UFDC Home  myUFDC Home  Help 



Full Text  
WIRELESS COOPERATIVE NETWORKS WITH DIFFERENTIAL MODULATION: PERFORMANCE ANALYSIS AND RESOURCE OPTIMIZATION By WOONG CHO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 2007 Woong Cho To my wife and family ACKNOWLEDGMENTS I want to express my tremendous gratitude to my supervisor, Dr. Liuqing Yang, for her tireless effort as well as for providing me with insights, inspiration, and encouragement without which I could not have performed this research. I deeply thank her for all of the patient guidance, invaluable advice, and numerous discussions we have shared throughout my graduate studies. I would like to thank all the members of my advisory committee, Dr. Jenshan Lin, Dr. Tao Li and Dr. Shigang C'!, for their valuable time and energy in serving on my supervisory committee. I would also like to thank Rui Cao, my friend and colleague at Signal processing, Communications, and Networking (SCaN) group, for the priceless discussions we shared, which generated many ideas for this research. I wish to extend my sincere thanks to all the members of the SCaN group, Huilin Xu, Fengzhong Qu, Dongliang Duan, and Wenshu Zhang, for their companionship and support throughout our time together. As ahv ~, I want to thank to my parents and parentsinlaw, for their unyielding support and love. Their encouragement and understanding through my studying periods have meant more than I can ever express. Last, I would like to express my greatest thanks and adoration to my loving wife. I want to thank her for supporting and understanding me in innumerable v ,i, particularly during all our time together in the United States, and throughout my Ph.D studies. TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. 4 LIST OF FIGURES .................................... 7 A BSTRA CT . . . . . . . . . . 10 CHAPTER 1 INTRODUCTION ...................... .......... 12 1.1 Cooperative Networks .................... ........ 12 1.2 Motivation ...................... ........... 13 2 SYSTEM M ODEL .................................. 17 2.1 Relaying Protocols and C('!h i,, I Modeling ........ ......... 17 2.1.1 DecodeandForward (DF) ......... ............. 18 2.1.2 AmplifyandForward (AF) ........... ........... 19 2.1.3 Distributed SpaceTime Coding (DSTC) .............. .. 19 2.2 Differential Demodulation and Decision Rules ............... .. 21 2.2.1 DF Protocol .................. ............ .. 21 2.2.2 AF Protocol .................. ............ .. 23 2.2.3 DSTC Protocol .................. .......... .. 24 3 PERFORMANCE ANALYSIS .................. ......... .. 26 3.1 SER for DF Protocol .................. ........... .. 26 3.2 SER for AF Protocol .................. ........... .. 30 3.3 CER for DSTC Protocol .................. ......... .. 38 4 OPTIMUM RESOURCE ALLOCATION ................. .. 45 4.1 Convexity of SER .................. ............ .. 46 4.2 Energy Optimization ............... ........... ..50 4.3 Relay Location Optimization. .............. ... .. 63 4.4 Joint Energy and Location Optimization ............... .. .. 70 5 SIMULATIONS AND DISCUSSIONS ................ .... 75 5.1 Benefits of Energy and Location Optimizations . . ..... 75 5.2 Benefits of Joint Optimization ................ ... .. 83 5.3 Conventional System vs STCbased System. ............... .. 87 6 CONCLUSIONS AND FUTURE WORK ............. .... .. 92 6.1 Conclusions .................. ................ .. 92 6.2 Future W ork .................. ................ .. 94 LIST OF REFERENCES ..................... .......... 96 BIOGRAPHICAL SKETCH ................... .......... 102 LIST OF FIGURES Figure 11 Simple cooperative network . ......... 21 Setup of the cooperative network . ...... 31 SER at different r,s, values (DF, L = 2, M = 2) . . 32 SER at different yd,, values (DF, L = 2, M = 2) .. 33 SER bound versus ~d,, and y, (DF, L = 2, M = 2). . 34 SER comparison between approximation and simulation ( Yd,r ). . . . . . . . 35 SER comparison between approximation and simulation (i Y d,r ) . . . . . . 36 SER comparison between coherent  I, i and differential 7 d,re) . . . . . . 37 The CER for the DFSTC protocol (L = 1, 2, and 3, SNR= 38 The CER for the AFSTC protocol (L = 1, 2, and 3, SNR AF ND, AF DL, system 39 The effect of unbalanced link SNR for the DFSTC protocol (L = 1, 2, al 310 The effect of unbalanced link SNR for the AFSTC protocol (L = 1, 2, ai 41 Network topologies: (a) Ellipse case; (b) Line case . ..... 42 Exact and approximate optimum energy allocations with different path nents (DF, L = 1, p 10dB) . ............... 43 SER versus energy allocation at the given relay location Ds,r (DF, L D = 1, v = 4) . . . . ... . . . SNR(dB) SNR(dB)= 7d,, (AF DL, SNR= nd 3). id 3). loss expo 2, p 10=dB, 44 Comparison of optimal energy allocation between the numerical search and simulated results at various L values (DF, p = 10dB, v = 4) .. ................ 45 Comparison of normalized optimum energy allocation at different p values (DF, L = 1). 46 Existence of the optimum solution (AF ND, p = 15dB, v = 3, L=2) .. ...... 47 SER versus energy allocation at the given relay location Ds,r (AF ND, L = 2, p = 15dB D = 1, v = 4) . . . . . ... . . 48 SER versus energy allocation at the given relay location Ds,r (AF DL, L = 2, p 15dB D = 1, v = 4) . . . . . ... . . page 13 17 29 29 30 36 37 . 37 . 42 . 42 . 43 . 44 . 46 S= d,r),, . S .. = d,r) . . ' 49 Optimum energy allocation (AF, ND and DL, D= 1, p 30dB, v = 4). . ... 62 410 Optimum energy allocation (DFSTC and AFSTC, D = 1.2, Ds,d = 1, p=15dB, v = 4). 62 411 Optimum location of relays (DF, p = 10dB and L 1). .. . . ...... 65 412 SER versus relay location distribution at the given energy allocation ps/p (DF, L = 2, p 10dB, D 1v = 4). ...... .......... .......... 66 413 SER versus relay location distribution at the given energy allocation ps/p (AF ND, L 2, p 15dB, D 1,v = 4) ...... ........... ... ...... 67 414 SER versus relay location distribution at the given energy allocation ps/p (AF DL, L 2, p 15dB, D 1,v = 4) ...... ........... ... ..... 68 415 Optimum relay location (AF, ND and DL, D = Ds,d = 1, p 30dB, v =4). ... . 69 416 Optimum relay location (DFSTC and AFSTC, D = 1.2, Ds,d = p 15dB, v = 4). 69 417 Iterative search: flow chart. .................. ........... .. 71 418 Performance surface versus ps/p and Ds,r (DF, p = 10dB, v = 4, L = 3, DBPSK). 72 419 Performance surface versus ps/p and Ds,r (AF, ND, p = 15dB, v = 4, L = 3, DBPSK). 72 420 Performance surface versus ps/p and Ds,r (DFSTC, p = 15dB, v = 4, L = 2). . 73 421 Performance surface versus ps/p and Ds,r (AFSTC, p = 15dB, v = 4, L 2). . 74 51 SER comparison between relay systems with and without energy optimization (DF, p = 15dB, D = 1.2, Ds,d = 1, v = 4) ........... . . 76 52 SER comparison between relay systems with and without relay location optimization (DF, p = 15dB, D = 1.2, Ds,d = 1, v 4). . . . . 76 53 SER comparison between relay systems with and without energy optimization (AF ND, p = 15dB, D = 1.2, Ds,d = 1, v = 4). . .. . . 78 54 SER comparison between relay systems with and without energy optimization (AF DL, p= 15dB, D = 1.2, Ds,d = 1, v = 4). . . . . 78 55 SER comparison between relay systems with and without relay location optimization (AF ND, p 15dB, D= 1.2, Ds,d = 1, v =4). .. . . 79 56 SER comparison between relay systems with and without relay location optimization (AF DL, p = 15dB, D = 1.2, Ds,d = 1, v = 4). .. . . 80 57 CER comparison between relay systems with and without energy optimization (DF STC, p = 15dB and 25dB, D 1.2, Ds,d 1, v = 4). ... . . 81 58 CER comparison between relay systems with and without relay location optimization (DFSTC, p 15dB and 25dB, D 1.2, Ds,d 1, = 4). . . 81 59 CER comparison between relay systems with and without energy optimization (AF STC, p 15dB and 25dB, D= 1.2, Ds,d 1, v =4). ... . . 82 510 CER comparison between relay systems with and without relay location optimization (AFSTC, p = 15dB and 25dB, D = 1.2, Ds,d = 1, v = 4). .. . . 82 511 The SER contour versus Ds,r and ps/p (AF ND, Ds,d = D = 1, p 15dB, L = 2). 84 512 The SER contour versus Ds,r and ps/p (AF DL, Ds,d = D = 1, p 15dB, L = 2). 84 513 The SER contour versus Ds,r and ps/p (DF, Ds,d = D = 1, p10dB, L = 3). . 85 514 The CER contour versus Ds,r and ps/p (DFSTC, D = 1.2, Ds,d = 1, p 15dB, L = 2). 86 515 The CER contour versus Ds,r and ps/p (AFSTC, D = 1.2, Ds,d = 1, p 15dB, L = 2). 86 516 Data rate comparison between the conventional system and STCbased system in terms of the required time slots per information symbol. ................. 87 517 BER comparison between the conventional systems and STCbased systems with same modulation size (DF vs DFSTC, SNR =,,, = 7d,r). .... . . 89 518 BER comparison between the conventional systems and STCbased systems with same modulation size (AF vs AFSTC, SNR 7,,, = d,r). .... . . 89 519 BER comparison between the conventional systems and STCbased systems with equal/similar transmission rate (DF vs DFSTC, SNR=,, = %d,r)yd. ................ 90 520 BER comparison between the conventional systems and STCbased systems with equal/similar transmission rate (AF vs AFSTC, SNR= r, = yd,r). ................ 91 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy WIRELESS COOPERATIVE NETWORKS WITH DIFFERENTIAL MODULATION: PERFORMANCE ANALYSIS AND RESOURCE OPTIMIZATION By Woong Cho December 2007 'C! ir: Liuqing Yang Major: Electrical and Computer Engineering In wireless cooperative networks, virtual antenna arrays formed by distributed network nodes can provide cooperative diversity. Obviating channel estimation, differential schemes have long been appreciated in conventional multiinput multioutput (MI\ !lO) communications. However, distributed differential schemes for general cooperative network setups have not been thoroughly investigated. In this dissertation, we develop and analyze distributed differential schemes using two conventional relaying protocols, decodeandforward (DF) and amplifyandforward (AF), and spacetime coding (STC)based relaying protocol with an arbitrary number of relays. For each protocol, we analyze the error performance and consider the resource allocation as a twodimensional optimization problem: energy optimization, location optimization, and joint energylocation optimization. We first derive an upper bound of the error performance for the DF system, the approximated error performance for the AF system, and an upper bound for the STCbased system at reasonably high SNR, respectively. Based on these results, we then develop the energy optimization and relay location optimization schemes that minimize the average system error. Analytical and simulated comparisons confirm that the optimized systems provide considerable improvement over unoptimized ones, and that the minimum error can be achieved via the joint energylocation optimization. We compare the results of optimization and the effects of different relaying protocols and obtain several interesting results. In addition, the comparison between the conventional system and STCbased system is addressed. CHAPTER 1 INTRODUCTION 1.1 Cooperative Networks Diversity techniques have been widely adopted to combat multipath fading in wireless communications. In particular, space diversity based on multiinput multioutput (\!l \ O) has emerged as an attractive research area due to its capacity and error performance enhancements. However, the antenna packing limitation of the MIMO technique renders practical implementation clumsy in wireless applications, such as cellular, sensor, or adhoc networks. To overcome this limitation while pertaining the diversity benefits, distributed modulation schemes have been i::I. 1. by using relay node(s). The basic idea is to create multiple paths over a network by using relay node(s), where each relay node is equipped with a single antenna. Then, the rJ l ,. .1 signals from each relay node are combined at the destination node, which provides cooperative diversity gain by forming the distributed network nodes into virtual antenna arrays. Therefore, by exploiting the cooperative networks, the antenna packing limitations can be eliminated while the spatial diversity gain can still be achieved [36, 48, 49]. Thanks to these advantages, cooperative networks can be applied in various scenarios to enhance the network performance. For example, wireless personal area network (WPAN) and local area network (WLAN) can extend their coverage areas by using relay nodes. In home environments, cooperative networks can help distribute the multimedia data from the central entertainment unit to devices anywhere in the house, by mounting relay nodes to the wall or even embedding them inside the walls. Furthermore, cooperative networks also find applications in intelligent transportation systems including intervehicle, intravehicle and vehicletoroad communications, to enable reliable distribution of the emergency information to certain groups of drivers via cooperation of multiple vehicles on the road. To illustrate the basic concept of cooperative networks, Fig. 11 represents a simple cooperative network in which the source node transmits a signal to the destination relay source destination Figure 11. Simple cooperative network. node via the relay node. The sourcerelaydestination and sourcedestination links are commonly referred to as the relay link and the direct link, respectively. There are two conventional relaying protocols which are widely considered in the existing literature. One is the decodeandforward (DF) protocol and the other is the amplifyandforward (AF) protocol. With DF relaying protocol, the relay node demodulates the received signal from the source node and remodulates that signal. Then the relay node transmits the remodulated signal to the destination node. With AF relaying protocol, the relay amplifies the signal from the source and then simply forwards it to the destination node. Another protocol is the distributed spacetime coding (DSTC) protocol which can support higher transmission rate. This protocol can also adopts DF and AF depending on the relay node operation. 1.2 Motivation Cooperative networks have received tremendous attention in wireless communications [9, 38, 42, 46, 53]. Many studies have been carried out to analyze the performance of cooperative networks. The outage probability of cooperative networks from the information theoretic perspective is presented in [35]. The error rate and outage probability of a cooperative network are analyzed in [26] without considering the direct link. In [28], a later study by the same authors, they consider both relay and direct links with a fixed gain at the relay. In [10], the multihop relay transmission is considered, and the exact symbol error rate is derived in [3]. A more general case of the relay networks is developed in [45]. In [4, 37, 47, 54, 62], the performance of STCbased cooperative networks is considered. All of these work on cooperative networks focuses on coherent demodulation based on the availability of the channel state information (CSI) at both the reliv and the destination. Accurate estimation of the CSI, however, can induce considerable communication overhead and transceiver complexity, which increases with the number of relay nodes employ, .1 In addition, CSI estimation may not be feasible when the channel is rapidly timevarying. To bypass channel estimation, cooperative networks obviating CSI have been recently introduced. These cooperative systems rely on noncoherent or differential modulations, including conventional fre'ii,l'v hift keying (FSK) and differential phaseshift keying (DPSK) as well as STCbased ones. In [12, 16], using noncoherent and differential modulations, it is shown that the log likelihood ratio can be combined by capturing the detection error at the relay nodes according to the sotermed transition probability if the partial CSI is known at the reliv, and destination. The performance of a single relay system with noncoherent and differential modulations is considered in [6, 13] and [29, 30, 57, 7173], respectively. In [16, 17, 56, 63, 64, 70], a distributed STC system for cooperative networks is introduced by using differential or noncoherent schemes. To improve the error performance and to enhance the energy efficiency of cooperative networks, optimum resource allocation recently emerged as an important problem attracting increasing research interests (see e.g., [5, 22, 27, 39, 65]). These work is based on different relaying protocols (amplifyandforward, decodeandforward and block Markov coding), under various optimization criteria (signaltonoise ratio (SNR) gain, SNR outage probability, energy efficiency and capacity), and with different levels of CSI (instantaneous CSI and channel statistics). However, all of them only consider the power allocation and mostly focus on a singlerelay setup. In [7, 14, 18, 30], optimum power allocation for multiple relay links is developed under various relaying protocols. [40] and [23, 24], respectively, consider the optimum energy and bandwidth allocation in Gaussian channel and multihop system, while [50] introduces the opportunistic relay selection. Recently, it has been noticed that the relay location is a critical factor influencing the relay network performance [38, 67]. However, [38] neglects the path loss effect which is closely related to the relay location such that the location optimization problem is erroneously formulated as an energy allocation problem. In [67], it is shown that given uniform energy allocation there is an optimum relay location which provides the optimum performance. However, they only consider the optimum location of the rel iv without appropriate energy allocation. With DF and AF protocols, the joint energy and location optimization for the cooperative network is introduced in [15, 1921]. In this research, we develop cooperative networks with an arbitrary number of relays by employing DF, AF, and DSTC protocols. Equally attractive is that our analysis is tailored for relay systems with differential modulation, which is known to reduce the receiver complexity by bypassing channel estimation. Notice that the DF and AF protocols generalize the standard differential modulation to a distributed scenario with an arbitrary number of rel i. The DSTC protocol relies on an increased level of user cooperation via the distributed counterpart of the differential spacetime codes [31, 32]; that is, for each data block, a spacetime codeword encoded across distributed relays is transmitted over a common relaydestination channel. Using these relaying protocols, we then derive the error performance and the optimum resource allocation of cooperative networks. Different from existing works on resource optimization, we tackle the problem from two angles: i) Optimizing the power allocation across relay and source nodes for any given sourcerelaydestination distances; and ii) Optimizing the relay location for any power distribution and sourcedestination distance. To the best of our knowledge, we are among the first to formulate the 2dimensional optimization problem. In addition, we are also the first to consider the joint powerandrelaylocation optimization for cooperative networks. To facilitate the resource optimization, we develop analytical expressions of the error performance for various relay protocols with arbitrary number of rel i . We first derive an upper bound of the overall symbol error rate (SER) for the DF protocol, the approximated SER for the AF protocol, and an upper bound of the overall codeword error rate (CER) for the DSTC protocol. Then, the optimum resource allocation which minimizes the system SER and CER is developed. For the DF and DSTC protocols, the multiple relay system with no direct link is considered for analytical tractability. For the AF protocol, we consider the resource allocation of the relay system both with and without the direct link. The former evaluates a scenario where there is a direct link between the source node and the destination node, while the latter takes into consideration a situation where obstacles disable a direct transmission resulting in no direct link. For each protocol, under the total energy and sourcerelaydestination distance constraints, we show that the optimum energy allocation can be achieved given the relay location, the optimum relay location can be obtained given the source and relay transmit energy, and the optimum error performance can be achieved through the joint energy and location optimization. The benefits of optimization are confirmed by numerical examples and simulations. We show that each optimum resource allocation has a different property depending on the relaying protocol. In addition, the comparison between the conventional cooperative networks and STCbased ones is addressed. CHAPTER 2 SYSTEM MODEL The system model is based on a network setup with one source node s, L relay nodes {r}k 1 and one destination node d, as depicted in Fig. 21. Each node is equipped with a switch that controls its transmit/receive mode to enable halfduplex communications. Multiplexing among the network nodes can be achieved via frequencydivision, timedivision or codedivision techniques. For notational convenience, we will consider the timedivision multiplexing (TDM). However, the presented analysis and results are readily applicable to freqi. i i division multiplexing (FDM) and codedivision multiplexing (CDM). For this study, we first consider two conventional relaying protocols, i.e., decodeandforward (DF) and amplifyandforward (AF), and then we develop the distributed spacetime coding (DSTC) protocol. 2.1 Relaying Protocols and Channel Modeling Let us first consider the conventional relaying protocols. With DF protocol, the relay nodes demodulate the signal from the source node, then remodulate and forward to the destination node. With AF protocol, the re1iv, amplify the signal from its source and then forward it to the destination. Using TDM, the relay transmission consists of two phases. In phase I, the source broadcasts a symbol to all rel i,. In phase II, each relay transmits the amplified signal to the destination during its distinct time slot. As a result, the L \\ \ '' destination d source s relays {r} Figure 21. Setup of the cooperative network. sourcetorelay (s rk) links share a common channel, while the L relaytodestination (rk d) links are mutually orthogonal. In order to bypass channel estimation and to cope with time variation of the wireless channel, differential modulation is employ, ,1 at the source node. Specifically, with the nth phaseshift keying (PSK) symbol being as s, = ej2c,"/M, Cn E {0, 1, ..., M 1}, the corresponding transmitted signal from the source is: 8 jX7_iSn, n> 1 Xn (21) 1, n = 0. In phase I, for both DF and AF protocols, the encoded signal is broadcast via a common channel. The received signals at the kth relay and the destination are given by yns VShs'x8 + zns, k 1, 2, ..., L, y d,s d s + _Lyd,s, (2'2) ysn V hSs'n n (2n2) where SE is the energy per symbol at the source, and we denote the fading coefficients of channels s rk and s d during the nth symbol duration as hr'' ~ CA(0, ok ) and h~' ~ CA(0, o),) and the corresponding noise components as z'k, CN(O, rI,s) and zf'" ~ C.A(O,A/d,,), respectively. Here, CNA'(p, u2) represents the complex Gaussian distribution with mean p and variance a2 2.1.1 DecodeandForward (DF) In phase II, the received signal from the source is differentially demodulated and remodulated independently at each relay rk. The demodulation step generates an estimate s^ from y/'~ in Eq. (22), using the decision rule that we will present in the next subsection. The remodulation step is carried out as in Eq. (21), but with s, replaced by its estimate and x replaced by s^ and xf. Then, the received signal at the destination corresponding to each relay node is given by yd,rk h ,r~ x + z ', k 1 2,..., (23) v/rk, fn + .; k t, 2, ..., L, (23) where S,, is the energy per symbol at the kth relay node, the fading coefficient of the channel between rk and d during the nth symbol duration is ht'r ~ CA(0, J,, ), and the noise component is z, ~ CN(0, N,rk). 2.1.2 AmplifyandForward (AF) For the AF protocol, the received signal at the destination corresponding to each relay is given by Ynd,rk V ,'S,rh d n + ,r k Ir d... ^Z ( ) hdk kr x + zd k= 1,2,...,L, (24) where SrC is the energy per symbol and xi denotes the nth transmitted symbol from the kth relay. The fading coefficients of the rk d channels and the noise component at the destination are h CA(O, ) and zT ~ CA/'(O,Ad,rA), respectively. At each of the reliv, xi can be represented as ; = A,,y'", k 1, 2, ...,L, (25) where A, is the amplification factor. To maintain a constant average power at the relay output, the amplification factor is given by Ark tk= 1,2,..., L. (26) This A, is reasonable for both differential and noncoherent modulations, since we can estimate the value of ar by averaging the received signals without knowing the instantaneous CSI [12, 71]. 2.1.3 Distributed SpaceTime Coding (DSTC) In the DSTC protocol, we will consider cooperative networks employing the differential unitary space time code (DUSTC) which does not require the exact channel estimation [31, 32]. For simplicity, we will use the diagonal design with the cyclic construction in [31]. Notice that each diagonal element of the codeword corresponds to a standard differential phaseshift keying (DPSK) _ii in.i where its modulation size increases as L increases. During the first L time slots of a transmission, the diagonal entries of the DUSTC symbol block are broadcast to the relv. Then, each relay node decodes (or amplifies) the corresponding Lth diagonal element of STC signal, and these signals are transmitted by a common rk d channels during the following L time slots, which is different from the conventional relaying protocol. With this protocol, both DF and AF schemes are considered for the cooperative network. We denote DFSTC and AFSTC as the decodeandforward spacetime coding and amplifyandforward spacetime coding, respectively. Denote the nth differentially encoded signal block from the source as X' X_, V(Q) with X = IL, where V(Q) is an L x L diagonal unitary matrix, Q, E {0, 1,..., M 1} with M = 2", and IL is an L x L identity matrix, where r represents the data rate of the original information which we set to 1. The matrix V(") has the form V() = V7" with (see [31]) C 6j(27/M)ui o V,= 0 . 