Space-Time Network Coding with Transmit Antenna ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2014.2381217, IEEE Transactions on Wireless Communications 1

Space-Time Network Coding with Transmit Antenna Selection and Maximal-Ratio Combining Kai Yang, Member, IEEE, Nan Yang, Member, IEEE, Chengwen Xing, Member, IEEE, Jinsong Wu, Senior Member, IEEE, and Zhongshan Zhang, Member, IEEE

Abstract—In this paper, we investigate space-time network coding (STNC) in cooperative multiple-input multiple-output networks, where U users communicate with a common destination D with the aid of R decode-and-forward relays. The transmit antenna selection with maximal-ratio combining (TAS/MRC) is adopted in user-destination and relay-destination links where a single transmit antenna that maximizes the instantaneous received signal-to-noise ratio is selected and fed back to transmitter by receiver and all the receive antennas are combined with MRC. In the presence of perfect feedback, we derive new exact and asymptotic closed-form expressions for the outage probability (OP) and the symbol error rate (SER) of STNC with TAS/MRC in independent but not necessarily identically distributed Rayleigh fading channels. We demonstrate that STNC with TAS/MRC guarantees full diversity order. To quantify the impact of delayed feedback, we further derive new exact and asymptotic OP and SER expressions in closed form. We prove that the delayed feedback degrades the full diversity order to (R + 1) ND , where ND is the antenna number of the destination D. Numerical and Monte Carlo simulation results are provided to demonstrate the accuracy of our theoretical analysis and evaluate the impact of network parameters on the performance of STNC with TAS/MRC. Index Terms—Cooperative communications, space-time network coding (STNC), multiple-input multiple-output (MIMO), delayed feedback.

I. I NTRODUCTION OOPERATIVE communications have recently attracted considerable attention in emerging wireless applications due to their advantages of network coverage extension and capacity expansion [1–3]. The inherent concept of cooperative communications is to provide spatial diversity by employing relay nodes to forward signals from the source to the destination. Against this background, various cooperative diversity schemes have been proposed and analyzed [4, 5]. We note that most research contributions in cooperative communications assumed perfect synchronization. Such an assumption may be difficult or impossible in practical multi-node wireless networks as it is very challenging to align all the signals from multiple sources at multiple destinations [6, 7]. High-accuracy

C

K. Yang and C. Xing are with the School of Information and Electronics, Beijing Institute of Technology, Beijing, China (email: [email protected], [email protected]). N. Yang is with the Research School of Engineering, Australian National University, Canberra, ACT 0200, Australia (email: [email protected]). J. Wu is with the Department of Electrical Engineering, Universidad de Chile, Santiago, Chile (email: [email protected]). Z. Zhang is with the Institute of Advanced Network Technology and New Services, and Beijing Engineering and Technology Research Center for Convergence Networks and Ubiquitous Services, University of Science and Technology Beijing, Beijing 100083, China (e-mail: [email protected]).

synchronization requires complicated control mechanisms and extra control messages, which leads to high system complexities and overheads [7]. When synchronization is imperfect, the performance of cooperative communications can be severely degraded. In order to overcome the problem caused by imperfect synchronization, the space-time network coding (STNC) scheme was proposed in [8], which uses time-division multiple access (TDMA) to deal with the imperfect synchronization issues. Importantly, this scheme achieves the full diversity order. STNC combines information from different sources at each relay node and transmits the combined signal in dedicated time slots, which jointly exploit the benefits of both network coding and space-time coding. To quantify the benefits offered by STNC, the symbol error rates (SERs) of STNC over Rayleigh and Nakagami-m fading channels were investigated in [8] and [9], respectively, and the outage probability (OP) of STNC in Rayleigh fading channel was analyzed in [10]. In [8, 9], it was assumed that the channel state information (CSI) is known at the receivers. To avoid the requirement of channel estimation, the differential space-time network coding (DSTNC) and distributed differential space-time-frequency network coding (DSTFNC) schemes were designed for narrowband and broadband cooperative communication systems, respectively, in [11]. Similarly to STNC, both the DSTNC and DSTFNC schemes provide the full diversity. By allowing each relay to exploit the overheard signals transmitted from not only the sources but also the previous relays, a new STNC scheme with overhearing relays was proposed in [12]. When there is no dedicated relay in the systems, users are required to help each other to exploit the cooperative diversity gain [13, 14]. In [13], a novel clustering based STNC scheme was proposed to achieve a better tradeoff between diversity gain and bandwidth efficiency. The core idea of this scheme is to divide the whole network into several small clusters and allow different clusters to help each other to relay signals. In [14], the STNC with optimal node selection scheme was presented to allow multiple users to exchange their data simultaneously. We note that in [8–14], each node in the network is equipped with a single antenna. Since multiple-input multiple-output (MIMO) technology, in which communication nodes are equipped with multiple transmit and/or receive antennas, can significantly increase communication reliability through the use of spatial diversity [15], we focus on multi-antenna STNC in this work. In this paper, we examine STNC in the cooperative MIMO network, where U users communicate with a common destination D with the assistance of R relays and all the

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nodes are equipped with multiple antennas. We consider the independent but not necessarily identically distributed (i.n.i.d.) Rayleigh fading channels and include the direct links between the users and destination. Particularly, we focus on decodeand-forward (DF) relaying protocol due to its application in the Third-Generation Partnership Project Long-Term Evolution and IEEE 802.16m [16, 17] such that the relays need to decode the received signal, reencode it, and forward it to the destination. For user-destination (u–D) and relay-destination (r– D) links, we adopt transmit antenna selection with maximalratio combining (TAS/MRC) [18, 19], where a single transmit antenna that maximizes the output signal-to-noise ratio (SNR) is selected and all the receive antennas are combined using MRC. As such, the transmitter can be easily implemented with a single front-end and an analog switch, and the receiver only needs to feed back the index of the selected transmit antenna. We first consider perfect feedback and derive new closedform expressions for the exact and asymptotic OP and SER. As TAS could be performed by using outdated CSI due to feedback delays, we then quantify the effect of delayed feedback in u–D and r–D links on the OP and SER. In doing so, we derive new closed-form expressions for the exact and asymptotic OP and SER by taking the delayed feedback into account. The primary analytical contributions of this paper are summarized as follows. 1) We integrate cascaded TAS/MRC into STNC as a solution to preserve full transmit and receive diversity with low computational complexity and reduced feedback overhead; 2) We derive new closed-form exact and asymptotic expressions for the OP and SER. These results are valid for general operating scenarios with arbitrary number of antennas and arbitrary number of relays; 3) We derive new closed-form exact and asymptotic expressions for the OP and SER with feedback delays to examine the impact of outdated CSI on the performance. Based on our results, it is demonstrated that the transmit diversity vanishes and that the diversity order is entirely independent of the number of transmit antennas. The rest of the paper is organized as follows. The system model is presented in Section II. In Section III, the exact and asymptotic performance of STNC with TAS/MRC in cooperative MIMO network in the presence of perfect feedback is analyzed. In Section IV, the impact of delayed feedback on the performance of STNC with TAS/MRC is quantified. Numerical and simulation results are presented in Section V. The conclusions are drawn in Section VI. II. S YSTEM M ODEL We consider a cooperative MIMO network where U users transmit their own information to a common destination D with the aid of R relays. In this MIMO network, user u, 1 ≤ u ≤ U , relay r, 1 ≤ r ≤ R, and destination D are equipped with Nu , Nr , and ND antennas, respectively. The channel coefficient between the jth antenna of transmitter µ and the −α ith antenna of receiver ν is defined as hνµ ij ∼ CN (0, dµν ), µ ∈ {u, r}, ν ∈ {r, D}, µ ̸= ν, where dµν and α denote

