Relay-Aided Amplify-and-Forward Powerline Communications

IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 1, MARCH 2013 265

Relay-Aided Amplify-and-ForwardPowerline Communications

Xilin Cheng, Student Member, IEEE, Rui Cao, Member, IEEE, and Liuqing Yang, Senior Member, IEEE

Abstract—Powerline communications (PLC) is a favorable tech-nique for many smart grid applications. By transmitting informa-tion over the existing powerline infrastructure, PLC has the ben-efit of low deployment cost. However, due to low transmit power,limited bandwidth, and harsh channel conditions, reliable long-distance and high-capacity PLC is challenging. Relay-aided (RA-)PLC is promising in addressing these issues. In this paper, we inves-tigate the performance of the amplify-and-forward (AF) RA-PLCsystem from an information-theoretic perspective. The capacity ofAF-based RA-PLC is analyzed for frequency-selective PLC chan-nels. The capacity bounds are derived, and the optimal power al-location between the transmitting nodes and the optimal powerdistribution over the signal frequency band are obtained. The ca-pacity benefits and features of AF-based RA-PLC are evaluatedwith two prevalent powerline channel models. Based on the signalattenuation model, the capacity of AF-based RA-PLC is comparedwith direct-link (DL-)PLC, and the effect of relay location is re-vealed. In addition, based on the transmission line (TL) model, theeffects of branch density and load impedance on the capacity ofAF-based RA-PLC are evaluated.

Index Terms—Amplify-and-forward relaying, channel capacity,powerline communications, signal attenuation model, transmis-sion line model.

I. INTRODUCTION

T HE SMART GRID (SG) uses two-way electricity andinformation flow to create a widely distributed and

automated energy delivery network in order to meet the effi-ciency and reliability requirements of the modern society. Datacommunications is the key technology in SG for informationexchange. In SG applications, the data communications needsto ensure reliability and broad coverage, guarantee securityand privacy, and support quality of service (QoS) [1]. Amongall communication architectures, powerline communications(PLC) [2] utilizes the existing powerline infrastructure totransmit information alongside the electric power, and providea solution with many advantages including cost-effectivenessand broad coverage in the SG. The communication can reachanywhere a powerline exists, especially places where radio

Manuscript received April 01, 2012; revised September 20, 2012; acceptedOctober 02, 2012. Date of publication January 14, 2013; date of current versionFebruary 27, 2013. Part of the results in this paper were presented at the IEEEInternational Symposium on Power Line Communications and Its Applications(ISPLC), Udine, Italy, 2011. Paper no. TSG-00175-2012.X. Cheng and L. Yang are with the Department of Electrical and Computer

Engineering, Colorado State University, Fort Collins, CO 80523 USA (e-mail:[email protected]; [email protected]).R. Cao is with the LSI Corporation, Milpitas, CA 95035 USA (e-mail: ray.

[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSG.2012.2225645

signal cannot propagate through, e.g., underground, under-water, and rooms with metal walls. In recent years, severalPLC-enabled applications have been deployed or proposed.The utility companies around the world have been using nar-rowband (NB) PLC (3–500 kHz) for automation and controlapplications. The broadband (BB) PLC systems which operatein the high frequency band (2–30 MHz) with data rates upto a few hundred Mbps have been advocated to provide In-ternet access for residential customers and facilitate local areanetworking within home/office [3]. PLC is believed to play asignificant role in future SG [4].However, as the powerline is originally designed for elec-

trical energy delivery but not for communication purposes,direct-link communications over the powerline has limitedcapacity and transmission distance. First, the signal attenuationis severe for long distance and high frequency signal propa-gation, resulting in limited bandwidth. Secondly, unlike manyother communication channels, the noise in the powerline isinherently complex and consists of several components, e.g.,colored background noise, narrowband noise, and impulsivenoise. Thirdly, the transmit power of PLC is restricted byelectromagnetic compatibility (EMC) regulations. Finally, thePLC channel is doubly-selective and also varies with location,network topology, and connected loads. These challenges moti-vate the PLC research towards enhanced capacity and extendedcommunication range.In terrestrial wireless environments, relay communications