0 (27) 0 .. j(27/M)uL where uz e {0,..., M 1}; = {1,..., L}. Then, the nth received signal block at the relJ', is given by Y" = VSSH,"X` + Z' (28) where Qs is the energy per symbol at the source, H'' := diag{h ,', hr2', ...,hr'L'} is the channel matrix between the source and rel , , and ZX := diag{zl'", z,, 2'~, .. z'L} is the noise matrix at the rel v. We use diag{al, a2,..., aL} as a diagonal matrix with [ai, a2, ..., aL] on its diagonal. Let us denote the nth transmitted signal block from the rel~,1 as Xr, then the corresponding received signal block at the destination is given by yd,= E1/2H,rXr + Z (29) n Er Hn n n+ , where Er : diag{S,,,S ...,",} is the energy per symbol matrix at the rel,v, Hnr : diag h'l ,hd,"2,...,h'rL/} is the channel matrix between the rel ,v and destination, and Z := diag {z', z'2, ..., z rL} is the noise matrix at the destination. Depending on the relaying protocols, X' has different forms at the rel . Throughout this dissertation, all fading coefficients are assumed to be independent. Without loss of generality, we also assume that all noise components are independent and identically distributed (i.i.d) with A,,j = Ao, i,j C {s, rk, d}. Accordingly, we can find the received instantaneous signaltonoise ratio (SNR) between the transmitter j and the receiver i as I h ,j 12 1 A/2 ; e {srFk,d}. It then follows that the average received SNR is ; (o jSj)/AMo. 2.2 Differential Demodulation and Decision Rules In this section, the differential demodulation and decision rules for the relay networks will be developed. We consider L relay links for the DF and DSTC protocols, and L relay links and the direct link for the AF protocol. 2.2.1 DF Protocol As mentioned before, differential demodulation is performed at the relay and destination nodes. To derive the demodulation, decision and diversity combining rules, let us begin with the received signal at the relay or the destination node y, = hnx, + zn, which is extracted from Eqs. (22) and (23) by dropping the superscripts. Using the differential encoding in Eq. (21), the received signal can be reexpressed as: yn = hn(xnslS) + Zn = n1in + z' , where z, = z,,_l,. For Mary PSK symbols, it follows that L[ s,] = 1. Hence, the conditional distribution of y, is complex Gaussian with mean ynls and variance 2/o0. As a result, we obtain the log likelihood function (LLF) of y, as: l(y>) := InpJlx(n s7) {u~ny1(sn)*} R{(ys)*y 1si} (210) where i,j e {s, rk, d, s = ej2 m/M and m c {0,1,..., M 1}. We use E[.] for expectation, (.)* for conjugate, and {.} for the real part. At the kth relay node, the differential demodulator is then straightforward: s e j2f n/M :h argmax l'rs(y,) argmax R{ (yS) *y s"5}. (211) At the destination node, however, there are L different LLF's corresponding to the L transmitted signals from the relays: lrk (y) Inpy1l (y ) {(y *y ds} k 1,2,...,L (212) If the channel state information is known at the relJi and the destination node, then it is possible to combine the LLF's by weighting them accordingly. However, keeping in mind that the differential modulation is considered in the first place because of its capability of bypassing channel estimation, we will focus on the scenario where no channel state information is available. In this case, the LLF's in Eq. (212) have to be combined with equal weights. As a result, the decision rule at the destination node can be readily obtained as: L S, ej2/ M : r argmax l '(y,). (2 13) k= 1 With no channel information assumed at either the relays or the destination node, this decision rule turns out to be the differential detection with equal gain combining (EGC) [55, C'!h Ipter 6.6]. 2.2.2 AF Protocol Now consider the AF protocol. Similar to the DF protocol, we can represent the received signal at the destination corresponding to each relay node as yt d hrkx, + zr = ysr + r)', k = 1, 2, ..., L, where hd,fr = A h hrk, A = zA, fh z + zT, and ( )r' = zT zr'(_s,. Then, conditioned on the channels, the received signal is y? ~ C(y drkl~", a2 ), where the variance of the .,.:regate noise is given by hk,ef = 2Aro(TAr K d, + 1), k= 1, 2,..., L. (214) The received signal at the destination corresponding to the source can be represented as y's = yis, + (z with y ~ C(yss,, 2f/o). As a result, we obtain the LLF corresponding to the L transmitted signals from the rel 1v and the transmitted signal from the source, given that x' is transmitted by the source: Idr(yr: In p (y d,rk m )ydrn,rk d },k rk2,.., Id, (y ) ds n n , S(yp: Inp (yd's') = RI(yds)*y }, (215) where s = ej2 r/M and m e {0, 1,..., M 1}. Notice that, although the LLF, ldr(y,), in Eqs. (212) and (215) has the same form, the one in (215) is obtained from a different transmitted signal. For the DF protocol, the LLF is obtained from the given relay transmitted signal which is a demodulated signal from the source transmitted signal. However, the LLF for the AF protocol is obtained from the source transmitted signal which is simply amplified and forwarded at the rel i without any demodulation. At the destination node, these (L + 1) signals can be combined to estimate the transmitted signal from the source. Using the multichannel communication results in [44, ('! Ilpter. 12] and the above LLFs, the decision rule at the destination node can be obtained as (see [30, 72]): L 1drrenr (2 16)( =ej2m/M : arg max Wd,s T un) + Wd,r n (216) me{0,1,...,M1}  k/ 1 where Wd,s and Wd,,r are combining weights which are given by 1/A/o and 2/o a ,h, respectively, and it is assumed that the variances of channels are available at the destination node. Interestingly, this decision rule is the same as the DF protocol in Eq. (213) except for the weighing terms. 2.2.3 DSTC Protocol Since the transmission signal is based on the differential spacetime code, we can apply the corresponding spacetime differential demodulation. Then, the maximum likelihood (il I) differential demodulation rule [31], given X' = Xn, is Qn = arg max r Y i + YV m (217) nme{0,1,...,Ml} where   represents the Frobenius norm. This decision rule is the general structure for DUSTC. Depending on the relaying protocol, the Frobenius norm part can have different values. In the DFSTC protocol, the received signal at the re!i, Y"', is decoded. Since each diagonal entry of the codeword X' is a DPSK signal and the kth relay demodulates and remodulates .:4,./ I". .l ';./.l, the corresponding kth entry of Y"', we can reencode X' using standard differential demodulation. The received signal block for the given relay transmitted signal X = X7' is ydr Hd,rXr V(m') + Zd yd,r V(m') 'd (218 n Hn n1 n + n nln + n,(2 o) where Z't = Z Z _V m '). Since VX"') is a unitary matrix, Z' has twice the variance of Z%. Then, given X' = X7', we can apply the ML decision rule in Eq. (217). For the AFSTC protocol, each entry of the received signal from the source, Y,", is amplified and forwarded to the destination. Therefore, the amplified signal block at the re1,i can be represented as Xr = AY", (219) where A:= diag{Al,, A2, ..., A, } is the amplification matrix, and Ar is the amplification factor which we defined in Eq. (26). Then, using the differential modulation, the received signal block at the destination can be represented as Yd,r HX +Z = Yr V(m) + (220) where f, = E l /2AH drHs, = E /2AH d,rZ + Z and d, = ,V(') rZn Ei AH*Z* + Zn and Z Zn The ML decision rule is the same as Eq. (217) given X, = X". Notice that the ML decision rule of both relaying protocols has the same form. However, the value of the Frobenius norm is different depending on the protocols. CHAPTER 3 PERFORMANCE ANALYSIS To facilitate our resource optimization, we will derive the analytical expressions of the error performance for the cooperative systems described in the preceding section. Symbol error probability of cooperative networks with relay transmissions has been derived in [2, 7, 45] for coherent detection, and in [29, 71, 72] for a differential scheme with a single relay, both employing the AF protocol. The performance of traditional STC systems is well analyzed in [5961, 68]. The differential and distributed STC systems are considered in [31, 32, 58] and [66], respectively. All these existing work considered the performance of cooperative systems to some extent. However, the error performance of general cases with differential modulation has not been thoroughly investigated. We will consider the error rate for a general Lrelay setup in the distributed scenario. First, we derive an upper bound of the symbol error rate (SER) for the DF protocol. Then, under high signaltonoise ratio (SNR) approximation, an approximated SER for the AF protocol is derived. Finally, we derive upper bounds of the codeword error rate (CER) for the DFSTC and AFSTC protocols. 3.1 SER for DF Protocol Let us denote the average SER at the kth relay node as P',r. For differential Aary PSK (DMPSK) Jii, ii the s rk link SER pD can be obtained as [52, C!i pter 8.2] pDF V9PSK / M ([ gpSK cos 0]) " 1e,r2 dO, 9PSK = sin (31) 27r J2/7 1 V/ PSK cos 0 M where MA(x) = 1/(1 x), Vx > 0, and y represents the average SNR. In particular, for M = 2 (DBPSK), Eq. (31) can be simplified as PDF 32) Cr" 2(1 + (3 At the destination, the signals from the L re1 i. are combined to make a decision. Conditioned on that the symbol s, is correctly demodulated and remodulated at all relay nodes, the conditional SER DdF can be obtained by applying the results for Ldiversity branch reception of Mphase signals in [44, Appendix C] as: pDF ) L 2)L (L I t 71M )1 e,d r(L 1)! 9bL1 b b 2 M p sin(7/M) ot cos(7/M) })13) b p2 COS2(T/M) b p_2 COS2(/M) b1 where = 7d,, /(1 + Yd,rc). For DBPSK, Eq. (33) can be simplified as PeDdF iF (14) d 2 1l 4 (34) Using the unconditional SER PDF at the relays and the conditional SER PDF at the destination, we formulate an upper bound on the overall average error performance, namely the unconditional SER pDF at the destination, as follows: Proposition 1. With PDF and P9DdF given by Eqs. (31) and (33), "1'.. ,:;, ; an upper bound on pDF can be found as: L pDF < pDF (_ p ,)(1 Pe,d). (35) k= Proof. To prove that Eq. (35) provides an upper bound on the exact SER pDF let us start with the probability of correct detection pDF 1 _ pDF. Counting the events that lead to the correct detection, pDF can be obtained as PDF Pr{[( k S, Vrk) n ( Sn)] U [(s^ :/ S for some k) n (s sn)]} = Pr{< s = Sn, Vrk} Pr{j Sn, Vrk + Pr{< SnS^ / S for some k} Pr{sf / sT, for some k}. (36) where s^ and s^ are the symbol estimates formed at the relay rk and the destination d, respectively. The first summand in Eq. (36) turns out to be C (1 PF)(F dF), which leads to the upper bound in Eq. (35). D Several remarks are due here on the second summand in Eq. (36), which corresponds to the gap between the true SER and its upper bound APDF : pF PDF and determines the tightness of the error bound in Proposition 1. For DBPSK with a single relay (L = 1, M = 2), this gap can be easily obtained as: ApDF pDFpDF (3 7) For practical pDF and pDF values (e.g., < 103), APF is negligible compared with pDF = PfF + P 2PDF pDF However, for L > 2, all possible errors have to be considered for both the s r and r d links, which renders AP[F analytically untractable. But intuitively, as L increases, APDF also increases since there is an increasing chance that detection errors at the relay nodes do not lead to a detection error at the destination node. In addition to this effect, the performance bound PDF and the gap APDF also depends on the quality of the s r and r d links. These effects are evident from the simulated examples in Figs. 31 and 32, where a relay network with L = 2 relay nodes using DBPSK signaling is considered at various 7,,, and 7d,r levels. In these simulated examples, the channels between the source and all relays have identical powers a2, = Vk, which implies that = r,, Vk. Accordingly, we have PDF pDF, Vk, and pDF < PDF 1 (1 _ Pf)L( PDF) from Proposition 1. Likewise, the SNRs between all the relay nodes and the destination have the same value, 7d,rm = 7d,r,Vk. In both Figs. 31 and 32, the bound P, closely captures the dependency of the system SER on the SNR levels r7,, and 'd,r. Specifically, we have the following observations: Fig. 31 reveals that, at any given value of 7,,,, the system SER exhibits an error floor as 7d,r increases. Intuitively, this error floor comes from the detection error at the re v, which heavily relies on the s r link quality 7r,s and can only be reduced by imposing sufficiently high 7,*,. 0 O a rW 102 >, '*. w :: !! : L, 3 y,,,=10dB (bound) < 10 ,,=20dB (bound) ...... y ,,=30dB (bound) ,, =40dB (bound 104 0 Y=10dB (true) > 7,=20dB (ture) Y 7,,=30dB (true) 7,=40dB(true) 5 10 15 Y, (dB) Figure 31. SER at different r,,, values (DF, L 10 UJ S10 10 Figure 32. SER at different 7y,r values (DF, L = 2, M * On the contrary, Fig. 32 shows that, at mediumtohigh 7d,r levels, the overall SER can alv,v be reduced by increasing the SNR of the s r link 7,,, and does not exhibit any error floor. 20 25 30 2, M =2). 5 10 15 20 25 30 r, (dB) E' F E : : : : :: : : : : : 1000 " ":" . 10 . . . .. ..... . :.. ... .. L 10 10 10 20 20 (dB) 30 30 40 Yd,. (dB) Figure 33. SER bound versus yd,, and 7r,s (DF, L = 2, M = 2). To better illustrate these points, we plot the SER bound PDF as a function of both Yd,r and y7,s in Fig. 33. Notice that the surface flattens along the Yd,r axis, but keeps descending along the ,,s axis. These observations si r. I that the overall error performance of the DF based cooperative system depends more on the s r link than the r d link. Such unbalanced effects of the relay links confirm that appropriate resource allocation is critical in achieving the optimum error performance. 3.2 SER for AF Protocol In this section, we will derive the analytical expression of the error performance for the system under high SNR approximation. While the SER upper bound for an Lrelay system using differential (de)modulation is considered in [30], we provide a more general yet, simple expression for the average SER. The SER is developed in two cases, the system with no direct link and the one with a direct transmission. The two cases are denoted as "ND" and "DL," respectively. At the destination, we evaluate the equivalent SNR from the source through the kth relay node as Yeq d,er (38) .1 + Yd,rk + 1 This allows us to deduce the SER using a multichannel model. We will first investigate the SER of the relay systems with no direct link, the result can be easily extended to the case with a direct link. In a Lrelay system with no direct link between the source and the destination, the received SNR is: L 7ND = e7q,rk. (39) k= 1 Proposition 2. At high SNR, with the SNR given by Eq. (39), the average SER PA FD can be found as: k 1 1 PND D C(L) 7J + In(7a,)j (310) k=1 7d,rke where C(L) is a constant depending on the number of relays, which is given by C(L) 1 1 Lk L) (311) 22L1 L t k n=0 k0 Proof. The conditional SER for multiple independent channel DMPSK is given in [51]. For the binary case, the SER conditioned on 7ND is given by [44, Chap. 12] L1 pAF 1 7 ( 1 PAFD e 7ND n D, (312) n=0 where 7ND is defined in Eq. (39), and 1 ln 2L (3 k (3 13) k=0 2 1) In Eq. (312), 77VD can be expressed as follows by expanding 7ND: L2 / T n2i C 1 / '= L1 7N D E n1 lri M2 1 2.. r L1 m=0 m2=0 mLl =0 L1 where L = n E ~ mi, then Eq. (312) becomes: 2 9i Te r L (314) )7eq,rL 1 ,eq,rL^ ;\ ^^^^^ ^^^^ ^^^^^ ^^^^ L pAF q,rlkTk eANoD q 1 e1 ', er,rk, n,m k=0 (315) where Ll n nm / 1 T, n .mi . S22L1 C2 M2 n,m n=0 m =0 mn20 L2 nf mi, n mL1=0 The average SER can be obtained by averaging the conditional SER with respect to the probability density function (PDF) of %Yq,rl, namely p(%Yq,r~), which is given as AF eND ' j / N P( eqrk) eqri l eqlr2l Teqr L where the PDF of Yq,rk is derived in [72], and which is given as P(7eq,rk) exp e Ko(/ Y ),r + 2 eqr + '7dr exp ( K qk), I *7d,r V (3 17) (318) where 2 := 2 , 'a Ko(.) and K1(.) denote the zerothorder modified Bessel function of the second kind and the first order modified Bessel function of the second kind, respectively. By substituting (315) into (317), we can get: (319) L eND eq,,r k P7q,rk) ) q,k. n,m k=1 0 EL2 ) EL 1 ) (316) Let us first evaluate the integral in Eq. (319), denoted as In,m,k, 0/oo + T qC e q,r, 2P+ =k  e qr k rk,! 2 qKo) d7eq, 2 m oo 2 Ye ae (r/3o q ,r )d7eq,rq 0 e,'3e j 7yIjg e aJK1 (/ '/7eqr) d'7eq,r _, (320) where 3 2 /(1 + 1/ /7d,rk and a = 1 1/ .. Each integration term in Eq. (3 20) can be computed by using the integration property of Bessel function, [25, Eqs. 6.631.3] 0 2 1 1i 1 +p+ V _2+p W 02 ) x "e K,(Ox)dxa 2 P ) eo ) w '('2 )8 (3 21) JO 23 2 2 (4a where Wm,,Q() is the Whittaker function Wm, (Z) ez/2+1/2 (1/2 + n T, 1 + 2n, z), with U(., .) denoting confluent hypergeometric function of the second kind. By further using the properties U(a, 1, l/x) U(a, 2, /x) a (see [1, Eqs. 13.5.9 and 13.5.7]), and a 1 + 1/, a 1 at high SNR, we can simplify Eq. (3 20) as In,m,k n! 1 + in 7d,] (3 22) 7d,rl Plugging the above result back into PCoAD, we get PAF L F 1 eA/rD nk InY 7,d,r (323) n,m k 1 7d,rk Now, we move to the coefficient part of Eq. (3 23), which is given by Z. 1 a n L 2 L. (2 M k =C E2 L T2 ,1 l n k !.(3 2 4 ) n,mk 1 n=0 mi=0 m2=0 m 0 k 1 L1 By using the fact (325) ) j ) ( L2 L n 1 (T, i=1 i k! (M M2 TL1I kI Ll the coefficient can be simplified as L n,mn k i Ll n nm\ nL mY 22LL n0 m 0 m2 20 mL1 0 Li Ll Lln / n1m n mi 2L t i 22Ln= k=I =0 m k n O k0 m'1 m 0 m2=0 m =L 1 0 11J (326) Denote the coefficient in Eq. (326) as C(L). By using mathematical induction, we can prove that: We know that, for L n + L 1 Lt )' 1, (1) =1 = ("ni ). Suppose for an arbitrary L (327) 1 > 1, ((1) (n" 1i), then, we have (1+ 1) that: nm E"o (n 1) when L 1+ 1. We can also show (328) m+l1 11 ( + 1). Therefore, equality (327) holds for any L > 1. Using the result in Eq. (327), we can simplify Eq. (326) to Eq. (311). Finally, the average SER yields Eq. (310). n nm1 n' Z=1 2m, (L) := . 15 m1=0 m2=0 mL =0 (n +1 (n +1)! (n +1 1>)! (n +1 1>)! (n +1 1)! (n 1 + 1 )! (1 )! n!(1 1)! (n 1)!(1 1)! 0!(1 1)! In the relay system with a direct link between the source and the destination, the received SNR will be calculated by adding the direct link SNR: L 7DL + 7eq,rk +7d,s. (329) k= 1 Following the same steps as the nodirectlink case, the average SER can be similarly evaluated by calculating the integral for the direct link as in Eq. (320): Id,s d= ee dyds = m + 1)!. (3 30) 0J 7d,s 7d,s Combining the result above with (322), we can evaluate the average SER for a system with a direct link as: L1 1I PeL CLzC(L +1)1 ln(7yd< (331) Yd,skl1 r. where C(.) is the the same function as in (311), which depends on the number of relays. For example, for L = 1, 2, 3, and 4, we have C(1) = 1/2, C(2) = 3/4, C(3) = 5/4, C(4) 35/16. It is worth stressing that the SER expressions in Eqs. (310) and (331) coincide with the average SER of the coherent system in [45] except for the log term, which leads to the coding gain loss compared with the coherent system. When = 7d,rk = 7d,s = V, Vk, and as y  oo, Eqs. (310) and (331) give rise to P, ND C L and C (L+1) (332) where C' and C" are both constants. From Eq. (332), it is clear that the diversity gain can be obtained using a differential scheme with AF protocol for sufficiently large SNR. In Figs. 34 and 35, we compare the approximated and simulated SER when L = 1, 2, and 3 for the systems with and without a direct link. The figures confirm that the diversity benefit increases in direct proportion to the number of relays, and demonstrate that the approximations are very tight compared with the simulations, especially when the SNR is high and for small L. From Fig. 35, it is certain that a direct 10 2 : i ... . . . . . . .. . . I : ::: .. .o .. : : . . S. . . . . . .. . . . 1 0 3 ', Approximation L=l : : '. : . Approximation L=2 . 4 ...... Approximation L=3. ... . S D Simulation L=;  O Simulation L=2: : : : : : : : : : : : : : : o Simulation L=3 Sim ulation L=3 .. .. .... .* .. .. 105 i i 0 5 10 15 20 25 30 SNR (dB) Figure 34. SER comparison between approximation and simulation (AF ND, SNR(dB) = d,r). transmission contributes to the diversity gain compared with the system with no direct link as in Fig. 34. As L increases, the quality of approximation decreases, since more approximation errors are accumulated as the number of relays increases. In Fig. 36, we compare the SER of the systems with coherent modulation and differential modulation when a direct link is present. We use the result in [45] for the coherent system. The only difference between the coherent system and differential system is the log term in Eqs. (3 10) and (3 31). The figure shows that there are coding gain differences between the systems, and these differences increase as the SNR increases, i.e., the required SER decreases. These increasing differences are due to the log term in the differential system. For example, approximately 2.2dB, 2.8dB, and 3.2dB more SNR are required in differential system to achieve 103, 104, and 10s of SER, respectively, compared with the coherent system. 100 .: : Approximation L=1 S.:: : : ! Approximation L=2 S'.... Approximation L=3 10 ..... .. Simulation L=1 O : : Simulation L=2 : : : : : : : + Simulation L=3 . . .. . . .. . .. . C* 102 . . + . . . .. . . . S103 0 5 10 15 20 25 30 SNR (dB) Figure 35. SER comparison between approximation and simulation (AF DL, SNR(dB)= 7, .. d ). ~ ~ ~ ~ .N.. . .. . . . 0 5 10 15. 20 25 30 .... ..... ~ ~ ~ ~ ~ ~ ~ N (dB)........ .. ... Figur 3. SE comarso bewe aprxmto an siuato (A ..DL, ........ ..... SN IcB .'Y .....d ..s. 0 5 10 15 20 25 30 SNR (dB) Figure 36. SER comparison between coherent , .i, and differential system (AF DL, SNR= ., d,r). 3.3 CER for DSTC Protocol In this section, we will analyze the error performance of the cooperative system employing the DUSTC. Under high SNR assumption, the upper bound of CER will be derived depending on the relaying protocols. Let us first consider the DFSTC protocol. Due to independent demodulation and remodulation of the corresponding diagonal entry at each relay, the s rk link SER, pDF can be obtained using Eq. (31). Since one symbol error at each relay can induce the codeword error, the CER PDF at s rk links is given by L PDFSTc PDF (333) k=1 At the destination, the received signals from the Lrelays reconstruct the transmitted STC signal. Conditioned on that the source transmitted signal block V, is correctly decoded at the rel i, and by dropping the superscripts for notational brevity, the CER at rk d links is given by P[V, VQY,_I] = Q d2(V, V,)/4 ) < exp [d2(V,, V,)/8Vo] (3 34) where d2(V,, V )  [V V]Y _12 Str{Y,1(V,V')(VTV')Y ,}. (335) At high SNR, we can make the following assumption Y, n E /2H H X,, (336) where where Er := diag{S,, ,2,,...,, L} is the energy per symbol matrix at the rel,v, H, := diag{h'dr, h'2,..., hr'rL} is the channel matrix between the relays and destination, and XX is the nth transmitted signal block from the rel ~1i. Then, Eq. (335) can be approximated as d2(V,, V') trf{Hd,rADF(H',)}, (337) where ADF = E /2X,_1(V, V')(V, V')Xf E 1/2. Since Af 7 is Hermitian, we can express Eq. (337) as d2(V,, V,) tr {H rU DF U(Hdr) }, (338) where U is a unitary matrix and DeF is diag{ADe f, A,..., ADF}. Each diagonal entry DF, f k 1,2, ..., L, represents the eigenvalue of ADF. Therefore, we can obtain the CER by averaging Eq. (334) with respect to the channel Hd,r. For simplicity, by assuming that the fading coefficient has unit variance, the conditional CER PDdFSTCat the destination is given by pDFSTC P[V V < 1 DF (39) c^d n k ( n\ e,k and under high SNR condition, this equation can be further simplified as L /DF\1 PDFSTC P[V, V ] <1 8./V) (340) Finally, using Eqs. (333) and (340), we can formulate the unconditional CER for DFSTC protocol as : PDFSTC < _(_ PDFSTC) DFSTC). (341) )e,r ( e,d (3 It is worth mentioning that if there is no error between the source and re i, the above equation boils down to the CER of multiinput singleoutput (\ ISO) system employing the DUSTC. However, as L increases, the CER of s rk links becomes worse because of the increasing modulation size at each diagonal entry. This will induce the performance degradation of the DFbased system. To provide better performance and pertain the diversity gain, s rk links have to maintain lower CER. For the AFSTC protocol, similar to the DFSTC protocol at rk d links, the CER can be found by calculating the pairwise CER between the source and the destination. The id covariance matrix of the .,. regate noise Z, in Eq. (2 20) is diag{o ,, j 2,h,  L, } where the kth diagonal entry of the covariance matrix is given by Eq. (214). To normalize the .,. regate noise variance, let us define the matrix G diag{gl,, g, ..., gL with gk A (S.rA 2 cr, + 1)1/2. Then, by multiplying G with the received signal block at the destination, we can rewrite Eq. (220) as yd,r dr d YrG = Y'V(')G + ZTG, (342) or equivalently, d,r d,r y (') ( Yn YnI n + Z~, (343) where Y =YG, V = VG, and Z = ZdG. Then, the CER for the AFSTC protocol can be achieved using Eq. (343). Following the same steps as Eq. (334) to (337), the CER can be obtained as: P[ V, i I Y, ] < exp d2(3VT V')/8A]0 (3 44) where d n YV,1(V, VT)GG (VT,VT )Y, 1}. (345) At high SNR, the code distance can be approximated as d2(V,, V') tr (Hd,r Hr,)AAF (H r Hr)}, (346) where AAF EE /2X,_1(V, V')(AG)(AG)H(V, V')XH YzE /2. Similar to the DFSTC protocol, we can express AAF as eAF U'eDAU'FU (3 47) where U' is a unitary matrix, and DAF is the L x L diagonal matrix in which each diagonal entry is AA, k = 1, 2, ..., L. Then, the CER for the DFSTC protocol can be achieved by averaging Eq. (344) with respect to the combined channel H H rH,". Let us define h : hd'h'"'8, then the PDF of a = Ih is given by [43] Pa Ko 4a (2 a2 P(a) Ko (2 2 2, (348) d,rk rks d,rk rk s where Ko(.) is the zeroth order modified Bessel function of the second kind. By assuming that each fading coefficient has unit variance, and using the properties of Bessel function and confluent hypergeometric function (see [25, Eq. 6.631.3] and [1, Eqs. 13.5.9]) at high SNR, the CER can be simplified as L AF AF1 PFSTc = P[Vrn V'] < ] ek (49) k= 1 \I / Notice that the CER of the AFSTC protocol has almost the same form as its counterpart of the DFSTC protocol at rk d links except for the log term which reflects the effect of the amplification and .r. .egate noise and this leads to coding gain loss. Eq. (349) confirms that AFSTC protocol provides full diversity gain. In Figs. 37 and 38, we plot the bounds and simulated CERs for the systems with DFSTC and AFSTC, respectively, when L = 1, 2, and 3. When L = 1, the STCbased cooperative system is reduced to the conventional cooperative network, thus we can use the SER formulas derived in [20, 21] as the CER bound of the STCbased system. Fig. 37 shows that the bounds are tight to the simulations, especially when L is small. Notice that the cardinality of the signal block at the rel i equals to ML because of the independent decoding at each relay. However, the bound at rk d links only considers M signals. Thus, as L increases, the gap between the bound and simulation increases. Fig. 37 also shows that no diversity gain is obtained by DFSTC protocol, since the CER at s rk links increases in direct proportion to the number of rel v, which induces the degradation of the overall error performance of DFbased system. These results confirm our analysis in 10 10 m lto : : : :.:s::m L=2, simulation i 310 0 5 10 15 SNR( Figure 37. The CER for the DFSTC protocol (L 1 0 1 . . . . . 