the distance between µ and ν and the path loss exponent, respectively. The transmission of STNC takes place over U + R time slots, which are divided into two consecutive phases [8]. In the first phase, the U users take turns to broadcast their symbols to the relays and destination in the first U time slots, i.e., user u broadcasts its symbol in time slot u while other users staying in silence. In the second phase, R relays combine the overheard symbols from multiple users during the first phase to a single symbol and then take turns to transmit it to destination in the last R time slots, i.e., relay r transmits its symbol in time slot U + r. We outline the STNC with TAS/MRC in the cooperative MIMO network as follows. In the first phase, TAS/MRC is applied between each user and the destination D. The optimal antenna amongst the Nu antennas at user u is selected to maximize the instantaneous SNR of the signal from user u to the destination D. Therefore, the index of the optimal transmit antenna, n∗u , is determined as n∗u = arg max ∥hnu D ∥F , where hnu D is the channel 1≤nu ≤Nu

vector between the nu th antenna at user u and the multiple antennas at the destination D, and ∥ · ∥F denotes the Frobenius norm. Since the multiple antennas at one node do not appear colocated and the MIMO links are close to independent and identically distributed (i.i.d.) Rayleigh fading in the rich-scattering scenario [20], we assume that hnu D has i.i.d. Rayleigh entries. The signals received at the destination D and relay r from user u in time slot u are √ yuD (t) = hn∗u D P0u xu su (t) + nuD (t) (1) and yur (t) = hn∗u r

√ P0u xu su (t) + nur (t)

(2)

respectively, where hn∗u D and hn∗u r denote the Rayleigh channel vectors between the n∗u th antenna at user u and multiple antennas at the destination D and between the n∗u th antenna at user u and multiple antennas at relay r respectively, nuD (t) and nur (t) are the additive white Gaussian noise (AWGN) vectors with zero mean and variances of N0 IND and N0 INr respectively, In , n ∈ {ND , Nr }, denotes the identity matrix of size n, P0u is the transmit power of xu , and xu and su (t) denote the symbol with unit energy transmitted by user u and the related spreading code, respectively. The cross correlation between su (t) and su′ (t) is defined as ρuu′ = ∫T ∆ ⟨su (t) , su′ (t)⟩, where ⟨f (t) , g (t)⟩ = T1 0 f (t) g ∗ (t) dt is the inner product between f (t) and g (t) with the symbol interval T , and g ∗ (t) is the complex conjugate of g (t). 2 Moreover, we assume that ρuu = ∥su (t)∥F = 1. After combining the received signal replicas from different antennas using MRC and applying matched-filtering, the received signals at the destination D and relay r are given by

√ u yuD = ⟨wuD yuD (t) , su (t)⟩ = hn∗u D F P0u xu + nuuD (3) and

√ u yur = ⟨wur yur (t) , su (t)⟩ = hn∗u r F P0u xu + nuur (4)



respectively, where weight vectors wuD = h†n∗u D /∥hn∗u D ∥F and wur = h†n∗u r /∥hn∗u r ∥F , (·)† denotes conjugate transpose, and nuuD and nuur are AWGN with zero mean and variance of N0 . In the second phase, TAS/MRC is applied between each relay and the destination D. The index of the optimal transmit antenna at relay r, n∗r , is determined as n∗r = arg max ∥hnr D ∥F , where hnr D is the channel vector between 1≤nr ≤Nr

the nr th antenna at relay r and the multiple antennas at the destination D with i.i.d. Rayleigh entries. Each relay linearly combines the overheard symbols during the first phase to a single encoded signal and then forwards it to the destination D in the second phase. The signal received at the destination D from relay r in time slot U + r is yrD (t) = hn∗r D

U ∑

βru

√ Pru xu su (t) + nrD (t)

(5)

u=1

where hn∗r D denotes the Rayleigh channel vector between the n∗r th antenna at relay r and multiple antennas at the destination D, nrD (t) is the AWGN vector with zero mean and variance of N0 IND , Pru is the transmit power of xu at relay r, and βru denotes the detection state of relay r on xu . If relay r correctly decodes xu , βru = 1; otherwise, βru = 0. The relay detection state can possibly be done by examining the included cyclicredundancy-check digits or the received SNR levels. Here, we assume that the destination D knows the detection states of the relays, which can be obtained through the indicators sent by the relays. By combining the received signal replicas from multiple antennas with MRC and applying matched-filtering, the received signal at the destination D from user u′ through relay r is ′

u yrD = ⟨wrD yrD (t) , su′ (t)⟩ U √

∑ ′

∗ = hnr D F βru Pru xu ρuu′ + nurD

(6)

where weight vector wrD = h†n∗r D /∥hn∗r D ∥F , and nurD is AWGN with zero mean and variance of N0 . Rewriting (6) into matrix form yields

˜ rD y˜rD = hn∗r D F RPr x + n (7) ′

   R= 

1 ρ12 .. .

ρ21 1 .. .

··· ··· .. .

ρ1U

ρ2U

···

1 2 [yrD , yrD ,··· √

ρU 1 ρU 2 .. .

    

1 U T , yrD ] ,

where n ˜ urD is the AWGN with zero mean and variance of N0 θu with θu being the uth diagonal element of matrix R−1 associated with symbol xu . By combining the information on xu from user u and R relays with MRC, the instantaneous end-to-end SNR of xu at the destination D is written as Υu = ΥuD +

R ∑

βru ΥrD

(9)

r=1

hn∗ D 2 P0u /N0 and ΥrD = where ΥuD = u F

hn∗ D 2 Pru / (N0 θu ). Similarly, the instantaneous SNR r F

2

of xu at relay r can be given by Υur = hn∗u r F P0u /N0 . By defining the active relay set for user u as Du = {r : βru = 1, r = 1, 2, · · · , R}, (9) is rewritten as ∑ Υu = ΥuD + ΥrD . (10) r∈Du

When all the relays fail to decode the symbol xu , i.e., Du = ϕ, it is obvious that the STNC with TAS/MRC reduces to the conventional single hop TAS/MRC MIMO scheme. We note that the optimal transmit antenna for the destination D corresponds to a random transmit antenna for relay r, which is due to the fact that the optimal antenna at user u is entirely determined by the channel state between user u and the destination D and that the channels from user u to the destination D and to relay r are mutually independent. Therefore, the output SNR at r after MRC combing, Υur , is the sum of the SNR at each antenna of r. As such, the probability density function (PDF) of Υur is the convolution of exponential PDFs, and is given by [22] γ

u=1

where

multiplying the both sides of (7) with R−1 , which yields √

u ˜ urD (8) y˜rD = hn∗r D F βru Pru xu + n

with y˜√ =√ Pr = rD diag{βr1 Pr1 , βr2 Pr2 , · · · , βrU PrU }, x = T ˜ rD = [n1rD , n2rD , · · · , nU [x1 , x2 , · · · , xU ]T , n is the rD ] AWGN vector with zero mean and variance matrix of N0 R, and [·]T denotes transpose. Here, the correlation amongst ˜ rD is from the matched-filtering operation. the entries of n Assuming that R is invertible with the inverse matrix R−1 [21], the soft symbol of xu from relay r is derived by

γ Nr −1 e− γ¯ur fΥur (γ) = Nr Γ (Nr ) γ¯ur

(11)

