IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 7, JULY 2010 2147

Analytical Performance ofAmplify-and-Forward MIMO Relaying with

Orthogonal Space-Time Block CodesPrathapasinghe Dharmawansa, Member, IEEE, Matthew R. McKay, Member, IEEE,

and Ranjan K. Mallik, Senior Member, IEEE

Abstract—This paper considers orthogonal space-time blockcoded transmission for a multiple-input multiple-output channel(MIMO) with non-coherent amplify-and-forward (AF) relayingin Rayleigh fading. We first characterize the statistical propertiesof the instantaneous signal-to-noise ratio (SNR) at the destination,by deriving new exact closed form expressions for the momentgenerating function, cumulants, and first and second moments.These results show a reciprocity relationship between the numberof antennas at the relay and destination. The probability densityfunction and cumulative distribution function of the SNR arealso derived for certain system configurations, and for variousasymptotic regimes. We then investigate the system performanceby presenting new analytical expressions for the symbol errorrate, outage probability, amount of fading, as well as the diversityorder and array gain. Our results indicate that the proposedscheme can achieve the maximum diversity order of the non-coherent AF MIMO relay channel.

Index Terms—Amplify-and-forward relaying, multiple-inputmultiple-output, orthogonal space-time block coding, perfor-mance analysis.

I. INTRODUCTION

IN recent years, wireless relaying techniques have receivedconsiderable attention as a means of significantly im-

proving coverage and performance [1]–[3]. Of the variousrelaying protocols which have been proposed, amplify-and-forward (AF) techniques present one of the most attractivemethods, since they offer significant gains without requiringsophisticated signal processing at the relay terminals [4]–[6].

Multiple-input multiple-output (MIMO) techniques havealso received huge interest in the past decade as a meansof substantially increasing capacity in both single-user andmulti-user communications. More recently, a major focus hasbeen on developing MIMO relaying technologies [7], [8]; with

Paper approved by M.-S. Alouini, the Editor for Modulation and DiversitySystems of the IEEE Communications Society. Manuscript received January21, 2009; revised May 27, 2009 and September 22, 2009.

This paper was presented in part at the IEEE International Conferenceon Communications, Dresden, Germany, June 2009. The work of Pratha-pasinghe Dharmawansa and Matthew R. McKay was supported by GrantRPC07/08.EG16. The work of Ranjan K. Mallik was supported in part byIDRC Research Grant RP02253.

P. Dharmawansa and M. R. McKay are with the Department of Electronicand Computer Engineering, Hong Kong University of Science and Tech-nology, Clear Water Bay, Kowloon, Hong Kong (e-mail: eesinghe@ust.hk;eemckay@ust.hk).

R. K. Mallik is with the Department of Electrical Engineering, IndianInstitute of Technology - Delhi, Hauz Khas, New Delhi 110016, India (e-mail: rkmallik@ee.iitd.ernet.in).

Digital Object Identifier 10.1109/TCOMM.2010.07.090038

such technologies currently being proposed as a next majorextension of the IEEE 802.16 WiMAX industry standard.

Prior work on MIMO relaying with AF (see e.g. [9]–[12]and references therein) has largely been based on the keyassumption that the instantaneous channel state information(CSI) is available at the relay terminals. In this case, the relaymust consume resources in estimating the channel parameters,and moreover, these parameters may be inaccurate due tochannel estimation errors. Furthermore, as discussed in [13],it may be inconvenient to forward the estimated CSI to thedestination via another wireless link.

An alternative approach, which has also been receivingconsiderable attention, is to employ a fixed gain at the relayterminals. This technique is commonly referred to as “non-coherent” relaying. In this case, the relay gain is typicallyselected based on the long-term statistical properties of thechannel. Although the performance of non-coherent relayingis inferior to that of CSI assisted relaying, the low complexitymakes it feasible for many practical scenarios [14]. Veryrecent contributions in the area of non-coherent MIMO relay-ing have focused mainly on characterizing the fundamentalinformation-theoretic limits [15]–[20].

In this paper, we consider space-time coding for the non-coherent AF MIMO relay channel. For the case of single-antenna AF relay networks, much work has been done inthis area; in particular, considering the design and analysis ofdistributed space-time coding techniques (see e.g. [21], [22]).For MIMO AF relay networks, however, there are relativelyfew prior contributions. In this respect, [23] proposed generalspace-time block code design criterion for the dual hop non-coherent AF MIMO relay channel, and established a lowerbound on the optimal diversity-multiplexing trade-off (DMT).The work of [23] was further extended in [24] to incorporatemulti-hop MIMO relays. A more comprehensive study of theDMT was also presented, as well as an upper bound onthe maximal achievable diversity. In addition to the DMT,specific space-time block code designs were proposed in[23],[24], which were shown to achieve the full diversity ofthe MIMO relay channel, however they required high decodingcomplexity.

For point-to-point MIMO systems, orthogonal space-timeblock coding (OSTBC) techniques transform the MIMO chan-nel into equivalent parallel sub channels. In the presence ofadditive white Gaussian noise (AWGN), these sub channels

0090-6778/10$25.00 c⃝ 2010 IEEE

2148 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 7, JULY 2010

are independent, and therefore permit very simple (scalar)maximum likelihood decoding [26]. For the non-coherent AFMIMO relay channel, we show that by employing a certainpre-whitening filter at the destination to remove the effectivecorrelation in the noise caused by the noise amplificationprocess at the relay, standard OSTBC techniques designedfor point-to-point MIMO channels can be directly applied.The major technical challenge is to analytically characterizethe performance of the proposed system, in order to quantifythe key performance measures (such as the diversity order),and to draw insights into the effect of the various systemparameters. This, in turn, requires a statistical characterizationof the signal-to-noise ratio (SNR) at the output of the OSTBCdecoder. We address this problem by employing the theory offinite-dimensional random matrices.

In particular, we first derive new exact expressions for themoment generating function (m.g.f.), higher-order cumulants,and first and second moments of the SNR. These results revealan interesting reciprocity relationship between the numberof antennas at the relay and destination. We also presentnew exact expressions for the probability density function(p.d.f.) and cumulative distribution function (c.d.f.) of theSNR for some particular antenna configurations, as well assome general asymptotic results as either the number ofrelay antennas or the relay gain grow large. Based on ourstatistical characterization of the SNR, we then investigate theperformance of the system by deriving analytical expressionsfor the symbol error rate (SER) with𝑀 -ary phase-shift keying(MPSK), as well as the outage probability, and the amountof fading (AoF). In addition, we present a comprehensiveinvestigation of the diversity order and array gain: the two keyparameters dictating performance in the high SNR regime1.Our results suggest that for most system configurations, theproposed scheme achieves the maximum possible diversityorder of the non-coherent AF MIMO relay channel, charac-terized in [23].

II. SYSTEM AND CHANNEL MODELS

We consider a dual hop MIMO communication systemwith source, relay, and destination terminals having 𝑁𝑆 , 𝑁𝑅,and 𝑁𝐷 antennas respectively. Half-duplex time-orthogonalrelaying [3] with non-regenerative AF is assumed2, with eachtransmission period divided into two time slots. During thefirst time slot, the source transmits to the relay. The relayamplifies its received signal subject to an average powerconstraint, then transmits to the destination in the second time-slot. We assume that there is no direct link between the sourceand destination.

Let H1 ∈ ℂ𝑁𝑅×𝑁𝑆 denote the channel matrix between the

source and relay, and H2 ∈ ℂ𝑁𝐷×𝑁𝑅 represent the channelmatrix between the relay and destination. Both channel ma-trices are assumed to undergo independent Rayleigh fading,with elements 𝒞𝒩 (0, 1). We adopt the common non-coherentrelaying assumption (as in [15]–[17]) where the destination

1We note that a related high SNR analysis was conducted independentlyin [25], which appeared whilst our paper was in the review process.

2Note that the analysis given in this paper can be easily extended to thecase of full duplex relaying, assuming that the self interference caused by thetransmit-receive echo is negligible or can be canceled perfectly by the relay.

has perfect knowledge3 of H1 and H2, whereas the relayand source have no knowledge. Perfect synchronization is alsoassumed amongst all terminals.

We assume that the source employs OSTBC encoding. Inparticular, groups of 𝑁 independent and identically distributed(i.i.d.) symbols 𝑠1, 𝑠2, . . . , 𝑠𝑁 are mapped to a row orthogonaltransmission matrix 퓧 ∈ ℂ

𝑁𝑆×𝑁𝑇 , where 𝑁𝑇 denotes thenumber of symbol periods used to send each OSTBC code-word. Note that the entries of 퓧 are linear combinations of{𝑠𝑖}𝑁𝑖=1 and {𝑠∗𝑖 }𝑁𝑖=1, with the exact construction dependingon the specific OSTBC used [26]. Since it takes 𝑁𝑇 symbolperiods to transmit 𝑁 symbols, the code rate is 𝑅 = 𝑁/𝑁𝑇 .

Let us write 퓧 = (x1, . . . , x𝑁𝑇 ) where x𝑖 ∈ ℂ𝑁𝑆×1.