10 L=1, bound 2 L=2, bound 4 10 '.. L=3, bound L=1, simulation L=2, simulation '0 *O L=3, simulation 10 3 0 5 10 SNR( Figure 37. The CER for the DFSTC protocol (L w\.......... 102 L=1, bound L=2, bound 104 ....... L=3, bound : :  L=1, simulation : ::: ; 6 L=2, simulation I ;; 0 L=3, simulation 1051 15 20 dB) 1, 2, and 3, SNR= SNR(dB) 25 30 Figure 38. The CER for the AFSTC protocol (L 1, 2, and 3, SNR= 10' "s,, 102 1 0 :. : : : : : ..: : : : : : :: : : : : : : 103 L=1 . L=2 : : : : :: : : : . L=3 104 ,s=dr=SNR * +: Yrs=SNRydr=SNR+20dB : :: : Sdr=SNR, yrs=SNR+20dB ' 105 0 5 10 15 20 25 30 SNR(dB) Figure 39. The effect of unbalanced link SNR for the DFSTC protocol (L 1, 2, and 3). the preceding section. In Fig. 38, though the bounds for AFSTC protocol are inaccurate when SNR is low because of the log term in the analytical expression, the bounds and simulations have tight values at high SNR. Furthermore, it is clear that AFSTC protocol provides full diversity gain. As we mentioned above, the link quality between the source and relays is critical to the performance of the DFbased cooperative system. To capture the effect of unbalanced link quality, we consider different average SNRs at s rk and rk d links for both DFSTC and AFSTC protocols in Figs. 39 and 310, respectively. We assume that = 7, and d,rr = 7d,r, Vk, and consider i) equal SNR for both s rk and rk d links, ii) higher SNR is assigned at s rk, and iii) higher SNR is assigned at rk d links with L = 1, 2 and 3. As shown in Fig. 39, when we assign high SNRs at s rk links, the overall CER decreases and the diversity gain begins to appear. For the extreme case, i.e, infinite SNR is assigned at s rk links, the DFSTC cooperative system behaves like a MISO system. The figure also shows that the coding gain is achieved by assigning higher SNR at rk d links. However, the diversity gain is dominant compared with the coding gain especially 10 1   10 2' :: :: l ._* ~ .. ..., : :: : .: : :: : :: : ::.. ... : . % .. . __10 .. i ii .. L=1 104 L=2 10 . ..... L=3 : :: :: ::: : Yrs dr=SNR 10o5 Trs=SNR, yd,r=SNR+20dB .. ; ' yd, r=SNR, yr,s=SNR+20dB : : 106 0 5 10 15 20 25 30 SNR(dB) Figure 310. The effect of unbalanced link SNR for the AFSTC protocol (L 1, 2, and 3). at medium to high SNR, which confirms that the s rk links are more critical to provide better performance. Fig. 310 shows that the coding gain is obtained by assigning higher SNR at both links. Increasing SNR at s rk and rk d links leads to decreasing the effect of .,.:regate noise and increasing SNR at the destination, respectively. Both scenarios induce the enhancement of coding gain. Notice that the effect of SNR at rk d links provides more coding gain than rk d links, which implies that increasing average power at the relay output is more crucial than reducing the effect of noise at the s rk links. CHAPTER 4 OPTIMUM RESOURCE ALLOCATION In this chapter, we will investigate the effects of resource allocation on the error performance. We will show that an optimum allocation of the limited resources is possible, and it achieves the optimum system error performance. The resource allocation which minimizes the average error rate will be addressed from three perspectives: 1) Given the relative distances among the source, relay and destination nodes, the path loss exponent of the wireless channel, and the total available energy per symbol, determine the optimum energy allocation among the source and relay nodes. 2) Given the sourcedestination distance, the path loss exponent of the wireless channel, and the energy per symbol at the source and relay nodes, determine the optimum location of the relay nodes. 3) Given the sourcedestination distance and the total available energy per symbol, determine the joint energy and location optimization. For analytical tractability, we consider an idealized Lrelay system with all relay nodes located at the same distance from the source and destination nodes; that is, Ds,r = Ds,r and Drm,d Dr,d, Vk. It is then reasonable to assign equal energies at all relay nodes 8,r = S, Vk. To carry out the optimization in the ensuing subsections, we will also make use of the relationship between the average power of channel fading coefficient ohij and the internode distance Dj, as follows: a = C D i,j e{s, r, d} (41) where v is the path loss exponent of the wireless channel and C is a constant which we henceforth set to 1 without loss of generality. For the conventional DF and AF protocols, we will present the analytical results of optimizations as well as simulated examples. For the DSTC protocol, due to the analytical intractability, the optimization results will be shown by simulations and compared with conventional systems. Relay(s) /D Source s'd Destination! (a) Ds,r ( Dr,d D D s,d (b) Figure 41. Network topologies: (a) Ellipse case; (b) Line case. We will consider two network topologies as depicted in Fig. 41. One is the ellipse case and the other one is the line case. For the ellipse case, Ds,r + Dr,d = D > Ds,d. The line case can be regarded as a special case of the ellipse case, i.e., D = D,d. By changing the value of D, we can solve the optimization problem at any point on a 2D plane. Therefore, optimum resource allocation for these idealized topologies can provide useful insights for understanding the effect of resource allocation in relay networks. 4.1 Convexity of SER Let us first consider the convergence of the error performance. The convergence of our optimization is guaranteed by showing the convexity of the error rate. With DF and DSTC protocols, it is cumbersome to prove the convexity analytically, due to their complex expressions of the error performance (see Eqs. (35), (341), and (349)). However, our simulations will show that the error rate is generally convex, which ensures convergence of the optimization. With the AF protocol, we can prove the convexity analytically and confirm by simulations. The proof of convexity for the AF protocol is given as below, which guarantees the convergence of the error performance as a function of energy and location. Lemma 1. Under high SNR, the average SERs in Eqs. (310) and (331) are convex functions of the energy and location, i'..1/,. ; Proof. Given location Dr,d, i.e. ar,, and energy constraint ps + Lpr p, the SER can be written as: AF 1 1 'eDN C(L) L 2 l(p,) (4 (p Lpr)2rs Prca,r L peL = C(L + 1) In (4 p Lpr (p Lpr)r,7 p dr, which are functions of the single variable pr e (0, p/L). The second derivatives of pFD and pAF are given as follows: For L > 2, 2 pAF e,ND C'(L) {(L 1)[f(pr)]L2 [g(r)]2 + [f(r )]L h(r) (4 apr 2 pAF e,DL ap SC"(L) [f ()]L2[g(p,)]2 {A()} (p Lp)3 1 + C"(L) [f(p,)]Ll[h(pr)], P Lpr for L = 1, a2 AF e,ND ap 2 pAF e,DL C"1 1 ,DL C1) (t )3 {2 f(p,) + 2pg(p,) + p h(p)} , where C'L) LCL) and C ) LCL + 1), and where C'(L) = LC(L) and C"(L) = LC(L + t), and f(Pr) 2) 3) 4) (45) (46) (47) (4 8) 1 1 ( Lpr) 'r pr (p Lp,)u, Pr2+ ,r C'(t) h (p,), g(pr) = Jr,s (P 1 1 Lp)2 2+ 2 (pr), Lpr) P "ar 2L2 1 1 h(pr) = 2 )3 + 321n(prr) 3], ,s (P Lpr) Pra A(p) 2L P [ "p 2 + 2L ()l + (L  p Lp [(P)] p Lpr [g(Pr) 2 Under high SNR approximation, pr dr > 1, we have f(pr) > 0 and notice that A(pr) is a quadratic function of 1 [ (P, and its quadratic A (2L)2 4(2L)(L 1) 4L(2 L) < 0, for L > 2. Thus, we have A(pr) > 0 when L > 2. Therefore, 02 pAF 2 pAF 0 "eND eDL ' > 0 and > 0, for L > 2. Op2 Op2 (411) h(pr) > 0. We discriminant is: (412) (413) When L = 1, the convexity of the system with no direct link is readily obtained from Eq. (46). For the system with a direct link, after some manipulation, Eq. (47) can be reexpressed as: = c (() pP {2[P (P + 1 ln(pr,1 ) 3 + C"//(1) P 2 p) + 6PS (414) (p Pr)3 pr pr s Using the inequality, (pr/ps)2 (r/ps) + 1 > 3/4, we have the lower bound of 2pDL/AF L 2. ad2 pAF1 13 2 2pe > C/(1) 3p ( )ln (Pr) 2} (415) Op, (p pr)3 7r 2 pr, On the condition of high SNR, In(prf) > 2, we have '2AF P /p > 0 for L = 1. Finally, both PAeND and PAFDL are convex functions of pr, and accordingly ps for any L> 1. (49) (410) Now, consider the average SER as a function of location. Plug ,, = Dr' and 2r = D into the formula of PAF, and use the constraint Ds,r + Dr,d = D. It follows ad,r r,d iNDu that: 'eND ( CL) [(D Dr) Dd In(prDr]) (416) (L ( Lpr) pr which is a function of a single variable Dr,d e (0, D). The second derivative of PAJD is 2 pAF 2,ND C'(L) {(L )[u(Dr,)]L2[v(Dr,)]2 + [u(Dr,d)]I [w(Dr,)]} (417) aD2r,d where (D D )V DV u(Dr,d) ( +D r, ln(prD ), (418) (p Lpr) Pr D"v v(Dr,d) ( (D Dr,d) + rd [ln(prDr) 1] (419) Ps Pr w(Dr,d) (= 1)(D D,) v2 + Dd {( 1) [ln(prD,) 2] 1}. (420) p Lpr Pr Under high SNR approximation, i.e., prd > 1, and v > 1, we have u(Dr,d) > 0, V(Dr,d) > 0 and w(Dr,d) > 0. Thus: aD2 ODr,d Similarly, for the system with a direct link, we have PL L = C(L 1 + )ln(prD ) (4 22) p Lpr (p Lpr) Pr pD (4 This equation is the same as Eq. (416) except for the constant term, C(L + l)/(p Lpr), therefore its convexity can be readily proved using the same steps in the above. D 4.2 Energy Optimization Now let us consider the energy optimization given the relative distances among the nodes. Problem Statement 1. For i,. ; given source, ,. Ir;, and destination node locations (Ds,r and Dr, or ('I;'. nlj.i and a,), and the total energy per symbol S, determine the optimum energy allocation S, and S, which minimize the average SER in Eq. (35) for DF and Eq. (310) or Eq. (331) for AF while .,7/i.f,, L s, + r, = S, + LS, S. (423) k=l By defining the total SNR, p := S/Ao, the transmit SNR at the source node ps := s/AMo and the transmit SNR at the relay nodes pr := S/A/o, the energy constraint can be reexpressed as the SNR constraint: p = ps + Lpr. (424) Using Eq. (41), the average received SNRs at the relay and destination nodes can be expressed in terms of the transmit SNRs as: 2 p D ir,s = Ps =h, PsD and 7d,, pr hl,, = prD (425) As a result, the total energy constraint, Eq. (424), can be further rewritten as P = + L7d/ ,, 7r,sD + L7d,rD"d (426) Let us consider the DF protocol first. To gain some insights, we start from a singlerelay setup and establish the following result: Proposition 3. With DF protocol, for a single,, AI.r; setup with L = 1, at given s r and r d distances Ds,r and Dr,d, and under the total energy constraint in Eq. (423), the optimum energy allocation s should if1' fy: rfo V~r1 Vi 5) n 2,(2D2vp2 in 1 s 2 o 2~~ 2 + D'ysDr (6Dr p + 5) + 2Ds,r (2Dr P2 + 3Drp + PS roV 4D5D(DD D"4)2 s,r r,d 8,r r,d 2Dr"p + 3 r' (427) 2(Dv Dr ') 8\s,r r,di and ,,, i, ,'.,,.:,i,, S = 7 so. Proof. Treating the SER bound pDF as a function of 7r,s and 7d,r, we have the first order conditions for the optimum solution 8PDF apF AD_ = 0, (428) QPDF 7d ADd = 0, (429) p (r,sDr + 7d,rDf,) 0, (430) where A is the Lagrange multiplier. Solving Eqs. (428) and (429), we obtain (1 + 27d,r)(1 + 7r,,) Drv rd (431) (1 + 27r,)(1 + 7,) D, ' which leads to the following relationship between 7d,r and 7,, 3DS,/2 + D S + 8D' (1 + 37r, + 7,) 7d,r 4Dv 2 (432) Substituting Eq. (432) into Eq. (430), we find po as in Eq. (427). E Although Eq. (427) is accurate for all S and A/o values and for all s r and r d distances, its complex form does not provide much intuition. Fortunately, for several special cases, it can be simplified without much loss of accuracy. Next we will consider some of such cases. Special Case 1 [Centered Relays]: When D,r = Dr,d, Eq. (431) simplifies to 27d2, + 37d,r + 1 S1. 2~2, + 3, + 1 By symmetry, we obtain 7r,, = 7d,r. Therefore, the optimum energy allocation amounts to assigning equal energies to both the source and relay nodes; that is, S = f = 8/2. Special Case 2 [High SNR]: In this case, expanding Eq. (431) as 27d,t + 37d,r + 1 Dr 272, + 3r, + 1 D, and neglecting the constant terms in both the numerator and the denominator, we obtain an approximate solution for the optimal energy allocation S(2D p + 3) + (2Dp + 3)(2Drp + 3) o o 2(DV, Dr) (433) This solution can be further simplified by neglecting its constant terms to Dv/2 Dv/2 0 r,d r ,d (434) SDv2 v+D 2 1 D8f2 (434) D,,/2 + Drd Ds,/ + Dr,d Interestingly, this solution coincides with the optimum power allocation obtained by minimizing the outage probability [27, (8)] with a singlerelay transmission. From Eq. (434), it readily follows that the energy allocation ratio between the source and the relay nodes is co D /2 Eq. (435) reveals explicitly that the optimum energy allocation heavily hinges upon the internode distances. In addition, the path loss exponent of the wireless channel, v also affects the optimal energy allocation. Interestingly, the S/7o ratio is linear in Ds,r/Dr,d only when v = 2. The optimum energy allocation favors the link with a larger node separation if v > 2 and vice versa, as we will show next with an example. Fig. 42 depicts the transmit SNR ps obtained from the optimum energy allocation. A onedimensional setup is considered; that is, Ds,r + Dr,d = D,d = D. The system parameters are: p = 10dB, L = 1, Ds,d = D = 1 and v = (1, 2, 3, 4). In Fig. 42, the E 7 .9 6v=4 6 v=3 S. Simulated= 0 0.2 0.4 0.6 0.8 1 E 4 E Figure 42. Exact and approximate optimum energy allocations with different path loss exponents (DF, L= p =ldB). simulated optimum is plotted together with the exact (46)analytical value in Eq. (4 and the approximations in Eqs. (433) and (434). These results are nearly identical in all cases with various v values. By closely inspecting the figure, we find that the Spproximation in Eq. (433) provides more accurate curves than (412)the one in Eq. (434), as 0 0 0.2 0.4 0.6 0.8 1 D s,r Figure 42. Exact and approximate optimum energy allocations with different path loss exponents (DF, L 1, p 10dB). simulated optimum p, is plotted together with the exact analytical value in Eq. (427) and the approximations in Eqs. (433) and (434). These results are nearly identical in all cases with various v values. By closely inspecting the figure, we find that the approximation in Eq. (433) provides more accurate curves than the one in Eq. (434), as expected. Although the approximate expressions in Eqs. (433) and (434) are obtained under high SNR assumption, they remain very accurate even at medium SNR of 10dB. From Fig. 42, we also observe that, for all v values, the source node energy E, increases as the relay moves towards the destination node. With v = 2, S, increases linearly with Ds,r. At higher values of the path loss exponent, v > 2, we observe that p < Ds,r/Ds,, when Ds,r < Ds,a/2 , (436) P > Ds,r/Ds,d, when Ds,r > D,d/2 . In other words, the optimum energy allocation favors the link with irj internode distance. When the path loss exponent v = 1, Fig. 42 shows the opposite of Eq. (436). So far, we have been focusing on the singlerelay case, where an analytical solution (427) can be obtained and a very accurate and insightful approximation is available under high SNR assumption. For L > 2, however, the first order conditions obtained by differentiating the SER bound PeDF have complicated forms, which render analytical solutions impossible. Fortunately, the SER bound PeDF as in Proposition 1 still allows for a numerical search, as opposed to Monte Carlo simulations needed otherwise. For example, with DBPSK and L = 2 we have pDF 1 3 ( d,r 1 d,r 37) e,d 2 4 1 + ,r 4 &1+. (,r/ and, accordingly, the SER bound is given by pDF 1 (2 +7,yd)(1+ 27d,)2 p = 1 (438) 4(1 + r,s)2( + 7d, )3 By using the first order conditions in Eq. (428) and the high SNR approximation, the optimum 7r,s and 7d,r should satisfy 4(1 + 7y,r)(2 + 7d,))(1 + 27y,r) Dj r (4 39) 37ry,(1 + 7q,<) (D4, Although an analytical solution is not readily available, one can resort to the numerical search. Let us consider some examples of optimum energy allocation. Fig. 43 represents the average SER for various energy allocations at the fixed relay location. Total transmit SNR p = 10dB and L = 2 are considered with D,r = 0.25, 0.5, and 0.75. For each case, the SER has one minimum point, and the corresponding energy allocation is the optimum value, i.e., pO/p. The figure shows that the p /p increases as the rei move towards the destination. This coincides with our analytical results and simulations in Fig. 42. The optimum energy allocation obtained from the numerical search is plotted in Fig. 44 and compared with the simulated results. The total SNR value of p = 10dB and a path loss exponent of v = 4 are considered with various L values. The results 10  103 0 0.2 0.4 0.6 0.8 Figure 43. SER versus energy allocation at the given relay location Ds,r (DF, L = 2, p = 10dB, D = 1, v =4). 0.2 0.4 0.6 0.8 Figure 44. Comparison of optimal energy allocation between the numerical search and simulated results at various L values (DF, p = 10dB, v = 4). 0.7 0 0.6 / v=4 2v=2 0.4 E E Z 0.3 p=OdB O p=5dB 0.2  p=lOdB 0.1 p=15dB p=20dB 0 1 0 0.2 0.4 0.6 0.8 1 D s,r Figure 45. Comparison of normalized optimum energy allocation at different p values (DF, L 1). show that the analytical values and simulated ones are nearly identical for L = 1. As the number of rel i L increases beyond 2, a gap between the numerical search and the simulated results can be observed from Fig. 44. This discrepancy arises from the fact that the numerical search is based on the SER bound, whereas the simulations generate the true SER, and that the SER bound is looser for larger L values, as we mentioned in the preceding section. Nevertheless, the numerical results still closely indicate the trend and relative distances corresponding to various L values. Notice that these curves exhibit a converging tendency as L increases. This implies that the optimum energy allocation curve may achieve an .,i'i:!l, 1 ic limit as the number of re!,v L grows. We have seen that the exact expression of the optimum energy allocation in Eq. (427) will give rise to a ps/p ratio that depends on the actual value of the total SNR p with DF protocol. However, the highSNR approximation in Eq. (434) results in a ps/p ratio which is independent of p (see Eq. (435)). As a result, the approximate solutions (433) and (434) are expected to differ from the exact solution (427), depending on different values of p. Fig. 42 shows that these solutions agree very well at p = 10dB. In Fig. 45, the optimum energy ratio ps/p obtained from the exact solution (427) is depicted at various SNR values p = (0, 5, 10, 15, 20)dB, and with two values of v (2 and 4). We observe that when v = 4, the curves corresponding to different p values are almost identical; whereas when v = 2, all curves coincide except for the p = OdB one. These observations confirm that the approximations (433) and (434) are both very accurate even for p as low as 5dB when v = 2 and for all SNR levels when v = 4. In other words, the optimum energy allocation ratio ps/p is almost independent of the actual energy level except for very low p values. It only depends on the location of the relays as in Eq. (435). Likewise, a similar result can be deduced for the optimum distance allocation; that is, the optimum distance allocation ratio Ds,r/D,,d is nearly independent of the actual sourcedestination distance Ds,d, as we derived in Eq. (460). Now let us consider the AF protocol. Similar to the DF protocol, by treating the approximated SER PD or P' DLS a function of ps and pr,, we can find an optimum solution. Proposition 4. With AF protocol, at given s r and r d distances Ds,r and Dr,d, and under the total energy constraint in Eq. (424), the optimum energy allocation pg and pO should ,/;f, a ,[ln(p ) l]p + porp ppo ,r 0 (440) and (L + 1) r, ln(p odr) L P + P 0'r L n(po ,r) P PP7 ,r } 0(4 ) for the system with no direct link and with a direct link, ,, "i' /, *;. /; Proof. We have the following first order conditions for the system with no direct link: 8pAF S,ND O, (442) OPs 8PAF ND A =0, (443) aprk p (p, + Lpr) = 0, (444) where A is the Lagrange multiplier. Eqs. (442) and (443) give us LpAF 2 ,ND Pr d,r o A 0, (445) Ps Pr ,r PsP ,s dpr ,r) LPAFD Ps2,s PJr) 1] ,ND [ P ln(p LA 0. (446) Pr Prr + Psrsaln(prr) Then, by substituting ps and pr in Eqs. (445) and (446) for the total energy constraint in Eq. (444), we have LpAF 2 p2lnp2) ePND Prad,r + r,s P ,r 2p. (447) A Pr ,r Ps lnrs (pr,7,r) 1] With Eqs. (445) and (447), we arrive at Eq. (440). Similarly, the first order conditions of the system with a direct link are given by ,DL 1 + L 2 Pd,r A 0, (448) Ps p Prd Pr,sPrdr) LPfL Pss [ln(pdr) 1] DL In n(pr, 1) LA 0. (449) Pr Prd ,r Ps ,sn prdr) By using the same steps as shown, we have Eq. (441). D In Eq. (441), the p71,,n(pi ,rd)/(L + 1) term mainly affects the energy allocation compared with the system with no direct link. This effect is obvious especially when the reh.v are located close to the the source. Notice that the log term in Eqs. (440) and (441) renders a closedform solution incalculable. Although an analytical solution is not " 150 , S,o' 100  00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Pr/P Figure 46. Existence of the optimum solution (AF ND, p = 15dB, v = 3, L=2). readily available, one could resort to the numerical search using Eqs. (440) and (441). The following Lemma 2 shows that optimum values have only one solution, which allows us the numerical search. Lemma 2. The optimum solutions in Eqs. (440) and (441) have one solution of p, and accordingly pO. Proof. Since the Eqs. (440) and (441) cannot be solved algebraically, we represent a solution graphically. Using Eq. (424), Eq. (440) is given by 2 o 2 + _2 l o ,o o o. ,s[.l(p,.o) ]?p2+ po,rp (p + Lp)pco,r =0. (450) This can be represented as o,[ln(por) 1]p Lp o ,r. (451) By squaring root both sides, we have v2 [n(p,r) 1]p pdr, (452) i ...... .. 1  0.2 0.4 0.6 0.8 Ps/P Figure 47. SER versus energy allocation p = 15dB, D= 1, v 4). 10 \ '. iii\ ng . .. .. ... ... g, 10 : :: g : : : :'.. : ::: ^ 102 105 10 104 io .. . at the given relay location Ds,r (AF ND, L = 2, 0.2 0.4 0.6 0.8 Figure 48. SER versus energy allocation at the given relay location Ds,r (AF DL, L = 2, p = 15dB, D= 1, v 4). 