Alamouti-OSTBC Wireless Cooperative Networks With ... - IEEE Xplore

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 4, APRIL 2018

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Alamouti-OSTBC Wireless Cooperative Networks With Mobile Nodes and Imperfect CSI Estimation Yazid M. Khattabi

, Member, IEEE and Mustafa M. Matalgah, Senior Member, IEEE

Abstract—This paper is concerned with investigating the impact of both nodes mobility and imperfect channel-state-information (CSI) estimation on the symbol-error-probability (SEP) performance of a multiple-relay amplify-and-forward cooperative diversity system employing Alamouti-type orthogonal-space-timeblock-code transmission at the source and classical Alamouti’s space-time-decoder (ALD) at the destination. Due to nodes mobility, the system’s links are characterized by time-selective fading channels, which are modeled by the first-order-autoregressive (AR1) process, and due to imperfect CSI estimation, the estimated channel gains are assumed to be corrupted by Gaussian errors. For such a system model, an approach is proposed to derive a tight approximate expression for the system’s conditional SEP. This approach is based on utilizing the AR1 model to derive exact expressions for the signal-to-interference-plus-noise ratios of the ALD decoder’s decision variables, and on benefiting from the central limit theorem to approximate some of the non-Gaussian interference and noise terms. The obtained conditional SEP expression is function of both the fading channel correlation parameters and the estimation error variances, and thus, it is valid for mobile as well as static nodes for imperfect as well as perfect CSI estimation processes. The system’s average SEP is evaluated semianalytically using the obtained conditional SEP expression. Numerical results along with Monte Carlo link-level simulations are provided to validate the accuracy of the presented analyses and to demonstrate the system performance under several realistic scenarios. Index Terms—Space-time-block-coding, cooperative-diversity, time-selective fading, imperfect estimation, error probability.

I. INTRODUCTION PACE-TIME-BLOCK-CODING (STBC) techniques are desirable for mobile terminals operating in wireless networks with multiple-antenna base-stations. This is because of the achieved transmit diversity-gain at these terminals, along with the low decoding computational complexity that results from the linear maximum likelihood (ML) decoder, even though they are equipped with single receiving antenna [1]–[3]. Cooperative communication techniques have been introduced in wireless networks to achieve spacial diversity-gain via the readiness of multiple users (via relays) to assist a source forwarding

S

Manuscript received May 15, 2017; revised August 19, 2017 and October 18, 2017; accepted November 28, 2017. Date of publication December 22, 2017; date of current version April 16, 2018. The review of this paper was coordinated by Dr. Tao Jiang. (Corresponding author: Yazid M. Khattabi.) Y. M. Khattabi is with the Department of Electrical Engineering, The University of Jordan, Amman 11942, Jordan (e-mail: [email protected]). M. M. Matalgah is with the Department of Electrical Engineering, The University of Mississippi, Oxford, MS 38677 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2017.2786471

its data to a final destination [4]. Cooperative communication techniques have also shown their capability in improving system reliability and spectrum efficiency, see [5] and references therein. The combination of STBC coding with cooperative systems has gained great interest in the research community in terms of design issues and performance analysis. This combination is possible in two different methods. In the first method, the space-time code is constructed without the need of replacing multiple-antenna at the transmission side by exploiting antennas of other users (relays) operating within the network; this is what so-called in literature as distributed-STBC [6]–[11]. In the second one, the space-time code is constructed conventionally at the transmitter using its co-located antennas and then the spacetime encoded signals are sent to the ultimate destination through the relaying network (we refer to this kind of combination as STBC-cooperative) [12], [15]. It should be noted that system performance analyses and results reported in [6]–[15] are based on the assumptions that the fading channels are quasi-static (i.e., their fading gains are assumed to be constant over a number of consecutive signaling periods) and the channel-state-information (CSI) estimation processes at the systems’ receivers (the relays and destination) are perfect. However, in practical wireless network applications, these assumptions are not fairly realistic. For example, nowadays, number of users using wireless terminals while they are riding high-speed public transportation vehicles (e.g., cars, buses, trains, subways, or airplanes) is increasing [16] and [17]. As a result of such high mobility wireless terminals, the assumption of time-selective (non quasi-static) fading is more realistic, see [18] and references therein. Furthermore, due to impairments associated with practical receiver implementation issues, it is more general to assume that estimated channel gains are corrupted by estimation error (i.e., channel estimation is imperfect) [19]. In the following, however, we provide some related literature work that consider the impact of such issues on the performance of combined STBC cooperative-diversity systems and then highlight the main contribution and motivation of this work. In [20], the impact of the time-selective fading on the performance of a point-to-point (non-cooperative) system with Alamouti-type orthogonal-STBC (OSTBC) and perfect CSI estimation has been investigated. It has been shown that, due to the time-selective fading, the Alamouti-OSTBC channel-gain matrix is no longer orthogonal, the Alamouti’s decoder is no longer optimal ML, the obtained decision variables are interfering, and the system error performance is degraded. In

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[21], a point-to-point quasi-static fading system with multi-level quadrature-amplitude-modulation (M-QAM) and AlamoutiOSTBC transmission has been considered under the effect of the imperfect CSI estimation. The authors in [21] have shown that the imperfect CSI estimation causes self-interference in the Alamouti decoder’s decision variables, and, starting from these variables, they have analyzed the system’s bit error rate (BER) performance. In [22], the authors have considered the same non-cooperative system model as in [20] but, along with the time-selective fading assumption, they have assumed imperfect CSI estimation. Under these considerations, they have provided analytical expression for the system’s BPSK BER performance. In [23] and [24], the error performance of distributed (cooperative-based) Alamouti-OSTBC system has been investigated under the influence of time-selective fading and imperfect CSI estimation, respectively. In [25]–[27], the effect of the imperfect CSI estimation on the performance of quasi-static fading AF multiple-input-multiple-output (MIMO) cooperative based systems with OSTBC transmission has been investigated considering different performance metrics. Recently, the authors in [28] and [29] have studied the timeselective fading (due to nodes’ mobility) impact on the symbol error probability (SEP) performance of a multiple-relay AF cooperative-diversity system with Alamouti-type OSTBC transmission. They have shown that the time-selective fading destroys the optimality of the classical Alamouti’s space-time decoder (called as ALD) and its SEP performance is degraded and experiences error floors. However, the SEP performance results reported in [28] and [29] have been obtained only via link-level simulation and no theoretical analyses have been provided. This link-level simulation has disadvantages that (i) it requires long running time to get the system average SEP plots (ii) it makes it difficult to get any insight of the system performance under the time-selective fading influence, and more specifically, it does not help to quantitatively explain the error floors phenomenon. In addition, results reported in [28] and [29] have been obtained without taking into account the effect of the more realistic assumption of imperfect CSI estimation. Therefore, motivated by this, in this work, we extend the study in [28] and [29] by adding two main contributions: (i) we generalize the system model in [28] and [29] by following, along with the general assumption of time-selective fading (mobile nodes), the assumption of imperfect CSI estimation; (ii) we consider this more general Alamouti-OSTBC based cooperative system model and, unlike the work in [28] and [29], we propose a theoretical approach to evaluate its SEP performance. This approach starts by exploiting the AR1 model to derive exact expressions for the ALD statistics’ SINRs. These SINRs are then exploited along with the central-limit-theorem (CLT) [30] to provide closed-form expression for the system’s conditional SEP performance. This conditional SEP expression is helpful by itself to (i) quantitatively determine the error floors that appear due to the influence of both imperfect CSI estimation and time-selective fading; (ii) evaluate the system average SEP performance via low running-time semianalytical computation; (iii) shed light on the system performance under several practical special cases. In order to make it easy for researchers and


