4068 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS,...

4068 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

Relay Selection for Decode-and-ForwardCooperative Network with Multiple AntennasXianglan Jin, Jong-Seon No, Senior Member, IEEE, and Dong-Joon Shin, Senior Member, IEEE

Abstract—In this paper, a new relay selection scheme fordecode-and-forward (DF) relay cooperative network with multi-ple antennas is proposed based on both channel state information(CSI) and transmission scheme by deriving the upper boundon the pairwise error probability (PEP) of the near-maximum-likelihood (near-ML) decoder. It is also proved that the proposedrelay selection which selects 𝑚 (1 ≤ 𝑚 ≤ 𝑁) relays from𝑁 relays achieves full diversity 𝑀𝑆𝑀𝐷 + 𝑁𝑀𝑅 min[𝑀𝑆 ,𝑀𝐷]regardless of the value of 𝑚 in the DF relay network consistingof one source, one destination, and 𝑁 relays with 𝑀𝑆 , 𝑀𝐷, and𝑀𝑅 antennas, respectively. Through Monte Carlo simulation, theerror correction performance of the proposed relay selection forvarious 𝑚 is shown for the uncoded single-antenna, Alamouticoded, and multiple-input multiple-output (MIMO) DF relaynetworks.

Index Terms—Decode-and-forward (DF), diversity, maximum-likelihood (ML), multiple antennas, pairwise error probability(PEP), relay selection.

I. INTRODUCTION

IN the wireless communication networks, deep fading oftencauses failure of reliable data transmission. In this case, re-

lays can be used to cooperatively assist the data transmission.Such a system is called a cooperative communication network,where the cooperative diversity can be achieved [1]-[3].

In [1] and [2], Sendonaris, Erkip, and Aazhang presentedan information theoretic model for cooperative communicationnetwork and analyzed the achievable rate region and outageprobability in the code division multiple access (CDMA) sys-tem. Laneman and Wornell [3] developed various cooperativediversity algorithms for a source and destination pair basedon relays amplifying or fully decoding and forwarding theirreceived signals. These algorithms are referred as amplify-and-forward (AF) and decode-and-forward (DF) relaying, re-spectively. Even though the relay operation for the AF relay

Manuscript received January 9, 2010; revised August 5, 2010 and January12, 2011; accepted August 15, 2011. The associate editor coordinating thereview of this paper and approving it for publication was D. Michelson.

X. Jin is with the Department of Information and CommunicationEngineering, Dongguk University-Seoul, Seoul 100-715, Korea (e-mail:[email protected]).

J.-S. No is with the Department of Electrical Engineering and ComputerScience, INMC, Seoul National University, Seoul 151-744, Korea (e-mail:[email protected]).

D.-J. Shin is with the Department of Electronic Engineering, HanyangUniversity, Seoul 133-791, Korea (e-mail: [email protected]).

This work was partly supported by the KCC (Korea CommunicationsCommission), Korea, under the R&D program supervised by the KCA (KoreaCommunications Agency) (KCA-2011-08913-04003), a Korea Science andEngineering Foundation (KOSEF) grant funded by the Korea government(MEST) (No. 2011-0000328), and the Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded by theMinistry of Education, Science and Technology (2011-0005909).

Digital Object Identifier 10.1109/TWC.2011.092911.100030

is simple, their transceivers require expensive radio frequencyamplifiers [4].

For the DF relay network, a maximum-likelihood (ML)decoder has been introduced and a suboptimal low-complexitydecoder, called 𝜆 maximum ratio combining (𝜆-MRC), wasproposed in [2] for a single-antenna system with binary phaseshift keying (BPSK). In [4], a cooperative MRC (C-MRC)was proposed and it was proven that C-MRC for the uncodedsingle-antenna DF relay network can achieve full diversity.For many DF relay networks with multiple antennas, MLdecoder becomes very complicated and thus cannot be usedfor the most cases, for example, multiple-input multiple-output(MIMO) DF relay network. In [5], Jin et al. proposed a near-ML decoder and proved that the near-ML decoder for the DFrelay network with multiple antennas achieves full diversity.

Recently, relay networks with relay selection have beenwidely investigated [6]-[14]. In [6], a relay selection criterionfrom the information theoretic aspect was proposed for DFrelay cooperative network, in which the relays are allowedto cooperate if their source-relay (SR) channel coefficientmagnitudes exceed a threshold. In [7], a nearest relay selectioncriterion, that is, selecting relays nearest to the source or to thedestination, was proposed. Some other works for relay selec-tion have been focused on maximizing the received signal tonoise ratio (SNR) at the destination [8]-[10], maximizing theminimum coefficient magnitude of SR and relay-destination(RD) channels [11], [12], and minimizing the upper boundon average symbol error probability [13]. All of these workswere performed under the single-antenna assumption.

For the multiple-antenna case, Fan and Thompson [14] useda selection criterion of choosing a relay which achieves thehighest network capacity for the optimal selective routing ofa two-hop network. However, this relay selection criterion ofmaximizing the network capacity may not achieve the besterror correction performance for most of the multiple-antennacases because it only considers the channel state information(CSI) but not the transmission scheme. Therefore, in thispaper, we focus on deriving a new relay selection criterionto minimize the pairwise error probability (PEP) of decodingthe received signal into x̃ when x is transmitted from thesource by assuming that there are only two symbols x andx̃ based on both CSI and transmission scheme for the DFrelay cooperative network with multiple antennas.

In this paper, one source, one destination, and 𝑁 relayswith the direct link between the source and destination areconsidered with 𝑀𝑆 , 𝑀𝐷, and 𝑀𝑅 antennas, respectively.For simplicity, it is assumed that all of the relays use thesame number of antennas and the same transmission scheme,

1536-1276/11$25.00 c⃝ 2011 IEEE

JIN et al.: RELAY SELECTION FOR DECODE-AND-FORWARD COOPERATIVE NETWORK WITH MULTIPLE ANTENNAS 4069

and all of the SR channels experience the independent andidentically distributed (i.i.d.) fading and so do all of theRD channels. We first extend the results of the near-MLdecoder in [5] to the multiple-relay cases and derive anupper bound on PEP for the DF relay network in high SNRregion. Using this upper bound, a new relay selection schemebased on both CSI and transmission scheme is proposed. Wealso prove that the proposed relay selection which selects 𝑚(1 ≤ 𝑚 ≤ 𝑁) relays from 𝑁 relays achieves full diversity𝑀𝑆𝑀𝐷 +𝑁𝑀𝑅min[𝑀𝑆,𝑀𝐷] regardless of the value of 𝑚.Through simulation, compared with the relay selection in [14]we show that the proposed relay selection has the similar errorperformance for uncoded single-antenna DF relay network andbetter error performance for Alamouti coded and MIMO DFrelay networks.

It is also interesting to find how many relays should beselected to satisfy the requirement of the networks. It is clearthat the proposed relay selection with 𝑚 = 1 is the bestin terms of bandwidth efficiency. However, it is not easyto determine analytically the number of selected relays toachieve the best error performance. Thus, the proposed relayselection for various 𝑚 is evaluated through simulation forthe uncoded single-antenna, Alamouti coded, and MIMO DFrelay networks.

This paper is organized as follows. In Section II, the systemmodel and the near-ML decoder are introduced. An upperbound on PEP for the DF relay network with multiple antennasis derived in Section III. Using the upper bound on PEP, anew relay selection scheme is proposed and its diversity isderived in Section IV. The discussion and simulation resultsare provided in Section V and the conclusion is given inSection VI.

The following notations are used in this paper: the capitalletter denotes a matrix; 𝐼𝑛 denotes the 𝑛× 𝑛 identity matrix;ℂ𝑛×𝑚 denotes a set of 𝑛×𝑚 complex matrices; ∥ ⋅ ∥ and tr(⋅)represent the Frobenius norm and the trace of a matrix, re-spectively; 𝐸[⋅] is the expectation; the superscript (⋅)† denotesthe complex conjugate transpose; Re(⋅) means the real partof a complex number. For 𝐴 ∈ ℂ

𝑛×𝑚, 𝐴 ∼ 𝒞𝒩 (0, 𝜎2𝐼𝑛𝑚)denotes that the elements of 𝐴 are i.i.d. circularly symmetricGaussian random variables with zero mean and variance 𝜎2.𝑃 (𝑎 = 𝑏) in probability, i.e., lim𝜎2→0 𝑃 (𝑎 = 𝑏) = 1, is

denoted by 𝑎𝑃= 𝑏 and similarly the notations

𝑃≤,𝑃≥,

𝑃≈,𝑃

⪅,

and𝑃

⪆ are also used.

II. NEAR-ML DECODER FOR DECODE-AND-FORWARD

RELAY COOPERATIVE NETWORK WITH MULTIPLE RELAYS

A. System Model

A DF relay network with one source, one destination, andmultiple relays using half-duplex transmission is shown inFig. 1. It is also assumed that the channels are frequency-flat quasi-static fading channels, the relays know the CSI ofthe corresponding SR channel, and the destination knows theCSIs of all SR, source-destination (SD), and RD channels.

In the first phase, the source with 𝑀𝑆 antennas broadcasts𝑀𝑆 ×𝑇1 codeword 𝑋𝑆(x) constructed from 𝐿-tuple messagevector x = (𝑥1, 𝑥2, . . . , 𝑥𝐿) ∈ 𝒜𝐿 to the 𝑁 relays and the

......

G

1K

NK

1F

NF...

NR

1R

S D

RM

SR RD

SD

...

RM

...DMSM

......Fig. 1. The DF relay network using multiple relays. The solid line denotesthe first phase transmission and the dashed line denotes the second phasetransmission.

destination, where 𝒜 is the set of message symbols from the𝑀 -ary signal constellation. Then, the received signals at the𝑖th relay with 𝑀𝑅 antennas in the first phase can be writtenas

𝑌𝑆𝑅𝑖 =√

𝑃𝑆𝐾𝑖𝑋𝑆(x) +𝑁𝑆𝑅𝑖 (1)

where 𝑃𝑆 is the average transmit power at the source,𝐾𝑖 ∈ ℂ

𝑀𝑅×𝑀𝑆 is the channel coefficient matrix of the𝑖th SR channel distributed as 𝐾𝑖 ∼ 𝒞𝒩 (0, 𝜎2

𝑆𝑅𝐼𝑀𝑅𝑀𝑆 ),and 𝑁𝑆𝑅𝑖 ∈ ℂ𝑀𝑅×𝑇1 is the noise matrix with distribution𝑁𝑆𝑅𝑖 ∼ 𝒞𝒩 (0, 𝜎2𝐼𝑀𝑅𝑇1). At the same time, the destinationalso receives the signal transmitted from the source as

𝑌𝑆𝐷 =√

𝑃𝑆𝐺𝑋𝑆(x) +𝑁𝑆𝐷 (2)

where 𝐺 ∈ ℂ𝑀𝐷×𝑀𝑆 is the channel coefficient matrix ofthe SD channel distributed as 𝐺 ∼ 𝒞𝒩 (0, 𝜎2

𝑆𝐷𝐼𝑀𝐷𝑀𝑆 ) and𝑁𝑆𝐷 ∈ ℂ𝑀𝐷×𝑇1 represents the noise matrix at the destinationwith distribution 𝑁𝑆𝐷 ∼ 𝒞𝒩 (0, 𝜎2𝐼𝑀𝐷𝑇1).

