Cooperative wireless networks based on distributed space-time coding

INTERNATIONAL WORKSHOP ON WIRELESS AD-HOC NETWORKS (IWWAN) 2004

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Cooperative wireless networks based on distributed space-time coding S. Barbarossa, L. Pescosolido, D. Ludovici, L. Barbetta, G. Scutari INFOCOM Dpt., Univ. of Rome ‘La Sapienza’, Rome, Italy E-mail: [email protected]

Abstract— The aim of this paper is to show how cooperation among nodes of a wireless network can be useful to reduce the overall radiated power necessary to guarantee reliable links among the network nodes. The basic idea is that if the links between cooperating nodes are sufficiently reliable, the cooperating nodes can transmit in a coordinated manner in order to emulate a virtual MIMO system that can yield considerable gains in terms of diversity or capacity. In this paper, we provide first a theoretical analysis of a single-user scenario showing how the cooperation gain is related to the spatial density of the cooperating nodes. Then, we compare alternative distributed space-time coding strategies aimed at achieving the promised advantages in a multi-user context1 .

I. I NTRODUCTION Routing packets through a network has been typically formulated as a problem of finding the best path from source and destination nodes. This way of thinking is perfectly reasonable, but it is somehow biased by years of research devoted to wired networks. A wireless network has certainly its specific problems, due primarily to the randomness of the links, but it offers, at the same time, great flexibility in designing routing strategies that are not necessarily based on the search for the optimal path. With a wireless network, in fact, we can think of the information emanating from the source as an information wave that propagates through a medium composed of many nodes that can, in principle, retransmit the received information at the same time. Thus, instead of one path, there could be many paths arriving at the final destination simultaneously. At first sight, it could appear that this would cause interference. However, if the cooperating nodes retransmit the information in a coordinate manner, the potential interference can be turned into a constructive interference that can improve the link quality considerably. The basic tool that guarantees that this coordinated transmission induces a real gain is distributed space-time coding (DSTC). Spacetime coding (STC) has received a huge attention in the last few years as a way to increase capacity and/or reduce the transmitted power necessary to achieve a target bit error rate (BER) using multiple antenna transceivers, see e.g. [1], [2], [3]. More recently, there have been several efforts dedicated to extending the STC idea to cooperative networks whose nodes might have a single antenna. In this case, it is the cooperation that induces a virtual array: If the information of one source is located through more nodes, these nodes can re-transmit 1 This work has been supported by the project IST-2001-32549 (ROMANTIK) funded by the European Community

together acting as the antennas of a potentially huge array [4], [6], [8]. While most previous works on DSTC concentrated on a single-user scenario, in this work we focus on a multi-user context. II. C OOPERATION CODING GAIN In this section we derive the coding gain arising from cooperation and distributed space-time coding. To simplify the analysis and arrive at closed form expressions, in this section we consider a single user scenario. The extension to a multiuser system will be carried out in the next section. We model the position of the radio nodes as a spatially homogeneous two-dimensional Poisson point process [5]. This model arises rather naturally in modelling points distributed uniformly over an infinite surface or as an approximation of the scenario where the points are confined to a limited region, provided that some conditions, to be specified next, are met. Let us consider, as an example, n radio nodes distributed uniformly and independently of each other over a circle of radius R centered around 0. We denote with C(x0 , r) the circle of radius r, centered around the point x0 . If the nodes are uniformly spaced within C(0, R), the probability of finding k dots within a circle C(x, r0 ) of radius r0 , included in C(0, R), is the Bernoulli distribution µ ¶ n pn (k) = pk (1 − p)n−k , (1) k where p = r02 /R2 is the probability of finding a node within a circle of radius r0 . In the limit, as R and n go to infinity, but the ratio ρ := n/πR2 remains constant, if k ¿ n and p ¿ 1 (i.e., r0 ¿ R), (1) tends to the Poisson distribution (ρπr02 )k −ρπr02 e . (2) k! We compute now the maximum distance between two nodes that guarantees a link between the nodes with a given reliability. We assume that the channels are Rayleigh flat fading, so that h is a complex Gaussian random variable with zero mean and variance σh2 = 1/rα ; r is the length of the link, whereas the exponent α is typically between 2 and 5, depending on the propagation environment. We say that two nodes are linked to each other if the bit error rate on their link is smaller than a given maximum value Pemax with a given out-of-service probability Pos . Using QAM constellations, the bit error rate is ! Ãs 2 2 Eb 2 g 2 |h| ≤ ce−gEb |h| /σn , (3) Pe = c Q σn pr0 (k) ≈

