On the Capacity of Wireless Powered Cognitive

arXiv:1712.04747v1 [] 13 Dec 2017

Relay Network with Interference Alignment

Sultangali Arzykulov∗ , Galymzhan Nauryzbayev, Theodoros A. Tsiftsis∗
and Mohamed Abdallah‡
∗
‡

School of Engineering, Nazarbayev University, Astana, Kazakhstan

Division of Information and Computing Technology, College of Science and Engineering,
Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar
Email: {sultangali.arzykulov, theodoros.tsiftsis}@nu.edu.kz, nauryzbayevg@gmail.com,
moabdallah@hbku.edu.qa
Abstract
In this paper, a two-hop decode-and-forward cognitive radio system with deployed interference alignment is

considered. The relay node is energy-constrained and scavenges the energy from the interference signals. In the
literature, there are two main energy harvesting protocols, namely, time-switching relaying and power-splitting
relaying. We first demonstrate how to design the beamforming matrices for the considered primary and secondary
networks. Then, the system capacity under perfect and imperfect channel state information scenarios, considering
different portions of time-switching and power-splitting protocols, is estimated.
Keywords
Decode-and-forward (DF), energy-harvesting (EH), capacity, cognitive radio (CR), interference alignment (IA).

I.

C

OGNITIVE

I NTRODUCTION

radio (CR) is a promising paradigm that can resolve the problem of the spectrum scarcity

[1]. Cognitive radio network (CRN) consists of primary users (PUs), who are licensed users of the

spectrum, and secondary users (SUs), who can access the spectrum only when they do not cause harmful
interference to PUs [2]. SUs can access the licensed spectrum by three paradigms such as interweave,
underlay and overlay [2]. Interference alignment (IA) is a very powerful technique potential to provide

higher degrees of freedom (DoFs) which means interference-free signaling dimensions at receivers [3]–
[7]. The promising performance of IA can be also attained in CR by suppressing interference at PUs. By
using this technique, SUs easily coexist with primary networks (PNs) by achieving higher data rates and
not causing harmful interference to PUs [8]. Many previous studies introduced the impact of IA on the
performance of the underlay CR [9]–[11]. A multiple-input multiple-output (MIMO) CRN with cooperative
relay was presented in [12], where DoFs for the network was increased by using IA. The authors in
[13] and [14] introduced energy-harvesting (EH) as a promising technology to prolong the life-time of
battery-powered wireless devices. There are two main EH architectures such as time-switching (TS) and
power-splitting (PS) protocols [15]. In [16], the authors proposed a common framework to jointly study IA
and simultaneous wireless information and power transfer (SWIPT) in MIMO networks, where the users
were dynamically chosen as EH terminals to improve the performance of wireless power transfer (WPT)
and information transmission. The work in [17] considered a spectrum sensing policy for the EH-based
CR to ensure that SUs can harvest the energy from the PUs signals in the TS mode. In [18], the same
system was investigated with the optimal information and energy cooperation methods between SUs and
PUs, where SUs harvest energy from PUs, then use that energy to transmit their own and PUs’ signals.
Both [17] and [18] assumed that all nodes in CR have a perfect channel state information (CSI), however,
in practice it is more common to deal with imperfect CSI due to channel estimation errors [19].
Unlike the existing studies and to the best of our knowledge, no work has been addressed to jointly
study these three important areas, namely, CRN, WPT and IA. Therefore, this work focuses on studying
the underlay IA-based CRN where an energy-restricted relay node operates in the time-switching relaying
(TSR) and power-splitting relaying (PSR) modes. The performance of primary receivers and the EH-based
relay of the secondary network (SN) under interference environment is analyzed. Interference at the PUs
and the relay is cancelled by the well-known IA technique. The network capacity for these protocols is
evaluated with different settings of TS/PS portions and under various CSI scenarios.
II.

S YSTEM M ODEL

We consider a system model consisting of two pairs of PUs and three SUs. Within the PN, each
transmitter communicates to its corresponding receiver and causes interference to another primary receiver
and secondary relay node. It is assumed that there is no temperature constraint at the SUs which may imply

N1

1

M1

1

...

...

T1

R1
M2

1

N2

1

...

