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Using a Bit Error Rate Criterion ... rate (BER), and is referred as joint bit error rate (JBER) detector. ... minimum bit error rate (MBER) are proposed for the NAF.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 4, APRIL 2014

Linear Transceiver Design for Nonorthogonal Amplify-and-Forward Protocol Using a Bit Error Rate Criterion Qasim Zeeshan Ahmed, Ki-Hong Park, Member, IEEE, Mohamed-Slim Alouini, Fellow, IEEE, and Sonia Aïssa, Senior Member, IEEE

Abstract—The ever growing demand of higher data rates can now be addressed by exploiting cooperative diversity. This form of diversity has become a fundamental technique for achieving spatial diversity by exploiting the presence of idle users in the network. This has led to new challenges in terms of designing new protocols and detectors for cooperative communications. Among various amplify-and-forward (AF) protocols, the half duplex nonorthogonal amplify-and-forward (NAF) protocol is superior to other AF schemes in terms of error performance and capacity. However, this superiority is achieved at the cost of higher receiver complexity. Furthermore, in order to exploit the full diversity of the system an optimal precoder is required. In this paper, an optimal joint linear transceiver is proposed for the NAF protocol. This transceiver operates on the principles of minimum bit error rate (BER), and is referred as joint bit error rate (JBER) detector. The BER performance of JBER detector is superior to all the proposed linear detectors such as channel inversion, the maximal ratio combining, the biased maximum likelihood detectors, and the minimum mean square error. The proposed transceiver also outperforms previous precoders designed for the NAF protocol. Index Terms—Cooperative diversity, nonorthogonal amplifyand-forward protocol, minimum mean square error (MMSE), bit error rate (BER).

on the principle of decode-and-forward (DF) or amplify-andforward (AF). The AF scheme is preferred because of its lower implementation complexity at the relay as compared to the DF scheme [1, 4]. The bit error rate (BER) performance of AF scheme is also similar to that of the DF scheme [2]. Therefore, in this contribution only AF scheme is considered. Three different time division multiple access protocols are introduced by Nabar et al. in [1]. Among these three protocols, protocol-I also known as non-orthogonal AF (NAF) relaying, provides maximum degrees of broadcasting and receive collision [1, 5, 6]. By employing the NAF protocol, different signals can be conveyed to the relay and the destination at the same time, thereby improving the spectral efficiency of the system. It is also shown in [1, 5] that protocol-I outperforms protocol-II and protocol-III in terms of error performance and capacity. This superiority of NAF protocol is achieved at the expense of a higher receiver complexity. Therefore, the development of efficient receivers for the NAF protocol becomes paramount. A. Previous Work

I. I NTRODUCTION OOPERATIVE diversity enhances the performance of wireless communication systems over fading channels [1, 2]. The users share their antennas in a distributed manner to form a virtual antenna array resulting in spatial diversity. This diversity mitigates the effect of fading and improves the transmission reliability [3]. An idle user may serve as a relay for an active source and destination. Usually, the relay operates

C

Manuscript received February 27, 2013; revised July 11 and November 6, 2013; accepted December 31, 2013. The associate editor coordinating the review of this paper and approving it for publication was M. Bhatnagar. This work was supported by a grant from the King Abdulaziz City of Science and Technology (KACST), SABIC post-doctoral fellowship, and an NPRP project from Qatar National Research Fund (QNRF) (A member of Qatar Foundation). Q. Z. Ahmed, K.-H. Park, and M.-S. Alouini are with the Computer, Electrical, and Mathematical Science and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Kingdom of Saudi Arabia (e-mail: {qasim.ahmed, kihong.park, slim.alouini}@kaust.edu.sa). Q. Z. Ahmed and M.-S. Alouini are also members of the KAUST Strategic Research Initiative (SRI) in Uncertainty Quantification in Computational Science and Engineering. S. Aïssa is with the Institut National de la Recherche Scientifique (INRS), University of Quebec, Montreal, QC, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2014.022114.130369

Maximum likelihood (ML) detector with complete channel state information (CSI) for NAF protocol is proposed in [1]. The ML detector achieves full diversity with complete and partial CSI as shown in [1] and [6], respectively. The computational complexity of the ML detector increases with the constellation size and becomes intractable for large constellations or codewords [7]. Therefore, due to its high computational complexity, the ML detector is not a preferred solution. In order to reduce the computational complexity, three different linear detectors have been alternatively proposed for NAF protocol in [8]. These detectors operate on the principle of channel inversion (CINV), maximal ratio combining (MRC), and biased ML (BML). From [8], it is concluded that the BER performance of the MRC detector is superior to that of the CINV detector but inferior to the BML detector. However, despite their low complexity, error floors are observed at high signal-to-noise ratio (SNR) for all these linear detectors [8]. In order to solve this problem, linear detectors based on the principles of minimum mean square error (MMSE) and the minimum bit error rate (MBER) are proposed for the NAF protocol in [9]. It is shown that the MMSE detector is not able to achieve full diversity, while the MBER detector is able to achieve similar BER performance as the ML detector. The ML