Moreover, let 𝜌 denote the total transmit power, spanningall 𝑁𝑆 source antennas (i.e., 𝐸

(∣∣x𝑘∣∣2) = 𝜌). The receivedsignal at the relay during the 𝑘th symbol period in the firsttime slot is given by

y𝑘 = H1x𝑘 + n𝑘 , 𝑘 = 1, 2, . . . , 𝑁𝑇 (1)

where n𝑘 ∼ 𝒞𝒩 (0, I𝑁𝑅) is the unit variance noise vectorat the relay. The relay, in the absence of instantaneous CSI,operates according to the common fixed-gain non-regenerativeAF protocol (see eg. [17]). In particular, the relay multipliesits received signal by the constant gain matrix G = 𝑎I𝑁𝑅 ,where

𝑎 =

√𝑏

𝑁𝑅 (1 + 𝜌), 𝑏 ∈ ℝ

+ (2)

prior to retransmitting the amplified signal to the destinationduring the second time slot. Note that the above selec-tion of parameter 𝑎 ensures that the total power constraint𝐸(∣∣Gy𝑘∣∣2

) ≤ 𝑏 is met. The received signal at the destinationduring the 𝑘th symbol period in the second time slot is thusgiven by

r𝑘 = 𝑎H2y𝑘 +w𝑘 , 𝑘 = 1, 2, . . . , 𝑁𝑇 (3)

where w𝑘 ∼ 𝒞𝒩 (0, I𝑁𝐷 ) is the unit variance noise vector atthe destination. Combining (1) and (3) yields the input-outputrelation of the overall system as

r𝑘 = 𝑎H2H1x𝑘 + 𝑎H2n𝑘 +w𝑘 , 𝑘 = 1, 2, . . . , 𝑁𝑇 . (4)

Since the noise variance is unity, the average SNR is definedas 𝛾 = 𝜌. It is important to note that the overall input-outputmodel (4) is equivalent to a conventional point-to-point MIMOsystem with channel matrix 𝑎H2H1, and colored Gaussiannoise with conditional covariance matrix

𝐸{(𝑎H2n𝑘 +w𝑘) (𝑎H2n𝑘 +w𝑘)

𝐻 ∣H2

}= K (5)

whereK = 𝑎2H2H

𝐻2 + I𝑁𝐷 . (6)

As such, to decode, we first apply a filter to whiten theeffective noise, and then employ standard linear OSTBCprocessing to the filter output. The noise whitening operationis achieved by

r𝑘 = K−1/2r𝑘 = Hx𝑘 + n𝑘, (7)

3In fact, as the following analysis shows, knowledge of H2H1 and H2 issufficient. This issue is also discussed in [15].

DHARMAWANSA et al.: ANALYTICAL PERFORMANCE OF AMPLIFY-AND-FORWARD MIMO RELAYING WITH ORTHOGONAL SPACE-TIME BLOCK . . . 2149

where H = 𝑎K−1/2H2H1, and n𝑘 = K−1/2 (𝑎H2n𝑘 +w𝑘)is the equivalent whitened noise ∼ 𝒞𝒩 (0, I𝑁𝐷).

It is convenient to write the full transmission equation fora given codeword 퓧 as

R = H퓧 + N (8)

where R = (r1, . . . , r𝑁𝑇 ) and N ∼ 𝒞𝒩 (0, I𝑁𝐷 ⊗ I𝑁𝑇 ).After linear OSTBC processing, the MIMO relation (8)

decomposes into a set of parallel non-interacting single-inputsingle-output relations as follows

𝑠𝑙 = ∣∣H∣∣2𝐹 𝑠𝑙 + 𝜂𝑙 , 𝑙 = 1, 2, . . . , 𝑁 (9)

where 𝜂𝑙 ∼ 𝒞𝒩(0, ∣∣H∣∣2𝐹

), with ∣∣H∣∣𝐹 denoting the Frobe-

nius norm of H. Thus, following [27], the instantaneous SNRfor the 𝑙th symbol can be written as

𝛾𝑙 = ∣∣H∣∣2𝐹𝐸{∣𝑠𝑙∣2} = 𝛼𝛾𝑎2Tr

(H𝐻1 H𝐻2 K−1H2H1

)(10)

where 𝛼 = 1/𝑅𝑁𝑆.Note that the right-hand side of (10) is independent of 𝑙.

Thus, without loss of generality, we will henceforth drop thesubscript 𝑙 and denote the instantaneous SNR as simply 𝛾.

To evaluate the performance of the proposed approach, wemust characterize the statistical properties of the SNR in (10).This problem will be addressed in the following section byapplying tools from finite-dimensional random matrix theory.It is convenient to introduce the following notation: 𝑝 =max (𝑁𝑅, 𝑁𝐷), 𝑞 = min (𝑁𝑅, 𝑁𝐷), and 𝜈𝑖𝑗 = 𝑖+𝑗+𝑝−𝑞−1.

III. STATISTICS OF THE SNR 𝛾

A. M.G.F., Cumulants, and Moments

We first present a new exact closed-form expression for them.g.f. of the SNR 𝛾. This will be employed to study the errorperformance in the following section.

Theorem 1: The m.g.f. of the SNR 𝛾 is given by

ℳ𝛾(𝑠) = 𝒦−1 det (I(𝑠)) (11)

where

𝒦 =

𝑞∏𝑖=1

Γ(𝑝− 𝑖+ 1)Γ(𝑞 − 𝑖+ 1)

and I(𝑠) is a 𝑞× 𝑞 Hankel matrix with (𝑖, 𝑗)th entry given by

𝐼𝑖𝑗(𝑠) =Γ (𝜈𝑖𝑗)

𝑎2𝜈𝑖𝑗 (1 + 𝛼𝛾𝑠)𝜈𝑖𝑗+𝑁𝑆

𝑁𝑆∑𝑙=0

(𝑁𝑆𝑙

)(𝛼𝛾𝑠)𝑙

× 𝑈(𝜈𝑖𝑗 ; 𝜈𝑖𝑗 + 1 − 𝑙; 1

𝑎2(1 + 𝛼𝛾𝑠)

), (12)

where Γ(⋅) is the gamma function, and 𝑈(⋅; ⋅; ⋅) is the conflu-ent hypergeometric function of the second kind.

Proof: See Appendix A.Interestingly, we see that the m.g.f. has a direct dependence

on 𝑁𝑆 , however the dependence on 𝑁𝑅 and 𝑁𝐷 is reciprocal(i.e., through their minimum and maximum), except for thescale factor 𝑎.

Next we present new closed-form expressions for the cu-mulants of 𝛾. These will be used to obtain expressions for thefirst and second moments which, in turn, will be employed to

characterize the AoF of the proposed OSTBC system in thefollowing section.

Theorem 2: The 𝑛th order cumulant of the SNR 𝛾 is givenby

𝜇𝑛 = 𝒦−1(𝑛− 1)!𝑁𝑆(𝛼𝛾𝑎2

)𝑛 𝑞∑𝑙=1

det(J(𝑙))

(13)

where J(𝑙) is a 𝑞 × 𝑞 matrix with (𝑖, 𝑗)th entry given by

𝐽𝑖𝑗(𝑙) =

{Γ(𝜈𝑖𝑗+𝑛)

𝑎2(𝜈𝑖𝑗+𝑛)𝑈(𝜈𝑖𝑗 + 𝑛; 𝜈𝑖𝑗 + 1; 1

𝑎2

)if 𝑙 = 𝑗

Γ (𝜈𝑖𝑗) if 𝑙 ∕= 𝑗.Proof: See Appendix B.

Corollary 1: The first moment of the SNR 𝛾 is given by

𝑚1 = 𝒦−1𝑁𝑆𝛼𝛾

𝑞∑𝑙=1

det (J(𝑙)) (14)

where J(𝑙) is a 𝑞 × 𝑞 matrix with (𝑖, 𝑗)th entry given by

𝐽𝑖𝑗(𝑙) =

{Γ(𝜈𝑖𝑗+1)

𝑎2𝜈𝑖𝑗𝑈(𝜈𝑖𝑗 + 1; 𝜈𝑖𝑗 + 1; 1

𝑎2

)if 𝑙 = 𝑗

Γ (𝜈𝑖𝑗) if 𝑙 ∕= 𝑗.Proof: The result follows by substituting 𝑛 = 1 in (13).

The second moment requires more manipulations, and is givenby the following corollary.

Corollary 2: The second moment of 𝛾 is given by

𝑚2 = 𝜇2 (𝑁𝑆 + 1)+𝒦−1

𝑞!𝑁2𝑆

(𝛼𝛾𝑎2

)2 (𝑞2 − 𝑞) 𝒯 (A) (15)

where A is a 𝑞 × 𝑞 × 𝑞 rank-3 tensor with (𝑖, 𝑗, 𝑘)th elementgiven by

𝑎𝑖𝑗𝑘 =

{Γ(𝜈𝑖𝑗+1)

𝑎2(𝜈𝑖𝑗+1)𝑈(𝜈𝑖𝑗 + 1; 𝜈𝑖𝑗 + 1; 1

𝑎2

)if 𝑘 = 1, 2

Γ (𝜈𝑖𝑗) if 𝑘 ∕= 1, 2,

and 𝒯 (⋅) is the rank-3 tensor operator defined as [28]

𝒯 (A) =∑𝜖

sgn(𝜖)∑𝛿

sgn(𝛿)𝑞∏𝑘=1

𝑎𝜖𝑘𝛿𝑘𝑘

with 𝜖 = 𝜖1, 𝜖2, . . . , 𝜖𝑞 and 𝛿 = 𝛿1, 𝛿2, . . . , 𝛿𝑞 being permuta-tions of the integers 1, 2, . . . , 𝑞, and sgn(⋅) denotes the signof the permutation.