INTERNATIONAL WORKSHOP ON WIRELESS AD-HOC NETWORKS (IWWAN) 2004

The out-of-service probability is defined as

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Average BER

where Eb is the energy per bit, σn2 is the noise variance and c and g are two coefficients that depend on the order M of the QAM constellation. A useful approximation for the BER of an M -QAM, given in [14], is obtained by setting √ M −1 3 , g= c = 4√ log2 M. (4) M −1 M · log2 M

2

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2 −gEb |h|2 /σn

Pos = P{Pe > Pemax } ≤ P{ce

k=1

> Pemax }, (5) k=2

where we have also used the bound in (3). Since the channel is Rayleigh, |h|2 is an exponential random variable. Hence Pos can be upper bounded as 2

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Pos ≤ 1 − e−σn log(c/pemax )/(gEb σh ) .

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A link is reliable and it is established if the out-of-service event occurs with a probability smaller than a given value. Setting σh2 = 1/rα in (6) and inverting (6), we find the coverage radius · ¸1/α gEb log(1 − Pos ) r0 = − 2 . (7) σn log(c/Pemax ) Let us now quantify the improvement resulting from cooperation, in terms of average BER. We consider a source (S) node wishing to communicate with a destination (D) node, using all potential relays (R). To avoid excessive complications, we consider a hopping strategy with no more than two hops. The number of relay nodes is a random variable that depends on the relay density and on the transmission rate. In formulas, denoting with pr0 (k) the probability for S to find k relays within a distance r0 , and with Pe (k+1; h) the error probability corresponding to a multiple transmit antenna system having k + 1 antennas (the k relays’ antenna plus the source itself), conditioned to a set of channels h, the error probability at the destination is ∞ X Pe (h) ≈ pr0 (k)Pe (k + 1; h). (8) k=0

This formula is approximated as it assumes that there are no errors at the relay nodes. On the other hand, the goodness of this approximation is kept under control by adopting a relay discovery strategy such that a node is chosen as a relay only if its distance from the source S is less than the value r0 given in (7). This guarantees that the error probability of a relay node is, with probability 1 − Pos , smaller than a given threshold Pemax . Assuming a spatial Poisson distribution for the relays, the probability pr0 (k) of finding k relays is then given by (2), with r0 given by (7). If the final destination has nR receive antennas, the error probability Pe (k + 1; h) is2  v u (k+1)nR u X Eb   |hi |2  . (9) Pe (k + 1; h) = c Q tg 2 σn (k + 1) i=1

coop

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Fig. 1.

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Average BER at the final destination in a cooperating network.

expected value of Pe (k + 1; h) := Eh {Pe (k + 1; h)} is √ µ ¶(k+1)nR 4 M −1 1−µ Pe (k + 1) = √ (10) 2 M log2 (M ) ¶µ ¶m (k+1)nR −1 µ X 1+µ (k + 1)nR + m − 1 · , m 2 m=0 where µ :=

s

3Eb log2 (M )σh2 . 3Eb log2 (M )σh2 + 2(M − 1)(k + 1)σn2

(11)

The average error probability, in case of cooperation is then P¯e =

∞ X

pr0 (k)Pe (k + 1),

(12)

k=0

with pr0 (k) and Pe (k + 1) given by (2) and (10), respectively. An example of performance is shown in Fig. 1, where we report the average BER curves Pe (k), for k = 0, . . . , 15 (solid lines) and the average BER (12) (dotted line). Clearly, as SN R tends to infinity, the diversity is given by the worst case, i.e. by the term with k = 0 in (8), which, at high SNR, it behaves as 1/SN R. Hence there is no diversity gain. However, as we can see from Fig.1, there is a considerable coding gain. More specifically, the asymptotic behavior of P¯e , at high SNR, is c c P¯e ∝ pr0 (0) = , (13) SN R Gc SN R where 2 Gc = eρπr0 (14) is the cooperation coding gain. Hence, the gain can be controlled by acting on the product ρr02 . For a given node density, from (7) we see that we can improve the gain by increasing the transmit power or decreasing the bit rate. III. C OOPERATION PROTOCOL