In the second phase, 𝑁 relays transmit the codewordsconstructed for their decoded symbols through 𝑁 orthogonalRD channels, where “orthogonal” means that the relays use theindependent channels, e.g., time division or frequency divisionchannel. Thus, the received signal at the destination throughthe 𝑖th orthogonal channel for the 𝑖th relay is given as

𝑌𝑅𝑖𝐷 =√

𝑃𝑅𝐹𝑖𝑋𝑅(x𝑅𝑖) +𝑁𝑅𝑖𝐷 (3)

where 𝑋𝑅(x𝑅𝑖) ∈ ℂ𝑀𝑅×𝑇2 is the codeword constructed fromthe 𝐿-tuple message vector x𝑅𝑖 = (𝑥𝑅𝑖

1 , 𝑥𝑅𝑖2 , . . . , 𝑥𝑅𝑖

𝐿 ) ∈ 𝒜𝐿

decoded by the 𝑖th relay in the first phase. 𝑃𝑅 is the av-erage transmit power at each relay, 𝐹𝑖 ∈ ℂ

𝑀𝐷×𝑀𝑅 is thechannel coefficient matrix of the 𝑖th RD channel distributedas 𝐹𝑖 ∼ 𝒞𝒩 (0, 𝜎2

𝑅𝐷𝐼𝑀𝐷𝑀𝑅), and 𝑁𝑅𝑖𝐷 ∈ ℂ𝑀𝐷×𝑇2 is thenoise matrix at the destination in the 𝑖th orthogonal channelwith 𝑁𝑅𝑖𝐷 ∼ 𝒞𝒩 (0, 𝜎2𝐼𝑀𝐷𝑇2).

B. Near-ML Decoder

Let 𝑃𝑆𝑅𝑖(x̂𝑅𝑖 ∣x) be the probability that the 𝑖th relay 𝑅𝑖

decodes the received signal to x̂𝑅𝑖 when the source transmitsthe codeword corresponding to the message vector x in thefirst phase. Considering the decoding error at the relay as in


[5], the ML decoder for DF relay network with 𝑁 relays canbe written as

x̂=arg maxx∈𝒜𝐿

𝑝 (𝑌𝑆𝐷, 𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷∣x)

=argmaxx∈𝒜𝐿

𝑝 (𝑌𝑆𝐷∣𝑋𝑆(x))

𝑁∏𝑖=1

∑x̂𝑅𝑖

∈𝒜𝐿

𝑝 (𝑌𝑅𝑖𝐷∣𝑋𝑅(x̂𝑅𝑖))𝑃𝑆𝑅𝑖(x̂𝑅𝑖∣x)

=arg maxx∈𝒜𝐿

[−∥∥𝑌𝑆𝐷−√

𝑃𝑆𝐺𝑋𝑆(x)∥∥2

𝜎2

+

𝑁∑𝑖=1

ln∑

x̂𝑅𝑖∈𝒜𝐿

exp

(−∥∥𝑌𝑅𝑖𝐷−√𝑃𝑅𝐹𝑖𝑋𝑅(x̂𝑅𝑖)∥∥2+𝜎2ln𝑃𝑆𝑅𝑖(x̂𝑅𝑖∣x)

𝜎2

)].

(4)

Since it is very difficult to derive 𝑃𝑆𝑅𝑖(x̂𝑅𝑖 ∣x) for the code-word 𝑋𝑆(x), the PEP at the 𝑖th relay 𝑃𝑆𝑅𝑖(x → x̂𝑅𝑖)will be used instead of 𝑃𝑆𝑅𝑖(x̂𝑅𝑖 ∣x) in (4). Although thePEP 𝑃𝑆𝑅𝑖(x → x̂𝑅𝑖) is not equal to 𝑃𝑆𝑅𝑖(x̂𝑅𝑖 ∣x), it canbe a good substitution for 𝑃𝑆𝑅𝑖(x̂𝑅𝑖 ∣x) to find the solutionof (4) as in [5].1 The widely-used max-log approximationln∑

𝑖 𝑒𝑥𝑖 ≈ max𝑖 𝑥𝑖 [15], [16], [17] is also used. Through

these two steps, the ML decoder can be simplified to the so-called near-ML decoder as in [5]. Then, the near-ML decoderfor (4) can be written as

x̂=arg minx∈𝒜𝐿

{∥∥𝑌𝑆𝐷−√𝑃𝑆𝐺𝑋𝑆(x)

∥∥2+

𝑁∑𝑖=1

minx̂𝑅𝑖

∈𝒜𝐿

[∥∥𝑌𝑅𝑖𝐷−√𝑃𝑅𝐹𝑖𝑋𝑅(x̂𝑅𝑖)

∥∥2−𝜎2ln𝑃𝑆𝑅𝑖(x→x̂𝑅𝑖)]}

.

(5)

Unlike the ML decoder, the near-ML decoder can be appliedto most of the DF relay networks with multiple antennas, suchas space-time code (STC) and MIMO DF relay networks.

In order to derive the relay selection criterion, the PEP ofthe near-ML decoder will be derived in the next section.

III. PAIRWISE ERROR PROBABILITY FOR DF RELAY

NETWORK WITH 𝑁 RELAYS

The following theorem is used to derive the PEP of thenear-ML decoder.

Theorem 1: [5] Let 𝐴 and 𝐵 be complex matrices sat-isfying ∥𝐵∥2 > ∥𝐴∥2 and 𝐶 be a random matrix of theentries with complex Gaussian distribution 𝒞𝒩 (0, 𝜎2). Then,for 𝜎2 → 0, ∥𝐵 + 𝐶∥2 ≥ ∥𝐴+ 𝐶∥2 in probability, i.e.,

lim𝜎2→0

𝑃(∥𝐵 + 𝐶∥2 ≥ ∥𝐴+ 𝐶∥2) = 1.

□

Since a relay may transmit any vector in the signal set 𝒜𝐿

due to the decoding error, the PEP between x and x̃ at thedestination should be written as

𝑃 (x → x̃)

=∑

x𝑅1∈𝒜𝐿

⋅ ⋅ ⋅∑

x𝑅𝑁∈𝒜𝐿

𝑃 (x → x̃∣x, x𝑅1 , . . . , x𝑅𝑁 )𝑁∏𝑖=1

𝑃𝑆𝑅𝑖(x𝑅𝑖 ∣x)

(6)

1𝑃𝑆𝑅𝑖(x → x̂𝑅𝑖

) is equal to 𝑃𝑆𝑅𝑖(x̂𝑅𝑖

∣x) for a single-antenna systemwith BPSK modulation.

where 𝑃 (x → x̃∣x, x𝑅1 , . . . , x𝑅𝑁 ) denotes the conditional PEPof decoding the received signals into x̃ at the destination whenx and x𝑅𝑖 for 𝑖 = 1, . . . , 𝑁 are transmitted from the sourceand the relays, respectively. The condition x will be omittedto simplify the expression as 𝑃 (x → x̃∣x𝑅1 , . . . , x𝑅𝑁 ).

Then, the conditional PEP in (6) can be written as

𝑃 (x→x̃∣x𝑅1 , . . . , x𝑅𝑁 )

=𝑃(𝑚([𝑌𝑆𝐷,𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x∣x, x𝑅1 , . . . , x𝑅𝑁

)>𝑚

([𝑌𝑆𝐷, 𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x̃∣x, x𝑅1, . . . , x𝑅𝑁

))(7)

where

𝑚([𝑌𝑆𝐷,𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x∣x,x𝑅1, . . . , x𝑅𝑁

)=∥∥𝑌𝑆𝐷−

√𝑃𝑆𝐺𝑋𝑆(x)

∥∥2+

𝑁∑𝑖=1

minx̂𝑅𝑖

∈𝒜𝐿


∥∥2−𝜎2ln𝑃𝑆𝑅𝑖(x→ x̂𝑅𝑖)]

and

𝑚([𝑌𝑆𝐷,𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x̃∣x,x𝑅1, . . . , x𝑅𝑁

)=∥∥𝑌𝑆𝐷−

√𝑃𝑆𝐺𝑋𝑆 (̃x)

∥∥2+

𝑁∑𝑖=1

minx̂𝑅𝑖

∈𝒜𝐿


∥∥2−𝜎2ln𝑃𝑆𝑅𝑖(x̃→ x̂𝑅𝑖)]

are the metrics in (5) to decide x and x̃ for the given xand x𝑅1 , . . . , x𝑅𝑁 transmitted from the source and relays,respectively.

As derived in [18], the PEPs for the 𝑖th SR channel withthe given 𝐾𝑖 can be written as

𝑃𝑆𝑅𝑖(x→ x̂𝑅𝑖)=𝑄(√ 𝑃𝑆

2𝜎2

∥∥𝐾𝑖(𝑋𝑆(x)−𝑋𝑆(x̂𝑅𝑖))∥∥2) (8)

and

𝑃𝑆𝑅𝑖(x̃ → x̂𝑅𝑖) = 𝑄(√ 𝑃𝑆

2𝜎2

∥∥𝐾𝑖(𝑋𝑆(x̃)−𝑋𝑆(x̂𝑅𝑖))∥∥2).