Proof: See Appendix C.Note that the higher-order moments can also be obtained

by exploiting general relationship between moments and cu-mulants.B. Exact Probability Distributions

In this subsection, we investigate the p.d.f. and c.d.f. of theSNR 𝛾.

1) Probability Density Function (P.D.F.): To evaluate thep.d.f. of 𝛾, a natural method is to attempt to directly invertthe m.g.f., given by Theorem 1. This approach, however,appears intractable for general system configurations, due tothe complexity of the m.g.f. expression. As such, to proceed,here we adopt an alternative derivation approach, and restrictattention to some specific system configurations, in which caseexact expressions can be obtained for the p.d.f. These are givenby the following theorem.

2150 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 7, JULY 2010

Theorem 3: The p.d.f. of the SNR 𝛾 is given for the casemin (𝑁𝑅, 𝑁𝐷) = 1 by

𝑓𝛾(𝛾) =2𝛾𝑁𝑆−1 exp

(− 𝛾𝛼𝛾

)(𝑎2𝛾𝛼)

𝑁𝑆 Γ (𝑝) Γ (𝑁𝑆)

𝑁𝑆∑𝑗=0

(𝑁𝑆𝑗

)𝑎2𝑗

×(𝛾

𝑎2𝛾𝛼

) 𝑗+𝑝−𝑁𝑆2

𝐾𝑗+𝑝−𝑁𝑆

(2

√𝛾

𝑎2𝛾𝛼

)(16)

and for the case min (𝑁𝑅, 𝑁𝐷) = 2 by

𝑓𝛾(𝛾) =2𝒦−1

2

√𝜋𝛾𝑁𝑆−1/2 exp

(− 𝛾𝛼𝛾

)(𝑎2𝛾𝛼)

𝑁𝑆+1/2Γ (𝑁𝑆)

×∞∑𝑛=0

𝑁𝑆∑𝑖,𝑗=0

2𝑛+2∑𝑙=0

(−1)𝑙(𝑁𝑆

𝑖

)(𝑁𝑆

𝑗

)(2𝑛+2𝑙

)𝑛!Γ (𝜁1 + 1)

𝑎2(2𝑁𝑆−𝑖−𝑗)

×(

𝛾

4𝑎2𝛾𝛼

)𝑛+𝜁1 ( 𝛾

2𝑎2𝛾𝛼

)𝑁𝑅−𝑛−(𝑖+𝑗)/2

×𝐾𝜁2(√

2𝛾

𝑎2𝛾𝛼

)𝐾𝜁3

(√2𝛾

𝑎2𝛾𝛼

)(17)

where 𝜁1 = 𝑛 + 𝑁𝑆 − 1/2, 𝜁2 = 𝑝 − 𝑖 − 2𝑛 − 1 + 𝑙, 𝜁3 =𝑝− 𝑗 − 𝑙 + 1, 𝒦2 = Γ(𝑝)Γ(𝑝− 1) and 𝐾𝑛(⋅) is the modifiedBessel function of the second kind and of order 𝑛.

Proof: See Appendix D-A.The expression (16), for the case min (𝑁𝑅, 𝑁𝐷) = 1, agrees

with a prior result derived via more complicated methodsand in a different context (i.e., considering the information-theoretic capacity of MIMO relay channels), in [29]. Weinclude this result, along with its simplified derivation, forcompleteness. The expression (17), on the other hand, is new.We note that, although (17) involves an infinite series, thisinfinite series converges rapidly, typically requiring less than6 terms to achieve very high accuracy.

Figs. 1(a) and 1(b) compare analytical p.d.f. curves andcorresponding Monte-Carlo simulated curves for a systemwith 𝑁𝐷 = 1 and 𝑁𝐷 = 2 respectively. The analytical resultsare based on (16) and (17) respectively. Results are shownfor the Alamouti code and 𝛾 = 10 dB with 𝑏 = 0 dB. Asexpected, in both cases there is an accurate match betweenthe simulations and analysis. Note that, in generating theanalytical curve in Fig. 1(b), the infinite series in (17) wastruncated to maximum of 5 terms, leading to an accuracy of3 significant figures.

2) Cumulative Distribution Function (C.D.F.): Given thep.d.f. expressions in Theorem 3, the most direct approach ofevaluating the corresponding c.d.f.s is via direct integration.However, this appears to be a difficult and tedious procedure,mainly due to the presence of the modified Bessel functionof the second kind. As such, to obtain simple and tractableexpressions, here we adopt an alternative method based onemploying the conditional m.g.f., as given by the followingtheorem. Note that we restrict our c.d.f. analysis to the casemin (𝑁𝐷, 𝑁𝑅) = 1. Obtaining tractable c.d.f. expressions formore general configurations appears to be a highly challengingtask, and remains an open problem.

Theorem 4: The c.d.f. of the SNR 𝛾 is given for the case

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ

f γ(γ

)

SimulationAnalytical

(a) P.d.f. of 𝛾 for the configuration (2, 5, 1)

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

γ

f γ(γ

)

SimulationAnalytical

(b) P.d.f. of 𝛾 for the configuration (2, 5, 2)

Fig. 1. Comparison of analytical and simulation p.d.f.s. Results are shownfor the Alamouti OSTBC and 𝛾 = 10 dB with 𝑏 = 0 dB.

min (𝑁𝑅, 𝑁𝐷) = 1 by

𝐹 (𝛾) = 1−2 exp

(− 𝛾𝛾𝛼

)Γ (𝑝)

𝑁𝑆−1∑𝑘=0

𝑘∑𝑙=0

(𝑘𝑙

)𝛾𝑘

𝑘!𝛼𝑘𝛾𝑘𝑎2𝑙

×(√

𝛾

𝑎2𝛾𝛼

)𝑝−𝑙𝐾𝑝−𝑙

(2

√𝛾

𝑎2𝛾𝛼

). (18)

Proof: See Appendix D-B.Fig. 2 compares the analytical c.d.f., based on (18), with

Monte-Carlo simulations, for a system with 𝑁𝐷 = 1. Resultsare shown for the Alamouti code, with 𝛾 = 10 dB and𝑏 = 0 dB. Clearly, there is a precise agreement between theanalysis and simulations.

C. Asymptotic Probability Distributions

As discussed above, it appears particularly challenging toobtain exact expressions for the p.d.f. and c.d.f. of the SNR 𝛾for general system configurations. It turns out, however, thatmuch simpler results can be derived for the p.d.f. and c.d.f.when either the number of relay antennas𝑁𝑅 or the relay gain𝑏 grows large. These results are presented in the followingtheorem.

DHARMAWANSA et al.: ANALYTICAL PERFORMANCE OF AMPLIFY-AND-FORWARD MIMO RELAYING WITH ORTHOGONAL SPACE-TIME BLOCK . . . 2151

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ

Fγ(γ

)

SimulationAnalytical

Fig. 2. Comparison of analytical and simulation c.d.f. in the case of (2, 5, 1).Results are shown for Alamouti code and 𝛾 = 10 dB with 𝑏 = 0 dB.

0 2 4 6 8 10 12 140

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

γ

f γ(γ

)

fasy,N

R

(γ)

(2,10,3)(2,8,3)

Fig. 3. Asymptotic behavior of the p.d.f of 𝛾 for large 𝑁𝑅. Results areshown for 𝑅 = 0.5, 𝛾 = 10 dB and 𝑏 = 2𝛾.

Theorem 5: The p.d.f. of the SNR 𝛾 is given, for large 𝑁𝑅(i.e., 𝑁𝑅 → ∞), by

𝑓asy,𝑁𝑅(𝛾) =1

Γ (𝑁𝑆𝑁𝐷)

(1 + 𝑏+ 𝛾

𝛼𝛾𝑏

)𝑁𝑆𝑁𝐷

𝛾𝑁𝑆𝑁𝐷−1

× exp

{−𝛾(𝑏+ 1 + 𝛾

𝛼𝛾𝑏

)}, (19)

and, for large 𝑏 (i.e., 𝑏→ ∞), by

𝑓asy,𝑏(𝛾) =1

Γ (𝑁𝑆𝑞)

𝛾𝑁𝑆𝑞−1

(𝛼𝛾)𝑁𝑆𝑞exp

(− 𝛾

𝛼𝛾

). (20)

Proof: See Appendix D-C.Note that the corresponding asymptotic c.d.f. results can

be derived, trivially, by integrating the respective p.d.f.s. Wealso note that it is possible to derive similar asymptotic resultsfor the cases 𝑁𝑆 → ∞ or 𝑁𝐷 → ∞ by applying a similarderivation method. Interestingly, Theorem 5 shows that aseither the number of relay antennas or the relay gain growslarge, the SNR 𝛾 becomes asymptotically Gamma distributed.

