Signal Transmission With Unequal Error Protection ... - Semantic Scholar

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE 2010

Signal Transmission With Unequal Error Protection in Wireless Relay Networks Ha X. Nguyen, Student Member, IEEE, Ha H. Nguyen, Senior Member, IEEE, and Tho Le-Ngoc, Fellow, IEEE

Abstract—This paper studies a relaying technique based on the signal-to-noise (SNR) threshold in cooperative networks using hierarchical modulation. Hierarchical modulation is useful in applications that require different protection classes of the information. In particular, a cooperative network with one source, one relay, and one destination is considered. Two different protection classes are modulated by a hierarchical 2/4-amplitude shift keying (ASK) constellation at the source. Based on the instantaneous received SNR at the relay, the relay decides to retransmit both classes by using a hierarchical 2/4-ASK constellation, or the more-protection class by using a 2-ASK constellation, or remains silent. Optimal thresholds are chosen to minimize the bit error rate (BER) of the less-protection class, whereas the BER of the more-protection class meets a given requirement. Numerical and simulation results are provided to verify the analysis. The results show that the optimal thresholds significantly improve the performance. Index Terms—Cooperative communications, cooperative diversity, fading channel, hierarchical modulation, relay networks, unequal error protection (UEP).

I. I NTRODUCTION

I

N COOPERATIVE systems, the transmission from a source to a destination is assisted by relay(s). Research on cooperative communication systems started with the introduction of the relay channel in [1]. Fundamental theorems on the bound of the capacity of relay networks were then established in [2] and [3]. Recently, cooperative (or relay) diversity has also been actively studied as a technique to combat fading experienced in wireless transmission [4]–[8]. In particular, the end-to-end (e2e) biterror-rate (BER) performance in a wireless network can be improved by having nodes (users) in the network cooperate with each other. The basic idea is that a source node cooperates with other nodes (or relays) in the network to form a virtual multipleantenna system [4]–[8], hence providing spatial diversity. In a relay network, it is common to consider the amplify-andforward (AF), decode-and-forward (DF), or compress-andforward (CF) protocol at the relays. With AF, relays receive noisy versions of the source’s messages, amplify, and retransmit to the destination. With DF, relays decode the source’s messages, reencode, and retransmit to the destination. On the other hand, the relays in the CF protocol forward the quantized/ Manuscript received May, 4, 2009; revised October 2, 2009 and December 29, 2009; accepted January 24, 2010. Date of publication February 8, 2010; date of current version June 16, 2010. The review of this paper was coordinated by Prof. A. M. Tonello. H. X. Nguyen and H. H. Nguyen are with the Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada (e-mail: [email protected]; [email protected]). T. Le-Ngoc is with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2010.2042627

compressed/estimated version of its received signals [2], [9]. A major drawback of the DF protocol is that cooperation does not achieve full diversity if the relays always retransmit the decoded message. This is due to possible retransmission of erroneously decoded bits of the message by the relays. With an adaptive version of the DF protocol, full diversity can be achieved since only the relay that successfully decodes the message from the source reencodes and retransmits the message to the destination [7], [10]–[13]. Consider a simple network with only one source–destination pair and a single relay. When the link between the source and the relay is error free, it is possible for the destination to combine two signals received from the source and the relay to achieve a diversity order of 2. However, it is impractical to have an error-free source–relay link. As mentioned before, there is no performance improvement when the relay retransmits the erroneous bits. Hence, dealing with detection (or decoding) errors at the relay becomes crucial. For example, an error-detection code can be added at the source. Based on the decoding result in the first phase, the relay can decide to retransmit or remain silent in the second phase [14], [15]. Another approach is based on the instantaneous SNR of the source–relay link. When the source–relay SNR is larger than a threshold, the probability of decoding error at the relay can be negligible, and hence, the relay retransmits the message [7], [10]–[13]. In [16], a receiver has been designed to eliminate errors at the relay, which allows the relay to always forward the received data. In particular, a cooperative maximum-ratio-combining (C-MRC) detector was proposed at the destination to collect the full diversity order by taking into consideration the instantaneous BER of the source–relay link. However, the use of error-detection codes increases the processing complexity of the relay, as well as the delay of the network. Using a C-MRC detector is also complicated because the destination requires knowledge of the instantaneous BER values of the source–relay link. Setting a threshold at the relay is clearly more practical. For this approach, [17] and [18] derive the optimal threshold value for binary phase-shift keying modulation to minimize the e2e BER. It was demonstrated that the e2e BER can significantly be reduced with the optimal threshold value. An important consideration in many communication systems is the quality of services under a wide range of channel conditions. Typically, data can be divided into different important classes, which require different degrees of protection. Under poor channel conditions, the receiver can recover the more important classes (known as basic or coarse data) with an acceptable BER, whereas the less-important classes (known as

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NGUYEN et al.: SIGNAL TRANSMISSION WITH UNEQUAL ERROR PROTECTION IN WIRELESS RELAY NETWORKS

refinement or enhancement data) are only recovered in better channel conditions [19], [20]. For example, multimedia data such as audio, images, and video exhibit unequal sensitivity for different bits. It is wasteful if all of the bits are equally protected. In contrast, the unequal error protection (UEP) scheme protects the data according to the system requirements. The bit stream of the source data is divided into two or more groups, and different protection levels are applied to these groups. A simple and practical way to achieve UEP is by using hierarchical modulations. Basically, hierarchical modulations are those constellations with nonuniformly spaced signal points [19]–[22]. The basic idea in using hierarchical modulations is that the information bits are mapped onto nonuniform constellation points according to their importance. With such modulations, the more important bits are decoded with fewer errors than the less important bits. As a result, hierarchical modulations can offer variable degrees of error protection for the information bits. In the literature, although a large amount of research work related to UEP in point-to-point communications exists [19], [20], [23], [24], only a few studies were devoted to UEP in cooperative networks. In particular, a relay communication system employing distributed turbo coding together with hierarchical modulation is studied in [25]. The relay adjusts the modulation order based on the cyclic redundancy check indication of the decoding result over the source–relay link. However, how to systematically choose the channel code, modulation, and allocate power between the source and the relay is not discussed. Hierarchical modulation is also adopted in [26] to broadcast the signals to multiple destinations. Higher destinations work as relays to lower destinations.1 C-MRC is applied at each destination, which also means that each destination requires knowledge of the BERs of all previous destinations. In [27], the optimal distance parameters of the constellation are analyzed to minimize the BERs. The technique in [26] is complicated. Moreover, the relay always reduces the size of the hierarchical constellation, which implies that the destination cannot exploit full diversity for the less-important class, although it can correctly be decoded at the relay. With regard to the approach in [27], since the relay always retransmits the decoded bits, error propagation to the destination can happen as discussed before. This paper is also concerned with a cooperative network with hierarchical modulation for UEP of information data. Two information classes with different levels of protection are transmitted from the source. The adaptive transmission protocol proposed at the relay is based on two thresholds as follows: After receiving the signal in the first phase from the source, the relay compares the instantaneous received SNR with the two thresholds to decide whether to retransmit both classes, or the more-important class, or remain silent in the second phase. As such, the key difference of the protocol in this paper, as compared with those in [26] and [27], is in the adaptive transmission at the relay in the second phase. In particular, how many and which bits are transmitted in the second phase in our protocol are determined by the channel quality of the 1 A higher destination means a destination with a higher reception capability, i.e., the destination is able to reliably detect most of the transmitted bits.