...

T2

R2
NS

1

NR

1

ND

1

...

...

...

S

R

D

direct links

interfering links

Fig. 1. The IA- and EH-based CRN with two PUs and one SU sharing the spectrum simultaneously.

interference at the primary receivers. The SN consists of a source (S), a relay (R) and a destination (D)
nodes. S communicates with D through the assistance of the energy-constrained R that operates in a halfduplex mode and decodes and forwards (DF) the signal of S to D within two time slots. To support this, R
harvests energy from the interference signals, while S and D are assumed to have external power sources.
Furthermore, D is assumed to be located far from PUs and does not receive any interference. Each node of
the network is assumed to be deployed with multiple antennas as shown in Fig.1, where solid lines indicate
the direct channels while the dotted lines denote the interfering links. We also assume that the channels
remain constant during a transmission block time T , but vary independently from one block to another.
[k]

Channel links between nodes are defined as follows. For the PN case, Hj,i ∈ CNj ×Mi , ∀i, j ∈ {1, 2}
denotes the channel between receiver jth (Rj ) and transmitter ith (Ti ), where superscript k indicates a
certain time period (TP) when the data transmission occurs. Nj and Mi are the numbers of antennas at
Rj and Ti , respectively. For the SN case, HR,S and HD,R denote the channel links related to the S-R and
R-D transmissions while the inter-network channels are given by Hj,R ∈ CNj ×NR , Hj,S ∈ CNj ×NS and
HR,i ∈ CNR ×Mi , where NS , NR and ND denote the numbers of antennas at S, R and D, respectively. Each
entry of any matrix H is assumed to be independent and identically distributed (i.i.d.) random variables
according to CN (0, 1), where CN (0, 1) denotes the complex normal distribution with zero mean and unit
variance. Also, note that each channel link is characterized by the corresponding distance and path-loss
exponent denoted by dm,n and τm,n , ∀m ∈ {1, 2, R, D}, ∀n ∈ A = {1, 2, S, R}, respectively. Moreover, a

global CSI is also assumed.
We assume that the SN and PN nodes exploit the IA technique to mitigate the interference at R and Ri ,
respectively. Thus, any transmit node l with power Pl utilizes a transmit BF matrix Vl ∈ C(Ml

or Nl )×fl

,

with trace{Vl VlH } = 1, ∀l ∈ A, where fl is the number of the transmitted data streams. At the same time,
all receive nodes (except D) exploit the receive BF matrix Ut ∈ CNt ×ft , ∀t ∈ {1, 2, R}, where ft is the
number of data streams to be decoded at the corresponding receiver. For simplicity, we assume that each
node is deployed with N antennas (Mi = Nj = NS = NR = ND = N).
The primary receivers, within two TPs, obtain the following signal
[k]
yj

=

s

s
Pj [k]H [k] [k]
Pi [k]H [k] [k]
[k]
+
T[k]
Hj,j Vj sj +
Hj,i Vi si +ñj , k
τj,j Uj
τj,i Uj
|{z}
dj,j
dj,i
{z
} interference from SN |
{z
}
|

[k]

[k] H

{1, 2}, (1)

interference from PN, i 6= j

desired signal

where ñj = Uj

∈

[k]

nj is the effective zero-mean additive white Gaussian noise (AWGN) vector at the
[k] [k]H

output of the beamformer, with E{ñj ñj

} = σñ2 j I, where E{·} denotes an expectation operator. Since we

assume that sl is the vector containing the symbols drawn from i.i.d. Gaussian input signalling and chosen
from a desired constellation, we have E{sl sH
l } = I, with l ∈ A. Meeting all these conditions satisfies the
average power constraint at each transmit node. Regarding the SN, its interference to PN can be defined as
r

[k]H
PS


 dτj,S Uj Hj,S VS sS , if k = 1,
j,S
(2)
T[k] = r

[k]H
P

R

 τj,R Uj Hj,R VR sR , if k = 2.
dj,R

During the S-R transmission, the received signal at R can be written as
s
s
2
PS H
Pi X H
[1]
yR =
UR HR,i Vi si +ñR ,
τR,S UR HR,S VS sS +
τR,i
dR,S
dR,i i=1
{z
} |
{z
}
|
desired signal

(3)

interference from PN

where ñR = UH
R nR is the effective noise after receive beamforming at the relay.

