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Use of Space Shift Keying (SSK) Modulation in Two-Way Amplify-and-Forward Relaying

Miaowen Wen, Xiang Cheng, Senior Member, IEEE, H. Vincent Poor, Fellow, IEEE, and Bingli Jiao, Senior Member, IEEE

AbstractSpace shift keying (SSK) modulation is a novel multiple-input-multiple-output technique, which activates only a single antenna for transmission at any time instant and uses the index of this active antenna to implicitly convey information. In this correspondence, a pragmatic communication strategy for the use of SSK modulation in two-way amplify-and-forward relaying is proposed. Specically, the transceiver sends the signal by employing SSK modulation and detects the signal based on the maximal-ratio combining principle. By considering a Nakagamim fading environment, upper bounded and asymptotic bit error probabilities (BEPs) are both derived in closed form. The optimal power allocation problem in the minimization of the average BEP is also addressed. Finally, Monte Carlo simulations are conducted to verify the accuracy of the analysis. Index TermsSpace shift keying (SSK), two-way relaying, amplify-and-forward (AF), MIMO.

has been proposed as a promising candidate for next generation wireless systems [12]-[15]. Recently, attentive to the potential of SSK modulation in MIMO systems, research regarding the use of SSK modulation in cooperative MIMO has been carried out. In view of the function of relays, exiting related work can be classied into the following two categories. 1) For the rst category, relays mainly serve as virtual antennas congured to the transmitter [16]-[18]. For instance, in [16], different relay-selection patterns were dened to implicitly convey information, similar to activating transmit antennas in the generalized SSK modulation [4]. Also, in [17], a matrix dispersion approach was proposed to activate relays. In [18], relays were used collaboratively to form space-time SSK to achieve transmit diversity. 2) For the second category, relays function primarily as repeaters for the reconstruction of distorted signals and message exchange [19]-[23]. Specically, a dual-hop SM scheme with the decode-and-forward (DF) protocol in a non-cooperative scenario is proposed in [19], where a signicant performance gain is expected compared with the non-cooperative DF system. In [20], dual-hop amplifyand-forward (AF) relaying using SSK was introduced, where the relay and the destination are both equipped with a single antenna. In addition, by proposing an optimum maximum-likelihood (ML) detector, the authors in [20] have investigated the upper bounded and asymptotic bit error probability (BEPs) achieved by the system. This work in [20] was later extended to a multiple-relay system incorporating the direct link from the source to the destination in [21]. Since as proved in [22], a single receive antenna SM or SSK system is suboptimal to a classical single-input-single-output system, a SM or SSK system with multiple receive antennas is more favorable. However, for both [20] and [21], the extension to the case of multiple antennas at the destination within the same framework is not straightforward and may be very complicated as the effective noise processes at different antennas are correlated. More recently, two-way denoiseand-forward (DNF) relaying with all nodes using SSK modulation was considered in [23], where the messages from both sources are exchanged with the aid of a multiple-antenna relay employing denoise mapping. In this correspondence, we shed light on a particular twoway relaying scenario, in which both sources adopt SSK modulation and a relay with a single antenna follows the

I. I NTRODUCTION Space shift keying (SSK) modulation is an appealing multiple-input-multiple-output (MIMO) technique, which was rst conceptualized in [1] and then further developed in [2] as spatial modulation (SM). In [3] and [4], the concept of SSK modulation was redened and has been clearly distinguished from SM. Henceforth, SM indicates a modulation scheme in which transmitted symbols are drawn from both the spatial constellation and the signal constellation [5]-[9], while SSK modulation refers to only the spatial constellation. Therefore, SSK modulation can be considered as a special realization of SM. Despite a slight reduction of spectral efciency, the elimination of the signal constellation in SSK modulation provides outstanding advantages over SM, e.g., simple transceiver framework, low detection complexity, and ease of integration within communication systems [3]. Consequently, more recently, a great deal of work has focused on the study of the performance of SSK modulation [10], [11]. Cooperative communications provide signicant improvements in overall system capacity and wireless coverage, which
Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. This work was in part supported by the National Natural Science Foundation of China (Grant No. 61101079), the Science Foundation for the Youth Scholar of Ministry of Education of China (Grant No. 20110001120129), the open research fund of National Mobile Communications Research Laboratory (Grant No. 2012D06), Southeast University, and the China Scholarship Council. M. Wen, X. Cheng, and B. Jiao are with School of Electronics Engineering & Computer Science, Peking University, Beijing, China. M. Wen and X. Cheng are also with the National Mobile Communications Research Laboratory, Southeast University, Nanjing, China (e-mail: {wenmiaowen, xiangcheng, jiaobl}@pku.edu.cn). H. V. Poor is with Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA (e-mail: poor@princeton.edu).