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Fig. 1. Simple wireless relay network.

source–relay links in the first phase. In contrast, the relays always retransmit a predefined number of bits in [26] and [27]. For simplicity, we concentrate on the use of a hierarchical 2/4-ASK constellation at the source. At the relay, depending on the received instantaneous SNR, a hierarchical 2/4-ASK or 2-ASK constellation is employed. The BER for each information class is first derived. The use of optimal thresholds at the relay is then discussed to minimize the BER of the less protection class, whereas the BER of the more-protection class satisfies a given requirement.2 It should be pointed out that the framework proposed here can be extended to a general hierarchical modulation [such as quadrature amplitude modulation (QAM)] with many different classes. As an example, Appendix C discusses the analysis with generalized hierarchical 4/16-QAM constellations.3 It should also be mentioned that, although our system and analysis do not consider explicit channel coding, the developed framework and results could be applied to the coded bits corresponding to each information class (i.e., the error probability of the coded bits is considered instead of the error probability of the information bits). The remainder of this paper is organized as follows: Section II describes the system model and presents the BER analysis for two information classes. Approximations of the BERs are presented in Section III, which are used to conveniently optimize the threshold values in Section IV. Numerical and simulation results are presented in Section V. Finally, Section VI concludes this paper. Notations: Eγ {x} is the expectation of x(γ) with respect to the random variable γ. The conditional probability distribution function (pdf) of γ given Θ is denoted by fγ|Θ (γ). CN (0, σ 2 ) denotes a circularly symmetric complex Gaussian random variable with variance σ 2 . P (·) denotes the probability measure of some probability space (Ω, B), where Ω is the finite set, and B is the sigma algebra by this set. The Q function is √ ∞generated 2 defined as Q(x) = x (1/ 2π)e−t /2 dt. II. S YSTEM M ODEL AND B IT E RROR R ATE A NALYSIS Fig. 1 illustrates a simple relay network with three nodes, in which a source node S sends information to a destination node D with the assistance of a relay node R. The information bits 2 It might also be of interest and straightforward to minimize the BER of the more-protection class under some constraint on the BER of the less-protection class. It should also be pointed out that our system model and analysis do not consider channel coding. If channel coding is employed, the BER expressions will be different and deserve further study. 3 The analysis extension to a nonsymmetric hierarchical modulation (such as the nonregular QAM used in asymmetric digital subscriber line and high bitrate digital subscriber line) is more tedious since the BER computation is much more complicated, and no general expressions exist for an arbitrary modulation order.

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Fig. 2. Generalized hierarchical 2/4-ASK constellation.

at S are modulated to symbol xs by a hierarchical 2/4-ASK constellation, as shown in Fig. 2, where bit i1 is more protected than bit i2 . The received signals at the relay and destination in the first phase can be written as (1) ysr = Es hsr xs + nsr ysd = Es hsd xs + nsd (2) where Es is the average symbol energy at the source; and hij is the fading channel coefficient between node i and node j, where i, j ∈ {s, r, d}. In this paper, we assume that the channel between any two nodes is Rayleigh flat fading, modeled as 2 ), where i, j refer to transmit and receive nodes, CN (0, σij respectively. The noise components (nsr , nsd ) at both the relay and the destination are modeled as independent identically distributed (i.i.d.) CN (0, N0 ) random variables. Therefore, the instantaneous received SNR for the transmission from node i to node j, which is denoted by γij , is given as γij = Ei |hi,j |2 /N0 . With Rayleigh fading, the pdf of γij is exponential and given by 2 2 2 )e−γij /σij , where σij is the average SNR of fij (γij ) = (1/σij the i−j link. Let the two thresholds employed at R be denoted by γ1th and γ2th , where γ1th < γ2th . In the second phase, if the instantaneous received SNR at R satisfies γsr > γ2th , R decodes both bits and remodulates to xr by the same hierarchical 2/4-ASK constellation as used in the source S. If γ1th < γsr < γ2th , R decodes the important bit and remodulates to symbol xr with a 2-ASK constellation. If γsr < γ1th , R remains silent. If the relay transmits, the received signal at the destination is given by (3) yrd = Er hrd xr + nrd where Er is the average energy per symbol sent by R, hrd is the fading channel coefficient between R and D, and nrd is the noise component at D in the second phase, also modeled as CN (0, N0 ). Depending on what R transmits in the second phase, D combines the received signals in two phases and decodes. It is assumed that the relay utilizes the channel state informa2 2 , σsd } to make the decision. Furthermore, the tion of {γsr , σrd destination exactly knows the behavior of the relay. In practice, this can be done by having the relay send one flag bit to the destination whenever there is a change in the reliability status of detection at the relay. It should be noted that the SNR-based selective relaying technique [17], [18] also needs one bit to inform the state of the relay to the destination. As a result, the efficiency of our proposed scheme is similar to the SNR-based selective relaying technique (referred to as the one-threshold method in the discussion and comparison with our scheme in Section V). In general, the rate at which the relay needs to

inform the destination about its new detection status depends on the fading rate of the source–relay channel. To compute the average BERs of both bits i1 (more-protected bit) and i2 (less-protected bit), we first classify all different cases that result in different conditional BERs at the destination. The average BER of each information bit is then obtained by a corresponding weighted sum of these conditional BERs. Three major cases can be classified and parameterized by variable Θ as follows: The first case, i.e., Θ = 1, is when γsr < γ1th . In this case, the destination simply uses the received signal in the first phase to decode both bits i1 and i2 . The second case, which is parameterized by Θ = 2, corresponds to γ1th < γsr < γ2th . In this case, the destination combines two received signals in two phases to decode bit i1 and uses the received signal in the first phase to decode bit i2 . Two different subcases, which are parameterized by Φ = {1, 2}, can further be classified under Θ = 2 depending on the correctness of the decoded bit i1 at the relay. Finally, the third case, i.e., Θ = 3, happens if γsr > γ2th . Under Θ = 3, the destination combines two received signals in two phases to decode both bits i1 and i2 . Since the relay may decode either i1 or i2 incorrectly, four different subcases, which are parameterized by Φ = {1, 2, 3, 4}, can further be separated under Θ = 3. All seven different cases resulting from the aforementioned three major cases are summarized in Fig. 3. Let4 P (εw , ik , Θ = j, Φ = l) denote the BER of bit ik at node w under case Θ = j and subcase Φ = l. With two given thresholds γ1th and γ2th , the average BER for bit ik , where k = 1, 2, can be written as BER γ1th , γ2th , ik = P (εd , ik , Θ = 1)+