2

where γ¯ur = E[∥hru ij ∥F ]P0u /N0 is the average per-antenna SNR between user u and relay r, and E[x] denotes the expectation of x. Since hn∗u D is the channel vector between the optimal transmit antenna n∗u at user u and ND receive antennas at the destination D, the PDF of ΥuD is ) Nu −1 ( Nu ∑ Nu − 1 Φg e−ξ0 γ fΥuD (γ) = g Γ (ND ) g =0 g0 (−1) 0 γ¯uD 0 ( )ϕg +ND −1 γ × (12) γ¯uD 2

where γ¯uD = E[∥hDu ij ∥F ]P0u /N0 is the average per-antenna SNR between user u and the destination D, and Φg , ϕg , and ξ0 are defined in Appendix A. Proof: The proof is presented in Appendix A. Following the similar procedures of deriving the PDF of



ΥuD , we can derive the the PDF of ΥrD as ) N∑ r −1 ( Nr − 1 Φr e−ξr,0 γ Nr fΥrD (γ) = h Γ (ND ) hr,0 (−1) r,0 γ¯rD hr,0 =0 ( )ϕr +ND −1 γ × (13) γ¯rD ∏ND −1 (∑hr,i−1 ( hr,i−1 )( 1 )hr,i −hr,i+1 ) , ϕr = where Φr = i=1 hr,i =0 i! hr,i ∑ND −1 γrD , and γ¯rD = i=1 hr,i , hr,ND = 0, ξr,0 = (hr,0 + 1) /¯ 2 E[∥hDr ∥ ]P /(N θ ) is the average per-antenna SNR beru 0 u ij F tween relay r and the destination D. In (12) and (13), the MIMO links between the same transmitter/receiver pair are assumed i.i.d. Rayleigh distributed. If they are i.n.i.d. Rayleigh distributed, we have fΥµD (γ) =

Nµ ∑ j=1

αj

ND ∑

−

βij e

i=1

∏

−αj ′

j ′ =1,j ′ ̸=j

∑

1m0

Ωm

(

Nµ

×

γ Dµ γ ¯ ij

ND ∑

βi′ j ′ γ¯iDµ ′j′ e

−

γ Dµ γ ¯ ′ ′ i j

In this section, we analyze the performance of STNC in the cooperative network without feedback delays. We first derive the new closed-form expressions for the exact OP and SER. We then present the asymptotic expressions for the OP and SER to provide the useful insights into the behavior of STNC with TAS/MRC in the high SNR regime. A. Exact Performance 1) OP: The outage probability is an important quality-ofservice measure as it characterizes the probability that the instantaneous end-to-end SNR falls below a predetermined threshold Υth . Here, we define Υth = 2R − 1, where R is the target transmission rate of the users. From (10), the OP of STNC associated with xu can be given by Pr ( Υu < Υth | Du ) Pr (Du )

Pr ( Υu < Υth | Du,v ) Pr (Du,v )

(16)

|Du |=0 v=1

where Du,v is the vth possible choice of |Du | relays from the R relays, and the probability Pr (Du,v ) is ∏ ∏ Pr (Du,v ) = (1 − Ps,ur ) Ps,ur (17) r ∈D / u,v

r∈Du,v

where Ps,ur is the SER of detecting xu at relay r. We will derive Ps,ur later in this subsection. In the following, we proceed to evaluate the conditional probability Pr ( Υu < Υth | Du,v ) and the probability Pr (Du,v ). Without loss of generality, we now re-number the indices of the active relays belonging to Du,v as {1, 2, · · · , ϱ}, where ϱ = |Du |. As such, the end-to-end SNR of xu conditioned on active relay set Du,v , Υ u|Du,v , is ϱ ∑

ΥrD .

(18)

r=1

III. P ERFORMANCE WITH P ERFECT F EEDBACK

|Du |=0

R ϑ ∑ ∑

Υ u|Du,v = ΥuD +

i′ =1

where µ ∈ {u, r}, Ωm is the set of nonnegative integers {m0 , m1 , · · · , mj−1 , mj+1 , · · · , mNµ } such that ∑Nµ ′ j ′ =0,j ′ ̸=j mj = Nµ − 1 with 0 ≤ m0 ≤ Nµ − 1 and Dµ mj ′ ∈ {0, 1}, 1 ≤ j ′ ≤ Nµ , and αj , βij , and γ¯ij are defined in Appendix B. Proof: The proof is presented in Appendix B. The PDF of ΥµD given by (14) is the summation of exponential functions. Therefore, (14) is in the same form as (12) and (13). As such, it is easy to generate our theoretical analysis in the following sections to the i.n.i.d. Rayleigh fading case.

Pout,u =

Pout,u =

) mj ′

(14)

R ∑

rewritten as

(15)

( ) where |Du | is the size of Du . As there are ϑ = |DRu | possible choices of the active relay set Du for a given |Du |, (15) is

Given that Υ u|Du,v is the summation of ϱ + 1 independent largest order statistics of Gamma distributed variables, we present the PDF of Υ u|Du,v in the following lemma. Lemma 1: The PDF of Υ u|Du,v is derived as  Nϱ −1 ϕg +ND N∑ u −1 N 1 −1 ∑ ∑ cu γ p−1 e−ξ0 γ ∑ fΥ u|Du,v (γ) = Ψ ··· Γ (p) g0 =0 h1,0 =0 p=1 hϱ,0 =0 ) ϱ ϕr∑ +ND ∑ cr γ p−1 e−ξr,0 γ + (19) Γ (p) r=1 p=1 where Ψ, cu , and cr are defined in Appendix C. Proof: The proof is presented in Appendix C. Integrating (19) from 0 to γ with the aid of the definite integral of the exponential function, which is given by [23, Eq. (3.351.1)], we obtain the corresponding cumulative distribution function (CDF) of Υ u|Du,v , as shown in (20). Based on (20), it is easy to derive the conditional probability Pr ( Υu < Υth | Du,v ) as Pr ( Υu < Υth | Du,v ) = FΥ u|Du,v (Υth )

(21)

where FΥ u|Du,v (Υth ) is the CDF of Υ u|Du,v evaluated at γ = Υth . We now derive the SER Ps,ur to calculate the Pr (Du,v ) according to (17). The closed-form expression for SER could be given directly in terms of the CDF of the received SNR, F (γ), as [24] √ ∫ a b ∞ F (γ) −bγ (22) Ps = √ e dγ, 2 π 0 γ where the parameters a and b are up to a specific used modulation scheme, which encompasses a variety of modulations such as binary phase-shift keying (BPSK) (a = b = 1), M ary phase-shift keying (a = 2, b = sin2 (π/M √ )), and /√M ary quadrature amplitude modulation (a = 4( M − 1) M , b = 3/(2(M − 1))) [22]. Integrating (11) with the aid of [23,



FΥ u|Du,v (γ) =

Nϱ −1

N∑ u −1 N 1 −1 ∑

···

g0 =0 h1,0 =0

=1−

∑

 Ψ

hϱ,0 =0

N∑ u −1 N 1 −1 ∑

Nϱ −1

···

g0 =0 h1,0 =0

∑

hϱ,0 =0

∑ cu

ϕg +ND

p=1



Ψ

p=1

Substituting (23) into (17), the probability of the active relay set Du,v is obtained. Therefore, we can derive the exact closed-form expression for the OP of STNC with TAS/MRC by substituting (17) and (21) into (16). 2) SER: Similar to (16), the SER of STNC associated with xu is given by Ps,u =

0

 p−1 −ξr,0 x x e dx 0  Γ (p)  ϱ ϕr∑ +ND ∑ p−1 k −ξr,0 γ ∑ cr γ e . + p−k k!ξr,0 r=1 p=1 k=0

ϱ ϕ +N xp−1 e−ξ0 x dx ∑ r∑ D cr + Γ (p) r=1 p=1

ϕg +ND p−1 ∑ ∑

Eq. (3.351.1)] and then substituting the outcome into (22) with the aid of identity [23, Eq. (3.371)] yield √ Nr −1 ( )−i− 12 a a b ∑ (2i − 1)!! 1 Ps,ur = − +b . (23) i 2 2 i=0 i!2i γ¯ur γ¯ur

R ϑ ∑ ∑

∫γ

cu γ k e−ξ0 γ

k=0

k!ξ0p−k

(24)

|Du |=0 v=1

where P s,u|Du,v denotes the SER of detecting xu at the destination D conditioned on active relay set Du,v . Substituting the CDF of end-to-end SNR associated with xu , (20), into (22) with the aid of identity [23, Eq. (3.371)], we derive the exact SER of xu conditioned on Du,v , as shown in (25). Substituting (17) and (25) into (24) yields the exact closed-form SER expression for STNC with TAS/MRC. The SER expression in (24) is valid for a variety of modulations and applies to arbitrary numbers of users and relays and arbitrary antenna numbers at the nodes. It is noticed that our result encompasses the SER expression for single-antenna STNC in [8] as a special case.