102 103 0 since pi > 0 and oaj > 0,Vi,j. Let us denote yl and y2 as the lefthand side and the righthand side of Eq. (452), respectively. Given that domain of ln(poa2,r) > 1 and using Eq. (41), we plot yl and Y2 in Fig. 46 for various relay locations Ds,r with D,d = 1. The line with no marker and with circle marker represent yl and y2, respectively. The figure shows only one crossing point, and that point provides us the optimum solution. Similar trends are observed in the system with a direct link. E Figs. 47 and 48 depict the average SER for various energy allocations at fixed relay location for the system with no direct link and with a direct link. We locate rel i at 0.25, 0.5, and 0.75 with p = 15dB and L = 2. Both figures show that the SER has one minimum point, and the corresponding energy allocation is the optimum value, which confirms Lemma 2. For the system with no direct link the optimum energy allocation increases as the relays move toward the destination. However, for the system with a direct link, optimum energy allocation l i, at the middle value, i.e., uniform energy allocation when the rel i, are located close to the source (see Fig. 48 with D,,r = 0.25 and 0.5). This result will be verified by the following numerical search results. For all cases, the SERs in Fig. 48 show better performance in Fig. 47 due to the direct link. The optimum energy allocation obtained from the numerical search for both the system with and without a direct link is plotted in Fig. 49. We consider the total SNR value of p = 30dB and a path loss exponent of v = 4 with various L values and D = Ds,d = 1. In the system with no direct transmission, for all L values, the optimum energy allocation at the source increases as the relay moves towards the destination. However, for the system with a direct link, a uniform energy allocation is optimum when the rel i, are located close to the source. Intuitively, this is because the direct transmission is present that the diversity gain is dominant over the coding gain. When the rel i are located close to the destination, much of the energy is assigned at the source to assure that a transmitted signal can reach the r el;: it is the same as in the system with no direct link. 0.2 0.4 0.6 0.8 Figure 49. Optimum energy allocation (AF, ND and DL, D 0.7 P 0.4 P 0 0.2 0.4 0.6 D s,r 1, p 30dB, v 0.8 1 1.2 Figure 410. Optimum energy allocation (DFSTC and v 4). AFSTC, D = 1.2, Ds,d 1, p 15dB, L=2 L=3 Fig. 410 represents the optimum energy allocation for both DFSTC and AFSTC protocols when L = 2 and 3. We use the total SNR value of p = 15dB and a path loss exponent of v = 4. with D = 1.2 and D,d = 1. The figure shows that the optimum energy allocation at the source increases as the rel i, move toward the destination for both cases, which is the same as the conventional cooperative networks with no direct link. The figure also shows that, in general, DFSTC protocol requires more energy than AFSTC protocol to decrease the error rate at the rel i,v. 4.3 Relay Location Optimization Consider now the optimum relay location with the sum distance between the sourcereli.,] and relaysdestination being D, and the sourcedestination distance Ds,d. If the transmit energies at the source and reli, are preset, where is the optimum location to place the rel iv? To answer this question, we treat the distance D as a fixed resource and formulate an optimization problem as follows: Problem Statement 2. For iwq ; given transmit energies at the source and luI.; nodes (S and or (,;it'',. ,i': ;1 ps and pr), and the path loss exponent v of the wireless channel, determine the optimum location of the relays, DQ,r, which minimizes the average SER in Eq. (35) for DF and Eq. (310) or Eq. (331) for AF while ,il flying: Ds,r + Dr,d = D, where 0 < D,r < D. (453) Starting with the singlerelay (L = 1) setup and applying the highSNR approximation with DF protocol, we establish the following result: Proposition 5. For DF protocol, with a single,' AI.r; setup where L = 1 and the source relay(s)destination distance D, and $s and S, denote the prescribed transmit energy levels at the source and ', 1.r; nodes, ', /'.. /i 1.:, the optimum location of the ', Ir;, is 1/(V1) Do, P D (454) PS + pr and accordingly, D,d = D Ds,r. Proof. Treating the SER bound pDF as a function of Ds,r and Dr,d, we have the first order conditions for the optimum solution 8pDF ^ aP 0 X A =0, (455) 07r,s ODs,r aPF ,r A 0, (456) 07d,r ODr,d D (D,,r + Dr,d) = 0 (457) where we used the fact that 7,,s (or 7d,r) is independent of Ds,r (or Dr,d). Using the definition of the average receiver SNRs in Eq. (425), we can reexpress Eqs. (455) and (456) as aPDF  07,s D,+1 aPEF V P_ r . A= 0, (458) 0d,r r+,d which leads to pr Dy,+1 PDF/d7r,s (1 + 27y,r)(1 + 7,) (459) Ps D+I PpDF/ d,r (1 + 27r,s)(1 + 7r,s) At high SNR, the constant terms on the righthand side of Eq. (459) can be ndglected. Consequently, we have [c.f. (425)] pr D 81 r p2 D 2v ,D+1 2 2 2V Pr_ s,r d,r Pr Ds,r s r,d r,s s r,d As a result, the optimum distances Ds,r and Dr,d should satisfy D_ (p)1( 1) (460) Dr,d Pr Sr which, together with the condition in Eq. (457), concludes the proof of Proposition 5. E Interestingly, Eq. (460) bears a very similar form as its counterpart for the optimum energy allocation in Eq. (435). In fact, when the path loss exponent v = 2, Eq. (460) is essentially identical to Eq. (435). For general v values, however, these two relationships 0.6 o 0.5 Sv=2 v3 v=4 E 0.4  0.3 0.2 Numerical search S Approxitmation .1....... Simulated 01 0 0.2 0.4 0.6 0.8 1 ps/P Figure 411. Optimum location of relays (DF, p = 10dB and L 1). are quite different. Such a discrepancy is actually very reasonable, because Eqs. (435) and (460) result from two distinct optimization problems: the former is obtained for arbitrary distances Ds,, and Dr,d under a total energy constraint; whereas the latter is obtained for prescribed S, and S, under a total distance constraint. With the SER bound pDF being a twodimensional function, the energy and location optimizations are carried out on uncorrelated dimensions. For general L values, the optimum location can be determined in a similar manner as we discussed in the above. Essentially, the path loss exponent v renders it impossible to derive an analytical solution for the optimum location problem, even with the high SNR approximation. One can resort to the numerical search using the SER bound in Proposition 1. In Fig. 411, the optimum distances obtained from the numerical search and the simulations are compared for different v values, at total SNR p = 10dB and with L = 1 relay node. Notice that, as its counterpart in Fig. 42, the optimum relay location is linear in Es/S only when v = 2. 10 : : : : '. ... .. . . .. 0) . . .. .. .... .... . ... . 0 0.2 0.4 0.6 0.8 1 L = 2, p = 10dB, D = 1 v = 4). Fig. 412 depicts the average SER versus relay locations at the given energy allocation when p = 10dB and L = 2. Notice that at the given prescribed energy, there exists only one minimum SER point, and the corresponding Ds,r is the optimum relay location. This figure is the counterpart of the optimum energy allocation in Fig. 43. We can see that the range of optimum relay location is smaller than the optimum energy allocation, which results in the flatness of location optimization. We will verify this phenomenon with a numerical example. Let us now consider the optimum relay location for the AF protocol. Lemma 3. For rn; given energy at the source, the optimum ', .r';X location which mini mizes the average SER is independent of a direct link between the source and the destina tion. Proof. This can be proven by the average SER expression in Eq. (331). The 1/Ys = D s/Ps has a fixed value given the s d distance Dsd and prescribed energy at the source. Therefore, a direct link does not affect location optimization. Hence, the location 10' 0 0.2 0.4 0.6 0.8 1 D s,r Figure 412. SER versus relay location distribution at the given energy allocation Ps/P (DF, L 2, p 0OdB, D v 4). Fig. 412 depicts the average SER versus relay locations at the given energy allocation when p 10dB and L 2. Notice that at the given prescribed energy, there exists only one minimum SER point, and the corresponding D,,, is the optimum relay location. This figure is the counterpart of the optimum energy allocation in Fig. 43. We can see that the range of optimum relay location is smaller than the optimum energy allocation, which results in the flatness of location optimization. We will verify this phenomenon with a numerical example. Let us now consider the optimum relay location for the AF protocol. Lemma 3. For ,,.,. given energy at the source, the optimum ,i location which mini mizes the average SER is iZndependent of a direct link between the source and the destina tion. Proof. This can be proven by the average SER expression in Eq. (3 31). The 1/7d,, Dv,dp/p has a fixed value given the s d distance D8,d and prescribed energy at the source. Therefore, a direct link does not affect location optimization. Hence, the location III 102 . . . .. ., .. .. ... . . . . . .. . .. . . o : :::: p:: p .::: ::: : ::^ : ::: : :,, : : : :::: :": S. .... . .... .... . .. : .. . S ps/p=0.8 104 104 0 0.2 0.4 0.6 0.8 1 D s,r Figure 413. SER versus relay location distribution at the given energy allocation ps/p (AF ND, L = 2, p = 15dB, D = 1, v = 4). optimization can be achieved without considering the direct link; i.e., the results of location optimization are the same in both systems with and without a direct link. D The optimum location can be found by treating the SER as a function of distance and solving the first order conditions. Proposition 6. For the AF protocol, with the given sourcedestination distance Ds,d, sourcerelay(s)destination distance D, and the prescribed transmit energy levels ps and pr Sthe optimum location of the '. 1.,;/ should ,.ri/fy vD,;1 r ,(D  o I I 06 vD pr vD D )p{ln[p(D D,)"] 1} 0, (461) and accordingly, Dd = D D . This solution can be obtained by using Lagrange multiplier as we discussed in the previous propositions; therefore, we omit the detailed derivation. Again, the log term and path loss exponent v make it difficult to find the closed form solution. By applying Lemma 2, one could resort to numerical search for the optimum solution. I I I I 10 '' 3 C, N S. .... . .o . .. . . .. . .. . . .. ...p./.p. 10 : : : : : : 1 4 .. .... .. .. ... . .. .:'. :.:::: : :  ps/p=0.2 ... . .... .. ^ .... ..... . S p /p=0. .. . ... . . .. . .. . 105 ..... ps/p=0.8 .. .. 10'5 0 0.2 0.4 0.6 0.8 1 D s,r Figure 414. SER versus relay location distribution at the given energy allocation ps/p (AF DL, L =2, p 15dB, D =1,v = 4). Figs. 413 and 414 present the average SER for the various relay locations at the given energy allocation. Both figures confirm that there exists only one minimum point which provides the optimum relay location for relay networks. It is interesting that the minimum points are the same for both figures although the SER is different. Hence, the existence of a direct link is independent on the optimum relay location, which confirms the Lemma 3. In both figures, the range of optimum locations is smaller than the energy optimization cases (see Figs. 47 and 48), which is the same as the DF case (see Fig. 412). Fig. 415 depicts the optimum relay location which is applicable to systems with and without a direct link. We consider the total SNR value of p = 30dB and a path loss exponent of v = 4 with various L values. One dimensional setup, Ds,r + D,d = D = Ds,d = 1, is considered. As more transmit energy is assigned at the source, the optimum location moves toward the destination. The figure shows that the optimized values change slowly compared with the energy optimization. I I I I 0.2 0.4 0.6 0.8 P,/P Figure 415. Optimum relay location (AF, ND and DL, D 0.9  0.6 H Ds,a = 1, p 30dB, v = 4). 0 0.2 0.4 0.6 0.8 Figure 416. Optimum relay location (DFSTC and AFSTC, D = 1.2, Ds,d = p 15dB, v = 4). o 0.8 C O 8 0.7 0.6 . 0.5  L=1  L=2 S.... L=3 L=2 SL=3 I Fig. 416 depicts the optimum relay location of DFSTC and AFSTC protocols when L = 2 and 3. We use the total SNR value of p = 15dB and a path loss exponent of v = 4. with D = 1.2 and D,d = 1. The figure shows that the optimum relay locations move toward the destination as the transmit energy at the source increases for both cases. The figure also shows that, in comparison with AFSTC protocol, the relay locates closer to the source for the DFSTC protocol. From Figs. 410 and 416, we can see that the location optimizations are much flatter than the energy optimizations. These results are the same as those of the conventional cooperative systems with no direct link. 4.4 Joint Energy and Location Optimization So far, we have been focusing on the energy optimization and location optimization separately. Now let us consider the joint optimization which satisfies both the energy and location optimization. The analytical solution can be obtained by using Eqs. (35), (310) or (331). First, by treating each equation as the function of transmit energy, we can find the solution for the first order conditions. Then, the same step is proceeded by the transmit energy that is replaced with the location. Finally, by equating two solutions, we can find the common solution. Consequently, this solution provides the global optimization which minimizes the error rate. With DF protocol, for L = 1, we can readily obtain the global solution from Eqs. (435) and (460), which gives us Ds, = Dr,d = 0.5 with ps/P = Pr/P = 0.5, V. However, the analytical solution cannot be easily obtained even with the idealized case as we have seen in the previous section. In general, the joint optimization can be obtained by carrying out a twodimensional numerical search iteratively. The searching steps are as follows : Step 1. (Initialization) Set the uniform energy allocation as the optimum, i.e. p p/(L + 1). Step 2. (Location Optimization) For a given energy allocation, find the optimum relay location, D,,new, which is error rateminimizing. If the difference between new optimum location and the original one is smaller than the threshold distance, ED, Figure 417. Iterative search: flow chart. i.e., ID,.new D ,r < ED, stop; otherwise, set the optimum location to the new one, Dr = Dre and continue to Step 3. Step 3. (Energy Optimization) For a given relay location, find the optimum energy allocation, ponw. If the difference between new optimum energy allocation and the original one is smaller than the threshold energy, Ep, i.e., pOnew PO < Ep, stop; otherwise set the optimum energy to the new one, pO = pOn and go back to Step 2. These iterative searching can be illustrated as the flow chart in Fig 417. Without considering a direct link, Figs. 418 and 419 depict the SER performance surface when L = 3 with Ds,d = D = 1, and v = 4, for both the DF and AF protocol. We use p = 10dB and p = 15dB for the DF and AF protocol, respectively. We can obtain the energy optimization and the location optimization by taking minimum value along the ! LU 10  10 1 00.8 0. 0.5 0 0.2 0 0 Ds, r P/P Figure 420. Performance surface versus ps/p and Ds,r (DFSTC, p = 15dB, v = 4, L 2). Ds,r axis and ps/p axis, respectively. Using the above steps, the global minimum can be obtained. This point provides the joint energy and location optimization. Similar to the conventional protocols, we plot the CER versus ps/p and Ds,r for DFSTC and AFSTC protocols in Figs. 420 and 421, respectively, when D = 1.2, Ds,d 1, and v = 4. These figures exhibit the same trends as in the DF and AF protocols. Notice that the two figures show almost the same shape. However, the axis values of the two figures are opposite. This implies that the systems employing DF and AF protocol have quite different relationships for the optimum values. Notice that above searching steps for joint energy and location optimization can be applicable for the DFSTC and AFSTC protocols by replace SER with CER though the analytical solutions are intractable. More detailed examples and comparisons are given in the following chapter. Notice that our simulations (Figs. 43, 412, 418, 420, and 421) confirm that the error rate is generally convex, which ensures convergence of the iterative strategy. CHAPTER 5 SIMULATIONS AND DISCUSSIONS In this chapter, we will discuss the performance of relay systems combined with differential demodulation and the optimum resource allocation. We will compare the performance of the systems with and without optimization. The benefits of the joint energy and location as well as the resource allocation comparison of different protocols are addressed. 5.1 Benefits of Energy and Location Optimizations To verify the advantages of the optimum energy allocation and relay location selection, the SERs of the relay systems with and without optimization are depicted in Fig. 51 through Fig. 56. We use the following system parameters: p = 15dB, D = 1.2, v = 4, and L = (1,2, 3) with DBPSK. In the system without energy optimization, a uniform energy allocation is emplo,' d: that is, ps = pr = p/(L + 1) at any Ds,,. In the system without location optimization, the re i,, are placed at the midpoint of the sourcedestination link. Figs. 51 and 52 illustrate the benefits of optimization of the DF system. In Fig. 51, we observe that, as L increases, the SER performance can get even worse unless the energy optimization is performed, and that the energyoptimized system universally outperforms the unoptimized one. These observations confirm our discussions in the preceding chapter. Interestingly, notice that the minima of the energyoptimized SER curves almost coincide with the unoptimized ones. This implies that the nearoptimum SER can be achieved even with the uniform energy allocation across the source and relay nodes, provided that the relay location is carefully selected. As shown in Fig. 51, the optimum relay location corresponding to the uniform energy allocation shifts from the midpoint for L = 1 to the source node as L increases. Intuitively, this is because the overall SER is more sensitive to the sourcerelay link quality, as we mentioned in C'! plter 3.  / P  r: Unoptimized (ND) S p =po: Optimized (ND) 0.2 0.4 0.6 D s,r 0.8 1 1.2 Figure 51. SER comparison between relay systems with and without energy optimization (DF, p = 15dB, D = 1.2, Ds,d = 1, v = 4).  =Dr,d: Unoptimized (ND)  D =D : Optimized (ND) ...... s,r s,r 0.2 0.4 0.6 0.8 ps/P Figure 52. SER comparison between relay systems with and without relay location optimization (DF, p = 15dB, D = 1.2, Ds,d = v = 4). 104 0 L=21 : : : , L..2 / In Fig. 52, we verify the advantage of the optimum relay location by comparing the SER with and without location optimization. Similar to the energy optimization case, Fig. 52 confirms the advantages of the location optimization, in which the locationoptimized system universally outperforms the unoptimized system. Different from the energy optimization case, however, as L increases, the SER performance ah,v improves even without any location optimization. The curves in Fig. 52 also exhibit more flatness compared with the ones in Fig. 51. This implies that the system SER is more sensitive to the location distribution than the energy distribution. In addition, the minima of the locationoptimized SER curves are far from those of the unoptimized ones, except for the L = 1 case (see Fig. 52). This indicates that placing the relay nodes at the midpoint cannot achieve the minimum SER even with careful allocation of the source and relay energies, for any L > 1. This is to be distinguished from the uniform energy case depicted in Fig. 51, as well as from the coherent relay systems in [45]. Figs. 53 and 54 depict the benefits of energy optimization for the AF system with no direct link and with a direct link, respectively. From Figs. 53 and 54, we observe that the energyoptimized system universally outperforms the unoptimized system as we expected. We also observe that, in the system with a direct link, the SERs of the optimized system and unoptimized system are almost identical when the rel i are located close to the source, since a uniform energy allocation is optimum. These observations coincide with our analysis in the preceding chapter. Both figures show that the unoptimized systems have the minimum SER almost at the midpoint, coinciding with the results in [38, 45, 67]. Notice that the minimum points of the energyoptimized SER curves move towards the destination except in the system without a direct link for L = 1, which is opposite compared with the DF case. From Figs. 53 and 54, it is clear that we cannot achieve optimum SER value without energy optimization except for the system L = 1 with no direct link. This is different from the DF case, as in Fig. 51. It is worth mentioning 101 Lu U) , 102 10 103 104 0 0.8 1 1.2 Figure 53. SER comparison between relay systems with and without energy optimization (AF ND, p = 15dB, D = 1.2, Ds,d = 1, v = 4). ps Pr: Unoptimized (DL) ps=ps: Optimized (DL) 0 0.2 0.4 0.6 D s.r 0.8 1 1.2 Figure 54. SER comparison between relay systems with and without energy optimization (AF DL, p = 15dB, D = 1.2, Ds,d 1, v = 4).  ps=Pr: Unoptimized (ND) Sp=p: Optimized (ND) 0.2 0.4 0.6 D s,r S . .. .. / . 104 I.. .I . ... D =D : Unoptimized (ND) .. . .. r r, d ...... .... D =Do :Optimized (ND) .s,r s,r 69I~~~~l iI ... .. .... .. .... ..   : : ..... / 102 \: . .... ... . .... .. ..... ... .. 0   0 0.2 0.4 0.6 0.8 1 Ps/P Figure 55. SER comparison between relay systems with and without relay location optimization (AF ND, p = 15dB, D = 1.2, Ds,d = 1, v = 4). that the optimum relay location corresponding to the uniform energy allocation alh,v keeps the midpoint regardless of the number of rel 1v, which is the same for the coherent systems in [45] with AF protocol, but different from the DF case. Next, let us consider the benefits of location optimization for the AF systems. Figs. 55 and 56 verify the advantages by comparing the SERs of the systems with and without location optimization. Similar to the energy optimization case, the locationoptimized system universally outperforms the unoptimized system. The figures show that the optimum SER can be achieved by assigning more energy to the source except for the system without a direct link with L = 1, which is the opposite of the DF case. The curves in Figs. 55 and 56 also show more flatness compared with the energy optimized curves, as we have observed for the DF case. Similar to the location optimization of the DF and the energy optimization of the AF, Figs. 55 and 56 show that optimum SER cannot be obtained without relay location optimization except for the system L = 1 with no direct .. L=1 / 10 S. .... : ....... D ,r=Dr,d: Unoptimized (DL) SD =D : Optimized (DL) s,r s,r 101 3 :L3 . . . . .. . . . 105 0 0.2 0.4 0.6 0.8 1 Ps/P Figure 56. SER comparison between relay systems with and without relay location optimization (AF DL, p = 15dB, D = 1.2, Ds,d = 1, v = 4). link. The benefit of a direct transmission is obvious in the SERs from Figs. 54 and 56 compared with Figs. 53 and 55, respectively. Figs. 57, 58, 59, and 510 depict the benefits of energy and location optimizations for DFSTC (Figs. 57 and 58) AFSTC (Figs. 59 and 510) protocols. For all cases, we consider D = 1.2, Ds,d 1, and v = 4 with p = 15dB and 25dB when L = 2 and 3 since the single relay setup (L = 1) is the same as the conventional cooperative system. Similar to the conventional case, we plot the CER for the system with and without resource allocations. We use the same parameters for the unoptimized systems. The figures confirm that the minimum CER can be achieved by the optimum energy and relay location selection. The figures also show that the trends of optimizations for DFSTC and AFSTC protocols are the same as those for DF and AF protocols, respectively. Notice that, at low p, the systems with more L may underperform the systems with less L. However, at high p, the error performance improves as L increases except some cases of DFSTCbased system as we have seen in DFbased system. 0.2 0.4 0.6 D s,r 0.8 1 1.2 Figure 57. CER comparison between relay systems with and without energy optimization (DFSTC, p = 15dB and 25dB, D = 1.2, D,d = 1, v = 4). 0.2 0.4 0.6 0.8 Figure 58. CER comparison between relay systems with and without relay location optimization (DFSTC, p = 15dB and 25dB, D = 1.2, Ds,d = v = 4). 102 105L 0 p=15dB . . ____ t  L=2,p up. UnoptinizEd S. . L=3, p=p:Unoptimized S p 25 dB eL=2,ps=Ps: Optimized . ... . ... . + L=3, ps=p: Optimized 0.2 0.4 0.6 D s,r  L=2,p =p: Optimized  + L=3, ps=ps: Optimized 0.8 1 1.2 Figure 59. CER comparison between relay systems with and without energy optimization (AFSTC, p = 15dB and 25dB, D = 1.2, Ds,d = 1, v = 4). 10 10 Cr w 10 10 10 10 10 0.2 0.4 0.6 0.8 Figure 510. CER comparison between relay systems with and without relay location optimization (AFSTC, p = 15dB and 25dB, D = 1.2, D,d = 1, v = 4). 101 102 S103 C) 106 0 ,* _* p15dB L=3, ps =Pr:Unoptimized SL=3, Ps=Pr:Unoptimized Summarizing, we confirm the advantages of energy and relay location optimization which minimize the error rate of the cooperative systems. With uniform energy allocation, the DFbased systems and AFbased systems exhibit an unbalanced and balanced effect on the error performance, respectively. Interestingly, for the DFbased systems, relay location optimization may be more critical than energy optimization. In other words, nearoptimum performance can be achieved by allocating a uniform energy at each node, but not by locating the re 1 i, at the midpoint between sourcedestination distance. For the AFbased systems, both energy and location optimization are critical in the sense that the optimum SER cannot be achieved without any optimization. Our results show that the optimum resource allocation has different optimum values depending on the protocols, which is confirmed in the next section. 5.2 Benefits of Joint Optimization In this section we will consider the joint energy and location optimization. Figs. 511 and 512 depict the SER contour of the relay systems for the AF protocol with no direct link and with a direct link, respectively, and Fig. 513 depicts the SER contour of the relay system for the DF protocol. With system parameter D,d = D = 1, L 2 is considered for the system with no direct link and with a direct link. We use p = 15dB and 10dB for the AF and DF protocol, respectively. In all figures, the vertical line and horizontal line represent the SERs of the systems with a uniform energy allocation and middistance allocation, respectively, i.e., unoptimized systems. We also plot lines for the energy optimization and location optimization. Notice that the crossing point of the two optimizations is the minimum SER of the system; accordingly, this point corresponds to the joint energy and location optimization. With AF protocol in Figs. 511 and 512, it is clear that the minimum SER of the unoptimized systems is far from the optimized ones; this indicates that we cannot obtain the minimum SER with a uniform energy allocation or middistance allocation. To achieve the minimum SER, we have to adapt a system via the energy optimization or 0.5  0.4 sr./ oc mized 0.3 s,=D rd: Unoptimized 0.2 PsPr: Unoptimized p: =p: Energy optimized 0.1 . 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PS/P Figure 511. The SER contour versus Ds,r and ps/p (AF ND, Ds,d S=D oca opti 0 i9 sr r 08 0 06 0 .5 0.4 0.3 energy optimize 0.2  Prp: Unoptimized 0.1  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PS/P Figure 512. The SER contour versus Ds,r and ps/p (AF DL, Ds,d 0.9 D = 1, p 15dB, L= 2). 0.9 D = 1, p 15dB, L 2). 0.4 0.2 1 s Unoptimized 0.1 ps =p : Energy Cpli~mn'iz.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ps/P Figure 513. The SER contour versus Ds,r and ps/p (DF, D,d = D = 1, p10dB, L = 3). location optimization. However, the DFbased system in Fig. 513 shows that the near optimum performance can be achieved with a uniform energy allocation, but cannot with a middistance allocation. All these results coincide with the simulation results in the previous section and theoretical analysis in C'! lpter 4. From Fig. 513, we can see that the uniform energy allocation is a very good starting point for the iterative optimization. Figs. 514 and 515 depict the CER contour for the DFSTC and AFSTC protocols, together with the optimum energy allocation and relay location optimization curves. We consider D = 1.2, Ds,d 1, and p=15dB when L = 2. These figures show that the CER contours for the DFSTC and AFSTC protocols have the same trends as the SER contours for the DF and AF protocols, respectively. It is interesting that the optimum resource allocation is different in the two protocols. For the AFbased protocol, we can achieve the minimum error rate by locating the rel,i, closer to the destination and assigning more energy at the source than the rel,i . This is due to the fact that the error rate decreases by allowing the source to transmit signals with more energy while reducing the effect of noise on the amplification factor, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 514. The CER contour versus Ds,r and ps/p (DFSTC, D = 1.2, Ds,d 1, p=15dB, L 2). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P,/P Figure 515. The CER contour versus Ds,r and ps/p (AFSTC, D = 1.2, Ds,d = p 15dB, L 2). 0 Conventional system 7 + STCbased system 0 6 .0' E S .0 0 E o 5 3 .0 z 2  + +  + . .. + . 