2

P (εd , ik , Θ = 2, Φ = l)

l=1

+

4

P (εd , ik , Θ = 3, Φ = l) = P (εd , ik |Θ = 1)P (Θ = 1)

l=1

+

2

P (εd , ik |Θ = 2, Φ = l)P (Θ = 2|Φ = l)P (Φ = l)

l=1

+

4

P (εd , ik |Θ = 3, Φ = l)P (Θ = 3|Φ = l)P (Φ = l). (4)

l=1

All the terms in (4) are computed as follows. A. Case 1 (Θ = 1): γsr < γ1th In this case, the instantaneous SNR of the S−R link is smaller than the threshold γ1th . Hence, R remains silent in the second phase. D uses only the received signal in the first phase

4 When a subcase does not exist under Θ = 1, Φ = l does not appear in the notation.



Fig. 3.

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Seven different possible cases that result in different BERs at the destination.

to decode both bits. Using the results in [20] and [26], the BERs conditioned on γsd for i1 and i2 are given, respectively, by P (εd , i1 |Θ = 1, γsd )

√ √ (α − 1) 2γsd (α + 1) 2γsd 1 √ √ = +Q (5) Q 2 α2 + 1 α2 + 1 P (εd , i2 |Θ = 1, γsd ) √ √ (2α + 1) 2γsd 2γsd 1 √ = −Q 2Q √ 2 α2 + 1 α2 + 1

√ (2α − 1) 2γsd √ +Q α2 + 1

(6)

where α = d1 /d2 , d1 and d2 are the two distance parameters of the hierarchical 2/4-ASK constellation, as shown in Fig. 2. The average BERs for i1 and i2 can be calculated by averaging the conditional BERs over γsd . They are found to be [20], [26]

In this case, since the instantaneous SNR of the S−R link is between thresholds γ1th and γ2th , the relay decodes the first bit and remodulates the decoded bit as a 2-ASK signal. This is based on the expectation that the relay is likely to correctly decode the first bit but to incorrectly decode the second bit, and hence, only the first bit should be remodulated and sent to the destination. Note that there is still a probability of error in decoding the first bit at the relay. The MRC method5 is used at D to decode bit i1 . Meanwhile, only the signal received in the first phase at D is used to decode bit i2 . When the first bit is correctly decoded at the relay, i.e., Φ = 1, the BERs of i1 and i2 at the destination can be calculated as (see Appendix A) P (εd , i1 |Θ = 2, Φ = 1)

α+1 α−1 1 , 1 + J1 √ ,1 = J1 √ 2 α2 + 1 α2 + 1 where

∞ ∞

P (εd , i1 |Θ = 1)

J1 (μ, ν) =

= Eγsd {P (εd , i1 |Θ = 1, γsd )}

⎡ ⎤ (α+1)2 2 (α−1)2 2 α2 +1 σsd α2 +1 σsd 1⎣ ⎦ = 2− − 2 2 2 2 4 1 + (α+1) σ 1 + (α−1) α2 +1 sd α2 +1 σsd

Q 0

0

μγ + νγrd sd (γsd + γrd )/2

(9)

1 2 2 −γsd /σsd e−γrd /σrd dγsd dγrd (10) 2 σ2 e σsd rd P (εd , i2 |Θ = 2, Φ = 1)

⎧ (2α+1)2 2 2 α21+1 σsd 1⎨ α2 +1 σsd = + 2−2 2 1 2 2 4⎩ 1 + α2 +1 σsd 1 + (2α+1) α2 +1 σsd

⎫ (2α+1)2 2 ⎬ σ 2 sd α +1 − . (11) 2 2 ⎭ 1 + (2α+1) 2 α +1 σsd ×

(7)

P (εd , i2 |Θ = 1) = Eγsd {P (εd , i2 |Θ = 1, γsd )}

⎡ (2α+1)2 2 2 α21+1 σsd 1⎣ α2 +1 σsd = 2−2 + 2 1 2 2 4 1 + α2 +1 σsd 1 + (2α+1) σsd 2

−

B. Case 2 (Θ = 2): γ1th < γsr < γ2th

α +1

1

(2α+1)2 2 α2 +1 σsd 2 2 + (2α+1) α2 +1 σsd

⎤ ⎦.

(8)

5 It should be pointed out that instead of MRC, equal-gain combining (EGC) or selection combining (SC) can also be employed. The comparison of MRC, EGC, and SC methods is beyond the scope of this paper; see [28] for a similar framework and analysis with SC.


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When the first bit is wrongly decoded at the relay, i.e., Φ = 2, the BER of i1 is given by (see Appendix A) P (εd , i1 |Θ = 2, Φ = 2)

α+1 α−1 1 , 1 + J2 √ ,1 = J2 √ 2 α2 + 1 α2 + 1

(12)

where

∞ ∞ J2 (μ, ν) =

Q 0

0

×

μγ + νγrd sd (γsd + γrd )/2

1 2 2 e−γsd /σsd e−γrd /σrd 2 2 σsd σrd

dγsd dγrd .

(13)

The BER of i2 does not change in this case, i.e., P (εd , i2 |Θ = 2, Φ = 1) = P (εd , i2 |Θ = 2, Φ = 2) = P (εd , i2 |Θ = 1).