Using the above PEPs, (2), and (3), the metrics in (7) can bewritten as

𝑚([𝑌𝑆𝐷, 𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x∣x, x𝑅1 , . . . , x𝑅𝑁

)=∥∥𝑁𝑆𝐷∥∥2+ 𝑁∑

𝑖=1

minx̂𝑅𝑖

∈𝒜𝐿

[∥∥√𝑃𝑅𝐹𝑖(𝑋𝑅(x𝑅𝑖)−𝑋𝑅(x̂𝑅𝑖)

)+𝑁𝑅𝑖𝐷

∥∥2

−𝜎2ln𝑄

(√𝑃𝑆2𝜎2

∥∥𝐾𝑖

(𝑋𝑆(x)−𝑋𝑆(x̂𝑅𝑖)

)∥∥2)] (9)

and

𝑚([𝑌𝑆𝐷, 𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x̃∣x, x𝑅1 , . . . , x𝑅𝑁

)=∥∥√𝑃𝑆𝐺

(𝑋𝑆(x)−𝑋𝑆(x̃)

)+𝑁𝑆𝐷

∥∥2+

𝑁∑𝑖=1

minx̂𝑅𝑖

∈𝒜𝐿

[∥∥√𝑃𝑆𝐹𝑖(𝑋𝑅(x𝑅𝑖)−𝑋𝑅(x̂𝑅𝑖)

)+𝑁𝑅𝑖𝐷

∥∥2

−𝜎2ln𝑄

(√𝑃𝑆2𝜎2

∥∥𝐾𝑖(𝑋𝑆(x̃)−𝑋𝑆(x̂𝑅𝑖))∥∥2)]. (10)

The conditional PEP in (7) is very difficult to simplifydue to the 𝑄 function. However, by using 𝑄(𝑥) ≈


𝑃 (x → x̃) =

𝑁∑𝑛𝑅=0

∑𝑆∈𝑆(𝑛𝑅)

∑x𝑅𝑖

∈𝒜𝐿

x𝑅𝑖∕=x,𝑖∈𝑆𝑁∖𝑆

𝑃(x→ x̃

∣∣x𝑅𝑗=x, 𝑗∈𝑆 and x𝑅𝑖 , 𝑖∈𝑆𝑁∖𝑆)⋅∏𝑗∈𝑆

𝑃𝑆𝑅𝑗 (x∣x)∏

𝑖∈𝑆𝑁∖𝑆𝑃𝑆𝑅𝑖(x𝑅𝑖 ∣x) (13)

minx̂𝑅𝑖

∈𝒜𝐿


)+𝑁𝑅𝑖𝐷

∥∥2+ 𝑃𝑆4

∥∥𝐾𝑖

(𝑋𝑆(x̃)−𝑋𝑆(x̂𝑅𝑖)

)∥∥2]𝑃≥ min

[∥𝑁𝑅𝑖𝐷∥2 +

𝑃𝑆4

∥∥𝐾𝑖

(𝑋𝑆(x̃)−𝑋𝑆(x)

)∥∥2, ∥∥√𝑃𝑅𝐹𝑖(𝑋𝑅(x)−𝑋𝑅(x

min𝐹𝑖

))+𝑁𝑅𝑖𝐷

∥∥2] (15)

1√2𝜋

exp(− 𝑥2

2

)(1− 1

𝜋 )𝑥+ 1𝜋

√𝑥2 + 2𝜋

, 0 < 𝑥 < ∞, in [19], the following

approximation can be obtained as

lim𝜎2→0

𝜎2 ln𝑄(√ 𝑃𝑆

2𝜎2

∥∥𝐾𝑖(𝑋𝑆(x)−𝑋𝑆(x̌))∥∥2)

≈ −𝑃𝑆4

∥∥𝐾𝑖(𝑋𝑆(x)−𝑋𝑆(x̌))∥∥2.

Thus, for high SNR, the metrics (9) and (10) can be approx-imated as


)≈∥∥𝑁𝑆𝐷∥∥2+ 𝑁∑

𝑖=1

minx̂𝑅𝑖

∈𝒜𝐿


)+𝑁𝑅𝑖𝐷

∥∥2+

𝑃𝑆4

∥∥𝐾𝑖


)∥∥2] (11)

and

𝑚([𝑌𝑆𝐷, 𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x̃∣x, x𝑅1 , . . . , x𝑅𝑁

)≈ ∥∥√𝑃𝑆𝐺


)+𝑁𝑆𝐷

∥∥2+

𝑁∑𝑖=1

minx̂𝑅𝑖

∈𝒜𝐿

[∥∥√𝑃𝑆𝐹𝑖(𝑋𝑅(x𝑅𝑖)−𝑋𝑅(x̂𝑅𝑖)

)+𝑁𝑅𝑖𝐷

∥∥2+

𝑃𝑆4

∥∥𝐾𝑖

(𝑋𝑆(x̃)−𝑋𝑆(x̂𝑅𝑖)

)∥∥2]. (12)

Since the minimization in (11) and (12) still makes it difficultto use, we further simplify two metrics (11) and (12) bysplitting x𝑅𝑖 in (6) into two cases, x𝑅𝑖 = x and x𝑅𝑖 ∕= x,for high SNR region. Let 𝑆𝑁 = {1, . . . , 𝑁} and 𝑆(𝑛𝑅) ={𝑆′∣𝑆′ ⊆ 𝑆𝑁 with ∣𝑆′∣ = 𝑛𝑅}, where 0 ≤ 𝑛𝑅 ≤ 𝑁 . Weassume that x𝑅𝑖 = x for 𝑛𝑅 relays in 𝑆 ∈ 𝑆(𝑛𝑅) andx𝑅𝑖 ∕= x for 𝑁− 𝑛𝑅 relays in 𝑆𝑁 ∖ 𝑆, where 𝑆𝑁 ∖ 𝑆 meansthe complement of 𝑆 in 𝑆𝑁 , i.e., 𝑛𝑅 relays have no decodingerror and 𝑁 − 𝑛𝑅 relays have decoding error. Then, the PEPin (6) can be rewritten as (13). Then, the PEP 𝑃 (x → x̃)can be obtained by deriving the summand in (13).

First, we consider the metrics (11) and (12) for the case ofx𝑅𝑖 = x, 𝑖 ∈𝑆 and x𝑅𝑖 ∕= x, 𝑖 ∈𝑆𝑁∖𝑆. For 𝑖 ∈ 𝑆, i.e., x𝑅𝑖 = x,the minimization in (11) results in ∥𝑁𝑅𝑖𝐷∥2 in probabilityfrom Theorem 1. For 𝑖 ∈𝑆𝑁∖𝑆, i.e., x𝑅𝑖 ∕= x, it is not easyto determine the value of the minimization term. However,we know that the value for any x̂𝑅𝑖 ∈ 𝒜𝐿 is greater thanthe minimization term. Thus, we have the upper bound on theminimization term when x̂𝑅𝑖 = x𝑅𝑖 . The minimization term

in (11) can be derived as

minx̂𝑅𝑖

∈𝒜𝐿


)+𝑁𝑅𝑖𝐷

∥∥2+

𝑃𝑆4

∥∥𝐾𝑖


)∥∥2]⎧⎨⎩𝑃=∥𝑁𝑅𝑖𝐷∥2 for 𝑖∈𝑆𝑃≤∥𝑁𝑅𝑖𝐷∥2+𝑃𝑆

4

∥∥𝐾𝑖

(𝑋𝑆(x)−𝑋𝑆(x𝑅𝑖)

)∥∥2 for 𝑖∈𝑆𝑁∖𝑆.

Thus, an upper bound on the metric in (11) can be obtainedas


)𝑃

⪅∥𝑁𝑆𝐷∥2+∑𝑖∈𝑆𝑁

∥𝑁𝑅𝑖𝐷∥2+∑𝑖∈𝑆𝑁∖𝑆

𝑃𝑆4

∥∥𝐾𝑖


)∥∥2.(14)

Similarly, the metric 𝑚([𝑌𝑆𝐷,𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x̃∣x,x𝑅1, . . . , x𝑅𝑁

)in (12) can also be derived. For 𝑖 ∈𝑆, using Theorem 1, wehave

∥√𝑃𝑅𝐹𝑖(𝑋𝑅(x𝑅𝑖)−𝑋𝑅(x̂𝑅𝑖))+𝑁𝑅𝑖𝐷∥2+𝑃𝑆4∥𝐾𝑖(𝑋𝑆(x̃)−𝑋𝑆(x̂𝑅𝑖))∥2{

= ∥𝑁𝑅𝑖𝐷∥2+ 𝑃𝑆4

∥∥𝐾𝑖


)∥∥2if x̂𝑅𝑖 =x

𝑃≥ ∥∥√𝑃𝑅𝐹𝑖

(𝑋𝑅(x)−𝑋𝑅(x

min𝐹𝑖

))+𝑁𝑅𝑖𝐷

∥∥2if x̂𝑅𝑖 ∕=x

where xmin𝐹𝑖

= argminx̂𝑅𝑖∕=x

∥∥𝐹𝑖(𝑋𝑅(x) −𝑋𝑅(x̂𝑅𝑖))∥∥2, and

thus we have an inequality (15). For 𝑖 ∈𝑆𝑁∖𝑆, from Theorem1, we also have

minx̂𝑅𝑖

∈𝒜𝐿


)+𝑁𝑅𝑖𝐷

∥∥2+

𝑃𝑆4

∥∥𝐾𝑖

(𝑋𝑆( x̃)−𝑋𝑆(x̂𝑅𝑖)

)∥∥2] 𝑃≥∥𝑁𝑅𝑖𝐷∥2. (16)

Combining (15) and (16), (12) can be lower bounded as

𝑚([𝑌𝑆𝐷,𝑌𝑅1𝐷, . . . , 𝑌𝑅𝑁𝐷], x̃∣x, x𝑅1 , . . . , x𝑅𝑁

)𝑃

⪆∥∥√𝑃𝑆𝐺(𝑋𝑆(x)−𝑋𝑆(x̃))+𝑁𝑆𝐷

∥∥2+ ∑𝑖∈𝑆𝑁∖𝑆

∥𝑁𝑅𝑖𝐷∥2

+∑𝑖∈𝑆

min[∥∥√𝑃𝑅𝐹𝑖


min𝐹𝑖

))+𝑁𝑅𝑖𝐷

∥∥2,∥𝑁𝑅𝑖𝐷∥2+

𝑃𝑆4

∥∥𝐾𝑖


)∥∥2]. (17)

Using 𝑃𝑆𝑅𝑖(x∣x) ≤ 1 for 𝑖 ∈ 𝑆, 𝑃𝑆𝑅𝑖(x𝑅𝑖 ∣x) ≤ 𝑃𝑆𝑅𝑖(x →x𝑅𝑖) for 𝑖 ∈𝑆𝑁∖𝑆, (8), and 𝑄(𝑥) ≤ exp(−𝑥2/2), 𝑥 > 0, the


𝑃(x → x̃

∣∣x𝑅𝑖 = x, 𝑖 ∈𝑆 and x𝑅𝑖 ∕= x, 𝑖 ∈𝑆𝑁∖𝑆)∏𝑖∈𝑆

𝑃𝑆𝑅𝑖(x∣x)∏

𝑖∈𝑆𝑁∖𝑆𝑃𝑆𝑅𝑖(x𝑅𝑖 ∣x)

𝑃

⪅𝑃

(∥∥𝑁𝑆𝐷

∥∥2+ 𝑁∑𝑖=1

∥∥𝑁𝑅𝑖𝐷

∥∥2+∑𝑖∈𝑆𝑁∖𝑆

𝑃𝑆4

∥∥𝐾𝑖(𝑋𝑆(x)−𝑋𝑆(x𝑅𝑖))∥∥2

>∑𝑖∈𝑆

min[∥∥√𝑃𝑅𝐹𝑖


min𝐹𝑖

))+𝑁𝑅𝑖𝐷

∥∥2, ∥∥𝑁𝑅𝑖𝐷

∥∥2+𝑃𝑆4

∥∥𝐾𝑖


)∥∥2]

+∥∥√𝑃𝑆𝐺


)+𝑁𝑆𝐷

∥∥2+∑𝑖∈𝑆𝑁∖𝑆

∥∥𝑁𝑅𝑖𝐷

∥∥2) exp(−∑𝑖∈𝑆𝑁∖𝑆

𝑃𝑆4𝜎2

∥∥𝐾𝑖


)∥∥2) (19)

product term in (13) is upper bounded as∏𝑖∈𝑆



≤ exp

(−∑

𝑖∈𝑆𝑁∖𝑆

𝑃𝑆4𝜎2

∥∥𝐾𝑖


)∥∥2). (18)

Thus, by using (14), (17), and (18), the summand in (13)can be upper bounded as (19). Plugging (19) into (13), thefollowing theorem can be derived.

