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Dual-hop MIMO relaying with OSTBC over doubly-correlated Nakagami-m fading channels

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Abstract

In this article, we investigate the performance of dual-hop amplify-and-forward multiple-input multiple-output relaying system with orthogonal space-time block code transmissions over doubly-correlated Nakagami-m fading channel, where the source, relay, and destination terminals are all equipped with multiple antennas. For two different CSI-assisted relaying schemes, which could be encompassed by a unified model, we provide the compact closed-form expressions for cumulative distribution function, probability density function, moment generating function, and generalized moment (GM) of the instantaneous end-to-end SNR. Besides, the exact analytical expressions for the outage probability (OP) and symbol error rate (SER) and approximate expression for ergodic capacity are also derived. Furthermore, we present the asymptotic expressions for OP and SER in the high SNR regime, from which we gain an insight into the system performance and derive the achievable diversity order and array gain. The analytical expressions are validated by Monte-Carlo simulations.

Introduction

Recently, relaying transmission has attracted great attention due to their considerable advantages over direct transmission, such as extending the coverage, increasing the reliability, and saving the power consumption [13], and has already been discussed as part of the LTE-Advanced study [4]. One of the most common relay protocols is amplify-and-forward (AF) (or non-regenerative) for its low complexity and cost-effectiveness. For dual-hop AF relaying with single-input single-output (SISO), its performance has been well studied over both Rayleigh [2, 3] and Nakagami [5] channels.

Multiple-input multiple-output (MIMO) technology has also received a significant interest in the past decade for its significant increasement in data throughput and link range without additional bandwidth or increased transmit power. Beamforming and space-time block code (STBC) techniques are two emerging technologies that can be employed with multiple antennas to provide diversity.

In [68], the performance of two-hop relay network over correlated Rayleigh fading channels with beamforming [maximum ratio transmission (MRT) at the source and maximal ratio combining (MRC) at the destination] has been analyzed. In [9, 10], the end-to-end performance of dual-hop relaying systems with beamforming over Nakagami-m fading channels has been investigated.

In [1115], orthogonal space-time block code (OSTBC) has been employed in dual-hop relaying system. In [11, 12], the end-to-end performance of dual-hop wireless communication systems employing transmit diversity with OSTBC over independent but not necessarily identically distributed (i.n.i.d.) Rayleigh and Nakagami-m fading channels has been well investigated respectively. Yan and Zhang [13] and Ferdinand et al. [14] extend the single-antenna relay in [11] to multiple-antenna relay over independent Rayleigh and asymmetric correlated Rayleigh-Rician fading channels, respectively. Yan and Zhang [13] have investigated the performance of a wireless relay network, where each of its nodes is equipped with N antennas and the same OSTBC scheme is adopted at its source and relay nodes for transmission. In [14], the performance of MIMO AF relay network with OSTBC over Rayleigh-Rician asymmetric channel with antenna correlation has been analyzed. Duong et al. [15] extended i.n.i.d. Nakagami-m fading channels to arbitrarily correlated and not necessarily identically distributed (c.n.i.d.) Nakagami-m fading channels by transforming the sum of c.n.i.d. Gamma random variables (RVs) to a sum of independent Gamma RVs, where the relay is equipped with a single antenna.

Antenna correlation occurs in many practical scenarios due to the limited antenna separation or the lack of local scatters [16]. Louie et al. [68, 15] only considered the one-sided correlated fading channels, where there was antenna correlation at either the transmitter or receiver. For a more general case of correlated fading at both the transmitter and receiver, which is referred to as doubly-correlated MIMO channels [17], it is important to quantify this correlation effect. In this article, we analyze the performance of two channel state information (CSI) assisted AF MIMO relaying systems over doubly-correlated Nakagami-m fading channels by building up the multiple-dimensional Gamma distributed correlated vector with order parameter m based on the sum of squared Gaussians [18], where all the source, relay, and destination terminals are equipped with multiple antennas and the channel fading at both transmission and reception ends is correlated with arbitrary correlation matrices. The source and relay terminals employ the OSTBC transmissions, which are not necessarily the same. It is noteworthy that it’s easy to derive the performance of the dual-hop AF MIMO relaying system with OSTBC transmissions over doubly-correlated Rayleigh fading channels based on our analysis results, since the Nakagami-m distribution equals the Rayleigh distribution for m=1. We assume that the instantaneous CSI is available at the relay terminal.