have been shown to enhance reliability and extend communi-cation range [5], [6]. Thus relay-aided (RA-)PLC is an attrac-tive option for long-range PLC. In the literature, RA-PLC re-search can be categorized into dual-hop [7]–[9] and multihop[10], [11], both of which mainly focus on the decode-and-for-ward (DF) relaying protocol. In [12], the MAC and routing is-sues are discussed for advanced control and automation systemsusing PLC. In a more recent work [13], multihop transmissionis utilized in low voltage NB PLC networks to combat the dis-tance-attenuation property of powerline channels. Relay-aidedschemes utilizing the cooperation between PLC and wirelesscommunications are also investigated in [14], [15].In this work, we will investigate a dual-hop amplify-and-for-

ward (AF) RA-PLC system from a unique information-theoreticperspective, and consider two prevalent PLC channel models.First, we analyze the capacity of the AF-based RA-PLC systemfor frequency-selective PLC channels. The system capacity isderived by jointly optimizing the transmit power allocation be-tween the source and the relay as well as the power distributionover the signal frequency band. Tight upper and lower bounds ofthe capacity are also obtained to facilitate closed-form solutionof the optimum power allocation factor. Secondly, the capacity

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266 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 1, MARCH 2013

Fig. 1. RA-PLC system with line topology.

benefits and features of AF-based RA-PLC are demonstratedusing two prevalent PLC channel models. With the signal at-tenuation model, the benefits of RA-PLC are evaluated by com-paring with direct link (DL-)PLC at various distances. The ef-fect of relay location on the capacity of AF-based RA-PLC isalso investigated. With the transmission line (TL) model, westudy the effects of branch density and load impedance. Numer-ical results show that larger system capacity can be achievedwith lower branch density and higher load impedance. This im-plies that AF-based RA-PLC is more beneficial for powerlinenetworks with more branches and lower load impedance.The rest of the paper is organized as follows. In Section II,

the RA-PLC system model is introduced. In Section III, we an-alyze the capacity of both DL-PLC and AF-based RA-PLC. Thecapacity benefits of AF-based RA-PLC and the effect of relaylocation are shown in Section IV based on the signal attenua-tion model. Then, the TL model is introduced and the effectsof branch density and load impedance are given in Section V.Summarizing remarks are given in Section VI.

II. RA-PLC SYSTEM MODEL

In this paper, we consider a dual-hop RA-PLC system withthe line topology as depicted in Fig. 1. The system consists ofthree nodes: one source node , one relay node , and one desti-nation node . In order to inject the PLC signal to the powerline,the coupling circuit which behaves as a high-pass filter is placedbetween each node and the powerline [16].For simple relay processing, we adopt the AF relaying pro-

tocol. The transmission of one packet in AF-based RA-PLC iscompleted in two time slots. In the first time slot, the sourcebroadcasts the information symbol to the relay. In the secondtime slot, the relay amplifies the received signal and retransmitsit to the destination. The equivalent end-to-end SNR [6] is com-puted as

(1)

where

(2)

represents the direct-link SNR from node to node ,, is the transmit power spectral density (PSD) at

node , is the signal power attenu-ation over distance at frequency , and is the back-ground noise PSD at node . The equivalent end-to-end SNRcan be further represented as

(3)

where and. After receiving the signal from the relay,

the receiver at the destination performs decoding.

III. SYSTEM CAPACITY ANALYSIS

System capacity is an important performance metric for com-munication systems, which reveals the maximum amount of in-formation that can be transmitted per channel use. We first an-alyze the system capacity of DL-PLC and then the system ca-pacity of AF-based RA-PLC is derived.

A. Direct-Link PLC

With the total power constraint , the system capacity of aDL-PLC system over a frequency-selective channel is given as

(4)

Subject to the total transmit power, the channel capacityis computed using the classical method of Lagrange mul-tipliers. The solution of the optimal power allocation overfrequency is given by the water-filling algorithm [17]. With

, the optimal power allocation result is

(5)

where is a “cutoff” value that satisfies the power constraint. Then, the signal bandwidth can be deter-

mined as . Substituting(5) into (4), the optimized capacity is

(6)

Clearly, the overall system capacity increases with transmitpower and is a function of distance . The dependency onwill also be illustrated by numerical results at a later point.