(20)

and fΥ∞ (γ) ≈ rD

Nr γ ND Nr −1 Nr −1 ND Nr γ¯rD

Γ (ND ) (Γ (ND + 1))

(28)

respectively. Performing the Laplace transforms of (27) and (28) with the aid of [23, Eq. (3.351.3)] to calculate the moment ∞ generating functions (MGFs) of Υ∞ uD and ΥrD and multiplying the MGFs together, we have the MGF of Υ∞ u as −Gd

MΥ∞ (s) ≈ Θ(s¯ γuD ) u

(29)

where Nu Γ (ND Nu )

Θ=

P s,u|Du,v Pr (Du,v )

∫γ

N −1

Γ (ND ) (Γ (ND + 1)) u R ∏ D Nr Nr Γ (ND Nr ) κN r × Nr −1 r=1 Γ (ND ) (Γ (ND + 1)) ) ( ∑R with κr = γ¯uD /¯ γrD and Gd = Nu + r=1 Nr ND . Performing the inverse Laplace transform of (29) and integrating the outcome produce the CDF of Υ∞ u as FΥ∞ (γ) ≈ u

Θ γ Gd . Gd Γ (Gd + 1) γ¯uD

(30)

By evaluating FΥ∞ (γ) at γ = Υth , the OP of the end-to-end u SNR associated with xu in the high SNR regime is given by ∞ = FΥ∞ (Υth ) . Pout,u u

(31)

B. Asymptotic Performance In this subsection, we derive the asymptotic OP and SER expressions in the high SNR regime to characterize the behavior of STNC with TAS/MRC. 1) OP: In the high SNR regime, the probability that relay r decodes the symbol xu correctly approaches one, i.e., lim βru = 1. In such a scenario,

γ ¯ur →∞

∞ Υ∞ u = ΥuD +

R ∑

Υ∞ rD .

(26)

r=1

In order to obtain the CDF of Υ∞ u , the first order expansions of the PDFs of ΥuD and ΥrD are needed. Applying the Taylor series expansion of the exponential function in (12) and (13) and retaining the first order term, we obtain fΥ∞ (γ) and uD fΥ∞ (γ) as rD fΥ∞ (γ) ≈ uD

Nu γ ND Nu −1 Nu −1 ND Nu γ¯uD

Γ (ND ) (Γ (ND + 1))

(27)

2) SER: We now derive the asymptotic SER based on FΥ ∞ (γ). Substituting (30) into (22) and calculating the reu sultant integral, we derive the asymptotic SER as −G

∞ Ps,u ≈ (Ga γ¯uD ) d (32) )− G1 ( d aΘ(2Gd −1)!! where the array gain Ga = b Γ(G . Based on Gd +1 +1)2 d (32), we demonstrate that STNC(with TAS/MRC )achieves the ∑R full diversity order of Gd = Nu + r=1 Nr ND in the cooperative MIMO network and the system performance is significantly improved through integrating multiple antennas in the nodes. The contributions of the u–D and r–D links to the diversity order are Nu ND and Nr ND respectively, and u Γ(ND Nu ) their contributions to the array gain are Γ(N N )(Γ(N +1))Nu −1 ND Nr

D

D

Nr Γ(ND Nr )κr and Γ(N Nr −1 respectively. In the special case where D )(Γ(ND +1)) all the nodes are equipped with a single antenna with Nu = Nr = ND = 1, the diversity order reduces to R + 1, which agrees with the result given in [8, 10].



P s,u|Du,v

  √ Nu −1 N1 −1 Nϱ −1 ϕg +ND p−1 ϱ ϕr∑ +ND ∑ p−1 k−p ∑ ∑ ∑ ∑ cu (2k − 1)!!ξ k−p ∑ c (2k − 1)!!ξ a a b ∑ r r,0 0 . = − ··· Ψ + k+ 1 k+ 1 2 2 g =0 k!2k (ξ0 + b) 2 k!2k (ξr,0 + b) 2 p=1 r=1 p=1 0

h1,0 =0

hϱ,0 =0

k=0

IV. P ERFORMANCE WITH D ELAYED F EEDBACK In the presence of perfect feedback, the optimal transmit antenna is selected based on accurate CSI. However, the feedback processes usually lead to delay, which results in that the TAS is performed based on outdated CSI. In this section, we examine the detrimental impact of delayed feedback on the performance of STNC with TAS/MRC.

˜ rD , the PDF of Υ ˜ u|D is ˜ uD and Υ Based on the PDFs of Υ u,v presented in the following lemma. ˜ u|D is derived as Lemma 3: The PDF of Υ u,v

fΥ ˜ u|D

u,v

(˜ γ) =

ϕg N1 −1 N∑ u −1 ∑ ∑

hDµ ij

between (t) and (t − τµ ). For Clarke’s fading spectrum, ρµ = J0 (2πfµ τµ ), where fµ is the Doppler frequency and J0 (·) is the zeroth-order Bessel function of the first kind ˜ µD , µ ∈ {u, r}, as the outdated [23, Eq. (8.402)]. Define Υ ˜ µD . SNR, and the following lemma gives the PDF of Υ ˜ ˜ Lemma 2: The PDFs of ΥuD and ΥrD are fΥ γ) = ˜ uD (˜

ϕg N∑ u −1 ∑ g0 =0 k=0

˜ 0,k γ˜ ND +k−1 ˜ Ψ e−ξ0 γ˜ Γ (ND + k)

(33)

and fΥ γ) = ˜ rD (˜

N∑ r −1

ϕr ∑ ˜ r ,k γ˜ ND +kr −1 ˜ Ψ 0 r e−ξr,0 γ˜ Γ (ND + kr )

(34)

hr,0 =0 kr =0

˜ 0,k , Ψ ˜ r ,k , ξ˜0 , and ξ˜r,0 are defined in respectively, where Ψ 0 r Appendix D. Proof: The proof is presented in Appendix D. ˜ µD = In the presence of perfect feedback, i.e., ρµ = 1, Υ ΥµD . In such a case, it is easy to demonstrate that (33) and (34) are equal to (12) and (13), respectively, by setting k = ϕg in (33) and kr = ϕr in (34). For the case of fully delayed ˜ µD reduces to a gamma distributed feedback, i.e., ρµ = 0, Υ variable with shape parameter ND and scale parameter γ¯µD , and we demonstrate that (33) and (34) reduce to the same form as that in (11) by setting k = 0 and kr = 0 in (33) and (34), respectively. 1) OP: Following the procedure outlined in Section III-A1, the end-to-end SNR of xu conditioned on active relay set Du,v with delayed feedback is ˜ u|D = Υ ˜ uD + Υ u,v

ϱ ∑ r=1

˜ rD . Υ

(35)

ϕϱ ∑ ∑

···

ϱ N∑ ˜ D +kr ∑ c˜r γ˜ p−1 e−ξr,0 γ˜ + Γ (p) r=1 p=1

where ND

c˜u = (−1)

