Joint relay selection and adaption of modulation, coding ... - IEEE Xplore

www.ietdl.org Published in IET Communications Received on 12th November 2013 Revised on 22nd February 2014 Accepted on 9th March 2014 doi: 10.1049/iet-com.2013.0948

ISSN 1751-8628

Joint relay selection and adaption of modulation, coding and transmit power for spectral efficiency optimisation in amplify-forward relay network Mehrdad Taki, Mohammad Sadeghi Electrical Engineering Department, University of Isfahan, Iran E-mail: [email protected]

Abstract: Three schemes are designed for joint relay selection and link adaption in amplify and forward relay network which are called RS-CAP (joint relay selection with continuous adaptation of powers), RS-DAP (joint relay selection with discrete adaptation of powers) and OTD-DAP (optimised time division with discrete adaptation of powers). In all schemes spectral efficiency is optimised using adaptive modulation, coding and transmit power subject to average power constraint of each node and a bit error rate constraint of detection. The main idea behind all designs is to save power of relays in inactive time to be utilised for active time. In RS-CAP and RS-DAP, optimised relay selection is done based on channel state information of all corresponding paths, however, in OTD-DAP just an optimised time percentage is allocated to each relay to cooperate in signal transmission. Considerable less channel estimation load is imposed on the receiver in the OTD-DAP scheme compared with the others. In RS-CAP power of nodes are continuously adapted, however, in RS-DAP and OTD-DAP, powers are discrete adaptive. Thus there is a considerable decrease in feedback rate in the RS-DAP and OTD-DAP schemes in comparison with RS-CAP. A trade-off between the performance and channel estimation load and feedback rate is seen in numerical evaluations. Noticeable improvement in achievable rates is seen by using proposed schemes compared with the previous works.

1

Introduction

Using relays to improve spectral efficiency of wireless links and expanding network coverage area is inevitable. Amplify and forward (AF) relays compared with other types at which signal detection is required, are less complicated and the implementation is less costly, hence there are utilisable widely [1]. When there is not precise synchronisation and there are multiple relays in a single hop network, received signals from different relays are to be orthogonal in order to prevent interference. In this case, to decrease performance degradation caused by signal space division, optimised selection of a limited set of relays is significant. If transmit powers are constant and relays are selected based on channel state information (CSI) of all corresponding paths, in almost all cases one or more specific relay(s) might be selected with respect to a network structure. However, relays often have independent power supplies with average power constraint, therefore, at inactive time power could be saved for active time, improving SNR of corresponding path. With power adaptation all relays may contribute in amplification, thus the resulted diversity improves the performance considerably. In this wireless system, link adaptation is an efficient tool employing rate and power adaptation to overcome fading [2]. In this article three new schemes are designed for joint relay selection and link adaptation with their own pros and IET Commun., 2014, Vol. 8, Iss. 11, pp. 1955–1964 doi: 10.1049/iet-com.2013.0948

cons. In all schemes, discrete rate adaptation is used which is implemented by adaptive modulation and coding (AMC). A bit error rate (BER) constraint of detection is also provisioned. The goal is to optimise overall spectral efficiency (average transmission rate per unit bandwidth) between the transmitter and the final receiver subject to independent average power constraints for all nodes (transmitter and relays). In the first scheme, transmission powers of all nodes are continuously adapted based on feedback SNRs of channels. In the second and the third schemes, power of the transmitter is constant and power of the relays are discrete adaptive which result in a considerable decrease in feedback rate in comparison with the first. In the first and the second schemes, relay selection is done based on CSI of all paths, however, in the third scheme relay selection is independent of CSIs and the active time percentage of each relay is optimised based on average SNRs and power constraints. The channel estimation load and feedback rate decrease from the first scheme to the third at the cost of decrease in overall average spectral efficiency. Performance comparison of designed schemes with the previous joint relay selection and link adaptation schemes shows a considerable raise by the first and the second schemes. The third scheme has a comparable performance with the previously presented schemes, however with a considerable decrease in channel estimation and feedback load. In the following, we first 1955

& The Institution of Engineering and Technology 2014

www.ietdl.org study related principles and review the previous works, then we present contributions and structure of the paper. 1.1

Review of previous related works

1.1.1 AF relay selection: Using multiple relays in a single hop network improves the received SNR and results in space diversity, however, to prevent interference, transmitted signals via relays are to be orthogonal unless there is a precise synchronisation. Dividing signal space into orthogonal subspaces substantially decreases the performance. Hence, selecting an optimised limited set of relays plays a significant role in improving network performance. In [3–15], AF relay selection schemes are presented, in [3–8] transmission powers are constant and in [9–15] powers are adapted in a way that sum of instantaneous transmitter’s and relays’ powers is lower than a specific level. In [15] relays’ powers are constant and transmitter’s power is continuously adapted. With constant powers, optimised relay selection based on the links’ SNRs in almost all cases would lead to select a specific number of relays according to the network structure. In practice, relays and transmitter are disjoint transceivers with independent power supplies subject to average constraints. Power of relays maybe saved during inactive time to be utilised in active time considering average power constraints. By power adaptation all relays may contribute to transmission. According to the author’s knowledge, no scheme has been yet presented for joint AF relay selection and adaptation of transmission powers considering independent average power constraint for each node. 1.1.2 Link adaptation: Link adaptation is an efficient tool for optimising performance of wireless links and overcoming the effect of multipath fading in which transmission rate is adapted based on channel’s SNR [2]. Link adaptation is in continuous or discrete form. Capacity achieving codes with tending to zero error probability are used for continuous link adaptation where it is assumed that transmission rate is adaptable continuously and instantaneously based on the links’ SNR [2]. Although continuous link adaptation gives a good view of system performance, however, it is not implementable in practical systems where low quantity of BER is accepted and transmission rate is chosen from a limited set of discrete rates. Discrete rate adaptation may be implemented by adaptive modulation (AM) [2] or using AMC [16]. with AM, transmit rate is estimated with a closed form continuous function, thus the design of transmission scenario becomes similar to continuous link adaptation [2]. However, utilisation of coding improves the achievable rates considerably at the cost of different and more complicated design. In parallel with rate adaptation, transmission power remains constant or may be adapted. In a point to point link, it is indicated that continuous power adaptation compared with constant power transmission improves the achievable spectral efficiency [2]. In [17] a transmission scheme is designed to optimise spectral efficiency in a AF-relay network using AMC and continuous power adaptation. In [18–20], the performance of network with a single AF relay is analysed when continuous rate adaptation is utilised. In [17] there is also a comprehensive review on the previous works in which link adaptation is employed for networks containing a single AF relay. 1956 & The Institution of Engineering and Technology 2014