B. AF-Based RA-PLC

With the equivalent SNR at the destination given in (3), thecapacity of AF-based RA-PLC is calculated as

(7)

with . The system capacity is halved,because each packet transmission is completed in two time slots.With the expression of SNR in (3), the closed-form solution

to the optimization problem in (7) is mathematically intractable.Thus we resort to the upper and lower bounds of (7) to charac-terize its features.It can be readily shown that the equivalent SNR is

upper bounded by

(8)

CHENG et al.: RELAY-AIDED AMPLIFY-AND-FORWARD POWERLINE COMMUNICATIONS 267

and lower bounded by

(9)

where in obtaining the lower bound , we used the fol-lowing inequality for positive and :

(10)

Accordingly, the following result can be established.Theorem 1: The upper bound of the capacity of AF-based

RA-PLC is

(11)

where is in (26) and is chosen such that (28) is satisfied;the lower bound of the capacity of AF-based RA-PLC is

(12)

where is in (31) and is chosen such that (33) is satisfied.Proof: See Appendix I.

As analyzed above, and represent the upper andlower bounds of the capacity of AF-based RA-PLC with thecorresponding optimal power allocation in (26),(27) and in (31), (32), respectively. It can bereadily verified that with the power allocation ,the resultant capacity

is between and , i.e., . For tightbounds of and , gives a very close approximationfor the system capacity of AF-based RA-PLC in (7), and thepower allocation can be treated as the optimalpower allocation for AF-based RA-PLC. The tightness ofand will be illustrated in Section IV-B.

IV. CAPACITY ANALYSIS BASED ON THE SIGNAL ATTENUATIONMODEL

In the literature, there are mainly two categories of powerlinechannel models: the multipath model and the TL model. In themultipath model [18], [19], signal propagation in the powerlineis described as being dominated by multiple signal reflectionsinduced by various powerline branches and impedance mis-match. The multipath model is obtained based on real channelmeasurements, and the parameters of each path, e.g., delay andattenuation, are estimated using the least square fitting algo-rithm. In contrast, the TL model is based on the TL theory withno need of measurements [20], [21]. The TL model is com-puted according to the information of the powerline link, e.g.,the topology, the cable types, the terminating impedance on eachbranch, etc.The signal attenuation model is a special multipath model,

which only considers the dominant path while omitting otherpaths [19]. Due to simplicity as well as its capability to capturethe basic attenuation characteristics of the powerline channel,

Fig. 2. Signal attenuation on the distribution powerline with different lengths,i.e., 100 m, 150 m and 200 m.

the signal attenuation model is utilized to compute and analyzethe capacity features of AF-based RA-PLC in this section.

A. Signal Attenuation Model

The frequency response of a powerline line link withlength can be approximated as

(13)

where is the attenuation factor with typical value between0.5 and 1; and are constants depending on the systemconfiguration; is in MHz throughout this paper.

B. Numerical Analysis

In this section, we compare the system capacity for bothDL-PLC and AF-based RA-PLC. The effects of transmissiondistance and relay location are also investigated.1) Setup: We adopt the following experimental values for the

signal attenuation model: , ,and , which are measured from German low-voltage(LV) power distribution networks [19].We consider BB PLC forpowerline BB access systems operating in the frequency bandfrom 2 MHz to 20 MHz. In practice, these systems are installedover the electricity distribution grid and provide services to endcustomers of utilities, such as broadband internet access, VoIP,and video services [2], [22]. The signal attenuation in (13) withthe above parameters is depicted in Fig. 2. It can be observedthat strong attenuation occurs at long transmission distances andhigh frequencies. Hence the available bandwidth and system ca-pacity are expected to be small for long-distance transmission.By adding a relay into the transmission system, a long link is di-vided into two short links. Shorter links have weaker attenuationand at the same time offer a greater bandwidth, thus motivatingRA-PLC.The colored background noise PSD can be modeled by a first-

order exponential function as follows [23]:

(14)

The parameters for the residential environment, i.e.,, , and [23] are used in

the numerical results.


Fig. 3. The capacity of DL-PLC with different powerline length under threeEMC regulations: FCC Part 15, EN55022 Class B, and German Law NB30.