ϱ ∑∏ +k−p

( (

˜ u r=1 Ω

c˜r =(−1)

ND +kr −p

∑

( (

˜r Ω

×

ϱ ∏

ND +kr +ir −1 ir

ξ˜r,0 − ξ˜0

(

r ′ =1,r ′ ̸=r

) (36)

)

)ND +kr +ir

ND +k+i0 −1 i0

ξ˜0 − ξ˜r,0 (

˜ Ψ

hϱ,0 =0 kϱ =0

(N +k ˜ D ∑ c˜u γ˜ p−1 e−ξ0 γ˜ × Γ (p) p=1

CN 0, d−α and ρµ is the normalized correlation coefficient µD hDµ ij

Nϱ −1

ϕ1 ∑

g0 =0 k=0 h1,0 =0 k1 =0

A. Exact Performance Due to the delay between the instants of channel estimation and data transmission, we assume that the transmit antennas of user u and relay r are selected based on the outdated CSI with τu and τr time delays, respectively. Dµ To model the relationship between hDµ ij (t) and hij (t − τµ ), µ ∈ {u, r}, we employ the time-varying channel feedback error model to express the channel coefficient as hDµ ij (t) = √ 2 e (t) [25, 26], where e (t) ∼ ρµ hDµ (t − τ ) + 1 − |ρ | µ µ µ µ ij ( )

(25)

k=0

)

)ND +k+i0

ND +kr′ +ir′ −1 ir ′

ξ˜r′ ,0 − ξ˜r,0

)

)ND +kr′ +ir′

˜ u is the set of nonnegative integers {i1 , i2 , · · · , iϱ }, with Ω ∑ϱ ˜ such that j=1 ij = ND + k − p, Ωr is the set of nonnegative integers {i0 , i1 , · · · , ir−1 , ir+1 , · · ·∏ , iϱ }, such that ∑ϱ ˜ =Ψ ˜ 0,k ϱ Ψ ˜ i = N + k − p, and Ψ j D r j=0,j̸=r r=1 r0 ,kr . Proof: Following the similar procedures outlined in Ap˜ u|D by multiplying the pendix C, we derive the MGF of Υ u,v ˜ ˜ ˜ uD and Υ ˜ rD MGFs of ΥuD and ΥrD . Here, the MGFs of Υ ˜ ˜ can be obtained based on the PDFs of ΥuD and ΥrD . Then performing the inverse Laplace transform with some algebraic ˜ u|D as (36). manipulations, we derive the PDF of Υ u,v Integrating (36) with the aid of [23, Eq. (3.351.1)], we ˜ u|D as obtain the CDF of Υ u,v FΥ ˜ u|D

u,v

(˜ γ ) =1 −

ϕg N1 −1 N∑ u −1 ∑ ∑

ϕ1 ∑

Nϱ −1

···

g0 =0 k=0 h1,0 =0 k1 =0

×

(N +k p−1 D ∑ ∑ c˜u γ˜ q e−ξ˜0 γ˜ p=1 q=0

+

ϕϱ ∑ ∑

q!ξ˜0p−q

ϱ N∑ p−1 ˜ D +kr ∑ ∑ c˜r γ˜ q e−ξr,0 γ˜ r=1

p=1

q=0

˜ Ψ

hϱ,0 =0 kϱ =0

p−q q!ξ˜r,0

) . (37)

In the presence of perfect feedback, i.e., ρu = ρr = 1 and ˜ u = Υu , it is easy to demonstrate that (37) is equal to (20) Υ by setting k = ϕg and kr = ϕr in (37).



As we mentioned previously that the optimal transmit antenna of user u for the destination D corresponds to a random transmit antenna for relay r in the first phase, the delayed feedback has no impact on the SNR of xu at relay r. Hence, ˜ u < Υth |Du,v ) = substituting the conditional CDF Pr(Υ FΥ (Υth ) and (17) into (16), we can derive the exact ˜ u|D u,v closed-form expression for the OP of STNC with TAS/MRC in the presence of delayed feedback. 2) SER: In the presence of delayed feedback, the SER of STNC associated with xu is given by P˜s,u =

R ϑ ∑ ∑

P˜ s,u|Du,v Pr (Du,v )

(38)

|Du |=0 v=1

where P˜ s,u|Du,v denotes the SER of detecting xu at the destination D conditioned on active relay set Du,v in the presence of delayed feedback. Substituting (37) into (22) and applying the identity [23, Eq. (3.371)] to solve the resultant integral, we derive the exact closed-form expression for P˜ s,u|Du,v as √ Nu −1 ϕg N1 −1 ϕ1 Nϱ −1 ϕϱ ∑ ∑ ∑ ∑ ∑ a a b ∑ ˜ ˜ ··· P s,u|Du,v = − Ψ 2 2 g =0 k=0 h1,0 =0 k1 =0 hϱ,0 =0 kϱ =0 0  ND +k p−1 q−p  ∑ ∑ c˜u (2q − 1)!!ξ˜0 × ( )q+ 12 p=1 q=0 q!2q ξ˜ + b 0  ϱ N∑ p−1 q−p D +kr ∑ ˜ ∑ c˜r (2q − 1)!!ξr,0  + ( )q+ 12  . (39) r=1 p=1 q=0 q!2q ξ˜ r,0 + b For the extreme case of perfect feedback with ρu = ρr = 1, it is easy to verify that (39) is equivalent to (25) by setting k = ϕg and kr = ϕr in (39). Substituting (17) and (39) into (38) yields the exact closed-form SER expression P˜s,u with delayed feedback.

×

( )ϕg g Nu Φg (−1) 0 Γ (ND + ϕg ) 1 − ρ2u 2

(Γ (ND )) (1 + (1 −

N +ϕ ND ρ2u ) g0 ) D g γ¯uD

(41) fΥ γ) ≈ ˜ ∞ (˜

N∑ r −1

rD

hr,0 =0

×

(

Nr − 1 hr,0

Nr Φr (−1)

)

hr,0

( )ϕr Γ (ND + ϕr ) 1 − ρ2r

2

ND +ϕr ND γ¯rD

(Γ (ND )) (1 + (1 − ρ2r ) hr,0 )

Performing the Laplace transforms of (41) and (42) with the aid of [23, Eq. (3.351.3)] and multiplying the resulted ˜∞ expressions together, we have the MGF of Υ u as ˜ γuD ) MΥ ˜ ∞ (s) ≈ Θ(s¯

−(R+1)ND

u

˜∞ ˜∞ Υ u = ΥuD +

R ∑

˜∞ Υ rD

(40)

r=1

˜ ∞ and Υ ˜ ∞ are the received SNRs of xu in the where Υ uD rD high SNR regime through the u–D and r–D links respectively, whose asymptotic PDFs can be derived by applying the Taylor series expansion of the exponential function in (33) and (34) and discarding the high order items as follows ) N∑ u −1 ( Nu − 1 fΥ γ) ≈ ˜ ∞ (˜ uD g0 g =0 0

(43)

where

( ) ) g0 2 ϕg N Φ (−1) Γ (N + ϕ ) 1 − ρ N − 1 u g D g u u ˜ = Θ N +ϕ g0 Γ (ND ) (1 + (1 − ρ2u ) g0 ) D g g0 =0 ) R N∑ r −1 ( ∏ Nr − 1 × hr,0 r=1 hr,0 =0 ( )ϕr h D Nr Φr (−1) r,0 Γ (ND + ϕr ) 1 − ρ2r κN r . × N +ϕ D r Γ (ND ) (1 + (1 − ρ2r ) hr,0 ) N∑ u −1 (

Performing the inverse Laplace transform of (43) yields the ˜ ∞ , from which we derive the asymptotic CDF of PDF of Υ u ˜∞ Υ in the high SNR regime as u FΥ γ) ≈ ˜ ∞ (˜ u

˜ Θ γ˜ (R+1)ND . Γ ((R + 1) ND + 1) γ¯ (R+1)ND

(44)

uD

Through evaluating FΥ γ ) at γ˜ = Υth , the asymptotic OP ˜ ∞ (˜ u of the end-to-end SNR associated with xu under delayed feedback conditions is given by u

In this subsection, we derive the asymptotic OP and SER expressions in the high SNR regime to quantify the detrimental impact of delayed feedback and offer useful insights into the behavior of STNC with TAS/MRC. 1) OP: In the presence of delayed feedback, the end-to-end SNR of xu in the high SNR regime is given by

γ˜ ND −1 . (42)

∞ = FΥ P˜out,u ˜ ∞ (Υth ) .