field engineers, the MATLB source script that semianalytically computes the system average SEP performance and its floors is uploaded online as an open and free source [31]. The remainder of this paper is organized as follows. Section II presents the system model under study. In Section III, the decision variables at the output of the destination’s ALD decoder are obtained and discussed. In Section IV, the decision variables’ conditional SINRs are derived and the system’s SEP performance is evaluated. Numerical and realistic linklevel simulation results are provided in Section V and finally conclusions are drawn in Section VI. II. SYSTEM MODEL A. Fading Link and Signal Model We consider a wireless cooperative network with M number of relays (R , ∈ {1, 2, . . . , M }) which are ready to assist a source S in forwarding its data to a destination D over orthogonal transmissions, which could be achieved either by TDMA, FDMA or CDMA. The direct path between S and D is assumed to be absent. The source is equipped with two transmit antennas (e.g., corresponding to a base station), while the relays and the destination each equipped with single antenna and work as mobile terminals. We consider an aggregate fading wireless channel model which takes into account small-scale fading and large-scale propagation models. Let hi, and h,d denote the complex channel gains for the links from the source ith transmit antenna (i ∈ {1, 2}) to the th relay’s receive antenna and from the th relay’s transmit antenna to the destination’s receive antenna, respectively. We assume that hi, and h,d have Rayleigh envelop and uniform phase, and thus, they are distributed as zero-mean-circularly-symmetric-complex-Gaussian 2 2 ) and h,d ∼ CN (0, σ,d ). (ZMCSCG), i.e., hi, ∼ CN (0, σi, To take into account the effect of the path-loss large-scale prop2 2 and σ,d are given as 1/dns, and 1/dn,d , respecagation, σi, tively, where ds, and d,d are the S-R and R -D distances and n is the path-loss exponent. All of the system’s fading links are characterized, in the small scale fading, as frequency-flat (i.e., single-tap fading links) and, due to nodes’ mobility, as timeselective. To characterize the time-selective fading, each fading channel (say the channel from antenna a to b) is described by AR1 process as [18] (1) ha,b (τ1 ) = ρa,b ha,b (τ2 ) + 1 − ρ2a,b ea,b (τ1 ) where the pair (a, b) ∈ {(i, ), (, d)}; τ1 and τ2 denote any two adjacent signaling period positions, the random process ea,b (k) is the varying-component of the associated channel with density 2 ), and CN (0, σa,b 2πfc νa,b ρa,b = J0 (2) Rs c is the channel’s correlation parameter [32] where νa,b is the relative speed between nodes a and b, Rs is the transmission symbol rate, fc is the carrier frequency, c is the speed of light and J0 (.) is the zeroth-order bessel function of the first kind. It is also assumed that ρ1, = ρ2, = ρs, .

KHATTABI AND MATALGAH: ALAMOUTI-OSTBC WIRELESS COOPERATIVE NETWORKS WITH MOBILE NODES

At the source, the modulated (using q-ary QAM) complex symbol sequence {xi } (each with energy Es /2) is parsed into code vectors x = [x1 , x2 ]T , and then transmitted over space and time as Alamouti-OSTBC matrix [1] x1 −x∗2 X= (3) x2 x∗1 where the first column’s symbols are transmitted simultaneously at the t1 th signaling period using the two antennas of S, while that of the second column are transmitted in the same manner but over the t2 th signaling period (t2 = t1 + 1). Throughout data transmission between S and D two phases are accomplished. In the first phase, S transmits the OSTBC matrix in (3) as described above, while each relay, say R , receives two signals over the t1 th and t2 th signaling periods, respectively, as ys, (t1 ) = h1, (t1 )x1 + h2, (t1 )x2 + ns, (t1 )

(4)

ys, (t2 ) = − h1, (t2 )x∗2 + h2, (t2 )x∗1 + ns, (t2 )

(5)

where ns, (t1 ) and ns, (t2 ) (∼ CN (0, No )) are the white noise samples. In the second transmission phase, the th relay amplifies its received signals in (4) and (5) by the fixed gain Er (6) G= Es + N o where Er = Es is the overall relay transmit energy [33]. After that, it sequentially transmits Gys, (t1 ) and Gys, (t2 ) towards the destination over the k1 th and k2 th periods (where k2 = k1 + 1; and indicates the th orthogonal channel allocated to R ), respectively, which results in the following received signals y,d (k1 ) = h,d (k1 )(Gys, (t1 )) + n,d (k1 )

(7)

y,d (k2 )

(8)

=

h,d (k2 )(Gys, (t2 ))

+

n,d (k2 )

where n,d (k1 ) and n,d (k2 ) are also white noise samples with density CN (0, No ). By substituting (4) and (5) into (7) and (8), respectively, and taking the complex-conjugate (∗) of y,d (k2 ), we can write all received signals at D from all relays as a 2M × 1 signal vector Yd as ⎡ ⎤ y1,d (k11 ) ⎢ y ∗ (k 1 ) ⎥ ⎢ 1,d 2 ⎥ ⎢ ⎥ .. ⎢ ⎥ = H x1 . ⎢ ⎥ x2 ⎢ ⎥ ⎣ yM ,d (k1M ) ⎦ ∗ M yM ,d (k2 )

x

Yd

⎤ Gh1,d (k11 )ns,1 (t1 ) + n1,d (k11 ) ⎢ Gh∗ (k 1 )n∗ (t2 ) + n∗ (k 1 ) ⎥ s,1 1,d 2 1,d 2 ⎥ ⎢ ⎥ ⎢ .. ⎥ +⎢ . ⎥ ⎢ ⎥ ⎢ ⎣ GhM ,d (k1M )ns,M (t1 ) + nM ,d (k1M ) ⎦ Gh∗M ,d (k2M )n∗s,M (t2 ) + n∗M ,d (k2M ) ⎡

Nd

(9)

where

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⎤ Gh2,1 (t1 )h1,d (k11 ) Gh1,1 (t1 )h1,d (k11 ) ⎢ Gh∗ (t2 )h∗ (k 1 ) −Gh∗1,1 (t2 )h∗1,d (k21 ) ⎥ 2,1 1,d 2 ⎥ ⎢ ⎥ ⎢ .. .. ⎥. ⎢ H=⎢ . . ⎥ ⎥ ⎢ ⎣ Gh1,M (t1 )hM ,d (k1M ) Gh2,M (t1 )hM ,d (k1M ) ⎦ Gh∗2,M (t2 )h∗M ,d (k2M ) −Gh∗1,M (t2 )h∗M ,d (k2M ) ⎡

is the system’s end-to-end channel-gain matrix. B. CSI Estimation Despite that the fading links in this work are assumed to be time-selective (i.e., rapidly time-varying), we assume that the relays and the destination tracking loops are capable of estimating the channel gains of their corresponding fading links over the individual signaling periods.1 However, unlike the work in [28], we assume that these channel estimation processes are imperfect (i.e., channel estimation error is significant). This imperfect channel estimation assumption is more realistic due to the fact that receivers tracking loops might suffer from practical impairments (related to implementation issues) that lead to errors in the channel estimation processes. Thus, we assume that the estimated channel gain over the τ th signaling period, say ˆ a,b (τ ), is related to the actual one ha,b (τ ) as [19] h ˆ a,b (τ ) = ha,b (τ ) + h (τ ) h a,b