1 2 3 4 5 6 Number of relays Figure 516. Data rate comparison between the conventional , I, and STCbased system in terms of the required time slots per information symbol. and enhancing the error performance of relaydestination links. On the contrary, for the DFbased protocol, the minimum error rate can be achieved by locating the relays closer to the source while assigning less energy at the source than the relays. This is due to the fact that the error of the sourcerelay links decreases, thereby causing a decreasing of the entire system error. 5.3 Conventional System vs STCbased System In this section, we will compare the conventional cooperative system and the STCbased cooperative systems. We first compare the data rate of two systems, and then the error rates of two systems are compared in two scenarios, i.e., the same modulation size is adopted and same/similar data rate is used. Assume that the original information symbols are equiprobable binary signal (q = 1) and there are two relay nodes (L = 2). In the conventional system, one symbol transmission takes three time slots: one for broadcasting the symbol to the relay nodes, and two time slots are used to transmit the remodulated/amplified signal from each relay I I I I to the destination. For the STCbased system, 2 consecutive symbols can be packed into each 2 x 2 unitary matrix. Since we use the diagonal design for STC, four time slots are needed to transmit 2 symbols from the source node: two time slots are used to transmit between the source and each relay, and two time slots for the transmission of the 2 x 2 matrix to the destination over the common relaydestination channel. Accordingly, the number of time slots required for the transmission of each symbol further decreases for the STCbased system, as the number of r1eliv increases. This trend is depicted in Fig. 516. We observe that the number of required time slots increases with the number of relays for the conventional system, but remains constant for the STCbased system. Therefore, we can increase transmission rate by using STCbased system. This implies that the STCbased system can potentially provide the differential benefit as well as high transmission rate regardless of the number of relays. For the STCbased system, the modulation size increases as the number of re v  increases. Whereas, we can choose the modulation size for the conventional system. By adopting the same modulation size, the comparison of bit error rate (BER) between the conventional system and STCbased system is depicted in Figs. 517 and 518 with DF/DFSTC and AF/AFSTC protocols, respectively. We consider DQPSK, D8PSK, and D16PSK for L=2, 3, and 4, respectively, for the conventional system. The figures show that STCbased system performs comparably with or better than the conventional system especially when L > 3. For the conventional system, the error performance decreases as L increases for both DF and AF protocols. However, for the STCbased system, the BER is not affected dramatically regardless of L and the diversity gain is alvbv guaranteed by decreasing error rate for the DFSTC and AFSTC protocol, respectively. As we mentioned in the above, the STCbased system provides higher transmission rate. If the conventional system adopts the same date rate as in the STCbased system, how does this affect the error performance? To answer this question, we compare the BER of the cooperative systems which have the same or similar data rate in Figs. 519 and 520 S"10i "t. .. . 102 2 wU 10 S L=2, DQPSK S10. L=4, D16PSK ... . ... ,, .......... L=4, D 16PS K :: : : : : : : :: : : : : : : ::: : : : : : ::4I  L=2, DUSTC 8 L=3, DUSTC + L=4, DUSTC 104 i i 0 5 10 15 20 25 30 SNR Figure 517. BER comparison between the conventional systems and STCbased systems with same modulation size (DF vs DFSTC, SNR=Br,s = d,r). 5 10 15 SNR 20 25 30 Figure 518. BER comparison between the conventional systems and STCbased systems with same modulation size (AF vs AFSTC, SNR ,r, = yd,r). 100 101 102 S103 104 105 106 > '''',,  L=2, DQPSK(2/3) L=3, DQPSK(1/2) . ''* , 10 .... L=4, D8PSK(3/5). .. .. ... .. . ..L=2, DUSTC(1/2) : : :' : e L=3, DUSTC(1/2). : : : :: : : : : : : :. S*+ L=4, DUSTC(1/2) 104 0 5 10 15 20 25 30 SNR Figure 519. BER comparison between the conventional systems and STCbased systems with equal/similar transmission rate (DF vs DFSTC, SNR=7r,s = 7d,r). for the DF/DFSTC and AF/AFSTC protocols, respectively. To keep equal data rate, we use DQPSK when L=2 and 3, and D8PSK for L=4. These modulation sizes correspond to the data rate of 2/3, 1/2, and 3/5 (bits/time slot) for L = 2, 3 and 4, respectively. The STCbased system alv keeps data rate as 1/2. The figures show that the STCbased system does not necessarily outperform the conventional system. The conventional system may provide better performance by properly choosing the modulation size. From Fig. 517 to Fig. 520, we can see that the modulation size dramatically affects the performance of the conventional system. Summarizing, we compared two cooperative systems with respect to the transmission rate and BER. Our results show that the STCbased system provides higher transmission rate compared with the conventional system. It is also shown that the conventional system may provide better performance than the STCbased one by keeping the same/higher data rate. In addition, the BER comparisons reveal that the modulation size is critical to determine the error performance of conventional system. 10 2 S1032. 104 L=2, DQPSK(2/3) L=3, DQPSK(1/2) S : : L=4, D8PSK(3/5): : :: S L=2, DUSTC(1/2) e L=3, DUSTC(1/2) ! *+ L=4, DUSTC(1/2) : : : : 106 0 5 10 15 20 25 30 SNR Figure 520. BER comparison between the conventional systems and STCbased systems with equal/similar transmission rate (AF vs AFSTC, SNR,,, = yd,r). CHAPTER 6 CONCLUSIONS AND FUTURE WORK 6.1 Conclusions In this research, we investigated cooperative networks with an arbitrary number of rel i employing differential (de)modulation. Two conventional relaying protocols, i.e., decodeandforward (DF) and amplifyandforward (AF), and two distributed spacetime coding (DSTC) protocols, i.e., decodeandforward spacetime coding (DFSTC) and AFSTC, are considered for the relay systems. We analyzed the error performances of cooperative networks, and based on these, we developed the optimum resource allocation which minimizes the error performance. We derived the upper bound of symbol error rate (SER) for the DF protocol, approximated SER for the AF protocol, and the upper bound of the codeword error rate (CER) for STCbased systems. The DFbased (DF and DFSTC) protocols showed an unbalanced error performance depending on the relay locations, and our analytical results and simulations sI .. I that it is mainly the sourcerelay links that determine the overall system performance. For the AF protocol, the SER was derived for two cases: the system with no direct link and the system with a direct link. This SER expression had a general but very simple form. The AFSTC protocol showed the same trends as the AF protocol with no direct link. Based on the error performance, the average SER and CER for the conventional DF/AF protocols and the DSTC protocol, respectively, we explored the resource allocation as a twodimension problem. We showed that: i) given the source, relay and destination locations, the average error rate can be minimized by appropriately distributing the prescribed total energy per symbol across the source and the re1 i: ii) given the source and relay energy levels, there is an optimum relay location which minimizes the average error rate; and iii) given the source, relay and destination locations, and total transmit energy, the minimum error rate can be achieved by the joint energylocation optimization. With DF and DFSTC protocols, for any L values, the optimum energy allocation at the source increases as the relay moves toward the destination. This is the same as the AF protocol with no direct link and the AFSTC protocol. However, a uniform energy allocation is optimum for the AF relay system with a direct link when the rel 1 are located near to the source. Our analysis reveals that the optimum energy allocation depends on a direct link, but the optimum relay location does not. For all cases, the optimum relay locations move towards the destination as more energy is assigned to the source, and the optimum relay locations show more flatness than the optimum energy allocations. Our simulations and numerical examples confirm that both the energy and location optimizations provide considerable error performance advantages. We have observed the following results for the DF and DFSTC protocols. (1) Without energy optimization, performance degradation is observed when more re 1 i are included in the system especially when the re1 i, are located close to the destination node. (2) For all cases, the optimized systems universally outperform the unoptimized ones. (3) The location optimization is more critical than the energy optimization. In other words, the differential relay system with uniform energy distribution can achieve nearoptimum error performance by appropriately choosing the relay location; while a system with re i. sitting at the midpoint between the source and the destination cannot approach the optimum error performance even with optimized energy distribution. For the AF and AFSTC protocols, we have observed following results. (1) For the system based on AF protocol with a direct link, the error rates of the system with a uniform energy allocation and optimum energy allocation are almost identical when the reli , are located near to the source, since the uniform energy allocation is optimum in such cases. (2) For all cases, the optimized systems universally outperform the unoptimized ones. (3) Energy and location optimizations are equally important, since minimum error performance cannot be achievable without either of them. We have also shown that the minimum error performance can be achieved by the joint energylocation optimization. In addition, we compared the cooperative systems with different protocols by considering both the energy distribution and the relay location selection. It is interesting that the optimum location and energy allocations are very different in the two protocols. In general, we can achieve the minimum error performance of the cooperative networks by locating relays closer to the source node with a less amount of the source transmit energy for the DFbased system, and by locating r1el i closer to the destination node with a large amount of the source transmit energy for the AFbased system. Finally, we compared the conventional system with STCbased system. In general, STCbased system can support higher transmission rates. However, the conventional system can achieve comparable performance in comparison with STCbased one by choosing appropriate modulation size. 6.2 Future Work In our research, we consider the conventional cooperative networks which are the distributed counterpart of standard differential scheme for a singleinput singleoutput (SISO) channel, and STCbased cooperative networks which are the distributed counterpart of the differential spacetime codes. We showed that the cooperative networks have different properties in their performance and optimum resource allocation depending on the relaying protocol. Recently, the hybrid scheme which selects the advantages of DF and AF protocols is i.. I. 1 for coherent system [8, 11, 33]. It will be interesting to develop a hybrid cooperative system and analyze its performance and resource optimizing schemes. It will be also valuable to consider multihop cooperative networks which can support reliable communications (see [34, 41, 69]). Finally, in this research, we mainly focused on physical l vr analysis. It will be helpful to consider higher l vr issues, i.e, medium access control (! AC) or network lv. i~, for improving the overall networking performance. LIST OF REFERENCES [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formu las, Graphs, and Mathematical Tables. Dover, New York, 1972. [2] P. A. Anghel and M. Kaveh, "Exact symbol error probability of a cooperative network in a rayleighfading environment," IEEE Trans. on Wireless Communica tions, vol. 3, no. 5, pp. 14161421, September 2004. [3] P. A. Anghel and M. Kaveh, "On the diversity of cooperative systems," in Proc. of Intl. Conf. on Acoustics, Speech and S.:g,,Ll Processing, vol. 4, Montreal, Quebec, Canada, M .i 1721, 2004, pp. 577580. [4] P. A. Anghel and M. Kaveh, "On the performance of distributed spacetime coding systems with one and two nonregeneratvie relays," IEEE Trans. on Wireless Com munications, vol. 5, no. 3, pp. 682692, March 2006. [5] P. A. Anghel, M. Kaveh, and Z. Q. Luo, "Optimal rml i, I power allocation in interferncefree nonregenerative cooperative systems," in Proc. of S.':,,rl Proc. Workshop on Advances in Wireless Communications, Lisbon, Portugal, July 1114, 2004, pp. 2125. [6] R. Annavajjala, P. C. Cosman, and L. B. Milstein, "On the performance of optimum noncoherent amplifyandforward reception for cooperative diversity," in Proc. of MILCOM Conf., vol. 5, Atlantic city, NJ, oct 1720, 2005, pp. 32803288. [7] R. Annavajjala, P. C. Cosman, and L. B. Milstein, "Statistical channel knowledgebased optimum power allocation for relayig protocols in the high snr regime," IEEE Journal on Selected Areas in Communications, vol. 25, no. 2, pp. 292305, February 2007. [8] X. Bao and J. Li, "Decodeamplifyforward (daf): A new class of forwarding strategy for wireless relay channels," in Proc. of S.:,1i'l Proc. Workshop on Advances in Wireless Communications, New York, NY, June 58, 2005, pp. 816820. [9] A. Bletsas and A. Lippman, Ii,!, i i i, i iiig cooperative diversity antenna arrays with commodity hardware," IEEE Communications M rl. ..:,. vol. 44, no. 12, pp. 3349, December 2006. [10] J. Bc.V r, D. D. Falconer, and H. Yanikomeroglu, '\!,ull !i'p diversity in wireless relaying channels," IEEE Trans. on Communications, vol. 52, no. 10, pp. 18201830, October 2004. [11] B. Can, H. Yomo, and E. D. Carvalho, "Hybrid forwarding scheme for cooperative relaying in ofdm based networks," in Proc. of International Conf. on Communica tions, vol. 10, Istanbul, Turkey, June 1115, 2006, pp. 45204525. [12] D. C'!, i, and J. N. Laneman, "Cooperative diversity for wireless fading channels without channel state information," in Proc. of Asilomar Conf. on S.:i,.l. S', and Computers, Monterey, CA, November 710, 2004, pp. 13071312. [13] D. C'!, i1 and J. N. Laneman, "Modulation and demodulation for cooperative diversity in wireless systems," IEEE Trans. on Wireless Communications, vol. 5, no. 7, pp. 17851794, July 2006. [14] W. Cho, R. Cao, and L. Yang, "Optimum energy allocation in cooperative networks: A comparative study," in Proc. of MILCOM Conf., Orlando, FL, Oct 2931, 2007. [15] W. Cho, R. Cao, and L. Yang, "Optimum resource allocation for amplifyandforward relay networks with differential modulation," IEEE Trans. on S.:,i'..l Processing, June 2007 (submitted). [16] W. Cho and L. Yang, "Differential modulation schemes for cooperative diversity," in Proc. of IEEE International Conference on Networking, Sensing and Control, Ft. Lauderdale, FL, April 2325, 2006, pp. 813818. [17] W. Cho and L. Yang, "Distributed differential schemes for cooperative wireless networks," in Proc. of Intl. Conf. on Acoustics, Speech and S.: ji..l Processing, vol. 4, Toulouse, France, May 1519, 2006, pp. 6164. [18] W. Cho and L. Yang, "Optimum energy allocation for cooperative networks with differential modulation," in Proc. of MILCOM Conf., Washington, DC, Oct 2325, 2006. [19] W. Cho and L. Yang, "Joint energy and location optimization for relay networks with differential modulation," in Proc. of Intl. Conf. on Acoustics, Speech and S.:,,..ld Processing, vol. 3, Honolulu, Hawaii, apr 1520, 2007, pp. 153156. [20] W. Cho and L. Yang, "Optimum resource allocation for relay networks with differential modulation," IEEE Trans. on Communications, 2007 (To appear). [21] W. Cho and L. Yang, "Resource allocation for amplifyandforward relay networks with differential modulation," in Proc. of Global Telecommunications Conf., Washington, D.C., November 2630, 2007 (To appear). [22] X. Deng and A. M. Haimovich, "Power allocation for cooperative relaying in wireless networks," IEEE Communications Letters, vol. 9, no. 11, pp. 994996, November 2005. [23] M. Dohler, A. Gkelias, and H. Aghvami, "Resource allocation for fdmabased regenerative multihop links," IEEE Trans. on Wireless Communications, vol. 3, no. 6, pp. 19891993, November 2004. [24] M. Dohler, A. Gkelias, and H. Aghvami, "A resource allocation strategy for distributed mimo multihop communication systems," IEEE Communications Letters, vol. 8, no. 2, pp. 99101, February 2004. [25] I. S. Gradshteyn and I. M. Ryzhik, Table of Integradls, Series, and Products, 6th ed. Academic Press, 2000. [26] M. O. Hasna and M. Alouini, "Endtoend performance of transmsiion systems with rel ,i over rayleighfading channels," IEEE Trans. on Wireless Communications, vol. 2, no. 6, pp. 11261131, November 2003. [27] M. O. Hasna and M. Alouini, "Optimal power allocation for r~l '1iv transmissions over rayleighfading channels," IEEE Trans. on Wireless Communications, vol. 3, no. 6, pp. 19992004, November 2004. [28] M. O. Hasna and M. Alouini, "A performance study of dualhop transmissions with fixed gain relays," IEEE Trans. on Wireless Communications, vol. 3, no. 6, pp. 19631968, November 2004. [29] T. Himsoon, W. Su, and K. J. R. Liu, "Differential transmission for amplifyandforward cooperative communications," IEEE S.:,ii/l Processing Let ters, vol. 12, no. 9, pp. 597600, September 2005. [30] T. Himsoon, W. Su, and K. J. R. Liu, "Differential modulation for multinode amplifyandforward wireless relay networks," in Proc. of Wireless Communications and Networking Conf., vol. 2, Las Vegas, NV, April 36, 2006, pp. 11951200. [31] B. M. Hochwald and W. Sweldens, "Differential unitary spacetime modulation," IEEE Trans. on Communications, vol. 48, no. 12, pp. 20412052, December 2000. [32] B. L. Hughes, "Differential spacetime modulation," IEEE Trans. on Information The(. .; vol. 46, no. 7, pp. 25672578, November 2000. [33] A. Kannan and J. R. Barry, "Spacedivison relay: a highrate cooperation scheme for fading multipleaccess channels," in Proc. of Global Telecommunications Conf., Washington, D.C., November 2630, 2007 (To appear). [34] G. K. Karagiannidis, "Performance bounds of multihop wireless communications with blind rel i. over generalized fadin channels," IEEE Trans. on Wireless Communica tions, vol. 5, no. 3, pp. 498503, March 2006. [35] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, "Cooperative diversity in wireless networks: Efficient protocols and outage behavior," IEEE Trans. on Information The(.':; vol. 50, no. 12, pp. 30623080, December 2004. [36] J. N. Laneman and G. W. Wornell, "Energyefficient antenna sharing and relaying for wireless networks," in Proc. of Wireless Communications and Networking Conf., vol. 1, Chicago, IL, September 2328, 2000, pp. 712. [37] J. N. Laneman and G. W. Wornell, "Distributed spacetimecoded protocols for exploiting cooperative diverstiy in wireless networks," IEEE Trans. on Information The(.;, vol. 49, no. 10, pp. 24152425, October 2003. [38] H. Li and Q. Zhao, "Distributed modulation for cooperative wireless communications," IEEE S.:..:'il Processing _I.ii..:,. vol. 23, no. 5, pp. 3036, September 2006. [39] Y. Liang and V. V. Veeravalli, "Gaussian orthogonal relay channels: Optimal resource allocation and capacity," IEEE Trans. on Information Th(. ,; vol. 51, no. 9, pp. 32843289, September 2005. [40] I. Maric and R. D. Yates, ,. i. vding strategies for gaussian rarallelrelay networks," in Proc. of Conf. on Info. Sciences and Sil1' i Princeton, NJ, March 1719, 2004. [41] T. Miyano, H. Murata, and K. Araki, "Space time coded cooperative relaying technique for multihop communications," in Proc. of Vehicular T . I, i. i/; Conf., vol. 7, Los Angeles, CA, September 2629, 2004, pp. 51405144. [42] R. Pabst, B. H. Walke, D. C. Schultz, R. Herhold, H. Yanikomeroglu, S. Mukherjee, H. Viswanathan, M. Lott, W. Zirwas, M. Dohler, H. Aghvami, D. D. Falconer, and G. P. Gettweis, "Relaybased deployment concepts for wireless and mobile broadband radio," IEEE Communications M''i .,, vol. 42, no. 9, pp. 8089, September 2004. [43] C. S. Patel, G. L. Stiiber, and T. G. Pratt, "Statistical properties of amplify and forward relay fading channel," IEEE Trans. on Vehicular Tech., vol. 55, no. 1, pp. 19, January 2006. [44] J. Proakis, D.:.l:l.d Communications, 4th ed. McGrawHill, New York, 2001. [45] A. Ribeiro, X. Cai, and G. B. Giannakis, "Symbol error probabilities for general cooperative links," IEEE Trans. on Wireless Communications, vol. 4, no. 3, pp. 12641273, A li 2005. [46] A. Scaglione, D. L. Goeckel, and J. N. Laneman, "Cooperative communications in mobile ad hoc networks," IEEE S.:l,.rl Processing MI rr..:,. vol. 23, no. 5, pp. 1829, September 2006. [47] G. Scutari and S. Barbarossa, "Distributed spacetime coding for regeneratvie relay networks," IEEE Trans. on Wireless Communications, vol. 4, no. 5, pp. 23872399, September 2005. [48] A. Sendonaris, E. Erkip, and B. A .. i i.i "User cooperation diversity, part I: system description," IEEE Trans. on Communications, vol. 51, no. 11, pp. 19271938, November 2003. [49] A. Sendonaris, E. Erkip, and B. A .. li i.i "User cooperation diversity, part II: implementation aspect and performance ,i1 ,ievi IEEE Trans. on Communications, vol. 51, no. 11, pp. 19391948, November 2003. [50] A. B. H. Shin and M. Z. Win, "Outageoptimal cooperative communications with regenerative rel,," in Proc. of Conf. on Info. Sciences and Sl'.. ii Princenton, NJ, Mar. 2224, 2006, pp. 632637. [51] M. K. Simon and M. S. Alouini, "A unified approach to the probability of error for noncoherent and differentially coherent modulations over generalized fading channels," IEEE Trans. on Communications, vol. 46, no. 12, pp. 16251638, December 1998. [52] M. K. Simon and M. S. Alouini, D.:,:l,1 Communication over Fading C'lh,,,. 2nd ed. Wiley, 2004. [53] V. Stankovi<, A. HostMadsen, and Z. Xiong, "Cooperative diversity for wireless ad hoc networks," IEEE S.:,j.irl Processing M'rLj' ..:,. vol. 23, no. 5, pp. 3749, September 2006. [54] A. Stefanov and E. Erkip, "Cooperative spacetime coding for wireless networks," IEEE Trans. on Communications, vol. 53, no. 11, pp. 18041809, November 2005. [55] G. L. Stiiber, Principles of Mobile Communication, 2nd ed. Springer, 2001. [56] K. T and B. S. R ii in "Partiallycohernet distributed spacetime codes with differential encoder and decoder," IEEE Journal on Selected Areas in Communi cations, vol. 25, no. 2, pp. 426433, February 2007. [57] P. Tarasak, H. Minn, and V. K. Bhargava, "Differential modulation for twouser cooperative diversity systems," IEEE Journal on Selected Areas in Communications, vol. 23, no. 9, pp. 18911900, September 2005. [58] V. Tarokh and H. Jafarkhani, "A differential detection scheme for transmit diversity," IEEE Journal on Selected Areas in Communications, vol. 18, no. 7, pp. 11691174, July 2000. [59] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Spacetime block codes from orthogonal designs," IEEE Trans. on In f.'i i,,l.:.n The(..,; vol. 45, no. 5, pp. 14561467, July 1999. [60] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Spacetime block coding for wireless communications: performance results," IEEE Journal on Selected Areas in Communications, vol. 17, no. 3, pp. 451460, March 1999. [61] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Spacetime codes for high data rate wireless communication: Performance criterion and code construction," IEEE Trans. on Information Th(..,i vol. 44, no. 2, pp. 744765, March 1998. [62] M. Uysal, O. Canpolat, and M. M. Fareed, "Asymptotic performance analysis of distributed spacetime codes," IEEE Communications Letters, vol. 10, no. 11, pp. 775777, November 2006. [63] G. Wang, Y. Z!i ii. and M. Amin, "Differential distributed spacetime modulation for cooperative networks," IEEE Trans. on Wireless Communications, vol. 5, no. 11, pp. 30973180, November 2006. [64] T. Wang, Y. Yao, and G. B. Giannakis, "Noncoherent distributed spacetime processing for multiuser cooperative transmissions," in Proc. of Global Telecommuni cations Conf., vol. 6, St. Louis, MO, November 28December 2, 2005, pp. 37383742. [65] Y. Yao, X. Cai, and G. B. Giannakis, "On energy efficiency and optimum resource allocation of relay transmissions in the lowpower regime," IEEE Trans. on Wireless Communications, vol. 4, no. 6, pp. 29172927, November 2005. [66] S. Yiu, R. Schober, and L. Lampe, "Performance and design of spacetime coding in fading channels," IEEE Trans. on Communications, vol. 54, no. 7, pp. 11951206, July 2006. [67] M. Yu, J. Li, and H. q ,li ilpour, "Amplifyforward and decodeforward: The impact of location and capacity contour," in Proc. of MILCOM Conf., vol. 3, Atlantic city, NJ, October 1720, 2005, pp. 16091615. [68] J. Yuan, Z. C'!. i1 B. Vucetic, and W. Firmanto, "Performance and design of spacetime coding in fading channels," IEEE Trans. on Communications, vol. 51, no. 12, pp. 19911996, December 2003. [69] J. Zhang and T. M. Lok, "Performance comparison of conventional and cooperaitve multihop transmission," in Proc. of Wireless Communications and Networking Conf., vol. 2, Las Vegas, NV, April 36, 2006, pp. 897901. [70] Y. Zhli ii, "Differential modulation schemes for decodeandforward cooperative diversity," in Proc. of Intl. Conf. on Acoustics, Speech and S.':,,',l Processing, vol. 4, Philadelphia, PA, March 1923, 2005, pp. 917920. [71] Q. Zhao and H. Li, "Performance of a differential modulation scheme with wireless reliv, in rayleigh fading channels," in Proc. of Asilomar Conf. on S.:j,'l, S, 1. and Computers, vol. 1, Monterey, CA, November 710, 2004, pp. 11981202. [72] Q. Zhao and H. Li, "Performance of differential modulation with wireless rel i., in rayleigh fading channels," IEEE Communications Letters, vol. 9, no. 4, pp. 343345, April 2005. [73] Q. Zhao and H. Li, "Differential modulation for cooperative wireless systems," IEEE Trans. on S.:j.i',1 Processing, vol. 55, no. 5, pp. 22732283, May 2007. BIOGRAPHICAL SKETCH Woong Cho was born in Tongyoung, South Korea. He received his B.S. degree in electronics engineering from University of Ulsan, Ulsan, South Korea, in 1997 and his M.S. degree in electronic communications engineering from Hanyang University, Seoul, South Korea, in 1999. From March 1999 to August 2000, we was a research engineer in division of mobile telecommunication at Hyundai Electronics. He received his M.S. degree form electrical engineering from University of Southern California, Los Angeles, CA, in 2003. Since August 2003, he has been a Ph.D student in electrical and computer engineering at University of Florida, Gainesville, FL. His research interests include communications, signal processing, and networking. He received his Ph.D degree in 2007. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 Iwanttoexpressmytremendousgratitudetomysupervisor,Dr.LiuqingYang,forhertirelesseortaswellasforprovidingmewithinsights,inspiration,andencouragementwithoutwhichIcouldnothaveperformedthisresearch.Ideeplythankherforallofthepatientguidance,invaluableadvice,andnumerousdiscussionswehavesharedthroughoutmygraduatestudies.Iwouldliketothankallthemembersofmyadvisorycommittee,Dr.JenshanLin,Dr.TaoLiandDr.ShigangChen,fortheirvaluabletimeandenergyinservingonmysupervisorycommittee.IwouldalsoliketothankRuiCao,myfriendandcolleagueatSignalprocessing,Communications,andNetworking(SCaN)group,forthepricelessdiscussionsweshared,whichgeneratedmanyideasforthisresearch.IwishtoextendmysincerethankstoallthemembersoftheSCaNgroup,HuilinXu,FengzhongQu,DongliangDuan,andWenshuZhang,fortheircompanionshipandsupportthroughoutourtimetogether.Asalways,Iwanttothanktomyparentsandparentsinlaw,fortheirunyieldingsupportandlove.TheirencouragementandunderstandingthroughmystudyingperiodshavemeantmorethanIcaneverexpress.Last,Iwouldliketoexpressmygreatestthanksandadorationtomylovingwife.Iwanttothankherforsupportingandunderstandingmeininnumerableways,particularlyduringallourtimetogetherintheUnitedStates,andthroughoutmyPh.Dstudies. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1CooperativeNetworks ............................. 12 1.2Motivation .................................... 13 2SYSTEMMODEL .................................. 17 2.1RelayingProtocolsandChannelModeling .................. 17 2.1.1DecodeandForward(DF) ....................... 18 2.1.2AmplifyandForward(AF) ....................... 19 2.1.3DistributedSpaceTimeCoding(DSTC) ............... 19 2.2DierentialDemodulationandDecisionRules ................ 21 2.2.1DFProtocol ............................... 21 2.2.2AFProtocol ............................... 23 2.2.3DSTCProtocol ............................. 24 3PERFORMANCEANALYSIS ............................ 26 3.1SERforDFProtocol .............................. 26 3.2SERforAFProtocol .............................. 30 3.3CERforDSTCProtocol ............................ 38 4OPTIMUMRESOURCEALLOCATION ...................... 45 4.1ConvexityofSER ................................ 46 4.2EnergyOptimization .............................. 50 4.3RelayLocationOptimization .......................... 63 4.4JointEnergyandLocationOptimization ................... 70 5SIMULATIONSANDDISCUSSIONS ....................... 75 5.1BenetsofEnergyandLocationOptimizations ............... 75 5.2BenetsofJointOptimization ......................... 83 5.3ConventionalSystemvsSTCbasedSystem .................. 87 6CONCLUSIONSANDFUTUREWORK ...................... 92 6.1Conclusions ................................... 92 6.2FutureWork ................................... 94 5 PAGE 6 ................................. 96 BIOGRAPHICALSKETCH ................................ 102 6 PAGE 7 Figure page 11Simplecooperativenetwork. ............................. 13 21Setupofthecooperativenetwork. .......................... 17 31SERatdierentr;svalues(DF,L=2;M=2). 29 32SERatdierentd;rvalues(DF,L=2;M=2). 29 33SERboundversusd;randr;s(DF,L=2;M=2). 30 34SERcomparisonbetweenapproximationandsimulation(AFND,SNR(dB)=rk;s=d;rk). 36 35SERcomparisonbetweenapproximationandsimulation(AFDL,SNR(dB)=d;s=rk;s=d;rk). 37 36SERcomparisonbetweencoherentsystemanddierentialsystem(AFDL,SNR=rk;s=d;rk). 37 37TheCERfortheDFSTCprotocol(L=1;2;and3;SNR=rk;s=d;rk). 42 38TheCERfortheAFSTCprotocol(L=1;2;and3;SNR=rk;s=d;rk). 42 39TheeectofunbalancedlinkSNRfortheDFSTCprotocol(L=1;2;and3). 43 310TheeectofunbalancedlinkSNRfortheAFSTCprotocol(L=1;2;and3). 44 41Networktopologies:(a)Ellipsecase;(b)Linecase. 46 42Exactandapproximateoptimumenergyallocationswithdierentpathlossexponents(DF,L=1;=10dB). 53 43SERversusenergyallocationatthegivenrelaylocationDs;r(DF,L=2,=10dB,D=1,=4). 55 44ComparisonofoptimalenergyallocationbetweenthenumericalsearchandsimulatedresultsatvariousLvalues(DF,=10dB,=4). 55 45Comparisonofnormalizedoptimumenergyallocationatdierentvalues(DF,L=1). 46Existenceoftheoptimumsolution(AFND,=15dB,=3,L=2). 59 47SERversusenergyallocationatthegivenrelaylocationDs;r(AFND,L=2,=15dB,D=1,=4). 60 48SERversusenergyallocationatthegivenrelaylocationDs;r(AFDL,L=2,=15dB,D=1,=4). 60 7 PAGE 8 62 410Optimumenergyallocation(DFSTCandAFSTC,D=1:2;Ds;d=1,=15dB,=4). 411Optimumlocationofrelays(DF,=10dBandL=1). 65 412SERversusrelaylocationdistributionatthegivenenergyallocations=(DF,L=2,=10dB,D=1=4). 66 413SERversusrelaylocationdistributionatthegivenenergyallocations=(AFND,L=2,=15dB,D=1,=4). 67 414SERversusrelaylocationdistributionatthegivenenergyallocations=(AFDL,L=2,=15dB,D=1,=4). 68 415Optimumrelaylocation(AF,NDandDL,D=Ds;d=1,=30dB,=4). 69 416Optimumrelaylocation(DFSTCandAFSTC,D=1:2;Ds;d=1,=15dB,=4). 69 417Iterativesearch:owchart. 71 418Performancesurfaceversuss=andDs;r(DF,=10dB,=4;L=3,DBPSK). 72 419Performancesurfaceversuss=andDs;r(AF,ND,=15dB,=4;L=3,DBPSK). 420Performancesurfaceversuss=andDs;r(DFSTC,=15dB,=4;L=2). 73 421Performancesurfaceversuss=andDs;r(AFSTC,=15dB,=4;L=2). 74 51SERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(DF,=15dB,D=1:2,Ds;d=1,=4). 76 52SERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(DF,=15dB,D=1:2,Ds;d=1,=4). 76 53SERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(AFND,=15dB,D=1:2,Ds;d=1,=4). 78 54SERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(AFDL,=15dB,D=1:2,Ds;d=1,=4). 78 55SERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(AFND,=15dB,D=1:2,Ds;d=1,=4). 79 56SERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(AFDL,=15dB,D=1:2,Ds;d=1,=4). 80 57CERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(DFSTC,=15dBand25dB,D=1:2,Ds;d=1,=4). 81 8 PAGE 9 81 59CERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(AFSTC,=15dBand25dB,D=1:2,Ds;d=1,=4). 82 510CERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(AFSTC,=15dBand25dB,D=1:2,Ds;d=1,=4). 82 511TheSERcontourversusDs;rands=(AFND,Ds;d=D=1,=15dB,L=2). 84 512TheSERcontourversusDs;rands=(AFDL,Ds;d=D=1,=15dB,L=2). 84 513TheSERcontourversusDs;rands=(DF,Ds;d=D=1,=10dB,L=3). 85 514TheCERcontourversusDs;rands=(DFSTC,D=1:2;Ds;d=1,=15dB,L=2). 515TheCERcontourversusDs;rands=(AFSTC,D=1:2;Ds;d=1,=15dB,L=2). 516DataratecomparisonbetweentheconventionalsystemandSTCbasedsystemintermsoftherequiredtimeslotsperinformationsymbol. 87 517BERcomparisonbetweentheconventionalsystemsandSTCbasedsystemswithsamemodulationsize(DFvsDFSTC,SNR=r;s=d;r). 89 518BERcomparisonbetweentheconventionalsystemsandSTCbasedsystemswithsamemodulationsize(AFvsAFSTC,SNR=r;s=d;r). 89 519BERcomparisonbetweentheconventionalsystemsandSTCbasedsystemswithequal/similartransmissionrate(DFvsDFSTC,SNR=r;s=d;r). 90 520BERcomparisonbetweentheconventionalsystemsandSTCbasedsystemswithequal/similartransmissionrate(AFvsAFSTC,SNR=r;s=d;r). 91 9 PAGE 10 Inwirelesscooperativenetworks,virtualantennaarraysformedbydistributednetworknodescanprovidecooperativediversity.Obviatingchannelestimation,dierentialschemeshavelongbeenappreciatedinconventionalmultiinputmultioutput(MIMO)communications.However,distributeddierentialschemesforgeneralcooperativenetworksetupshavenotbeenthoroughlyinvestigated.Inthisdissertation,wedevelopandanalyzedistributeddierentialschemesusingtwoconventionalrelayingprotocols,decodeandforward(DF)andamplifyandforward(AF),andspacetimecoding(STC)basedrelayingprotocolwithanarbitrarynumberofrelays.Foreachprotocol,weanalyzetheerrorperformanceandconsidertheresourceallocationasatwodimensionaloptimizationproblem:energyoptimization,locationoptimization,andjointenergylocationoptimization. WerstderiveanupperboundoftheerrorperformancefortheDFsystem,theapproximatederrorperformancefortheAFsystem,andanupperboundfortheSTCbasedsystematreasonablyhighSNR,respectively.Basedontheseresults,wethendeveloptheenergyoptimizationandrelaylocationoptimizationschemesthatminimizetheaveragesystemerror.Analyticalandsimulatedcomparisonsconrmthattheoptimizedsystemsprovideconsiderableimprovementoverunoptimizedones,andthattheminimumerrorcanbeachievedviathejointenergylocationoptimization.Wecomparetheresultsofoptimizationandtheeectsofdierentrelayingprotocolsandobtainseveral 10 PAGE 11 11 PAGE 12 36 48 49 ].Thankstotheseadvantages,cooperativenetworkscanbeappliedinvariousscenariostoenhancethenetworkperformance.Forexample,wirelesspersonalareanetwork(WPAN)andlocalareanetwork(WLAN)canextendtheircoverageareasbyusingrelaynodes.Inhomeenvironments,cooperativenetworkscanhelpdistributethemultimediadatafromthecentralentertainmentunittodevicesanywhereinthehouse,bymountingrelaynodestothewallorevenembeddingtheminsidethewalls.Furthermore,cooperativenetworksalsondapplicationsinintelligenttransportationsystemsincludingintervehicle,intravehicleandvehicletoroadcommunications,toenablereliabledistributionoftheemergencyinformationtocertaingroupsofdriversviacooperationofmultiplevehiclesontheroad. Toillustratethebasicconceptofcooperativenetworks,Fig. 11 representsasimplecooperativenetworkinwhichthesourcenodetransmitsasignaltothedestination 12 PAGE 13 Simplecooperativenetwork. nodeviatherelaynode.Thesourcerelaydestinationandsourcedestinationlinksarecommonlyreferredtoastherelaylinkandthedirectlink,respectively.Therearetwoconventionalrelayingprotocolswhicharewidelyconsideredintheexistingliterature.Oneisthedecodeandforward(DF)protocolandtheotheristheamplifyandforward(AF)protocol.WithDFrelayingprotocol,therelaynodedemodulatesthereceivedsignalfromthesourcenodeandremodulatesthatsignal.Thentherelaynodetransmitstheremodulatedsignaltothedestinationnode.WithAFrelayingprotocol,therelayampliesthesignalfromthesourceandthensimplyforwardsittothedestinationnode.Anotherprotocolisthedistributedspacetimecoding(DSTC)protocolwhichcansupporthighertransmissionrate.ThisprotocolcanalsoadoptsDFandAFdependingontherelaynodeoperation. 9 38 42 46 53 ].Manystudieshavebeencarriedouttoanalyzetheperformanceofcooperativenetworks.Theoutageprobabilityofcooperativenetworksfromtheinformationtheoreticperspectiveispresentedin[ 35 ].Theerrorrateandoutageprobabilityofacooperativenetworkareanalyzedin[ 26 ]withoutconsideringthedirectlink.In[ 28 ],alaterstudybythesameauthors,theyconsiderbothrelayanddirectlinkswithaxedgainattherelay.In[ 10 ],themultihoprelaytransmissionisconsidered,andtheexactsymbolerrorrateisderivedin[ 3 ].Amoregeneralcaseoftherelaynetworksisdevelopedin[ 45 ].In[ 4 37 47 54 62 ],theperformanceofSTCbasedcooperativenetworksisconsidered.Alloftheseworkoncooperativenetworksfocusesoncoherent 13 PAGE 14 AccurateestimationoftheCSI,however,caninduceconsiderablecommunicationoverheadandtransceivercomplexity,whichincreaseswiththenumberofrelaynodesemployed.Inaddition,CSIestimationmaynotbefeasiblewhenthechannelisrapidlytimevarying.Tobypasschannelestimation,cooperativenetworksobviatingCSIhavebeenrecentlyintroduced.Thesecooperativesystemsrelyonnoncoherentordierentialmodulations,includingconventionalfrequencyshiftkeying(FSK)anddierentialphaseshiftkeying(DPSK)aswellasSTCbasedones.In[ 12 16 ],usingnoncoherentanddierentialmodulations,itisshownthattheloglikelihoodratiocanbecombinedbycapturingthedetectionerrorattherelaynodesaccordingtothesotermedtransitionprobabilityifthepartialCSIisknownattherelaysanddestination.Theperformanceofasinglerelaysystemwithnoncoherentanddierentialmodulationsisconsideredin[ 6 13 ]and[ 29 30 57 71 { 73 ],respectively.In[ 16 17 56 63 64 70 ],adistributedSTCsystemforcooperativenetworksisintroducedbyusingdierentialornoncoherentschemes. Toimprovetheerrorperformanceandtoenhancetheenergyeciencyofcooperativenetworks,optimumresourceallocationrecentlyemergedasanimportantproblemattractingincreasingresearchinterests(seee.g.,[ 5 22 27 39 65 ]).Theseworkisbasedondierentrelayingprotocols(amplifyandforward,decodeandforwardandblockMarkovcoding),undervariousoptimizationcriteria(signaltonoiseratio(SNR)gain,SNRoutageprobability,energyeciencyandcapacity),andwithdierentlevelsofCSI(instantaneousCSIandchannelstatistics).However,allofthemonlyconsiderthepowerallocationandmostlyfocusonasinglerelaysetup.In[ 7 14 18 30 ],optimumpowerallocationformultiplerelaylinksisdevelopedundervariousrelayingprotocols.[ 40 ]and[ 23 24 ],respectively,considertheoptimumenergyandbandwidthallocationinGaussianchannelandmultihopsystem,while[ 50 ]introducestheopportunisticrelayselection. 14 PAGE 15 38 67 ].However,[ 38 ]neglectsthepathlosseectwhichiscloselyrelatedtotherelaylocationsuchthatthelocationoptimizationproblemiserroneouslyformulatedasanenergyallocationproblem.In[ 67 ],itisshownthatgivenuniformenergyallocationthereisanoptimumrelaylocationwhichprovidestheoptimumperformance.However,theyonlyconsidertheoptimumlocationoftherelayswithoutappropriateenergyallocation.WithDFandAFprotocols,thejointenergyandlocationoptimizationforthecooperativenetworkisintroducedin[ 15 19 { 21 ]. Inthisresearch,wedevelopcooperativenetworkswithanarbitrarynumberofrelaysbyemployingDF,AF,andDSTCprotocols.Equallyattractiveisthatouranalysisistailoredforrelaysystemswithdierentialmodulation,whichisknowntoreducethereceivercomplexitybybypassingchannelestimation.NoticethattheDFandAFprotocolsgeneralizethestandarddierentialmodulationtoadistributedscenariowithanarbitrarynumberofrelays.TheDSTCprotocolreliesonanincreasedlevelofusercooperationviathedistributedcounterpartofthedierentialspacetimecodes[ 31 32 ];thatis,foreachdatablock,aspacetimecodewordencodedacrossdistributedrelaysistransmittedoveracommonrelaydestinationchannel.Usingtheserelayingprotocols,wethenderivetheerrorperformanceandtheoptimumresourceallocationofcooperativenetworks.Dierentfromexistingworksonresourceoptimization,wetackletheproblemfromtwoangles:i)Optimizingthepowerallocationacrossrelayandsourcenodesforanygivensourcerelaydestinationdistances;andii)Optimizingtherelaylocationforanypowerdistributionandsourcedestinationdistance.Tothebestofourknowledge,weareamongthersttoformulatethe2dimensionaloptimizationproblem.Inaddition,wearealsothersttoconsiderthejointpowerandrelaylocationoptimizationforcooperativenetworks. Tofacilitatetheresourceoptimization,wedevelopeanalyticalexpressionsoftheerrorperformanceforvariousrelayprotocolswitharbitrarynumberofrelays.Werstderivean 15 PAGE 16 16 PAGE 17 Thesystemmodelisbasedonanetworksetupwithonesourcenodes,LrelaynodesfrkgLk=1andonedestinationnoded,asdepictedinFig. 21 .Eachnodeisequippedwithaswitchthatcontrolsitstransmit/receivemodetoenablehalfduplexcommunications.Multiplexingamongthenetworknodescanbeachievedviafrequencydivision,timedivisionorcodedivisiontechniques.Fornotationalconvenience,wewillconsiderthetimedivisionmultiplexing(TDM).However,thepresentedanalysisandresultsarereadilyapplicabletofrequencydivisionmultiplexing(FDM)andcodedivisionmultiplexing(CDM).Forthisstudy,werstconsidertwoconventionalrelayingprotocols,i.e.,decodeandforward(DF)andamplifyandforward(AF),andthenwedevelopthedistributedspacetimecoding(DSTC)protocol. Setupofthecooperativenetwork. 17 PAGE 18 Inordertobypasschannelestimationandtocopewithtimevariationofthewirelesschannel,dierentialmodulationisemployedatthesourcenode.Specically,withthenthphaseshiftkeying(PSK)symbolbeingassn=ej2cn=M,cn2f0;1;:::;M1g,thecorrespondingtransmittedsignalfromthesourceis: InphaseI,forbothDFandAFprotocols,theencodedsignalisbroadcastviaacommonchannel.Thereceivedsignalsatthekthrelayandthedestinationaregivenby whereEsistheenergypersymbolatthesource,andwedenotethefadingcoecientsofchannelssrkandsdduringthenthsymboldurationashrk;snCN(0;2rk;s)andhd;snCN(0;2d;s),andthecorrespondingnoisecomponentsaszrk;snCN(0;Nrk;s)andzd;snCN(0;Nd;s),respectively.Here,CN(;2)representsthecomplexGaussiandistributionwithmeanandvariance2. 2{2 ),usingthedecisionrulethatwewillpresentinthenextsubsection.TheremodulationstepiscarriedoutasinEq.( 2{1 ),butwithsnreplacedbyitsestimateandxsnreplacedby^srknandxrkn.Then,thereceivedsignalatthedestinationcorrespondingtoeachrelaynodeisgivenby 18 PAGE 19 whereErkistheenergypersymbolandxrkndenotesthenthtransmittedsymbolfromthekthrelay.Thefadingcoecientsoftherkdchannelsandthenoisecomponentatthedestinationarehd;rknCN(0;2d;rk)andzd;rknCN(0;Nd;rk),respectively.Ateachoftherelays,xrkncanberepresentedas whereArkistheamplicationfactor.Tomaintainaconstantaveragepowerattherelayoutput,theamplicationfactorisgivenby ThisArkisreasonableforbothdierentialandnoncoherentmodulations,sincewecanestimatethevalueof2rk;sbyaveragingthereceivedsignalswithoutknowingtheinstantaneousCSI[ 12 71 ]. 31 32 ].Forsimplicity,wewillusethediagonaldesignwiththecyclicconstructionin[ 31 ].Noticethateachdiagonalelementofthecodewordcorrespondstoastandarddierentialphaseshiftkeying(DPSK)signaling,whereitsmodulationsize 19 PAGE 20 DenotethenthdierentiallyencodedsignalblockfromthesourceasXsn:=Xsn1V(Qn)withXs0=IL,whereV(Qn)isanLLdiagonalunitarymatrix,Qn2f0;1;:::;M1gwithM=2L,andILisanLLidentitymatrix,whererepresentsthedatarateoftheoriginalinformationwhichwesetto1.ThematrixV(Qn)hastheformV(Qn)=VQn1with(see[ 31 ]) whereul2f0;:::;M1g;l=f1;:::;Lg.Then,thenthreceivedsignalblockattherelaysisgivenby whereEsistheenergypersymbolatthesource,Hr;sn:=diagfhr1;sn;hr2;sn;:::;hrL;sngisthechannelmatrixbetweenthesourceandrelays,andZrn:=diagfzr1;sn;zr2;sn;:::;zrL;sngisthenoisematrixattherelays.Weusediagfa1;a2;:::;aLgasadiagonalmatrixwith[a1;a2;:::;aL]onitsdiagonal.LetusdenotethenthtransmittedsignalblockfromtherelaysasXrn,thenthecorrespondingreceivedsignalblockatthedestinationisgivenby 20 PAGE 21 Throughoutthisdissertation,allfadingcoecientsareassumedtobeindependent.Withoutlossofgenerality,wealsoassumethatallnoisecomponentsareindependentandidenticallydistributed(i.i.d)withNi;j=N0;i;j2fs;rk;dg.Accordingly,wecanndthereceivedinstantaneoussignaltonoiseratio(SNR)betweenthetransmitterjandthereceiveriasi;j=jhi;jnj2Ej 2{2 )and( 2{3 )bydroppingthesuperscripts.UsingthedierentialencodinginEq.( 2{1 ),thereceivedsignalcanbereexpressedas: 21 PAGE 22 wherei;j2fs;rk;dg,smn=ej2m=Mandm2f0;1;:::;M1g.WeuseE[]forexpectation,()forconjugate,and PAGE 23 Esp Thereceivedsignalatthedestinationcorrespondingtothesourcecanberepresentedasyd;sn=yd;sn1sn+(~zd;sn)0withyd;snCN(yd;sn1sn;2N0).Asaresult,weobtaintheLLFcorrespondingtotheLtransmittedsignalsfromtherelaysandthetransmittedsignalfromthesource,giventhatxmnistransmittedbythesource: wheresmn=ej2m=Mandm2f0;1;:::;M1g.Noticethat,althoughtheLLF,ld;rkm(yn),inEqs.( 2{12 )and( 2{15 )hasthesameform,theonein( 2{15 )isobtainedfromadierenttransmittedsignal.FortheDFprotocol,theLLFisobtainedfromthegivenrelaytransmittedsignalwhichisademodulatedsignalfromthesourcetransmittedsignal.However,theLLFfortheAFprotocolisobtainedfromthesourcetransmittedsignalwhichissimplyampliedandforwardedattherelayswithoutanydemodulation.Atthedestinationnode,these(L+1)signalscanbecombinedtoestimatethetransmittedsignalfromthesource.Usingthemultichannelcommunicationresultsin[ 44 ,Chapter.12]and 23 PAGE 24 30 72 ]): ^sn=ej2^m=M:^m=argmaxm2f0;1;:::;M1g"wd;sld;sm(yn)+LXk=1wd;rkld;rkm(yn)#; wherewd;sandwd;rkarecombiningweightswhicharegivenby1=N0and2=2hk;eff,respectively,anditisassumedthatthevariancesofchannelsareavailableatthedestinationnode.Interestingly,thisdecisionruleisthesameastheDFprotocolinEq.( 2{13 )exceptfortheweighingterms. 31 ],givenXsn=Xmn,is ^Qn=argmaxm2f0;1;:::;M1gkYd;rn1+Yd;rnV(m)nHk; wherekkrepresentstheFrobeniusnorm.ThisdecisionruleisthegeneralstructureforDUSTC.Dependingontherelayingprotocol,theFrobeniusnormpartcanhavedierentvalues. IntheDFSTCprotocol,thereceivedsignalattherelays,Yr;sn,isdecoded.SinceeachdiagonalentryofthecodewordXsnisaDPSKsignalandthekthrelaydemodulatesandremodulatesindependentlythecorrespondingkthentryofYr;sn,wecanreencodeXrnusingstandarddierentialdemodulation.ThereceivedsignalblockforthegivenrelaytransmittedsignalXrn=Xm0nis whereZ0dn=ZdnZdn1V(m0)n.SinceV(m0)nisaunitarymatrix,Z0dnhastwicethevarianceofZdn.Then,givenXrn=Xm0n,wecanapplytheMLdecisionruleinEq.( 2{17 ). FortheAFSTCprotocol,eachentryofthereceivedsignalfromthesource,Yr;sn,isampliedandforwardedtothedestination.Therefore,theampliedsignalblockatthe 24 PAGE 25 whereA:=diagfAr1;Ar2;:::;ArLgistheamplicationmatrix,andArkistheamplicationfactorwhichwedenedinEq.( 2{6 ).Then,usingthedierentialmodulation,thereceivedsignalblockatthedestinationcanberepresentedas where~Hn=p 2{17 )givenXsn=Xmn.NoticethattheMLdecisionruleofbothrelayingprotocolshasthesameform.However,thevalueoftheFrobeniusnormisdierentdependingontheprotocols. 25 PAGE 26 Tofacilitateourresourceoptimization,wewillderivetheanalyticalexpressionsoftheerrorperformanceforthecooperativesystemsdescribedintheprecedingsection.Symbolerrorprobabilityofcooperativenetworkswithrelaytransmissionshasbeenderivedin[ 2 7 45 ]forcoherentdetection,andin[ 29 71 72 ]foradierentialschemewithasinglerelay,bothemployingtheAFprotocol.TheperformanceoftraditionalSTCsystemsiswellanalyzedin[ 59 { 61 68 ].ThedierentialanddistributedSTCsystemsareconsideredin[ 31 32 58 ]and[ 66 ],respectively.Alltheseexistingworkconsideredtheperformanceofcooperativesystemstosomeextent.However,theerrorperformanceofgeneralcaseswithdierentialmodulationhasnotbeenthoroughlyinvestigated.WewillconsidertheerrorrateforageneralLrelaysetupinthedistributedscenario.First,wederiveanupperboundofthesymbolerrorrate(SER)fortheDFprotocol.Then,underhighsignaltonoiseratio(SNR)approximation,anapproximatedSERfortheAFprotocolisderived.Finally,wederiveupperboundsofthecodeworderrorrate(CER)fortheDFSTCandAFSTCprotocols. 52 ,Chapter8.2] 1p M; whereM(x)=1=(1x),8x>0,andrepresentstheaverageSNR.Inparticular,forM=2(DBPSK),Eq.( 3{1 )canbesimpliedas 2(1+rk;s): Atthedestination,thesignalsfromtheLrelaysarecombinedtomakeadecision.Conditionedonthatthesymbolsniscorrectlydemodulatedandremodulatedatallrelay 26 PAGE 27 44 ,AppendixC]as: M(M1)sin(=M) where=d;rk=(1+d;rk).ForDBPSK,Eq.( 3{3 )canbesimpliedas 2"1L1Xk=02kk12 UsingtheunconditionalSERPDFe;rkattherelaysandtheconditionalSERPDFe;datthedestination,weformulateanupperboundontheoverallaverageerrorperformance,namelytheunconditionalSERPDFeatthedestination,asfollows: 3{1 )and( 3{3 ),respectively,anupperboundonPDFecanbefoundas: 3{5 )providesanupperboundontheexactSERPDFe,letusstartwiththeprobabilityofcorrectdetectionPDFc=1PDFe.Countingtheeventsthatleadtothecorrectdetection,PDFccanbeobtainedas where^srknand^sdnarethesymbolestimatesformedattherelayrkandthedestinationd,respectively.TherstsummandinEq.( 3{6 )turnsouttobeQLk=1(1PDFe;rk)(1PDFe;d),whichleadstotheupperboundinEq.( 3{5 ). 27 PAGE 28 3{6 ),whichcorrespondstothegapbetweenthetrueSERanditsupperboundPDFe:=PDFePDFeanddeterminesthetightnessoftheerrorboundinProposition1.ForDBPSKwithasinglerelay(L=1,M=2),thisgapcanbeeasilyobtainedas: PDFe=PDFe;rPDFe;d: ForpracticalPDFe;randPDFe;dvalues(e.g.,<103),PDFeisnegligiblecomparedwithPDFe=PDFe;r+PDFe;d2PDFe;rPDFe;d.However,forL2,allpossibleerrorshavetobeconsideredforboththesrandrdlinks,whichrendersPDFeanalyticallyuntractable.Butintuitively,asLincreases,PDFealsoincreasessincethereisanincreasingchancethatdetectionerrorsattherelaynodesdonotleadtoadetectionerroratthedestinationnode.Inadditiontothiseect,theperformanceboundPDFeandthegapPDFealsodependsonthequalityofthesrandrdlinks. TheseeectsareevidentfromthesimulatedexamplesinFigs. 31 and 32 ,wherearelaynetworkwithL=2relaynodesusingDBPSKsignalingisconsideredatvariousr;sandd;rlevels.Inthesesimulatedexamples,thechannelsbetweenthesourceandallrelayshaveidenticalpowers2rk;s=2r;s,8k,whichimpliesthatrk;s=r;s,8k.Accordingly,wehavePDFe;rk=PDFe;r,8k,andPDFePDFe=1(1PDFe;r)L(1PDFe;d)fromProposition1.Likewise,theSNRsbetweenalltherelaynodesandthedestinationhavethesamevalue,d;rk=d;r;8k.InbothFigs. 31 and 32 ,theboundPecloselycapturesthedependencyofthesystemSERontheSNRlevelsr;sandd;r.Specically,wehavethefollowingobservations: 31 revealsthat,atanygivenvalueofr;s,thesystemSERexhibitsanerroroorasd;rincreases.Intuitively,thiserroroorcomesfromthedetectionerrorattherelays,whichheavilyreliesonthesrlinkqualityr;sandcanonlybereducedbyimposingsucientlyhighr;s. 28 PAGE 29 32 showsthat,atmediumtohighd;rlevels,theoverallSERcanalwaysbereducedbyincreasingtheSNRofthesrlinkr;sanddoesnotexhibitanyerroroor. 29 PAGE 30 33 .Noticethatthesurfaceattensalongthed;raxis,butkeepsdescendingalongther;saxis.TheseobservationssuggestthattheoverallerrorperformanceoftheDFbasedcooperativesystemdependsmoreonthesrlinkthantherdlink.Suchunbalancedeectsoftherelaylinksconrmthatappropriateresourceallocationiscriticalinachievingtheoptimumerrorperformance. 30 ],weprovideamoregeneralyet,simpleexpressionfortheaverageSER.TheSERisdevelopedintwocases,thesystemwithnodirectlinkandtheonewithadirecttransmission.Thetwocasesaredenotedas\ND"and\DL,"respectively. 30 PAGE 31 ThisallowsustodeducetheSERusingamultichannelmodel.WewillrstinvestigatetheSERoftherelaysystemswithnodirectlink,theresultcanbeeasilyextendedtothecasewithadirectlink. InaLrelaysystemwithnodirectlinkbetweenthesourceandthedestination,thereceivedSNRis: 3{9 ),theaverageSERPAFe;NDcanbefoundas: rk;s+1 d;rkln(d;rk); 22L1L1Xn=0n+L1L1L1nXk=02L1k: 51 ].Forthebinarycase,theSERconditionedonNDisgivenby[ 44 ,Chap.12] 22L1eNDL1Xn=0cnnND; whereNDisdenedinEq.( 3{9 ),and 31 PAGE 32 3{12 ),nNDcanbeexpressedasfollowsbyexpandingND: {z }L1; wheremL=nPL1i=1mi,thenEq.( 3{12 )becomes: where 22L1L1Xn=0cnnXm1=0nm1nm1Xm2=0nm1m2nPL2i=1miXmL1=0nPL2i=1mimL1 {z }L1: TheaverageSERcanbeobtainedbyaveragingtheconditionalSERwithrespecttotheprobabilitydensityfunction(PDF)ofeq;rk,namelyp(eq;rk),whichisgivenas PAFe;ND=Z10Z10Z10 {z }LPejNDLYk=1p(eq;rk)deq;r1deq;r2deq;rL; wherethePDFofeq;rkisderivedin[ 72 ],andwhichisgivenas (3{18) +2 rk;sd;rks Bysubstituting( 3{15 )into( 3{17 ),wecanget: Pe;ND=Xn;mLYk=1Z10eeq;rkmkeq;rkP(eq;rk)deq;rk: 32 PAGE 33 3{19 ),denotedasIn;m;k, 2eq;rkeeq;rkK1(p where=2p 3{20 )canbecomputedbyusingtheintegrationpropertyofBesselfunction,[ 25 ,Eqs.6.631.3] 21 211++ 8W whereWm;n()istheWhittakerfunction (a),U(a;2;1=x)x 1 ,Eqs.13.5.9and13.5.7]),and=1+1=rk;s1athighSNR,wecansimplifyEq.( 3{20 )as rk;s+1 d;rklnd;rk: PluggingtheaboveresultbackintoPAFe;ND,weget PAFe;ND=Xn;mLYk=1mk!1 rk;s+1 d;rklnd;rk: Now,wemovetothecoecientpartofEq.