Assuming that the channels are statistically independent, the 2 In case of coordinated transmission from k + 1 nodes, we normalize the transmit power of each node by k + 1, so that the overall radiated power is independent of k.

Cooperation occurs through a series of steps. We may individuate three main phases: 1) relay discovery (RD), 2) transmission from source to the relays (S2R), and 3) coordinated transmission from source and relays to the destination

INTERNATIONAL WORKSHOP ON WIRELESS AD-HOC NETWORKS (IWWAN) 2004

(SR2D). All these steps require a proper resource allocation (RA) strategy. We start describing these steps with the RA phase that encompasses all the others. A. Resource allocation (RA) We may distinguish between coordinated and uncoordinated multihop networking. In coordinated multihop networking there are time slots dedicated to the different phases, i.e. RD, S2R and SR2D transmission. Conversely, in an uncoordinated strategy, the different phases occur without having any specific resource allocated to relaying. In this work, we will concentrate on coordinated networking. As it will appear through the paper, to avoid an excessive waste of resources, as proposed in [7], it is useful to allow the reuse of the same time slot for the simultaneous exchange of data between each source and its own relays. Besides time slots, the other major resource to be considered is the power allocated to exchange of data between sources and relays and in the final coordinated transmission. This aspect will treated in detail in the performance section. B. Resource discovery (RD): A source wishing to find relays starts sending sounding signals to verify whether there are available neighbors. The sounding signal is a pseudo-noise code identifying the source. A potential relay may receive the sounding signals from more than one source. The radio nodes available to act as relays compute the signal-to-noise plus interference ratio (SNIR) for each source. This step requires the node to be able to separate the signals coming from different sources. This is made possible by the use of orthogonal codes. The potential relays retransmit an acknowledgment signal back only to those sources whose SNIR exceeds a certain threshold. The source receives then the acknowledgments and the relative SNIR from all potential relays and it decides which relays to use. This phase insures that the relay, once chosen, is sufficiently reliable. Given the variability of the wireless channel, this operation has to be repeated at least once every channel coherence time. To avoid excessive complications, a node may act as a relay for no more than one source. The basic philosophy we follow to discover relays is that source and relays should be as close as possible. This is justified by the following concurring reasons: i) less power is wasted in the S2R slot; ii) there are less synchronization problems in the final SR2D slot; iii) there is less interference between different source/relay sets. In summary, this RD phase creates many spatially separated micro-cells where each source acts as a local base station broadcasting to its relays, who may get interference from other micro-cells (sources). The only difference with conventional cellular structure is that if a radio node does not have a sufficient SNIR, it does not act as a relay. C. Source to relays (S2R) transmission Once a source has found its own relays, it transmits its information to them in a dedicated S2R slot. To limit the rate reduction resulting from the insertion of this slot, each source would like to use a constellation order as high as