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AF protocol. Therefore, our work can be attributed to the aforementioned second category. With respect to all current related work, our contributions are summarized as follows. The system setup removes the limitation in two-way DNF relaying that the number of antennas at the relay must not be less than the maximum number of antennas at both sources [23]. Moreover, in our work the channel is considered to be Nakagami-m faded with an integer m rather than Rayleigh faded (m = 1) as all current related works assume, which helps capture more unique characteristics of the overall system performance [5], [10]. The optimum ML detector in theory and a pragmatic detector based on the maximal ratio combining (MRC) principle are proposed. It is shown that the pragmatic detector expresses much lower computational complexity, while exhibiting performance very close to that of the ML detector. Moreover, the proposed near-ML detector, also called the MRC detector in this correspondence, facilitates the performance analysis. The exact BEP at either source when equipped with two antennas is derived and a tight upper bound on the BEP for more than two antennas is presented. The asymptotic BEP is also derived, from which the effects of different system parameters on the overall system performance are clearly revealed. Additionally, the optimum power allocation (PA) strategy in the minimization of the average BEP is exploited. The remainder of this correspondence is organized as follows. In Section II, the system model is introduced and the optimum ML detector is derived. A pragmatic detector is proposed and an in-depth analysis of its performance is provided in Section III. Section IV presents some simulation results. Finally, conclusions are drawn in Section V. T 1 Notation: () , () , and ()H denote transpose, inversion, and Hermitian operations, respectively. | |, F , and x+1) (x)m = (1)m (( xm+1) denote the absolute value, the Frobenius norm, and the Pochhammer symbol [28, 6.1.22], respectively. Bold capital letter denotes matrix, while lowercase bold letter denotes vector. (), U (), (, ), Q(), W, (), Kv (), and p Fq (), respectively, denote the Dirac delta function, the Heaviside step function, the incomplete gamma function, the Gaussian Q function, the Whittaker W function, the modied Bessel function of the second kind, and the generalized hypergeometric function (see [27] for denitions). II. S IGNAL F LOWCHART
AND

#
k S 1

#
q S 2

#
R R N N S 1

#
S 2

1 g

#
k S 1 h

#
g q q S 2

g N

S N 1

#
R R N N S 1

#
S 2

Fig. 1.

The schematic of the two-way relaying with SSK modulation.

|hi | , i = 1, . . . , NS1 and |gl | , l = 1, . . . , NS2 , are assumed to follow Nakagami-m distributions with parameters (h , mh ) and (g , mg ), respectively. We further presume that mg and mh are integers, the channels are mutually independent and reciprocal, and each node is subjected to additive white Gaussian noise (AWGN) of variance N0 . Each message exchange between two source nodes takes place in two phases. In the rst phase, both sources simultaneously send the information and the following superimposed signal is received at the relay node yR = PS1 hk + PS2 gq + nR (1)

where PI is the transmit power of node I (I {S1 , S2 , R}), nR is the AWGN at the relay, and without any loss of generality, it is assumed that the k -th indexed antenna of S1 and the q -th indexed antenna of S2 are activated for transmission. In the second phase, the relay rst processes the received signal by multiplying an amplication factor G and then forward it to the two source nodes. Due to symmetry, in what follows only the signal owchart at S1 will be detailed. The received signal at the i-th antenna of S1 can be expressed by yS1 ,i = + PS1 PR Ghi hk + PS2 PR Ghi gq (2)

PR Ghi nR + nS1 ,i

O PTIMUM ML D ETECTION

We consider a bidirectional relay network, which is comprised of two source nodes, denoted by S1 and S2 , and a relay node, denoted by R, as depicted in Fig. 1. We assume that both source nodes adopt SSK modulation such that the numbers of antennas at these nodes, NS1 and NS2 , are no less than 2, and the relay node with a single antenna serves as a non-regenerative repeater. The channel vectors for the S1 R and S2 R links, respectively, are represented by h and g, where the magnitudes of the entries of h and g,

where nS1 ,i is the AWGN at the i-th antenna of S1 and G = (PS1 h + PS2 g + N0 )1/2 is chosen to x the average transmit power of relay. Since the source acquires the knowledge of the activated antenna index for the previous transmission phase and the channel state information (CSI) during the pilot transmission phase, it is able to eliminate the self-interference as yS1 ,i = PS2 PR Ghi gq + PR Ghi nR + nS1 ,i .
Ef f ective noise