1.2

Paper contributions and structure

In this article a network including a transmitter, a receiver and several AF relays is considered in which there is no direct link between the transmitter and the receiver. The goal is to maximise average spectral efficiency between the transmitter and the receiver. To that end, three new schemes for joint relay selection and link adaptation are designed. In all schemes, discrete rate adaptation using AMC is utilised at the transmitter with a detection BER constraint. In the first scheme, power of the relays and of the transmitter are continuously adapted jointly with optimised relay selection. Independent average power constraints for the relays and for the transmitter are provisioned. The optimisation is done at the receiver based on the SNRs of all paths. The index of selected relay, the index of transmission rate and the power of transmitter are sent to the relays and to the transmitter. A function is derived for the optimised power of each relay based on the SNRs of its channels towards the transmitter and towards the receiver, thus there is no need to have feedback on relay power. This scheme is called joint relay selection with continuous adaptation of powers (RS-CAP) in the followings. Decreasing the feedback rate because of the continuous adaptation of transmitter’s power, the second scheme is presented in which discrete adaptive power for the relays and constant power for the transmitter are employed. The optimised relay selection in this scheme is performed based on CSI of all corresponding paths. Two power levels corresponding to active and inactive time are set for each relay. The power levels are optimised at the beginning of transmission and just the index of transmission mode and the index of selected relay is to be sent back during the transmission. This scheme is called joint relay selection with discrete adaptation of powers (RS-DAP) in the followings. It should be noted CSIs of all paths are needed for optimised relay selection. To decrease the channel estimation load and also benefit from the saved power during inactive time, the third scheme is designed in which relay selection is independent of CSIs, however, the active time percentage for each relay is optimised based on average SNRs of the channels and average power constraints. Power of the transmitter is constant and power of the relays are discrete adaptive as in RS-DAP scheme. At the start of transmission, power levels and active time percentages are optimised at the receiver and they are sent back to the relays. During transmission only the index of transmission mode is sent back towards the transmitter. This scheme is called optimised time division with discrete adaptation of powers (OTD-DAP). To set a comparison base, we consider schemes in [3–8] which are using AM with constant powers and are applicable in practical systems. Similar to these schemes, a new scheme is designed at which powers are constant, AMC is utilised and relay selection is done based on all CSIs. This scheme is called relay selection with constant powers (RS-CP). Numerical evaluations show designed RS-CAP and RS-DAP schemes have considerable outperformance compared with RS-CP scheme. Also OTD-DAP has a comparable performance with RS-CAP, however with considerable less channel estimation load. The rest of this paper is organised as follows; Section 2 is devoted to preliminaries where notation procedure, system model, relaying schemes, available schemes for AF relay IET Commun., 2014, Vol. 8, Iss. 11, pp. 1955–1964 doi: 10.1049/iet-com.2013.0948

www.ietdl.org selection and link adaptation method are explained. Newly designed schemes RS-CAP, RS-DAP and OTO-DAP are respectively presented in Sections 3–5. In Section 6, to acquire a comparison basis, RS-CP scheme is studied. Numerical evaluations are presented in Sections 7 and 8 consists of the paper’s conclusion.

2 2.1

Preliminaries Notations

In this paper, lowercase letters denote variables, for example, z and constants are shown with uppercase letters, for example, N. Joint PDF of random variables x1, x2, … is specified with fx1, x2,... (x1 , x2 , . . . ) and statistical average of function g(x1, x2, …) is shown by Ex1, x2,... g x1 , x2 , . . . that is computed as follows: Ex1, x2,... g x1 , x2 , . . . = g x1 , x2 , . . . fx1, x2,... (x1 , x2 , . . . ) dx1 dx2 . . .

2.2

(1)

System description and channel model

Fig. 1 shows the considered network; transmitter (Tx) sends data to destination (Dx). According to unavailability of direct link between Tx and Dx, one or a number of relays, Rxi‘1 ≤ i ≤ N, are to be utilised. To prevent interference, transmitter’s and relays’ signal are to be orthogonal. Moreover, as precise phase or frequency synchronisation between signals of different relays is impossible in practice, in order to prevent interference, relays’ signals are to be orthogonal. Therefore the signal space is to be divided into Na + 1 orthogonal subspaces where Na is the number of active relays. Channels between the relays and the transmitter/receiver are supposed to be time variant with quasistatic Rayleigh fading in which channel gain remains constant during a block (a number of codewords) and changes from one block to the other one independently. More complex characteristics for the channels as in [21, 22] may be applied with a difference in mathematics. Powers of the transmitter and of the relays are expressed by p0 and pi, 1 ≤ i ≤ N, respectively. It is assumed that complex channel coefficient between the ith relay and the transmitter/receiver is expressed in term of hi /h′i ; the variance of noise at the receiver and at the relays is denoted by s02 . Thus normalised SNR of channel (normalised respect to transmission power in a channel) between the ith relay and |h |2 |h′ |2 the transmitter/receiver equals to si = i 2 /s′i = i 2 . The s0 s0 total bandwidth of the system is assumed 2 Hz which is