Fig. 3 investigates the asymptotic behavior of the p.d.f. of 𝛾with respect to 𝑁𝑅. The exact p.d.f. curves are presented forthe relaying configurations (2, 8, 3) and (2, 10, 3), generatedbased on (16), in addition to the asymptotic p.d.f. curve based

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

γ

f γ(γ

)

fasy,b

(γ)

b = 20dBb = 30dB

Fig. 4. Asymptotic behavior of the p.d.f of 𝛾 for large 𝑏. Results are shownfor 𝑅 = 1, 𝛾 = 10 dB and the antenna configuration (2, 5, 1).

on (19). The results are shown for full-rate OSTBC with 𝛾 = 5dB and 𝑏 = 2𝛾. The accuracy of the asymptotic expression(19) is clearly seen. Fig. 4 investigates the asymptotic behaviorof the p.d.f. of 𝛾 with respect to 𝑏. The exact p.d.f. curves arepresented for 𝑏 = 20 dB and 𝑏 = 30 dB, generated basedon (16), in addition to the asymptotic p.d.f. curve based on(19). We clearly see that as the relay gain 𝑏 grows large,the p.d.f. of 𝛾 converges to the asymptotic distribution. Theresults in Figs. 3 and 4 suggest that the simple approximationspresented in Theorem 5 are particularly useful since, althoughformally defined in asymptotic regimes, they are quite accuratefor system parameters of practical interest.

IV. PERFORMANCE MEASURES

Here we investigate the performance of the proposed OS-TBC MIMO relaying approach with the help of the statisticalresults developed in the previous section.

A. Exact and Approximate SER of MPSK

The exact SER of MPSK is given by4 [30]

𝑃MPSK =1

𝜋

∫ Θ

0

ℳ𝛾

(𝑔MPSK

sin2 𝜃

)𝑑𝜃 (21)

where Θ = 𝜋(𝑀 − 1)/𝑀 and 𝑔MPSK = sin2(𝜋/𝑀). Substi-tuting (11) into (21) gives an exact expression for the SER.In the most general case, evaluating the finite-range integralin closed-form seems intractable; however, the expressioncan be easily computed more efficiently than via Monte-Carlo simulations. Alternatively, following [31], we can obtainan approximation for the SER by substituting (11) into thefollowing expression

𝑃MPSK ≈(

Θ

2𝜋− 1

6

)ℳ𝛾 (𝑔MPSK) +

1

4ℳ𝛾

(4𝑔MPSK

3

)

+

(Θ

2𝜋− 1

4

)ℳ𝛾

(𝑔MPSK

sin2 Θ

). (22)

Fig. 5 presents analytical and simulated SER curves for QPSKmodulation. Results are shown for the Alamouti code and rate

4Similar m.g.f.-based expressions are also available for other modulationformats, e.g., QAM.

2152 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 7, JULY 2010

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 5 10 15 20 25 30

SNR (dB)

SER

Analytical (Exact)Analytical (Approx.)SimulationCoherent (Simu.)

(2,1,1)

(2,2,1)

(2,3,2)

(4,5,3)

(4,6,5)

Fig. 5. SER of QPSK versus average SNR 𝛾; comparison of analysisand simulations. Results are shown for the Alamouti OSTBC, and rate 1/2orthogonal design, for different relay and receiver antenna configurations(𝑁𝑆 , 𝑁𝑅, 𝑁𝐷), and with 𝑏 = 𝛾.

1/2 orthogonal design, with different antenna configurations.The “Analytical (Exact)” curves are generated by numericallysolving the definite-integral expression (21), and are seen tomatch precisely with the simulations. The “Analytical (Ap-prox.)” curves are generated based on the expression obtainedby substituting (11) into (22), and are seen to be very accurate.The “Coherent (Simu.)” curves are simulated SER resultsbased on the coherent relay model given in [24]. We seethat for low to moderate SNR values both non-coherent andcoherent relaying exhibit similar performance, whereas at highSNR coherent relaying exhibits a performance gain. A similarobservation has been made in [14], which considered dualhop single input single output relaying. Thus, it is clear thatour analytical performance results for non-coherent relayingalso give good approximations for coherent relaying, whichotherwise would appear to be analytically intractable.

B. Exact Closed-Form BER of BPSK

For BPSK, the exact SER coincides with the bit error rate(BER), and (21) becomes

𝑃BPSK =1

𝜋

∫ 𝜋/20

ℳ𝛾

(1

sin2 𝜃

)𝑑𝜃. (23)

In this case, for system with min (𝑁𝑅, 𝑁𝐷) = 1, a closed-form BER solution can be obtained, as we now show. Westart by applying the integral form of the m.g.f. of 𝛾, givenby (33) in Appendix A, to arrive at

𝑃BPSK =1

Γ (𝑝)

∫ ∞

0

𝜆𝑝−1 exp(−𝜆)

× 1

𝜋

∫ 𝜋/20

𝑑𝜃(1 + 𝑎2𝛼𝛾𝜆/ (1 + 𝑎2𝜆) sin2 𝜃

)𝑁𝑆𝑑𝜆 .

(24)

Following [30], the inner integral can be solved to give

1

𝜋

∫ 𝜋/20

𝑑𝜃(1 + 𝑎2𝛼𝛾𝜆/ (1 + 𝑎2𝜆) sin2 𝜃

)𝑁𝑆=

1

2

(1−

𝑁𝑆−1∑𝑘=0

(2𝑘𝑘

)√𝑎2𝛼𝛾𝜆

(1 + 𝑎2𝜆

)𝑘4𝑘 (1 + 𝑎2 (1 + 𝛼𝛾) 𝜆)

𝑘+1/2

). (25)

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 5 10 15 20 25

SNR (dB)

BE

R

Analytical

Analytical (High SNR)

Simulation

(2,3,1)

(4,2,1)

(2,2,2)

Fig. 6. BER of BPSK versus average SNR 𝛾; comparison of analysis andsimulations with the asymptotic high SNR. Results are shown for differentrelay and destination antenna configurations (𝑁𝑆 , 𝑁𝑅, 𝑁𝐷), and with 𝑏 = 𝛾.

Now, substituting (25) back into (24), followed by the use ofthe binomial expansion yields

𝑃BPSK =1

2

(1− 1

Γ (𝑝)

𝑁𝑆−1∑𝑘=0

𝑘∑𝑙=0

(2𝑘𝑘

)(𝑘𝑙

)𝑎2𝑙+1

√𝛼𝛾

4𝑘

×∫ ∞

0

𝜆𝑝+𝑙−1/2

(1 + 𝑎2 (1 + 𝛼𝛾)𝜆)𝑘+1/2exp(−𝜆)𝑑𝜆

).

Finally, using [32, Eq. (3.383.5)], we obtain the BER inclosed-form as follows

𝑃BPSK=1

2

(1− 1

Γ (𝑝)

𝑁𝑆−1∑𝑘=0

𝑘∑𝑙=0

(2𝑘𝑘

)(𝑘𝑙

)Γ (𝑝+𝑙+1/2)

√𝛼𝛾

4𝑘𝑎2𝑝 (1 + 𝛼𝛾)𝑝+𝑙+1/2

× 𝑈(𝑝+ 𝑙 + 1/2; 𝑝+ 𝑙 − 𝑘 − 1;

1

𝑎2(1 + 𝛼𝛾)

)).

(26)

Fig. 6 presents analytical and simulated BER curves forBPSK modulation. Results are shown for different antennaconfigurations, employing the real full-rate OSTBC designs in[26]. The “Analytical” curves for the relaying configurations(2, 3, 1) and (4, 2, 1) were generated based on the closed-form expression (26), whereas the corresponding curve for theconfiguration (2, 2, 2) was generated by numerically solvingthe definite-integral expression (23). As expected, a perfectmatch is observed between the analysis and simulations.

C. High SNR Analysis of SER: Diversity Order and ArrayGain

Here we investigate the high SNR SER performance ofthe proposed OSTBC MIMO relaying approach. In the highSNR regime, the two key parameters which dictate the systemperformance are the diversity order and array gain. Classically,the diversity order is defined as the slope of the SER versusSNR curve (plotted on log-log scale), as SNR grows large.However, for the proposed relaying approach, when evaluatingthe diversity order, care must be taken in interpreting themeaning of SNR, since there are multiple power gains appliedin the system; i.e., at both the source and the relay. Inparticular, if the relay gain 𝑏 is kept fixed and the source

DHARMAWANSA et al.: ANALYTICAL PERFORMANCE OF AMPLIFY-AND-FORWARD MIMO RELAYING WITH ORTHOGONAL SPACE-TIME BLOCK . . . 2153

power 𝜌 is made large, then the average SNR 𝛾 (= 𝜌) alsobecomes large; i.e., we are operating in the “high SNR”regime. However, in this high SNR regime, the instantaneousreceived SNR in (10) is seen to approach a finite limit givenby

lim𝛾→∞

𝛾 = lim𝛾→∞

𝛼𝑏𝛾

𝑁𝑅(1 + 𝛾)Tr(H𝐻1 H𝐻2 K−1H2H1

)=𝑏𝛼

𝑁𝑅Tr(H𝐻1 H𝐻2 H2H1

)<∞ .