(10)

where ha,b (τ ) is the estimation error, which is assumed to be ZMCSCG with variance σe2a , b (i.e., ∼ CN (0, σe2a , b ). In the next section, we consider the system model described above and employ the conventional Alamouti’s space-timedecoder (ALD) at the system’s destination D and obtain its outputs’ decision variables. III. ALD AND DECISION VARIABLES Employing the ALD at the destination D needs first the estimated version of H, which can be obtained (by substituting (10) into H above) as T bM b11,1 b12,1 · · · bM 1,1 2,1 ˆ = (11) H b11,2 b12,2 · · · bM bM 1,2 2,2 where b1,1 = G h1, (t1 )h,d (k1 ) + h1, (t1 )h,d (k1 ) + h1, (t1 ) h,d (k1 ) + h1, (t1 )h,d (k1 ) ], b1,2 = G[h2, (t1 )h,d (k1 ) + h2, (t1 )h,d (k1 ) + h2, (t1 )h, d (k1 ) + h2, (t1 )h, d (k1 ) ], b2,1 = G[h∗2, (t2 )h∗,d (k2 ) + h∗2, (t2 )h ∗,d (k2 ) + h ∗2, (t2 )h∗,d (k2 ) + h ∗2, (t2 )h ∗,d (k2 ) , and b2,2 = −G[h∗1, (t2 )h∗,d (k2 ) + h∗1, (t2 ) h ∗,d (k2 ) + h ∗1, (t2 )h∗,d (k2 ) + h ∗1, (t2 )h ∗,d (k2 )]. Now, applying the ALD at D is just by multiplying the received ˆ (H ˆ H ), where signal vector Yd in (9) by the hermitian of H the resulted two elements are the ALD’s decision variables corresponding to the two transmitted symbols x1 and x2 , which 1 Several algorithms have been proposed to track and estimate time-selective (time-varying) fading channels for space-time block coding [34] and [35].

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can be written after doing some simplifications, respectively, as

(12)

is the average SNR of the input decision-variable ∞ −x2 2 dx is the Q− function. However, due and Q(u) = u e to the complicated form of the obtained decision variables y1 in (12) and y2 in (13) as well as due to the fact that several terms inside their overall-interference terms (I1 and I2 ) and overall 2 ) are not exact complex-Gaussian, white-noise-terms ( χ1 and χ deriving exact expression for the system’s SEP from scratch is not easily tractable. Thus, to avoid complexity, we propose to directly use the AWGN SEP expression in (14) to evaluate the system’s SEP. This use is basically stands on (i) exploiting the central-limit-theorem (CLT) to approximate the terms I1 , I2 , 2 as complex-Gaussian (ii) deriving explicit and exact χ 1 and χ closed-form expressions for the SINRs of the decision variables y1 and y2 , which we do in the following.

(13)

A. Decision Variables SINRs

β1 x1

+

overall-interference1 I1

ζx2

overall-white-noise1 χ 1

nodes-mobility-interference

+

+

n 1

background-noise

+

ϑ1 x1 + ξ1 x2

υ 1

imperfect-CSI-noise

imperfect-CSI-interference desired-signal ofx 2

β2 x2

y2 = +

overall-interference2 I2

ζ ∗ x1

overall-white-noise2 χ 2

nodes-mobility-interference

+

Es No

√1 2π

desired-signal ofx 1

y1 =

where γ =

ϑ2 x2 + ξ2 x1

imperfect-CSI-interference

+

n 2

background-noise

+

υ 2

imperfect-CSI-noise

where the coefficients β1 , ζ, ϑ1 ξ1 , υ 1 , n 1 , β2 , ϑ2 , ξ2 , n 2 and υ 2 are described in Appendix A. Observe that because of the more practical assumptions of time-selective fading and imperfect CSI estimation the obtained decision variables y1 and y2 are (i) nonseparable for x1 and x2 , respectively (see the imposed interference terms I1 and I2 as well as the additional 2 ) (ii) statistically correlated because noise components υ 1 and υ 2 are correlated (see [36, their effective noise terms χ 1 and χ Eq.(8.3.15)]). If the system operates under the special case of quasi-static fading (i.e., ρa,b = 1 ∀(a, b)) and perfect CSI estimation (i.e., ha,b (τ ) = 0 ∀(a, b)) y1 and y2 reduce to be separable and independent. Under this situation, we can say that the ALD is under its originally designed optimal version. However, in the following section, we analyze the SEP performance of this system considering the more general scenarios of both time-selective fading (i.e., ρa,b < 1 ∀(a, b)) and imperfect estimation (ha,b (τ ) = 0), which has not been reported before in the literature. IV. SINRS AND SYSTEM SEP PERFORMANCE At the destination, the two decision variables y1 and y2 are used by the demodulator to make decisions about the two transmitted symbols x1 and x2 , respectively, and, however, our target here is to analyze the probability of making error in these decisions (i.e., evaluating the system’s SEP). First of all, in q-ary QAM AWGN point-to-point non-cooperative systems, the average SEP is given by [37, eqs. (5.2-79) and (5.2-78)] 2 3 1 AWGN Pe =1− 1−2 1− √ Q γ (14) q q−1

It is clear from (12) and (13) that both of the decision variables y1 and y2 are functions of the channel gains in the set S = {h1, (t1 ), h1, (t2 ), h2, (t1 ), h2, (t2 ), h,d (k1 ), h,d (k2 ), h1, (t1 ), h1, (t2 ), h2, (t1 ), h2, (t2 ), h,d (k1 ), h,d (k2 ) : ∀ = 1, 2 . . . , M }. However, in order to simplify the derivation of their conditional SINRs (say γ1 and γ2 , respectively), we propose to derive γ1 conditioned on the channel gains in the set S1 = {h1, (t1 ), h2, (t2 ), h,d (k1 ), h,d (k2 ) : ∀ = 1, 2 . . . , M }, and to derive γ2 conditioned on the channel gains in the set S2 = {h1, (t2 ), h2, (t1 ), h,d (k1 ), h,d (k2 ) : ∀ = 1, 2 . . . , M }.2 The following theorem and corollary provide the ultimate exact expressions for γ1 |S1 and γ2 |S2 , respectively. Theorem 4.1: The exact expression for γ1 |S1 is given by (15), shown at the bottom of this page, where A = ρs, h∗1, (t1 )h2, (t2 ) |h,d (k1 )|2 − |h,d (k2 )|2 B = |h1, (t1 )|2 |h,d (k1 )|2 σe2 , d |h1, (t1 )|2 + [σe21, 2 + (1 − ρ2s, ) × σ2, ]|h,d (k1 )|2 + σe21, σe2 , d C = σe2 , d |h1, (t1 )|2 |h,d (k1 )|2 + σe21, |h,d (k1 )|4 + σe21, σe2 , d 2 × |h,d (k1 )|2 ρ2s, |h2, (t2 )|2 + (1 − ρ2s, )σ2, 1 2 2 2 D = |h1, (t1 )| |h,d (k1 )| |h,d (k1 )| + 2 G 2 + |h,d (k1 )| + 1 × |h1, (t1 )|2 σe2 , d + σe21, |h,d (k1 )|2 + σe21, σe2 , d

2 The channel gains in the sets S and S are the ones that appear as coefficients 1 2 of the desired-signal terms in y 1 in (12) and y 2 in (13), respectively.