equally divided for orthogonal transmission transmitter and the relays. 2.3

of

the

Link adaptation

Using continuous link adaptation, instantaneous transmission rate is continuously adapted based on the instantaneous value of the received SNR. Transmission rate per unit bandwidth equals log2(1 + γeq) (bit/sec/Hz), where γeq = pseq denotes the received SNR and seq is the normalised SNR of equivalent link. In practical systems because of the hardware limitation, a limited set of transmission rates are available and a low level of BER is acceptable. Various rates may be implemented by AMC in which there are M + 1 transmission modes, each that is implemented by a modulation and a coding scheme. Transmissions modes depending on their rate are sorted as 0 = R0 < R1 < R2… < RM where ‘0’ mode states the case that no information is transmitted. Coding significantly improves the performance. An example of AMC utilisation is in IEEE802.11 standard where BER in mth mode may be estimated as follows pe pseq , Rm = Am exp −A′m × pseq ,

0 ≤ seq

(2)

That Am , A′m are mode dependent constant. In mth transmission mode, the minimum SNR required to guarantee that maximum BER is B0, is computed as follows pe pseq , Rm ≤ B0 ⇒ pseq ≥ gB0 Rm gB0 Rm def − 1/A′m × ln B0 /Am , B0 ≤ Am 2.4

(3)

AF relaying and relay selection algorithms

In AF relaying with the aid of ith relay, the received SNR using the maximum ratio combining is obtained as follows [24]

gi =

p0 si pi s′i p0 si pi s′i ≃ p0 si + pi s′i + 1 p0 si + pi s′i

(4)

The above approximation is valid while p0 si ≫ 1 or pi s′i ≫ 1 [24]. Hence the total received SNR is computed as follows,

geq =

N

ai gi

(5)

i=1

where variable ai shows the ith relay activity, that is, if the mentioned relay is active ai = 1 otherwise ai = 0. Using continuous rate adaptation, transmissible rate in equal link between the transmitter and the receiver is upper bounded as follows 1 1 log(1 + geq ) ≤ log 1 + Na gmax ; Na + 1 Na + 1

(6)

gmax = max gi i

Fig. 1 Network structure IET Commun., 2014, Vol. 8, Iss. 11, pp. 1955–1964 doi: 10.1049/iet-com.2013.0948

where

1 / (Na + 1)

coefficient

is

added

to

consider 1957


www.ietdl.org orthogonality of signals. Considering presented upper bound in above equation it could be concluded that maximum achievable rate is a descending function of Na, in other words, even if adding a relay cause maximum increase in total SNR, total rate will be decreased because of signal space division. For this reason when it is impossible to precisely synchronise between signals of relays, just one relay with the most corresponding SNR path, is associated in signal amplification. With constant powers, relay selection criterion is considered as follows i∗ = max i

ai∗ = 1;

p0 si pi s′i ′ p0 si + pi si

aj = 0,

(7) ∗

j=i ,

1≤j≤N

3.1 Relay’s power allocation and relay selection strategy In the light of results in appendix, power of relays are assigned as follows pi = q′i p0 ;

k;ai ,pi ,0≤i≤N

i

k e R0 , . . . , RM ,

i∗ = arg max ci ; i

3.2

p0 =

N i=1

ai

N si q′i s′i = ai ci ; si + q′i s′i i=1

1958 & The Institution of Engineering and Technology 2014

i = i∗

ai = 0,

(11)

gB0 (k)

(12)

seq

Problem reformulation and solution

According to assigned powers and rate in the previous sections, average spectral efficiency of equal link and average power of the transmitter and of the relays are respectively computed as follows

(8)

where C(N + 3) constraint assures at most one of the relays is active all the time. The above problem is a constrained mixed integer optimisation problem which its solution is not straightforward. Therefore the same problem with continuous rate adaptation is solved in appendix. Acquired results give an idea to solve (8). The problem is solved in three steps as follows. At first, a general appropriate function for the power of relays and a relay selection strategy are determined, next rate and power assignment strategy for the transmitter are presented and finally the problem is reformulated and its solution is presented.

ai∗ = 1,

Power and rate assignment for the Transmitter

E{k} =

N M i∗ =1 m=1

i=1

seq =

(9)

Inspired from what is done in [16] for a point to point link, to determine the transmission rate, seq axis is divided into M + 1 nonoverlapping adjacent intervals using thresholds 0 = L0 , L1 . . . , Lm , 1 and if seq e [Lm , Lm+1 ) transmission rate is set as k = Rm. To provide the BER constraint with respect to (3), it is required that p0seq ≤ gB0(k) where because of the limited power, the constraint is satisfied with equality, as follows

3.3 subject to :

⎧ 0 C(1):Esi ,s′ ;1≤i≤N p0 ≤ P ⎪ ⎪ i ⎪ ⎪ ⎪ i ⎪ C(2) . . . C(N + 1):Esi ,s′ ;1≤i≤N pi ≤ P ⎪ i ⎨ C(N + 2):pe geq , k ≤ B0 ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ ⎪ ⎩ C(N + 3): ai = 1

1≤i≤N

With respect to the goal of spectral efficiency maximisation with only one relay, appropriate relay is one that has a maximum corresponding normalised SNR, as follows