As such, if the relay power gain is kept fixed, whilst the sourcepower is made large, we expect the SER to converge to afinite value at high SNR (i.e., an error floor), and as such, thediversity order is zero.

The alternative “high SNR” scenario is if the relay gain𝑏 were to scale in proportion to the source power 𝜌. In thiscase, we would expect the diversity order to be non-zero. Toexamine this scenario in more detail, let us define 𝑏 = 𝛽𝜌,where 𝛽 is a fixed constant. For sufficiently high 𝛾 (= 𝜌), theinstantaneous received SNR (10) then becomes 𝛾 ≈ 𝛾𝑦 , where𝑦 = 𝛼��2Tr

(H𝐻1 H𝐻2 K−1H2H1

), K = ��2HH𝐻 + I𝑁𝐷 , and

��2 = 𝛽/𝑁𝑅.In what follows, we will derive the diversity order and array

gain by focusing on the BER with BPSK modulation. For highSNR, the average BER of BPSK can be well approximatedwith 𝑃ASY

BPSK ≈ 𝐺𝑎/𝛾𝑑𝑣 , where 𝑑𝑣 is the diversity order and𝐺𝑎 is the array gain. These two parameters are directly relatedto the asymptotic expansion of the m.g.f. of 𝑦, i.e., ℳ𝑌 (𝑠),for large 𝑠. Specifically, if, as 𝑠→ ∞, the m.g.f. of 𝑦 can beapproximated as

ℳ𝑌 (𝑠) ≈ 𝑐

𝑠𝑑(27)

for some constant 𝑐, then 𝑑𝑣 = 𝑑 defines the diversity order,and the array gain 𝐺𝑎 is given by

𝐺𝑎 =𝑐Γ(𝑑+ 1

2

)2√𝜋Γ (𝑑+ 1)

. (28)

Evaluating such an asymptotic expansion for the m.g.f. of 𝑦for arbitrary system configurations (i.e., arbitrary combinationsof 𝑁𝑆 , 𝑁𝑅, and 𝑁𝐷) appears prohibitively complicated. Forsystems with 𝑞 = 1 or 𝑞 = 2, however, tractable expressionscan be obtained, which clearly then directly lead to thediversity order and array gain. These results are presented inthe following theorem.

Theorem 6: The diversity order and array gain of theproposed MIMO OSTBC relaying system are given formin (𝑁𝑅, 𝑁𝐷) = 1 by

𝑑𝑣 = 𝑝 , 𝐺𝑎 =Γ(𝑁𝑆−𝑝)Γ(𝑝+ 1

2 )2√𝜋(��2𝛼)𝑝Γ(𝑁𝑆)Γ(𝑝+1)

if 𝑝 < 𝑁𝑆

𝑑𝑣 = 𝑁𝑆 , 𝐺𝑎 =Γ(𝑁𝑆+ 1

2 )𝑐12√𝜋𝛼𝑁𝑆Γ(𝑝)Γ(𝑁𝑆+1)

if 𝑝 > 𝑁𝑆 ,

and for min (𝑁𝑅, 𝑁𝐷) = 2 by

𝑑𝑣 = 2𝑝 , 𝐺𝑎1 if 𝑁𝑆 = 𝑝+ 1𝑑𝑣 = 2𝑝 , 𝐺𝑎2 if 𝑁𝑆 > 𝑝+ 1𝑑𝑣 = 2𝑁𝑆 − 1 , 𝐺𝑎3 if 𝑁𝑆 = 𝑝𝑑𝑣 = 2𝑁𝑆 , 𝐺𝑎4 if 𝑁𝑆 < 𝑝− 1

where

𝐺𝑎1 =Γ(2𝑝+ 1

2

)2√𝜋𝑝2 (��2𝛼)

2𝑝Γ(𝑝)Γ(𝑝− 1)Γ(2𝑝+ 1)

𝐺𝑎2 =Γ(2𝑝+ 1

2

)𝑐2

2√𝜋 (��2𝛼)2𝑝 Γ2 (𝑁𝑆) Γ(𝑝)Γ(𝑝− 1)Γ(2𝑝+ 1)

𝐺𝑎3 =𝑁𝑆Γ

(2𝑁𝑆 − 1

2

)∑𝑁𝑆

𝑙=0

(𝑁𝑆

𝑙

)Γ (𝑁𝑆 − 𝑙 + 1) ��−2𝑙

2√𝜋𝛼2𝑁𝑆−1��2(𝑁𝑆−1)Γ(𝑝)Γ(𝑝− 1)Γ (2𝑁𝑆)

𝐺𝑎4 =Γ(2𝑁𝑆 + 1

2

)𝑐3

2√𝜋𝛼2𝑁𝑆Γ(𝑝)Γ(𝑝− 1)Γ (2𝑁𝑆 + 1)

𝑐1 =

𝑁𝑆∑𝑙=0

(𝑁𝑆

𝑙

)Γ(𝑝− 𝑙)𝑎2𝑙

𝑐2 = Γ(𝑝+ 1)Γ(𝑝− 1)Γ (𝑁𝑆 − 𝑝+ 1)Γ (𝑁𝑆 − 𝑝− 1)

− Γ2(𝑝)Γ2 (𝑁𝑆 − 𝑝)

𝑐3 =

𝑁𝑆∑𝑙=0

𝑁𝑆∑𝑚=0

(𝑁𝑆

𝑙

)��2(𝑙+𝑚)

(𝑁𝑆𝑚

)Γ(𝑝− 𝑙 − 1)Γ (𝑝+ 1 −𝑚)

−(𝑁𝑆∑𝑙=0

(𝑁𝑆

𝑙

)��2𝑙

Γ(𝑝− 𝑙))2

.

Proof: See Appendix E.Recalling that 𝑝 = max(𝑁𝑅, 𝑁𝐷) and 𝑞 = min(𝑁𝑅, 𝑁𝐷),

these results show that the diversity order achieved by theMIMO OSTBC relaying system under consideration is given

by min

(𝑁𝑅𝑁𝐷, 𝑁𝑆 min (𝑁𝑅, 𝑁𝐷)

). It is important to note

that although this conclusion is made based on the specificcases 𝑞 = 1 and 𝑞 = 2, our numerical results have confirmedthe accuracy of this claim more for general configurations.We also have the important result that this diversity ordercorresponds to the maximum achievable diversity order of theMIMO AF relay channel, derived in [24, Eq. (22)]. Fig. 6depicts the high SNR SER approximations, based on Theorem6, for different (𝑁𝑆 , 𝑁𝑅, 𝑁𝐷) configurations. The array gainand diversity order predicted by our analysis are clearly seento be accurate.

D. Outage Probability

The outage probability is an important measure in deter-mining quality of service. It is defined as the probabilitythat the received SNR drops below a predefined threshold𝛾th; i.e., 𝑃out (𝛾th) = Pr (𝛾 ≤ 𝛾th) = 𝐹 (𝛾th). Thus, for thecase min(𝑁𝑅, 𝑁𝐷) = 1, the outage probability is obtained inclosed-form directly from (18). Fig. 7 presents analytical andsimulated outage probability curves for the Alamouti OSTBC,with different antenna configurations. The analytical curveswere generated based on (18), and are seen to match preciselywith the simulations. Moreover, as expected, the figure demon-strates that increasing the number of relay antennas leads toan improved performance through a lower outage probability.

E. Amount of Fading (AoF)

The AoF is an important performance measure whichquantifies the severity of the fading channel. For our system,using the definition from [30], we obtain a closed-form

2154 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 7, JULY 2010

−5 0 5 10 1510

−2

10−1

100

γth

(dB)

F(γ

th)

AnalyticalSimulation

(2,5,1)

(2,1,1)

Fig. 7. Outage probability of dual hop MIMO Rayleigh channel with OSTBCversus the SNR threshold 𝛾tℎ; comparison of analysis and simulations. Resultsare shown for the Alamouti OSTBC, for different relay antenna configurations(𝑁𝑆 , 𝑁𝑅, 𝑁𝐷), and with 𝛾 = 𝑏 = 10 d𝐵.

0

0.3

0.6

0.9

1.2

0 5 10 15 20 25 30b (dB)

AoF

Analytical

Simulation

(2,2,1)

(2,5,1)

(2,8,1)

(2,2,2)(2,5,2)

(2,8,2)

Fig. 8. AoF versus 𝑏 (in dB) for different antenna configurations(𝑁𝑆 , 𝑁𝑅, 𝑁𝐷); comparison of analysis and simulations. Results are shownfor the Alamouti OSTBC, and average SNR 𝛾 = 10 dB.

expression for the AoF by substituting (14) and (15) intoAoF =

(𝑚2 −𝑚2

1

)/𝑚2

1 . Fig. 8 compares the analytical AoFcurves and their simulated counterparts for different antennaconfigurations. Once again, results are shown for the Alamouticode. We see that the AoF, being the normalized varianceof the instantaneous SNR, reduces when either 𝑁𝑅 or 𝑁𝐷increases. This reduction in effective channel fluctuations, or“channel-hardening” phenomenon, is due to the additionalspatial diversity provided by the extra antennas. Furthermore,for all curves, we see an asymptotic floor as the relay gain 𝑏becomes large.