(14)

C. Case 3 (Θ = 3): γsr > γ2th In this case, the instantaneous SNR of the S−R link is larger than a threshold γ2th . The relay decodes both bits and remodulates by using a hierarchical 2/4-ASK constellation as done in the source. Since γsr > γ2th , it is likely that both bits are correctly decoded. However, errors can still happen. All subcases are considered as follows. In the first subcase (Φ = 1), the relay incorrectly decodes the first bit but correctly decodes the second bit, and the BERs of i1 and i2 are given, respectively, by (see Appendix B) P (εd , i1 |Θ = 3, Φ = 1) α+1 1 α+1 = ,√ J2 √ 2 α2 + 1 α2 + 1

α−1 α−1 + J2 √ ,√ α2 + 1 α2 + 1 P (εd , i2 |Θ = 3, Φ = 1) 1 1 2α + 1 = ,√ J2 √ 2 α2 + 1 α2 + 1 2α + 1 1 − J2 √ ,√ α2 + 1 α2 + 1 1 2α − 1 + J1 √ ,√ α2 + 1 α2 + 1

2α − 1 1 + J1 √ ,√ . α2 + 1 α2 + 1

(15)

(16)

(17)

(18)

In the third subcase (Φ = 3), the relay correctly decodes the first bit but incorrectly decodes the second bit, and the BERs of i1 and i2 are P (εd , i1 |Θ = 3, Φ = 3) α+1 α−1 1 ,√ = J1 √ 2 α2 + 1 α2 + 1

α−1 α+1 + J1 √ ,√ α2 + 1 α2 + 1 P (εd , i2 |Θ = 3, Φ = 3) 1 1 1 ,√ = 2J2 √ 2 α2 + 1 α2 + 1 2α + 1 2α − 1 + J1 √ ,√ α2 + 1 α2 + 1

2α − 1 2α + 1 + J1 √ ,√ . α2 + 1 α2 + 1

(19)

(20)

Finally, in the fourth subcase (Φ = 4), both bits are correctly decoded at the relay, and the BERs of i1 and i2 are P (εd , i1 |Θ = 3, Φ = 4) α+1 α+1 1 ,√ = J1 √ 2 α2 + 1 α2 + 1

α−1 α−1 + J1 √ ,√ α2 + 1 α2 + 1

In the second subcase (Φ = 2), the relay incorrectly decodes both bits, and the BERs of i1 and i2 are P (εd , i1 |Θ = 3, Φ = 2) α+1 α−1 1 ,√ = J2 √ 2 α2 + 1 α2 + 1

α−1 α+1 ,√ + J2 √ α2 + 1 α2 + 1

P (εd , i2 |Θ = 3, Φ = 2) 2α − 1 1 1 ,√ = J2 √ 2 α2 + 1 α2 + 1 2α + 1 1 ,√ − J1 √ α2 + 1 α2 + 1 1 2α + 1 + J1 √ ,√ α2 + 1 α2 + 1

2α − 1 1 + J2 √ ,√ . α2 + 1 α2 + 1

P (εd , i2 |Θ = 3, Φ = 4) 1 1 1 ,√ = 2J2 √ 2 α2 + 1 α2 + 1 2α + 1 2α + 1 + J1 √ ,√ α2 + 1 α2 + 1

2α − 1 2α − 1 ,√ . + J1 √ α2 + 1 α2 + 1

(21)

(22)

Furthermore, the expectations over random variables γsd and γrd in (21) and (22) can explicitly be evaluated (see (66) of Appendix B).



D. Other Computations Since the channel between S and R is Rayleigh flat fad2 ing, γsr is an exponential random variable with mean σsr . One has th 2 (23) P (Θ = 1) = P γsr ≤ γ1th = 1 − e−γ1 /σsr th P (Θ = 2) = P γ1 < γsr ≤ γ2th = e−γ1 /σsr − e−γ2 /σsr th 2 P (Θ = 3) = P γsr > γ2th = e−γ2 /σsr . th

2

th

2

(24) (25)

To compute (4), we also need to calculate the BERs of i1 and i2 at the relay. It can be verified that fγsr |Θ=1 (γsr ) = fγsr |Θ=2 (γsr ) =

e−γsr /σsr 2 σsr 2

1 1 − e−γ1

th /σ 2 sr

2 −γsr /σsr

e

1 th 2 e−γ1 /σsr

(26)

th 2 e−γ2 /σsr

− 1 γ2th /σsr 2 2 fγsr |Θ=3 (γsr ) = 2 e e−γsr /σsr . σsr

2 σsr

(27) (28)

Using [29] and [30, eq. (3.361.1)],6 the average BERs of i1 at the relay when Θ = 2 can be found as follows: P (εr , i1 |Θ = 2) =

1 th th 2 2 e−γ1 /σsr −e−γ2 /σsr

1 2 2σsr

√ √ γ2 (α−1) 2γsr (α+1) 2γsr √ √ +Q × Q α2 +1 α2 +1 th

γ1th 2 −γsr /σsr

×e =

2171

Similarly, the average BERs of i1 and i2 at the relay when Θ = 3 can be found as P (εr , i1 |Θ = 3) th

2

eγ2 /σsr = 2 2σsr

√ ∞ (α + 1) 2γsr √ Q α2 + 1 γ2th

√ (α − 1) 2γsr 2 √ +Q e−γsr /σsr dγsr α2 + 1 th 2 2(α + 1)2 2 th eγ2 /σsr , σ = , γ I sr 2 2 α2 + 1 2(α − 1)2 2 th , σ −I , γ (31) sr 2 α2 + 1 P (εr , i2 |Θ = 3) th

2

eγ2 /σsr = 2 4σsr

√ ∞ √ (2α + 1) 2γsd 2γsd √ √ − 2Q 4Q α2 + 1 α2 + 1 γ2th

√ (2α − 1) 2γsd 2 √ e−γsr /σsr dγsr + 2Q 2 α +1 2 γ2th /σsr 2 e 2 th , σ = , γ 2I 2 α2 + 1 sr 2 2(2α + 1)2 2 th , σsr , γ2 −I α2 + 1 2(2α − 1)2 2 th , σsr , γ2 +I . (32) α2 + 1 Finally, the remaining computations required in (4) are as follows:

dγsr

1 1 th 2 2 2 e−γ1th /σsr −e−γ2 /σsr 2(α+1)2 2 th 2(α−1)2 2 th , σ , σ × I , γ , γ −I sr 1 sr 2 α2 +1 α2 +1 2(α+1)2 2 th 2(α−1)2 2 th , σsr , γ1 −I , σsr , γ2 +I α2 +1 α2 +1 (29)

P (Φ = 1|Θ = 2) = 1 − P (εr , i1 |Θ = 2)

(33)

P (Φ = 2|Θ = 2) = P (εr , i1 |Θ = 2)

(34)

P (Φ = 1|Θ = 3) = P (εr , i1 |Θ = 3) × [1 − P (εr , i2 |Θ = 3)]

(35)

P (Φ = 2|Θ = 3) = P (εr , i1 |Θ = 3) × P (εr , i2 |Θ = 3)

(36)