When transmission powers are constant, relay selection based on (7) results in selection of a specific relay in almost all cases. Adapting power of relays considering average constraint could lead to relays saving their power during inactive time and improving SNR of their corresponding path during active time, thus all relays may be associated in signal amplification. Designing transmission scheme to maximise the average spectral efficiency between the Tx and Dx by using joint continuous power adaptation of all nodes, discrete adaptation of transmission rate and optimised relay selection is the point of this section. For each set of s′i , si , 1 ≤ i ≤ N , transmission rate (k), power of transmitter ( p0), power of relays ( pi, 1 ≤ i ≤ N) and active relay (ai, 1 ≤ i ≤ N) are set in a way that detection BER is lower than B0, average power of transmitter is lower 0 and average power of ith relay is lower than than P Pi , 1 ≤ i ≤ N . The problem is formulated as follows Esi ,s′ ;1≤i≤N {k},

si , s′i

1 bi

where βi is a constant which is determined in optimisation process. Based on the assigned powers to the relays, the normalised SNR of equal link between Tx and Dx (normalised to the transmit power) is as follows (see (10))

3 Spectral efficiency optimisation using RS-CAP

max

q′i def

=

N M i∗ =1 m=1

=

N M i∗ =1 m=1

Rm Pr Lm ≤ ci∗ , Lm+1 ; ci∗ . cj , j = i∗ x N

Lm+1 Rm

Lm

fci∗ (x)

j=1,j= i∗

Lm+1 Rm

Lm

fci∗ (x)

N j=1,j=i∗

0

fcj (y) dy dx

Fcj (x) dx (13)

N M gB0 Rm E p0 = i∗ =1 m=1

Lm+1 Lm

N 1 fci∗ (x) Fcj (x) dx x j=1,j= i∗

(14)

ci def

si si = 1 + b2i q′i 1 + bi si /s′i

(10)

IET Commun., 2014, Vol. 8, Iss. 11, pp. 1955–1964 doi: 10.1049/iet-com.2013.0948

www.ietdl.org M E pi = gB0 Rm m=1

Lm+1 1 Lm

0

N y fci ,q′ (x, y) Fcj (x) dy dx i x j=1,j= i

(15)

In above equations, Pr{A} specifies the probability of event A and Fcj (x) shows the cumulative density function of random variable cj, that is, Pr{cj ≤ x}. Thus, (8) is reformulated as follows max Esi ,s′ ;1≤i≤N {k},

subject

i

tm ,1≤m≤M

to:

0 C(1):Esi ,s′ ;1≤i≤N p0 ≤ P

i i C(2) . . . C(N + 1):Esi ,s′ ;1≤i≤N pi ≤ P

(16)

equality. Finding lis is not straightforward, however, it can be inferred from (18) and (36) that there is a direct relevance between βis and lis, so as to maintain mentioned constraints, βis may be set. Also to solve the problem, Fci (.) and fci ,q′ (., .) are needed, that are computed in i appendix. Remark 1: In the RS-CAP scheme, the optimisation problem may be solved at the receiver before the start of transmission and βi (0 ≤ i ≤ N) are sent back to the relays and the transmitter. During the transmission to assign instantaneous transmission rate and power of the transmitter based on Section 3.2, it is required that normalised SNR of equal link is sent back to the transmitter. The index of active relay is determined based on (11) at the receiver and is sent to the relays. However, each relay sets its power based on SNRs of the channels from itself to the transmitter and to the receiver based on (9).

i

where C(N + 2) and C(N + 3) constraints in (8), are now considered in the power assignment (12) and relay selection strategy (11). Problem (16) is a constrained optimisation problem for which the Lagrangian multipliers method is applicable. The Lagrangian is formed as follows L=

N M

Lm+1 1 Rm

i=1 m=1

− l0

Lm

N M

0

i

gB0 Rm

i=1 m=1

×

N

N

fci ,q′ (x, y)

Fcj (x) dy dx

j=1,j=i

Lm+1 1 Lm

0

1 f ′ (x, y) x ci ,qi

0 Fcj (x) dy dx − P

j=1,j=i

−

N

li

i=1

×

N

M

gB0 Rm

Lm+1 1 Lm

m=1

0

y f ′ (x, y) x ci ,qi

4 Spectral efficiency optimisation using RS-DAP To decrease the feedback load because of continuous adaptation of transmitter’s power in the RS-CAP scheme, a new scheme is designed in this section in which power of transmitter is constant and power of relays are discrete adaptive. To benefit from the saved power during inactive time, each relay operates with two power levels corresponding to active and inactive portions of the time. Power levels are optimised at the beginning of transmission based on the average SNRs and power constraints. To overcome fading, transmission rate is discretely adapted and relay selection is performed at the receiver based on CSI of all paths. During transmission, just the index of active relay and the index of transmission mode are to be provided feedback for. In the following, at first the problem is formulated, next the solution is presented. 4.1

i Fcj (x) dy dx − P

j=1,j=i

(17) Optimum points happen where ∇L = 0 or where ∇L is undefined (like the beginning and the end of acceptable intervals for each variable): ∂L = 0; 1 ≤ m ≤ M ⇒ (18) ∂Lm gB0 Rm − gB0 Rm−1 ; Lm = b0 Rm − Rm−1 1 N yfci ,q′ tm , y dy 0 i ; 1≤m≤M b0 = l0 + li 1 f ′ t , y dy i=1 0 ci ,q m ∇L = 0 ⇒

Problem formulation

The goal is to set discrete transmission rate (k) and active relay (ai, 1 ≤ i ≤ N) based on channel’s SNRs to maximise average spectral efficiency. Power of ith relay is assumed to be Pi when it contributes to transmission and is zero when it is inactive. Power levels are optimised somehow that i . The optimisation problem average power is lower than P is formulated as follows max

k; ai ,Pi ,0≤i≤N

Esi ,s′ ;1≤i≤N {k}, i

k e R0 , . . . , RM ,

subject to:

⎧ C(1) . . . C(N ):Pi Fi ≤ P ⎪ ⎪ i ⎪ ⎪ ⎨ C(N + 1):pe g , k ≤ B eq 0 ⎪ N ⎪ ⎪ ⎪ ai = 1 ⎩ C(N + 2):

(19)

i=1

i

where Φi is probability of ith relay is being active. To find final solution, Lagrangian multipliers li, 0 ≤ i ≤ N are to be set. Because the spectral efficiency is increasing by increasing the powers, all power constraints are active. Therefore Lagrangian multipliers are set such that average power constraints C(2)…C(N + 1) in (16) are satisfied with IET Commun., 2014, Vol. 8, Iss. 11, pp. 1955–1964 doi: 10.1049/iet-com.2013.0948

4.2

Problem solution

As the goal is maximising average spectral efficiency, the selected relay is which with the maximum SNR of the 1959


www.ietdl.org sent to the relays. During the transmission, the index of active relay and transmission mode is to be sent back to the relays and to the transmitter.

corresponding path, that is, 0 si Pi s′i P ; i = arg max gi = i P0 si + Pi s′i ∗

ai∗ = 1, ai = 0, i = i∗ (20)

5 Spectral efficiency optimisation with OTD-DAP

Thus Φi is computed as follows

Fi = Prob gi ≥ gj , ∀j = i =

1 0

fgi (x)

x N

0

j=1,j=i

In the presented RS-CAP and RS-DAP schemes, optimised relay selection is done based on CSI of all paths. To decrease the channel estimation load at the receiver, a new scheme is designed in this section in which relay section is independent of CSIs, however, active time percentage for each relay is optimised based on average SNRs and power constraints. Similar to RS-DAP scheme, power of the transmitter is constant and power of the relays are discrete adaptive. Assigned power levels to the relays are optimised at the start of transmission based on average SNRs and average power constraints. Thus the idea of utilising saved power during inactive time is also implemented in this scheme. To overcome fading, transmission rate is discretely adapted and during transmission, just the index of transmission mode is to be sent back. In the following, at first the problem is formulated, next the solution is presented.

(21)

fgj (y) dy dx

If ith relay is active, to satisfy BER constraint with respect to (3), transmission rate is to set as gB0 Rm ≤ gi , gB0 Rm+1 ⇒ k = Rm

(22)

Thus, the average spectral efficiency is computed as

Esi ,s′ ;1≤i≤N {k} = i

N x j=1,j=i

0

N M

gB (Rm+1 ) 0 Rm

i=1 m=1

fgj (y) dy dx

gB 0 (R m )

fgi (x)

5.1

Instantaneous transmission rate when ith relay is active, ki, is to be set based on channels’ SNRs to maximise average spectral efficiency. The ith relay is active during ti percentage of time with power of Pi′ , 1 ≤ i ≤ N and is inactive in the remaining time. The design goal is to maximise average spectral efficiency in a way that average i , 1 ≤ i ≤ N and detection power of ith relay is lower than P BER is lower than B0. The problem is formulated as follows

(23)

With Rayleigh fading channels that si , s′i ; 1 ≤ i ≤ N are exponentially distributed random variables with si = E si and s′i = E s′i , fgi (x) is computed as follows [24] 1 1 ′ −x + ′ 2x P + P s s i i i P0si Pisi 0 fgi (x) = e P0si Pis′i P0si Pis′i 2x 2x × K1 ′ + 2K0 ′ P0si Pisi P0si Pisi

Problem formulation

max ′

ti ,Pi ,1≤i≤N

eq

5.2

i=1 m=1

C(1). . .C(N ):Pi

gB0 (Rm )

0

Pi′ =

fgi (x)

fgi (x)

j=1,j=i


N x j=1,j=i

N x

1

0

0

To satisfy average power constraints (C(2), …, C(N + 1)) and full utilisation of available resources, power levels are set as

gB (Rm+1 ) 0 Rm

i

(26)

Problem solution

Remark 2: In RS-CAP scheme in this section, the optimisation problem is solved at the receiver before the start of transmission and the power levels Pi(1 ≤ i ≤ N) are

max

i

i C(2) . . . C(N + 1):Pi′ ti ≤ P ⎪ ⎪ ⎪ ⎪ ⎩ C(N + 2):pe g , k ≤ B , 1 ≤ i ≤ N

where C(N + 1) and C(N + 2) in (19) are now considered in the rate assignment (22) and relay selection strategy (20). The optimisation problem (25) may be solved using the active-set method in which a line search is performed by updating the Hessian matrix of the Lagrangian [23].

k;ai ,Pi ,0≤i≤N

i=1

ti E{si ,s′ } ki subject to :

⎧ N ⎪ ⎪ ⎪ C(1): ti = 1 ⎪ ⎨ i=1

(24)

Thus (19) is reformulated as follows (see (25))

N M

N

0

fgj (y) dy dx,

i fgj (y) dy dx ≤ P

i P , ti

1≤i≤N

subject

(27)

to: (25)


www.ietdl.org To provide BER constraint with respect to (3), if ith relay is active, transmission rate is set as s P ′ s′ P gB0 Rm ≤ g′i = 0 i i ′i ′ , gB0 Rm+1 ⇒ ki = Rm P 0 si + P i si (28) Thus, the average spectral efficiency is computed as follows N P

i ′ P i i=1

N Pi M Esi ,s′ ki = Rm i P′ i=1 i m=1

gB (Rm+1 ) 0 gB 0 (R m )

fg′ (x) dx i

(29)

where fg′ (x) is computed as in (24) in which Pi is replaced i with Pi′ . Finally, (26) is reformulated as follows N P

i ′ P Pi ,1≤i≤N i=1 i

max ′

C(1) :