(3)

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Finally, based on the ML principle, the optimum detection for the transmit antenna index can be given by q = arg max fyS1 (yS1 |gq , h)
1qNS2

and (5) as Pr (q q |h, g) = Pr hH yS1 / h

2 2 F 2 2

= arg min yS1 dgq

1qNS2

gq

> hH yS1 / h

2 F

gq

1 C nef f,S1 yS1 dgq

(4)

= Pr =Q

|gq gq | < (gq gq ) neq 2 RS1 S2 R RS1 + CS1

H

where fyS1 (yS1 |gq , h) =( )NS1 Cnef f,S1

1/2 H 1 C nef f,S1 yS1 dgq

(6)

exp yS1 dgq

yS1 , of dimension NS1 , is the received signal vector, whose i-th entry is given by (3), dgq , of dimension NS1 , is the useful signal vector, whose i-th entry is PS2 PR Ghi gq , and Cnef f,S1 , of dimension NS1 NS1 , is the covariance matrix of the effective noise, whose (i, i )-th entry is N0 PR G2 h i hi + (i i ) . III. N EAR -ML D ETECTION AND P ERFORMANCE A NALYSIS A. Near-ML Detector As indicated in (4), the optimum ML detector invokes an inverse of a positive denite matrix Cnef f,S1 , resulting in a cubic increase in the dimension of the matrix instead of the linear increase with the number of received antennas in previous SSK modulation schemes [3], [20], [21]. This is due to the fact that by incurring the relay retransmission, the effective noise at different antennas of either source become highly correlated. To solve this problem, we propose a suboptimal detector, which enables not only a largely reduced detection but also a near-ML performance as will be validated in Section IV. The idea is motivated by the discovery of the underlying spatial constellation for SSK modulation in [3] and can be realized based on the MRC principle [24, p. 821823] as follows, q = arg min hH yS1 / h
1qNS2 2 F

h nS 1 nR 1 , where CS1 = NS1 G2 N0 , neq = PS + PS PR G h 2 is the F 2 2 equivalent noise after MRC, nS1 is the noise vector at S1 , 2 PS |gq gq | PR h 2 F and RS1 = N and S2 R = 2 2N0 are dened S1 N0 as the instantaneous received signal-to-noise ratio (SNR) for the R S1 link and the S2 R link, respectively. From (6), we see that to obtain the PEP dispensing with the channels, it is necessary to characterize the distributions of RS1 and S2 R rst. By considering an integer-fading-parameter Nakagami-m environment, the probability density function (PDF) of RS1 can be expressed from [25, Eq. (7)] as

fRS1 ( ) =

mh N S 1 1 m h N S 1 (mh NS1 ) RS1

mh N S 1

mh N S

1 R S 1

(7)

h is the average received SNR for the where RS1 = PR N0 R S1 link. On the other hand, since the PDF of |gq gq | is given by [26, Eq. (8)] of a form

f|gq gq | (u) = g
mg 1 mg 1

+1 u 2 z z g

+1 mz g

2z +1

mg u2 2g

(8)

where the operator g is dened as g =

i0 =0 i1 =0 i0 +i1

(i0 + i1 )!( (mg 1))i0 ( (mg 1))i1 (i0 !)2 (i1 !)2 2i0 +i1

( (i0 + i1 ))z (z !)2

(9)

z =0

gq .

(5)

from basic probability theory [24, p. 3032] we have fS2 R ( ) = = g

P

It is clear that the evaluation of (5) requires (4NS1 +2NS2 1) 3 complex operations, which is much less than the O NS2 NS 1 required in (4). It is worth noting that the powers of the effective noise processes at all antennas are simply treated as equal while performing MRC in (5), which greatly helps relieve the burden on the source nodes in estimating the noise power at different antennas.