M 2 2 2 E s =1 |h1, (t1 )h,d (k1 )| + |h2, (t2 )h,d (k2 )| No γ1 |S1 = M 2 E Es M M s B A + + B + C + C + 2 No =1 N o =1 =1 D + D

(15)


are obtained directly from B , C and D , respecB , C , and D tively, by replacing the index 1, by 2, , the index 2, by 1, , t1 by t2 , t2 by t1 , k1 by k2 , and k2 by k1 wherever they appear. Proof: See Appendix B. Corollary 4.2: The exact expression for γ2 |S2 is the duality of γ1 |S1 , i.e., it is obtained from (15) just by replacing the index 1, by 2, and the index 2, by 1, wherever they appear. Proof: See Appendix C. Notice that the conditional SINRs obtained above are functions of important and more practical parameters, which are the channels correlation parameters ρa,b (which indicate the nodes mobility level, either slow or fast), the estimation errors variances σe2a , b (which indicate the level of the tracking loops’ impairment), and the per-symbol average signalEs . Thus, these expressions are general and to noise ratio N o valid for different realistic scenarios of mobile as well as static nodes for imperfect as well as perfect CSI estimation processes. For example, in the following, we shed light on these SINRs under two special cases of very-slow moving and very-fast moving both with perfect CSI estimation assumption. 1) Very-Slow Moving and Perfect CSI Estimation: Under the ideal situation of non-moving (or very slow moving) nodes (i.e, ρa,b = 1 and, according to (1), ha,b (τ2 ) = ha,b (τ1 ) = ha,b ) ˆ a,b (τ ) = and perfect CSI estimation (i.e., σe2a , b = 0, and thus, h ha,b (τ )), the SINRs γ1 |S1 and γ2 |S2 reduce to 2 M 2 2 2 h |h | s, ,d =1 Es G γ slow = (16) M 2 2 2 2 2No =1 (G |h,d | + 1) h |h,d | where hs, 2 = |h1, |2 + |h2, |2 . Further, if we substitute M = 1 in (16) (i.e., the case of single-relay network) and after doing some manipulations, γ slow reduces to slow γsingle-relay

Es = 2

h 2 |h , d |2 No No |h , d |2 1 N o + G2N o

(17)

which is know in the literature for a system model of OSTBC transmission over single-relay quasi-static fading network [12, eq. (7)]. We conclude from this that the obtained γ1 |S1 and γ2 |S2 generalize [12, eq. (7)] for a multiplerelay network with mobile nodes and imperfect channel estimation. 2) Very-Fast Moving and Perfect CSI Estimation: Under the case of extremely high mobility, severe timeselective fading (i.e., ρa,b → 0) could be assumed, where, according to (1), ha,b (τ2 ) = ea,b (τ2 ) which is completely independent of ha,b (τ1 ). Under this case along with perfect CSI estimation assumption, the SINR γ1 |S1 in (15) reduces to γ1fast |S1 M 2 2 2 E s =1 |h1, (t1 )h,d (k1 )| + |h2, (t2 )h,d (k2 )| No = Es M M ! =1 W + W N o + 2 =1 V + V (18)

where

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2 |h1, (t1 )|2 |h,d (k1 )|4 + 1 W = σ2, 1 V = |h1, (t1 )|2 |h,d (k1 )|2 |h,d (k1 )|2 + 2 G

! and γ2 |S2 reduces to γ2fast |S2 which is the duality of γ1fast |S1 . W and V are obtained directly from W and V , respectively, by replacing the index 1, by 2, , the index 2, by 1, , t1 by t2 , t2 by t1 , k1 by k2 , and k2 by k1 wherever they appear. In order to give useful insight about the system’s SEP performance under the effect of these two special cases, more discussions are provided in Section IV-C. It is clear that the SINR γ1 |S1 in (35) is random variable in terms of the channel gains in the set S1 = {h1, (t1 ), h2, (t2 ), h,d (k1 ), h,d (k2 ) : ∀ = 1, 2 . . . , M }, and, however, because of its very complicated form, obtaining its statistics in terms of either the probability-density-function (pdf) or moment-generating-function (mgf) is intractable. Same thing can be stated about γ2 |S2 . B. System Conditional and Average SEP Without loss of generality, and by assuming equiprobable symbol transmissions for x1 and x2 , we can express the overall SEP at the output of the q-QAM demodulator, conditioned on all channel gains in the sets S1 and S2 , as 1 1 (19) Pe |S1 ∪ S2 = Pey1 |S1 + Pey2 |S2 2 2 where Pey1 |S1 and Pey2 |S2 are the conditional probabilities of symbol decision errors made by the demodulator in estimating x1 from y1 (conditioned on S1 ) and in estimating x2 from y2 (conditioned on S2 ), respectively. Now, by directly using the AWGN-system’s SEP expression in (14) to evaluate Pey1 |S1 (by replacing γ by γ1 |S1 )3 and Pey2 |S2 (by replacing γ by γ2 |S2 ),4 we can obtain Pe |S1 ∪ S2 as in the following (approximate) closed-form expression ⎛$ %%2 $ 1 3 1⎝ 1− 2 1 − √ Q γ1 |S1 Pe |S1 ∪ S2 = 1 − 2 q q−1 $ −

1 1−2 1− √ q

$

Q

3 γ2 |S2 q−1

%%2 ⎞ ⎠.

(20)

The system’s average SEP, say P e , can be obtained from the conditional SEP Pe |S1 ∪ S2 in (20) as ⎛ ⎡$ %%2 ⎤ $ 3 1 1⎝ ⎣ ⎦ ES1 1 − 2 1 − √ Q γ1 |S1 Pe = 1 − 2 q q−1 ⎡$ %%2 ⎤⎞ $ 3 1 ⎦⎠ − ES2 ⎣ 1 − 2 1 − √ γ2 |S2 Q q q−1 (21) 3 This is valid because, conditioned on S , the terms I and χ 1 in (12) are 1 1 tight CLT-based approximate complex Gaussian variables. 4 This is valid because, conditioned on S , the terms I and χ 1 in (13) are 2 1 tight CLT-based approximate complex Gaussian variables.