Then, R decodes the desired signal sS and forwards the information signal to D. Hence, D receives the

following signal
yD =

s

PR
τD,R HD,R VR sR + nD ,
dD,R

(4)

2
where nD is the AWGN noise vector, with E{nD nH
D } = σD I.

The interference is assumed to be completely eliminated if the following conditions are satisfied at Rj
as [20], [21]
[k]H

[k]

[k]

Hj,i Vi = 0, ∀i, j ∈ {1, 2}, ∀i 6= j,


Hj,S VS , if k = 1,
[k]H [k]
[k]
Uj J = 0, where J =

H V , if k = 2,
j,R R


[k]H [k] [k]
rank Uj Hj,j Vj = fj , ∀j ∈ {1, 2},

Uj

(5a)

(5b)

(5c)

and at R as
[1]

UH
R HR,i Vi = 0, ∀i ∈ {1, 2},

rank UH
R HR,S VS = fS .

(6a)
(6b)

A. Beamforming Design
[1]

[2]

The existing interfering signals need to be orthogonalized to Uj and Uj at Rj during two TPs to
satisfy conditions (5). Then, the interference at R needs to be orthogonalized to UR , as R receives the
interference only during the S-R transmission. To decode the desired signal from the received signal, the
interference space should be linearly independent from the desired signal space. Thus, the precoding matrices
need to be designed in such a way that all interference in each receiver span to each others. Thereby,
[1]

[1]

the interference at R1 , R2 and R, in the first TP, can be spanned as span(H1,2 V2 ) = span(H1,S VS ),
[1]

[1]

[1]

[1]

span(H2,1 V1 ) = span(H2,S VS ) and span(HR,1 V1 ) = span(HR,2 V2 ), where span(A) denotes the
[1]

[1]

vector space spanned by the column vectors of A. From these definitions, precoding matrices V1 , V2

and VS can be derived as [22]
[1]

[1]

V2 = (HR,2 )−1 HR,1 V1 ,
[1]

(7a)

[1]

VS = (H2,S )−1 H2,1 V1 ,
[1]

[1]

(7b)
[1]

where V1 = eig (A) and A = (HR,1 )−1 HR,2 (H1,2 )−1 H1,S (H2,S )−1 H2,1 ; eig (A) are the eigenvectors of
A. Now, we derive the corresponding receive BF matrices as


[k]
[k] [k]
Uj = null [Hj,i Vi ]H , j 6= i,


[1]
UR = null [HR,1 V1 ]H .

(8a)
(8b)

During the R-D transmission, S stays silent while R establishes its own communication. Therefore, it
is clear that the BF matrices design follows the same steps given by (7)–(8), and hence they are omitted
for the sake of brevity.
B. Imperfect CSI
The model for imperfect CSI can be written as [4], [23]
Ĥ = H + E,

(9)

where Ĥ is the observed mismatched channel, H ∼ CN (0, I) represents the real channel matrix and E is
the error matrix which represents an inaccuracy degree in the estimated CSI. It is also assumed that E is
independent of H. Considering the nominal signal-to-noise ratio (SNR), θ, E is described as
E ∼ CN (0, λI) with λ = ψθ−κ ,

(10)

where λ is an error variance, κ ≥ 0 and ψ > 0 determine various CSI scenarios. Finally, the real channel
matrix, conditioning on Ĥ, [24], can be described as
H=
λ
I) is independent of Ĥ.
where H̃ ∼ CN (0, 1+λ

1
Ĥ + H̃,
1+λ

(11)

III.