N0 f 2PS2 |gq gq | mg S 2 R
z +1

2 N0 PS2 ze

mg S 2 R

(10)

2 g where S2 R = S is the average received SNR for N0 the S2 R link. Accordingly, given the PDF in (10), the cumulative distribution function (CDF) of S2 R can be readily derived as

B. Pairwise Error Probability (PEP) The PEP Pr (q q ), which is dened as the probability of the error event that occurs when the q -th antenna of S2 is activated but antenna index q is detected, is derived in closed form in the following. When conditioned upon the instantaneous CSI, the PEP can be readily expressed from (3) FS2 R ( ) = g (z + 1) z + 1, = 1 g z !e

mg S 2 R

mg S 2 R mg S 2 R
m

z m=0

m m! (11)

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where the property g (z + 1) = 1 and the series representation of the incomplete gamma function in [27, Eq. (8.352.2)] are used in obtaining the last equation. Therefore, by analogy with the mathematics in [20], the CDF of the equivalent received SNR in (6), i.e., eq = R S 1 S 2 R RS +CS , can be readily derived as
1 1

identity in [27, Eq. (6.631.3)], yielding

S 2 R 1 e Pr (q q ) = 2 2 2DS1 (mh NS1 )

DS1 /

2mg

+1

z !DS1 2 (m n)!n! m=0 n=0 mg

m n

mh N S + n 1

Feq ( ) =
0

FS2 R

mg

1+

C x

mg

fRS1 (x) dx
mh N S 1

S R 2 e mh N S 1 =1 (mh NS1 ) RS1

z! C x

m m! m=0
m

S2 R 1 +m 2 W

S2 R 1 + mh N S 1 + m n 2
mg S2

1 + 2

mh NS n+2m 1 2

mg S2 R
mg C S R 2

m 0

xmh NS1 1 1 +
x

mh NS n+2m mh NS n 1 1 , 2 2

DS1 + R

1 2

(15) (12) which with the help of [28, Eqs. (13.1.10) and (13.1.33)] can be also expressed in terms of 2 F0 as 1 1 ( mh N S 1 ) Pr (q q ) = g 2 2 2 mg
m n
2 z !DS1 (m n)!n! m=0 n=0

mh N S 1 1 x RS 1

dx .

nm 1

and Next, by resorting to the binomial expansion of 1 + C x the integral result in [27, Eq. (3.471.9)], (12) can be further simplied as

S R 2 2e Feq ( ) =1 g (mh NS1 ) mh N S + n 1 2

mg

z! ( m n)!n! m=0 n=0

m n

1 + mh N S 1 + m n S2 R 2 1 1 + m 2 F0 mh NS1 + m n + , 2 2 1 mg 1 m + ;; + /DS1 . (16) 2 S2 R 2 For a Rayleigh fading environment, where mg = mh = 1, (16) reduces to Pr (q q ) =

DS1

mg S2 R

mh N S n 1 +m 2

K mh N S 1 n 2

DS1

(13)

1 N S1 + 1 2 2 4 (NS1 )

2 RS1 S2 R S1 R + S2 R + 1 . (17)

where DS1 =
PS1 h N0

mg mh C S 1 N S 1 S 2 R R S 1

= mg mh

S1 R + S2 R +1 S 2 R R S 1

and

S 1 R = is the average received SNR for the S1 R link. Finally, by using the partial integral, the PEP can be expressed as

RS1 + 1 RS1 S2 R 1 1 2 2 F0 NS1 + , ; ; 2 2 S1 R + S2 R + 1 C. Upper-Bounded and Asymptotic BEPs

1 Pr (q q ) = 2 2

+ 0

1 e 2 Feq ( ) d

When NS2 = 2, one can readily determine that the exact BEP at S1 , i.e., Pb,S1 , turns out to be (16). As for NS2 > 2, the union bounding technique [24, p. 261262] is used here to derive an upper bound on the BEP at S1 . That is Pb,S1 1 NS2 log2 (NS2 )
NS2 NS2

1 1 = g 2 2 (mh NS1 ) DS1 e

mh N S + n 1 2

mg S2 R

m n + 0

z! ( m n)!n! m=0 n=0

mh NS n1+2m 1 2

q=1 q =q+1

2N (q, q ) Pr (q q ) (18)

mg S R 2

1 +2

K mh N S 1 n 2

where N (q, q ) is the Hamming distance between the bit sequences mapped to antenna indexes q and q . In the case in DS1 d . which NS2 is equal to a power of 2, which is of great interest be supposed in the following analysis, it (14) in practice andNwill NS2 S2 1 2 follows that q=1 q=q+1 2N (q, q ) = 2 NS2 log2 (NS2 ) and (18) becomes [11] Pb,S1 N S2 Pr (q q ) . 2 (19)