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where EU · denotes the expectation operator with respect to U. Evaluating the last two expectations in (21) requires first deriving the pdfs of both γ|S1 and γ|S2 , which is as mentioned before too hard to accomplish. Therefore, we propose to compute P e semianalytical based on the sampling mean concept as Pe =

N 1 ( (j ) (j ) Pe |S1 ∪ S2 N

(22)

j =1

(j )

(j )

where Pe |S1 ∪ S2 is the generated conditional SEP in the jth realization obtained by (20) for random values of γ1 |S1 and γ2 |S2 , and N is the number of realizations in the simulation. The MATLAB source script that computes (22) is available online as an open and free source at [31]. In the numerical results section, we provide realistic link-level simulations to validate and show the tightness of the semianalytical results obtained based on (22). C. Nodes Mobility and Imperfect CSI Estimation Effects As an impact of the nodes mobility (i.e., the time-selective fading) and the imperfect CSI estimation, the system’s SEP performance experiences degradation especially at high values Es . This degradation is mainly represented by irreducible of N o conditional symbol error floors, which can be evaluated from Pe |S1 ∪ S2 in (20) as PeF |S1 ∪ S2 = lim Pe |S1 ∪ S2 Es No

→∞

⎛$ %%2 $ 1 3 1⎝ F 1−2 1− √ γ |S1 =1− Q 2 q q−1 1 %%2 ⎞ $ 3 1 ⎠ γ F |S2 − 1−2 1− √ Q q q−1 2

Fig. 1. Average SEP versus E s /N o for M = 1, 4-ary and 64-ary QAM constellations, transmission data-rate R s = 44 ksps, carrier frequency fc = 1.9 GHz, path-loss exponent n = 2, normalized nodes distances ds , = 1 and d , d = 2.

It is worthwhile to mention that the limit of Peslow in (26) as Es N o → ∞ is zero. Thus, under this special scenario, the system error performance does not experience error floors (as expected), which is due to the fact that the ALD decoder operates in its optimal version when the fading channels are quasi-static and the CSI estimation processes are perfect. On the other hand, under the case of very fast moving nodes considered in Section IV-A2, the system conditional SEP in (20) reduces to ⎛$ %%2 $ 3 1 1⎝ fast fast 1−2 1− √ γ |S1 Pe = 1 − Q 2 q q−1 1

$

$ (23)

−

1 1−2 1− √ q

$

Q

%%2 ⎞ 3 ⎠. γ fast |S2 q−1 2

where

(27)

γ1F |S1 =

lim γ1 |S1

Es No

→∞

M 2 2 2 =1 |h1, (t1 )h,d (k1 )| + |h2, (t2 )h,d (k2 )| = M 2 M =1 A + =1 B + B + C + C (24) and γ2F |S2 is the duality of γ1F |S1 . The average of PeF |S1 ∪ S2 can be computed semianalytical as F Pe

N 1 ( F (j ) (j ) = Pe |S1 ∪ S2 . N

(25)

j =1

If we consider now the special case of very-slow moving nodes and perfect CSI estimation considered in Section IV-A1, the system conditional SEP in (20) reduces to $ %%2 $ 3 1 γ slow . Peslow = 1 − 1 − 2 1 − √ Q q q−1 (26)

which, because of the very high mobility, suffers from relatively enormous conditional error floors of lim E s →∞ Pefast . No

V. NUMERICAL AND SIMULATION RESULTS In this section, we present numerical results along with realistic link-level simulations to validate the accuracy of the derived expressions and to show the effects of both the nodes mobility (the time-selective fading) and the imperfect CSI estimation on the system average SEP performance. These results are provided in terms of the QAM constellation size q, the number of relays M , the estimation error variances σe2a , b , the nodes’ relative speeds va,b in mile-per-hour (mph), the transmission symbol-rate Rs in kilo-symbol-per-second (ksps), the carrier frequency fc in GHz (va,b , Rs and fc are related to ρa,b by (2)), the path loss exponent n, the cooperating nodes normalized distances da,b . Further, moving-nodes and static-nodes cases are related to time-selective and quasi-static fading cases, respectively. In all figures below we set Er = Es = 1 (i.e., normalized energy). Fig. 1 shows numerical plots for the system’s average Es with M = 1 (single relay network), and 4-ary SEP versus N o


Fig. 2. Average SEP versus E s /N o for M = 1 and 2, 16-ary QAM constellation, transmission data-rate R s = 44 ksps, carrier frequency fc = 1.9 GHz, path-loss exponent n = 2, normalized nodes distances ds , = 1 and d , d = 2.

and 64-ary QAM constellations. From this figure, we can observe that the system average SEP results plotted using (20) and (22) provide a perfect match with the exact link-level simulation, which corroborates the correctness of the derived exact SINRs, γ1 |S1 and γ2 |S2 , and the tightness of the CLT-based approximate conditional SEP expression in (20). Further, this figure shows that, as compared to the system’s average SEP performance under the static-nodes (0 mph) and perfect CSI estimation (σe2s , = σe2 , d = 0) scenario, the high nodes mobility (e.g., 65 mph) degrades the system’s performance (especially, at Es ) and limits it by floors. This SEP performance high values of N o is further degraded if the CSI estimation is imperfect (e.g., with σe2s , = σe2 , d = 0.001). In Fig. 2, we plot the system’s average Es SEP versus N with 16-QAM constellation showing the effect o of the increase in the variances of the CSI estimation errors for different values of the number of relays M . It is clear that the system’s average SEP performance, as expected, is improved by increasing M as a result of the increase in the diversity-gain achieved via the relaying process. However, the harmful impact of both the nodes mobility and the imperfect CSI estimation still exists for any value of M , where, it is clear that the small increase in the estimation error variance, for e.g., from 0.0001 to 0.0005, causes a noticeable performance deterioration. We can also notice the high agreement between the numerical and simulation results. In Fig. 3, we plot the system’s average SEP versus the transmission data rate Rs in ksps with perfect as well as imperfect CSI estimations and different nodes speeds in mph. We can notice from this figure that the system’s average SEP is getting higher with the increase in the nodes speeds. However, for fixed Rs and speed, the system average SEP performance is getting worse when the CSI estimation is imperfect even though the estimation error variance is relatively small (for e.g., σe2s , = σe2 , d = 0.001). On the other hand, for fixed estimation error variance and speed, the system average SEP performance is improved by increasing Rs . This is due to the fact that increasing Rs increases ρ, which reduces the likelihood of the time-selective fading to occur.

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Fig. 3. Average SEP versus transmission data-rate in ksps with E s /N o = 25 dB, different nodes relative speeds of 10, 75 and 120 mph, M = 1, 16 − QAM constellation, carrier frequency fc = 2.4 GHz, path-loss exponent n = 2, normalized nodes distances ds , = 1 and d , d = 2.

Fig. 4. Simulation results for the system average SEP versus E s /N o for both cases of fixed and variable gains. M = 1, 4-ary QAM constellations, transmission data-rate R s = 25 ksps, carrier frequency fc = 1.9 GHz, pathloss exponent n = 2, normalized nodes distances ds , = 1 and d , d = 2. In case of mobile nodes the relative speed is 80 mph. In case of imperfect CSI estimation σ e2 a , b = 0.001.