N P

i ′ P i=1 i

Esi ,s′ ki i

(30)

≤1

where C(2),…, C(N + 2) constraints in (26) are now considered in determination of power levels and rate assignment strategy. However, still solving (30) is not straightforward. To simplify the solution, average transmission rate using each of relays is described with a closed form function based on average SNRs of corresponding channels and average powers as follows Esi ,s′ ki i

M

s P ′ s′ P Rm Pr gB0 Rm ≤ 0 i i ′i ′ , gB0 Rm+1 s + P s P 0 i i i m=1 ′ ′ s P s P ≃ A1 log 1 + A2 0 i i ′i ′ , 1 ≤ i ≤ N P0si + Pisi =

Fig. 2 Average spectral efficiency for different values of S where 1 = P 0 = 1; Comparison of Exact and approximate s1 = s′1 = S, P values in (31)

(31)

where parameters A1, A2 depend on {Rm, 1 ≤ m ≤ M} and B0 that are obtained by curve fitting. Our extensive numerical evaluations which part of them is seen in Fig. 2 show that the results of the above approximate function are close to the exact values. Using the closed form function in (31), the problem V may be simply solved by the active-set method. Remark 3: In OTD-DAP scheme, the power levels Pi′ (1 ≤ i ≤ N ) are determined by solving (30) before the start of transmission and are sent to the relays. During the transmission just the index of transmission mode is to be sent back. Moreover, no channel estimation is required to set the active relay and each relay is activated during its optimised assigned time percentage. Remark 4: The OTD-DAP scheme may be compared with a scheme in which time is equally shared among relays, that is, ti = 1 / N, 1 ≤ i ≤ N. The scheme is called as Time Division with constant powers (TD-CP) by which average spectral efficiency is computed as follows N N M 1 1 Esj ,s′ kj = R j N j=1 N j=1 m=1 m

gB (Rm+1 ) 0 gB0 (Rm )

fg′′ (x) dx


j

(32)

0 sj P j s′j P where gj′′ = j s′j and fgj′′ (x) is computed as in (24) in P 0 sj + P j , 1 ≤ j ≤ N. which Pj is replaced with P

6 Spectral efficiency optimisation using RS-CP The previous schemes which are closed to the presented ones in this paper are those in [3–8] at which transmission powers are constant, adaptive modulation is utilised and relay selection is done based on CSI of all paths as in (7). To have a comparison base, these schemes are redesigned by using AMC instead of AM. It is assumed that transmitter’s 0 , power of ith relay is always constant power is P Pi , 1 ≤ i ≤ N , maximum BER is B0 and relay selection is done based on (7). The scheme is called RS-CP and its resulted average spectral efficiency is as follows N M

0 si Pi s′i P Rm Pr gB0 Rm ≤ . i 0 si + Pi s′i P i=1 m=1 ! 0 sj Pj s′j 0 si Pi s′i P P . ∀j = i , gB0 Rm+1 , 0 si + Pi s′i P 0 sj + Pj s′j P

Esi ,s′ ;1≤i≤N {k} =

(33) Remark 5: In RS-CP scheme, during the transmission the index of transmission mode is to be sent back to the transmitter. Moreover, the index of active relay is set based on CSIs of all paths and is sent to the relays.

7

Numerical evaluations

The goal of this section is to evaluate the performance of designed schemes; RS-CAP, RS-DAP, OTD-DAP and RS-CP. For discrete rate adaptation, transmission modes are selected from IEEE 802.11a standard. There are eight modes which are set up using different convolutional codes and QAM modulations resulting in a set of discrete rates as {0, 0.5, 0.75, 1, 1.5, 2, 3, 4}. Parameters of BER estimation 1961


www.ietdl.org in (2) are extracted from [17]. Maximum acceptable BER is considered as B0 = 10−5. Distance between the ith relay and the transmitter or the receiver are respectively denoted by di or di′ , 1 ≤ i ≤ N. A large scale path loss model with path loss exponent 3 is adopted such that si = E si = K0 /di3 and s′i = E s′i = K0 /di′3 . To examine achievable spectral efficiencies by different schemes and evaluate proficiency of proposed schemes, several setups are considered based on the network structure in Fig. 1, however a limited set of simulations are reported here as follows. In Fig. 3, a network with three relays is considered in which i = 1, 1 ≤ i ≤ 3 and P 0 = 1. The spectral si = s′i = S0 , P efficiency is plotted against different values of S0. The goal of evaluations is to analyse the performance of different schemes for a wide range of average SNRs. In Fig. 4, a network containing two relays is studied 1 = P 2 = 1 and s1 = s′1 = 6 dB. The overall where P0 = P

Fig. 3 Comparison of achievable spectral efficiencies by different schemes in a network with 3 relays for different values of S0 where i = 1, 1 ≤ i ≤ 3 and P 0 = 1 si = s′i = S0 , P

Fig. 4 Comparison of achievable rates by different schemes in a dual-relay network for different values of S2 where s2 = s′2 = S2 , 1 = P 2 = 1 and s1 = s′1 = 6 dB P0 = P 1962 & The Institution of Engineering and Technology 2014

spectral efficiency is plotted for different values of s2 = s′2 = S2 . The evaluations are done to examine the performance of different schemes in a network with heterogeneous paths. In Fig. 5, a network with N relays is considered in which i = 1, si = s′i = 3 dB for 1 ≤ i ≤ N and P0 = 1. The P proficiency of designed schemes is evaluated for different number of relays. In Fig. 6, for a dual-relay network it is assumed that sum of distances between each relay to the transmitter and towards the receiver is fixed; that is, d1 + d1′ = d2 + d2′ = D and also d1 = d2 = D0. The spectral efficiency is plotted for different values of D0. The goal is to find the best position of the relays in a fixed-length path. Considering numerical results in Figs. 3–6 the following insightful points are concluded.