( 3{23 ),whichisgivenby 22L1L1Xn=0cnnXm1=0nm1nm1Xm2=0nm1m2nPL2i=1miXmL1=0nPL2i=1mimL1 {z }L1LYk=1mk!: 33 PAGE 34 {z }L1LYk=1mk!=n!; thecoecientcanbesimpliedas 22L1L1Xn=0cnnXm1=0nm1Xm2=0nPL2i=1miXmL1=0 {z }L1n!=1 22L1L1Xn=0L1nXk=02L1knXm1=0nm1Xm2=0nPL2i=1miXmL1=01 {z }L1: DenotethecoecientinEq.( 3{26 )asC(L).Byusingmathematicalinduction,wecanprovethat: {z }L1=n+L1L1: Weknowthat,forL=1,(1)=1=n+1111.SupposeforanarbitraryL=l1,(l)=n+l1l1,then,wehave(l+1)=Pnm=0nm+l1l1whenL=l+1.Wecanalsoshowthat: (n1)!(l1)!...=(n+l1)! (n1)!(l1)!++(l1)! 0!(l1)!=nXm=0nm+l1l1=(l+1): Therefore,equality( 3{27 )holdsforanyL1.UsingtheresultinEq.( 3{27 ),wecansimplifyEq.( 3{26 )toEq.( 3{11 ).Finally,theaverageSERyieldsEq.( 3{10 ). 34 PAGE 35 Followingthesamestepsasthenodirectlinkcase,theaverageSERcanbesimilarlyevaluatedbycalculatingtheintegralforthedirectlinkasinEq.( 3{20 ): d;s(m+1)!: Combiningtheresultabovewith( 3{22 ),wecanevaluatetheaverageSERforasystemwithadirectlinkas: PAFe;DLC(L+1)1 d;sLYk=11 rk;s+1 d;rkln(d;rk); whereC()isthethesamefunctionasin( 3{11 ),whichdependsonthenumberofrelays.Forexample,forL=1;2;3;and4,wehaveC(1)=1=2;C(2)=3=4;C(3)=5=4;C(4)=35=16.ItisworthstressingthattheSERexpressionsinEqs.( 3{10 )and( 3{31 )coincidewiththeaverageSERofthecoherentsystemin[ 45 ]exceptforthelogterm,whichleadstothecodinggainlosscomparedwiththecoherentsystem. Whenrk;s=d;rk=d;s=~;8k,andas~!1,Eqs.( 3{10 )and( 3{31 )giveriseto PAFe;ND/C0~LandPAFe;DL/C00~(L+1); whereC0andC00arebothconstants.FromEq.( 3{32 ),itisclearthatthediversitygaincanbeobtainedusingadierentialschemewithAFprotocolforsucientlylargeSNR. InFigs. 34 and 35 ,wecomparetheapproximatedandsimulatedSERwhenL=1;2;and3forthesystemswithandwithoutadirectlink.Theguresconrmthatthediversitybenetincreasesindirectproportiontothenumberofrelays,anddemonstratethattheapproximationsareverytightcomparedwiththesimulations,especiallywhentheSNRishighandforsmallL.FromFig. 35 ,itiscertainthatadirect 35 PAGE 36 34 .AsLincreases,thequalityofapproximationdecreases,sincemoreapproximationerrorsareaccumulatedasthenumberofrelaysincreases. InFig. 36 ,wecomparetheSERofthesystemswithcoherentmodulationanddierentialmodulationwhenadirectlinkispresent.Weusetheresultin[ 45 ]forthecoherentsystem.TheonlydierencebetweenthecoherentsystemanddierentialsystemisthelogterminEqs.( 3{10 )and( 3{31 ).Thegureshowsthattherearecodinggaindierencesbetweenthesystems,andthesedierencesincreaseastheSNRincreases,i.e.,therequiredSERdecreases.Theseincreasingdierencesareduetothelogterminthedierentialsystem.Forexample,approximately2.2dB,2.8dB,and3.2dBmoreSNRarerequiredindierentialsystemtoachieve103,104,and105ofSER,respectively,comparedwiththecoherentsystem. 36 PAGE 38 LetusrstconsidertheDFSTCprotocol.Duetoindependentdemodulationandremodulationofthecorrespondingdiagonalentryateachrelay,thesrklinkSER,PDFe;rk,canbeobtainedusingEq.( 3{1 ).Sinceonesymbolerrorateachrelaycaninducethecodeworderror,theCERPDFe;ratsrklinksisgivenby Atthedestination,thereceivedsignalsfromtheLrelaysreconstructthetransmittedSTCsignal.ConditionedonthatthesourcetransmittedsignalblockVniscorrectlydecodedattherelays,andbydroppingthesuperscriptsfornotationalbrevity,theCERatrkdlinksisgivenby where AthighSNR,wecanmakethefollowingassumption wherewhereEr:=diagfEr1;Er2;:::;ErLgistheenergypersymbolmatrixattherelays,Hd;rn:=diagfhd;r1n;hd;r2n;:::;hd;rLngisthechannelmatrixbetweentherelaysanddestination,andXrnisthenthtransmittedsignalblockfromtherelays.Then,Eq.( 3{35 )canbe 38 PAGE 39 whereDFe=E1=2rXn1(VnV0n)(VnV0n)XHn1E1=2r.SinceDFeisHermitian,wecanexpressEq.( 3{37 )as whereUisaunitarymatrixandDDFeisdiagfDFe;1;DFe;2;:::;DFe;Lg.EachdiagonalentryDFe;k;k=1;2;:::;L;representstheeigenvalueofDFe.Therefore,wecanobtaintheCERbyaveragingEq.( 3{34 )withrespecttothechannelHd;r.Forsimplicity,byassumingthatthefadingcoecienthasunitvariance,theconditionalCERPDFSTCe;datthedestinationisgivenby 8N0DFe;k1; andunderhighSNRcondition,thisequationcanbefurthersimpliedas Finally,usingEqs.( 3{33 )and( 3{40 ),wecanformulatetheunconditionalCERforDFSTCprotocolas: Itisworthmentioningthatifthereisnoerrorbetweenthesourceandrelays,theaboveequationboilsdowntotheCERofmultiinputsingleoutput(MISO)systememployingtheDUSTC.However,asLincreases,theCERofsrklinksbecomesworsebecauseoftheincreasingmodulationsizeateachdiagonalentry.ThiswillinducetheperformancedegradationoftheDFbasedsystem.Toprovidebetterperformanceandpertainthediversitygain,srklinkshavetomaintainlowerCER. 39 PAGE 40 2{20 )isdiagf2h1;eff;2h2;eff;:::;2hL;effg,wherethekthdiagonalentryofthecovariancematrixisgivenbyEq.( 2{14 ).Tonormalizetheaggregatenoisevariance,letusdenethematrixG:=diagfg1;g2;:::;gLgwithgk=(ErkA2rk2d;rk+1)1=2.Then,bymultiplyingGwiththereceivedsignalblockatthedestination,wecanrewriteEq.( 2{20 )as orequivalently, ~Yd;rn=Yd;rn1~V(m0)n+~Zn; where~Y=YG;~V=VG,and~Z=~Z0dG.Then,theCERfortheAFSTCprotocolcanbeachievedusingEq.( 3{43 ).FollowingthesamestepsasEq.( 3{34 )to( 3{37 ),theCERcanbeobtainedas: where AthighSNR,thecodedistancecanbeapproximatedas whereAFe=EsE1=2rXn1(VnV0n)(AG)(AG)H(VnV0n)HXHn1E1=2r.SimilartotheDFSTCprotocol,wecanexpressAFeas AFe=U0HDAFeU0; 40 PAGE 41 3{44 )withrespecttothecombinedchannelHd;rnHr;sn.Letusdeneh:=hd;rkhrk;s,thenthePDFof=jhjisgivenby[ 43 ] 2d;rk2rk;sK02s whereK0()isthezerothordermodiedBesselfunctionofthesecondkind.Byassumingthateachfadingcoecienthasunitvariance,andusingthepropertiesofBesselfunctionandconuenthypergeometricfunction(see[ 25 ,Eq.6.631.3]and[ 1 ,Eqs.13.5.9])athighSNR,theCERcanbesimpliedas NoticethattheCERoftheAFSTCprotocolhasalmostthesameformasitscounterpartoftheDFSTCprotocolatrkdlinksexceptforthelogtermwhichreectstheeectoftheamplicationandaggregatenoiseandthisleadstocodinggainloss.Eq.( 3{49 )conrmsthatAFSTCprotocolprovidesfulldiversitygain. InFigs. 37 and 38 ,weplottheboundsandsimulatedCERsforthesystemswithDFSTCandAFSTC,respectively,whenL=1;2;and3.WhenL=1,theSTCbasedcooperativesystemisreducedtotheconventionalcooperativenetwork,thuswecanusetheSERformulasderivedin[ 20 21 ]astheCERboundoftheSTCbasedsystem.Fig. 37 showsthattheboundsaretighttothesimulations,especiallywhenLissmall.NoticethatthecardinalityofthesignalblockattherelaysequalstoMLbecauseoftheindependentdecodingateachrelay.However,theboundatrkdlinksonlyconsidersMsignals.Thus,asLincreases,thegapbetweentheboundandsimulationincreases.Fig. 37 alsoshowsthatnodiversitygainisobtainedbyDFSTCprotocol,sincetheCERatsrklinksincreasesindirectproportiontothenumberofrelays,whichinducesthedegradationoftheoverallerrorperformanceofDFbasedsystem.Theseresultsconrmouranalysisin 41 PAGE 43 38 ,thoughtheboundsforAFSTCprotocolareinaccuratewhenSNRislowbecauseofthelogtermintheanalyticalexpression,theboundsandsimulationshavetightvaluesathighSNR.Furthermore,itisclearthatAFSTCprotocolprovidesfulldiversitygain. Aswementionedabove,thelinkqualitybetweenthesourceandrelaysiscriticaltotheperformanceoftheDFbasedcooperativesystem.Tocapturetheeectofunbalancedlinkquality,weconsiderdierentaverageSNRsatsrkandrkdlinksforbothDFSTCandAFSTCprotocolsinFigs. 39 and 310 ,respectively.Weassumethatrk;s=r;s,andd;rk=d;r;8k,andconsideri)equalSNRforbothsrkandrkdlinks,ii)higherSNRisassignedatsrk,andiii)higherSNRisassignedatrkdlinkswithL=1;2and3.AsshowninFig. 39 ,whenweassignhighSNRsatsrklinks,theoverallCERdecreasesandthediversitygainbeginstoappear.Fortheextremecase,i.e,inniteSNRisassignedatsrklinks,theDFSTCcooperativesystembehaveslikeaMISOsystem.ThegurealsoshowsthatthecodinggainisachievedbyassigninghigherSNRatrkdlinks.However,thediversitygainisdominantcomparedwiththecodinggainespecially 43 PAGE 44 310 showsthatthecodinggainisobtainedbyassigninghigherSNRatbothlinks.IncreasingSNRatsrkandrkdlinksleadstodecreasingtheeectofaggregatenoiseandincreasingSNRatthedestination,respectively.Bothscenariosinducetheenhancementofcodinggain.NoticethattheeectofSNRatrkdlinksprovidesmorecodinggainthanrkdlinks,whichimpliesthatincreasingaveragepowerattherelayoutputismorecrucialthanreducingtheeectofnoiseatthesrklinks. 44 PAGE 45 Inthischapter,wewillinvestigatetheeectsofresourceallocationontheerrorperformance.Wewillshowthatanoptimumallocationofthelimitedresourcesispossible,anditachievestheoptimumsystemerrorperformance.Theresourceallocationwhichminimizestheaverageerrorratewillbeaddressedfromthreeperspectives: 1) Giventherelativedistancesamongthesource,relayanddestinationnodes,thepathlossexponentofthewirelesschannel,andthetotalavailableenergypersymbol,determinetheoptimumenergyallocationamongthesourceandrelaynodes. 2) Giventhesourcedestinationdistance,thepathlossexponentofthewirelesschannel,andtheenergypersymbolatthesourceandrelaynodes,determinetheoptimumlocationoftherelaynodes. 3) Giventhesourcedestinationdistanceandthetotalavailableenergypersymbol,determinethejointenergyandlocationoptimization. Foranalyticaltractability,weconsideranidealizedLrelaysystemwithallrelaynodeslocatedatthesamedistancefromthesourceanddestinationnodes;thatis,Ds;rk=Ds;randDrk;d=Dr;d,8k.ItisthenreasonabletoassignequalenergiesatallrelaynodesErk=Er,8k.Tocarryouttheoptimizationintheensuingsubsections,wewillalsomakeuseoftherelationshipbetweentheaveragepowerofchannelfadingcoecient2hi;jandtheinternodedistanceDj;iasfollows: whereisthepathlossexponentofthewirelesschannelandCisaconstantwhichwehenceforthsetto1withoutlossofgenerality.FortheconventionalDFandAFprotocols,wewillpresenttheanalyticalresultsofoptimizationsaswellassimulatedexamples.FortheDSTCprotocol,duetotheanalyticalintractability,theoptimizationresultswillbeshownbysimulationsandcomparedwithconventionalsystems. 45 PAGE 46 (b) 41 .Oneistheellipsecaseandtheotheroneisthelinecase.Fortheellipsecase,Ds;r+Dr;d=DDs;d.Thelinecasecanberegardedasaspecialcaseoftheellipsecase,i.e.,D=Ds;d.BychangingthevalueofD,wecansolvetheoptimizationproblematanypointona2Dplane.Therefore,optimumresourceallocationfortheseidealizedtopologiescanprovideusefulinsightsforunderstandingtheeectofresourceallocationinrelaynetworks. 3{5 ),( 3{41 ),and( 3{49 )).However,oursimulationswillshowthattheerrorrateisgenerallyconvex,whichensuresconvergenceoftheoptimization.WiththeAFprotocol,wecanprovetheconvexityanalyticallyandconrmbysimulations.TheproofofconvexityfortheAFprotocolisgivenasbelow,whichguaranteestheconvergenceoftheerrorperformanceasafunctionofenergyandlocation. 46 PAGE 47 3{10 )and( 3{31 )areconvexfunctionsoftheenergyandlocation,respectively. Proof. PAFe;ND=C(L)"1 (Lr)2r;s+1 PAFe;DL=C(L+1)1 (Lr)2r;s+1 whicharefunctionsofthesinglevariabler2(0;=L). ThesecondderivativesofPAFe;NDandPAFe;DLaregivenasfollows: ForL2, (Lr)3[f(r)]L2[g(r)]2f(r)g+C00(L)1 forL=1, (r)32f(r)+2sg(r)+2sh(s); whereC0(L)=LC(L)andC00(L)=LC(L+1),and (Lr)2r;s+1 47 PAGE 48 2r;s1 (Lr)2+1 (Lr)3+1 [g(r)]2+2L1 [g(r)]+(L1): UnderhighSNRapproximation,r2d;r1,wehavef(r)>0andh(r)>0.Wenoticethat(r)isaquadraticfunctionof1 [g(r)],anditsquadraticdiscriminantis: =(2L)24(2L)(L1)=4L(2L)0;forL2: Thus,wehave(r)0whenL2.Therefore, WhenL=1,theconvexityofthesystemwithnodirectlinkisreadilyobtainedfromEq.( 4{6 ).Forthesystemwithadirectlink,aftersomemanipulation,Eq.( 4{7 )canbereexpressedas: (r)31 (r)32 Usingtheinequality,(r=s)2(r=s)+13=4,wehavethelowerboundof@2PAFe;DL=@2r: (r)31 2s OntheconditionofhighSNR,ln(r2r)>2,wehave@2PAFe;DL=@2r>0forL=1. Finally,bothPAFe;NDandPAFe;DLareconvexfunctionsofr,andaccordinglysforanyL1. 48 PAGE 49 PAFe;ND=C(L)(DDr;d) whichisafunctionofasinglevariableDr;d2(0;D).ThesecondderivativeofPAFe;NDis where s(DDr;d)1+D1r;d UnderhighSNRapproximation,i.e.,r2rd1,and>1,wehaveu(Dr;d)>0,v(Dr;d)>0andw(Dr;d)>0.Thus: Similarly,forthesystemwithadirectlink,wehave PAFe;DL=C(L+1)1 ThisequationisthesameasEq.( 4{16 )exceptfortheconstantterm,C(L+1)=(Lr),thereforeitsconvexitycanbereadilyprovedusingthesamestepsintheabove. 49 PAGE 50 3{5 )forDFandEq.( 3{10 )orEq.( 3{31 )forAFwhilesatisfying: BydeningthetotalSNR,:=E=N0,thetransmitSNRatthesourcenodes:=Es=N0andthetransmitSNRattherelaynodesr:=Er=N0,theenergyconstraintcanbereexpressedastheSNRconstraint: UsingEq.( 4{1 ),theaveragereceivedSNRsattherelayanddestinationnodescanbeexpressedintermsofthetransmitSNRsas: r;s=s2hr;s=sDs;randd;r=r2hd;r=rDr;d: Asaresult,thetotalenergyconstraint,Eq.( 4{24 ),canbefurtherrewrittenas LetusconsidertheDFprotocolrst.Togainsomeinsights,westartfromasinglerelaysetupandestablishthefollowingresult: 4{23 ),the PAGE 51 4Ds;rDr;d(Ds;rDr;d)22Dr;d+3 2(Ds;rDr;d); Proof. whereistheLagrangemultiplier.SolvingEqs.( 4{28 )and( 4{29 ),weobtain (1+2d;r)(1+d;r) (1+2r;s)(1+r;s)=Dr;d whichleadstothefollowingrelationshipbetweend;randr;s: d;r=3D=2s;r+q SubstitutingEq.( 4{32 )intoEq.( 4{30 ),wendosasinEq.( 4{27 ). AlthoughEq.( 4{27 )isaccurateforallEandN0valuesandforallsrandrddistances,itscomplexformdoesnotprovidemuchintuition.Fortunately,forseveralspecialcases,itcanbesimpliedwithoutmuchlossofaccuracy.Nextwewillconsidersomeofsuchcases. 4{31 )simpliesto22d;r+3d;r+1 22r;s+3r;s+1=1: PAGE 52 4{31 )as22d;r+3d;r+1 22r;s+3r;s+1=Dr;d 2(Ds;rDr;d): Thissolutioncanbefurthersimpliedbyneglectingitsconstanttermsto Interestingly,thissolutioncoincideswiththeoptimumpowerallocationobtainedbyminimizingtheoutageprobability[ 27 ,(8)]withasinglerelaytransmission.FromEq.( 4{34 ),itreadilyfollowsthattheenergyallocationratiobetweenthesourceandtherelaynodesis Eq.( 4{35 )revealsexplicitlythattheoptimumenergyallocationheavilyhingesupontheinternodedistances.Inaddition,thepathlossexponentofthewirelesschannel,,alsoaectstheoptimalenergyallocation.Interestingly,theEos=EorratioislinearinDs;r=Dr;donlywhen=2.Theoptimumenergyallocationfavorsthelinkwithalargernodeseparationif>2andviceversa,aswewillshownextwithanexample. Fig. 42 depictsthetransmitSNRsobtainedfromtheoptimumenergyallocation.Aonedimensionalsetupisconsidered;thatis,Ds;r+Dr;d=Ds;d=D.Thesystemparametersare:=10dB,L=1,Ds;d=D=1and=(1;2;3;4).InFig. 42 ,the 52 PAGE 53 4{27 )andtheapproximationsinEqs.( 4{33 )and( 4{34 ).Theseresultsarenearlyidenticalinallcaseswithvariousvalues.Bycloselyinspectingthegure,wendthattheapproximationinEq.( 4{33 )providesmoreaccuratecurvesthantheoneinEq.( 4{34 ),asexpected.AlthoughtheapproximateexpressionsinEqs.( 4{33 )and( 4{34 )areobtainedunderhighSNRassumption,theyremainveryaccurateevenatmediumSNRof10dB. FromFig. 42 ,wealsoobservethat,forallvalues,thesourcenodeenergyEsincreasesastherelaymovestowardsthedestinationnode.With=2,EsincreaseslinearlywithDs;r.Athighervaluesofthepathlossexponent,>2,weobservethat Inotherwords,theoptimumenergyallocationfavorsthelinkwithlargerinternodedistance.Whenthepathlossexponent=1,Fig. 42 showstheoppositeofEq.( 4{36 ). 53 PAGE 54 4{27 )canbeobtainedandaveryaccurateandinsightfulapproximationisavailableunderhighSNRassumption.ForL2,however,therstorderconditionsobtainedbydierentiatingtheSERboundPDFehavecomplicatedforms,whichrenderanalyticalsolutionsimpossible.Fortunately,theSERboundPDFeasinProposition1stillallowsforanumericalsearch,asopposedtoMonteCarlosimulationsneededotherwise. Forexample,withDBPSKandL=2,wehave 23 4d;r 4d;r and,accordingly,theSERboundisgivenby PDFe=12r;s(2+d;r)(1+2d;r)2 ByusingtherstorderconditionsinEq.( 4{28 )andthehighSNRapproximation,theoptimumr;sandd;rshouldsatisfy 4(1+d;r)(2+d;r)(1+2d;r) 3r;s(1+r;s)=Dr;d Althoughananalyticalsolutionisnotreadilyavailable,onecanresorttothenumericalsearch. Letusconsidersomeexamplesofoptimumenergyallocation.Fig. 43 representstheaverageSERforvariousenergyallocationsatthexedrelaylocation.TotaltransmitSNR=10dBandL=2areconsideredwithDs;r=0:25;0:5;and0.75.Foreachcase,theSERhasoneminimumpoint,andthecorrespondingenergyallocationistheoptimumvalue,i.e.,os=.Thegureshowsthattheos=increasesastherelaysmovetowardsthedestination.ThiscoincideswithouranalyticalresultsandsimulationsinFig. 42 TheoptimumenergyallocationobtainedfromthenumericalsearchisplottedinFig. 44 andcomparedwiththesimulatedresults.ThetotalSNRvalueof=10dBandapathlossexponentof=4areconsideredwithvariousLvalues.Theresults 54 PAGE 56 44 .ThisdiscrepancyarisesfromthefactthatthenumericalsearchisbasedontheSERbound,whereasthesimulationsgeneratethetrueSER,andthattheSERboundislooserforlargerLvalues,aswementionedintheprecedingsection.Nevertheless,thenumericalresultsstillcloselyindicatethetrendandrelativedistancescorrespondingtovariousLvalues.NoticethatthesecurvesexhibitaconvergingtendencyasLincreases.ThisimpliesthattheoptimumenergyallocationcurvemayachieveanasymptoticlimitasthenumberofrelaysLgrows. WehaveseenthattheexactexpressionoftheoptimumenergyallocationinEq.( 4{27 )willgiverisetoas=ratiothatdependsontheactualvalueofthetotalSNRwithDFprotocol.However,thehighSNRapproximationinEq.( 4{34 )resultsinas=ratiowhichisindependentof(seeEq.( 4{35 )).Asaresult,theapproximatesolutions( 4{33 )and( 4{34 )areexpectedtodierfromtheexactsolution( 4{27 ),dependingon 56 PAGE 57 42 showsthatthesesolutionsagreeverywellat=10dB.InFig. 45 ,theoptimumenergyratios=obtainedfromtheexactsolution( 4{27 )isdepictedatvariousSNRvalues=(0;5;10;15;20)dB,andwithtwovaluesof(2and4).Weobservethatwhen=4,thecurvescorrespondingtodierentvaluesarealmostidentical;whereaswhen=2,allcurvescoincideexceptforthe=0dBone.Theseobservationsconrmthattheapproximations( 4{33 )and( 4{34 )arebothveryaccurateevenforaslowas5dBwhen=2andforallSNRlevelswhen=4.Inotherwords,theoptimumenergyallocationratios=isalmostindependentoftheactualenergylevelexceptforverylowvalues.ItonlydependsonthelocationoftherelaysasinEq.( 4{35 ).Likewise,asimilarresultcanbededucedfortheoptimumdistanceallocation;thatis,theoptimumdistanceallocationratioDs;r=Ds;disnearlyindependentoftheactualsourcedestinationdistanceDs;d,aswederivedinEq.( 4{60 ). NowletusconsidertheAFprotocol.SimilartotheDFprotocol,bytreatingtheapproximatedSERPAFe;NDorPAFe;DLasafunctionofsandrk,wecanndanoptimumsolution. 4{24 ),theoptimumenergyallocationosandorshouldsatisfy: (4{40) L+1o2s+or2d;r2r;s (4{41) PAGE 58 whereistheLagrangemultiplier.Eqs.( 4{42 )and( 4{43 )giveus Then,bysubstitutingsandrinEqs.( 4{45 )and( 4{46 )forthetotalenergyconstraintinEq.( 4{44 ),wehave WithEqs.( 4{45 )and( 4{47 ),wearriveatEq.( 4{40 ).Similarly,therstorderconditionsofthesystemwithadirectlinkaregivenby Byusingthesamestepsasshown,wehaveEq.( 4{41 ). InEq.( 4{41 ),the2r;sln(or2d;r)=(L+1)termmainlyaectstheenergyallocationcomparedwiththesystemwithnodirectlink.Thiseectisobviousespeciallywhentherelaysarelocatedclosetothethesource.NoticethatthelogterminEqs.( 4{40 )and( 4{41 )rendersaclosedformsolutionincalculable.Althoughananalyticalsolutionisnot 58 PAGE 59 4{40 )and( 4{41 ).ThefollowingLemma2showsthatoptimumvalueshaveonlyonesolution,whichallowsusthenumericalsearch. 4{40 )and( 4{41 )haveonesolutionofos,andaccordinglyor. Proof. 4{40 )and( 4{41 )cannotbesolvedalgebraically,werepresentasolutiongraphically.UsingEq.( 4{24 ),Eq.( 4{40 )isgivenby Thiscanberepresentedas Bysquaringrootbothsides,wehave 59 PAGE 61 4{52 ),respectively.Giventhatdomainofln(or2d;r)>1andusingEq.( 4{1 ),weploty1andy2inFig. 46 forvariousrelaylocationsDs;rwithDs;d=1.Thelinewithnomarkerandwithcirclemarkerrepresenty1andy2,respectively.Thegureshowsonlyonecrossingpoint,andthatpointprovidesustheoptimumsolution.Similartrendsareobservedinthesystemwithadirectlink. Figs. 47 and 48 depicttheaverageSERforvariousenergyallocationsatxedrelaylocationforthesystemwithnodirectlinkandwithadirectlink.Welocaterelaysat0:25;0:5;and0:75with=15dBandL=2.BothguresshowthattheSERhasoneminimumpoint,andthecorrespondingenergyallocationistheoptimumvalue,whichconrmsLemma2.Forthesystemwithnodirectlinktheoptimumenergyallocationincreasesastherelaysmovetowardthedestination.However,forthesystemwithadirectlink,optimumenergyallocationstaysatthemiddlevalue,i.e.,uniformenergyallocationwhentherelaysarelocatedclosetothesource(seeFig. 48 withDs;r=0:25and0.5).Thisresultwillbeveriedbythefollowingnumericalsearchresults.Forallcases,theSERsinFig. 48 showbetterperformanceinFig. 47 duetothedirectlink. TheoptimumenergyallocationobtainedfromthenumericalsearchforboththesystemwithandwithoutadirectlinkisplottedinFig. 49 .WeconsiderthetotalSNRvalueof=30dBandapathlossexponentof=4withvariousLvaluesandD=Ds;d=1.Inthesystemwithnodirecttransmission,forallLvalues,theoptimumenergyallocationatthesourceincreasesastherelaymovestowardsthedestination.However,forthesystemwithadirectlink,auniformenergyallocationisoptimumwhentherelaysarelocatedclosetothesource.Intuitively,thisisbecausethedirecttransmissionispresentthatthediversitygainisdominantoverthecodinggain.Whentherelaysarelocatedclosetothedestination,muchoftheenergyisassignedatthesourcetoassurethatatransmittedsignalcanreachtherelays;itisthesameasinthesystemwithnodirectlink. 61 PAGE 63 410 representstheoptimumenergyallocationforbothDFSTCandAFSTCprotocolswhenL=2and3.WeusethetotalSNRvalueof=15dBandapathlossexponentof=4.withD=1:2andDs;d=1.Thegureshowsthattheoptimumenergyallocationatthesourceincreasesastherelaysmovetowardthedestinationforbothcases,whichisthesameastheconventionalcooperativenetworkswithnodirectlink.Thegurealsoshowsthat,ingeneral,DFSTCprotocolrequiresmoreenergythanAFSTCprotocoltodecreasetheerrorrateattherelays. 3{5 )forDFandEq.( 3{10 )orEq.( 3{31 )forAFwhilesatisfying: Startingwiththesinglerelay(L=1)setupandapplyingthehighSNRapproximationwithDFprotocol,weestablishthefollowingresult: PAGE 64 whereweusedthefactthatr;s(ord;r)isindependentofDs;r(orDr;d).UsingthedenitionoftheaveragereceiverSNRsinEq.( 4{25 ),wecanreexpressEqs.( 4{55 )and( 4{56 )as D+1s;rs=0;@PDFe D+1r;dr=0; whichleadsto (1+2r;s)(1+r;s): AthighSNR,theconstanttermsontherighthandsideofEq.( 4{59 )canbendglected.Consequently,wehave[c.f.( 4{25 )]r which,togetherwiththeconditioninEq.( 4{57 ),concludestheproofofProposition5. Interestingly,Eq.( 4{60 )bearsaverysimilarformasitscounterpartfortheoptimumenergyallocationinEq.( 4{35 ).Infact,whenthepathlossexponent=2,Eq.( 4{60 )isessentiallyidenticaltoEq.( 4{35 ).Forgeneralvalues,however,thesetworelationships 64 PAGE 65 4{35 )and( 4{60 )resultfromtwodistinctoptimizationproblems:theformerisobtainedforarbitrarydistancesDs;randDr;dunderatotalenergyconstraint;whereasthelatterisobtainedforprescribedEsandErunderatotaldistanceconstraint.WiththeSERboundPDFebeingatwodimensionalfunction,theenergyandlocationoptimizationsarecarriedoutonuncorrelateddimensions. ForgeneralLvalues,theoptimumlocationcanbedeterminedinasimilarmanneraswediscussedintheabove.Essentially,thepathlossexponentrendersitimpossibletoderiveananalyticalsolutionfortheoptimumlocationproblem,evenwiththehighSNRapproximation.OnecanresorttothenumericalsearchusingtheSERboundinProposition1.InFig. 411 ,theoptimumdistancesobtainedfromthenumericalsearchandthesimulationsarecomparedfordierentvalues,attotalSNR=10dBandwithL=1relaynode.Noticethat,asitscounterpartinFig. 42 ,theoptimumrelaylocationislinearinEs=Eonlywhen=2. 65 PAGE 66 412 depictstheaverageSERversusrelaylocationsatthegivenenergyallocationwhen=10dBandL=2.Noticethatatthegivenprescribedenergy,thereexistsonlyoneminimumSERpoint,andthecorrespondingDs;ristheoptimumrelaylocation.ThisgureisthecounterpartoftheoptimumenergyallocationinFig. 43 .Wecanseethattherangeofoptimumrelaylocationissmallerthantheoptimumenergyallocation,whichresultsintheatnessoflocationoptimization.Wewillverifythisphenomenonwithanumericalexample. LetusnowconsidertheoptimumrelaylocationfortheAFprotocol. Proof. 3{31 ).The1=d;s=Ds;d=shasaxedvaluegiventhesddistanceDs;dandprescribedenergyatthesource.Therefore,adirectlinkdoesnotaectlocationoptimization.Hence,thelocation 66 PAGE 67 TheoptimumlocationcanbefoundbytreatingtheSERasafunctionofdistanceandsolvingtherstorderconditions. 67 PAGE 68 413 and 414 presenttheaverageSERforthevariousrelaylocationsatthegivenenergyallocation.Bothguresconrmthatthereexistsonlyoneminimumpointwhichprovidestheoptimumrelaylocationforrelaynetworks.ItisinterestingthattheminimumpointsarethesameforbothguresalthoughtheSERisdierent.Hence,theexistenceofadirectlinkisindependentontheoptimumrelaylocation,whichconrmstheLemma3.Inbothgures,therangeofoptimumlocationsissmallerthantheenergyoptimizationcases(seeFigs. 47 and 48 ),whichisthesameastheDFcase(seeFig. 412 ). Fig. 415 depictstheoptimumrelaylocationwhichisapplicabletosystemswithandwithoutadirectlink.WeconsiderthetotalSNRvalueof=30dBandapathlossexponentof=4withvariousLvalues.Onedimensionalsetup,Ds;r+Dr;d=D=Ds;d=1,isconsidered.Asmoretransmitenergyisassignedatthesource,theoptimumlocationmovestowardthedestination.Thegureshowsthattheoptimizedvalueschangeslowlycomparedwiththeenergyoptimization. 68 PAGE 70 416 depictstheoptimumrelaylocationofDFSTCandAFSTCprotocolswhenL=2and3.WeusethetotalSNRvalueof=15dBandapathlossexponentof=4.withD=1:2andDs;d=1.Thegureshowsthattheoptimumrelaylocationsmovetowardthedestinationasthetransmitenergyatthesourceincreasesforbothcases.Thegurealsoshowsthat,incomparisonwithAFSTCprotocol,therelaylocatesclosertothesourcefortheDFSTCprotocol.FromFigs. 410 and 416 ,wecanseethatthelocationoptimizationsaremuchatterthantheenergyoptimizations.Theseresultsarethesameasthoseoftheconventionalcooperativesystemswithnodirectlink. 3{5 ),( 3{10 )or( 3{31 ).First,bytreatingeachequationasthefunctionoftransmitenergy,wecanndthesolutionfortherstorderconditions.Then,thesamestepisproceededbythetransmitenergythatisreplacedwiththelocation.Finally,byequatingtwosolutions,wecanndthecommonsolution.Consequently,thissolutionprovidestheglobaloptimizationwhichminimizestheerrorrate.WithDFprotocol,forL=1,wecanreadilyobtaintheglobalsolutionfromEqs.( 4{35 )and( 4{60 ),whichgivesusDs;r=Dr;d=0:5withs==r==0:5;8.However,theanalyticalsolutioncannotbeeasilyobtainedevenwiththeidealizedcaseaswehaveseenintheprevioussection.Ingeneral,thejointoptimizationcanbeobtainedbycarryingoutatwodimensionalnumericalsearchiteratively.Thesearchingstepsareasfollows: Step1. (Initialization)Settheuniformenergyallocationastheoptimum,i.e.os==(L+1). Step2. (LocationOptimization)Foragivenenergyallocation,ndtheoptimumrelaylocation,Dos;r;new,whichiserrorrateminimizing.Ifthedierencebetweennewoptimumlocationandtheoriginaloneissmallerthanthethresholddistance,"D, 70 PAGE 71 Step3. (EnergyOptimization)Foragivenrelaylocation,ndtheoptimumenergyallocation,os;new.Ifthedierencebetweennewoptimumenergyallocationandtheoriginaloneissmallerthanthethresholdenergy,",i.e.,jos;newosj<",stop;otherwisesettheoptimumenergytothenewone,os=os;newandgobacktoStep2. TheseiterativesearchingcanbeillustratedastheowchartinFig 417 Withoutconsideringadirectlink,Figs. 418 and 419 depicttheSERperformancesurfacewhenL=3withDs;d=D=1,and=4,forboththeDFandAFprotocol.Weuse=10dBand=15dBfortheDFandAFprotocol,respectively.Wecanobtaintheenergyoptimizationandthelocationoptimizationbytakingminimumvaluealongthe 71 PAGE 73 Similartotheconventionalprotocols,weplottheCERversuss=andDs;rforDFSTCandAFSTCprotocolsinFigs. 