Theorem 2: For the DF relay network with 𝑁 relays andmultiple antennas, the PEP of near-ML decoder between𝐿-tuple message vectors x and x̃ from the 𝑀 -ary signalconstellation can be upper bounded as

𝑃 (x→x̃)𝑃

⪅2(𝑀𝐿+1)𝑁exp(− 1

4𝜎2𝑃𝑆∥𝐺(𝑋𝑆(x)−𝑋𝑆(x̃))∥2

)exp

(− 1

4𝜎2

∑𝑖∈𝑆𝑁

min[𝑃𝑆2

∥∥𝐾𝑖

(𝑋𝑆(x)−𝑋𝑆(x

min𝐾𝑖

))∥∥2,

𝑃𝑅∥∥𝐹𝑖(𝑋𝑅(x)−𝑋𝑅(x

min𝐹𝑖 ))∥∥2]) (20)

where xmin𝐾𝑖

=argmin x̌ ∕=x

∥∥𝐾𝑖

(𝑋𝑆(x)−𝑋𝑆( x̌)

)∥∥2 and xmin𝐹𝑖

=

argminx̂𝑅𝑖∕=x

∥∥𝐹𝑖(𝑋𝑅(x)−𝑋𝑅(x̂𝑅𝑖))∥∥2. 𝑃𝑆 , 𝑃𝑅, 𝐺, 𝐾𝑖, 𝐹𝑖,

𝑋𝑆 , 𝑋𝑅, and 𝜎2 are introduced in Section II.Proof: See the Appendix A. □

In the next section, a relay selection criterion for DF relaynetwork with multiple antennas is derived from Theorem 2.

IV. RELAY SELECTION FOR DECODE-AND-FORWARD

RELAY NETWORK

A. A New Relay Selection Scheme

In general, it is very difficult to derive the optimal crite-rion for the relay selection in the DF relay networks withmultiple antennas. However, the maximum PEP can be usedfor deriving a criterion of selecting good relays because themaximum PEP is a dominant term for the union bound on biterror probability (BEP) [20]. It is also difficult to derive theexact PEP in the DF relay network with multiple antennas asshown in Section III. Thus the upper bound on PEP is derivedin Theorem 2, which can be separated into two parts, one isthe SD direct path and the other is the source-relay-destination(SRD) path. The SRD path is also separated into 𝑁 SRD paths

corresponding to 𝑁 relays. The upper bound on PEP is veryuseful for finding a new relay ordering.

From the upper bound on the PEP in (20), we define a relaypath metric 𝛾𝑖 as

𝛾𝑖 =min[𝑃𝑆2

minx,x̌∕=x

∥∥𝐾𝑖

(𝑋𝑆(x)−𝑋𝑆(x̌)

)∥∥2,𝑃𝑅 min

x,x̌ ∕=x

∥∥𝐹𝑖(𝑋𝑅(x)−𝑋𝑅(x̌))∥∥2]. (21)

Based on the fact that the upper bound on PEP in (20)decreases as 𝛾𝑖 increases, we propose the following selectionscheme of 𝑚 relays (1 ≤ 𝑚 ≤ 𝑁 ) for the DF relay network:

1) The destination finds 𝑚 relays which have 𝑚 largest relaypath metrics 𝛾𝑖;

2) The destination sends the indices of the selected 𝑚 relaysto 𝑁 relays;

3) The selected relays transmit the signal.

Next, we discuss the complexity and overhead for theproposed relay selection scheme.

i) Complexity:

The DF relay network considered in this paper includes oneSD and multiple SRD links and thus the error probability isrelated to the channel coefficients and transmission schemesof SD link and SRD links. To select 𝑚 relays among 𝑁 relays,we have to consider the total error probability, i.e., all of theSD and SRD links and compare

(𝑁𝑚

)possible sets of the

relays. However, our relay selection scheme considers eachrelay separately through the metric 𝛾𝑖 (SRD link), but not SDlink and thus only needs to compare 𝑁 SRD links. Therefore,our new relay selection scheme has less complexity.

ii) Overhead:

To feedback the indices of the selected relays, the followingtwo methods for the feedback message can be used:

∙ 𝑁 bits, where each bit indicates whether the correspond-ing relay transmits signal or not;

∙ 𝑚⌈log2 𝑁⌉ bits for the indices of the 𝑚 selected relays.

Since the first and second methods need total 𝑁 bits and𝑚⌈log2 𝑁⌉ bits, respectively, the number of required bits forthe feedback message is min

[𝑁,𝑚⌈log2 𝑁⌉].

As mentioned in Section I, the relay selection with 𝑚 = 1is the best in terms of bandwidth efficiency. However, it isnot easy to determine how many relays should be selected toachieve the best error correction performance. To compare theerror performance for various 𝑚, we set up the relay selectionfor arbitrary 𝑚 (1 ≤ 𝑚 ≤ 𝑁 ). The diversity for various 𝑚 is


derived in the next subsection and the simulated BEP is givenin Section V.

B. Diversity Analysis for the New Relay Selection Scheme

Clearly, there are(𝑁𝑚

)sets of the selected relays. Let

𝑆𝑛, 𝑛 = 1, . . . ,(𝑁𝑚

), be the sets of possible selected relay

indices with ∣𝑆𝑛∣ = 𝑚. Then, using (20), the PEP for DFrelay network with 𝑚 relay selection out of 𝑁 relays can bederived as

𝐸[𝑃𝑅𝑆𝑒𝑙𝑒𝑐𝑡(x → x̃)]𝑃

⪅ 2(𝑀𝐿+1)𝑚𝐸[exp(− 𝑃𝑆∥𝐺(𝑋𝑆(x)−𝑋𝑆(x̃))∥2

4𝜎2

)]𝐸

[exp

(− 1

4𝜎2max

𝑛∈{1,...,(𝑁𝑚)}

∑𝑖∈𝑆𝑛

𝛾𝑖

)]. (22)

Let 𝑟𝑆 and 𝑟𝑅 be the minimum ranks among the ranksof(𝑋𝑆(x) − 𝑋𝑆(x̌)

)(𝑋𝑆(x) − 𝑋𝑆(x̌)

)†and

(𝑋𝑅(x) −

𝑋𝑅(x̌))(

𝑋𝑅(x) − 𝑋𝑅(x̌))†

for all x̌ ∕= x, respectively. Wedefine 𝑀𝐷×𝑟𝑆 matrix 𝐺′, 𝑀𝑅×𝑟𝑆 matrix 𝐾 ′

𝑖, and 𝑀𝐷×𝑟𝑅matrix 𝐹 ′

𝑖 , 𝑖 = 1, . . . , 𝑁 with [𝐺′]𝑙 = [𝐺𝑈 ]𝑙 for 𝑙 = 1, . . . , 𝑟𝑆 ,[𝐾 ′

𝑖]𝑙 = [𝐾𝑖𝑈 ]𝑙 for 𝑙 = 1, . . . , 𝑟𝑆 , and [𝐹 ′𝑖 ]𝑙 = [𝐹𝑖𝑉 ]𝑙 for 𝑙 =

1, . . . , 𝑟𝑅, respectively, where [𝐴]𝑙 means the 𝑙th column ofthe matrix A, 𝑈 and 𝑉 are the unitary matrices whose columnsare the eigenvectors of (𝑋𝑆(x)−𝑋𝑆(x̌))(𝑋𝑆(x)−𝑋𝑆(x̌))

† and(𝑋𝑅(x)−𝑋𝑅(x̌))(𝑋𝑅(x)−𝑋𝑅(x̌))

† for any x̌ ∕= x, respec-tively. Since the multiplication of the unitary matrix does notchange the statistical distribution of the matrix with circularlysymmetric complex Gaussian entries, the entries of 𝐺′, 𝐾 ′

𝑖,and 𝐹 ′

𝑖 have the same distribution as the entries of 𝐺,𝐾𝑖,and 𝐹𝑖, respectively. Therefore, by Fact 1 in Appendix B, theupper bound on the average PEP in (22) can be rewritten as


⪅ 2(𝑀𝐿+1)𝑚𝐸[exp(− 𝑃𝑆𝜔min∥𝐺′∥2

4𝜎2

)]𝐸

[exp

(− 1

4𝜎2max

𝑛∈{1,...,(𝑁𝑚)}

∑𝑖∈𝑆𝑛

min[𝑃𝑆2

𝜔min∥𝐾 ′𝑖∥2, 𝑃𝑅𝜇min∥𝐹 ′

𝑖∥2])]

(23)

where 𝜔min and 𝜇min are the minimum values amongnonzero eigenvalues of (𝑋𝑆(x)−𝑋𝑆(x̌))(𝑋𝑆(x)−𝑋𝑆(x̌))

†

and (𝑋𝑅(x) − 𝑋𝑅(x̌))(𝑋𝑅(x) − 𝑋𝑅(x̌))† for all x̌ ∕= x,

respectively. Let 𝛾′𝑖 = min

[𝑃𝑆

2 𝜔min∥𝐾 ′𝑖∥2, 𝑃𝑅𝜇min∥𝐹 ′

𝑖∥2],

𝑦max = max𝑛∈{1,...,(𝑁𝑚)}∑

𝑖∈𝑆𝑛𝛾′𝑖, and 𝑦𝐺 = 𝑃𝑆𝜔min∥𝐺′∥2.

Then, we need to know the distribution of the randomvariables 𝑦max and 𝑦𝐺. Note that 𝑦𝐺 is an 𝑟𝑆𝑀𝐷-Erlangrandom variable with rate parameter 𝑃𝑆𝜔min𝜎

2𝑆𝐷 . However,

the cumulative distribution function (CDF) or the probabilitydensity function (PDF) of the random variable 𝑦max is verydifficult to derive and thus we have to find another way toderive the upper bound on the average PEP in (23). Since

𝐸[exp

(− 𝑦max

4𝜎2

)]=

1

4𝜎2

∫ ∞

0

exp(− 𝑦

4𝜎2

)𝑃𝑦max(𝑦)𝑑𝑦 (24)

in the right-hand side (RHS) of (23) where 𝑃𝑦max(𝑦) is theCDF of the random variable 𝑦max, the upper bound on theaverage PEP in (23) can be derived by calculating the upper

bound on the CDF. The upper bound on the CDF of 𝑦max isderived as in the following theorem.

Theorem 3: The CDF 𝑃𝑦max(𝑦) of 𝑦max can be upperbounded as

𝑃𝑦max(𝑦)

≤[1−exp

(−( 2

𝑃𝑆𝜔min𝜎2𝑆𝑅

+1

𝑃𝑅𝜇min𝜎2𝑅𝐷

)𝑦)]𝑁min[𝑟𝑆𝑀𝑅,𝑟𝑅𝑀𝐷 ]

.