V. CONCLUSION

This paper has investigated OSTBC transmission over aMIMO dual-hop Rayleigh fading channel with non-coherentAF relaying. By employing tools from finite-dimensional ran-dom matrix theory, we have investigated the statistical proper-ties of the instantaneous SNR at the destination. In particular,we have presented new exact closed-form expressions for them.g.f., higher-order cumulants, and first and second moments.These statistical measures reveal an interesting reciprocityrelationship between the relay and destination antennas. We

have also investigated the p.d.f. and c.d.f. of the SNR, andhave shown that when either the number of relay antennas orthe relay power amplification factor becomes large, the SNRbehaves as a gamma random variable. Moreover, numericalresults have demonstrated that this gamma distribution is quiteaccurate for various practical non-asymptotic scenarios. Wehave also studied the performance of the scheme in terms ofthe SER, outage probability, and AoF, as well as the diversityorder and array gain. This analysis reveals that the proposedscheme can achieve the maximum diversity order of the non-coherent AF MIMO relay channel.

APPENDIX A: PROOF OF THEOREM 1

By definition, the m.g.f. of 𝛾 is given by

ℳ𝛾(𝑠) = 𝐸𝛾 {exp(−𝑠𝛾)}= 𝐸H1,H2

{exp(−𝛼𝛾𝑎2𝑠Tr

(H𝐻1 H𝐻2 K−1H2H1

))}= 𝐸H2

{ℳ𝛾∣H2(𝑠)}.

Since H1 has a matrix-variate complex Gaussian distribution,ℳ𝛾∣H2

(𝑠) is obtained from well-known results as [30]

ℳ𝛾∣H2(𝑠) =

1

det(I𝑁𝑅 + 𝑎2𝛾𝛼𝑠H𝐻2 K−1H2

)𝑁𝑆. (29)

The major challenge is to integrate ℳ𝛾∣H2(𝑠) with respect

to H2. To proceed, we use the singular value decomposition(SVD) to express H2 = UΣV𝐻 , where U ∈ ℂ

𝑁𝐷×𝑁𝐷 , andV ∈ ℂ𝑁𝑅×𝑁𝑅 are unitary matrices, and Σ ∈ ℝ+𝑁𝐷×𝑁𝑅 isa diagonal matrix with singular values 𝜎1 > 𝜎2 > . . . > 𝜎𝑞along its main diagonal. Substituting (6) into (29) and usingthe SVD factorization of H2 yields

ℳ𝛾∣ΣΣ𝑇 (𝑠) =1

det(I𝑁𝐷+𝛼𝛾𝑎

2𝑠ΣΣ𝑇 (I𝑁𝐷+𝑎2ΣΣ𝑇 )

−1)𝑁𝑆

=

𝑞∏𝑖=1

⎛⎝ 1

1 +𝛼𝛾𝑎2𝑠𝜎2𝑖1+𝑎2𝜎2𝑖

⎞⎠𝑁𝑆

. (30)

At this point, it is convenient to deal with the eigenvaluesrather than the singular values. Thus, (30) can be rewritten as

ℳ𝛾∣Λ(𝑠) =

𝑞∏𝑖=1

(1

1 + 𝛼𝛾𝑎2𝑠𝜆𝑖1+𝑎2𝜆𝑖

)𝑁𝑆

(31)

where 𝜆𝑖 = 𝜎2𝑖 , 𝑖 = 1, 2, . . . , 𝑞, are the non-zero orderedeigenvalues of H𝐻2 H2, and Λ = diag (𝜆1, 𝜆2, . . . , 𝜆𝑞). Thejoint distribution of the 𝜆𝑖’s can be written as [33, Eq. (2.22)]

ℎ(Λ) = 𝒦−1

𝑞∏𝑖<𝑗

(𝜆𝑗 − 𝜆𝑖)2𝑞∏𝑘=1

𝜆𝑝−𝑞𝑘 exp (−𝜆𝑘) (32)

where 𝜆1 > . . . > 𝜆𝑞 > 0 and 𝒦 =∏𝑞𝑖=1 Γ(𝑝− 𝑖 + 1)Γ(𝑞 −

𝑖+1). The m.g.f. of 𝛾 can now be obtained by averaging (31)with respect to (32) to yield

ℳ𝛾(𝑠) = 𝒦−1

∫. . .

∫𝒟

𝑞∏𝑖<𝑗

(𝜆𝑗 − 𝜆𝑖)2

×𝑞∏𝑘=1

𝜆𝑝−𝑞𝑘 exp(−𝜆𝑘)(

1

1 + 𝛼𝛾𝑎2𝑠𝜆𝑘1+𝑎2𝜆𝑘

)𝑁𝑆

𝑑Λ

(33)

DHARMAWANSA et al.: ANALYTICAL PERFORMANCE OF AMPLIFY-AND-FORWARD MIMO RELAYING WITH ORTHOGONAL SPACE-TIME BLOCK . . . 2155

where 𝒟 = {𝜆1 ≥ 𝜆2 ≥ . . . ≥ 𝜆𝑞 > 0}. The multiple integralover the domain 𝒟 can be solved using [28, Corollary 2] toobtain (11), with

𝐼𝑖𝑗(𝑠) =

∫ ∞

0

𝜆𝜈𝑖𝑗−1

(1 + 𝑎2𝜆

1 + 𝑎2(1 + 𝛼𝛾𝑠)𝜆

)𝑁𝑠

exp(−𝜆)𝑑𝜆 .

The result (12) is now obtained by applying some algebraicmanipulations, and integrating using [32, Eq. (3.383.5)].

APPENDIX B: PROOF OF THEOREM 2

By definition, the 𝑛th cumulant of 𝛾 is given by

𝜇𝑛 = (−1)𝑛𝑑𝑛

𝑑𝑠𝑛lnℳ𝛾(𝑠)

∣∣∣∣𝑠=0

. (34)

Now (34) can be written as

𝜇𝑛 = (𝑛− 1)!𝑁𝑆(𝛼𝛾𝑎2

)𝑛𝐸H2

{Tr(H𝐻2 K−1H2

)𝑛}(35)

which, after applying the SVD, becomes

𝜇𝑛 = (𝑛− 1)!𝑁𝑆(𝛼𝛾𝑎2

)𝑛𝐸Λ

{𝑞∑𝑙=1

(𝜆𝑙

1 + 𝑎2𝜆𝑙

)𝑛}. (36)

The remaining expectation can be written in integral form as

𝜇𝑛 = 𝒦−1(𝑛− 1)!𝑁𝑆(𝛼𝛾𝑎2

)𝑛 ∫. . .

∫𝒟

𝑞∏𝑖<𝑗

(𝜆𝑗 − 𝜆𝑖)2

×𝑞∏𝑙=1

𝜆𝑝−𝑞𝑙 exp(−𝜆𝑙)𝑞∑𝑙=1

(𝜆𝑙

1 + 𝑎2𝜆𝑙

)𝑛𝑑Λ .

Finally, integrating using [28, Theorem 3], we arrive at (13).

APPENDIX C: PROOF OF COROLLARY 2

By definition, the second moment of 𝛾 can be written as

𝑚2 =(𝛼𝛾𝑎2

)2𝐸H2H1

{Tr2(H𝐻1 H𝐻2 K−1H2H1

)}. (37)

We can express this in terms of the second cumulant of 𝛾 as

𝑚2 = 𝜇2 +𝑁2𝑆

(𝛼𝛾𝑎2

)2𝐸H2

{Tr2(H𝐻2 K−1H2

)}. (38)

Now, using the SVD, we obtain

𝑚2 = 𝜇2 +𝑁2𝑆

(𝛼𝛾𝑎2

)2𝐸Λ

⎧⎨⎩(𝑞∑𝑖=1

𝜆𝑖1 + 𝑎2𝜆𝑖

)2⎫⎬⎭

= 𝜇2 +𝑁2𝑆

(𝛼𝛾𝑎2

)2𝐸Λ

{𝑞∑𝑖=1

(𝜆𝑖

1 + 𝑎2𝜆𝑖

)2}

+𝑁2𝑆

(𝛼𝛾𝑎2

)2 𝑞∑𝑖,𝑗=1𝑖∕=𝑗

𝐸Λ

{𝜆𝑖𝜆𝑗

(1 + 𝑎2𝜆𝑖) (1 + 𝑎2𝜆𝑗)

}.

(39)

Evaluating the second term with the help of (36), and exploit-ing the symmetry of the third term, we can simplify (39) asfollows

𝑚2 = 𝜇2 (𝑁𝑆 + 1) +𝑁2𝑆

(𝛼𝛾𝑎2

)2 (𝑞2 − 𝑞)

× 𝐸𝜆,𝜔{

𝜆𝜔

(1 + 𝑎2𝜆) (1 + 𝑎2𝜔)

}(40)

where (𝜆, 𝜔) is a pair of eigenvalues, randomly selectedfrom 𝜆1, . . . , 𝜆𝑞 . Let us assume, for the sake of notationalconvenience, that (𝜆, 𝜔) ≡ (𝜆1, 𝜆2). Now, the remainingexpectation can be solved by employing the unordered eigen-value distribution of a complex central Wishart matrix [33] toyield

𝑚2 = 𝜇2 (𝑁𝑆 + 1) +𝒦𝑞!𝑁2𝑆

(𝛼𝛾𝑎2

)2 (𝑞2 − 𝑞)

×∫ ∞

0

. . .