420 and 421 ,respectively,whenD=1:2;Ds;d=1,and=4.TheseguresexhibitthesametrendsasintheDFandAFprotocols.Noticethatthetwoguresshowalmostthesameshape.However,theaxisvaluesofthetwoguresareopposite.ThisimpliesthatthesystemsemployingDFandAFprotocolhavequitedierentrelationshipsfortheoptimumvalues.NoticethatabovesearchingstepsforjointenergyandlocationoptimizationcanbeapplicablefortheDFSTCandAFSTCprotocolsbyreplaceSERwithCERthoughtheanalyticalsolutionsareintractable.Moredetailedexamplesandcomparisonsaregiveninthefollowingchapter.Noticethatoursimulations(Figs. 43 412 418 420 ,and 421 )conrmthattheerrorrateisgenerallyconvex,whichensuresconvergenceoftheiterativestrategy. 73 PAGE 75 Inthischapter,wewilldiscusstheperformanceofrelaysystemscombinedwithdierentialdemodulationandtheoptimumresourceallocation.Wewillcomparetheperformanceofthesystemswithandwithoutoptimization.Thebenetsofthejointenergyandlocationaswellastheresourceallocationcomparisonofdierentprotocolsareaddressed. 51 throughFig. 56 .Weusethefollowingsystemparameters:=15dB,D=1:2,=4,andL=(1;2;3)withDBPSK.Inthesystemwithoutenergyoptimization,auniformenergyallocationisemployed;thatis,s=r==(L+1)atanyDs;r.Inthesystemwithoutlocationoptimization,therelaysareplacedatthemidpointofthesourcedestinationlink. Figs. 51 and 52 illustratethebenetsofoptimizationoftheDFsystem.InFig. 51 ,weobservethat,asLincreases,theSERperformancecangetevenworseunlesstheenergyoptimizationisperformed,andthattheenergyoptimizedsystemuniversallyoutperformstheunoptimizedone.Theseobservationsconrmourdiscussionsintheprecedingchapter.Interestingly,noticethattheminimaoftheenergyoptimizedSERcurvesalmostcoincidewiththeunoptimizedones.ThisimpliesthatthenearoptimumSERcanbeachievedevenwiththeuniformenergyallocationacrossthesourceandrelaynodes,providedthattherelaylocationiscarefullyselected.AsshowninFig. 51 ,theoptimumrelaylocationcorrespondingtotheuniformenergyallocationshiftsfromthemidpointforL=1tothesourcenodeasLincreases.Intuitively,thisisbecausetheoverallSERismoresensitivetothesourcerelaylinkquality,aswementionedinChapter3. 75 PAGE 77 52 ,weverifytheadvantageoftheoptimumrelaylocationbycomparingtheSERwithandwithoutlocationoptimization.Similartotheenergyoptimizationcase,Fig. 52 conrmstheadvantagesofthelocationoptimization,inwhichthelocationoptimizedsystemuniversallyoutperformstheunoptimizedsystem.Dierentfromtheenergyoptimizationcase,however,asLincreases,theSERperformancealwaysimprovesevenwithoutanylocationoptimization. ThecurvesinFig. 52 alsoexhibitmoreatnesscomparedwiththeonesinFig. 51 .ThisimpliesthatthesystemSERismoresensitivetothelocationdistributionthantheenergydistribution.Inaddition,theminimaofthelocationoptimizedSERcurvesarefarfromthoseoftheunoptimizedones,exceptfortheL=1case(seeFig. 52 ).ThisindicatesthatplacingtherelaynodesatthemidpointcannotachievetheminimumSERevenwithcarefulallocationofthesourceandrelayenergies,foranyL>1.ThisistobedistinguishedfromtheuniformenergycasedepictedinFig. 51 ,aswellasfromthecoherentrelaysystemsin[ 45 ]. Figs. 53 and 54 depictthebenetsofenergyoptimizationfortheAFsystemwithnodirectlinkandwithadirectlink,respectively.FromFigs. 53 and 54 ,weobservethattheenergyoptimizedsystemuniversallyoutperformstheunoptimizedsystemasweexpected.Wealsoobservethat,inthesystemwithadirectlink,theSERsoftheoptimizedsystemandunoptimizedsystemarealmostidenticalwhentherelaysarelocatedclosetothesource,sinceauniformenergyallocationisoptimum.Theseobservationscoincidewithouranalysisintheprecedingchapter.BothguresshowthattheunoptimizedsystemshavetheminimumSERalmostatthemidpoint,coincidingwiththeresultsin[ 38 45 67 ]. NoticethattheminimumpointsoftheenergyoptimizedSERcurvesmovetowardsthedestinationexceptinthesystemwithoutadirectlinkforL=1,whichisoppositecomparedwiththeDFcase.FromFigs. 53 and 54 ,itisclearthatwecannotachieveoptimumSERvaluewithoutenergyoptimizationexceptforthesystemL=1withnodirectlink.ThisisdierentfromtheDFcase,asinFig. 51 .Itisworthmentioning 77 PAGE 79 45 ]withAFprotocol,butdierentfromtheDFcase. Next,letusconsiderthebenetsoflocationoptimizationfortheAFsystems.Figs. 55 and 56 verifytheadvantagesbycomparingtheSERsofthesystemswithandwithoutlocationoptimization.Similartotheenergyoptimizationcase,thelocationoptimizedsystemuniversallyoutperformstheunoptimizedsystem.TheguresshowthattheoptimumSERcanbeachievedbyassigningmoreenergytothesourceexceptforthesystemwithoutadirectlinkwithL=1,whichistheoppositeoftheDFcase.ThecurvesinFigs. 55 and 56 alsoshowmoreatnesscomparedwiththeenergyoptimizedcurves,aswehaveobservedfortheDFcase.SimilartothelocationoptimizationoftheDFandtheenergyoptimizationoftheAF,Figs. 55 and 56 showthatoptimumSERcannotbeobtainedwithoutrelaylocationoptimizationexceptforthesystemL=1withnodirect 79 PAGE 80 54 and 56 comparedwithFigs. 53 and 55 ,respectively. Figs. 57 58 59 ,and 510 depictthebenetsofenergyandlocationoptimizationsforDFSTC(Figs. 57 and 58 )AFSTC(Figs. 59 and 510 )protocols.Forallcases,weconsiderD=1:2,Ds;d=1,and=4with=15dBand25dBwhenL=2and3sincethesinglerelaysetup(L=1)isthesameastheconventionalcooperativesystem.Similartotheconventionalcase,weplottheCERforthesystemwithandwithoutresourceallocations.Weusethesameparametersfortheunoptimizedsystems.TheguresconrmthattheminimumCERcanbeachievedbytheoptimumenergyandrelaylocationselection.TheguresalsoshowthatthetrendsofoptimizationsforDFSTCandAFSTCprotocolsarethesameasthoseforDFandAFprotocols,respectively.Noticethat,atlow,thesystemswithmoreLmayunderperformthesystemswithlessL.However,athigh,theerrorperformanceimprovesasLincreasesexceptsomecasesofDFSTCbasedsystemaswehaveseeninDFbasedsystem. 80 PAGE 83 511 and 512 depicttheSERcontouroftherelaysystemsfortheAFprotocolwithnodirectlinkandwithadirectlink,respectively,andFig. 513 depictstheSERcontouroftherelaysystemfortheDFprotocol.WithsystemparameterDs;d=D=1,L=2isconsideredforthesystemwithnodirectlinkandwithadirectlink.Weuse=15dBand10dBfortheAFandDFprotocol,respectively.Inallgures,theverticallineandhorizontallinerepresenttheSERsofthesystemswithauniformenergyallocationandmiddistanceallocation,respectively,i.e.,unoptimizedsystems.Wealsoplotlinesfortheenergyoptimizationandlocationoptimization.NoticethatthecrossingpointofthetwooptimizationsistheminimumSERofthesystem;accordingly,thispointcorrespondstothejointenergyandlocationoptimization. WithAFprotocolinFigs. 511 and 512 ,itisclearthattheminimumSERoftheunoptimizedsystemsisfarfromtheoptimizedones;thisindicatesthatwecannotobtaintheminimumSERwithauniformenergyallocationormiddistanceallocation.ToachievetheminimumSER,wehavetoadaptasystemviatheenergyoptimizationor 83 PAGE 85 513 showsthatthenearoptimumperformancecanbeachievedwithauniformenergyallocation,butcannotwithamiddistanceallocation.AlltheseresultscoincidewiththesimulationresultsintheprevioussectionandtheoreticalanalysisinChapter4.FromFig. 513 ,wecanseethattheuniformenergyallocationisaverygoodstartingpointfortheiterativeoptimization. Figs. 514 and 515 depicttheCERcontourfortheDFSTCandAFSTCprotocols,togetherwiththeoptimumenergyallocationandrelaylocationoptimizationcurves.WeconsiderD=1:2;Ds;d=1,and=15dBwhenL=2.TheseguresshowthattheCERcontoursfortheDFSTCandAFSTCprotocolshavethesametrendsastheSERcontoursfortheDFandAFprotocols,respectively. Itisinterestingthattheoptimumresourceallocationisdierentinthetwoprotocols.FortheAFbasedprotocol,wecanachievetheminimumerrorratebylocatingtherelaysclosertothedestinationandassigningmoreenergyatthesourcethantherelays.Thisisduetothefactthattheerrorratedecreasesbyallowingthesourcetotransmitsignalswithmoreenergywhilereducingtheeectofnoiseontheamplicationfactor, 85 PAGE 87 Assumethattheoriginalinformationsymbolsareequiprobablebinarysignal(=1)andtherearetworelaynodes(L=2).Intheconventionalsystem,onesymboltransmissiontakesthreetimeslots:oneforbroadcastingthesymboltotherelaynodes,andtwotimeslotsareusedtotransmittheremodulated/ampliedsignalfromeachrelay 87 PAGE 88 516 .Weobservethatthenumberofrequiredtimeslotsincreaseswiththenumberofrelaysfortheconventionalsystem,butremainsconstantfortheSTCbasedsystem.Therefore,wecanincreasetransmissionratebyusingSTCbasedsystem.ThisimpliesthattheSTCbasedsystemcanpotentiallyprovidethedierentialbenetaswellashightransmissionrateregardlessofthenumberofrelays. FortheSTCbasedsystem,themodulationsizeincreasesasthenumberofrelaysincreases.Whereas,wecanchoosethemodulationsizefortheconventionalsystem.Byadoptingthesamemodulationsize,thecomparisonofbiterrorrate(BER)betweentheconventionalsystemandSTCbasedsystemisdepictedinFigs. 517 and 518 withDF/DFSTCandAF/AFSTCprotocols,respectively.WeconsiderDQPSK,D8PSK,andD16PSKforL=2,3,and4,respectively,fortheconventionalsystem.TheguresshowthatSTCbasedsystemperformscomparablywithorbetterthantheconventionalsystemespeciallywhenL3.Fortheconventionalsystem,theerrorperformancedecreasesasLincreasesforbothDFandAFprotocols.However,fortheSTCbasedsystem,theBERisnotaecteddramaticallyregardlessofLandthediversitygainisalwaysguaranteedbydecreasingerrorratefortheDFSTCandAFSTCprotocol,respectively. Aswementionedintheabove,theSTCbasedsystemprovideshighertransmissionrate.IftheconventionalsystemadoptsthesamedaterateasintheSTCbasedsystem,howdoesthisaecttheerrorperformance?Toanswerthisquestion,wecomparetheBERofthecooperativesystemswhichhavethesameorsimilardatarateinFigs. 519 and 520 88 PAGE 90 517 toFig. 520 ,wecanseethatthemodulationsizedramaticallyaectstheperformanceoftheconventionalsystem. Summarizing,wecomparedtwocooperativesystemswithrespecttothetransmissionrateandBER.OurresultsshowthattheSTCbasedsystemprovideshighertransmissionratecomparedwiththeconventionalsystem.ItisalsoshownthattheconventionalsystemmayprovidebetterperformancethantheSTCbasedonebykeepingthesame/higherdatarate.Inaddition,theBERcomparisonsrevealthatthemodulationsizeiscriticaltodeterminetheerrorperformanceofconventionalsystem. 90 PAGE 92 Wederivedtheupperboundofsymbolerrorrate(SER)fortheDFprotocol,approximatedSERfortheAFprotocol,andtheupperboundofthecodeworderrorrate(CER)forSTCbasedsystems.TheDFbased(DFandDFSTC)protocolsshowedanunbalancederrorperformancedependingontherelaylocations,andouranalyticalresultsandsimulationssuggestedthatitismainlythesourcerelaylinksthatdeterminetheoverallsystemperformance.FortheAFprotocol,theSERwasderivedfortwocases:thesystemwithnodirectlinkandthesystemwithadirectlink.ThisSERexpressionhadageneralbutverysimpleform.TheAFSTCprotocolshowedthesametrendsastheAFprotocolwithnodirectlink. Basedontheerrorperformance,theaverageSERandCERfortheconventionalDF/AFprotocolsandtheDSTCprotocol,respectively,weexploredtheresourceallocationasatwodimensionproblem.Weshowedthat:i)giventhesource,relayanddestinationlocations,theaverageerrorratecanbeminimizedbyappropriatelydistributingtheprescribedtotalenergypersymbolacrossthesourceandtherelays;ii)giventhesourceandrelayenergylevels,thereisanoptimumrelaylocationwhichminimizestheaverageerrorrate;andiii)giventhesource,relayanddestinationlocations,andtotaltransmitenergy,theminimumerrorratecanbeachievedbythejointenergylocationoptimization. 92 PAGE 93 Oursimulationsandnumericalexamplesconrmthatboththeenergyandlocationoptimizationsprovideconsiderableerrorperformanceadvantages.WehaveobservedthefollowingresultsfortheDFandDFSTCprotocols. (1) Withoutenergyoptimization,performancedegradationisobservedwhenmorerelaysareincludedinthesystemespeciallywhentherelaysarelocatedclosetothedestinationnode. (2) Forallcases,theoptimizedsystemsuniversallyoutperformtheunoptimizedones. (3) Thelocationoptimizationismorecriticalthantheenergyoptimization.Inotherwords,thedierentialrelaysystemwithuniformenergydistributioncanachievenearoptimumerrorperformancebyappropriatelychoosingtherelaylocation;whileasystemwithrelayssittingatthemidpointbetweenthesourceandthedestinationcannotapproachtheoptimumerrorperformanceevenwithoptimizedenergydistribution. FortheAFandAFSTCprotocols,wehaveobservedfollowingresults. (1) ForthesystembasedonAFprotocolwithadirectlink,theerrorratesofthesystemwithauniformenergyallocationandoptimumenergyallocationarealmostidenticalwhentherelaysarelocatedneartothesource,sincetheuniformenergyallocationisoptimuminsuchcases. 93 PAGE 94 Forallcases,theoptimizedsystemsuniversallyoutperformtheunoptimizedones. (3) Energyandlocationoptimizationsareequallyimportant,sinceminimumerrorperformancecannotbeachievablewithouteitherofthem. Wehavealsoshownthattheminimumerrorperformancecanbeachievedbythejointenergylocationoptimization. Inaddition,wecomparedthecooperativesystemswithdierentprotocolsbyconsideringboththeenergydistributionandtherelaylocationselection.Itisinterestingthattheoptimumlocationandenergyallocationsareverydierentinthetwoprotocols.Ingeneral,wecanachievetheminimumerrorperformanceofthecooperativenetworksbylocatingrelaysclosertothesourcenodewithalessamountofthesourcetransmitenergyfortheDFbasedsystem,andbylocatingrelaysclosertothedestinationnodewithalargeamountofthesourcetransmitenergyfortheAFbasedsystem. Finally,wecomparedtheconventionalsystemwithSTCbasedsystem.Ingeneral,STCbasedsystemcansupporthighertransmissionrates.However,theconventionalsystemcanachievecomparableperformanceincomparisonwithSTCbasedonebychoosingappropriatemodulationsize. Recently,thehybridschemewhichselectstheadvantagesofDFandAFprotocolsissuggestedforcoherentsystem[ 8 11 33 ].Itwillbeinterestingtodevelopahybridcooperativesystemandanalyzeitsperformanceandresourceoptimizingschemes.Itwillbealsovaluabletoconsidermultihopcooperativenetworkswhichcansupportreliable 94 PAGE 95 34 41 69 ]).Finally,inthisresearch,wemainlyfocusedonphysicallayeranalysis.Itwillbehelpfultoconsiderhigherlayerissues,i.e,mediumaccesscontrol(MAC)ornetworklayers,forimprovingtheoverallnetworkingperformance. 95 PAGE 96 [1] M.AbramowitzandI.A.Stegun,HandbookofMathematicalFunctionswithFormulas,Graphs,andMathematicalTables.Dover,NewYork,1972. [2] P.A.AnghelandM.Kaveh,\Exactsymbolerrorprobabilityofacooperativenetworkinarayleighfadingenvironment,"IEEETrans.onWirelessCommunications,vol.3,no.5,pp.1416{1421,September2004. [3] P.A.AnghelandM.Kaveh,\Onthediversityofcooperativesystems,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.4,Montreal,Quebec,Canada,May1721,2004,pp.577{580. [4] P.A.AnghelandM.Kaveh,\Ontheperformanceofdistributedspacetimecodingsystemswithoneandtwononregeneratvierelays,"IEEETrans.onWirelessCommunications,vol.5,no.3,pp.682{692,March2006. [5] P.A.Anghel,M.Kaveh,andZ.Q.Luo,\Optimalrelayedpowerallocationininterferncefreenonregenerativecooperativesystems,"inProc.ofSignalProc.WorkshoponAdvancesinWirelessCommunications,Lisbon,Portugal,July1114,2004,pp.21{25. [6] R.Annavajjala,P.C.Cosman,andL.B.Milstein,\Ontheperformanceofoptimumnoncoherentamplifyandforwardreceptionforcooperativediversity,"inProc.ofMILCOMConf.,vol.5,Atlanticcity,NJ,oct1720,2005,pp.3280{3288. [7] R.Annavajjala,P.C.Cosman,andL.B.Milstein,\Statisticalchannelknowledgebasedoptimumpowerallocationforrelayigprotocolsinthehighsnrregime,"IEEEJournalonSelectedAreasinCommunications,vol.25,no.2,pp.292{305,February2007. [8] X.BaoandJ.Li,\Decodeamplifyforward(daf):Anewclassofforwardingstrategyforwirelessrelaychannels,"inProc.ofSignalProc.WorkshoponAdvancesinWirelessCommunications,NewYork,NY,June58,2005,pp.816{820. [9] A.BletsasandA.Lippman,\Im;lementingcooperativediverisityantennaarrayswithcommodityhardware,"IEEECommunicationsMagazine,vol.44,no.12,pp.33{49,December2006. [10] J.Boyer,D.D.Falconer,andH.Yanikomeroglu,\Multihopdiversityinwirelessrelayingchannels,"IEEETrans.onCommunications,vol.52,no.10,pp.1820{1830,October2004. [11] B.Can,H.Yomo,andE.D.Carvalho,\Hybridforwardingschemeforcooperativerelayinginofdmbasednetworks,"inProc.ofInternationalConf.onCommunications,vol.10,Istanbul,Turkey,June1115,2006,pp.4520{4525. 96 PAGE 97 D.ChenandJ.N.Laneman,\Cooperativediversityforwirelessfadingchannelswithoutchannelstateinformation,"inProc.ofAsilomarConf.onSignals,Systems,andComputers,Monterey,CA,November710,2004,pp.1307{1312. [13] D.ChenandJ.N.Laneman,\Modulationanddemodulationforcooperativediversityinwirelesssystems,"IEEETrans.onWirelessCommunications,vol.5,no.7,pp.1785{1794,July2006. [14] W.Cho,R.Cao,andL.Yang,\Optimumenergyallocationincooperativenetworks:Acomparativestudy,"inProc.ofMILCOMConf.,Orlando,FL,Oct2931,2007. [15] W.Cho,R.Cao,andL.Yang,\Optimumresourceallocationforamplifyandforwardrelaynetworkswithdierentialmodulation,"IEEETrans.onSignalProcessing,June2007(submitted). [16] W.ChoandL.Yang,\Dierentialmodulationschemesforcooperativediversity,"inProc.ofIEEEInternationalConferenceonNetworking,SensingandControl,Ft.Lauderdale,FL,April2325,2006,pp.813{818. [17] W.ChoandL.Yang,\Distributeddierentialschemesforcooperativewirelessnetworks,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.4,Toulouse,France,May1519,2006,pp.61{64. [18] W.ChoandL.Yang,\Optimumenergyallocationforcooperativenetworkswithdierentialmodulation,"inProc.ofMILCOMConf.,Washington,DC,Oct2325,2006. [19] W.ChoandL.Yang,\Jointenergyandlocationoptimizationforrelaynetworkswithdierentialmodulation,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.3,Honolulu,Hawaii,apr1520,2007,pp.153{156. [20] W.ChoandL.Yang,\Optimumresourceallocationforrelaynetworkswithdierentialmodulation,"IEEETrans.onCommunications,2007(Toappear). [21] W.ChoandL.Yang,\Resourceallocationforamplifyandforwardrelaynetworkswithdierentialmodulation,"inProc.ofGlobalTelecommunicationsConf.,Washington,D.C.,November2630,2007(Toappear). [22] X.DengandA.M.Haimovich,\Powerallocationforcooperativerelayinginwirelessnetworks,"IEEECommunicationsLetters,vol.9,no.11,pp.994{996,November2005. [23] M.Dohler,A.Gkelias,andH.Aghvami,\Resourceallocationforfdmabasedregenerativemultihoplinks,"IEEETrans.onWirelessCommunications,vol.3,no.6,pp.1989{1993,November2004. [24] M.Dohler,A.Gkelias,andH.Aghvami,\Aresourceallocationstrategyfordistrubutedmimomultihopcommuniationsystems,"IEEECommunicationsLetters,vol.8,no.2,pp.99{101,February2004. 97 PAGE 98 I.S.GradshteynandI.M.Ryzhik,TableofIntegradls,Series,andProducts,6thed.AcademicPress,2000. [26] M.O.HasnaandM.Alouini,\Endtoendperformanceoftransmsiionsystemswithrelaysoverrayleighfadingchannels,"IEEETrans.onWirelessCommunications,vol.2,no.6,pp.1126{1131,November2003. [27] M.O.HasnaandM.Alouini,\Optimalpowerallocationforrelayedtransmissionsoverrayleighfadingchannels,"IEEETrans.onWirelessCommunications,vol.3,no.6,pp.1999{2004,November2004. [28] M.O.HasnaandM.Alouini,\Aperformancestudyofdualhoptransmissionswithxedgainrelays,"IEEETrans.onWirelessCommunications,vol.3,no.6,pp.1963{1968,November2004. [29] T.Himsoon,W.Su,andK.J.R.Liu,\Dierentialtransmissionforamplifyandforwardcooperativecommunications,"IEEESignalProcessingLetters,vol.12,no.9,pp.597{600,September2005. [30] T.Himsoon,W.Su,andK.J.R.Liu,\Dierentialmodulationformultinodeamplifyandforwardwirelessrelaynetworks,"inProc.ofWirelessCommunicationsandNetworkingConf.,vol.2,LasVegas,NV,April3{6,2006,pp.1195{1200. [31] B.M.HochwaldandW.Sweldens,\Dierentialunitaryspacetimemodulation,"IEEETrans.onCommunications,vol.48,no.12,pp.2041{2052,December2000. [32] B.L.Hughes,\Dierentialspacetimemodulation,"IEEETrans.onInformationTheory,vol.46,no.7,pp.2567{2578,November2000. [33] A.KannanandJ.R.Barry,\Spacedivisonrelay:ahighratecooperationschemeforfadingmultipleaccesschannels,"inProc.ofGlobalTelecommunicationsConf.,Washington,D.C.,November2630,2007(Toappear). [34] G.K.Karagiannidis,\Performanceboundsofmultihopwirelesscommunicationswithblindrelaysovergeneralizedfadinchannels,"IEEETrans.onWirelessCommunications,vol.5,no.3,pp.498{503,March2006. [35] J.N.Laneman,D.N.C.Tse,andG.W.Wornell,\Cooperativediversityinwirelessnetworks:Ecientprotocolsandoutagebehavior,"IEEETrans.onInformationTheory,vol.50,no.12,pp.3062{3080,December2004. [36] J.N.LanemanandG.W.Wornell,\Energyecientantennasharingandrelayingforwirelessnetworks,"inProc.ofWirelessCommunicationsandNetworkingConf.,vol.1,Chicago,IL,September2328,2000,pp.7{12. [37] J.N.LanemanandG.W.Wornell,\Distributedspacetimecodedprotocolsforexploitingcooperativediverstiyinwirelessnetworks,"IEEETrans.onInformationTheory,vol.49,no.10,pp.2415{2425,October2003. 98 PAGE 99 H.LiandQ.Zhao,\Distributedmodulationforcooperativewirelesscommunications,"IEEESignalProcessingMagazine,vol.23,no.5,pp.30{36,September2006. [39] Y.LiangandV.V.Veeravalli,\Gaussianorthogonalrelaychannels:Optimalresourceallocationandcapacity,"IEEETrans.onInformationTheory,vol.51,no.9,pp.3284{3289,September2005. [40] I.MaricandR.D.Yates,\Forwardingstrategiesforgaussianrarallelrelaynetworks,"inProc.ofConf.onInfo.SciencesandSystems,Princeton,NJ,March1719,2004. [41] T.Miyano,H.Murata,andK.Araki,\Spacetimecodedcooperativerelayingtechniqueformultihopcommunications,"inProc.ofVehicularTechnologyConf.,vol.7,LosAngeles,CA,September26{29,2004,pp.5140{5144. [42] R.Pabst,B.H.Walke,D.C.Schultz,R.Herhold,H.Yanikomeroglu,S.Mukherjee,H.Viswanathan,M.Lott,W.Zirwas,M.Dohler,H.Aghvami,D.D.Falconer,andG.P.Gettweis,\Relaybaseddeploymentconceptsforwirelessandmobilebroadbandradio,"IEEECommunicationsMagazine,vol.42,no.9,pp.80{89,September2004. [43] C.S.Patel,G.L.Stuber,andT.G.Pratt,\Statisticalpropertiesofamplifyandforwardrelayfadingchannel,"IEEETrans.onVehicularTech.,vol.55,no.1,pp.1{9,January2006. [44] J.Proakis,DigitalCommunications,4thed.McGrawHill,NewYork,2001. [45] A.Ribeiro,X.Cai,andG.B.Giannakis,\Symbolerrorprobabilitiesforgeneralcooperativelinks,"IEEETrans.onWirelessCommunications,vol.4,no.3,pp.1264{1273,May2005. [46] A.Scaglione,D.L.Goeckel,andJ.N.Laneman,\Cooperativecommunicationsinmobileadhocnetworks,"IEEESignalProcessingMagazine,vol.23,no.5,pp.18{29,September2006. [47] G.ScutariandS.Barbarossa,\Distributedspacetimecodingforregeneratvierelaynetworks,"IEEETrans.onWirelessCommunications,vol.4,no.5,pp.2387{2399,September2005. [48] A.Sendonaris,E.Erkip,andB.Aazhang,\Usercooperationdiversity,partI:systemdescription,"IEEETrans.onCommunications,vol.51,no.11,pp.1927{1938,November2003. [49] A.Sendonaris,E.Erkip,andB.Aazhang,\Usercooperationdiversity,partII:implementationaspectandperformancealanysis,"IEEETrans.onCommunications,vol.51,no.11,pp.1939{1948,November2003. [50] A.B.H.ShinandM.Z.Win,\Outageoptimalcooperativecommunicationswithregenerativerelays,"inProc.ofConf.onInfo.SciencesandSystems,Princenton,NJ,Mar.2224,2006,pp.632{637. 99 PAGE 100 M.K.SimonandM.S.Alouini,\Auniedapproachtotheprobabilityoferrorfornoncoherentanddierentiallycoherentmodulationsovergeneralizedfadingchannels,"IEEETrans.onCommunications,vol.46,no.12,pp.1625{1638,December1998. [52] M.K.SimonandM.S.Alouini,DigitalCommunicationoverFadingChannels,2nded.Wiley,2004. [53] V.Stankovic,A.HstMadsen,andZ.Xiong,\Cooperativediversityforwirelessadhocnetworks,"IEEESignalProcessingMagazine,vol.23,no.5,pp.37{49,September2006. [54] A.StefanovandE.Erkip,\Cooperativespacetimecodingforwirelessnetworks,"IEEETrans.onCommunications,vol.53,no.11,pp.1804{1809,November2005. [55] G.L.Stuber,PrinciplesofMobileCommunication,2nded.Springer,2001. [56] K.TandB.S.Rajan,\Partiallycohernetdistributedspacetimecodeswithdierentialencoderanddecoder,"IEEEJournalonSelectedAreasinCommunications,vol.25,no.2,pp.426{433,February2007. [57] P.Tarasak,H.Minn,andV.K.Bhargava,\Dierentialmodulationfortwousercooperativediversitysystems,"IEEEJournalonSelectedAreasinCommunications,vol.23,no.9,pp.1891{1900,September2005. [58] V.TarokhandH.Jafarkhani,\Adierentialdetectionschemefortransmitdiversity,"IEEEJournalonSelectedAreasinCommunications,vol.18,no.7,pp.1169{1174,July2000. [59] V.Tarokh,H.Jafarkhani,andA.R.Calderbank,\Spacetimeblockcodesfromorthogonaldesigns,"IEEETrans.onInformationTheory,vol.45,no.5,pp.1456{1467,July1999. [60] V.Tarokh,H.Jafarkhani,andA.R.Calderbank,\Spacetimeblockcodingforwirelesscommunications:performanceresults,"IEEEJournalonSelectedAreasinCommunications,vol.17,no.3,pp.451{460,March1999. [61] V.Tarokh,N.Seshadri,andA.R.Calderbank,\Spacetimecodesforhighdataratewirelesscommunication:Performancecriterionandcodeconstruction,"IEEETrans.onInformationTheory,vol.44,no.2,pp.744{765,March1998. [62] M.Uysal,O.Canpolat,andM.M.Fareed,\Asymptoticperformanceanalysisofdistrubutedspacetimecodes,"IEEECommunicationsLetters,vol.10,no.11,pp.775{777,November2006. [63] G.Wang,Y.Zhang,andM.Amin,\Dierentialdistributedspacetimemodulationforcooperativenetworks,"IEEETrans.onWirelessCommunications,vol.5,no.11,pp.3097{3180,November2006. 100 PAGE 101 T.Wang,Y.Yao,andG.B.Giannakis,\Noncoherentdistributedspacetimeprocessingformultiusercooperativetransmissions,"inProc.ofGlobalTelecommunicationsConf.,vol.6,St.Louis,MO,November28December2,2005,pp.3738{3742. [65] Y.Yao,X.Cai,andG.B.Giannakis,\Onenergyeciencyandoptimumresourceallocationofrelaytransmissionsinthelowpowerregime,"IEEETrans.onWirelessCommunications,vol.4,no.6,pp.2917{2927,November2005. [66] S.Yiu,R.Schober,andL.Lampe,\Performanceanddesignofspacetimecodinginfadingchannels,"IEEETrans.onCommunications,vol.54,no.7,pp.1195{1206,July2006. [67] M.Yu,J.Li,andH.Sadjadpour,\Amplifyforwardanddecodeforward:Theimpactoflocationandcapacitycontour,"inProc.ofMILCOMConf.,vol.3,Atlanticcity,NJ,October1720,2005,pp.1609{1615. [68] J.Yuan,Z.Chen,B.Vucetic,andW.Firmanto,\Performanceanddesignofspacetimecodinginfadingchannels,"IEEETrans.onCommunications,vol.51,no.12,pp.1991{1996,December2003. [69] J.ZhangandT.M.Lok,\Performancecomparionofconventionalandcooperaitvemultihoptransmission,"inProc.ofWirelessCommunicationsandNetworkingConf.,vol.2,LasVegas,NV,April36,2006,pp.897{901. [70] Y.Zhang,\Dierentialmodulationschemesfordecodeandforwardcooperativediversity,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.4,Philadelphia,PA,March1923,2005,pp.917{920. [71] Q.ZhaoandH.Li,\Performanceofadierentialmodulationschemewithwirelessrelaysinrayleighfadingchannels,"inProc.ofAsilomarConf.onSignals,Systems,andComputers,vol.1,Monterey,CA,November710,2004,pp.1198{1202. [72] Q.ZhaoandH.Li,\Performanceofdierentialmodulationwithwirelessrelaysinrayleighfadingchannels,"IEEECommunicationsLetters,vol.9,no.4,pp.343{345,April2005. [73] Q.ZhaoandH.Li,\Dierentialmodulationforcooperativewirelesssystems,"IEEETrans.onSignalProcessing,vol.55,no.5,pp.2273{2283,May2007. 101 PAGE 102 WoongChowasborninTongyoung,SouthKorea.HereceivedhisB.S.degreeinelectronicsengineeringfromUniversityofUlsan,Ulsan,SouthKorea,in1997andhisM.S.degreeinelectroniccommunicationsengineeringfromHanyangUniversity,Seoul,SouthKorea,in1999.FromMarch1999toAugust2000,wewasaresearchengineerindivisionofmobiletelecommunicationatHyundaiElectronics.HereceivedhisM.S.degreeformelectricalengineeringfromUniversityofSouthernCalifornia,LosAngeles,CA,in2003.SinceAugust2003,hehasbeenaPh.DstudentinelectricalandcomputerengineeringatUniversityofFlorida,Gainesville,FL.Hisresearchinterestsincludecommunications,signalprocessing,andnetworking.HereceivedhisPh.Ddegreein2007. 102 