The analytical results obtained in this paper so far are for the system with the fixed-amplification-gain G in (6) employed at the relay nodes. Another amplification gain that could be employed in amplify-and-forward cooperative based systems is the variable-amplification-gain, which requires CSI knowledge at the relay nodes [38]. For the system model under study here in this work, the variable-gain computed at the th relay over the τ th signaling period (where τ ∈ {t1 , t2 }) can be computed by Er G (τ ) = (28) Es Es 2 2 2 |h1, (τ )| + 2 |h2, (τ )| + No In Fig. 4, we provide link-level simulation results to show the average SEP performance of the variable-gain based system

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and compare it to that of the fixed-gain based system. It is clear from this figure that when all nodes are static and CSI estimation is perfect, the variable-gain system provides better error performance than that of the fixed-gain system especially at the medium and high SNR regions. On the other hand, in the case of high nodes mobility (for example at 80 mph) and imperfect CSI estimations system, the gap between the two systems’ SEP performance is getting narrower and, at the high SNR region, they provide matched performance. VI. CONCLUSION In this work, we have investigated the impact of both nodesmobility and imperfect CSI estimation on the SEP performance of an M -relay AF cooperative system employing AlamoutiOSTBC coding and decoding. We have first shown that the Alamouti decoder is no longer optimal as a result of its output’s interfering-terms and correlated decision-variables. We have then derived new exact expressions for effective SINRs associated with these decision variables. Benefiting from the CLT theorem, we have provided a tight approximate closed-form expression for the system’s conditional SEP by directly using the obtained SINRs along with the already known SEP expression of the AWGN channel. Using this conditional SEP expression, we have computed the system’s average SEP semianalytical and verified the results via Mote Carlo simulations. By this study, we have quantitatively shown that the nodes-mobility and/or the imperfect CSI estimation degrades the system SEP performance by irreducible error floors. As a future work, the analyses conducted in this paper could be repeated considering the variableamplification-gain employed at the system relays, and/or considering the case of generalized quasi-orthogonal STBCs.

M (

G 2 |h1, (t1 )h,d (k1 )|2 + |h2, (t2 )h,d (k2 )|2

(29)

=1

ζ=

M (

G 2 h∗1, (t1 )h2, (t1 )|h,d (k1 )|2

=1

− h2, (t2 )h∗1, (t2 )|h,d (k2 )|2 ϑ1 =

M (

G 2 h2, (t1 )h,d (k1 )h ∗,d (k1 )[h∗1, (t1 ) + h ∗1, (t1 )]

=1

− h∗1, (t2 )h∗,d (k2 )h ,d (k2 )[h2, (t2 ) + h 2, (t2 )] + h2, (t1 ) h ∗1, (t1 )|h,d (k1 )|2 − h∗2, (t2 )h 2, (t2 )|h,d (k2 )|2 υ 1 =

M (

(32)

[G 2 h,d (k1 )ns, (t1 ) + Gn,d (k1 )][h∗1, (t1 )h ∗,d (k1 )

=1

+ h ∗1, (t1 )h∗,d (k1 ) + h ∗1, (t1 )h ∗,d (k1 )] + [G 2 h∗,d (k2 )n∗s, (t2 ) + Gn∗,d (k2 )][h2, (t2 )h ,d (k2 ) + h 2, (t2 )h,d (k2 ) + h 2, (t2 )h ,d (k2 )] n 1 =

M (

(33)

G 2 h∗1, (t1 )|h,d (k1 )|2 ns, (t1 )

=1

+Gh∗1, (t1 )h∗,d (k1 )n,d (k1 )+G 2 h2, (t2 )|h,d (k2 )|2 n∗s, (t2 ) +Gh2, (t2 )h,d (k2 )n∗,d (k2 ) . (34)

β2 , ϑ2 , ξ2 are obtained from β1 , ϑ1 , ξ1 , respectively, just by replacing the index 1, by 2, and 2, by 1, , while n 2 and 1 and υ 1 , respectively, just by replacing υ 2 are obtained from n h1, (t1 ) by h2, (t1 ), h2, (t2 ) by −h1, (t2 ), h 1, (t1 ) by h 2, (t1 ), and h 2, (t2 ) by −h 1, (t2 ).

APPENDIX B THEOREM 4.1 PROOF

APPENDIX A COEFFICIENTS OF (13) AND (14)

β1 =

ξ1 =

M (

(30)

G 2 h,d (k1 )h ∗,d (k1 )[|h1, (t1 )|2 + h1, (t1 )h ∗1, (t1 )]

=1

+ h∗,d (k2 )h ,d (k2 )[|h2, (t2 )|2 + h∗2, (t2 )h 2, (t2 )]

It is clear from (12) that γ1 |S1 can be expressed as (35), shown at the bottom of this page, where P denotes the power operator. In the following we evaluate P(ζ|S1 ), n1 |S1 ) and P( υ1 |S1 ). It is obvious P(ϑ1 |S1 ), P(ξ1 |S1 ), P( from (30) that ζ|S1 is random in terms of h1, (t2 ) and h2, (t1 ) ∀. However, in order to evaluate P(ζ|S1 ), we first utilize the AR1 model in (1) to write h∗1, (t2 ) in terms of h1, (t1 ) and h2, (t1 ) in terms of h2, (t2 ), respectively, as follows

h∗1, (t2 ) = ρs, h∗1, (t1 ) + h2, (t1 ) = ρs, h2, (t2 ) +

1 − ρ2s, e∗1, (t2 )

(36)

1 − ρ2s, e2, (t1 ).

(37)

+ h1, (t1 )

h ∗1, (t1 )|h,d (k1 )|2 + h∗2, (t2 )h 2, (t2 )|h,d (k2 )|2 (31)

γ1 |S1 =

By substituting (36) and (37) into (30), we can then expand ζ as

|β1 |2 (Es /2) P(ζ|S1 )(Es /2) + P(ϑ1 |S1 )(Es /2) + P(ξ1 |S1 )(Es /2) + P( n1 |S1 ) + P( υ1 |S1 )

(35)


ζ = G2

M (

ρs, h∗1, (t1 )h2, (t2 ) |h,d (k1 )|2 − |h,d (k2 )|2

=1

μ1

+ G2

M (

1 − ρ2s, h∗1, (k)|h,d (k1 )|2 e2, (t1 ) −

1 − ρ2s,

× h2, (t2 )|h,d (k2 )|2 e∗1, (t2 )

(38)

2 ) and Given that e∗1, (t2 ) and e2, (t1 ) in (38) are CN 0, σ1, 2 CN 0, σ2, ), respectively, P(ζ|S1 ) can be obtained by

P(ζ | S1 ) = |μ1 |2 +G 4

2 (1−ρ2s, ) σ2, |h1, (t1 )|2 |h,d (k1 )|4

=1

+

2 σ1, |h2, (t2 )|2 |h,d (k2 )|4

on S1 , is ZMCSCG and has the following power P( n1 |S1 ) =

M (

No G 2 |h1, (t1 )h,d (k1 )|2 (G 2 |h,d (k1 )|2 + 1)

=1

=1

M (

3455

.