(25)

Proof: See the Appendix C. □

Using the PDF of 𝑦𝐺 and the upper bound on the CDFof 𝑦max derived in Theorem 3, the following theorem for theachievable diversity can be established.

Theorem 4: The new relay selection scheme with thenear-ML decoder achieves the diversity 𝑟𝑆𝑀𝐷 + 𝑁 min[𝑟𝑆𝑀𝑅, 𝑟𝑅𝑀𝐷] regardless of the number 𝑚 of the selectedrelays in the DF relay network consisting of one source, onedestination, and 𝑁 relays with 𝑀𝑆 , 𝑀𝐷, and 𝑀𝑅 antennas,respectively. The full diversity 𝑀𝑆𝑀𝐷+𝑁𝑀𝑅min[𝑀𝑆,𝑀𝐷]is achieved when 𝑟𝑆 = 𝑀𝑆 and 𝑟𝑅 = 𝑀𝑅.

Proof: See the Appendix D. □

V. DISCUSSION AND SIMULATION RESULTS

In this section, the application of the proposed relay selec-tion to the following three cases are considered and evaluated.For simplicity, it is assumed that 𝑀 -ary symbols are normal-ized to the unit power.

i) Uncoded single-antenna DF relay network: 𝑀𝑆 = 𝑀𝑅 =𝑀𝐷 = 𝐿 = 1,

𝑋𝑆(x) = 𝑥 and 𝑋𝑅(x𝑅𝑖) = 𝑥𝑅𝑖 ,

and 𝜔min = 𝜇min = min𝑥,�̌�∕=𝑥 ∣𝑥− �̌�∣2. This is a specialcase of the multiple-antenna systems. Since the sourceand relays use the same modulation, the relay path metricin (21) becomes

𝛾𝑖 = min𝑥,�̌�∕=𝑥

∣𝑥− 𝑥∣2 min[𝑃𝑆2∣𝑘𝑖∣2, 𝑃𝑅∣𝑓𝑖∣2

]. (26)

The maximum achievable diversity order using the pro-posed relay selection scheme for total 𝑁 relays is 𝑁 +1for the case of 𝑀𝑆 = 𝑀𝑅 = 𝑀𝐷 = 1 from Theorem 4.

ii) Alamouti coded DF relay network: Alamouti scheme [21]is used at the source and relay, i.e., 𝑀𝑆 = 𝑀𝑅 = 𝑀𝐷 =2,

𝑋𝑆(x)=1√2

[𝑥1 −𝑥∗

2

𝑥2 𝑥∗1

]and 𝑋𝑅(x𝑅𝑖)=

1√2

[𝑥𝑅𝑖1 −𝑥𝑅𝑖∗

2

𝑥𝑅𝑖2 𝑥𝑅𝑖∗

1

],

and 𝜔min = 𝜇min = 12 min𝑥1,�̌�1 ∕=𝑥1 ∣𝑥1 − �̌�1∣2. Similarly

to the uncoded single-antenna case, the source and relaysuse the same modulation, and thus, the relay path metricbecomes

𝛾𝑖=1

2min

𝑥1,�̌�1∕=𝑥1∣𝑥1−�̌�1∣2min

[𝑃𝑆2∥𝐾𝑖∥2, 𝑃𝑅∥𝐹𝑖∥2

]. (27)

The maximum achievable diversity order using the pro-posed relay selection scheme for total 𝑁 relays is4(𝑁 + 1) for the case of 𝑀𝑆 = 𝑀𝑅 = 𝑀𝐷 = 2 fromTheorem 4.


0 10 20 30 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

ρ (dB)

Ave

rage

BE

P

Uncoded single-antenna with MLUncoded single-antenna with near-MLUncoded single-antenna with C-MRCρ−2

Alamouti with MLAlamouti with near-MLρ−8

MIMO2×2 with near-MLρ−4

Fig. 2. Average BEP comparison among the ML, near-ML, C-MRC decodersfor uncoded single-antenna, Alamouti coded, and MIMO DF relay networkwith (𝜎2

𝑆𝑅, 𝜎2𝑆𝐷, 𝜎2

𝑅𝐷) = (1, 1, 1).

0 5 10 15 2010

−6

10−5

10−4

10−3

10−2

10−1

ρ (dB)

Ave

rage

BE

P

N=4, m=1, random RSN=4, m=1, new RSN=4, m=1, MAX-CAP RSN=4, m=2, random RSN=4, m=2, new RSN=4, m=2, MAX-CAP RSN=4, m=4Uncoded single-antennaAlamouti codedMIMO

Fig. 3. Average BEP comparison between the proposed relay selection andMAX-CAP relay selection for DF relay network with (𝜎2


𝑅𝐷) =(1, 1, 1).

iii) MIMO DF relay network: The MIMO system is used atthe source and relay, i.e., the case of 𝑇1 = 𝑇2 = 1 and𝑀𝑆 = 𝑀𝑅 = 𝐿. In this paper, we consider the case of𝑀𝑆 = 𝑀𝑅 = 𝑀𝐷 = 𝐿 = 2. Then,

𝑋𝑆(x) =1√2

[𝑥1

𝑥2

]and 𝑋𝑅(x𝑅𝑖) =

1√2

[𝑥𝑅𝑖1

𝑥𝑅𝑖2

],

and 𝜔min = 𝜇min = 12 min𝑥1,�̌�1 ∕=𝑥1 ∣𝑥1 − 𝑥1∣2. Even

if the source and relays use the same modulation, therelay path metric cannot be simplified and the originalrelay path metric in (21) should be used. The MIMODF relay network with the proposed relay selection fortotal 𝑁 relays achieves the diversity order 2(𝑁+1) fromTheorem 4 .

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

ρ (dB)

Ave

rage

BE

P

Uncoded single-antenna, N = 4, m = 1Uncoded single-antenna, N = 4, m = 2Uncoded single-antenna, N = 4, m = 4Uncoded single-antenna, slope in (45)Alamouti coded, N = 4, m = 1Alamouti coded, N = 4, m = 2Alamouti coded, N = 4, m = 4Alamouti coded, slope in (45)MIMO, N = 4, m = 1MIMO, N = 4, m = 2MIMO, N = 4, m = 4MIMO, slope in (45)

Fig. 4. Average BEP for DF relay network with the proposed relay selectionscheme for (𝜎2


𝑅𝐷) = (1, 1, 1).

0 5 10 1510

−6

10−5

10−4

10−3

10−2

10−1

ρ (dB)

Ave

rage

BE

P

Uncoded single-antenna, N = 4, m = 1Uncoded single-antenna, N = 4, m = 2Uncoded single-antenna, N = 4, m = 4Alamouti coded, N = 4, m = 1Alamouti coded, N = 4, m = 2Alamouti coded, N = 4, m = 4MIMO, N = 4, m = 1MIMO, N = 4, m = 2MIMO, N = 4, m = 4



𝑅𝐷) = (1, 1, 10).

Next, we evaluate the above three cases by Monte Carlosimulation and compare their error correction performance.For the simulation, quadrature phase shift keying (QPSK) andpower allocation of 𝑃𝑆 = 1 and 𝑃𝑅 = 1/𝑚 are used.

First, we compare the average BEPs of ML, near-ML, andC-MRC decoders for the DF relay network with one relayin Fig. 2. For uncoded single-antenna DF relaying, the ML,near-ML, and C-MRC decoders have similar BEP performanceand the same diversity order 2; for Alamouti coded DF relaynetwork, the ML and near-ML decoder have almost the sameBEP performance and achieve the same diversity order 8; forMIMO DF relay network, the near-ML decoder achieves thediversity order 4.

Second, we compare the proposed relay selection with theconventional relay selection scheme. Unlike the proposed relay


0 5 10 1510

−6

10−5

10−4

10−3

10−2

10−1

ρ (dB)

Ave

rage

BE

P

Uncoded single-antenna, N = 4, m = 1Uncoded single-antenna, N = 4, m = 2Uncoded single-antenna, N = 4, m = 4Alamouti coded, N = 4, m = 1Alamouti coded, N = 4, m = 2Alamouti coded, N = 4, m = 4MIMO, N = 4, m = 1MIMO, N = 4, m = 2MIMO, N = 4, m = 4



𝑅𝐷) = (10, 1, 1).

selection scheme, the conventional scheme does not considerthe transmission scheme such as the optimal selective routingscheme [14] of selecting single relay which achieves thehighest two-hop network capacity (without considering thedirect link between the source and destination), that is calledMAX-CAP relay selection in this paper. In Fig. 3, we comparethe BEPs of the proposed relay selection and MAX-CAP relayselection for the uncoded single-antenna, Alamouti coded, andMIMO DF relay networks. We compare the above two relayselection schemes for 𝑚 = 1 and 𝑚 = 2 and note that theMAX-CAP scheme selects single relay as explained in [14].Fig. 3 shows that compared to MAX-CAP relay selection, theproposed relay selection has similar performance for uncodedsingle-antenna DF relay network, and better performance forAlamouti coded and MIMO DF relay networks. Therefore, theproposed relay selection scheme has better BEP performancethan MAX-CAP relay selection even though the complexityincreases for high order modulations because the proposedrelay selection scheme is required to search the minimumvalues of

∥∥𝐾𝑖

(𝑋𝑆(x)−𝑋𝑆(x̌)

)∥∥2 and∥∥𝐹𝑖(𝑋𝑅(x)−𝑋𝑅(x̌)

)∥∥2for different set of x and x̌.

Third, to show the achievable diversity derived in Theorem4, we compare the slope of the upper bound on the averagePEP of (45) in the proof of Theorem 4 and Monte Carlosimulated BEP for the DF relay network with relay selectionin Fig. 4. As shown in Fig. 4, the slope of the upper boundon the average PEP in (45) is almost the same as that of thesimulated BEP for uncoded single-antenna DF relay networkin high SNR region. For the Alamouti coded and MIMO DFrelay networks, the slopes of the upper bound on the averagePEP in (45) become closer to those of the simulated BEPas the SNR increases, and those slopes seem to become thesame in very high SNR region. This means that the DF relaynetwork with relay selection achieves the same diversity asthe upper bound in (45) regardless of the value of 𝑚.