2) Effect of Transmit Power and EMCRegulations: Transmitpower is a critical system parameter for PLC, which needs tocomply with the EMC regulations. The higher EMC regulatedpower is, the higher capacity can be achieved. EMC regula-tions enforced in different regions have different impact onthe transmit power. For example, Federal CommunicationsCommission (FCC) Part 15, EN55022 Class B, and GermanLaw NB30 suggest PLC transmit PSDs of ,

, and , respectively [24], andthe corresponding enforced regions are United States, EuropeanUnion, and Germany. Thus different capacity can be achievedin different regions. Fig. 3 shows the capacity of DL-PLC fordifferent powerline length under the three EMC regulations.It can be observed that DL-PLC has the highest capacity underFCC Part 15 and the smallest capacity under German LawNB30. In the following numerical results, the EMC regulationFCC Part 15 is adopted without loss of generality.3) Capacity Benefits: AF-based RA-PLC can be utilized

to improve the performance of DL-PLC. The total transmitpower is chosen to be with the average PSD

over the signal frequency band .After optimal power allocation, we plot the SNR for AF-basedRA-PLC and DL-PLC in Fig. 4. For AF-based RA-PLC,and are utilized to get , , and while for DL-PLC,

is used to calculate . From Fig. 4, we have the fol-lowing observations. First, the SNR upper and lower boundsand are very tight which implies tight bounds on the capacityof the AF-based RA-PLC. Secondly, for a given transmissiondistance, the SNR and bandwidth of AF-based RA-PLC aregreater than those of DL-PLC. This suggests that AF-basedRA-PLC will have higher system capacity than DL-PLC underthe same power and distance constraints. Finally, for bothsystems, the SNR decreases and the corresponding frequencyband shrinks for long transmission distances.The system capacity for both AF-based RA-PLC and

DL-PLC with uniform or optimal power allocation are shownin Fig. 5. The relay node is located in the middle of thesource and the destination. With the uniform (Unif) powerallocation, i.e., for DL-PLC, and

for AF-based RA-PLC,

Fig. 4. , , , for the distribution powerline with lengths 150 m,175 m and 200 m.

Fig. 5. The system capacity comparison between DL-PLC and AF-basedRA-PLC with different powerline length .

it can be observed that AF-based RA-PLC has higher capacitythan DL-PLC at long transmission distances. With the optimalpower allocation obtained in Section III, the performance ofAF-based RA-PLC can be further improved. In Fig. 5, theupper bound (UB) and lower bound (LB) of the capacity ofAF-based RA-PLC are plotted, and it is notable that theirdifference is negligible. As the upper and lower bounds aretight, and can be treated as the optimal powerallocation for AF-based RA-PLC. By optimizing the powerallocation, the AF-based RA-PLC outperforms DL-PLC at alldistances with higher system capacity. In other words, with agiven capacity requirement, the optimized AF-based RA-PLCcan save transmit power at all distances. In addition, Fig. 5shows that system capacity of both DL-PLC and AF-basedRA-PLC decreases significantly with distances.4) Effect of Relay Location: As shown in (1), the relay loca-

tion affects the end-to-end SNR and further influences the ca-pacity of AF-based RA-PLC and the optimal S-R power allo-cation. To facilitate the analysis of the relay location effect, wedefine the relay location ratio as the ratio of S-R distance

to source-destination distance , i.e., and. Likewise, the power allocation ratio is

defined as the portion of source transmit power out of total


Fig. 6. The AF-based RA-PLC system capacity with respect to the relay loca-tion ratio for a source-destination distance .

Fig. 7. The optimal power allocation ratio curve for the AF-based RA-PLCsystem capacity with respect to the relay location ratio .

transmit power , i.e., and where, and .

For a communication distance and total transmitpower , we calculate the capacity of AF-basedRA-PLC for different relay location ratios and plot the resultsin Fig. 6. Note that the highest system capacity is achieved with

. In addition, to gain some insights on the optimalpower allocation for RA-PLC with different relay locations, weplot the optimal power allocation ratio with respect toin Fig. 7. It can be seen that the optimal power allocation ratio

is a monotonically increasing function of the relay locationratio . Notice from (1), the end-to-end SNR of an AF re-laying system is dominated by the worse SNR between the S-Rlink and the R-D link. Thus the balanced link quality providesthe optimal system performance. When the relay node is closerto the source than the destination, more power needs to be al-located to the relay to compensate for the more severe signalattenuation on the R-D link, and vice versa.In summary, based on the signal attenuation model, the

AF-based RA-PLC achieves prominent capacity gain at alldistances compared with DL-PLC. In addition, the highestcapacity is achieved with relay placed in the middle point.