The rest of the article is organized as follows. The system and channel models studied are presented in Section “System and channel models”. In Section “Metrics of end-to-end SNR”, the compact closed-form expressions for the cumulative distribution function (CDF), probability density function (PDF), moment generating function (MGF), and generalized moment (GM) of the instantaneous end-to-end SNR are derived. In Section “Exact system performance analysis”, the exact analytical expressions are also derived for the outage probability (OP) and symbol error rate (SER). In Section “Asymptotic analysis”, the approximate expression for ergodic capacity is given firstly, and then the asymptotic expressions for OP and SER in the high SNR regime are presented. Numerical results that confirm our analysis are presented in Section “Numerical results”, which is followed by the conclusion in Section “Conclusions”.

System and channel models

Consider a dual-hop MIMO relaying network as shown in Figure 1. The source equipped with NS antennas communicates with the destination equipped with NDantennas via an NR-antenna relay. Note that the results in the article can be easily extended to a more general case, where the relay uses different number of transmit and receive antennas. Under the assumption that the direct link between the source and destination is in deep fading, the direct link is ignored.

Figure 1
figure1

System model of dual-hop AF relaying network.

System model

We assume that the source and relay with multiple antennas employ OSTBC transmissions. An OSTBC with NTx transmission antennas is defined by an NTx×K transmission matrix X, where K denotes the block length of an OSTBC and X is constructed by a set of transmitted symbols x, l=1,…,L. Since there is a total of L symbols transmitted over a period of K symbols, the code rate of OSTBCs is Rc=L/K.

During the first hop, the NR×K1signal matrix received at the relay could be written as

Y R = H 1 X 1 + N 1 ,
(1)

where H 1 C N R × N S denotes the spatially correlated Nakagami-m distributed channel matrix from the source to relay, X 1 C N S × K 1 is the OSTBC codeword matrix with code rate Rc,1=L/K1, and N 1 C N R × K 1 is the noise matrix whose elements are independent and identically distributed (i.i.d.) complex Gaussian RVs with zero mean and variance N0 / 2. OSTBC decouples the space-time channel into parallel scalar channels each with [19]

x R , l = H 1 F 2 x l + ñ R , l , l = 1 , , L ,
(2)

where the subscript F denotes the Frobenius norm, and ñ R , l is the filtered noise with zero mean and variance H 1 F 2 N 0 .

In the second hop, the output signal at the relay xR,lis amplified by the gain of the relay, G, encoded by an OSTBC matrix X 2 C N R × K 2 with code rate Rc,2=L/K2 and then sent to the destination. The ND×K2signal matrix received at the destination could be written as

Y D = H 2 X 2 + N 2 ,
(3)

where H 2 C N D × N R denotes the spatially correlated Nakagami-m distributed channel matrix between the relay and destination, and N 2 C N D × K 2 is the noise matrix whose elements are i.i.d. complex Gaussian RVs with zero mean and variance N0/2. The combined signal xD,l at the destination could be given by

x D , l = G H 2 F 2 x R , l + ñ D , l , l = 1 , , L ,
(4)

where ñ D , l is the filtered noise with zero mean and variance H 2 F 2 N 0 .

Substituting (2) into (4) yields

x D , l = G H 1 F 2 H 2 F 2 x l + G H 2 F 2 ñ R , l + ñ D , l .
(5)

Hence, the end-to-end SNR could be written as

γ = E S N 0 H 1 F 4 H 2 F 2 H 1 F 2 H 2 F 2 + 1 G 2 ,
(6)

where E S =E x l 2 . It is clear from (6) that the choice of the relaying gain defines the equivalent end-to-end SNR of the dual-hop relaying system.

Usually, there are two choices for the gain G to normalize the received signal [2]

G 1 = E R H 1 F 4 E S + H 1 F 2 N 0 , G 2 = E R H 1 F 4 E S ,
(7)

where E R =E x R , l 2 . The two choices are referred as the channel noise assisted and channel assisted gains respectively [5], and could be encompassed by the unified model

G = E R H 1 F 4 E S + β H 1 F 2 N 0 ,
(8)

where β is equal to 0 or 1 corresponding to the channel assisted gain G2 and channel noise assisted gain G1, respectively. By substituting (8) into (6), we have

γ = γ 1 γ 2 γ 1 + γ 2 + β ,
(9)

where γ 1 = ρ 1 H 1 F 2 , ρ1=ES/N0, γ 2 = ρ 2 H 2 F 2 , and ρ2=ER/N0.