Finally, we discuss and compare the performance of theproposed relay selection for various 𝑚. It is clear that the

proposed relay selection has better bandwidth efficiency asthe number of selected relays 𝑚 decreases. However, it is noteasy to determine the error performance and power efficiencyfor various 𝑚. Since we use the same total transmit power,i.e., 𝑃𝑆 = 1 and 𝑃𝑅 = 1/𝑚, good BEP performance meansgood power efficiency. For (𝜎2


𝑅𝐷) = (1, 1, 1), Fig.4 compares the BEP of the relay selection with 𝑚 = 1, 2, 4for 𝑁 = 4. for uncoded single-antenna and Alamouti codedDF relay networks, the case of 𝑚 = 2 has the best errorcorrection performance. That is, the required power for thecase of 𝑚 = 2 is about 0.3dB and 0.6dB less than that for thecase of 𝑚 = 1 at BEP= 10−5 for uncoded single-antenna andAlamouti coded DF relay networks, respectively. For MIMODF relay network, the case of 𝑚 = 4 has the best average BEPand the required power for the case of 𝑚 = 4 is about 1.4dBless than that for the case of 𝑚 = 1 at BEP= 10−5. We alsoconsider different channels (𝜎2


𝑅𝐷) = (1, 1, 10) and(𝜎2


𝑅𝐷) = (10, 1, 1) in Figs 5 and 6, respectively. Inthe case of (1, 1, 10), the BEP improves as 𝑚 increases, i.e.,𝑚 = 4 has the best error performance and 𝑚 = 1 has the worsterror performance. As an example, we focus on the dotted line,i.e., for MIMO DF network, the required power for the caseof 𝑚 = 4 is about 2dB and 3.6dB less than that for the casesof 𝑚 = 2 and 𝑚 = 1 at BEP= 10−5, respectively. In the caseof (10, 1, 1), the above three DF relay networks have differenttendencies from one another. For the uncoded single-antennaand Alamouti coded DF relay networks, the BEP degradesas 𝑚 increases but for the MIMO DF relay network, theBEP has no clear tendency. These results can be explainedby considering the extreme cases for the channel states. In thecase of 𝜎2

𝑆𝑅 << 𝜎2𝑅𝐷 , the error performance is determined by

the SR channels and since 𝑃𝑆 = 1, the BEP clearly improvesas the number of the selected relays 𝑚 increases. In the caseof 𝜎2

𝑆𝑅 >> 𝜎2𝑅𝐷 , the error performance is determined by the

RD channels and since 𝑃𝑅 = 1/𝑚, the equivalent averagetotal power for the system is the same regardless of the valueof 𝑚. Thus, the error performance depends on both channelsand codewords but it is difficult to determine which one hasthe best error performance through analysis.

VI. CONCLUSION

In this paper, we proposed a new relay selection schemefor arbitrary 𝑚 (1 ≤ 𝑚 ≤ 𝑁 ) based on both CSI andtransmission scheme by deriving the upper bound on the PEPof the near-ML decoder for the DF relay network with multipleantennas. We also proved that the proposed relay selectionscheme which selects 𝑚 relays from 𝑁 relays can achievefull diversity 𝑀𝑆𝑀𝐷 + 𝑁𝑀𝑅min[𝑀𝑆,𝑀𝐷] regardless ofthe value of 𝑚 in the DF relay network consisting of onesource, one destination, and 𝑁 relays with 𝑀𝑆 , 𝑀𝐷, and 𝑀𝑅

antennas, respectively. We compared the error performanceof the proposed relay selection with that of the MAX-CAPrelay selection. The simulation results show that the proposedrelay selection has better error performance than the MAX-CAP relay selection for the multiple-antenna cases. We alsodiscussed and simulated the error performance of the proposedrelay selection scheme for various 𝑚 in the uncoded single-antenna, Alamouti coded, and MIMO DF relay networks. As


a further work, the study on the relay selection for the non-orthogonal DF relay network with multiple antennas is alsovery interesting.

APPENDIX APROOF OF THEOREM 2

Let 𝑠= 2√𝑃𝑆Re

{tr(𝐺(𝑋𝑆(x)−𝑋𝑆(x̃)

)𝑁 †𝑆𝐷

)}, 𝑡𝑖 = 2

√𝑃𝑅

Re{tr(𝐹𝑖(𝑋𝑅𝑖(x)−𝑋𝑅𝑖(x

min𝐹𝑖

))𝑁 †𝑅𝑖𝐷

)}, 𝑞𝑖 = 𝑃𝑆∥𝐾𝑖(𝑋𝑆(x)−

𝑋𝑆(x̃))∥2, 𝑞′𝑖=𝑃𝑆∥𝐾𝑖(𝑋𝑆(x)−𝑋𝑆(x𝑅𝑖))∥2, 𝑤=𝑃𝑆∥𝐺(𝑋𝑆(x)−𝑋𝑆(x̃))∥2, and ℎ𝑖 = 𝑃𝑅

∥∥𝐹𝑖(𝑋𝑅𝑖(x) − 𝑋𝑅𝑖(xmin𝐹𝑖

))∥∥2. Then,

𝑠 ∼ 𝒩 (0, 2𝑤𝜎2) and 𝑡𝑖 ∼ 𝒩 (0, 2ℎ𝑖𝜎2). The RHS of (19) can

be rewritten as

𝑃

( ∑𝑖∈𝑆𝑁∖𝑆

𝑞′𝑖4

>𝑤+𝑠+∑𝑖∈𝑆

min[ℎ𝑖+𝑡𝑖,

𝑞𝑖4

])exp

(−∑𝑖∈𝑆𝑁∖𝑆

𝑞′𝑖4𝜎2

).

(28)

Let 𝑆′(𝑝) = {𝑆′∣𝑆′ ⊆ 𝑆 with ∣𝑆′∣ = 𝑝}, where 0 ≤ 𝑝 ≤ 𝑛𝑅.We assume that 𝑡𝑖 <

𝑞𝑖4 −ℎ𝑖 for 𝑝 relays in 𝑆′ ∈ 𝑆′(𝑝) and

𝑡𝑖 >𝑞𝑖4 −ℎ𝑖 for 𝑛𝑅 − 𝑝 relays in 𝑆 ∖𝑆′. Then the event of∑

𝑖∈𝑆𝑁∖𝑆𝑞′𝑖4 > 𝑤 + 𝑠 +

∑𝑖∈𝑆 min

[ℎ𝑖 + 𝑡𝑖,

𝑞𝑖4

]can be written

as ∑𝑖∈𝑆𝑁∖𝑆

𝑞′𝑖4

> 𝑤 + 𝑠 +∑

𝑖∈𝑆∖𝑆′

𝑞𝑖4+∑𝑖∈𝑆′

(ℎ𝑖 + 𝑡𝑖).

Since 𝑡𝑖’s and 𝑠 are independent, the probability in (28) canbe rewritten as

𝑃


𝑞′𝑖4

> 𝑤 + 𝑠 +∑𝑖∈𝑆

min[ℎ𝑖 + 𝑡𝑖,

𝑞𝑖4

])

=

𝑛𝑅∑𝑝=0

∑𝑆′∈𝑆′(𝑝)

𝑃(𝑡𝑖>

𝑞𝑖4−ℎ𝑖, 𝑖∈𝑆∖𝑆′

)

𝑃

(𝑡𝑖<

𝑞𝑖4−ℎ𝑖, 𝑖∈𝑆′,

∑𝑖∈𝑆𝑁∖𝑆

𝑞′𝑖4>𝑤+𝑠+

∑𝑖∈𝑆∖𝑆′


ℎ𝑖+∑𝑖∈𝑆′

𝑡𝑖

)

≤𝑛𝑅∑𝑝=0

∑𝑆′∈𝑆′(𝑝)

𝑃

(𝑡𝑖<

𝑞𝑖4−ℎ𝑖, 𝑖∈𝑆′,






𝑡𝑖

).

(29)

In addition, when 𝑡𝑖 <𝑞𝑖4 −ℎ𝑖 for 𝑖 ∈𝑆′ is true,

∑𝑖∈𝑆′𝑡𝑖 <∑

𝑖∈𝑆′𝑞𝑖4 −∑𝑖∈𝑆′ℎ𝑖 must be true. Therefore, (28) can be upper

bounded as

𝑃


𝑞′𝑖4


min[ℎ𝑖+𝑡𝑖,

𝑞𝑖4

])exp


𝑞′𝑖4𝜎2

)

≤𝑛𝑅∑𝑝=0

∑𝑆′∈𝑆′(𝑝)

𝑃

(∑𝑖∈𝑆′

𝑡𝑖<∑𝑖∈𝑆′

𝑞𝑖4−∑𝑖∈𝑆′

ℎ𝑖,






𝑡𝑖

)exp


𝑞′𝑖4𝜎2

). (30)

Then, for 𝑝 = 0, i.e., 𝑆′ is an empty set, the summand in (30)can be upper bounded as

𝑃


𝑞′𝑖4


𝑞𝑖4

)exp


𝑞′𝑖4𝜎2

)

≤

⎧⎨⎩exp(−(𝑤+

∑𝑖∈𝑆

𝑞𝑖4−


𝑞′𝑖4

)24𝑤𝜎2 −

∑𝑖∈𝑆𝑁∖𝑆𝑞

′𝑖

4𝜎2

)if 𝑤+

∑𝑖∈𝑆

𝑞𝑖4>∑𝑖∈𝑆𝑁∖𝑆

𝑞′𝑖4

exp(−

∑𝑖∈𝑆𝑁∖𝑆 𝑞

′𝑖

4𝜎2

)if 𝑤+

∑𝑖∈𝑆

𝑞𝑖4≤∑𝑖∈𝑆𝑁∖𝑆

𝑞′𝑖4

≤ exp

(− 𝑤 +

∑𝑖∈𝑆

𝑞𝑖2 +


𝑞′𝑖2

4𝜎2

). (31)

Let 𝑞𝑥 =∑

𝑖∈𝑆′𝑞𝑖4 , 𝑞𝑦 =


𝑞𝑖4 , 𝑞𝑧 =


𝑞′𝑖4 , 𝑡 =∑

𝑖∈𝑆′𝑡𝑖, and ℎ =∑

𝑖∈𝑆′ℎ𝑖. Then, for 𝑝 = 1, . . . , 𝑛𝑅, thesummand in (30) can be rewritten as

𝑃(𝑡<𝑞𝑥−ℎ, 𝑠<−𝑤−ℎ−𝑡−𝑞𝑦+𝑞𝑧

)exp

(− 𝑞𝑧

𝜎2

). (32)

Since 𝑠 ∼ 𝒩 (0, 2𝑤𝜎2) and 𝑡 ∼ 𝒩 (0, 2ℎ𝜎2), the probabilityin (32) can be derived as


)

=

∫ 𝑞𝑥−ℎ

−∞𝑄(𝑤 + ℎ+ 𝑡 + 𝑞𝑦 − 𝑞𝑧√

2𝑤𝜎2

)exp(− 𝑡2

4𝜎2ℎ)√4𝜋𝜎2ℎ

𝑑𝑡

≤

⎧⎨⎩

∫−𝑤−ℎ−𝑞𝑦+𝑞𝑧−∞

exp(− 𝑡2

4𝜎2ℎ)√

4𝜋𝜎2ℎ𝑑𝑡

+∫ 𝑞𝑥−ℎ−𝑤−ℎ−𝑞𝑦+𝑞𝑧exp

(− (𝑤+ℎ+𝑡+𝑞𝑦−𝑞𝑧)24𝑤𝜎2

) exp(− 𝑡2

4𝜎2ℎ)√

4𝜋𝜎2ℎ𝑑𝑡 if 𝑤+𝑞𝑥+𝑞𝑦>𝑞𝑧∫ 𝑞𝑥−ℎ

−∞exp(− 𝑡2

4𝜎2ℎ)√

4𝜋𝜎2ℎ𝑑𝑡 if 𝑤+𝑞𝑥+𝑞𝑦≤𝑞𝑧

≤

⎧⎨⎩

1+exp(− (𝑤+ℎ+𝑞𝑦−𝑞𝑧)2

4(𝑤+ℎ)𝜎2

)if 𝑤+𝑞𝑥+𝑞𝑦>𝑞𝑧> 𝑤+ℎ+𝑞𝑦

exp(− (𝑤+ℎ+𝑞𝑦−𝑞𝑧)2

4ℎ𝜎2

)+ exp

(− (𝑤+ℎ+𝑞𝑦−𝑞𝑧)24(𝑤+ℎ)𝜎2

)if 𝑤+𝑞𝑥+𝑞𝑦>𝑞𝑧and𝑤+ℎ+𝑞𝑦>𝑞𝑧

1 if 𝑤+𝑞𝑥+𝑞𝑦≤𝑞𝑧and ℎ ≤ 𝑞𝑥

exp(− (ℎ−𝑞𝑥)2

4𝜎2ℎ

)if 𝑤+𝑞𝑥+𝑞𝑦≤𝑞𝑧and ℎ > 𝑞𝑥.