Fig. 5 Comparison of achievable rates by different schemes in a i = 1, N-relay network for different number of relays, where P si = s′i = 3 dB for 1 ≤ i ≤ N and P0 = 1

Fig. 6 Overall spectral efficiency between the transmitter and the receiver for different values of D0 (distance between the transmitter and each of the relays) where d1 + d1′ = d2 + d2′ = D, d1 = d2 = D0 i = 1, si = s′i = 3 dB for 1 ≤ i ≤ N and P0 = 1 P IET Commun., 2014, Vol. 8, Iss. 11, pp. 1955–1964 doi: 10.1049/iet-com.2013.0948

www.ietdl.org † The efficiency of designed RS-CAP, RS-DAP, OTD-DAP and RS-CP schemes is proved. † As can be observed the best performance is achieved by RS-CAP scheme. Considerable performance gains is obtained if relay selection is done based on the SNRs of all paths and powers are continuously adapted. † The second place in performance is devoted to RS-DAP scheme with a quite less feedback rate compared with RS-CAP. The channel estimation load and feedback rate in this scheme is similar to RS-CP, however, a considerable gain is achieved by utilising the saved power during inactive time. † The OTD-DAP scheme provides a comparable performance with RS-CP, without the need to SNR estimation of all paths, that is, utilising the saved powers during inactive time may compensate imperfect relay selection. † The worst performance is achieved by TD-CP scheme. † As seen in Fig. 5, the spectral efficiency between the transmitter and the receiver raises by increasing the number of relays due the increase of diversity. † Considering Fig. 6, it could be concluded that the best place for the relays in a fixed-length path is at the middle.

8

Conclusion

8 9 10 11 12 13 14 15 16 17

Joint relay selection and link adaptation schemes in an AF relay network was designed. As relays and transmitter have independent power supplies with average power constraints, each device may save its power during inactive time to be utilised in active time. Considering this idea, three novel schemes were presented called as RS-CAP, RS-DAP and OTD-DAP schemes. A trade-off between channel estimation load, feedback rate and the achievable spectral efficiency is observed in the designed schemes. Outperformance of the proposed designs compared with the previous schemes was shown by several numerical evaluations. Other fruitful research directions include design of practical schemes in the relay network based on imperfect CSI and more complex channel characteristics.

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25

Acknowledgment

19 20 21 22 23 24

distributed cooperative paths’, IEEE Trans. Wirel. Commun., 2009, 8, (4), pp. 1953–1961 Zhi, H., Yang, L., Zhu, H.: ‘Jointing adaptive modulation relay selection protocols for two-way opportunistic relaying systems with amplify-and-forward policy’, Wirel. Pers. Commun., 2012, pp. 1–21 Zhao, Y., Adve, R., Lim, T.J.: ‘Improving amplify-and-forward relay networks: optimal power allocation versus selection’, IEEE Trans. Wirel. Commun., 2007, 6, (8), pp. 3114–3123 Ahmed, I., Ikhlef, A., Schober, R., Mallik, R.: ‘Joint power allocation and relay selection in energy harvesting AF relay systems’, IEEE Wirel. Commun. Lett., 2013, 2, (2), pp. 239–242 Ng, T.C.Y., Yu, W.: ‘Joint optimisation of relay strategies and resource allocations in cooperative cellular networks’, IEEE J. Sel. Areas Commun., 2007, 25, (2), pp. 328–339 Ma, Y., Yi, N., Tafazolli, R.: ‘Bit and power loading for OFDM-based three-node relaying communications’, IEEE Trans. Signal Process., 2008, 56, (7), pp. 3236–3247 Cai, J., Shen, X., Mark, J.W., Alfa, A.S.: ‘Semi-distributed user relaying algorithm for amplify-and-forward wireless relay networks’, IEEE Trans. Wirel. Commun., 2008, 7, (4), pp. 1348–1357 Song, L.: ‘Relay selection for two-way relaying with amplify-and-forward protocols’, IEEE Trans. Veh. Technol., 2011, 60, (4), pp. 1954–1959 Torabi, M., Haccoun, D.: ‘Capacity of amplify-and-forward selective relaying with adaptive transmission under outdated channel Information’, IEEE Trans. Veh. Technol., 2011, 60, (5), pp. 2416–2422 Hole, K.J., Holm, H., Øien, G.E.: ‘Adaptive multidimensional coded modulation over flat fading channels’, IEEE J. Sel. Commun., 2000, 18, (7), pp. 1153–1158 Taki, M.: ‘Spectral efficiency optimisation in amplify and forward relay network with diversity using adaptive rate and adaptive power transmission’, IET Commun., 2013, 7, (15), pp. 1656–1664 Hasna, M.O., Alouini, M.S.: ‘Optimal power allocation for relayed transmissions over Rayleigh-fading channels’, IEEE Trans. Wirel. Commun., 0000, 3, (6), pp. 1999–2004 Beaulieu, N.C., Farhadi, G., Chen, Yunfei.: ‘A precise approximation for performance evaluation of amplify-and-forward multihop relaying systems’, IEEE Trans. Wirel. Commun., 2011, 10, (12), pp. 3985–3989 Beaulieu, N.C., Farhadi, G.: ‘Power-optimised amplify-and-forward multi-hop relaying systems’, IEEE Trans. Wirel. Commun., 2009, 8, (9), pp. 4634–4643 Jakes, W.C.: ‘Microwave mobile communications’ (Wiley Interscience, New York, 1974) Salo, J., Vuokko, L., El-Sallabi, H.M., Vainikainen, P.: ‘An additive model as a physical basis for shadow fading’, IEEE Trans. Veh. Technol., 2007, 56, (1), pp. 13–26 Nocedal, J., Wright, S.J.: ‘Numerical optimisation’ (Springer Verlag, Berlin, New York, 2006, 2nd edn.) Hasna, M.O., Alouini, M.-S.: ‘Harmonic mean and end-to-end performance of transmission systems with relays’, IEEE Trans. Commun., 2004, 52, (1), pp. 130–135 Papoulis, A.: ‘Probability, random variables, and stochastic processes’ (McGraw-Hill, New York, 1991, 3rd ed.)