Closed-form solution exists by changing the integral variable with t = in the last equation of (14) and applying the

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Furthermore, greater insight BEP about the effects of the different system parameters can be gained from the following asymptotic expression for the BEP: Pb,S1 mg N S 2 (1 g z !U (z 1) ) 4 1 mh S1 R + S2 R + 1 + S2 R mh N S 1 1 RS1 S2 R

10

mg=mh=10

.
BER

10

Pe,S (MRC), simulation

1

(20)

P P

e,S e,S e,S e,S e,S e,S e,S

(ML), simulation (MRC), simulation (ML), simulation (MRC), upper bound (MRC), asymptotic (MRC), upper bound (MRC), asymptotic

See the appendix for a proof. Specically, for a Rayleigh fading environment, we have Pb,S1 N S2 4 1 S 2 R S 1 R + S 2 R + 1 1 + N S1 1 RS1 S2 R . (21)

10

P P P P

m =m =3
g h

10

10

20

From (20) and (21), it is clear that an increase in the number of received antennas in two-way relay systems with SSK modulation always achieves unit diversity order but results in an improvement of the coding gain. This is different from the case of conventional SSK modulation, where NS1 diversity order can be gained [3]. When relay retransmission is introduced, the received signal versions at all antennas become dependent and completely unresolvable. D. PA Strategy By excluding all parameters irrelevant to PA and after some manipulation, the PA problem based on the minimization of the average BEP Pe = (Pe,S1 + Pe,S2 ) /2, subject to total and individual power constraints, can be formulated as
, PS PS , PR 1 2

30 40 Ptot/N0 (dB)

50

60

Fig. 2.

BER performance in the case of NS1 = 8 and NS2 = 2.

10 10 10 BER 10 10 10

Pe,S , mg=mh=1
1

e,S

, m =m =1
g h

Pe,S , mg=10, mh=1

1

P
5

e,S e,S

, m =10, m =1
g g h

, m =m =10
h

min {PS1 ,PR ,PS2 } 1 mh PS1 h + PS2 g + PS2 mh N S 1 1 PR PS2 h 1 h z !U (z 1) + m g N S 2 h 1 mg PS1 h + PS2 g + PS1 mg N S 2 1 PR PS1 g s.t. PS1 + PR + PS2 Ptot & PS1 , PR , PS2 > 0 (22) where Pe,S2 is the BEP at S2 , h is the operator dened similar to (9) but with all related parameters extracted from the S1 R link, Ptot is the upper bound on the system power, and PS , PR , PS are the optimum values of {PS1 , PR , PS2 }, 1 2 which satisfy the constrained optimization problem in (22). A careful inspection of (22) reveals the convex property of the objective function with respect to the decision variables, i.e., PS1 , PR , and PS2 , and thus unique solutions, i.e., PS , PR , PS , exist for the above non-linear programming 1 2 problem. Unfortunately, to the best of our knowledge, closed form expressions for PS , PR , PS are unavailable. To this 1 2 end, numerical searching algorithms should be relied on, and this can be easily realized by standard mathematical software platforms, e.g., Matlab, Mathematica, etc.

1 g z !U (z 1) m h N S 1 g

Pe,S , mg=mh=10

10

10

20

30 40 Ptot/N0 (dB)

50

60

Fig. 3.

Effects of fading parameters on the BER performance.

IV. S IMULATION R ESULTS

AND

A NALYSIS

In this section, bit error rate (BER) simulations are conducted to validate the analysis given in Section III. Without any loss of generality, in the simulations we let h = g = 1 and the number of antennas at both sources be powers of 2. Unless otherwise specied, the transmit powers of all three nodes are set equal and the proposed MRC detector is applied. Fig. 2 depicts the BER performance for two different channel scenarios, where we choose NS1 = 8 and NS2 = 2. It is clear that the proposed MRC detector expresses nearly the same performance as the optimum ML detector. Moreover, as expected, the analytical exact, upper bounded, and asymptotic BEPs are all in perfect agreement with their simulation counterparts. On the other hand, as predicted in (20), the systems always gain unit diversity order. Fig. 3 shows the effects of fading parameters on the BER performance, where we choose NS1 = 32 and NS2 = 2.