Fig. 8. A two-port network (2PN).

When the relay moves closer to the source, more power needsto be allocated to the relay to achieve higher capacity.

V. CAPACITY ANALYSIS BASED ON THE TRANSMISSION LINEMODEL

Powerline cables are often connected to many branches.The branch density is usually high in the distribution grid orindoor environments, but low in the transmission grid. As thebranches can cause signal attenuation as well as reflection, thePLC channel is highly dependent on the branch configuration:for a fixed transmission distance, the powerline network withhigher branch density has stronger attenuation; the branchwith very low load impedance acts like a short circuit whichstrongly attenuates the transmit signal power. To quantify theeffects of branch density and load impedance upon DL-PLCand AF-based RA-PLC, the exact channel response for a givenpowerline network has to be calculated. Since the TL modelcan provide a closed-form expression of the transfer functionas long as the detailed powerline configuration is available, itis adopted to analyze the effects of branch density and loadimpedance. Thus in this section, the TL model is introduced[13], [20]. Then, the effects of branch configuration will beevaluated through numerical analysis.

A. Transmission Line Model

In the TL model, every uniform TL can be modeled as atwo-port network (2PN), as shown in Fig. 8, and a powerlinenetwork can be represented as a series of 2PNs. A common wayto represent a 2PN in TL theory is to use the ABCD matrix:

(15)

We can calculate the following important quantities based onthe ABCD parameters:

(16)

(17)

where is the transfer function which is defined as the ratioof the voltage on the load to the source voltage, and is theequivalent impedance.Some elementary 2PN ABCD matrices are given as follows.1) Transmission Line: The ABCD matrix for a two-con-

ductor transmission line with length is given as

(18)


Fig. 9. A PLC network architecture with multiple branches.

where is the charac-teristic impedance, andis the propagation constant. , , , and are TL parame-ters: is the resistance per unit length, including losses causedby the skin effect; represents the inductance per unit length;stands for the capacity per unit length; is the conductance

per unit length between the two wires, and it is mainly causedby dielectric losses of the insulating material between two con-ductors [2].2) Shunt Impedance: The ABCD matrix for a shunt with

impedance is

(19)

3) Series Impedance: The ABCD matrix for a series withimpedance is:

(20)

The transfer function of a powerline network at frequencycan be calculated in three steps. First, the entire powerline

is analyzed and divided into sections of transmission line,shunt impedance, and series impedance. The ABCD matrix foreach section is obtained using the elementary ABCD matrices.Secondly, the overall ABCD matrix of the end-to-end power-line network is computed by multiplying all sectional ABCDmatrices together sequentially. Finally, the transfer function iscomputed from the overall ABCD matrix using (16).

B. Numerical Results

In this section, the capacity of the AF-based RA-PLC systemand DL-PLC are computed based on the TL model. The effectsof branch density and load impedance are investigated.1) Setup: Consider a powerline network depicted in Fig. 9.

The source and destination nodes are connected by a powerlinecable which has many branches. NAYY150SE is used for thepowerline main cable, and it is a four-conductor power supplycable used for three-phase voltage supply in Europe [13]. Weassume the transmitted signal is injected into two phases withthe neutral conductor as return (see Figure 2.26 in [2]). Ac-cording to this setup, the line parameters can be approximatedas , , ,and , where , , , , andare vacuum permeability, vacuum permittivity, relative permit-tivity, dielectric loss angle of the insulation, and specific resis-tance of the conductor, respectively. The values of these param-eters are , ,

, and [13]. and represent the radius ofthe cable and the thickness of the insulation, and their numericalvalues for NAYY150SE are and .Source and destination impedances are chosen as

Fig. 10. The system capacity of AF-based RA-PLC and DL-PLCwith differentbranch densities and load impedances.