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possible, in this slot, in order to reduce its duration. On the other hand, as already mentioned in Section II, increasing the bit rate determines a reduction of the coverage, for a given source power budget and a required out-of-service probability at the relay. Limiting the coverage excessively would reduce considerably the probability of finding a relay and then it would nullify the whole relaying process. Hence, the choice of the constellation order has to result as a trade-off between rate reduction and probability of finding a relay. This trade-off will be quantified in Section IV. D. Source-relays to destination (SR2D) transmission Finally, each set of source and relative relays transmit towards the destination using dedicated resources, e.g. time slots, frequency bands or codes, so that SR2D transmissions from different sets do not interfere with each other. Each SR2D transmission occurs using a distributed spacetime coding strategy. In this paper, we consider only the case where the relays decode the received symbols and retransmit them. Alternatively, to simplify the relay structure, the relay could simply amplify and forward the received packet. The difference between amplify and forward and decode and forward strategies was considered in [8]. In all the cases described below, we assume for simplicity that each source interacts with only one relay, but the extension to more lays is straightforward. IV. A LTERNATIVE D ISTRIBUTED STC STRATEGIES The choice of the right space-time coding technique depends on several factors. Primarily, we can choose between the following classes of STC techniques: i) Orthogonal STC (OSTC) [3], as a strategy that maximizes the diversity gain and it minimizes the receiver complexity; ii) full-rate/full diversity codes (FRFD) [11], [12], as codes that yield maximum diversity gain and transmission rate, but with high receiver complexity; iii) V-BLAST codes [10], as a technique that maximizes the rate, sacrificing the diversity gain, but with limited receiver complexity. Alternatively, one could use the trace-orthogonal design [13] as a flexible way to trade complexity, bit rate and bit error rate. The optimal trade-off among these alternative strategies, in the distributed case, requires a specific attention, and it does not coincide, necessarily, with conventional spacetime coding. We denote with TS2R and TSR2D the duration of the S2R and SR2D time slots. Ts is the symbol duration in all slots. For a given bit rate, the durations depend on the constellation order used in the different slots. We denote with Q and M the constellation orders used in the S2R and in the SR2D slots, respectively. We assume also that the S2R slot is shared among N source-relay pairs. The frame containing both S2R and the N SR2D links has then a duration TF = TS2R +N TSR2D . The rate reduction factor, with respect to the non-cooperative case, in a TDMA context, is then η=

N TSR2D . TS2R + N TSR2D

(15)

Clearly, the rate loss can be reduced by decreasing TS2R , i.e. by increasing Q, or by increasing N . In the first case, the

INTERNATIONAL WORKSHOP ON WIRELESS AD-HOC NETWORKS (IWWAN) 2004

relay needs a higher SNIR, in the second case there is an SNIR increase at the relay. In both cases, it is less likely to discover a relay with sufficient SNIR. Hence, the right choice has to result from a trade-off between rate and performance. We discuss now in detail the alternative DSTC choices. In all cases, we denote with s(n) the sequence of symbols sent by S during the SR2D slot, whereas sˆ(n) indicates the estimate of s(n) performed at the relay.

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and the bit rate is R=

4N log2 M b/s/Hz. 2N + 2 log2 M/ log2 Q

(21)

Comparing the transmission rates of all the distributed schemes, for a given choice of the constellation orders Q and M , we see that D-BLAST has the highest transmission rate. V. P ERFORMANCE AND CONCLUDING REMARKS

A. Distributed Orthogonal STC (D-OSTC) D-OSTC guarantees maximum receiver simplicity and full diversity and it can be implemented also when the final destination has a single antenna. D-OSTC was proposed in [4], [6], where the errors in the relay node were not considered explicitly. The effects of the decision errors in the relay node, as well as the ways to compensate their effect was considered in [9]. D-OSTC transmits 2 symbols over two successive time periods, so that TSR2D = 2Ts and TS2R = 2 log2 (M )Ts / log2 (Q). The sequence transmitted by the source-relay pair is · ¸ s(n) −s∗ (n + 1) . (16) sˆ(n + 1) sˆ∗ (n) The first row of this matrix contains the symbols transmitted by the source, whereas the second row refers to the symbols transmitted by the relay (different columns refer to successive time instants). The overall bit rate, incorporating also the rate loss, is 2N log2 M R= b/s/Hz. (17) 2N + 2 log2 M/ log2 Q B. Distributed full rate/full diversity (D-FRFD) If the final destination has 2 antennas, there is a virtual 2×2 MIMO, with the possibility of increasing the rate. This can be achieved, for example, using distributed-FRFD or distributedBLAST. With D-FRFD, the pair S-R transmits 4 symbols over two consecutive time periods. The transmitted matrix is · ¸ s(n) + ϕs(n + 1) θ (s(n + 2) + ϕs(n + 3)) , θ (ˆ s(n + 2) − ϕˆ s(n + 3)) sˆ(n) − ϕˆ s(n + 1) (18) where ϕ = ej/2 , θ = ej/4 are two rotation parameters (see, e.g. [11] or [12], for the choice of ϕ and θ). The bit rate is R=