(33)

Then (32) can be upper bounded as


)exp

(− 𝑞𝑧

𝜎2

)≤ 2 exp

(− 𝑤 + ℎ+ 2𝑞𝑦 + 2𝑞𝑧

4𝜎2

). (34)

Plugging (31) and (34) into (30), the upper bound on (28) canbe derived as (35) at the top of the next page.

Since ℎ𝑖 = 𝑃𝑅∥∥𝐹𝑖(𝑋𝑅𝑖(x) − 𝑋𝑅𝑖(x

min𝐹𝑖

))∥∥2, 𝑞𝑖 =

𝑃𝑆∥𝐾𝑖(𝑋𝑆(x) − 𝑋𝑆(x̃))∥2 ≥ 𝑃𝑆∥∥𝐾𝑖


min𝐾𝑖

))∥∥2,

and 𝑞′𝑖 = 𝑃𝑆∥𝐾𝑖(𝑋𝑆(x) − 𝑋𝑆(x𝑅𝑖))∥2 ≥ 𝑃𝑆∥∥𝐾𝑖

(𝑋𝑆(x)−

𝑋𝑆(xmin𝐾𝑖

))∥∥2, we have

∑𝑖∈𝑆′

ℎ𝑖+∑

𝑖∈𝑆∖𝑆′

𝑞𝑖2+


𝑞′𝑖2

≥∑𝑖∈𝑆𝑁

min[𝑃𝑆2

∥∥𝐾𝑖


min𝐾𝑖

))∥∥2, 𝑃𝑅∥∥𝐹𝑖

(𝑋𝑅𝑖(x)−𝑋𝑅𝑖(x

min𝐹𝑖

))∥∥2

].

(36)

Plugging (35) and (36) into (19), the summand in (13) can beupper bounded as (37). Then, the PEP in (13) can be derivedas (20).


𝑃


𝑞′𝑖4

> 𝑤+𝑠 +∑𝑖∈𝑆

min[ℎ𝑖 + 𝑡𝑖,

𝑞𝑖4

])exp


𝑞′𝑖4𝜎2

)

≤exp

(− 𝑤 +

∑𝑖∈𝑆

𝑞𝑖2 +


𝑞′𝑖2

4𝜎2

)+

𝑛𝑅∑𝑝=1

∑𝑆′∈𝑆′(𝑝)

2 exp

(− 𝑤+

∑𝑖∈𝑆′ ℎ𝑖+


𝑞𝑖2 +∑

𝑖∈𝑆𝑁∖𝑆𝑞′𝑖2

4𝜎2

)

≤2 exp(− 𝑤

4𝜎2

) 𝑛𝑅∑𝑝=0

∑𝑆′∈𝑆′(𝑝)

exp

(−∑

𝑖∈𝑆′ ℎ𝑖+∑

𝑖∈𝑆∖𝑆′𝑞𝑖2 +∑

𝑖∈𝑆𝑁∖𝑆𝑞′𝑖2

4𝜎2

)(35)

𝑃(x → x̃

∣∣x𝑅𝑖 = x, 𝑖 ∈𝑆 and x𝑅𝑖 ∕= x, 𝑖 ∈𝑆𝑁∖𝑆)∏𝑖∈𝑆



𝑃

⪅2 exp(−𝑃𝑆∥𝐺(𝑋𝑆(x)−𝑋𝑆(x̃))∥2

4𝜎2

) 𝑛𝑅∑𝑝=0

∑𝑆′∈𝑆′(𝑝)

exp

(−∑

𝑖∈𝑆𝑁min[𝑃𝑆2

∥∥𝐾𝑖


min𝐾𝑖

))∥∥2, 𝑃𝑅∥∥𝐹𝑖(𝑋𝑅𝑖(x)−𝑋𝑅𝑖(x

min𝐹𝑖

))∥∥2]

4𝜎2

)(37)

APPENDIX BFACT 1

Fact 1: [5] For an 𝑛×𝑚 matrix 𝐴, there exist a unitary ma-trix 𝑈 and a real diagonal matrix Λ such that 𝐴𝐴† = 𝑈Λ𝑈 †,where Λ = diag(𝜆1, . . . , 𝜆𝑛), 𝜆1, . . . , 𝜆𝑛 are the eigenvaluesof 𝐴𝐴†, and the columns of 𝑈 are the corresponding eigenvec-tors. Suppose that 𝜆1, . . . , 𝜆𝑟 ∕= 0, 𝜆𝑟+1 = ⋅ ⋅ ⋅ = 𝜆𝑛 = 0, and𝜆min is the minimum nonzero eigenvalue. Then, the followinginequality holds for any 𝑙 × 𝑛 matrix 𝐵 as

∥𝐵𝐴∥2 ≥ 𝜆min∥𝐵′∥2

where 𝐵′ is an 𝑙×𝑟 matrix constructed by using the 𝑖th columnof 𝐵𝑈 as its 𝑖th column, 𝑖 = 1, . . . , 𝑟. □

APPENDIX CPROOF OF THEOREM 3

The CDF of 𝑦max can be written as

𝑃𝑦max(𝑦) = 𝑃(∑𝑖∈𝑆1

𝛾′𝑖 ≤ 𝑦, . . . ,

∑𝑖∈𝑆

(𝑁𝑚)

𝛾′𝑖 ≤ 𝑦

). (38)

Since the random variables∑

𝑖∈𝑆𝑗𝛾′𝑖, 𝑗 = 1, . . . ,

(𝑁𝑚

)are not

statistically independent, the CDF of 𝑦max is very difficult tobe derived. However, we can find that max𝑖∈𝑆𝑗 𝛾′

𝑖 ≤∑

𝑖∈𝑆𝑗𝛾′𝑖

and 𝛾′𝑖, 𝑖 = 1, . . . , 𝑁 are i.i.d., and thus, the CDF of 𝑦max is

upper bounded as

𝑃(∑𝑖∈𝑆1

𝛾′𝑖 ≤ 𝑦, . . . ,

∑𝑖∈𝑆

(𝑁𝑚)


)

≤ 𝑃(max𝑖∈𝑆1

𝛾′𝑖 ≤ 𝑦, . . . , max

𝑖∈𝑆(𝑁𝑚)


)=[𝑃 (𝛾′

1 ≤ 𝑦)]𝑁

.

(39)

Plugging (39) into (24), the upper bound on the RHS in (24)can be derived. Since ∥𝐾 ′

1∥2 and ∥𝐹 ′1∥2 are 𝑟𝑆𝑀𝑅 - Erlang

and 𝑟𝑅𝑀𝐷 - Erlang random variables with rate parameters𝜎2𝑆𝑅 and 𝜎2

𝑅𝐷 , the expression of 𝑃 (𝛾′1 ≤ 𝑦) is very com-

plicated and thus, the expectation in (24) cannot be upperbounded by a closed-form expression. Let 𝑛𝑢 = 𝑟𝑆𝑀𝑅 and𝑛𝑣 = 𝑟𝑅𝑀𝐷. Using the fact that the summation of 𝑛 i.i.d.

exponential random variables is an 𝑛-Erlang random variable,12𝑃𝑆𝜔min∥𝐾 ′

1∥2 and 𝑃𝑅𝜇min∥𝐹 ′1∥2 can be rewritten as

1

2𝑃𝑆𝜔min∥𝐾 ′

1∥2 =

𝑛𝑢∑𝑗=1

𝑢𝑗 and 𝑃𝑅𝜇min∥𝐹 ′1∥2 =

𝑛𝑣∑𝑗=1

𝑣𝑗 ,

respectively, where 𝑢𝑗, 𝑗 = 1, . . . , 𝑛𝑢, are i.i.d. exponentialrandom variables with parameter 𝜆𝑢 = 2/𝑃𝑆𝜔min𝜎

2𝑆𝑅, and

𝑣𝑗 , 𝑗 = 1, . . . , 𝑛𝑣, are i.i.d exponential random variables withparameter 𝜆𝑣 = 1/𝑃𝑅𝜇min𝜎

2𝑅𝐷 . In addition, we have

min[ 𝑛𝑢∑𝑗=1

𝑢𝑗 ,

𝑛𝑣∑𝑗=1

𝑣𝑗

]≥

min[𝑛𝑢,𝑛𝑣]∑𝑗=1

[𝑢𝑗, 𝑣𝑗 ]

≥ max𝑗∈{1,...,min[𝑛𝑢,𝑛𝑣]}

min[𝑢𝑗 , 𝑣𝑗 ]

from the nonnegativity of 𝑢𝑗 and 𝑣𝑗 . Therefore, the upperbound on the CDF of 𝑦max can be rewritten as

𝑃𝑦max(𝑦)

≤[𝑃(

max𝑗∈{1,...,min[𝑛𝑢,𝑛𝑣]}

min[𝑢𝑗 , 𝑣𝑗 ] < 𝑦)]𝑁

=[𝑃(min[𝑢1, 𝑣1] < 𝑦

)]𝑁 min[𝑛𝑢,𝑛𝑣]

=[1− 𝑃

(min[𝑢1, 𝑣1] ≥ 𝑦

)]𝑁 min[𝑛𝑢,𝑛𝑣]

=[1−exp

(−( 2


+1


)𝑦)]𝑁min[𝑟𝑆𝑀𝑅,𝑟𝑅𝑀𝐷 ]

.