c 2014 IEEE 1536-1276/14$31.00 

AHMED et al.: LINEAR TRANSCEIVER DESIGN FOR NONORTHOGONAL AMPLIFY-AND-FORWARD PROTOCOL USING A BIT ERROR RATE CRITERION

and MBER detectors are able to achieve diversity order of 2 for one transmitted symbol, while the diversity order achieved by the other transmitted symbol is 1. An optimal precoder is required in order to exploit full diversity for both transmitted symbols [10–14]. Precoder for NAF protocol was first designed by Ding et al. in [11]. This unitary precoder is designed based on pairwise error probability (PEP). As the precoder is fixed, the diversity order is obtained by averaging the PEP over the channel. The diversity order calculated is equivalent to ρ−2 ln ρ where ρ is the SNR of the system. The simulation results in [11] and [12] indicate that the use of such a precoder provides superior performance as compared to the system with no precoder. Interestingly, the proposed precoder also outperforms systems that employ distributed space-time codes [15]. However, the receiver operates on the principles of ML decoding [11, 12]. B. Motivation and Contributions Inspired by the aforementioned observations, this paper is concerned with the proposal of a joint generalized scheme for precoder and receiver design, for the NAF protocol. This new approach is based on minimizing the error probability or BER. Initially, the BER is calculated as a function of the transformation matrix of the precoder at the source and weight vectors at the destination. In order to minimize the BER, the first-order derivatives with respect to the weight vectors and transformation matrix are calculated. These weight vectors and transformation matrix are then updated using a steepest descent (SD) approach. Simulations indicate that the transceiver is able to exploit full diversity for both transmitted symbols. Our proposed scheme not only significantly improves the BER performance as compared to systems with no precoding, but also outperforms systems that employ an optimal precoder such as the ones in [11, 12, 16, 17]. The diversity order of the proposed detector is difficult to evaluate due to the numerical SD approach. As precoder and weights are a function of channel and their probability density functions (pdf) are not known, evaluating the diversity order is difficult. However, the simulation results attest that the slope of the proposed detector is similar to the slope of the optimal precoder as proposed in [11, 12]. In order to measure the maximum achievable information rate, channel capacity of the system is also calculated. It is observed that the proposed scheme is superior to all the previous schemes proposed for systems which implement NAF protocol. The remainder of this paper is organized as follows. Section II describes the general structure of the system and the basic assumptions under consideration. In Section III, the joint BER (JBER) detector is developed. In Section IV, simulation results for the JBER detectors are presented and discussed. Finally, the paper is concluded in Section V. Throughout this paper, the following notations are used. Bold upper-case letters denote matrices and bold lower-case letters denote vectors. For an arbitrary matrix, A, A∗ , AT , AH and A−1 denote the complex conjugate, transpose, Hermitian and the inverse of a matrix A, respectively. E[·] denotes expectation, diag(·) stands for diagonal matrix, | · | denotes the absolute value and T r(·) represents the trace of a matrix.

Relay

R

hSR

Phase-I Phase-II

hRD

hSD

S

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D

Destination Source Fig. 1. Schematic of the cooperative communication system under consideration.

II. C OOPERATIVE C OMMUNICATION M ODEL Consider the cooperative communication system shown in Fig. 1. Data is transmitted from the source terminal S to the destination terminal D with the assistance of the relay terminal R. All terminals are equipped with single antennas. The terminals cannot transmit and receive the message simultaneously. The data transmission takes place in two phases, namely phase-I and phase-II. When employing the NAF protocol S broadcasts data b1 to both R and D in phase-I. Both R and S communicate with D in phase-II. During this phase, terminal R simply amplifies and forwards the received signal of the transmission of b1 in phase-I, while S transmits b2 to D directly. As b1 is received with the help of two time slots, the maximum diversity order achieved by b1 is 2. Similarly, the maximum diversity order for b2 is 1. As b2 looses diversity order, we propose designing a precoder T such that both b1 and b2 are transmitted together. Now, S broadcasts data which is a combination of b1 and b2 to both R and D in phase-I. During phase-II, terminal R simply amplifies and forwards the received signal which is received in phase-I, while S transmits a combination of b1 and b2 to D directly. Now, both b1 and b2 are received with the help of two time slots and the maximum diversity order achieved by b1 and b2 will be the same. A power constraint on the precoder T is placed such that T r(TH T) = Es . This constraint indicates that the transmit power of both bits does not exceed the maximum transmit energy of the source, Es . For ease of analysis, we assume that the modulation scheme is binary phase shift keying (BPSK), that is, bi = ±1 for i = 1, 2, and the maximum transmit energy of the source will be Es = 2.0. It is straightforward to extend the system model utilizing BPSK to other modulation schemes which will be discussed in Section III-B. Let us now look at the joint transmission of bits b1 and b2 in more detail. A. Phase-I: Transmission From Source The precoder T multiplies the transmit signals b1 and b2 , such that the transmitted signal vector is represented as   x x1 x2

=  Tb t11 = t21

t12 t22

    b1 t b + t12 b2 = 11 1 . b2 t21 b1 + t22 b2

(1)

The signal received at terminal D is given by yD1

= hSD x1 + nD1 = hSD t11 b1 + hSD t12 b2 + nD1 ,

(2)

where hSD is a complex-valued gain of the channel between S and D and nD1 is complex additive white Gaussian noise 2 (AWGN) with zero mean and variance σD .

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 4, APRIL 2014

The signal received at terminal R is given by yR

= =

hSR x1 + nR hSR t11 b1 + hSR t12 b2 + nR ,

y (3)

where hSR is a complex-valued channel gain of the link between S and R and nR is complex AWGN with mean zero 2 and variance σR . B. Phase-II: Transmission From Relay and Source

This ensures that the relay output satisfies an average or 2 = 1, but allows long-term power constraint of E[|yR |2 ]/ζR the instantaneous transmit power to be much larger than the average [4, 18, 19]. The received signal at D, during phase-II, is given by = = +

yR + nD2 ζR hSD t21 b1 + hSD t22 b2 + nD2 hRD hSR t11 b1 + hRD hSR t12 b2 + hRD nR  , (5) 2 |hSR |2 |t11 |2 + |hSR |2 |t12 |2 + σR hSD x2 + hRD

where hRD is a complex valued channel gain between nodes R and D, and nD2 is complex AWGN with mean zero and 2 variance σD . C. Receiver Structure The effective input-output relation for the two phases can be summarized as y

= HTb + n,

(6)

where y = [yD1 , yD2 ]T is the received signal vector and H is the effective 2 × 2 channel matrix given by     hSD 0 (7) H= = h1 , h2 , αhSR hRD hSD where α = ζ1R and h1 and h2 represent the first and second columns of the channel matrix H corresponding to x1 and x2 , respectively. The noise vector can be represented as  nD1  .(8) n = 1 h n + nD2 |hSR |2 |t11 |2 +|hSR |2 |t12 |2 +σ2 RD R R

The noise conditioned on the channel knowledge H can be represented as a complex Gaussian noise with mean zero and variance Σ which can be expressed as  2 σD , 0 2 Σ= . (9) |hRD |2 σR 2 0 , |hSR |2 |t11 |2 +|h 2 + σD 2 2 SR | |t12 | +σ R

w2

Fig. 2.

z2

(z1 ) (z 2 )

zR

bˆ1

zR

bˆ2

1

2

Receiver schematic block diagram for the NAF protocol.