4N log2 M b/s/Hz. 2N + 4 log2 M/ log2 Q

(19)

C. Distributed BLAST (D-BLAST) We consider here the version of BLAST where two independent streams of data are transmitted from the two antennas. In its distributed version, D-BLAST requires that the relay receives only half of the bits to be transmitted. This implies an advantage with respect to D-FRFD, as it allows us to reduce the duration of the S2R time slot. The price paid with respect to D-FRFD is that D-BLAST is not full diversity. The transmitted matrix in the D-BLAST case, is · ¸ s(n) s(n + 2) (20) sˆ(n + 1) sˆ(n + 3)

In this section, we compare the performance of the proposed distributed coding schemes with a conventional noncooperative link. To make a fair comparison, we enforce all systems to transmit with the same overall energy. More specifically, if E is the energy radiated by the non-cooperative case, we denote with αE the energy radiated by S in the S2R slot and with (1 − α)E/2 the energy radiated by S and R in the SR2D slot, with α < 1. The overall radiated energy is then always E. Our choice implies that S and R transmit with the same energy. This is justified by our set-up where S and R have to be close to each other. In all our simulations, we assume that the constellation used in the S2R link is 16-QAM, i.e. Q = 16. In Fig. 2, we report the average BER, averaged over 16, 000 independent channel realizations. Solid lines refers to the noncooperative case, whereas dashed lines refer to the cooperative scenario. The number of source-relay pairs is N = 10. The destination has two real antennas. To have a global view of both BER and information bit rate, in Fig. 3, we report, for the same cases analyzed in Fig. 2, the average information rate computed by averaging the capacity of the equivalent binary channel, i.e. C(h) = 1 − H(BER(h)), over the channel statistics3 . The results shown in Figs. 2 and 3 refer to the situation where the density of the relay nodes is high enough to guarantee that the cooperative case always finds a relay available. In particular, the SNIR is considered fixed and equal to 20 dB. The more realistic case where the source might not be able to find a relay with a sufficient SNIR will be considered in Fig. 4. The percentage of energy devoted to the S2R link is α = 0.1. Observing Figs. 2 and 3, we may distinguish between two groups of systems, achieving, approximately, rates 2 and 4 b/s/Hz. The different rates are achieved using different constellations in the SR2D link, as detailed in the legend reported in each figure. In case of rate R = 2, D-OSTC outperforms the non-cooperative system, as it requires less (total) power to achieve the desired BER. This gain is a consequence of the greater diversity gain (4 instead of 2), as it can be checked from the slopes of the average BER curves at high SNR. As far as the schemes with 4 b/s/Hz are concerned, we see that D-FDFR, with 4-QAM achieves the lowest average BER. D-OSTC, with 16-QAM, is very close to D-FDFR. D-BLAST is slightly worse than D-FDFR, as it cannot achieve full diversity, however, it is a little better in terms of bit rate, because it suffers from less insertion losses. As anticipated before, in a more realistic scenario, the fading in the S2R link and the interference at the relay might prevent the source to find a relay with sufficient SNIR. In such a 3 We

use the standard notation H(p) = −p log2 (p) − (1 − p) log2 (1 − p)

INTERNATIONAL WORKSHOP ON WIRELESS AD-HOC NETWORKS (IWWAN) 2004

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networks. We have considered here only the case with two hops, but further advantages are expected in the multiple hop case. The price paid for this advantage is the additional signaling required to coordinate the transmission of source and relay nodes, an important issue which is currently under investigation.

No Coop 16−QAM V−BLAST 4−QAM Orthogonal 16−QAM Full Rate−Full Div 4−QAM No Coop 4−QAM Orthogonal 4−QAM

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V−BLAST 4−QAM, SNIR at the Relay =12.5 dB Full Rate−Full Div 4−QAM, SNIR at the Relay =12.5 dB No Coop 16−QAM Orthogonal STC 16−QAM, SINR at the Relay =12.5 dB Full Rate−Full Div 4−QAM, SINR at the Relay =15 dB Orthogonal STC 16−QAM, SINR at the Relay =15 dB V−BLAST 4−QAM, SNIR at the Relay =15 dB

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Final average BER: Comparison of alternative coding strategies.

