Low Complexity Affine MMSE detector for Iterative ... - CiteSeerX

Low Complexity Affine MMSE detector for Iterative Detection-Decoding MIMO OFDM system Daniel N. Liu and Michael P. Fitz Department of Electrical Engineering University of California Los Angeles, Los Angeles, CA, 90095 Email: {daniell and fitz}@ee.ucla.edu

Abstract— Iterative turbo processing between detection and decoding shows near-capacity performance on a multiple-antenna system. Combining iterative processing with optimum front-end detection is particularly challenging because the front-end maximum a posteriori (MAP) algorithm has a computational complexity that is exponential in the throughput. Sub-optimum detector such as the soft interference cancellation linear minimum mean square error (SIC-LMMSE) detector with near front-end MAP performance has been proposed. The asymptotic computational complexity of SIC-LMMSE remains O(n2t nr + nt n3r + nt Mc 2Mc ) per detection-decoding cycle where nt is number of transmit antenna, nr is number of receive antenna, and Mc is modulation size. A lower complexity detector is the hard interference cancellation LMMSE (HIC-LMMSE) detector. HIC-LMMSE has asymptotic complexity of O(n2t nr + nt Mc 2Mc ) but suffers extra performance degradation. In this paper, we introduce a frontend detection algorithm that achieves asymptotic computational complexity of O(nt Mc 2Mc ). Simulation results demonstrate that the proposed low complexity detection algorithm offers exactly same performance as their full complexity counterpart in an iterative receiver while being computational more efficient.

I. I NTRODUCTION Ever since Berrou and Glavieux published their landmark paper on iterative decoding between two parallel concatenated convolutional codes (turbo-codes) [1], [2], it has been generally accepted that iterative (turbo) processing techniques have great value. As pointed out in [3] the “Turbo Principle” not only can be used with traditional concatenated channel coding schemes, but also generally applies to many detectiondecoding algorithms. Of late, multiple-input multiple-output (MIMO) systems receive tremendous amount of attention due to the information theoretic studies done by Telatar, Foschini and Gans [4], [5]. To approach channel capacity in a computationally efficient manner, it seems quite natural to apply the “Iterative(Turbo) Paradigm” to MIMO systems. Therefore, many of iterative detection-decoding algorithms have successfully been generalized to MIMO enviroment [6]–[8], especially multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems [9]. The complexity of optimum front-end MIMO detection motivates the search for a low complexity suboptimal detector. In fact, the optimal front-end MAP detector has complexity that grows exponentially with the modulation size and the number of antennas. Thus, it is important to seek a detector that has reasonable performance while keeping manageable complexity. To address complexity issues, suboptimal detec-

tors/decoders such as: soft interference cancellation linear minimum mean square error (SIC-LMMSE) detector [6], [9], [10], hard interference cancellation LMMSE (HIC-LMMSE) detector [8], [11] and “list” sphere decoder [12] are proposed in the literature. However, the asymptotic computational complexity of this SIC-LMMSE detector is O(n2t nr + nt n3r + nt Mc 2Mc ) per detection-decoding cycle (i.e. turbo iteration) [8], [13], where nt is number of transmit antenna, nr is number of receive antenna and Mc is modulation size. Despite the SICLMMSE detector having a linear growth in the number of antennas, it’s computational complexity remains high even with moderate number of nt , nr and Mc . Further reduced complexity detection such as the HIC-LMMSE detector is also advocated in [8], [11]. HIC-LMMSE has asymptotic computational complexity of O(n2t nr +nt Mc 2Mc ) at the price of performance degradation. The novelty of our SIC-AMMSE detector lies in the detection process. The SIC-LMMSE detector is given as
 ¯ (−) (1) x ˆ = w† y − y ¯ (−) denotes the mean of observation vector given where y the transmitted data symbols other than the one we try to detect. With no a priori information available, it is natural to assume that x ¯ ≡ Ex = 0. Therefore, the optimal detector w, minimizes the mean square error with  constraint to zero  (MSE) ¯ (−) = x ¯ = 0. But in an bias [14]. That is: Eˆ x = w† E y − y iterative detection and decoding receiver, a priori information about the current detection estimate x does become available and x ¯ = 0 after the first turbo iteration. Thus, a priori information should also be taken into account in the detection algorithm. Different from conventional SIC-LMMSE detection algorithm, AMMSE detector forms its detection estimate as, ¯) + b x ˆ = w† (y − y