Channel model

The correlated channel matrix H i , i=1,2, can be factorized in the form [20]

H i = R Rx, i 1 2 H iid., i R Tx, i 1 2 T ,
(10)

where RTx,i and RRx,iare the transmission and reception correlation matrices of i-th hop respectively, Hi.i.d.,iis a matrix of i.i.d. RVs, and the superscript T denotes the matrix transpose operation. The entry of RTx,i(RRx,i), ρ Tx, i j , k ρ Rx, i j , k , is the correlation coefficient between j-th and k-th transmission (reception) antennas of i-th hop. Based on the Kronecker product of the transmission and reception correlation matrices, we have

vec H i = R i 1 2 vec H i.i.d., i ,
(11)

where vec(·) denotes the matrix vectorization operation, R i = R Tx, i T R Rx, i , and denotes the Kronecker product.

By building up the multiple-dimensional Gamma distributed correlated vector with order parameter m based on the sum of squared Gaussians, the MGF of H i F 2 could be given by [18]

M H i F 2 s = I i + s m i D γ ̄ , i R i m i ,
(12)

where mi is the Nakagami-m fading parameter of i-th hop, Iiis the identity matrix of size NTx,i×NRx,i, D γ ̄ , i =diag γ ̄ i , 1 , γ ̄ i , 2 , , γ ̄ i , N Tx, i × N Rx, i , and γ ̄ i , j , j=1,…,NTx,i×NRx,i, is the average SNR for the j-th branch in vec(Hi). By using an eigenvalue factorization, the matrix D γ ̄ , i R i can be diagonalized as

D γ ̄ , i R i = U i Λ i U i 1 ,
(13)

where

Λ i = diag { λ i , 1 , , λ i , j , , λ i , j v i , j , , λ i , J i } ,
(14)

and λi,j, j=1,…,Ji, is the distinct eigenvalue of D γ ̄ , i R i with multiplicity vi,jsuch that j = 1 J i v i , j = N Tx, i × N Rx, i . Substituting (13) into (12) yields

M H i F 2 s = I i + s m i Λ i m i = j = 1 J i 1 + s m i λ i , j m i v i , j .
(15)

In particular, for the case of balanced branches, i.e. γ ̄ i , 1 = γ ̄ i , 2 == γ ̄ i , N Tx, i × N Rx, i , (12) can be expressed in terms of the eigenvalues associated with Ri as follows:

M H i F 2 s = j = 1 J i 1 + s γ ̄ i , 1 m i ϕ i , j m i u i , j ,
(16)

where ϕi,j, j=1,, J i , denotes the distinct eigenvalue of Ri with multiplicity ui,j such that j = 1 J i u i , j = N Tx, i × N Rx, i , and is the product of the eigenvalues of RTx,iand RRx,i.

By expanding in poles and residuals, (14) can be decomposed into the following partial fractions ([21], eq. (2.102))

M H i F 2 ( s ) = j = 1 J i k = 1 m i v i , j c j , k i 1 + s m i λ i , j k ,
(17)

where

c j , k i = 1 m i v i , j k ! m i λ i , j m i v i , j k d m i v i , j k d s m i v i , j k × M H i F 2 ( s ) 1 + s m i λ i , j m i v i , j s = m i λ i , j .
(18)

From the relationship between MGF and PDF, based on the linearity of the inverse of the Laplace transform and the identity for inverse Laplace transformation, given in ([22], eq. (5.2.17)), we could therefore write for the PDF of H i F 2 =z, where z>0, as

f ( z ) = j = 1 J i k = 1 m i v i , j c j , k i λ i , j m i k z k 1 Γ ( k ) exp m i z λ i , j ,
(19)

where Γ(·) is the gamma function defined as Γ(x)= 0 t x 1 e t dt[21].

By integrating the PDF with respect to z with the help of ([21], eq. (3.351.1)), the CDF of z could be given by

F ( z ) = 1 j = 1 J i k = 1 m i v i , j l = 0 k 1 c j , k i l ! exp m i z λ i , j m i z λ i , j l ,
(20)

where the above deduction uses the property of j = 1 J i k = 1 m i v i , j c j , k i =1, which can be derived by substituting s=0 into (12) and (16) and comparing the results.

Metrics of end-to-end SNR

In this section, the compact closed-form expressions for CDF of the end-to-end SNR γ are presented firstly. Then the exact analytical expressions for the PDF, MGF, and GM of γ are derived relying on CDF.