P (Φ = 3|Θ = 3) = [1 − P (εr , i1 |Θ = 3)]

where ∞ 2

I(a, σ , x) =

√ e−t/σ2 Q( at) dt σ2

x

√ 2 = e−x/σ Q( ax) 2 a x +a . (30) − Q 2 σ2 σ2 + a

6 We should mention here that the average BERs of i and i at the relay 1 2 given Θ = 1 do not need to be calculated.

× P (εr , i2 |Θ = 3)

(37)

P (Φ = 4|Θ = 3) = [1 − P (εr , i1 |Θ = 3)] × [1 − P (εr , i2 |Θ = 3)]

(38)

where P (Φ = l|Θ = k) denotes the probability of occurrence of subcase Φ = l, given the case Θ = k. To summarize, the final expressions of the average e2e BERs of two information classes have been formulated in (4). These expressions can be computed by substituting in (7)–(25) and (29)–(38). Note that all the expressions can analytically be calculated, except the integrals in (15)–(20), which need to


2172


be numerically evaluated. Based on the average e2e BERs, the optimal threshold values shall be chosen to minimize the average e2e BER of the second class while making sure that the average e2e BER of the first class meets a given requirement. The optimization problem can be extended to include the distance parameters, and this is illustrated in Section IV. While the aforementioned optimization problems are interesting and appealing, finding the solutions is challenging due to the complicated expressions of the e2e BERs. To overcome this difficulty, we propose to work with the asymptotic approximations of the BERs, as presented in the next section. III. A PPROXIMATIONS OF A SYMPTOTIC B IT E RROR R ATES The approximate expressions are given to calculate the probability of errors given in (9) and (12) and from (15)–(20). First, we deal with the two expressions J1 (μ, ν) and J2 (μ, ν) in (10) and (13), respectively, where μ > 0 and ν > 0. The expression in (10) is invoked when the relay forwards a correctly decoded bit to the destination. The destination combines two signals using MRC and decodes. There is no error propagation in this case. With a high SNR, it is expected that the destination decodes the bit with a very small probability of error. The second expression in (13) applies when the relay forwards an incorrect bit to the destination. Normally, the R−D link has a stronger impact on the decision at the destination than the S−D link. Therefore, one can approximate the error probability in (13) by the probability of (μγsd − νγrd ) < 0 [17]. Since γsd and γrd are independent, one has ν μ γrd

∞ J2 (μ, ν) ≈ 0

=

0

1 2 2 −γsd /σsd e−γrd /σrd dγsd dγrd 2 σ2 e σsd rd

2 νσrd 2 . + νσrd

(39)

2 μσsd

On the other hand, (12) can be approximated as P (εd , i1 |Θ = 2, Φ = 2)

α+1 α−1 1 , 1 + J2 √ ,1 = J2 √ 2 α2 + 1 α2 + 1 2 2 σrd 1 σrd ≈ . + α−1 2 2 2 √ 2 √α+1 σ 2 + σrd σ + σrd α2 +1 sd α2 +1 sd

2 σrd 2 + σ2 σsd rd 2 (2α + 1)σrd 1 P (εd , i2 |Θ = 3, Φ = 1) ≈ 2 + (2α + 1)σ 2 2 σsd rd 2 σrd − 2 + σ2 (2α + 1)σsd rd

2 (α + 1)σrd + 2 2 (α − 1)σsd + (α + 1)σrd

(43)

P (εd , i2 |Θ = 3, Φ = 2) ≈

2 (2α − 1)σrd 1 2 2 2 σsd + (2α − 1)σrd

+ P (εd , i2 |Θ = 3, Φ = 3) ≈

2 σrd 2 + σ2 (2α − 1)σsd rd

2 σrd 2 . + σrd

2 σsd

(44) (45)

Next, we shall approximate P (εd , i1 |Θ = 2, Φ = 1) and P (εd , i1 |Θ = 3, Φ = 3) to compute the average BER of i1 in (4). Consider the following terms corresponding to the BER expression of i1 in (4): K2 =

2

P (Θ = 2)P (Φ = l|Θ = 2)

l=1

× P (εd , i1 |Θ = 2, Φ = k) K3 =

4

(46)

P (Θ = 3)P (Φ = l|Θ = 3)

l=1

× P (εd , i1 |Θ = 3, Φ = k).

(47)

For K2 , with a sufficient high SNR, it can be seen that7 P (εd , i1 |Θ = 2, Φ = 1) = O(SNR−2 ), P (Φ = 2|Θ = 2) = O(SNR−1 ), while P (εd , i1 |Θ = 2, Φ = 2) is given as in (40). Therefore, the dominant component in K2 is P (Θ = 2)P (Φ = 2|Θ = 2)P (εd , i1 |Θ = 2, Φ = 2), and we can ignore P (εd , i1 |Θ = 2, Φ = 1). Similarly, for the computation of K3 , one can approximate P (εd , i1 |Θ = 3, Φ = 3) ≈ 0. The accuracy of the aforementioned approximations shall be verified in Section V by comparing them with the exact expressions obtained in Section II. IV. O PTIMUM S IGNAL - TO -N OISE R ATIO T HRESHOLDS

(40)

Similarly, since J1 (μ, ν) is very small compared with J2 (μ, ν) and can be neglected, other BER expressions of i1 and i2 can be approximated as follows: P (εd , i1 |Θ = 3, Φ = 1) ≈

2 (α − 1)σrd 1 P (εd , i1 |Θ = 3, Φ = 2) ≈ 2 2 2 (α + 1)σsd + (α − 1)σrd

(41)

As mentioned before, with the approximate expressions of the average e2e BERs of two bits i1 and i2 , one can choose the optimal thresholds to minimize the BER of the less protected bit i2 when the BER of the more-protected bit i1 meets a given requirement. This type of UEP design is considered in different broadcasting and multimedia applications [22], [24], [27]. The optimization problem can be set up as follows: th th 2 = arg min BER γ1th , γ2th , i2 γ 1 , γ th th (γ1 ,γ2 ) BER γ1th , γ2th , i1 ≤ BER1 (48) subject to 0 ≤ γ1th ≤ γ2th

(42)

7 With two positive real functions f (x) and g(x), we say f (x) = O(g(x)) if lim supx→∞ (f (x)/g(x)) < ∞.



2173

where BER1 is the BER requirement of the more-protected information class. The distance parameters can also be jointly considered with the thresholds in the optimization problem. In this case, the optimization problem can be stated as follows: th th 2 , α = arg min BER γ1th , γ2th , α, i2 γ 1 , γ (γ1th ,γ2th ,α) BER γ1th , γ2th , α, i1 ≤ BER1 subject to (49) 0 ≤ γ1th ≤ γ2th . The foregoing problems can be solved by some optimization techniques such as the augmented Lagrange method [31] since the average e2e BER formulas of two bits i1 and i2 have been set up. Here, the optimization problems in (48) and (49) are solved by using the MATLAB Optimization Toolbox.8 Since it is difficult to prove the cost functions to be convex or not, to have good confidence in the numerical results, the optimization problems are solved with many initial conditions, and the best value is retained. Furthermore, it should be noted that the average BERs formulated in (4) only require information on the average SNRs of the source–destination, source–relay, and relay–destination links. The optimization problems can therefore be solved offline for typical sets of average SNRs, and the obtained optimal thresholds and/or distance parameters are stored in a lookup table.