B. Asymptotic Performance

γ˜ ND −1

(45)

2) SER: Based on FΥ γ ), we now derive the asymptotic ˜ ∞ (˜ u SER of STNC with TAS/MRC in the high SNR regime. Substituting (44) into (22) and calculating the resultant integral, we derive the asymptotic SER as ( )−G˜ d ∞ ˜ a γ¯uD P˜s,u ≈ G (46) ˜ = (R + 1) ND and array gain where the diversity order G ( )− ˜1 d G ˜ ˜ d ˜ a = b aΘ(2Gd −1˜ )!! G . ˜ d +1)2Gd +1 Γ(G Comparing (46) with (32), it is evident that delayed feedback has a severely detrimental effect on the ( ) SER and degrades ∑R the diversity order from Nu + r=1 Nr ND to (R + 1) ND , which indicates that the contribution of the multiple antennas at the transmit end to the diversity vanishes due to the delayed feedback. In the extreme case of fully delayed feedback, the optimal transmit antenna selected based on the fully outdated CSI actually corresponds to a random transmit antenna. The significant impact of delayed feedback implies that a high



0

0

10

10

Exact OP Simulation Asymptotic OP

−1

10

−2

10

Outage Probability

Outage Probability

10

−3

10

−4

10

R=1, N =2, N =1 (G =3) u r d R=1, N =1, N =2 (G =3) u r d R=2, N =2, N =1 (G =4) u r d R=3, N =2, N =1 (G =5) u r d R=2, N =2, N =2 (G =6)

−5

10

u

−6

10

0

r

−2

10

−3

10

−4

10

ρu=ρr=0 (Gd=3) ρu=ρr=0.5 (Gd=3) ρ =ρ =0.9 (G =3) u r d ρu=ρr=1 (Gd=6)

−5

10

d

5

Exact OP Simulation Asymptotic OP

−1

−6

10

15

20

10

25

0

5

10

Pu /N0 (dB)

Fig. 1. Outage probability of STNC for a threshold Υth = 10 dB in the presence of perfect feedback (ND = 1).

−1

25

−1

10

Exact SER Simulation Asymptotic SER

−2

10

−2

−3

10

−4

10

−5

10

−6

10

R=1, Nu=2, Nr=1 (Gd=3) R=1, Nu=1, Nr=2 (Gd=3) R=2, N =2, N =1 (G =4) u r d R=3, Nu=2, Nr=1 (Gd=5) R=2, N =2, N =2 (G =6) u

r

−3

10

−4

10

−5

10

−6

10

d

ρ =ρ =0 (G =3) u r d ρ =ρ =0.5 (G =3) u r d ρ =ρ =0.9 (G =3) u r d ρ =ρ =1 (G =6) u

−7

−5

Exact SER Simulation Asymptotic SER

10

Symbol Error Rate

Symbol Error Rate

20

Fig. 3. Outage probability of STNC for a threshold Υth = 10 dB in the presence of delayed feedback (R = 2, Nu = Nr = 2, ND = 1).

10

10

15

Pu /N0 (dB)

r

d

−7

0

5

10

15

Pu /N0 (dB)

Fig. 2. Symbol error rate of BPSK modulation of STNC in the presence of perfect feedback (ND = 1).

feedback rate may be required in practice in order to attain the full benefits of transmit antenna selection. V. N UMERICAL R ESULTS In this section, numerical and simulation results are presented to show the validity of our analysis and the impacts of the network parameters on the OP and SER of STNC with TAS/MRC. We assume that the coordinations of the destination D, relay r, and user u are (0, 0), ((d+r∆d) cos(rψ), (d+ r∆d) sin(rψ)), and (cos(uψ ′ ), sin(uψ ′ )) respectively, which implies that the distances from the destination D to relay r and to user u are d + r∆d and 1 respectively. In the simulations, d = 0.4, ∆d = 0.1, and ψ = ψ ′ = π/18. The cross correlations between different spread codes are set to be zero, and the value of the path-loss exponent α is 3.5 [27]. We also assume equal transmit power at each node. Figs. 1 and 2 plot the OP and SER of STNC versus transmit SNR Pu /N0 for different antenna configurations in

10

−5

0

5

10

15

20

Pu /N0 (dB)

Fig. 4. Symbol error rate of BPSK modulation of STNC in the presence of delayed feedback (R = 2, Nu = Nr = 2, ND = 1).

the presence ∑R of perfect feedback, respectively, where Pu = P0u + r=1 Pru . The arrows in the figures are used to link each curve with corresponding network parameters. It is observed that the simulation results perfectly match with exact theoretical results and exact curves approach asymptotic curves in the high SNR regime, which validate the accuracy of our theoretical analysis in Section III. In the presence of perfect feedback, STNC with ( ) TAS/MRC provides full diversity ∑R order of Nu + r=1 Nr ND , as indicated by (32). Notably, the diversity order increases when the antenna number and relay number increase, which in turns leads to a pronounced SNR gain. For the single relay case in Figs. 1 and 2, it is shown that the antenna configuration of Nu = 1 and Nr = 2 exhibits a better performance than that of Nu = 2 and Nr = 1, although both of them have the same diversity order. This is due to the fact that the relay is involved in both transmission phases. Fig. 3 plots the OP of STNC with different delay correlation



A PPENDIX A P ROOF OF E QUATION (12)

0

10

−1

10

Symbol Error Rate

We denote γu as the instantaneous SNR of the channel from a random transmit antenna at user u to the ND antennas at the destination D. The PDF and CDF of γu , fγu (γ) and Fγu (γ), are given by [22]

16QAM

−2

10

−3

10

−5

10

fγu (γ) = ρ =ρ =0 u

10

r

ρ =ρ =0.5 u

−6

−

QPSK

−4

10

−7

ρu=ρr=0.9 u

−5

(47)

and

r

Fγu (γ) = 1 − e

ρ =ρ =1 10 −10

γ

γ ND −1 e γ¯uD ND Γ (ND ) γ¯uD

r

0

5

10

15

− γ¯ γ

N∑ D −1

uD

i=0

20

Pu /N0 (dB)

Fig. 5. Symbol error rates of QPSK and 16QAM modulations of STNC in the presence of delayed feedback (R = 2, Nu = Nr = ND = 2).

coefficients versus transmit SNR Pu /N0 . Figs. 4 and 5 plot the SERs of STNC with different modulations and different delay correlation coefficients versus transmit SNR Pu /N0 . The simulation results are marked by “•” in Fig. 5. The precise agreement between the theoretical and the simulation results in Figs. 3 and 4 verifies our theoretical analysis in Section IV. In Fig. 5, the theoretical and simulation curves are consistent in the medium and high SNR regime; whereas there exists a very minor gap between the theoretical and the simulation curves in the low SNR regime, especially for 16QAM. This is due to the fact that the closed-form SER expression (22) is an approximation for higher degree of modulation, e.g., QPSK and 16QAM. We find that the performance of the STNC improves as the delay correlation coefficients ρu and ρr increase. It is shown that, in the presence of delayed feedback, the asymptotic curves in Figs. 3 and 4 keep the same slope of −3, which means that the delayed feedback reduces the full diversity order to (R + 1) ND regardless of the values of ρu and ρr . This indicates that the diversity advantage from transmit end vanishes and only the one from receive end remains due to the delayed feedback. In Fig. 5, it is shown that fully delayed feedback incurs an SNR loss of approximately 3 dB compared to perfect feedback at an SER of 10−6 .