. The colored background noise PSD is given in (14) withparameters , , and . The totaltransmit power is over the signal frequencyband . The length of the main cable is 1000 m.2) Effects of Branch Density and Load Impedance: Con-

sider first a uniform branch configuration, where all branchspacings are the same, and each branch has the same loadimpedance. Following the optimal relay location result ob-tained in Section IV, the relay node is placed in the middle of thepowerline cable. In Fig. 10, the capacity of AF-based RA-PLCand DL-PLC are plotted for different branch configurations.Observe that the capacity for both AF-based RA-PLC andDL-PLC decreases with the branch density but increases withthe load impedance. This confirms the strong impact of branchdensity and load impedance on the signal attenuation and thusthe system capacity. In addition, AF-based RA-PLC provideshigher capacity than DL-PLC for the power system with higherbranch density and lower load impedance. For example, withload impedance of , AF-based RA-PLC provides highercapacity when the branch number is greater than 11. This isbecause the signal experiences more attenuation on powerlineswith higher branch density and lower load impedance. Ourproposed AF-based RA-PLC system is more beneficial in suchsevere communication channels.3) The Effect of Non-Uniform Load Density on the Optimal

Relay Location: In real power systems, the branch configura-tion may not be uniform, thus the optimal relay location will notbe in the middle. To investigate the effect of load distribution onthe relay location selection, we consider a powerline cable at-tached with 20 equally-spaced branches, but with non-uniformload impedances. In the first 10 branches, the load impedance is

while in the next 10 branches, the load impedance is .In this configuration, the signal attenuation is stronger in the first

of the main cable but weaker for the remaining .As indicated in (1), the highest system capacity of AF-basedRA-PLC is achieved with balanced S-R and R-D links, thus weexpect the best relay position to be closer to the source thanthe destination. Fig. 11 shows the capacity of RA-PLC with therelay attached to different branches. The capacity of DL-PLC isalso plotted as a reference. In the figure, the highest RA-PLC


Fig. 11. The system capacity of AF-based RA-PLC with different relay loca-tions when the load distribution is nonuniform.

capacity is achieved when the relay is placed at branch 8, whichis closer to the source than the destination.In summary, we have seen that the branch density and the load

impedance are important factors that contribute to the signal at-tenuation on powerline, and AF-based RA-PLC is more bene-ficial for PLC networks with higher branch density and lowerload impedance.

VI. CONCLUSIONS

In this paper, we studied the capacity of the AF-basedRA-PLC. First, the capacity of AF-based RA-PLC was an-alyzed over frequency-selective PLC channels. By jointlyoptimizing the power allocation between the transmitting nodesand over the available frequency band, the system capacityhas been computed. Due to the mathematical intractability ofclosed-form solutions, tight upper and lower bounds were de-rived, and the optimum power allocation results were obtained.Secondly, the capacity benefits and features of the AF-basedRA-PLC were evaluated using two prevalent PLC channelmodels. With the signal attenuation model, the RA-PLC systemcapacity was computed and compared with that of DL-PLC.Numerical results show that through the optimal power alloca-tion, the AF-based RA-PLC can provide higher capacity thanDL-PLC. Besides, the study of the relay location indicates thatthe topology with balanced S-R and R-D link provides thehighest capacity. In addition, with the TL model, the effects ofbranch density and load impedance were revealed. The results

show that the AF-based RA-PLC is more beneficial for thePLC system with more branches and lower load impedances.

APPENDIXPROOF OF THE BOUNDS OF THE CAPACITY OF AF-BASED

RA-PLC

Upper Bound of the Capacity of AF-Based RA-PLC: Theupper bound of the capacity of AF-based RA-PLC is

(23)

with .The optimal transmit PSDs and is obtained

using the method of Lagrange multipliers. The Lagrangian isformed as follows:

(24)

where is the Lagrange multiplier.By differentiating the Lagrangian with respect to and, we can achieve (21) and (22) at the bottom of the page.

By setting the two derivatives equal to zero, the relationshipbetween and is shown as follows:

(25)

After substituting (25) into (21) and solvingwith the constraint

, we can compute the optimal and as:

(26)

and

(27)

(21)

(22)


The Lagrange multiplier is chosen such that the power con-straint is satisfied:

(28)

Then the upper bound of the system capacity can be computedby substituting and into (23):

(29)

Lower Bound of the Capacity of AF-Based RA-PLC: Thelower bound of the capacity of AF-based RA-PLC is given as

(30)

with .Following the same procedure as the upper bound calculation,

the optimal and for can be obtained as

(31)

and

(32)

The Lagrange multiplier is chosen such that

(33)

The resultant lower bound of the system capacity is

(34)

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Documents

Relay-Aided Amplify-and-Forward Powerline Communications