CDF of γ

The CDF of end-to-end SNR γ could be written as

F γ ( γ ) = 1 2 j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 × h = 0 l q 1 g l h c j , k 1 c p , q 2 Γ l + 1 Γ ( q ) × m 1 ρ 1 λ 1 , j θ + l 2 ρ 2 λ 2 , p m 2 θ l 2 q × exp m 1 γ ρ 1 λ 1 , j m 2 γ ρ 2 λ 2 , p × γ g + h + l 1 2 + q γ + β g h + l + 1 2 K θ l 2 γ ,
(21)

where γ = m 1 m 2 γ γ + β ρ 1 λ 1 , j ρ 2 λ 2 , p , θ=g + h + 1, and Kv(·) denotes the vth-order modified Bessel function of the second kind.

Proof

The proof is provided in Appendix 1. □

PDF of γ

The close-form expression for the PDF of γ can be given by

f γ ( γ ) = 2 j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 × h = 0 l q 1 g l h c j , k 1 c p , q 2 Γ l + 1 Γ q m 1 ρ 1 λ 1 , j θ + l 2 × ρ 2 λ 2 , p m 2 θ l 2 q × exp m 1 γ ρ 1 λ 1 , j m 2 γ ρ 2 λ 2 , p × γ g + h + l 3 2 + q γ + β g h + l 1 2 × γ 2 γ + β K θ l + 1 2 γ + φ K θ l 2 γ ,
(22)

where φ= m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p γ γ + β θ + q γ h + q β.

Proof

Taking the derivative of (19) with respect to γ and using the expression for the derivative of the modified Bessel function, given in ([21], eq. (8.486.13)), yields (20). □

MGF of γ

Moment generating function is useful in evaluating the system performance over fading channels. The MGF of γ can be presented as

M γ ( s ) = 1 s j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 h = 0 l × μ = 0 g + l + 2 q 1 g l h g + l + 2 μ × c j , k 1 c p , q 2 Γ θ + 1 Γ ( q ) β μ + h + 1 1 q g + μ 1 ρ 1 λ 1 , j m 1 ρ 2 λ 2 , p m 2 θ q + 1 × d q g + μ 1 d s ~ q g + μ 1 2 F 0 θ + 1 , l + 1 ; ; ω s ~ + s ~ × 2 F 0 θ + 1 , l + 1 ; ; ω s ~ s ~ s ~ = s ,
(23)

where s ~ = s ~ 2 2 βω , ω= ρ 1 λ 1 , j ρ 2 λ 2 , p 2 β m 1 m 2 , s =s+ m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p , and 2F0(a,b;;z) denotes the generalized hypergeometric function.

Proof

The proof is provided in Appendix 2. □

Although (21) appears complicated, it is in closed form since higher order derivatives of arbitrary order are known for the generalized hypergeometric function pFq(ap);(bq);z [23]. Having the MGF in closed form as in (21) and using the MGF-based approach for the performance evaluation of digital modulations over fading channels [24], the average bit and SERs for some types of modulation can be derived from the MGF of the instantaneous fading SNR directly (e.g., differentially coherent detection of phase-shift-keying (PSK) or noncoherent detection of orthogonal frequency-shift-keying (FSK)).

For β=0, (21) reduces to

M γ ( s ) = 1 2 s j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 h = 0 l q 1 g l h × c j , k 1 c p , q 2 Γ l + 1 Γ ( q ) m 1 ρ 1 λ 1 , j θ + l 2 ρ 2 λ 2 , p m 2 θ l 2 q × Π 2 ϖ θ l s + ϖ θ + q + 1 Γ θ + q + 1 Γ 2 l + q g h Γ l + q + 3 2 × 2 F 1 θ + q + 1 , θ l + 1 2 ; l + q + 3 2 ; s ϖ s + ϖ ,
(24)

where ϖ=2 m 1 m 2 ρ 1 λ 1 , j ρ 2 λ 2 , p .

Proof

By setting β=0, (43) can be reduced to a manipulable form. By performing some algebraic manipulations with the help of expression for the definite integral of Bessel function, given in ([21], eq. (6.621.3)), the MGF of γ with β=0 can be derived as shown in (22). □

GM of γ

We now derive the GMs of γ that can efficiently be applied to uncover other statistical measures of the system performance. For instance, the average SNR, amount of fading, and ergodic capacity can be evaluated with the first-order and second-order moments.

The GM of γ can be expressed as

E γ n = n j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 h = 0 l × μ = 0 g + l + 2 q 1 g l h g + l + 2 μ × c j , k 1 c p , q 2 Γ θ + 1 Γ ( q ) β μ + h + 1 1 q g + μ + n 2 ρ 1 λ 1 , j m 1 × ρ 2 λ 2 , p m 2 θ q + 1 × d q g + μ + n 2 d s ~ q g + μ + n 2
(25)
× 2 F 0 θ + 1 , l + 1 ; ; ω s ~ + × 2 F 0 θ + 1 , l + 1 ; ; ω s ~ s ~ = m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p ,
(26)

where s ~ + = s ~ + s ~ 2 4 m 1 m 2 ρ 1 λ 1 , j ρ 2 λ 2 , p , and s ~ = s ~ s ~ 2 4 m 1 m 2 ρ 1 λ 1 , j ρ 2 λ 2 , p .