III. D ETECTORS FOR NAF P ROTOCOL

During this phase, the received signal at R is normalized by a factor ζR , which is equivalent to  2. (4) ζR = |hSR |2 |t11 |2 + |hSR |2 |t12 |2 + σR

yD2

w1

z1

A. BPSK Constellation For JBER detection, the detector consists of a complex valued matrix W = [w1 , w2 ], where wk ∈ C2×1 . The first and second columns of W, i.e., w1 and w2 , represent the weight vector for b1 and b2 , respectively. The output of the detector is characterized by z =   z1 = z2

WH y   H   H w1 Ht2 b2 + w1H n w1 Ht1 b1 + , (10) w2H Ht2 b2 w2H Ht1 b1 + w2H n







Signal-Part

Noise-Part

where t1 and t2 is the first and second column of the precoder T. As our transmitted signal vector is modulated using BPSK, only the real part is of interest. Therefore, zR1 = (z1 ) and zR2 = (z2 ) are sent to the demodulator to detect bits b1 and b2 , as shown in Fig. 2. As n is assumed to be Gaussian, wkH n is also Gaussian with variance wkH Σwk . From (10), it can be observed that for detection of b1 , w1H Ht2 b2 acts as interference. Therefore, the noise part is modelled as Gaussian with mean (w1H Ht2 b2 ) and variance w1H Σw1 . Similarly, for b2 , the noise part is modelled as Gaussian with mean (w2H Ht1 b1 ) and variance w2H Σw2 . The pdf of zR1 and zR2 can be represented as 1 p(zRk | bk , bl ) =  (11) 2πwkH Σwk   2  zRk − (wkH Htk bk + wkH Htl bl ) , × exp − 2wkH Σwk where k, l = 1, 2 and k = l. In order to minimize the BER performance of the detector, we need to determine the probability of error PE . First, PE (wk , T) for bk is given by  0 PER (wk , T) = p(zRk |bk = +1)P (bk = +1)dzRk −∞  ∞ + p(zRk |bk = −1)P (bk = −1)dzRk , 0

k = 1, 2. (12)

For equally likely symbols, (12) reduces to  0 PER (wk , T) = p(zRk |bk = +1)dzRk −∞  0 = p(zRk |bk = +1, bl = +1)P (bl = +1)dzRk −∞  0 + p(zRk |bk = +1, bl = −1)P (bl = −1)dzRk −∞

AHMED et al.: LINEAR TRANSCEIVER DESIGN FOR NONORTHOGONAL AMPLIFY-AND-FORWARD PROTOCOL USING A BIT ERROR RATE CRITERION

 1 0 = p(zRk |bk = +1, bl = +1)dzRk 2 −∞  0 1 + p(zRk |bk = +1, bl = −1)dzRk , 2 −∞

(13)

where k, l = 1, 2 and k = l and the conditional pdf can be calculated as  0 p(zRk |bk = +1, bl = +1)dzRk −∞  0 1  = −∞ 2πwH Σw k k   2 zRk − (wkH H(tk + tl )) × exp − dzRk 2wkH Σwk ⎞ ⎛ H H(t + t )) (w k l ⎠ k  , k, l = 1, 2, k = l (14) = Q⎝ H wk Σwk and Q(·) is the standard Gaussian Q-function, defined as [20]  2  ∞ 1 v Q(u) = √ exp − dv. (15) 2 2π u Similarly, we have  0 P (zRk |bk = +1, bl = −1)dzRk −∞ ⎞ ⎛ H (w H(t − t )) k l k ⎠ , k, l = 1, 2, k = l.  (16) = Q⎝ H wk Σwk Substituting (14) and (16) in (13), we get ⎛ ⎞ 1 ⎝ (wkH H(tk + tl )) ⎠  PER (wk , T) = Q 2 wkH Σwk ⎛ ⎞ 1 ⎝ (wkH H(tk − tl )) ⎠  + Q , 2 wkH Σwk ⎞ ⎛ 1  ⎝ (wkH H(tk bk + tl bl )) ⎠  , Q = 2 wH Σw bl ∈B

k

k, l = 1, 2,

k

k = l, (17)

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Step 2: Determine suitable step-sizes μT and μwk , where the subscript represents the respective precoder or the weight vector. A larger μ will lead to divergence of the algorithm while a smaller μ will lead to slower convergence. Step 3: Once the weights are chosen, determine the gradient of the probability of error based on (17). The gradients, of PER (wk , T) is represented in (18). Step 4: The updated weights of the SD-based detector are determined as ∂PER (wk , T) , k = 1, 2, (19) wk (iter + 1) = wk (iter) − μwk ∂wk where ‘iter’ represents the iterations number. After completion of the iterations, iter, the weight vector, wk (iter + 1), is obtained and is normalized to satisfy the constraint wkH (iter + 1)wk (iter + 1) = 1. Step 5: After updating the weights w1 and w2 , the precoder matrix T is required to be updated. In order to update T, the partial derivative of PER is required which is a combination of PER (w1 , T) and PER (w2 , T). Therefore, for precoder T, PER can be given as PER (w1 , w2 , T)

= PER (w1 , T) + PER (w2 , T) (20)

and the gradient of PER (w1 , w2 , T) is represented in (21). Step 6: The updated precoder T is determined as ∂PER (w1 , w2 , T) . (23) ∂T In order to satisfy the constraint that the T r(TH T) = 2.0, the precoder is normalized after each iteration to satisfy the constraint. The steps of the JBER detector are summarized in the algorithm-I. Next, the above analysis is extended for higher modulation, followed by an example and then computational complexity of the algorithm is calculated. T(iter + 1) = T(iter) − μT