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Averages over: 5 realizations of the Terminals’ deployment 10( x 5 ) realizations of the Active Sources locations 20( x 10 x 5 ) channels realizations (for each Source) Block Length: Nb = 10

Average information rate as a function of SNR.

case, the source transmits without relaying. This possibility has been incorporated in a simulation program whose results are reported in Fig. 4. The scenario is composed of 10 active sources and 190 potential relays, all scattered uniformly and independently of each other within a circular cell of radius equal to 200 meters. As an example, requiring a SNIR of 12.5 dB, a relay has been found with probability pr0 (1) = 0.72, whereas at 15 dB, pr0 (1) = 0.65. The average BER reported in Fig. 4 takes into account both situations where the relay has been found or not. Clearly, increasing the target SNIR, it decreases the probability of finding a relay, but, at the same time there are less decision errors at the relay. The overall performance is then a combination of these two aspects. We can check from Fig. 4 that indeed, increasing the SNIR from 12.5 to 15, even though pr0 (1) decreases, the floor on the BER decreases by more than a decade. Of course, this result is also a consequence of the relay density. Finally, in the case of a SNIR of 15 dB, we can observe a gain of approximately 3 dB at BER = 1.e − 4. In conclusion, distributed space-time coding can be an important tool to reduce the overall radiated power in wireless

[1] Hottinen, A., Tirkkonen, O., Wichman, R., Multi-antenna - Transceiver techniques for 3G and beyond, West Sussex, UK: John Wiley & Sons Ltd., 2003. [2] Paulraj, A., Nabar, R., Gore, D., Introduction to space-time wireless communications, Cambridge, UK: Cambridge University Press, 2003. [3] Larsson, E., G., Stoica, P., Space-time block coding for wireless communications, Cambridge, UK: Cambridge University Press, 2003. [4] J. N., Laneman, G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks”, IEEE Trans. on Information Theory, pp. 2415–2425, Oct. 2003. [5] Cressie, N., A., C., Statistics for Spatial Data, John Wiley & Sons, New York, 1993. [6] P. A. Anghel, G. Leus, M. Kaveh, “Multi-user space-time coding in cooperative networks”, Proc. of ICASSP 2003, Vol. 4, pp. IV-73–IV-76, April 6-10, 2003. [7] G. Scutari, S. Barbarossa, D. Ludovici, “Cooperation diversity in multihop wireless networks using opportunistic driven multiple access”, Proc. of IEEE Signal Processing Advances in Wireless Communications, SPAWC 2003, Rome, Italy, June 2003. [8] S. Barbarossa, G. Scutari, “Distributed space-time coding strategies for wideband multihop networks: Regenerative vs. non-regenerative relays”, Proc. of ICASSP 2004, Montreal, Canada, May 2004. [9] S. Barbarossa, G. Scutari, “Distributed space-time coding for multihop networks”, Proc. of ICC 2004, Paris, France, June 2004. [10] Foschini, G., J., “Layered space-time architecture for wireless communication in a fading environment when using mutiple antennas”, Bell Lab. Tech. J., Vol. 1, 1996, pp. 41–59. [11] El Gamal, H., Damen, M., O., “Universal space-time coding”, IEEE Transactions on Information Theory, Vol. 49, May 2003, pp. 1097–1119. [12] Ma, X., Giannakis, G.B., “Full-diversity full-rate complex-field spacetime coding”, IEEE Transactions on Signal Processing, Vol. 49, Nov. 2003, pp. 2917–2930. [13] S. Barbarossa, “Trace-orthogonal design of MIMO systems with simple scalar detectors, full diversity and (almost) full rate”, Proc. of IEEE Signal Processing Advances in Wireless Communications, SPAWC 2004, Lisbon, Portugal, July 2004. [14] Simon, M. K., Alouini, M.-S., Digital communications over fading channels: A unified approach to performance analysis, New York, NY: John Wiley & Sons, 2000.