Proof

The GM of γ can be obtained from

E γ n = n 0 γ n 1 1 F γ ( γ ) dγ.
(27)

By substituting (19) into (24) and performing the similar algebraic manipulations as in Appendix 2, the closed-form expression for GM of γ can be derived as shown in (23). □

For β=0, (23) deduces to

E γ n = 2 n j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 × h = 0 l q 1 g l h c j , k 1 c p , q 2 Γ l + 1 Γ q m 1 ρ 1 λ 1 , j θ + l 2 × ρ 2 λ 2 , p m 2 θ l 2 q × Π 2 ϖ θ l m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p + ϖ θ + n + q × Γ θ + n + q Γ 2 l + n + q θ Γ l + n + q + 1 2 × 2 F 1 × θ + n + q , θ l + 1 2 ; l + n + q + 1 2 ; m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p ϖ m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p + ϖ .
(28)

Proof

Substituting (19) with β=0 into (24) and performing some algebraic manipulations with the help of expression for the definite integral of Bessel function, given in ([21], eq. (6.621.3)), the GM of γ with β=0 can be derived as shown in (25). □

Exact system performance analysis

In this section, the exact analytical expressions are derived for the OP and SER based on the expressions presented in last section.

Outage probability

The OP is an important quality of service measure, which is defined as the probability that the instantaneous end-to-end SNR falls below a predetermined threshold γthand could be given by

P out = P γ < γ th = F γ γ th ,
(29)

where Fγth) is the CDF of γ evaluated at γ=γth. Consequently, the OP could be evaluated by substituting (19) into (26).

Symbol error rate

With the aid of M γ (s) presented in Section “MGF of γ”, the average SER of some types of modulation can be easily obtained via the unified MGF-based approach [24]. However, for some cases (e.g., coherent detection of M-PSK, M-FSK, and M-QAM), the intractability arises in seeking a closed-form solution to the finite-range integration of the MGF expressions, which involve the generalized hypergeometric function.

In this section, we adopt an alternative approach to give the closed-form expression for SER directly in terms of the CDF of γ as [25]

P s = a 2 b Π 0 F ( γ ) γ exp ,
(30)

where the parameters a and b are up to a specific modulation scheme to be used, e.g., for BPSK, a=b=1.

For mathematical tractability, we set β=0. Substituting (19) into (27), and performing some algebraic manipulations with the help of the expression for the definite integral of Bessel function, given in ([21], eq. (6.621.3)), yields

P s = a 2 a b j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 h = 0 l q 1 g l h × c j , k 1 c p , q 2 Γ l + 1 Γ q m 1 ρ 1 λ 1 , j θ + l 2 ρ 2 λ 2 , p m 2 θ l 2 q × 2 ϖ θ l Γ θ + q + 1 2 Γ 2 l + q + 1 2 θ b ~ + ϖ θ + q + 1 2 Γ l + q + 1 × 2 F 1 θ + q + 1 2 , θ l + 1 2 ; l + q + 1 ; b ~ ϖ b ~ + ϖ ,
(31)

where b ~ =b+ m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p .

Asymptotic analysis

In this section, the approximate expression for ergodic capacity is given firstly. The asymptotic expressions for OP and SER in high SNR regime are then derived in order to gain insight into the system performance, and the achievable diversity order and array gain of the relaying system are also presented.

Ergodic capacity

The ergodic capacity is determined by the expected value of the instantaneous mutual information between the source and destination, which could be expressed in a second-order approximation as [26]

C erg log 2 e 2 ln 1 + E ( γ ) E γ 2 E 2 ( γ ) 2 1 + E ( γ ) 2 .
(32)