APPENDIX DPROOF OF THEOREM 4

Using the upper bound on the CDF of 𝑦max in (25), theupper bound on the second expectation in the RHS of (23) isderived. Let 𝑎 = 1/4𝜎2, 𝑏 = 2/𝑃𝑆𝜔min𝜎

2𝑆𝑅+1/𝑃𝑅𝜇min𝜎

2𝑅𝐷 ,

and 𝑁𝑝 = 𝑁min[𝑟𝑆𝑀𝑅, 𝑟𝑅𝑀𝐷]. From (24), the upper boundon the second expectation in the RHS of (23) can be rewritten


𝐸[exp(− 𝑦max

4𝜎2

)] ≤ (𝑁min[𝑟𝑆𝑀𝑅, 𝑟𝑅𝑀𝐷])!(2


+ 1𝑃𝑅𝜇min𝜎2

𝑅𝐷)𝑁min[𝑟𝑆𝑀𝑅,𝑟𝑅𝑀𝐷 ]

∏𝑁min[𝑟𝑆𝑀𝑅,𝑟𝑅𝑀𝐷 ]𝑖=1

(1

4𝜎2 + ( 2𝑃𝑆𝜔min𝜎2

𝑆𝑅+ 1


)𝑖)

≤ (𝑁min[𝑟𝑆𝑀𝑅, 𝑟𝑅𝑀𝐷])!( 8𝜎2


+4𝜎2


)𝑁min[𝑟𝑆𝑀𝑅,𝑟𝑅𝑀𝐷 ]

(43)

as

𝐸[exp

(− 𝑦max

4𝜎2

)]≤ 𝑎

∫ ∞

0

exp(−𝑎𝑦)[1− exp(−𝑏𝑦)]𝑁𝑝𝑑𝑦

= 𝑎

∫ ∞

0

exp(−𝑎𝑦)

𝑁𝑝∑𝑘=0

(𝑁𝑝

𝑘

)(−1)𝑘 exp(−𝑘𝑏𝑦)𝑑𝑦

= 𝑎

𝑁𝑝∑𝑘=0

(𝑁𝑝

𝑘

)(−1)𝑘

1

𝑎 + 𝑘𝑏

=

∑𝑁𝑝

𝑘=0

(𝑁𝑝

𝑘

)(−1)𝑘

[∑𝑁𝑝

𝑛=0 𝑎𝑁𝑝−𝑛𝑏𝑛∑

0≤𝑙1<...<𝑙𝑛≤𝑁𝑝𝑙1, ..., 𝑙𝑛 ∕=𝑘

𝑙1 ⋅ ⋅ ⋅ 𝑙𝑛]

∏𝑁𝑝

𝑖=1(𝑎+ 𝑏𝑖).

Let 𝑃𝑛 =∑

𝑙1<⋅⋅⋅<𝑙𝑛 𝑙1 ⋅ ⋅ ⋅ 𝑙𝑛 for 𝑛 ≥ 1 and 𝑃0 = 1. Then wehave ∑

𝑙1<⋅⋅⋅<𝑙𝑛𝑙1, ..., 𝑙𝑛 ∕=𝑘

𝑙1 ⋅ ⋅ ⋅ 𝑙𝑛 = 𝑃𝑛 − 𝑘∑

𝑙1<⋅⋅⋅<𝑙𝑛−1𝑙1, ..., 𝑙𝑛−1 ∕=𝑘

𝑙1 . . . 𝑙𝑛−1

=

𝑛∑𝑠=0

(−1)𝑠𝑘𝑠𝑃𝑛−𝑠 . (40)

Using the result in (40), we have

𝐸[exp

(− 𝑦max

4𝜎2

)]

≤∑𝑁𝑝

𝑘=0

(𝑁𝑝

𝑘

)(−1)𝑘

∑𝑁𝑝

𝑛=0 𝑎𝑁𝑝−𝑛𝑏𝑛∑𝑛

𝑠=0(−1)𝑠𝑘𝑠𝑃𝑛−𝑠∏𝑁𝑝

𝑖=1(𝑎 + 𝑏𝑖)

(𝑏)=

(−1)𝑁𝑝𝑁𝑝! 𝑏𝑁𝑝(−1)𝑁𝑝𝑃0∏𝑁𝑝

𝑖=1(𝑎 + 𝑏𝑖)

=𝑁𝑝!𝑏

𝑁𝑝∏𝑁𝑝

𝑖=1(𝑎+ 𝑏𝑖)(41)

where (𝑏) is established from the following equality in [22]

𝑁∑𝑘=0

(−1)𝑘(𝑁

𝑘

)𝑘𝑛 =

{0 if 0 ≤ 𝑛 < 𝑁

(−1)𝑁𝑁 ! if 𝑛 = 𝑁.

Therefore, the second expectation in the RHS of (23) canbe upper bounded as (43). Since 𝑦𝐺 = 𝑃𝑆𝜔min∥𝐺′∥2is an 𝑟𝑆𝑀𝐷-Erlang random variable with rate parameter𝑃𝑆𝜔min𝜎

2𝑆𝐷 , the first expectation in the RHS of (23) can be

rewritten as

𝐸[exp(− 𝑃𝑆𝜔min∥𝐺′∥2

4𝜎2

)]=

( 4𝜎2

𝑃𝑆𝜔min𝜎2𝑆𝐷

4𝜎2


+ 1

)𝑟𝑆𝑀𝐷

≤( 4𝜎2


)𝑟𝑆𝑀𝐷

. (44)

Plugging (43) and (44) into (23), the average PEP of the newrelay selection scheme can be upper bounded as


⪅2(𝑀𝐿+1

)𝑚(𝑁 min[𝑟𝑆𝑀𝑅, 𝑟𝑅𝑀𝐷]

)!(

4𝜎2


)𝑟𝑆𝑀𝐷(

8𝜎2


+4𝜎2


)𝑁min[𝑟𝑆𝑀𝑅,𝑟𝑅𝑀𝐷 ]

.

(45)

From the upper bound on the average PEP, the theorem isproved.

REFERENCES

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[2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part II: implementation aspects and performance analysis,” IEEE Trans.Commun., vol. 51, no. 11, pp. 1939–1948, Nov. 2003.

[3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity inwireless networks: efficient protocols and outage behavior,” IEEE Trans.Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.

[4] T. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, “High-performance cooperative demodulation with decode-and-forward relays,”IEEE Trans. Commun., vol. 55, no. 7, pp. 1427-1438, July 2007.

[5] X. Jin, D.-S. Jin, J.-S. No, and D.-J. Shin, “Diversity analysis of MIMOdecode-and-forward relay network by using near-ML decoder,” IEICETrans. Commun., vol. E94-B, no. 10, Oct. 2011.

[6] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded pro-tocols for exploiting cooperative diversity in wireless networks,” IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003.

[7] V. Sreng, H. Yanikomeroglu, and D. D. Falconer, “Relayer selectionstrategies in cellular networks with peer-to-peer relaying,” in Proc. IEEEVTC, pp. 1949–1953, Oct. 2003.

[8] Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selectionamplify-and-forward relay systems,” IEEE Commun. Lett., vol. 10, no.11, pp. 757–759, Nov. 2006.

[9] S. S. Ikki and M. H. Ahmed, “Performance of multiple-relay cooperativediversity systems with best relay selection over Rayleigh fading chan-nels,” EURASIP J. Adv. Signal Process., vol. 2008, article ID 580368,2008.

[10] Y. Jing and H. Jafarkhani, “Single and multiple relay selection schemesand their diversity orders,” IEEE Trans. Wireless Commun., vol. 8, no. 3,pp. 1414–1423, Mar. 2009.

[11] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple coop-erative diversity method based on network path selection,” IEEE J. Sel.Areas Commun., vol. 24, no. 3, pp. 659–672, Mar. 2006.

[12] Z. Yi and I.-M. Kim, “Diversity order analysis of the decode-and-forward relay networks with relay selection,” IEEE Trans. WirelessCommun., vol. 7, no. 5, pp. 1792–1799, May 2008.

[13] A. S. Ibrahim, A. Sadek, W. Su, and K. J. R. Liu, “Cooperativecommunications with relay selection: when to cooperate and whom tocooperate with?” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp. 2814–2827, July 2008.

[14] Y. Fan and J. S. Thompson, “MIMO configurations for relay channels:theory and practice,” IEEE Trans. Wireless Commun., vol. 6, no. 5,pp. 1774–1786, May 2007.

[15] B. M. Hochwald and S. T. Brink, “Achieving near-capacity on amultiple-antenna channel,” IEEE Trans. Commun., vol. 51, pp. 389–399,Mar. 2003.

[16] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals andApplications, 2nd edition. Pearson Prentice Hall, 2004.


[17] M. C. Ju and I. M. Kim, “ML performance analysis of the decode-and-forward protocol in cooperative deversity networks,” IEEE Trans.Wireless Commun., vol. 8, no. 7, pp. 3855–3867, July 2009.

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[19] P. O. B𝑜rjesson and C. E. W. Sundberg, “Simple approximations ofthe error function 𝑄(𝑥) for communications applications,” IEEE Trans.Commun., vol. 27, no. 3, pp. 639–643, Mar. 1979.

[20] S. Haykin, Communication Systems, 4th edition. John Wiley & Sons,Inc., 2001.

[21] S. Alamouti, “A simple transmit diversity technique for wireless commu-nications,” IEEE J. Sel. Areas Commum., vol. 16, no. 8, pp. 1451–1458,Oct. 1998.

[22] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, andProducts, 6th edition. Academic Press, 2000.

Xianglan Jin received the B.S. and M.S. degrees inTelecommunication Engineering from Beijing Uni-versity of Posts and Telecommunications, Beijing,China, in 1999 and 2002, respectively, and Ph.D.degree in the Department of EECS from SeoulNational University, Seoul, Korea, in 2008. Cur-rently, she is an assistant professor at the Departmentof ICE, Dongguk University-Seoul. Her researchinterests include MIMO, space-time codes, and in-formation theory.

Jong-Seon No received the B.S. and M.S.E.E. de-grees in Electronics Engineering from Seoul Na-tional University, Seoul, Korea, in 1981 and 1984,respectively, and the Ph.D. degree in Electrical Engi-neering from the University of Southern California,Los Angeles, in 1988. He was a Senior MTS withHughes Network Systems, Germantown, MD, fromFebruary 1988 to July 1990. He was an AssociateProfessor with the Department of Electronic Engi-neering, Konkuk University, Seoul, from September1990 to July 1999. He joined the faculty of the

Department of EECS, Seoul National University, in August 1999, where heis currently a Professor. His research interests include error-correcting codes,sequences, cryptography, space-time codes, and wireless communicationsystems.

Dong-Joon Shin received the B.S. degree in elec-tronics engineering from Seoul National University,Seoul, Korea, the M.S. degree in electrical engineer-ing from Northwestern University, Evanston, USA,and the Ph.D. degree in electrical engineering fromUniversity of Southern California, Los Angeles,USA. From 1999 to 2000, he was a member oftechnical staff in Wireless Network Division andSatellite Network Division, Hughes Network Sys-tems, Maryland, USA. Since September 2000, hehas been an Associate Professor in the Department

of EE at Hanyang University, Seoul, Korea. His current research interestsinclude error correcting codes, sequences, and discrete mathematics.

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