In this paper, we have analyzed the performance of STNC in cooperative MIMO network in terms of OP and SER, where multi-antenna diversity is guaranteed via TAS/MRC. We have derived new closed-form expressions of the exact and asymptotic OP and SER for both perfect and delayed feedback. In the presence of perfect feedback, we have confirmed that STNC with TAS/MRC preserves the full diversity order of ( ) ∑R Nu + r=1 Nr ND . In the presence of delayed feedback, we have quantified the detrimental effect of delayed feedback on the OP and SER of STNC. It is shown that the diversity advantage from the transmit end vanishes and the diversity order is degraded to (R + 1) ND due to delayed feedback.

(48)

respectively. Based on (47) and (48), we have )Nu −1 ( N∑ − γ D −1 Nu γ ND −1 e γ¯uD γi − γ¯ γ fΥuD (γ) = 1 − e uD i ND i!¯ γuD Γ (ND ) γ¯uD i=0 γ ) Nu −1 ( − g γ Nu γ ND −1 e γ¯uD ∑ Nu − 1 g − 0 = (−1) 0 e γ¯uD ND g0 Γ (ND ) γ¯uD g0 =0 )g0 (N −1 D ∑ γi × . (49) i i!¯ γuD i=0 According to [28, Eq. (9)], we have (N −1 )g0 ( )ϕg D ∑ γi γ = Φg (50) i γ¯uD i!¯ γuD i=0 ∏ND −1 (∑gi−1 ( gi−1 )( 1 )gi −gi+1 ) where Φg = , ϕg = gi =0 i=1 gi i! ∑ND −1 i=1 gi , and gND = 0. Substituting (50) into (49) yields (12), where ξ0 = (g0 + 1) /¯ γuD .

A PPENDIX B P ROOF OF E QUATION (14) Dµ Using γij , µ ∈ {u, r}, 1 ≤ i ≤ ND , and 1 ≤ j ≤ Nµ , to denote the i.n.i.d. distributed SNR between the jth antenna of transmitter µ and the ith antenna of destination D, we have

fγ Dµ (γ) = ij

VI. C ONCLUSIONS

γi i i!¯ γuD

1

e Dµ

−

γ Dµ γ ¯ ij

(51)

γ¯ij

Dµ Dµ Dµ where γ¯ij = E[γij ]. Based on (51), the MGF of γij is Dµ Mγ Dµ (s) = 1/(s¯ γij + 1). If we MRC combine the ND ij signal replicas from the jth antenna of transmitter µ at the Dµ destination D, the total SNR, γµ , is the summation of γij , 1 ≤ i ≤ ND . As such, the MGF of γµ is )−1 ( ND ND ∏ ∏ 1 1 = αj s + Dµ (52) Mγµ (s) = Dµ γij γ¯ij +1 i=1 i=1 s¯ ( )−1 ∏ND 1 ∏ND 1 where αj = . Expanding s + in Dµ Dµ i=1 γ i=1 ¯ γ ¯ ij

ij

poles and residuals with the aid of partial fraction decomposition [23, Eq. (2.102)] and performing inverse Laplace



transform on (52), we obtain ( )−1 γ ND ND ∏ ∑ − Dµ 1 1 γ ¯ ij . fγµ (γ) = αj − e Dµ γ¯iDµ γ¯ij ′j i=1 i′ =1,i′ ̸=i Defining βij =

1

i′ =1,i′ ̸=i

γ ¯iDµ ′j

fΥµD (γ) =

Nµ ∑

αj

ND ∑

−

Dµ βij γ¯ij e

γ Dµ γ ¯ ij

×

. Hence, PDF

γ Dµ γ ¯ ij

ND ∑

1 − αj ′

βi′ j ′ γ¯iDµ ′j′ e

−

γ Dµ γ ¯ ′ ′ i j

)

Based on (54), (14) can be derived directly by expanding the product term. A PPENDIX C P ROOF OF L EMMA 1 Here, we first derive the MGF of Υ u|Du,v , and then obtain the PDF of Υ u|Du,v based on the MGF. From (12) and (13), the MGFs of ΥuD and ΥrD are obtained by performing the Laplace transform with the aid of the definite integral of the exponential function [23, Eq. (3.351.3)] as N∑ u −1

−(ϕg +ND )

Ψ0 (ξ0 + s)

(55)

g0 =0

and MΥrD (s) =

N∑ r −1

−(ϕr +ND )

Ψr,0 (ξr,0 + s)

( ) Nu Γ(ϕg +ND )Φg respectively, where Ψ0 = Nug0−1 ϕ +ND , and Γ(ND )(−1)g0 γ ¯uDg ( ) Nr Γ(ϕr +ND )Φr −1 Ψr,0 = Nhrr,0 ϕ +ND . Γ(ND )(−1)hr,0 γ ¯rDr Since the MGF of the sum of multiple independent random variables is equal to the product of the MGFs of the random variables [22], the MGF of Υ u|Du,v is N∑ u −1

×

−(ϕg +ND )

ϱ ∏ r=1

 

N∑ r −1

 −(ϕr +ND ) 

Ψr,0 (ξr,0 + s)

′ r ′ =1,r ′ ̸=r (ξr ,0

− ξr,0 )

)

ϕr′ +ND +ir′

Denoting γ˜u as the outdated SNR relative to the original SNR γu , the joint PDF of γ˜u and γu is [29] γ ˜ +γ ( )ND +1 ( ) ND2−1 − 1 γ˜ γ e (1−ρ2u )γ¯uD fγ˜u ,γu (˜ γ , γ) = γ¯uD ρ2 Γ (N ) (1 − ρ2u ) ( √u ) D 2 ρ2u γ˜ γ × IND −1 (58) (1 − ρ2u ) γ¯uD

where In (·) stands for the nth-order modified Bessel function of the first kind [23, Eq. (8.406.1)]. According to the order ˜ uD , is the induced statistics, the time-delayed SNR after TAS, Υ order statistics of the original ordered SNR after TAS, ΥuD ˜ uD is derived as [30]. Therefore, the PDF of Υ ∫ ∞ fγ˜u ,γu (˜ γ , γ) fΥ γ) = fΥuD (γ) dγ ˜ uD (˜ f (γ) 0 ∫ ∞ γu N −1 =Nu fγ˜u ,γu (˜ γ , γ) (Fγu (γ)) u dγ. (59) Substituting (58) and the CDF of γu , (48), into (59) and performing some algebraic manipulations with the aid of [23, Eq. (6.643.2)] yield ND ) N∑ u −1 ( Nu − 1 Nu Φg Γ (ND + ϕg ) γ˜ 2 −1 fΥ γ) = ˜ uD (˜ ND 2 g g0 (Γ (N )) (−1) 0 (ρ2 γ¯ ) 2 g =0 0

.

hr,0 =0

(57) Expanding (57) in poles and residuals with the aid of partial fraction decomposition [23, Eq. (2.102)] and Faa di Bruno Formula and performing inverse Laplace transform yield (19), where ( ) ϕr +ND +ir −1 ϱ ∑ ∏ ir ϕ +N −p cu = (−1) g D ϕr +ND +ir Ωu r=1 (ξr,0 − ξ0 )