T IME -S WITCHING R ELAYING

The time used for the S-D information transmission is given by T , and the time fraction devoted for EH
purposes is given by αT , with 0 ≤ α ≤ 1. The remaining time (1−α)T is formed by two equal time phases
to support the S-R and R-D transmissions [15]. At the same time, the PN adopts its one-hop transmission
policy according to the SN time frame architecture as follows. The first data transmission occurs during the
(1 + α)T /2 time because within this TP the network transmission is performed by the primary transmitters
and the source node only. The remaining time (1 − α)T /2 is dedicated for the second PN transmission
when the source node is replaced by the relay in the network transmission.
Hence, the received signal at R during the EH phase can be written as
s
s
2
X
PS
Pi
[1]
yR =
τR,S HR,S VS sS +
τR,i HR,i Vi si + nR .
dR,S
dR,i
i=1

(12)

Now, neglecting the power harvested from the noise, the harvested energy at R is derived as [15]
!
2
X
2
Pi
PS
[1]
2
T SR
,
(13)
HR,i Vi
EH = ηαT
τR,S ||HR,S VS || +
τ
dR,S
d R,i
i=1 R,i
where k · k denotes the Euclidean norm.
T SR
The relay transmit power relates to the harvested energy as PRT SR = EH
/((1 − α)T /2) and can be

further rewritten as
PRT SR

2αη
=
1−α

2

X Pi
PS
2
||H
V
||
+
R,S S
τR,S
τ
dR,S
d R,i
i=1 R,i

[1]

HR,i Vi

2

!

,

(14)

where η (0 < η < 1) is the EH conversion efficiency.
During the S-R information transmission, taking into account imperfect CSI given by (11) and receive
BF matrices, the information signal at the relay can be written as
s
s




2
X
PS H
Pi H
1
1
[1]
IT
ĤR,S + H̃R,S VS sS +
ĤR,i + H̃R,i Vi si + ñR .
yR =
τR,S UR
τR,i UR
1+λ
1+λ
dR,S
dR,i
i=1

(15)

After some manipulation, the corresponding signal-to-interference-noise ratio (SINR) at R can be expressed

as
PS
2
||UH
τR,S
R ĤR,S VS ||
(1+λ)2
dR,S

γR =

where IP N =

Pi
τR,i
dR,i

noise power.

P2

i=1

PS
τR,S
dR,S

2
2
||UH
R H̃R,S VS || + IP N + σñR

,

(16)

[1]

2
2
||UH
R H̃R,i Vi || indicates the interference from the PN, and σñR stands for the

Since it is assumed that the PN does not interfere the destination node, the signal received at D can be
written as
yD =

s

PRT SR
τD,R
dD,R




1
ĤD,R + H̃D,R VR sR + nD .
1+λ

(17)

Then, the respective received SINR at D is calculated as
τ

γD =

T SR
PR

D,R
(1+λ)2
dD,R
T SR
PR
τD,R
dD,R

||ĤD,RVR ||2

2
||H̃D,R VR ||2 + σD

,

(18)

2
where σD
is the noise power. Finally, the received SINR for the primary receivers Rj is derived as

τ

[k]
γj

where B [k] =

Pj
τ

j,j
dj,j

[k]H

||Uj

[k]

[k]

H̃j,j Vj ||2 +

=

Pj

j,j
dj,j
(1+λ)2

[k]H

||Uj

[k]

[k]

Ĥj,j Vj ||2

B [k] + F [k] + σñ2 j

[k]H [k] [k] 2
Pi
H̃j,i Vi ||i6=j
τj,i ||Uj
dj,i

[k]

,

(19)

is the intra-network interference of the PN

due to the CSI mismatch while the inter-network interference from the SN is expressed by


[k]H

S
 Pτj,S
||Uj H̃j,S VS ||2 , if k = 1,
d
j,S
F [k] =
T SR

P

 Rτj,R ||U[k]H
H̃j,RVR ||2, if k = 2.
j

(20)

dj,R

IV.