By substituting n=1 and n=2 into (25), we have

E γ = 2 j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 h = 0 l q 1 g l h × c j , k 1 c p , q 2 Γ l + 1 Γ q m 1 ρ 1 λ 1 , j θ + l 2 ρ 2 λ 2 , p m 2 θ l 2 q × Π 2 ϖ θ l m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p + ϖ θ + q + 1 × Γ θ + q + 1 Γ 2 l + q θ + 1 Γ l + q + 3 2 × 2 F 1 θ + q + 1 , θ l + 1 2 ; l + q + 3 2 ; m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p ϖ m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p + ϖ .
(33)
E γ 2 = 4 j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 h = 0 l q 1 g l h × c j , k 1 c p , q 2 Γ l + 1 Γ q m 1 ρ 1 λ 1 , j θ + l 2 ρ 2 λ 2 , p m 2 θ l 2 q × Π 2 ϖ θ l m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p + ϖ θ + q + 2 × Γ θ + q + 2 Γ 2 l + q θ + 2 Γ l + q + 5 2 × 2 F 1 θ + q + 2 , θ l + 1 2 ; l + q + 5 2 ; m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p ϖ m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p + ϖ .
(34)

Obviously, we can evaluate the ergodic capacity of the MIMO relaying systems by substituting (30) and (31) into (29).

Outage probability

By using Taylor series expansion of the exponential function at z equal to zero, from (18), we have

F ( z ) = 1 j = 1 J i k = 1 m i v i , j l = 0 k 1 t = 0 1 t c j , k i t ! l ! m i z λ i , j l + t = D i z N Tx, i N Rx, i m i + o z N Tx, i N Rx, i m i + 1 ,
(35)

where

D i = j = 1 J i k = 1 m i v i , j l = 0 k 1 N Tx, i N Rx, i m i l c j , k i m i λ i , j N Tx, i N Rx, i m i N Tx, i N Rx, i m i ! 1 l 1 .
(36)

Hence, the asymptotic expression for OP at high SNR can be written as [27]

P out = D 1 γ th ρ 1 m ~ 1 + o γ th ρ 1 m ~ 1 + 1 , m ~ 1 < m ~ 2 D 2 γ th ρ 1 m ~ 2 + o γ th ρ 1 m ~ 2 + 1 , m ~ 1 > m ~ 2 D 1 + D 2 γ th ρ 1 m ~ 1 + o γ th ρ 1 m ~ 1 + 1 , m ~ 1 = m ~ 2
(37)

where

D 1 = j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 m ~ 1 l c j , k 1 m 1 λ 1 , j m ~ 1 m ~ 1 ! 1 l 1 , D 2 = j = 1 J 2 k = 1 m 2 v 2 , j l = 0 k 1 m ~ 2 l c j , k 2 m 2 κ λ 2 , j m ~ 2 m ~ 2 ! 1 l 1 ,
(38)

and m ~ 1 = N S N R m 1 , m ~ 2 = N R N D m 2 , κ=ρ21.

It should be noted that the value of β has no impact on the asymptotic expression of OP in high SNR regime.

Symbol error rate

Based on the relationship between CDF and OP, substituting (33) into (27) and after some algebraic manipulations with the help of definite integral of exponential function, given in ([21], eq. (3.381.10)), the asymptotic SER can be derived as [10, 28]

P s = G a ρ 1 G d + o ρ 1 G d ,
(39)

where the diversity order Gdis

G d = min m ~ 1 , m ~ 2 ,
(40)

and the array gain Gais

G a = b a D 1 2 Π Γ m ~ 1 + 1 2 1 m ~ 1 , m ~ 1 < m ~ 2 b a D 2 2 Π Γ m ~ 2 + 1 2 1 m ~ 2 , m ~ 1 > m ~ 2 b a D 1 + D 2 2 Π Γ m ~ 1 + 1 2 1 m ~ 1 , m ~ 1 = m ~ 2
(41)

From (35), we observe that the diversity order is equal to the minimum of the product of the number of source and relay antennas and the first hop Nakagami-m fading parameter and the product of the number of relay and destination antennas and the second hop Nakagami-m fading parameter, which means the diversity order is entirely determined by the weaker hop. It also shows that the diversity order is only determined by the antenna configuration and Nakagami-m fading parameter and is independent of the correlation.

Numerical results

In this section, the theoretical analysis developed in the previous sections is validated via numerical examples, and we also compare the performance of the CSI-assisted relaying system with that of the fixed gain relaying system [29]. We consider doubly-correlated Nakagami-m fading channels. Without loss of generality, the exponential correlation model is adopted, where the correlation model at both transmitting and receiving ends of both hops are assumed to be identical. For exponential correlation model, the components of the correlation matrix can be given by [17, 3032]

ρ ij = ρ | i j | , 0 ρ 1 ,
(42)

where ρ is the correlation coefficient of the neighboring antennas. Obviously, (37) may be not an accurate model for some real-world scenarios but this is a simple single-parameter model which allows us to study the effect of correlation on the MIMO relaying system in an explicit way and to get some insight. And this model is physically reasonable in the sense that the correlation between a pair of signals decreases as the separation between them increases.