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10

10

10 BER

benet about 2 dB SNR reduction with respect to the equal power distribution strategy at some specic level of Pe , e.g., Pe = 103 . We note that as the proposed PA strategy focuses on the minimization of average BER for a moderate-to-high SNR regime, it may worsen the average BER compared with the equal PA strategy for a very low SNR regime, where Pe > 101 . V. C ONCLUSIONS By introducing a pragmatic detector with low complexity and near-optimum performance, we have derived a closedform upper bound on BEP achieved by a two-way AF relay system adopting SSK modulation. We have also provided a simple asymptotic BEP expression, based on which the effect of the different system parameters on the overall system performance is revealed and a transmit power optimization strategy has been proposed. In our future work, we plan to consider the scenarios in which the relay is equipped with multiple antennas and in which there are multiple relays. A PPENDIX According to [29] and [30], the asymptotic error probability is closely related to the rst non-zero higher order derivative of the CDF of eq at the origin, which is contained in the rst term of the Taylors series of this CDF. By using the series representation of Kmh NS1 n 2 DS1 in (13) according to [27, Eq. (8.446)], one can determine that two parts in (13) contribute to the aforementioned rst term, where the rst part corresponds to the case when m = 0, given by
(1) F ( ) =1 eq

10

Pe, NS =4, NS =2, equal Power

1 2

10

P , N =4, N =2, optimum Power

e e e S S S
1

S S S

P , N =8, N =2, equal Power

1 2

10

P , N =8, N =2, optimum Power

1 2

10

20

30 40 Ptot/N0 (dB)

50

60

Fig. 4. BER performance comparison between equal power allocation and optimum power allocation.

For the purpose of gure clarity, the analytical curves are removed. We see that as the fading of the link between one source and the relay becomes less severe, i.e., the value of the corresponding parameter mh/g becomes larger, the BER performance of this source will be somewhat improved, whereas that of the other source will deteriorate. The improvement in the BER performance is due to the improvement of the channel quality, which fulls our intuition. However, deterioration of the BER performance seems counter-intuitive at rst but can be explained reasonably after further consideration as follows: the improvement of the channel quality at one link will result in the shrinking of the spatial constellation for the source at this link, and thus will lead to the detection at the other source becoming harder. Moreover, from Fig. 3, it is clear that compared with the increase of the BER, the decrease of the BER is more serious. This can be easily accounted for from the positions of the two fading parameters in the BEP expression in (20), which implies the dominant effect of the shrinking of the spatial constellation over the improvement of channel quality. Therefore, it is not surprising to see from Figs. 2 and 3 that the improvement of the fading severities for both links produces a reverse effect on the overall system performance, i.e., results in the deterioration of the overall system performance. Fig. 4 illustrates the feasibility of adopting the proposed PA strategy in lowering the average BEP, where the fading parameters are xed as mg = 1 and mh = 10. In Fig. 4, we consider two different system situations, i.e., 1) NS1 = 4, NS2 = 2; and 2) NS1 = 8 and NS2 = 2. The optimum PA strategies, according to (22), for the above two system settings in order are 1) = 0.1570Ptot; and PS = 0.4113Ptot, PR = 0.4317Ptot, PS 1 2 = 0.1144Ptot. 2) PS1 = 0.4398Ptot, PR = 0.4457Ptot, PS 2 Both BER values for the equal PA and optimum PA strategies are depicted in Fig. 4, while their analytical results are not shown for the purpose of gure clarity. One can observe that by adopting the proposed PA strategy, the system can even

and the second part corresponds to the case when m = 1, yielding

(2) F eq

mg 2 1 + O ( ) (mh NS1 ) S2 R (mh NS1 ) (mh NS1 1) DS1 + O ( ) 2 2 mg DS1 = + + O ( ) (23) S 2 R mh N S 1 1

( ) =

[g z !U (z 1) (mh NS1 ) mg (mh NS1 ) (mh NS1 1) + DS1 2 S2 R 2 +O ( )] mg DS1 = g z !U (z 1) + S2 R mh N S 1 1 + O ( ) . (24)

2 1

mg S2 R

+ O ( )

Then, by incorporating (23) and (24), we obtain the rst nonzero higher order derivative of Feq ( ) at the origin as Feq ( = 0) = feq (0) = (1 z !U (z 1) ) DS1 mg + S 2 R mh N S 1 1 (25)

where feq ( ) is the PDF of eq . Finally, the application of the identity in [29, Eq. (10)] completes the proof.

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