∫ ∞

0

𝑞∏𝑖<𝑗

(𝜆𝑗 − 𝜆𝑖)2𝑞∏𝑘=1

𝜉𝑘 (𝜆𝑘) 𝑑Λ

where

𝜉𝑘 (𝜆𝑘) =

{𝜆𝑝−𝑞+1𝑘

(1+𝑎2𝜆𝑘)exp (−𝜆𝑘) if 𝑘 = 1, 2

𝜆𝑝−𝑞𝑘 exp (−𝜆𝑘) if 𝑘 ∕= 1, 2 .

Finally, we use the multi-dimensional integration formula [28,Theorem 2] to obtain (15).

APPENDIX D: PROOF OF THEOREMS 3, 4 AND 5

A. Proof of Theorem 3

∙ Proof of (16): We start by inverting the conditional m.g.f.(31), to obtain the conditional p.d.f.

𝑓𝛾∣Λ(𝛾) = ℒ−1

⎧⎨⎩(

1

1 + 𝛼𝛾𝑎2𝜆𝑠1+𝑎2𝜆

)𝑁𝑆

⎫⎬⎭

=𝛾𝑁𝑆−1 exp

(− 𝛾𝛼𝛾

)Γ (𝑁𝑆) (𝑎2𝛾𝛼)

𝑁𝑆

(1 + 𝑎2𝜆

)𝑁𝑆

𝜆𝑁𝑆

× exp

(− 𝛾

𝑎2𝛼𝛾𝜆

)(41)

where ℒ−1 denotes the inverse Laplace transform opera-tor. Averaging (41) with respect to the density of 𝜆, givenby ℎ(𝜆) = 1

Γ(𝑝)𝜆𝑝−1 exp(−𝜆), yields

𝑓𝛾(𝛾) =𝛾𝑁𝑆−1 exp

(− 𝛾𝛼𝛾

)(𝑎2𝛾𝛼)

𝑁𝑆 Γ (𝑁𝑆) Γ (𝑝)

𝑁𝑆∑𝑗=0

(𝑁𝑆𝑗

)𝑎2𝑗

×∫ ∞

0

𝜆𝑗+𝑝−𝑁𝑆−1 exp

(−𝜆− 𝛾

𝑎2𝛾𝛼𝜆

)𝑑𝜆.

Solving the remaining integral using [32, Eq. (3.478.4)]leads to (16).

∙ Proof of (17): Again, we start by inverting the conditionalm.g.f. (31), which in this case yields the followingconditional p.d.f. 𝑓𝛾∣Λ(𝛾) = 𝑔(𝛾, 𝜆1) ∗ 𝑔(𝛾, 𝜆2) , where∗ denotes the convolution operation and

𝑔(𝛾, 𝜆)=

(1+𝑎2𝜆

𝑎2𝛼𝛾𝜆

)𝑁𝑆 𝛾𝑁𝑆−1

Γ (𝑁𝑆)exp

{− 𝛾

𝛼𝛾

(1+

1

𝑎2𝜆

)}.

The convolution is evaluated as

𝑓𝛾∣Λ(𝛾) =2∏𝑖=1

(1 + 𝑎2𝜆𝑖𝑎2𝛼𝛾𝜆𝑖

)𝑁𝑆 exp{− 𝛾𝛼𝛾

(1 + 1

𝑎2𝜆1

)}Γ2 (𝑁𝑆)

×∫ 𝛾0

𝑥𝑁𝑆−1(𝛾 − 𝑥)𝑁𝑆−1 exp

{−𝑥(𝜆1 − 𝜆2𝑎2𝛼𝛾𝜆1𝜆2

)}𝑑𝑥,

2156 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 7, JULY 2010

which can be solved using [34, Eq. (2.3.6.2)] to yield(42) at the bottom. It should be noted that (42) is welldefined ∀𝜆1, 𝜆2 ∈ [0,∞). Now, in order to obtain thep.d.f., we need to integrate (42) with respect to theunordered joint eigenvalue distribution5 ℎ (𝜆1, 𝜆2) =(𝒦−1

2 /2)𝜆𝑝−21 𝜆𝑝−2

2 (𝜆1 − 𝜆2)2 exp(−𝜆1 − 𝜆2) , where𝒦2 = Γ (𝑝) Γ (𝑝− 1). However, the resultant double inte-gral has no known closed form solution. Thus, to proceed,we express the Bessel function with its equivalent seriesexpansion and apply some algebraic manipulations toyield (43) at the bottom, where 𝜁1, 𝜁2 and 𝜁3 are definedin (17). The remaining integrals are solved using [32, Eq.(3.478.4)] to yield the final result.

B. Proof of Theorem 4

By definition, the c.d.f. can be written as

𝐹 (𝛾) =

∫ 𝛾0

𝑓𝛾(𝑥)𝑑𝑥 =

∫Λ

ℎ(Λ)

∫ 𝛾0

𝑓𝛾∣Λ (𝑥∣Λ) 𝑑𝑥𝑑Λ ,

(44)

which, in turn, may be expressed in terms of the conditionalm.g.f. of 𝛾 as

𝐹 (𝛾) =

∫Λ

ℒ−1

(ℳ𝛾∣Λ(𝑠)

𝑠

)ℎ(Λ)𝑑Λ . (45)

For the case min (𝑁𝑅, 𝑁𝐷) = 1, we may use (31) to write

ℒ−1

(ℳ𝛾∣Λ(𝑠)

𝑠

)= ℒ−1

⎧⎨⎩

1

𝑠(1 + 𝛼𝛾𝑎2𝜆𝑠

1+𝑎2𝜆

)𝑁𝑆

⎫⎬⎭

=1

Γ (𝑁𝑆)

(1 + 𝑎2𝜆

𝑎2𝛾𝛼𝜆

)𝑁𝑆 ∫ 𝛾0

𝑥𝑁𝑆−1

× exp

{−𝑥(

1

𝛾𝛼+

1

𝑎2𝛾𝛼𝜆

)}𝑑𝑥

= 1− exp

{− 𝛾

𝛼𝛾

(1 +

1

𝑎2𝜆

)}

×𝑁𝑆−1∑𝑘=0

𝛾𝑘

𝑘!𝛼𝑘𝛾𝑘

(1+

1

𝑎2𝜆

)𝑘. (46)

5This can be obtained from the ordered eigenvalue p.d.f. in (32), bysubstituting 𝑞 = 2, and applying a normalizing factor of 2.

Substituting (46) into (45), noting that ℎ(𝜆) = 𝜆𝑝−1

Γ(𝑝) exp (−𝜆),and applying some algebra yields

𝐹 (𝛾) = 1−exp(− 𝛾𝛾𝛼

)Γ (𝑝)

𝑁𝑆−1∑𝑘=0

𝑘∑𝑙=0

(𝑘𝑙

)𝛾𝑘

𝑘!𝛼𝑘𝛾𝑘𝑎2𝑙

×∫ ∞

0

𝜆𝑝−𝑙−1 exp

(−𝜆− 𝛾

𝑎2𝛼𝛾𝜆

)𝑑𝜆 .

Finally, we use [32, Eq. (3.478.4)] to arrive at (18).

C. Proof of Theorem 5

To calculate (19), we start by rewriting the conditionalm.g.f. (30) as follows

ℳ𝛾∣Λ(𝑠) =

𝑞∏𝑖=1

⎛⎝ 1

1 +𝛼𝛾𝑏𝑠𝜎2𝑖 /𝑁𝑅(1+𝛾)

1+𝑏𝜎2𝑖 /𝑁𝑅(1+𝛾)

⎞⎠𝑁𝑆

. (47)

Now, noting that as 𝑁𝑅 → ∞, by the law of large numbers,we have

lim𝑁𝑅→∞

H2H𝐻2

𝑁𝑅= I𝑁𝐷

which implies that

lim𝑁𝑅→∞

𝜎2𝑖𝑁𝑅

= 1, 𝑖 = 1, 2, . . . , 𝑁𝐷 . (48)

Applying (48) in (47) gives

ℳasy,𝑁𝑅,𝛾∣Λ(𝑠) =1(

1 + 𝛼𝛾𝑏𝑠𝑏+1+𝛾

)𝑁𝑆𝑁𝐷(49)

where we have used the fact that 𝑞 = 𝑁𝐷. The result (19)follows by calculating the inverse Laplace transformation of(49).

To derive (20), we again start with the conditional m.g.f.(30). In this case, noting that as 𝑏→ ∞ then 𝑎2 → ∞, we canwrite (30) as ℳasy,𝑏,𝛾∣Λ(𝑠) = 1/(1 + 𝛼𝛾𝑠)

𝑁𝑆𝑞 , the Laplaceinversion of which gives (20).

APPENDIX E: PROOF OF THEOREM 6

To derive the high SNR performance, we require an asymp-totic expansion of the m.g.f. of 𝑦, obtained via (11), in thegeneral form (27). Subsequently, we can then directly extractthe diversity order and array gain.