P OWER -S PLITTING R ELAYING

The PSR protocol exploits the time T divided into two equal parts to support the S-R and R-D
information transmissions [25]. In the first TP, R utilizes a portion of the received signal power for EH
purposes, ρ, while the rest of the power, (1 − ρ), is dedicated for the S-R data transmission [26]. Thus,

IT
yR

p
= 1−ρ

s

PS
H
τR,S UR
dR,S



!
s



2
X
1
1
Pi
[1]
H
ĤR,S + H̃R,S VS sS +
ĤR,i + H̃R,i Vi si + ñR .
τR,i UR
1+λ
1+λ
dR,i
i=1

the relay utilizes the energy harvested from the received signal given by
s
s
2
X
ρPS
ρPi
√
[1]
EH
yR =
ρnR .
τR,S HR,S VS sS +
τR,i HR,i Vi si +
dR,S
dR,i
i=1

(24)

(21)

EH
By assuming that the received signal yR
at R is used only for WPT and the noise power is neglected,

the instantaneous harvested energy can be expressed as [15]
P SR
EH

ηρT
=
2

2

X Pi
PS
2
τR,S ||HR,S VS || +
τ
dR,S
d R,i
i=1 R,i

[1]
HR,i Vi

2

!

,

(22)

where 0 < ρ < 1 is the signal power portion dedicated for EH purposes. The relay transmit power as a
P SR
function of the harvested energy is given by PRP SR = 2EH
/T , and it can be then written as [15]
!
2
X
2
P
P
i
S
[1]
2
.
HR,i Vi
PRP SR = ηρ
τR,S ||HR,S VS || +
τ
dR,S
d R,i
i=1 R,i

(23)

Using (11), the same signal, but with the power portion of (1 − ρ) and with the receive BF matrix UR
applied, can be received at the information decoder terminal as shown in (24) at the top of the next page.
Now, by using (24), the SINR of the S-R link can be derived as

γR =

where IP N =

Pi (1−ρ)
τR,i
dR,i

P2

i=1

PS (1−ρ)
2
||UH
τR,S
R ĤR,S VS ||
(1+λ)2
dR,S
PS (1−ρ)
H
2
τR,S ||UR H̃R,S VS ||
dR,S

+ IP N + σñ2 R

,

(25)

[1]

2
||UH
R H̃R,i Vi || denotes the interference from the PN.

Then, the corresponding SINR at D is calculated as

γD =

2
where σD
is the noise power.

P SR
PR
τD,R
dD,R (1+λ)2
P SR
PR
τD,R
D,R
dD,R

||ĤD,RVR ||2

||H̃

2
VR ||2 + σD

,

(26)

Finally, the SINR of the primary users can be calculated as
τ

[k]

γj =
where B [k] =

Pj
τ

j,j
dj,j

[k]H

||Uj

[k]

[k]

H̃j,j Vj ||2 +

Pj

j,j
dj,j
(1+λ)2

[k]H

||Uj

[k]

[k]

Ĥj,j Vj ||2

B [k] + F [k] + σñ2 j

[k]H [k] [k] 2
Pi
H̃j,i Vi ||i6=j
τj,i ||Uj
dj,i

[k]

,

(27)

is the intra-network interference of the PN

due to the CSI mismatch while the inter-network interference from the SN is expressed by


[k]H

S
 Pτj,S
||Uj H̃j,S VS ||2 , if k = 1,
dj,S
[k]
F =
P SR


 PRτj,R ||U[k]H
H̃j,R VR ||2 , if k = 2.
j

(28)

dj,R

V.

E RGODIC C APACITY

For the PN receivers, the general derivation of the instantaneous capacity can be written as [27]
Cj = log2 (1 + γj ),

(29)

where γj is the instantaneous SINR at Rj . Regarding the SN operating in the DF mode, the end-to-end
capacity at D is related to the weakest link of the S-R and R-D transmissions and can be written as [28]
CD =

1
log2 (1 + min (γR , γD )) ,
2

(30)

where γR and γD denote the instantaneous SINR at R and D, respectively.
Now, we derive an expression for the capacity of the TSR-based system. Using (16), (18) and (30), the
capacity of the destination node can be written as
CD =

1−α
log2 (1 + min (γR , γD )) .
2

(31)

Regarding the PN, the capacity of Rj can be written as
Cj =

where E [k] =



 1+α , when k = 1,
2

 1−α , otherwise,
2

2
X
k=1

[k]

E [k] log2 (1 + γj ), j ∈ {1, 2},

with k ∈ {1, 2}.