Figure 2 plots the PDF of the received end-to-end SNR γ for different terminal antenna configurations with different correlation coefficient ρ. It is observed that the analysis results of (20) match perfectly with the simulation results. The higher the correlation, the lower the end-to-end SNR. This phenomenon is more significant for system with larger number of antennas. It also shows that the end-to-end SNR γ with β=1 can be tightly upper bounded by γ with β=0, which is consistent with (9). For brevity, we consider only the end-to-end SNR γ with β=0 in the following.

Figure 2
figure2

PDF of the end-to-end SNR γ for different terminal antenna configurations (N S ,N R ,N D ). (m1 = m2 = 2, ρ1 = ρ2 = 1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1)

Figures 3 and 4 depict the system outage probabilities variation with different antenna configurations and different correlation coefficients ρ with a threshold value γth=10 dB. Figures 5 and 6 show the BER of BPSK modulation with different correlation coefficients ρ. As expected, the comparison shows an excellent agreement between analytical and simulation results, which validates the accuracy of our analytical results. The correlation deteriorates the OP and BER and the larger number of antennas improves the OP and BER. From Figures 3, 4, 5 and 6, it can be concluded that the effect of correlation could be negligible when the correlation between two adjacent antennas are less than 0.5 but it becomes significant for ρ>0.5, which is in agreement with previous results on the effect of spatial correlation [31, 32].

Figure 3
figure3

OP of the AF MIMO relaying system for a threshold γ th = 10 dB with terminal antenna configuration (N S ,N R ,N D ) = (2,2,2). (m1=m2=2, κ=1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1).

Figure 4
figure4

OP of the AF MIMO relaying system for a threshold γ th = 10 dB with terminal antenna configuration (N S ,N R ,N D ) = (4,4,4). (m1 = m2 = 2, κ = 1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1).

Figure 5
figure5

Bit error rate of BPSK for AF MIMO relaying systems with terminal antenna configuration (N S ,N R ,N D ) = (2,2,2). (m1 = m2 = 2, κ = 1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1).

Figure 6
figure6

Bit error rate of BPSK for AF MIMO relaying systems with terminal antenna configuration (N S ,N R ,N D ) = (4,4,4). (m1 = m2 = 2, κ = 1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1).

The asymptotic results are also drawn in the Figures 3, 4, 5 and 6. The asymptotic curves in each figure are parallel, i.e., they have the same slope, but are shifted to the right as ρ increases from 0 to 0.9, which implies that the correlation has no impact on the achievable diversity order. This is consistent with the works in [7, 15]. The asymptotic curves in Figures 3 and 5 are linear with a slope of 8, whereas the asymptotic curves in Figures 4 and 6 are very steep with a slope of 32. These observations agree with the result in Section “Symbol error rate”, namely, the diversity order is min m ~ 1 , m ~ 2 .

Figure 7 plots the BER of CSI-assisted and fixed gain relaying systems with different correlation coefficients. It’s shown that the performance of fixed gain relaying system deteriorates seriously, especially for high correlation coefficient, which means that fixed gain relaying system is much more influenced by the correlation than CSI-assisted relaying system.

Figure 7
figure7

Comparison of BER of CSI-assisted and fixed gain relaying systems for different terminal antenna configurations (N S ,N R ,N D ). (m1 = m2 = 2, κ = 1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1).

Figure 8 compares the BER of CSI-assisted and fixed gain relaying systems when correlation exists only at one terminal with ρ=0. 9. Terms “source”, “relay”, and “destination” in the legend mean that correlation exists only at the source, relay, and destination, respectively. For CSI-assisted relaying system, the performance of correlation at source and destination is almost the same, which is better than that of correlation at relay. This can be explained by the fact that the correlation at relay has negative influence on both hops, whereas correlation at source or destination has negative influence only on single hop. For fixed gain relaying system, the performance of correlation at relay is worst. The performance of correlation at source is inferior to that of correlation at destination, and the gap between them is relatively large. The reason is that fixed gain relaying does not take the CSI of the first hop into account and the error in first hop has a dominant influence.

Figure 8
figure8

Comparison of BER of CSI-assisted and fixed gain relaying systems when correlation exists only at one terminal. (m1 = m2 = 2, κ = 1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1, ρ = 0.9).

