t P μσ tr ζ(2i + 1)2v, κ t P μ



2k−1−i 2 k−1

2

(14) Here we define

ζ (β, a) =

∞

0

Q√

βx

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

ζ (β, a) = 1

√

β

2√

2π

∞

0

e−βx2

(x + a)√x d x

√

βe βa2

2√

2π

∞

a

e−βy2

y√

y − a d y

βπ 2a

e βa2

2

1− er f

βa

2

!

where er f (x) = √ 2

π x



0

e −t2

dt is the error function [27] and the closed-form expression of the integral in the second equality is borrowed from [27, eq (3.363.2)]

Given the set of the average BERs of all hops{R e (1), , R e (N)}, the exact closed-form

average BER of the underlay cognitive DF multi-hop networks is expressed as [25]

R e=

N

n=1

⎡

j =n+1

(1 − 2R e ( j))

⎤

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.2is applied to all following results

We consider the path-loss exponent of α = 3 and the CSI error variance of σ tr =

1/L p B t ,training η tr /N0+ 1[19] The value of B t ,trainingis selected such that the

aver-age received power at P does not exceed I T (i.e., B t ,training η tr ≤ I T).5 As a result, for

illustration purpose we select B t ,training = I T /η t P

5The 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.

Fig 2 Network topology

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10−4

10−3

10−2

10−1

100

ρ

P I

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

L

p =1

L

p =3

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

Figure3plots the interference probability versus the back-off power control coefficient

for I T /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, P I decreases with the decrease ofρ Nevertheless, this reduction of P Idegrades the BER performance of secondary networks as seen in the following results Moreover,

these results are reasonable in the sense that P I 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 P I

Figures4and5compares 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 I T This is obvious since I Timposes a constraint on the transmit power and

0 5 10 15 20

10−2

10−1

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)

3 hops

2 hops

I

T /N

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

10−2

10−1

I

T /N

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)

3 hops

2 hops

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

the higher I T, the higher the transmit power, eventually enhancing communication reliability Moreover, the BER performance is deteriorated with the decrease ofρ and the lack of CSI It

is recalled that P Iis proportional toρ Therefore, the trade-off between BER and P Ishould

be noted in system design

Figure6investigates 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 over-head 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

1 2 3 4 5 6 7 8 9 10

10−1

L

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

p

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, and6illustrate that 3-hop communication is worst than 2-hop communication for any set{ρ, L p , α, I T , 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 P I 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 P Iand 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.

This appendix derives P tin (7) Let x= t P t P | Then the joint pdf of x and y

is expressed as [26]

f x ,y (x, y) = 4x ye

−ηx ηy ηy x2+ηx y2 (1−ρxy)

η x η y



1− ρ x y

 I0

 2√ρ x y x y

√η x η y



1− ρ x y

whereη x = Ex2

= η t P − σ t P , η y = Ey2

= η t P, andρ x yis the power correlation coefficient

Using (2) and the definition ofρ x y= covx2, y2 &

var

x2 var

y2 , we finally obtain

ρ x y= 1 −σ t P

The joint pdf of z = x2andw = y2can 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

f z ,w (z, w) = e

−ηt P z+ ( ηt P −σt P ( ηt P −σt P ) σt P ) w

(η t P − σ t P ) σ t P

I0



2√

z w

σ t P



where z , w > 0 and I0() is the zeroth-order modified Bessel function of the first kind [27,

eq (8.431.1)]

We express P tas

P t = Pr {ρw > z} =

∞

0

ρy

0

f z ,w (x, y) dxdy =

∞

0

ρy

0

e−ηt P x+ ( ηt P −σt P ) y

( ηt P −σt P ) σt P

(η t P − σ t P ) σ t P

I0



2√x y

σ t P



d xd y

(21)

After changing the variable of t=√x and applying [28, eq (10)], we simplify (21) as

η t P

∞

0

e− y



2(η t P − σ t P )

η t P σ t P

√

y ,



2ρη t P (η t P − σ t P ) σ t P

√

y d y , (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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radios: An information theoretic perspective Proceedings of the IEEE, 97, 894–914.

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Figure6investigates 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... 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... 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