(32)

Using (25), (26) and (30), the capacity of the destination node for the PSR-based system can be written
as
CD =

1
log2 (1 + min (γR , γD )) .
2

(33)

Regarding the PN, the capacity of Rj can be written using (27) as
2

Cj =

1X
[k]
log2 (1 + γj ), j ∈ {1, 2}.
2 k=1

(34)

We evaluate the capacity of the optimized TSR and PSR-based systems as a function of SNR. To begin
with, we find the optimal values of α and ρ given by α∗ and ρ∗ with η = 0.8 for the corresponding TSR
and PSR protocols by solving the following dCD /dα = 0 and dCD /dρ = 0.
VI.

S IMULATION R ESULTS

In this section, we present numerical examples for the capacity expressions derived above. The adopted
system parameters are as follows: the channel distances and path loss exponents are identical and given by
d = 3 m and τ = 2.7, respectively; the EH conversion efficiency η = 0.8; the fixed transmit powers are
assumed to be equal (P1 = P2 = PS ). The calculated optimal EH time fraction α and power-splitting factor
ρ are taken as 0.19 and 0.75, accordingly. Finally, the values of (κ, ψ) such as (1.5, 15), (1, 10), (0, 0.001)
are considered to describe various CSI mismatch scenarios.
Fig. 2 presents an insight into how the CSI mismatch affects the obtainable capacity of the PUs and
the SU of the considered system model. When κ = 0, it can be noticed that ψ has a significant impact on
the system capacity. At ψ = 0.001 and SNR of 20 dB, the capacity loss of the PUs equals 1.85 bit/s/Hz
while the capacity losses of the destination node for the TSR and PSR protocols are given by 0.09 and 0.17
bit/s/Hz, respectively. Hence, it is obvious that the CSI mismatch severely degrades the performance of the
PUs which can be explained by the higher level of the interference due to the number of the transmitting
nodes. In general, the capacity of the destination node of the PSR-based system always outperforms the
capacity of the TSR protocol even the former relaying method experiences worse performance degradation
than the latter one. When κ 6= 0, it can be noticed that the capacity demonstrates a poor performance in
the low SNR region. On the other side, the dependence on the SNR results in the capacity growth when

2,6

4

8

perfect CSI

2,4

PU

Capacity (bits/s/Hz)

Capacity (bits/s/Hz)

=

TSR

2,2

2,0

6
18,0

18,5

4

19,0

19,5

20,0

=

PU
TSR
PSR

PSR
3

=
2

1

2

0

0
0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

SNR (dB)

10

12

14

16

18

20

SNR (dB)

(a) Perfect CSI vs. CSI mismatch (κ = 0, ψ = 0.001).

(b) CSI mismatch: (κ = 1.5, ψ = 15) vs. (κ = 1, ψ =
10).

Fig. 2. Capacity vs. SNR of the primary user and the destination node operating in the TSR and PSR
protocols under different CSI scenarios.

8

Perfect CSI

Capacity (bit/s/Hz)

7

PSR

6

5

4

3

2

1

TSR
0
0,0

0,2

0,4

0,6

0,8

1,0

EH time fraction
Power splitting factor

Fig. 3. Capacity vs. the EH time fraction and power splitting factor for the TSR and PSR at 20 dB,
respectively.

SNR increases. Being κ large makes the curve slope sharper.
Fig. 3 illustrates some simulated results for the capacities as a function of α and ρ for various CSI
acquisition scenarios. The results for the TSR and PSR systems are obtained from (31) and (33), respectively.
For the case of the PSR-based system, no sufficient portion of power allocated for EH will consequently
result in poor capacity. At the other extreme, being ρ too large which results in too much unnecessarily
harvested energy at the cost of the power level of the received signal leading to poor capacity. Similarly,

this justification applies to α in the TSR-based system.
VII.

C ONCLUSION

In this paper, we analyzed the capacity of different wireless powered IA-based DF CRN considering
the EH protocols, namely, TSR and PSR. The three special scenarios of the imperfect CSI given by
(1.5, 15), (1, 10), (0, 0.001) were studied to analyse the impact of the CSI quality on the system capacity.
The presented results demonstrated that an optimal selection of the power splitting factor in the PSR protocol
and the EH time factor in the TSR protocol were found to be important in achieving the best performance.
Finally, the optimized PSR-based system was shown to have the best performance over the optimized TSR
system.
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