Comparison of ergodic capacity of CSI-assisted and fixed gain relaying systems is shown in Figure 9. Both approximate solution (29) and simulation results are plotted. It’s observed that the CSI-assisted relaying system has a higher capacity, and this phenomenon is not obvious for independent channel fading (ρ=0). The capacity is higher for high SNR, and becomes lower at higher correlation values. It is also shown that the closed-form approximation matches the simulation results very well, which verifies the accuracy of the approximation solution (29).

Figure 9
figure9

Comparison of ergodic capacity of CSI-assisted and fixed gain relaying systems. (m1 = m2 = 2, κ = 1, γ ̄ 1 , j = γ ̄ 2 , j =1, Rc,1 = Rc,2 = 1).

Conclusions

In this article, the performance of dual-hop AF MIMO relaying system with OSTBC transmissions over doubly-correlated Nakagami-m fading channel has been investigated. The compact closed-form expressions for CDF, PDF, MGF, and GM of the end-to-end SNR are derived. Besides, the exact analytical expressions for OP and SER and approximate expression for ergodic capacity are also derived. In order to gain more insight into system performance, we present the asymptotic expression for OP and SER in the high SNR regime, and derive the achievable diversity order and array gain of the AF MIMO relaying system. The analytical expressions have been analytically proved and verified through extensive simulations. All the cases investigated reveal an excellent agreement between the results from analysis and simulation. It is observed that system performance increases with the decrease of antenna correlation and the increase of antenna number. Furthermore, we compare the performance of CSI-assisted and fixed gain relaying systems in terms of BER and ergodic capacity. CSI-assisted relaying outperforms fixed gain relaying at the cost of complexity.

Appendices

Appendix 1

From [6], the CDF of end-to-end SNR γ expressed by (9) could be derived as

F γ ( γ ) = 1 0 P γ 1 x ~ x + γ f γ 2 x + γ dx
(43)

where x ~ = x + γ + β γ/x. From (17) and (18), we have

f γ 2 x + γ = p = 1 J 2 q = 1 m 2 v 2 , p c p , q 2 Γ ( q ) m 2 ρ 2 λ 2 , p q x + γ q 1 × exp m 2 ρ 2 λ 2 , p x + γ ,
(44)
P γ 1 x ~ x + γ = j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 c j , k 1 l ! m 1 x ~ ρ 1 λ 1 , j l × exp m 1 x ~ ρ 1 λ 1 , j .
(45)

Substituting (39) and (40) into (38), and performing some algebraic manipulations, with the help of definite integral of exponential function, given by ([21], eq. (3.471.9)), and the binomial theorem, given by ([21], eq. (1.111)), the closed-form expression for the CDF can be derived as shown in (19).

Appendix 2

MGF M γ (s) associates with the fading PDF f γ (γ) and is defined by [24]

M γ ( s ) = 0 e f γ ( γ ) = 1 s 0 e 1 F γ γ
(46)

Substituting (19) into (41) yields

M γ ( s ) = 1 2 s j = 1 J 1 k = 1 m 1 v 1 , j l = 0 k 1 p = 1 J 2 q = 1 m 2 v 2 , p g = 0 q 1 h = 0 l q 1 g l h × c j , k 1 c p , q 2 Γ l + 1 Γ ( q ) m 1 ρ 1 λ 1 , j θ + l 2 ρ 2 λ 2 , p m 2 θ l 2 q × μ = 0 g + l + 2 g + l + 2 μ β g + l μ + 2 Δ ,
(47)

where

Δ = γ q g + μ 1 0 exp s + m 1 ρ 1 λ 1 , j + m 2 ρ 2 λ 2 , p γ × γ θ + l 2 γ + β θ + l + 2 2 K θ l 2 m 1 m 2 γ γ + β ρ 1 λ 1 , j ρ 2 λ 2 , p = Γ θ + 1 Γ l + 1 2 β θ + l + 2 1 q g + μ 1 m 1 m 2 ρ 1 λ 1 , j ρ 2 λ 2 , p θ + l + 2 2 × d q g + μ 1 d s ~ q g + μ 1 2 F 0 θ + 1 , l + 1 ; ; ω s ~ + s ~ × 2 F 0 θ + 1 , l + 1 ; ; ω s ~ s ~ s ~ = s .
(48)

It is deduced by performing some algebraic manipulations using the identities for Laplace transformation, given in ([22], eqs. (4.17.20) and (4.1.6)), and the relationship between Whittaker functions and generalized hypergeometric functions, given in [33]. Substituting (43) into (42) gives (21).

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Keywords

  • Amplify-and-forward (AF) relaying
  • Multiple-input multiple-output (MIMO)
  • Orthogonal space-time block code (OSTBC)