1. Initialize the precoder and weights of the filter with random values such that T r(TH T) = 2 and w1H w1 = w2H w2 = 1. 2. Find appropriate step sizes μT , μw1 and μw2 . for iter = 1 : Niter, where Niter is the total number of iterations. (wk ,T) 3. Evaluate ∂PE∂w using (18). k 4. Update the weights wk (iter) using (19). Normalize the weight vectors to satisfy the constraint. 1 ,w2 ,T) 5. Evaluate ∂PE (w∂T using (21). 6. Update the precoder T(iter) using (23) and normalize the updated precoder to satisfy the energy constraint. end

where bk = +1 and B contains all the available constellation for bl = {±1}. From (17), it can be observed that the JBER detector does not have a closed-form expression. Therefore, the weights and precoder need to be determined iteratively with the help of SD algorithm. The SD-based algorithm involves six steps which are explained as follows: Step 1: Initialize the precoder and the weights of the filter Algorithm I randomly such that T r(TH T) = 2 and wkH wk = 1, where Joint Bit Error Rate Detector k = 1, 2.     ∗ H H ∂PER (wk , T) 1   wkH H(tk bk + tl bl ) wkH Σ − wkH Σwk (b∗k tH k + bl tl )H √ = 3 ∂wk 2 2π b ∈B 2(wkH Σwk ) 2  l   2  wkH H(tk bk + tl bl ) × exp − , k, l = 1, 2, k = l 2wkH Σwk

(18)

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 4, APRIL 2014

B. Higher Modulation

carried out two different realizations of the algorithm for two different SNR values. The algorithm was stopped after 100 iterations. From Table I, following observations can be made:

For higher modulation, the focus of the algorithm will shift towards minimizing the symbol error rate rather than the bit error rate. The symbol error rate for higher modulation such as QPSK and M -QAM is given as [21] PE (wk , T) = −



PER (wk , T) + PEI (wk , T)



PER (wk , T)PEI (wk , T), k = 1, 2.(24)



where PER is related to the probability of real part while PEI is the probability of imaginary part. 1) QPSK Modulation: As BPSK is a special case of QPSK modulation, the analysis for real part remains the same as carried out for BPSK. The PER for transmitted bit bk = +1+j can be given as ⎞ ⎛ H  1 (wk H(tk bk + tl bl )) ⎠  PER = Q⎝ , (25) 4 wH Σw bl ∈B

k

D. Computational Complexity

k

where k, l = 1, 2, k = l and B contains all the available constellation for bl = {±1 ± j}. The derivative with w and T will remain the same as mentioned in (18) and (21) for the real case. However, for the imaginary part the real sign  can be changed to  and the probability of error for imaginary symbol is ⎛ ⎞ 1  ⎝ (wkH H(tk bk + tl bl )) ⎠  Q . (26) PEI = 4 wH Σw bl ∈Bl

k

The weights wi and the precoder ti have different values even for same SNRs. There are no zero element in the precoder or weights. The magnitude of the precoders are found to be very close to 1. It is observed from the table that the difference in these values is less than 3% at 5 dB SNR while the difference is less than 0.1% for 15 dB SNR. From the simulation, it is observed that the same optimized value is reached even though the algorithm is initialized with different initial values for W and T.

In this section we assume that conjugate, transpose, and hermitian operators require no effective computational operations. We ignore the amount of computation required for channel estimation in the ideal ML and MMSE detector. For calculation of the computational complexity the following assumptions are employed based on • •

k

The derivatives with wk and T are calculated and mentioned in (27) and (28). Now the proposed algorithm can be applied to determine the weights w and the precoder T. For higher modulation the probability of error is calculated based on decision boundaries and the algorithm can be easily extended.



Multiplication of an (M × N ) matrix with an (N × L) requires M (N −1)L additions and M N L multiplications. Computing the inverse of an (M × M ) matrix by using Cholesky decomposition requires M 3 /6 additions and M 3 /6 multiplications. For the joint maximal likelihood (JML) and BML detectors the complexity of inverting the noise covariance matrix is ignored.

From Table II, it can be observed that the JML detector has the highest computational complexity. The complexity of the JML detector increases as a squared power of M, where M represents the number of constellation points in the modulation scheme. For the JBER detector the computational complexity increases linearly with the modulation scheme. The T L and F L are the abbreviation used for training length and frame length. However, for the MMSE, BML, CINV and MRC detectors the complexity is a function of log2 (M).

C. Example For better understanding of the algorithm, we carried out an example where we fixed the channel gain between the S and D, S and R, and R and D to the following values hSD = −0.199 − 0.33i, hSR = 0.32 + 0.28i and hRD = 0.67 + 0.13i. The transmitted bits are also fixed to b1 = b2 = +1. We

   2  2    wkH H(tk bk + tl bl ) ∂PER (w1 , w2 , T) 1  =− Tk,l exp − , ∂T 2wkH Σwk 4 2πwH Σw b ∈B k=1

where



Tk,l

∗ wk1 hSD b1 +

k

2 ∗ σR wk2 hRD hSR b1 3 ζR

k

(21)

l l=k

⎢ |h |2 (|t |2 +2|t |2 )w∗ h h b 11 12 k2 RD SR 1 ⎢ + SR 3 ⎢ 2ζR ⎢ ∗ ⎢ wk2 hRD hSR t12 b2 |hSR |2 t∗ 11 ⎢ − 3 2ζR = ⎢ ⎢ (b∗1 t∗11 +b∗2 t∗12 )h∗RD h∗SR wk2 |hSR |2 t∗11 ⎢ − 3 2ζR ⎢ ⎢ ⎣ ∗ wk2 hSD b1

∗ wk1 hSD b2 +

+ −

2 ∗ σR wk2 hRD hSR b2 3 ζR

∗ |hSR |2 (2|t11 |2 +|t12 |2 )wk2 hRD hSR b2 3 2ζR ∗ wk2 hRD hSR t11 b1 |hSR |2 t∗ 12 3 2ζR

2 ∗ (b∗ t∗ +b∗ t∗ )h∗ h∗ SR wk2 |hSR | t12 − 1 11 2 12 RD 3 2ζR

∗ wk2 hSD b2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(22)