𝑓𝛾∣Λ(𝛾) =

√𝜋𝛾𝑁𝑆−1/2 exp

(− 𝛾𝛼𝛾

)(𝑎2𝛼𝛾)𝑁𝑆+1/2 Γ (𝑁𝑆)

2∏𝑖=1

(1 + 𝑎2𝜆𝑖𝜆𝑖

)𝑁𝑆(𝜆1𝜆2𝜆1 − 𝜆2

)𝑁𝑆−1/2

exp

{− 𝛾

2𝑎2𝛼𝛾

(1

𝜆1+

1

𝜆2

)}

× 𝐼𝑁𝑆−1/2

[𝛾

2𝑎2𝛼𝛾

(𝜆1 − 𝜆2𝜆1𝜆2

)]. (42)

𝑓𝛾(𝛾) =𝒦−1

2

√𝜋𝛾𝑁𝑆−1/2 exp

(− 𝛾𝛼𝛾

)2 (𝑎2𝛾𝛼)

𝑁𝑆+1/2Γ (𝑁𝑆)

∞∑𝑛=0

𝑁𝑆∑𝑖,𝑗

2𝑛+2∑𝑙=0

(−1)𝑙(𝑁𝑆

𝑖

)(𝑁𝑆

𝑗

)(2𝑛+2𝑙

)𝑛!Γ (𝜁1 + 1)

𝑎2(2𝑁𝑆−𝑖−𝑗)(

𝛾

4𝑎2𝛾𝛼

)𝑛+𝜁1

×2∏𝑖=1

∫ ∞

0

𝜆𝜁𝑖+1−1𝑖 exp

(− 𝛾

2𝑎2𝛾𝛼𝜆𝑖− 𝜆𝑖

)𝑑𝜆𝑖. (43)

DHARMAWANSA et al.: ANALYTICAL PERFORMANCE OF AMPLIFY-AND-FORWARD MIMO RELAYING WITH ORTHOGONAL SPACE-TIME BLOCK . . . 2157

Before going to the proof, it is useful to quote the followingasymptotic properties of the confluent hypergeometric functionof the second kind [35]

𝑈(𝑎; 𝑏; 𝑧) ≈

⎧⎨⎩

Γ(𝑏−1)Γ(𝑎) 𝑧

1−𝑏 if 𝑏 ≥ 2

− 1Γ(𝑎) (ln 𝑧 + 𝜓(𝑎)) if 𝑏 = 1

Γ(1−𝑏)Γ(1+𝑎−𝑏) if 𝑏 ≤ 0

(50)

valid for small 𝑧, where 𝜓(⋅) is the digamma function and𝑎 > 0.

We will present an explicit proof for the result (6), pertain-ing to the case min (𝑁𝑅, 𝑁𝐷) = 1. The corresponding prooffor the result (6), pertaining to the case min (𝑁𝑅, 𝑁𝐷) = 2,follows in a similar way and is therefore omitted.

∙ The case of 𝑝 < 𝑁𝑆: Using (11), the exact m.g.f. of 𝑦 inthis case can be written as

ℳ𝑌 (𝑠) =

∑𝑝−1𝑙=0 Φ(𝑙, 𝑠) + Φ(𝑝, 𝑠) +

∑𝑁𝑆

𝑙=𝑝+1 Φ(𝑙, 𝑠)

(1 + 𝛼𝑠)𝑁𝑆+𝑝

��2𝑝

(51)

where Φ(𝑙, 𝑠) =(𝑁𝑆

𝑙

)(𝛼𝑠)

𝑙𝑈(𝑝; 𝑝+ 1 − 𝑙; 1

��2(1+𝛼𝑠)

).

Taking 𝑠 large, we may apply (50) to obtain the followingapproximations

𝑝−1∑𝑙=0

Φ(𝑙, 𝑠) ≈ (𝛼𝑠)𝑝

Γ(𝑝)

𝑝−1∑𝑙=0

(𝑁𝑆𝑙

)Γ(𝑝− 𝑙)��2(𝑝−𝑙)

Φ(𝑝, 𝑠) ≈ − (𝛼𝑠)𝑝

(𝑁𝑆

𝑝

)Γ(𝑝)

(𝜓(𝑝) − ln

(��2𝛼)− ln 𝑠

)𝑁𝑆∑𝑙=𝑝+1

Φ(𝑙, 𝑠) ≈𝑁𝑆∑𝑙=𝑝+1

(𝑁𝑆𝑙

)Γ(𝑙 − 𝑝)

Γ(𝑙)(𝛼𝑠)

𝑙. (52)

Substituting (52) into (51), then taking 𝑠 large, we pickup the lowest order term of 1

𝑠 to yield

ℳ𝑌 (𝑠) ≈(

Γ (𝑁𝑆 − 𝑝)(��2𝛼)

𝑝Γ (𝑁𝑆)

)1

𝑠𝑝. (53)

The diversity order and array gain now follow from (27)and (28).

∙ The case of 𝑝 > 𝑁𝑆: Using (11), the exact m.g.f. of 𝑦 inthis case can be written as

ℳ𝑌 (𝑠) =

∑𝑁𝑆

𝑙=0

(𝑁𝑆

𝑙

)(𝛼𝑠)

𝑙𝑈(𝑝; 𝑝+ 1 − 𝑙; 1

��2(1+𝛼𝑠)

)(1 + 𝛼𝑠)

𝑁𝑆+𝑝 ��2𝑝.

Taking 𝑠 large, we may apply the properties (50), onceagain, to obtain the following asymptotic approximationfor the m.g.f.

ℳ𝑌 (𝑠) ≈(

1

𝛼𝑁𝑆Γ(𝑝)

𝑁𝑆∑𝑙=0

(𝑁𝑆𝑙

)Γ(𝑝− 𝑙)��2𝑙

)1

𝑠𝑁𝑆.

The diversity order and array gain follow directly from(27) and (28).

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Prathapasinghe Dharmawansa (S’05-M’09) re-ceived the B.Sc. and M.Sc. degrees in electronic andtelecommunication engineering from the Universityof Moratuwa, Moratuwa, Sri Lanka in 2003 and2004 respectively, and the D.Eng. degree in Infor-mation and Communications Technologies from theAsian Institute of Technology, Thailand in 2007.

Currently, he is a Postdoctoral Research Fellowwith the Department of Electronic and ComputerEngineering, Hong Kong University of Science andTechnology, Hong Kong. His research interests are

in MIMO systems, random matrix theory, and multivariate statistics.

Matthew R. McKay (S’03-M’10) received the com-bined B.E. degree in electrical engineering and B.IT.degree in computer science from the QueenslandUniversity of Technology, Australia, in 2002, andthe Ph.D. degree in electrical engineering from theUniversity of Sydney, Australia, in 2006. He thenworked as a Research Scientist at the Common-wealth Science and Industrial Research Organization(CSIRO), Sydney, prior to joining the faculty at theHong Kong University of Science and Technology(HKUST) in 2007, where he is currently an Assis-

tant Professor. He is also a member of the Center for Wireless InformationTechnology at HKUST. His research interests include communications andsignal processing; in particular, the analysis and design of MIMO systems,random matrix theory, information theory, and wireless ad hoc and sensornetworks.

Dr. McKay was awarded a 2006 Best Student Paper Award at IEEEICASSP06, and was jointly awarded the 2006 Best Student Paper Awardat IEEE VTC06-Spring. He was also awarded the University Medal upongraduating from the Queensland University of Technology.

Ranjan K. Mallik (S’88-M’93-SM’02) receivedthe B.Tech. degree from the Indian Institute ofTechnology, Kanpur, in 1987 and the M.S. and Ph.D.degrees from the University of Southern Califor-nia, Los Angeles, in 1988 and 1992, respectively,all in electrical engineering. From August 1992 toNovember 1994, he was a scientist at the DefenceElectronics Research Laboratory, Hyderabad, India,working on missile and EW projects. From Novem-ber 1994 to January 1996, he was a faculty memberof the Department of Electronics and Electrical

Communication Engineering, Indian Institute of Technology, Kharagpur. FromJanuary 1996 to December 1998, he was with the faculty of the Department ofElectronics and Communication Engineering, Indian Institute of Technology,Guwahati. Since December 1998, he has been with the faculty of theDepartment of Electrical Engineering, Indian Institute of Technology, Delhi,where he is currently a Professor. His research interests are in diversitycombining and channel modeling for wireless communications, space-timesystems, cooperative communications, multiple-access systems, differenceequations, and linear algebra.

Dr. Mallik is a member of Eta Kappa Nu. He is also a member ofthe IEEE Communications, Information Theory, and Vehicular TechnologySocieties, the American Mathematical Society, and the International LinearAlgebra Society, a fellow of the Indian National Academy of Engineering,The National Academy of Sciences, India, Allahabad, The Institution ofEngineering and Technology, U.K., and The Institution of Electronics andTelecommunication Engineers, India, a life member of the Indian Societyfor Technical Education, and an associate member of The Institution ofEngineers (India). He is an Area Editor for the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS and an Editor for the IEEE TRANSACTIONS

ON COMMUNICATIONS. He is a recipient of the Hari Om Ashram PreritDr. Vikram Sarabhai Research Award in the field of Electronics, Telematics,Informatics, and Automation, and of the Shanti Swarup Bhatnagar Prize inEngineering Sciences.