D

(

× ×e

Ψ0 (ξ0 + s)

g0 =0

ϕr′ +ND +ir′ −1 ir ′

0

(56)

hr,0 =0

MΥ u|Du,v (s) =

ϱ ∏

(ξ0 − ξr,0 ) (

A PPENDIX D P ROOF OF L EMMA 2

. (54)

MΥuD (s) =

)

∏ϱ with Ψ = Ψ0 r=1 Ψr,0 , Ωu is the∑set of nonnegaϱ tive integers {i1 , i2 , · · · , iϱ }, such that j=1 ij = ϕg + ND − p, and Ωr is the set of nonnegative integers ∑ϱ {i0 , i1 , · · · , ir−1 , ir+1 , · · · , iϱ }, such that j=0,j̸=r ij = ϕr + ND − p.

i′ =1

j ′ =1,j ′ ̸=j

ϕg +ND +i0 −1 i0

ϕg +ND +i0

Ωr

(53)

, the CDF of γµ is

Dµ γ ¯ij

(

∏ Nµ

×

1

−

i=1

βij e

i=1

j=1

−

∑ND

given by Fγµ (γ) = 1 − αj of ΥµD is given by

ϕr +ND

)−1

(

∏ND

cr =(−1)

(

∑ −p

)−( N2D +ϕg )

u uD

1 + g0 1 − ρ2u 2(1+(1−ρ2 ˜ −ρ2 ˜ u )g0 )γ uγ −

2(1−ρ2 ¯uD (1+(1−ρ2 u )γ u )g0 )

(

× M−( ND +ϕg ), ND −1 2

2

) ( )−1 2 1 − ρ2u ρu γ˜ γ¯uD (1 + (1 − ρ2u ) g0 ) (60)

where Ma,b (·) is the Whittaker function [23, Eq. (9.220.2)]. With the aid of the finite series expansion of Whittaker ˜ uD given in (33), where function [26], we derive the PDF of Υ ( )( ) ϕg ˜ 0,k = Nu − 1 Ψ g0 k ( ) g0 2 ϕg −k Nu Φg (−1) Γ (ND + ϕg ) ρ2k u 1 − ρu × N +ϕ +k ND +k Γ (ND ) γ¯uD (1 + (1 − ρ2u ) g0 ) D g



g0 +1 and ξ˜0 = (1+(1−ρ . Following the similar procedures 2 )g )¯ 0 γuD u ˜ uD , we derive the the PDF of Υ ˜ rD , of deriving the PDF of Υ as shown in (34), where ( )( ) ϕr ˜ r ,k = Nr − 1 Ψ 0 r hr,0 kr ( )ϕr −kr hr,0 r Nr Φr (−1) Γ (ND + ϕr ) ρ2k 1 − ρ2r r × N +ϕ +k ND +kr Γ (ND ) γ¯rD (1 + (1 − ρ2r ) hr,0 ) D r r

and ξ˜r,0 =

1+hr,0 (1+(1−ρ2r )hr,0 )¯ γrD .

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Kai Yang (M’12) received his B.E. degree and Ph.D. degree from National University of Defense Technology and Beijing Institute of Technology, China, in 2005 and 2010 respectively, both in communications engineering. From Jan. 2010 to July 2010, he was with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University. From 2010 to 2013, he was with Alcatel-Lucent Shanghai Bell, Shanghai, China. In 2013, he joined the Laboratoire de Recherche en Informatique, University Paris Sud 11, Orsay, France. Now, he is with the School of Information and Electronics, Beijing Institute of Technology, Beijing, China. His current research interests include convex optimization, cooperative communications, MIMO systems, resource allocation, and interference mitigation.

Nan Yang (S’09–M’11) received the B.S. degree in electronics from China Agricultural University in 2005, and the M.S. and Ph.D. degrees in electronic engineering from the Beijing Institute of Technology in 2007 and 2011, respectively. He is currently a Future Engineering Research Leadership Fellow and Lecturer in the Research School of Engineering at the Australian National University. Prior to this he was a Postdoctoral Research Fellow at the University of New South Wales (UNSW) from 2012 to 2014; a Postdoctoral Research Fellow at the Commonwealth Scientific and Industrial Research Organization from 2010 to 2012; and a visiting Ph.D. student at UNSW from 2008 to 2010. He received the IEEE Communications Society Asia-Pacific Outstanding Young Researcher Award in 2014, the Exemplary Reviewer Certificate of the IEEE Communications Letters in 2012 and 2013, and the Best Paper Award at the IEEE 77th Vehicular Technology Conference in 2013. He serves as an editor of Transactions on Emerging Telecommunications Technologies. His general research interests lie in the areas of communications theory and signal processing, with specific interests in collaborative networks, network security, resource management, massive multi-antenna systems, millimeter wave communications, and molecular communications.



Chengwen Xing (S’08–M’10) received the B. Eng. degree from Xidian University, Xi ’ an, China, in 2005 and the Ph. D. degree from the University of Hong Kong, Hong Kong, in 2010. Since Sep. 2010, he has been with the School of Information and Electronics, Beijing Institute of Technology, Beijing, China, where he is currently an Associate Professor. From Sep. 2012 to Dec. 2012, he was a visiting scholar at the University of Macau. His current research interests include statistical signal processing, convex optimization, multivariate statistics, combinatorial optimization, massive MIMO systems and high frequency band communication systems. Dr. Xing is currently serving as an Associate Editor for IEEE Transactions on Vehicular Technology, KSII Transactions on Internet and Information Systems, Transactions on Emerging Telecommunications Technologies and China Communications.

Jinsong Wu (SM’11) is the Founder and Founding Chair of Technical Committee on Green Communications and Computing (TCGCC), IEEE Communications Society. He is the Founder and Editor of Series on Green Communication and Computing networks of IEEE Communications Magazine. He has served as guest editors for IEEE Systems Journal, IEEE Access journal, IEEE J. Select. Areas on Communications (JSAC), IEEE Communications Magazine, IEEE Wireless Communications, Elsevier Computer Networks. He is an Area Editor of the incoming JSAC Series on Green Communications and Networking starting in 2015. He has been in editorial boards in a number of technical journals, such as IEEE Communications Surveys & Tutorials, IEEE Systems Journal, IEEE Access, International Journal of Big Data Intelligence. He was the leading Editor and a co-author of the comprehensive book “Green Communications: Theoretical Fundamentals, Algorithms, and Applications,” by CRC Press in 2012. He currently is an IEEE Senior Member.

Zhongshan Zhang received the B.E. and M.S. degrees in computer science from the Beijing University of Posts and Telecommunications (BUPT) in 1998 and 2001, respectively, and received Ph.D. degree in electrical engineering in 2004 from BUPT. From Aug. 2004 he joined DoCoMo Beijing Laboratories as an associate researcher, and was promoted to be a researcher in Dec. 2005. From Feb. 2006, he joined University of Alberta, Edmonton, AB, Canada, as a postdoctoral fellow. From Apr. 2009, he joined the Department of Research and Innovation (R&I), Alcatel-Lucent, Shanghai, as a Research Scientist. From Aug. 2010 to Jul. 2011, he worked in NEC China Laboratories, as a Senior Researcher. He served or is serving as a Guest Editor and/or an editor for several technical journals, such as the IEEE Communications Magazine and KSII Transactions on Internet and Information Systems. He is currently a professor of the School of Computer and Communication Engineering in the University of Science and Technology Beijing (USTB). His main research interests include statistical signal processing, self-organized networking, cognitive radio, and cooperative communications.