AHMED et al.: LINEAR TRANSCEIVER DESIGN FOR NONORTHOGONAL AMPLIFY-AND-FORWARD PROTOCOL USING A BIT ERROR RATE CRITERION

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TABLE I J OINT B IT E RROR R ATE D ETECTOR FOR FIXED CHANNEL 5 dB

SNR H

 −0.199 − 0.33i 0.261 + 0.335i 

T w1 T w2 tT 1 tT 2 Σ



−0.5 − 0.612i

0.387 + 0.474i

−0.638 + 0.539i

 0.825 − 0.181i 

0 + 0i −0.199 − 0.33i





0.379 − 0.396i

 

−0.534 − 0.0216i

−0.034 − 0.907i  0.158 0

0.074 + 0.414i 0 0.316







15 dB −0.199 − 0.33i 0.258 + 0.333i

0 + 0i −0.199 − 0.33i

0.056 − 0.778i

−0.0703 + 0.622i

 −0.778 − 0.073i  0.727 + 0.513i



−0.343 + 0.404i

0.158 0

0 0.313



−0.199 − 0.33i 0.304 + 0.391i



 −0.804 + 0.133i



 0.282 + 0.804i

0.621 + 0.0694i −0.406 − 0.207i

 0.533 − 0.659i





 



0.255 − 0.937i

 −0.930 − 0.346i





0.0158 0

0 + 0i −0.199 − 0.33i





−0.199 − 0.33i 0.364 + 0.468i

0 + 0i −0.199 − 0.33i



 −0.464 + 0.523i

0.692 − 0.182i





0.562 − 0.136i −0.244 − 0.464i 0.160 + 0.172i



0.0484 + 0.112i 0 0.0373

 

−0.225 − 0.841i

0.130 − 0.908i



0.813 + 0.241i





0.0158 0

−0.058 + 0.395i −0.500 + 0.177i 0 0.0466

0.0998

0.0971

3.385e-005

3.388e-005

PE (w2 , T) R

0.0981

0.0997

3.348e-005

3.343e-005

JML detector JBER detector

Ideal MMSE detector

(16×M+18×2×M)T L+(F L−T L)∗2∗log 2 (M) FL

(25×M+30×2×M)T L+(F L−T L)∗4∗log 2 (M) FL

(23 /6+18)+(2×log 2 (M)×F L) FL

(23 /6+32)+(4×log 2 (M)×F L) FL

2 FL

BML detector

4 FL

+ 2 × log2 (M)





∂PEI (wk , T) ∂wk

+ 4 × log2 (M)

2 FL

2 × log2 (M) 2 × log2 (M)

CINV detector MRC detector

+ 4 × log2 (M) 4 × log2 (M)

    ∗ H H 1  j wkH H(tk bk + tl bl ) wkH Σ + wkH Σwk (b∗k tH k + bl tl )H √ = 3 2 2π b ∈B 2j(wkH Σwk ) 2  l   2  wkH H(tk bk + tl bl ) × exp − , k, l = 1, 2, k = l 2wkH Σwk

   2  2    wkH H(tk bk + tl bl ) ∂PEI (w1 , w2 , T) 1  =− Tk,l exp − , ∂T 2wkH Σwk 4 2πwH Σw b ∈B k=1



∗ wk1 hSD b1 +

k

2 ∗ σR wk2 hRD hSR b1 3 ζR

k

(27)

(28)

l l=k

⎢ |h |2 (|t |2 +2|t |2 )w∗ h h b 11 12 ⎢ + SR k2 RD SR 1 3 ⎢ 2ζR ⎢ w∗ h h t b |h |2 t∗ 1⎢ ⎢ − k2 RD SR2ζ123 2 SR 11 R ⎢ j ⎢ (b∗1 t∗11 +b∗2 t∗12 )h∗RD h∗SR wk2 |hSR |2 t∗11 ⎢ + 3 2ζR ⎢ ⎢ ⎣ ∗ wk2 hSD b1

∗ wk1 hSD b2 +

+ − +

2 ∗ σR wk2 hRD hSR b2 3 ζR

∗ |hSR |2 (2|t11 |2 +|t12 |2 )wk2 hRD hSR b2 3 2ζR ∗ wk2 hRD hSR t11 b1 |hSR |2 t∗ 12 3 2ζR

∗ ∗ ∗ ∗ ∗ 2 ∗ (b∗ 1 t11 +b2 t12 )hRD hSR wk2 |hSR | t12 3 2ζR

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(29)

∗ wk2 hSD b2

 √   √ ⎤    π −1 −1 3− 3 3− 3 exp j sin sin cos sin 6 12 6 ⎥ ⎢ ⎥.   √    √  T=⎢   ⎦ ⎣ −1 π 3− 3 3− 3 − exp −j 12 sin sin−1 cos sin 6 6





TABLE II C OMPUTATIONAL COMPLEXITY. Number of operations per iteration Additions Multiplications M2 × 7 M2 × 10

Algorithm

Tk,l =



0.316 + 0.377i

PE (w1 , T) R

where





(34)

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IV. S IMULATION R ESULTS AND D ISCUSSION

0

10

−1

10

Bit Error Rate

In this section, to validate the performance of our proposed algorithms BER performance is compared with that of the previously proposed linear detectors such as CINV, MRC, BML, MMSE, MBER, multiple-input multiple-output (MIMO) and Ding et al. The weights vectors for the CINV, MRC, BML detectors are given as [8] ⎧ ai , for CINV ⎨ hi , for MRC i = 1, 2, (30) wi = ⎩ −1 Σ hi , for BML

−2

10

−3

10

where

−10 dB 0 dB 6 dB 13 dB 20 dB

−4

10

= [(hSD )−1 , (αhSR hRD )−1 ]T ,

(31)

a2

= [0 , (hSD )−1 ]T .