This work was supported by grant 910914 as a research project of the University of Isfahan.

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11.1

References

1 Liu, K.J.R., Sadek, A.K., Su, W., Kwasinski, A.: ‘Cooperative communications and networking’ (Cambridge University Press, 2009) 2 Chung, S.T., Goldsmith, A.J.: ‘Degrees of freedom in adaptive modulation: a unified view’, IEEE Trans. Commun., 2001, 49, (9), pp. 1561–1571 3 Altubaishi, E.S., Shen, X.: ‘Spectrally efficient variable-rate best-relay selection scheme for adaptive cooperative system’. Proc. IEEE Int. Conf. Global Telecommunications (GLOBECOM), 2011, pp. 1–5 4 Torabi, M., Haccoun, D., Frigon, J.-F.: ‘Relay selection in AF cooperative systems’, IEEE Veh. Technol. Mag., 2012, 6, (4), pp. 104–113 5 Hwang, K.S., Ko, Y.C., Alouini, M.S.: ‘Performance analysis of two-way amplify and forward relaying with adaptive modulation over multiple relay network’, IEEE Trans. Commun., 2011, 59, (2), pp. 402–406 6 Bin Sediq, A., Yanikomeroglu, H.: ‘Performance analysis of selection combining of signals with different modulation levels in cooperative communications’, IEEE Trans. Veh. Technol., 2011, 60, (4), pp. 1880–1887 7 Hwang, K.S., Ko, Y.C., Alouini, M.S.: ‘Performance analysis of incremental opportunistic relaying over identically and non-identically IET Commun., 2014, Vol. 8, Iss. 11, pp. 1955–1964 doi: 10.1049/iet-com.2013.0948

Appendix To solve (8) with continuous rate adaptation

In this section to obtain insights to solve (8), the same problem is considered with continuous rate adaptation as ! N p0 si pi s′i max Esi ,s′ ;1≤i≤N log 1 + ai , i pi ,0≤i≤N p0 si + pi s′i i=1 subject

to ⎧ 0 C(1):Esi , s′ ;1≤i≤N p0 ≤ P ⎪ ⎪ i ⎪ ⎪ ⎨ C(2) . . . C(N + 1):E ′ si ,si ;1≤i≤N pi ≤ Pi ⎪ N ⎪ ⎪ ⎪ ⎩ C(N + 2): ai = 1, ai [ {0, 1}

(34)

i=1

The above problem is a constrained optimisation problem which is solved using Lagrangian multipliers method where 1963


www.ietdl.org the Lagrangian is set up as follows ! N p0 si pi s′i ai L = Esi ,s′ ;1≤i≤N log 1 + i p0 si + pi s′i i=1 0 − l0 Esi ,s′ ;1≤i≤N p0 − P i N N i − m li Es ,s′ ;1≤i≤N pi − P ai − 1 − i

i=1

i

i=1

(35) As ∂2 L/∂p20 , 0 and ∂2 L/∂p2i , 0, therefore the Lagrangian is a convex function of p0 and pi and thus the optimum values of p0 and pi are obtained by solving ∂L/∂p0 = 0 and ∂L/∂pi = 0, 1 ≤ i ≤ N [23]

To complete the solution, Lagrangian variables, li, 0 ≤ i ≤ N are to be determined. As is could be seen in (36) if each of lis is set to zero, that is, constraint C(i + 1) is ignored, Esi ,s′ ;1≤i≤N pi tends to infinity. This means that all i constraints are active. Therefore lis, 0 ≤ i ≤ N are set such that C(1), …, C(N + 1) are satisfied with equality [23].

11.2 Computation of Fci (.) and fci ,q ′ (., .) used in i Section 3.3

(36)

In this section Fci (.) and f ci ,q′ (., .) are computed for a case i ′ that si and si have exponential ′distribution, namely fsi si = 1/si exp −si /si and fs′ si = 1/s′i exp −s′i /s′i , i where si = E si and s′i = E s′i . As ci and q′i are functions of si and s′i based on the scheme in [25] (Section 6.3), we have:

Considering C(N + 2) constraint, it is obvious that at most one of ais is to be set as 1. Because the goal is to maximise the Lagrangian, the selected relay is which has the maximum corresponding SNR, namely

si 1 si ′ = x; qi = =y⇒ ci = bi s′i 1 + bi si /s′i f x 1 + b2i y fs′ x 1 + b2i y /b2i y2 fci ,q′ x, y = s # ∂c ∂c # i # i i# # # # ∂si ∂s′i # # # ′ # ∂q ∂q′ # (38) i# # i # # ∂si ∂s′i

∂L ∂L = 0, =0⇒ ∂p0 ∂pi 1 qi s′i si + qi s′i − , l0 ln 2 si + qi s′i qi s′i si " l0 si pi = p0 qi , qi def li s′i

p0 =

p0 si pi s′i i p0 si + pi s′i s q s′ = arg max i i i ′ ; ai∗ = 1, ai = 0, i = i∗ i si + qi si

=

i∗ = arg max


(37)

2 2 1 x 1 + b2i y ′ 2 3 sisi bi y

1 1 1 2 exp −x 1 + bi y + si s′i b2i y2