Fig. 4. BERs of i1 and i2 when λsr = λrd = λsd = 1. Simulation results are shown in lines, and exact analytical values are shown as marker symbols.

V. S IMULATION R ESULTS This section presents analytical and simulation results to confirm the analysis of the average e2e BERs of two different classes. Moreover, the BER performances with approximations are provided to illustrate the accuracy of the approximations made. In all simulations, transmitted powers are set to be the same for the source and the relay (i.e., Es = Er ). The noise components at both the source and the relay are modeled as i.i.d. CN (0, 1) random variables. The average SNR of link 2 = λij Ei /N0 , where λij is a scaling i−j is represented by σij factor to reflect different distances among nodes. We also use th the normalized thresholds λth 1 and λ2 to represent the actual th th thresholds as γ1 = λ1 Es /N0 and γ2th = λth 2 Es /N0 . Unless stated otherwise, we simply refer to normalized thresholds as thresholds in the following discussion. Figs. 4 and 5 plot the BERs of i1 and i2 at the destination for th different values of thresholds λth 1 and λ2 . Here, α = 1/0.3, λsr = λrd = λsd = 1 for Fig. 4, and λsr = λrd = 10λsd = 1 for Fig. 5. The figures show that the exact analytical and simulation results are basically the same. The exact analytical results are obtained with numerical integrations of (9), (12), and (15)–(20). Similarly, Fig. 6 compares the BER performances at various channel conditions with approximate analytical and 8 Specifically, we have used the routine “fmincon,” which is designed to find the minimum of a given constrained nonlinear multivariate function. The exact complexity analysis of this routine requires expressions of the gradients and Hessian of the objective function, which are unfortunately not available. However, in general, the complexity would be O(n3 ), where n is the number of variables. Since our optimization problems involve only small numbers of variables, the complexity of the numerical optimization procedure is low.

Fig. 5. BERs of i1 and i2 when λsr = λrd = 10λsd = 1. Simulation results are shown in lines, and exact analytical values are shown as marker symbols.

simulation results. Observe that over the whole SNR range, the approximations are very accurate. Therefore, the approximations provide a useful tool to calculate the e2e BERs for both bits and in optimizing the relaying thresholds and distance parameters. Next, Fig. 7 shows the BERs of i1 and i2 obtained by simulation for one- and two-threshold methods. With the onethreshold method, the relay retransmits both bits received in the first phase during the second phase when the instantaneous received SNR at the relay is larger than a threshold value. Otherwise, the relay remains silent. The MATLAB Optimization Toolbox is also used to obtain the optimal threshold for this method. The obtained optimal threshold values are {λth 1 = th th = 0.055} and {λ = 0.009, λ = 0.055} for the oneand λth 2 1 2 two-threshold methods, respectively. The figure clearly shows that, while the BERs of i2 in both methods are the same, the BER of i1 obtained with the two-threshold method is significantly better than the BER obtained with the one-threshold


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th Fig. 6. BERs of i1 and i2 when λth 1 = 0.05, λ2 = 0.1, and α = 1/0.3. Simulation results are shown in solid lines, and approximate analytical values are shown in dashed lines.

th Fig. 7. BERs of i1 and i2 for optimal one-threshold ({λth 1 = λ2 = th = 0.055}) methods when = 0.009, λ 0.055}) and two-threshold ({λth 1 2 λsr = λrd = 10λsd = 1, and α = 1/0.3.

method. In particular, an SNR gain of about 5 dB is observed at the BER level of 10−5 for bit i1 . Fig. 8 shows performance improvement (also obtained with simulation) of the less-protected bit when we set BER1 in (48) to be 4 × 10−4 at SNR = 20 (dB). The scaling factors of Rayleigh fading channels are set to be identical with λsr = λrd = λsd = 1. With two other sets of threshold valth th th ues, namely, {λth 1 = 0.1, λ2 = 1.5} and {λ1 = 0.076, λ2 = 0.802}, the BER of the more protected bit also satisfies the requirement. However, with the optimal thresholds, namely, th {λth 1 = 0.071, λ2 = 0.161}, the BER of the less-protected bit is significantly better than that provided by the other threshold values. Note also that, although the optimality of the obtained thresholds is only guaranteed and can be confirmed at SNR = 20 (dB), Fig. 8 shows that such a set of thresholds also works very well at other SNR values. Finally, Fig. 9 illustrates the usefulness of jointly optimizing relaying thresholds and distance parameters. Note that, for the

Fig. 8. BER comparison of i1 and i2 when λsr = λrd = λsd = 1. The th optimal thresholds are {λth 1 = 0.071, λ2 = 0.161}.

Fig. 9. BER comparison of i1 and i2 when λsr = λrd = 8λsd = 1. The jointly optimal thresholds and distance parameter are provided in Table I.

case of the hierarchical 2/4-ASK constellation under consideration, the distance parameters can be optimized with a single variable α = d1 /d2 . For each SNR value in Fig. 9, the optimal thresholds and distance parameters obtained by both exact and approximated BER expressions are provided in Table I. Note that the constraint on BER1 at each SNR is also listed in Table I, and we set λsr = λrd = 8λsd = 1. The first observation from Fig. 9 is that the results obtained with approximated and exact BER expressions are very close (the curves are not clearly distinguishable). More importantly, it is seen from Fig. 9 that the jointly optimal thresholds and distance parameters yield a much better BER performance compared with other arbitrary values of the thresholds and distance parameter. VI. C ONCLUSION In this paper, we have obtained the average e2e BERs for two different protection classes in data transmission over a cooperative network, which consists of a source, a relay, and