(32)

The optimum precoder of a MIMO system which uses two transmit antennas and one receive antenna is designed in [16, 17]   1 1 exp(jφ) , (33) T= √ 2 1 − exp(jφ) where for BPSK modulation, the optimum value which gives full diversity is φ = π/4 as shown in [17]. An optimal unitary precoder for a NAF protocol is designed in [11, 12], which is represented in (34). Initially, the convergence curve of the algorithm is plotted to choose an appropriate parameter for the proposed algorithm. After this, the BER performance versus SNR per bit is plotted for these algorithms. Finally, the rate of all these algorithms is compared. In our simulations, the channel gains are assumed to follow the Rayleigh distribution. The destination and the relay have the same noise variance. It is assumed that the receiver has the perfect channel knowledge and that the terminals (S, R, D) are synchronized. The source and destination have complete information about the channel matrix H, the weight decoder W and precoder T. With the assumption of ideal feed-forward/feed-back channel, the precoder and the weights design can also be carried out at the transmitter/receiver side. The limited feed-back strategy has been studied in the literature [22] and references therein. The loss due to feed back is introduced in the system according to the operating SNR. However, it is out of scope for this paper. Fig. 3 shows the influence of the step-size μ on the BER performance of the cooperative communications. All the stepsize are kept same, therefore, μw1 = μw2 = μT = μ. The training length is fixed to 100. From the figure it can be observed that for lower SNR values any step-size will be appropriate. For higher SNR values a step-size between 10−2 till 10−3 seems suitable. This result also explains why μ = 0.005 is used in the sequel. It can also be observed in the simulation that for a smaller value of μ the step-size does not have much influence as the algorithm will require longer training. Fig. 4 shows the learning curves of the JBER algorithm with different step sizes at Eb /N0 = 14 dB. The ensemble average was taken over 100, 000 independent realizations of the channel. The plots in the figure show that the step-size μ determines the BER performance and convergence of the algorithm. With smaller step-size, the algorithm converges

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Fig. 3. sizes.

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Step−size, μ

−2

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Influence on cooperative communication system with different step

μ=0.005 μ=0.0015 μ=0.0005 μ=0.00015 ML

Bit Error Rate

a1

−2

10

−3

10

0

200

400 600 Number of iterations

800

1000

Fig. 4. Learning curves of the cooperative communication system with different step sizes.

slowly as compared to bigger step-size but to a lower BER performance. It can also be observed that μ = 0.005 yields much faster convergence. On the other hand, μ = 0.00015 yields slower convergence. Therefore, the step-size is fixed to μ = 0.005 and Niter = 100 in the sequel. Fig. 5 shows the overall BER performance as a function of the SNR for different linear detectors. The overall BER performance of the JBER detector is obtained by using (20), which is calculated by combining the BER of both bits b1 and b2 . It can be observed that the overall BER performance of the system is determined by the bit b2 as it achieves the worst BER performance as compared to b1 when operating without precoder T. For low SNR values, all the detectors have equivalent BER performance. For the CINV, MRC and BML detectors, error floors are observed at high SNR values. Among these three detectors, the MRC detector is superior in BER performance as compared to the CINV detector, while BML is the best in performance. The BER performance of the MMSE detector is far more superior to the CINV, MRC and BML detectors. For the MBER detector [9], the performance is superior to the MMSE detector but with a similar slope. The slope for MMSE and MBER detector is equivalent to

AHMED et al.: LINEAR TRANSCEIVER DESIGN FOR NONORTHOGONAL AMPLIFY-AND-FORWARD PROTOCOL USING A BIT ERROR RATE CRITERION

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Bit Error Rate

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CINV MRC BML MMSE MBER MIMO DING JBER DIV −5

0.7 0.6 0.5

CINV MRC BML MMSE MBER MIMO DING JBER

0.4 0.3 0.2 0.1

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Eb/N0 (dB) Fig. 5. Overall BER performance of different cooperative communication detectors in the presence of Rayleigh fading.

1 as b2 only achieved diversity order of 1 without using a precoder. When deploying a precoder, it can be observed that significant gain in terms of BER performance can be observed as compared to a system without a precoder. It can be observed that the best performance is achieved by our proposed JBER. The diversity order of JBER is difficult to analyze as it is an iterative detector. The diversity order of Ding et al. detector is only a function of channel statistics H as precoder T is fixed and there are no linear weights filters. The diversity order of Ding detector is obtained from the pair-wise error probability (PEP) by averaging over the channel H. In our proposed scheme, the diversity order is a function of H, T, w1 and w2 . As T, w1 and w2 are all functions of the channel and the pdf of the weights and precoder is not known, the diversity order is difficult to calculate. However, it can be observed that this detector has a similar slope as that of the MIMO or Ding et al.. The diversity order of MIMO and Ding et al. based precoders is calculated to be equivalent to ρ−2 ln ρ, where ρ is the SNR of the system. In the figure, we have also plotted the diversity order of ρ−2 ln ρ. From Fig. 5, it can be observed that, for overall BER of 2×10−3, the proposed JBER detector is 12 dB and 16 dB superior to the BER and MMSE detector, respectively. Furthermore, a gain of more than 1 dB and 1.5 dB can be observed as compared to Ding and MIMO based precoder. As our aim is to minimize the BER performance of the system, not maximizing the ideal capacity from an informationtheoretic perspective we do not need to consider the capacity formula for discrete message encoded by continuous Gaussian code book. Fig. 6 shows the sum capacity achieved by the detectors as a function of SNR per bit. The effective channel from BPSK modulator at the transmitter to the receiver for input symbols b1 and b2 can be modeled as a binary symmetric channel and its capacity can be calculated as

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0

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30

Eb/N0 (dB)

Fig. 6. Sum capacity of different cooperative communication detectors in the presence of Rayleigh fading.