(39)

+ |h2, (t2 )h,d (k2 )|2 (G 2 |h,d (k2 )|2 + 1) . (43) 1 in (33) is random with respect to ns, (t1 ), Conditioned on S1 , υ n,d (k1 ), n∗s, (t2 ), n∗,d (k2 ) as well as to h1, (t1 ), h2, (t2 ), h,d (k1 ) and h,d (k2 ). Given that ns, (t1 ), n,d (k1 ), n∗s, (t2 ), n∗,d (k2 ) are CN (0, No ); and ha,b (τ ) is CN (0, σe2a , b ) for all 1 |S1 can be given by (a, b) and τ ∈ {t1 , t2 }, the power of υ P( υ1 |S1 ) = G 2 No

M ( 2 G |h,d (k1 )|2 + 1 |h1, (t1 )|2 σe2 , d + =1

|h,d (k1 )|2 σe21, + σe21, σe2 , d + G 2 |h,d (k2 )|2 + 1 × |h2, (t2 )|2 σe2 , d + |h,d (k2 )|2 σe22, + σe22, σe2 , d

From (31) the power of ϑ1 |S1 , which is random in terms of h1, (t1 ), h2, (t2 ), h,d (k1 ), h,d (k2 ), can be evaluated as P(ϑ1 |S1 ) =

M (

G 4 |h1, (t1 )|2 |h,d (k1 )|2 |h1, (t1 )|2 σe2 , d +σe21,

=1

|h,d (k1 )|4 + σe21, σe2 , d + |h2, (t2 )|2 |h,d (k2 )|2 |h2, (t2 )|2 σe2 , d + |h,d (k2 )|4 σe22, + σe22, σe2 , d ]. (40) As can be seen from (32), ξ1 |S1 is random with respect to h1, (t2 ), h2, (t1 ), h1, (t1 ), h2, (t2 ), h,d (k1 ) and h,d (k2 ) ∀. Now, In order to simplify the evaluation of P(ξ1 |S1 ), we first substitute (36) and (37) into (32), which yields expanding ξ1 as ξ1 =

M (

G 2 ρs, h2, (t2 ) + 1 − ρ2s, e2, (t1 ) h ∗1, (t1 )

=1

|h,d (k1 )|2 + h,d (k1 )h ∗,d (k1 )[h∗1, (t1 ) + h ∗1, (t1 )] − ρs, h∗1, (t1 ) + 1 − ρ2s, e∗1, (t2 h 2, (t2 )|h,d (k2 )|2 + h∗,d (k2 )h ,d (k2 )[h2, (t2 ) + h 2, (t1 )] (41) Since e2, (t1 ), e1, (t2 ), h1, (t1 ), h2, (t2 ), h,d (k1 ), and h,d (k2 ) ∀ are complex Gaussian, P(ξ1 |S1 ) can be given as P(ξ1 |S1 ) =

M (

G 4 [|h1, (t1 )|2 |h,d (k1 )|2 σe2 , d +|h,d (k1 )|4 σe21,

=1

+ |h,d (k1 )|2 σe21, σe2 , d ][ρ2s, |h2, (t2 )|2

2 + (1 − ρ2s, )σ2, ]

+ [|h2, (t2 )|2 |h,d (k2 )|2 σe2 , d + σe22, |h,d (k2 )|4

2 + σe22, |h,d (k2 )|2 σe2 , d ][ρ2s, |h1, (t1 )|2 + (1 − ρ2s, )σ1, ] . (42) Because ns, (t1 ), n,d (k1 ), n∗s, (t2 ) and n∗,d (k2 ) for all are in1 in (34), conditioned dependent and distributed as CN (0, No ), n

(44) Finally, by substituting (39), (40), and (42)–(44) into (35), and after doing some simplifications, we obtain γ1 |S1 as in (15). APPENDIX C COROLLARY 4.2 PROOF By writing h1, (t1 ) in terms of h1, (t2 ) as h1, (t1 ) = ρs, h1, (t2 ) + 1 − ρ2s, e1, (t1 ) and writing h∗2, (t2 ) in terms of h2, (t1 ) as h∗2, (t2 ) = ρs, h∗2, (t1 ) + 1 − ρ2s, e∗2, (t2 ), and following the same steps as in Appendix B to derive γ1 |S1 , we can obtain γ2 |S2 as described in Corollary 4.2. REFERENCES [1] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [3] V. B. Pham, “Spacetime block code design for LTE-advanced systems,” Trans. Emerg. Telecommun. Technol., vol. 26, no. 5, pp. 918–928, May 2015. [4] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in the wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [5] X. Li, T. Jiang, S. Gui, J. An, and Q. Zhang, “Cooperative communications based on rateless network coding in distributed MIMO systems,” IEEE Wireless Commun., vol. 17, no. 3, pp. 60–67, Jun. 2010. [6] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless network,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [7] R. U. Nabar, H. Blcskei, and F. W. Kneubhler, “Fading relay channels: Performance limits and space-time signal design,” IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004. [8] Y. Jing and H. Jafarkhani, “Using orthogonal and quasi-orthogonal designs in wireless relay networks,” IEEE Trans. Inf. Theory, vol. 53, no. 11, pp. 4106–4118, Nov. 2007. [9] Y. Jing and H. Jafarkhani, “Distributed space-time coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524–3536, Dec. 2006. [10] S. Yiu, R. Schober, and L. Lampe, “Distributed space-time coding,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1195–1206, Dec. 2006.