2175

TABLE I BER C ONSTRAINT, O PTIMAL T HRESHOLDS , AND D ISTANCE PARAMETER

a destination. Each node is equipped with a single antenna, and the channels are Rayleigh fading. The two classes are modulated by a hierarchical 2/4-ASK constellation at the source. At the relay, based on the instantaneous received SNR and two threshold values, these classes can be modulated with a hierarchical 2/4-ASK or 2-ASK constellation or are not transmitted to the destination. Moreover, optimal thresholds are chosen to minimize the BER for the less-protection class, whereas the BER of the more protection class satisfies a given requirement. Numerical and simulation results were presented to corroborate the analysis. Performance comparison reveals that the optimal thresholds significantly improve the error performance. A PPENDIX A B IT-E RROR R ATES FOR C ASE 2 (Θ = 2) A. Subcase Φ = 1 In this subcase, the relay correctly decodes i1 . The received signals at the destination in the first and second phases are written, respectively, as (50) ysd = Es hsd (αs1 + s2 ) + nsd yrd = Er hrd ( s1 ) + nrd (51) where sm is the symbol corresponding to bit im at the source, m = 1, 2, and s1 is the symbol corresponding to bit√i1 at the relay. In this subcase, one has s1 , s2 = ±d2 = ±(1/ α2 + 1) and s1 = (1/d2 )s1 . The destination combines two received signals using MRC and produces the following sufficient statistic: yd = Es |hsd |2 (αs1 + s2 ) + Er |hrd |2 s1 + h∗sd nsd + h∗rd nrd ! = α|hsd |2 Es + α2 + 1|hrd |2 Er s1 + |hsd |2 Es s2 + h∗sd nsd + h∗rd nrd . (52) The BER for the first bit i1 in this subcase is given by P (εd , i1 |Θ = 2, Φ = 1) 1" P (εd , i1 |Θ = 2, Φ = 1, 00 sent) = 4 + P (εd , i1 |Θ = 2, Φ = 1, 01 sent) + P (εd , i1 |Θ = 2, Φ = 1, 10 sent)# + P (εd , i1 |Θ = 2, Φ = 1, 11 sent) $ $ α+1 √ γ + γrd 1 α2 +1 sd Q = Eγsd ,γrd 2 (γsd + γrd )/2 α−1 %% √ γ + γrd α2 +1 sd + Q . (γsd + γrd )/2

For the second bit i2 , only the signal received in the first phase is used to decode. The average BER of i2 is

⎧ 1 2 ⎨ σsd 1 2 P (εd , i2 |Θ = 2, Φ = 1) = 2 − 2 α +11 2 4⎩ 1 + α2 +1 σsd

⎫ (2α+1)2 2 (2α−1)2 2 ⎬ σ σ sd sd α2 +1 α2 +1 + − . (54) 2 (2α−1)2 2 ⎭ 2 1 + (2α+1) σ 1 + σ 2 2 α +1

sd

α +1

sd

B. Subcase Φ = 2 In this subcase, the relay incorrectly decodes i1 . The received signals at the destination in two phases are similar to (50) and (51). However, s1 = −(1/d2 )s1 . The sufficient statistic after combining two received signals using MRC is ! yd = α|hsd |2 Es − α2 + 1|hrd |2 Er s1 + |hsd |2 Es s2 + h∗sd nsd + h∗rd nrd . (55) Similar to the cooperation error, the BER of i1 is as follows: $ $ α+1 √ γ − γrd 1 α2 +1 sd P (εd , i1 |Θ = 2, Φ = 2) = Eγsd ,γrd Q 2 (γsd + γrd )/2 α−1 %% √ γ − γrd α2 +1 sd + Q . (56) (γsd + γrd )/2 A PPENDIX B B IT-E RROR R ATES FOR C ASE 3 (Θ = 3) A. Subcases Φ = 1, 2, 3 In these subcases, the received signals at the destination in two phases are ysd = Es hsd (αs1 + s2 ) + nsd (57) yrd = Er hrd (α s1 + s2 ) + nrd (58) where sm and sm are the symbols corresponding to bit im , m = 1, 2, at the √ source and the relay, respectively, i.e., s1 , s2 = ±d2 = ±(1/ α2 + 1), s1 = ±s1 , and s2 = ±s2 . Using MRC to combine both signals gives yd = Es |hsd |2 (αs1 + s2 ) + Er |hrd |2 (α s1 + s2 )

(53)

+ h∗sd nsd + h∗rd nrd ! =α Es |hsd |2 s1 + Er |hrd |2 s1 ! + Es |hsd |2 s2 + Er |hrd |2 s2 + h∗sd nsd + h∗rd nrd .


(59)

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With the first subcase (Φ = 1), the first bit is wrongly decoded at the relay; however, the second bit is correctly decoded, i.e., s1 = −s1 and s2 = s2 . Equation (59) becomes ! Es |hsd |2 − Er |hrd |2 s1 yd = α ! + Es |hsd |2 + Er |hrd |2 s2 + h∗sd nsd + h∗rd nrd . (60) The average BERs of i1 and i2 can be calculated as P (εd , i1 |Θ = 3, Φ = 1) 1" = P (εd , i1 |Θ = 3, Φ = 1, 00 sent) 4 + P (εd , i1 |Θ = 3, Φ = 1, 01 sent)

(61) P (εd , i2 |Θ = 3, Φ = 1) $ $ 1 √ γ − √2α+1 γ 1 α2 +1 sd α2 +1 rd = Eγsd ,γrd Q 2 (γsd + γrd )/2 2α+1 √ γ − √α12 +1 γrd α2 +1 sd −Q (γsd + γrd )/2 1 √ γ + √2α−1 γ α2 +1 rd α2 +1 sd +Q (γsd + γrd )/2 %% 2α−1 √ γ + √α12 +1 γrd α2 +1 sd . +Q (γsd + γrd )/2 (62) Similar to the first subcase, one can verify the BERs for the second and third cases as given in (17)–(20). B. Subcase Φ = 4 Similar to the previous subcases, the combination of two received signals by using MRC is s1 + s2 ) yd = Es |hsd |2 (αs1 + s2 ) + Er |hrd |2 (α + h∗sd nsd + h∗rd nrd ! Es |hsd |2 + Er |hrd |2 s1 =α ! + Es |hsd |2 + Er |hrd |2 s2 √ where s1 , s2 = ±d2 = ±(1/ α2 + 1).