detector as calculated in (17). The overall capacity is achieved by simply adding both the quantities Cb1 and Cb2 . It can be observed that by adopting this NAF protocol, the sum capacity achieved is unity which is equivalent to the maximum capacity achieved by BPSK system. The loss of a factor of 1/2 due to the dual-phase transmission is recovered due to the multiplexing gain achieved by deploying NAF protocol. Furthermore, it can be observed from Fig. 6 that the capacity achieved by our proposed JBER detector is much superior to all the previously proposed detectors. Indeed, it can be observed that at a capacity of 0.9 bps/Hz, the proposed JBER is 1dB superior to the MIMO based or Ding et al. based precoder. Moreover, a gain of more than 7dB and 9dB is observed as compared to the BER and the MMSE detector without precoder, respectively. The CINV, MRC and BML detectors do not have the ability to achieve the full capacity of BPSK system. Among these three detectors, the BML detector is able to achieve a maximum capacity of 0.75 bps/Hz which is 0.02 bps/Hz and 0.05 bps/Hz superior to the MRC and CINV detectors, respectively. Fig. 7 shows the BER versus SNR performance of Nabar et al. and the JBER detector. It can be observed that the diversity order (DIV) achieved by b1 is of order 2, while b2 achieves the diversity order of 1 when employing the Nabar detector. It is observed from Fig. 7 that both bits, b1 and b2 , achieve similar BER performance with the help of a precoder T in (23). However, the diversity order is less than 2 for both bits. It can be observed that the slope of the JBER is parallel to the slope of ρ−2 ln ρ. Fig. 8 shows the SER versus SNR performance of the JBER detector when using QPSK and 16-QAM modulation. It is observed from Fig. 8 that both symbols, b1 and b2 , achieve similar BER performance with the help of a precoder T. It can be observed that the slope of the JBER is parallel to the −2 Cbi = 1 + [PER (wi , T) log2 PER (wi , T) (35) slope of the slope of ρ ln ρ. (24) is used to determine the analytical performance of the system. It can be observed from + (1 − PER (wi , T)) log2 (1 − PER (wi , T))] , i = 1, 2. the figure that the simulated and analytical results are matched. In Fig. 9, the number of multiplications is plotted with Note from (35) that the channel capacity is determined by the transition probability, which depends on the BER of the respect to the constellation size M, as multiplications is more

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450 −1

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NAF, b2 JBER, b1 , b2 NAF, b1 DIV = 1 DIV = ρ −2 ln ρ DIV = 2

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Fig. 7. BER performance of different cooperative communication detectors in the presence of Rayleigh fading.

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QPSK, (24) QPSK, Simulated 16−QAM, (24) 16−QAM, Simulated Diversity order

−1

Symbol Error Rate

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350 300 250 200 150 100 50

Eb/N0 (dB)

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SNR (dB) Fig. 8. SER performance of cooperative communication detector in the presence of Rayleigh fading when using QPSK and 16-QAM.

complex as compared to additions. In our simulation, the training length is fixed to 100 and the frame length is fixed to 1000 [23]. It can be observed that the complexity of the JML algorithm increases with the squared power of M. The complexity of JBER detector increases linearly. While the complexity of all the other detectors are similar. V. C ONCLUSION This paper proposed a linear transceiver for the uncoded NAF protocol. This transceiver operates on the principle of minimizing the BER of the system. Simulation results were provided and showed that the transceiver is able to exploit full diversity for both transmitted symbols, when assuming BPSK or QPSK modulation. The BER performance of the proposed JBER detectors is superior to all known linear detectors known, such as CINV, MRC, BML and MMSE. This proposed transceiver also outperforms the previously proposed precoders designed for the NAF protocol.

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Constellation Size, M

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Fig. 9. BER performance of different cooperative communication detectors in the presence of Rayleigh fading.

R EFERENCES [1] R. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relay channels: performance limits and space-time signal design,” IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004. [2] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [3] D. Chen and J. N. Laneman, “Modulation and demodulation for cooperative diversity in wireless systems,” IEEE Trans. Wireless Commun., vol. 5, no. 7, pp. 1785–1794, July 2006. [4] C. S. Patel and G. L. Stuber, “Channel estimation for amplify and forward relay based cooperation diversity systems,” IEEE Trans. Wireless Commun., vol. 6, no. 6, pp. 2348–2355, June 2007. [5] K. Azarian, H. E. Gamal, and P. Schniter, “On the achievable diversitymultiplexing tradeoff in half-duplex cooperative channels,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4152–4172, Dec. 2005. [6] B. Gedik and M. Uysal, “Impact of imperfect channel estimation on the performance of amplify and forward relaying,” IEEE Trans. Wireless Commun., vol. 8, no. 3, pp. 1468–1478, Mar. 2009. [7] S. Verdu, Multiuser Detection. Cambridge University Press, 1998. [8] A. Chowdhery and R. K. Mallik, “Linear detection fot the nonorthogonal amplify and forward protocol,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 826–835, Feb. 2009. [9] Q. Z. Ahmed, K.-H. Park, M.-S. Alouini, and S. Aissa, “Optimal linear detectors for non-orthogonal amplify-and-forward protocol,” in Proc. 2013 IEEE Int. Conf. Commun., pp. 1–5. [10] G. M. Kraidy, J. J. Boutros, and A. G. Guillen, “Approaching the outage probability of the amplify-and-forward relay fading channel,” IEEE Commun. Lett., vol. 11, no. 10, pp. 808–810, Oct. 2007. [11] Y. Ding, J.-K. Zhang, and K. M. Wong, “The amplify-and-forward half-duplex cooperative system: pairwise error probability and precoder design,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 605–617, Feb. 2007. [12] Y. Ding, J.-K. Zhang, and K. M. Wong, “Optimal precoder for amplifyand-forward half-duplex relay system,” IEEE Trans. Wireless Commun., vol. 7, no. 8, pp. 2890–2894, Aug. 2008. [13] L. J. Rodriguez, N. H. Tran, and T. Le-Ngoc, “Jointly optimal precoder and power allocation for an amplify-and-forward half-duplex relay system,” REV J. Electron. Commun., vol. 1, no. 1, pp. 38–44, Jan.Mar. 2011. [14] L. J. Rodriguez, N. H. Tran, and T. Le-Ngoc, “Bandwidth-efficient bitinterleaved coded modulation over NAF relay channels: Error performance and precode design,” IEEE Trans. Veh. Tech., vol. 60, no. 5, pp. 2086–2100, June 2011. [15] H. Mheidat and M. Uysal, “Non-coherent and mismatched-coherent receivers for distributed STBCs with amplify-and-forward relaying,” IEEE Trans. Wireless Commun., vol. 6, no. 11, pp. 4060–4070, Nov. 2007. [16] Y. Xin, Z. Wang, and G. B. Giannakis, “Space-time diversity systems based on linear constellation precoding,” IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 294–309, Mar. 2003.