3456

[11] T. Unger and A. Klein, “On the performance of distributed space-time block codes in cooperative relay networks,” IEEE Commun. Lett., vol. 11, no. 5, pp. 411–413, May 2007. [12] S. Chen, W. Wang, and X. Zhang, “Performance analysis of OSTBC transmission in amplify-and-forward cooperative relay networks,” IEEE Trans. Veh. Technol., vol. 59, no. 1, pp. 105–113, Jan. 2010. [13] T. Q. Duong, G. C. Alexandropoulos, H.-J. Zepernick, and T. A. Tsiftsis, “Orthogonal space-time block codes with CSI-assisted amplifyand-forward relaying in correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol., vol. 60, no. 3, pp. 882–889, Mar. 2011. [14] L. Jayasinghe, N. Rajatheva, P. Dharmawansa, and M. Latva-aho, “Non-coherent amplify-and-forward MIMO relaying with OSTBC over Rayleigh-Rician fading channels,” IEEE Trans. Veh. Technol., vol. 62, no. 4, pp. 1610–1622, May 2013. [15] P. Jayasinghe, L. Jayasinghe, M. Juntti, and M. Latva-Aho, “Performance analysis of optimal beamforming in fixed-gain AF MIMO relaying over asymmetric fading channels,” IEEE Trans. Commun., vol. 62, no. 4, pp. 1201–1217, Apr. 2014. [16] O. Trullols-Cruces, M. Fiore, and J. M. Barcelo-Ordinas, “Cooperative download in vehicular environments,” IEEE Trans. Mobile Comput., vol. 11, no. 4, pp. 663–678, Apr. 2012. [17] P. Surez-Casal, J. Rodrguez-Pieiro, J. A Garca-Naya, and L. Castedo, “Experimental evaluation of the WiMAX downlink physical layer in highmobility scenarios,” EURASIP J. Wireless Commun. Netw., vol. 2015, pp. 1–21, 2015. [18] Y. Khattabi and M. M. Matalgah, “Performance analysis of multiple-relay AF cooperative systems over Rayleigh time-selective fading channels with imperfect channel estimation,” IEEE Trans. Veh. Technol., vol. 65, no. 1, pp. 427–434, Jan. 2016. [19] V. Tarokh, A. Naguib, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communications: Performance criteria in the presence of channel estimation errors, mobility and multiple paths,” IEEE Trans. Commun., vol. 147, no. 2, pp. 199–207, Feb. 1999. [20] Z. Liu, X. Ma, and G. B. Giannakis, “Space-time coding and Kalman filtering for time-selective fading channels,” IEEE Trans. Commun., vol. 50, no. 2, pp. 183–186, Feb. 2002. [21] H. Zhu, B. Xia, and Z. Tan, “Performance analysis of Alamouti transmit diversity with QAM in imperfect channel estimation,” IEEE J. Sel. Areas Commun., vol. 29, no. 6, pp. 1242–1248 Jun. 2011. [22] S. Ohno, “Impact of time-selective fading on orthogonal space-time block coding,” in Proc. IEEE Global Telecommun. Conf., Nov. 2004, vol. 4, pp. 2620–2624. [23] H. T. Cheng and T. M. Lok, “Detection schemes for distributed spacetime block coding in time-varying wireless cooperative systems,” in Proc. IEEE TENCON 2005, Nov. 2005, pp. 289–293. [24] W. Jaafar, W. Ajib, and D. Haccoun, “Performance evaluation of distributed STBC in wireless relay networks with imperfect CSI,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun., Sep. 2009, pp. 2335–2339. [25] R. Mo, Y. H. Chew, and C. Yuen, “Information rate and relay precoder design for amplify-and-forward MIMO relay networks with imperfect channel state information,” IEEE Trans. Veh. Technol., vol. 61, no. 9, pp. 3958–3968, Nov. 2012. [26] B. Zafar, S. Gherekhloo, F. Roemer, and M. Haardt, “Impact of synchronization errors on Alamouti-STBC-based cooperative MIMO schemes,” in Proc. 2012 IEEE 7th Sens. Array Multichannel Signal Process. Workshop, Jun. 2012, pp. 81–84. [27] M. K. Arti, R. K. Mallik, and R. Schober, “Performance analysis of OSTBC relaying in AF MIMO relay systems with imperfect CSI,” IEEE Trans. Veh. Technol., vol. 64, no. 7, pp. 3291–3298, Jul. 2016. [28] Y. M. Khattabi and M. M. Matalgah, “Improved error performance ZFSTD for high mobility relay-based cooperative systems,” Electron. Lett., vol. 52, no. 4, pp. 323–325, Feb. 2016. [29] Y. M. Khattabi and M. M. Matalgah, “A low-complexity sub-optimal decoder for OSTBC-based mobile cooperative systems,” in Proc. 2016 IEEE Wireless Commun. Netw. Conf., Doha, Qatar, Apr. 2016, pp. 1–6. [30] A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. New York, NY, USA: McGraw-Hill, 2002. [31] [Online]. Available: https://engineering.olemiss.edu/∼mustafa/students/ index.html [32] W. C. Jakes and D. C. Cox, Eds., Microwave Mobile Communications. New York, NY, USA: Wiley-IEEE Press, 1994. [33] M. O. Hasna and M.-S. Alouini, “A performance study of dual-hop transmission with fixed Gain relays,” IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 1963–1968, Nov. 2004.


[34] M. Enescu and V. Koivunen, “Time-varying channel tracking for spacetime block coding,” in Proc. IEEE Veh. Technol. Conf., Birmingham, AL, USA, May 2002, vol. 1, pp. 294–297. [35] M. L. Ku and C. C. Huang, “A refined channel estimation method for STBC/OFDM systems in high-mobility wireless Channels,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4312–4320, Nov. 2008. [36] Y. M. Khattabi, “Performance evaluation and improvement of wireless amplify-and-forward cooperative-based systems under nodes mobility and imperfect CSI estimation impacts,” Ph.D. dissertation, The Department of Electrical Engineering. The University of Mississippi, Oxford, MS, USA, Jul. 2016. [37] J. G. Proakis, Digital Communications, 4th ed. New York, NY, USA: McGraw-Hill, 2001. [38] M. O. Hasna and M.-S. Alouini, “End-to-End performance of transmission systems with relays over Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. 1126–1131, Nov. 2003. Yazid M. Khattabi (M’14) received the Bachelor’s degree in electrical engineering with emphasis in electronics and communications, the Master’s degree in electrical engineering with emphasis in wireless communications, both from Jordan University of Science and Technology, Irbid, Jordan, in 2008 and 2010, respectively, and the Ph.D. degree in electrical engineering from the University of Mississippi, Oxford, MS, USA. From March 2011 to December 2012, he was a Telecommunications and Electronics Design Engineer with King Abdullah II Design and Development Bureau, Amman, Jordan. During 2013–2016, he was a Research Assistant with the Center for Wireless Communications, The University of Mississippi, where he received several research awards. Since August 2016, he has been with The University of Jordan, Amman, Jordan, where he is currently an Assistant Professor in electrical engineering. His research interests include wireless communications with emphasis on the performance evaluation and optimization of high-mobility wireless cooperative communication systems over fading channels. He was a Reviewer for several refereed international journals and conferences. Mustafa M. Matalgah received the Bachelor’s and Master’s degrees from Jordan and the Ph.D. degree from the University of Missouri, Columbia, MO, USA, all in electrical engineering. He has a wide range of academic and industry experiences in the electrical engineering field with emphasis on communication engineering. From 1996 to 2002, he was with Sprint, Kansas City, MO, USA, where he held various technical positions leading a wide range of projects dealing with optical communication systems deployment and the evaluation and assessment of wireless communication emerging technologies for 3G and the next generation networks. Since August 2002, he has been with The University of Mississippi, Oxford, MS, USA, where he is currently a Full Professor in electrical engineering. In Summer 2008, he was a Visiting Professor with Chonbuk National University, Jeonju, South Korea. He was also a Visiting Professor and program evaluator with Misr International University, Cairo, Egypt, in Summers 2009, 2010, and 2012. From 2014 to 2015, he also held an academic position with King Faisal University, Saudi Arabia. His current technical and research experience is in the performance evaluation and optimization of wireless communication systems in fading channels. He also previously published in the fields of signal processing and optical information processing. He has authored or coauthored more than 150 archival publications (including journals, conference proceedings, book chapters, and patents) in addition to more than two dozens of industry technical reports in these areas. He was a Research Supervisor and on defense committees of several M.S. and Ph.D. students. He is on the Editorial Board of four international journals, served as a Member and the Chair of several university committees, the Chair of several international conferences sessions and workshops, a Member of several international conferences technical program and organizing committees, a reviewer for several funding proposals in USA and Canada, and was the Project Manager of several projects in industry. He served on the Faculty Senate of the University of Mississippi for four years. He received several certificates of recognition for his work accomplishments in the industry and academia. He was the recipient/corecipient of the Best Paper Award on several international and regional conferences and workshops such as the IEEE ISCC 2005 Conference, La Manga del Mar Menor, Spain, and the IEEE RWS 2011 Conference, Phoenix, AZ, USA. He was also the recipient of the 2006 School of Engineering Junior Faculty Research Award at The University of Mississippi.