P (εd , i1 |Θ = 3, Φ = 4) $ $ (α + 1) 2(γsd + γrd ) 1 √ Q = Eγsd ,γrd 2 α2 + 1 %% (α − 1) 2(γsd + γrd ) √ + Q α2 + 1 (64)

+ P (εd , i1 |Θ = 3, Φ = 1, 10 sent) # + P (εd , i1 |Θ = 3, Φ = 1, 11 sent) $ $ α+1 √ γ − √α+1 γ 1 α2 +1 sd α2 +1 rd Q = Eγsd ,γrd 2 (γsd + γrd )/2 α−1 %% √ √α−1 γrd γ − sd 2 2 α +1 α +1 + Q (γsd + γrd )/2

+ h∗sd nsd + h∗rd nrd

The probabilities of error of the subcase Φ = 4 of i1 and i2 are given, respectively, as

(63)

P (εd , i2 |Θ = 3, Φ = 4) $ $ 2(γsd + γrd ) 1 √ 4Q = Eγsd ,γrd 4 α2 + 1 (2α + 1) 2(γsd + γrd ) √ − 2Q α2 + 1 %% (2α − 1) 2(γsd + γrd ) √ + 2Q . α2 + 1 (65) Based on [32], it can be verified that & ' () Eγsd ,γrd Q 2β(γsd + γrd ) ⎧ 2 2 2 ⎪ βσrd βσrd 1 1 ⎪ 1 + 2 1+βσ2 , ⎪ 2 ⎪ 2 1 − 1+βσrd ⎪ rd ⎪ ⎞ ⎨ ⎛ =

⎜ ⎪ ⎪ 1⎜ ⎪ ⎪ 2 ⎝1 − ⎪ ⎪ ⎩

2 σsd

βσ 2 sd −σ 2 rd 1+βσ 2 sd 2 −σ 2 σsd rd

βσ 2 rd 1+βσ 2 rd

⎟ ⎟, ⎠

2 2 = σsd if σrd

2 2 if σrd = σsd .

(66) Therefore, the BERs of i1 and i2 in (64) and (65) can analytically be calculated.

E2E

A PPENDIX C B IT-E RROR R ATES FOR G ENERALIZED H IERARCHICAL 4/16-Q UADRATURE A MPLITUDE M ODULATION C ONSTELLATIONS

In this Appendix, we consider the case that the source and the relay use a hierarchical 4/16-QAM rectangular constellation, as illustrated in Fig. 10 [20]. It means that there are also two classes, and each class contains two bits. Two bits from the first class are assigned to the most significant bits in the in-phase (I) and quadrature (Q) channels. Similarly, two bits from the second class are assigned to the least significant bits in the I and (I) (I) Q channels. We have two distance vectors d(I) = [d1 , d2 ] (Q) (Q) and d(Q) = [d1 , d2 ]. As shown in [20] and [33], the average BER of the first class is BER(1) (γ1th , γ2th ) =

BER(γ1th , γ2th , i1 ) + BER(γ1th , γ2th , q1 ) 2 (67)



Fig. 10. Generalized hierarchical 4/16-QAM constellation.

where BER(γ1th , γ2th , i1 ) and BER(γ1th , γ2th , q1 ) are the BERs of the more protected bits of the in-phase and quadrature channels, respectively. Likewise, the average BER of the second class is BER(2) (γ1th , γ2th ) =

BER(γ1th , γ2th , i2 ) + BER(γ1th , γ2th , q2 ) 2 (68)

where BER(γ1th , γ2th , i1 ) and BER(γ1th , γ2th , q1 ) are the BERs of the less protected bits of the in-phase and quadrature channels, respectively. With rectangular constellations, the average BER computation of the first and second classes reduces to finding the BER of the in-phase bits only. The average e2e BERs for the in-phase bits i1 and i2 are similar to (4). The only differences are scaling factors. Similarly, we can find the average e2e BERs for the quadrature bits q1 and q2 . Therefore, the average e2e BERs of the first and second classes of a hierarchical 4/16-QAM constellation can be formulated. Based on such a formulation, the two optimum thresholds can numerically be found. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful comments, which have greatly improved the presentation of this paper. R EFERENCES [1] E. C. van der Meulen, “Three-terminal communication channels,” Adv. Appl. Probab., vol. 3, no. 1, pp. 120–154, 1971. [2] T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, vol. IT-25, no. 5, pp. 572–584, Sep. 1979. [3] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacity theorems for relay networks,” IEEE Trans. Inf. Theory, vol. 51, no. 9, pp. 3037–3063, Sep. 2005. [4] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity, Part I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [5] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity, Part II: Implementation aspects and performance analysis,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1939–1948, Nov. 2003.

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Ha X. Nguyen (S’08) received the B.Eng. degree in electrical engineering from Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam, in 2003 and the M.Sc. degree in electrical engineering from Information and Communications University, Daejeon, Korea, in 2007. He is currently working toward the Ph.D. degree with the Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada. His research interests span the areas of unequal error protection in digital communications and cooperative wireless networks.

Ha H. Nguyen (M’01–SM’05) received the B.Eng. degree in electrical engineering from Hanoi University of Technology, Hanoi, Vietnam, in 1995, the M.Eng. degree in electrical engineering from the Asian Institute of Technology, Bangkok, Thailand, in 1997, and the Ph.D. degree in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2001. He joined the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK, Canada, in 2001, where he is currently a Full Professor. He holds adjunct appointments with the Department of Electrical and Computer Engineering, University of Manitoba, and TRLabs, Saskatoon. He was a Senior Visiting Fellow with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia, during October 2007–June 2008. He is a coauthor (with E. Shwedyk) of A First Course in Digital Communications (Cambridge, U.K.: Cambridge University Press, 2009). His research interests include digital communications, spreadspectrum systems, and error-control coding. Dr. Nguyen is a Registered Member of the Association of Professional Engineers and Geoscientists of Saskatchewan. He is currently an Associate Editor of the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS and the IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY.

Tho Le-Ngoc (F’97) received the B.Eng. degree (with distinction) in electrical engineering and the M.Eng. degree in microprocessor applications from McGill University, Montreal, QC, Canada in 1976 and 1978, respectively, and the Ph.D. degree in digital communications from the University of Ottawa, Ottawa, ON, Canada, in 1983. From 1977 to 1982, he was with Spar Aerospace Ltd., where he was involved in the development and design of satellite communications systems. From 1982 to 1985, he was an Engineering Manager of the Radio Group with the Department of Development Engineering, SRTelecom Inc., where he developed the new point-to-multipoint DA-TDMA/TDM Subscriber Radio System SR500. From 1985 to 2000, he was a Professor with the Department of Electrical and Computer Engineering, Concordia University, Montreal. Since 2000, he has been with the Department of Electrical and Computer Engineering, McGill University. His research interest is in the area of broadband digital communications, with special emphasis on modulation, coding, and multiple-access techniques. Dr. Le-Ngoc is a Senior Member of the Ordre des Ingénieur du Quebec, a Fellow of the Engineering Institute of Canada, and a Fellow of the Canadian Academy of Engineering. He was the recipient of the 2004 Canadian Award in Telecommunications Research and the 2005 IEEE Canada Fessenden Award.