AHMED et al.: LINEAR TRANSCEIVER DESIGN FOR NONORTHOGONAL AMPLIFY-AND-FORWARD PROTOCOL USING A BIT ERROR RATE CRITERION

[17] M. O. Damen, H. E. Gamal, and N. C. Beaulieu, “Systematic construction of full diversity algebraic constellations,” IEEE Trans. Inf. Theory, vol. 49, no. 12, pp. 3344–3349, Dec. 2003. [18] Q. Z. Ahmed, K.-H. Park, M. S. Alouini, and S. Aissa, “Compression and combining based on channel shortening and rank reduction techniques for cooperative wireless sensor networks,” IEEE Trans. Veh. Technol., vol. 63, no. 1, pp. 72–81, Jan. 2014. . [19] D. Chen and J. N. Laneman, “Cooperative diversity for wireless fading channels without channel state information,” in Proc. 2004 IEEE Asilomar Conf. Signals, Syst., Comput., pp. 1307–1312. [20] J. G. Proakis and M. Salehi, Digital Communications, 4th ed. McGrawHill, 2008. [21] S. Chen, S. Tan, L. Xu, and L. Hanzo, “Adaptive minimum error-rate filtering design: a review,” Elsevier Signal Process., vol. 88, no. 7, pp. 1671–1679, 2008. [22] X. Wang, W-S. Lu, and A. Antoniou, “Limited feedback-based multiantenna relay broadcast channels with block diagonalization,” IEEE Trans. Wireless Commun., vol. 12, no. 8, pp. 4092–4101, Aug. 2013. [23] Q. Z. Ahmed and L.-L. Yang, “Normalised least mean-square aided decision-directed adaptive detection in hybrid direct-sequence timehopping UWB systems,” in Proc. 2008 IEEE Veh. Tech. Conf. — Fall, pp. 1236–1241. Qasim Zeeshan Ahmed received his B.Eng. degree in Electrical Engineering from the National University of Sciences and Technology (NUST), Rawalpindi, Pakistan in 2001, MSc degree from the University of Southern California (USC) LosAngeles, USA in 2005 and his Ph.D. degree from the University of Southampton, UK in 2009. He worked as an assistant professor at the National University of Computer and Emerging Sciences (NUCES-FAST) Islamabad, Pakistan. Since June 2011, he has been a postdoctoral fellow with Computer, Electrical and Mathematical Sciences and Engineering Division at King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. His research interests include mainly low-complexity transceiver design, adaptive signal processing and cooperative-communications. Ki-HongPark (S’06, M’11) received his B.Sc. degree in Electrical, Electronic, and Radio Engineering from Korea University, Seoul, Korea, in 2005 and his M.S. and Ph.D. degrees in the School of Electrical Engineering from Korea University, Seoul, Korea, in 2011. Since April 2011, he has been a postdoctoral fellow of Electrical Engineering in the Division of Computer, Electrical and Mathematical Sciences and Engineering at King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. He is a recipient of a SABIC Post-Doctoral Fellowship in 2012. His research interests are broad in communication theory and its application to the design and performance evaluation of wireless communication systems and networks. On-going research includes the application to MIMO diversity/beamforming systems, cooperative relaying systems, and physical layer secrecy.

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Mohamed-Slim Alouini (S’94, M’98, SM’03, F’09) was born in Tunis, Tunisia. He received the Ph.D. degree in Electrical Engineering from the California Institute of Technology (Caltech), Pasadena, CA, USA, in 1998. He served as a faculty member in the University of Minnesota, Minneapolis, MN, USA, then in the Texas A&M University at Qatar, Education City, Doha, Qatar before joining King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Saudi Arabia as a Professor of Electrical Engineering in 2009. His current research interests include the modeling, design, and performance analysis of wireless communication systems. Sonia Aïssa (S’93-M’00-SM’03) received her Ph.D. degree in Electrical and Computer Engineering from McGill University, Montreal, QC, Canada, in 1998. Since then, she has been with the Institut National de la Recherche Scientifique-Energy, Materials and Telecommunications Center (INRS-EMT), University of Quebec, Montreal, QC, Canada, where she is a Professor of Telecommunications. From 1996 to 1997, she was a Researcher with the Department of Electronics and Communications of Kyoto University, and with the Wireless Systems Laboratories of NTT, Japan. From 1998 to 2000, she was a Research Associate at INRS-EMT, Montreal. In 2000-2002, while she was an Assistant Professor, she was a Principal Investigator in the major program of personal and mobile communications of the Canadian Institute for Telecommunications Research, leading research in radio resource management for wireless networks. From 2004 to 2007, she was an Adjunct Professor with Concordia University, Montreal. In 2006, she was Visiting Invited Professor with the Graduate School of Informatics, Kyoto University, Japan. Her research interests lie in the area of wireless and mobile communications, and include radio resource management, cross-layer design and optimization, design and analysis of multiple antenna (MIMO) systems, cognitive and cooperative transmission techniques, and performance evaluation, with a focus on Cellular, Ad Hoc, and Cognitive Radio networks. Dr. Aïssa is the Founding Chair of the IEEE Women in Engineering Affinity Group in Montreal, 2004-2007; acted or is currently acting as TPC Leading Chair or Cochair of the Wireless Communications Symposium at IEEE ICC in 2006, 2009, 2011 and 2012; PHY/MAC Program Cochair of the 2007 IEEE WCNC; TPC Cochair of the 2013 IEEE VTC-spring; and TPC Symposia Cochair of the 2014 IEEE Globecom. Her main editorial activities include: Editor, IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS, 2004-2012; Technical Editor, IEEE Wireless Communications Magazine, 2006-2010; and Associate Editor, Wiley Security and Communication Networks Journal, 20072012. She currently serves as Technical Editor for the IEEE Communications Magazine. Awards to her credit include the NSERC University Faculty Award in 1999; the Quebec Government FQRNT Strategic Faculty Fellowship in 2001-2006; the INRS-EMT Performance Award multiple times since 2004, for outstanding achievements in research, teaching and service; and the Technical Community Service Award from the FQRNT Centre for Advanced Systems and Technologies in Communications in 2007. She is co-recipient of five IEEE Best Paper Awards and of the 2012 IEICE Best Paper Award; and recipient of NSERC Discovery Accelerator Supplement Award. She is a Distinguished Lecturer of the IEEE Communications Society (ComSoc) and an Elected Member of the ComSoc Board of Governors.