(2)

¯ is the mean of observation vector given every where y transmitted data symbols and {w, b} are constants to be determined. More importantly, the affine formulation in (2) no longer assumes zero-mean random variable of x, the current detection estimate, throughout the whole iterative detection and decoding process. Indeed, in the absence of a priori information about x AMMSE detector has exactly same form as LMMSE detector [14]. Hence, AMMSE formulation in (2) is really just a generalization of LMMSE formulation in (1).

4654 U.S. Government work not protected by U.S. copyright This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

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Again, as the number of turbo iterations increase, a priori information about the current detection estimate converges to the true detection estimate and much less computational power is needed in the detection process. Hence, SIC-AMMSE detector achieves asymptotic computational complexity of O(nt Mc 2Mc ) which is linear in nt , but more importantly without any performance degradation as compared to conventional SIC-LMMSE detector. The remainder of the paper is organized as follows: section II presents the system model and introduces our notation. Section III introduces SIC-AMMSE detector. Section IV presents several numerical examples for different number of receive and transmit antennas on standard wireless local area network (WLAN) channel model. Section V concludes the paper. II. S YSTEM M ODEL A. Transmitter We consider a multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) system with nt transmit and nr receive antennas. The transmission scheme is detailed in the upper half of Fig.1. Let vector b with size Nb be source information bits entering the rate Rc LDPC channel encoder. We denote c the vector of encoded bits; which is not only grouped into blocks of Mc bits where Mc is number of bits per constellation symbol, but also multiplexed to nt sub-streams. We will consider a linear model at the kth frequency subcarrier in which received vector y(k) = T [y1 (k), . . . , ynr (k)] ∈ Cnr ×1 depends on transmitted vector T x(k) = [x1 (k), . . . , xnt (k)] ∈ Cnt ×1 via y(k) = H(k)x(k) + n(k)

c

(3)

where H(k) ∈ Cnr ×nt is complex channel matrix, known perfectly by receiver, n(k) ∈ Cnr ×1 is a vector of independent zero-mean complex Gaussian noise entries with variance σ 2 =

N0 /2 per each real component and k = 1, 2, . . . , K where K refers to total number of frequency subcarriers. We assume the average symbol energy Es ≡ E|xi (k)|2 = 1 where i = 1, 2, . . . , nt and symbols are equally likely chosen from a complex constellation X with cardinality |X | = 2Mc . The spectral efficiency R is then defined as R = nt Mc Rc bits per channel use (BPCU). We also define the signal-to-noise ratio (SNR) as Eb /N0 , where Eb is the energy per transmitted information bit per receive antenna. Notice that each receive antenna collects total energy of nt Es which carries nt Mc Rc information bits, therefore Eb can be expressed as Eb = Es /(Mc Rc ). We assume that the data model (3) is used repeatedly for each frequency subcarrier k to transmit a continuous stream of information bits. During each application of data model (3), the channel matrix H(k) is a “snapshot” of the frequency response of MIMO propagation channel between all transmit and receive antennas. More specifically, H(k) is fully described as   (4) H(k) = h1 (k) h2 (k) · · · hnt (k) T

where hi (k) = [h1,i (k), h2,i (k), . . . , hnr ,i (k)] and hj,i (k) represents the complex channel coefficient from transmit antenna i to receive antenna j, j = 1, 2, . . . , nr , at kth frequency subcarrier. B. Iterative Receiver Structure The iterative receiver structure is depicted in the lower half of Fig. 1. The MIMO detector takes the channel observation y(k) and a priori log-likelihood ratio (LLR) LA (cl ) to compute the extrinsic information LE (cl ) for each of nt Mc bits per received vector y(k). With cl = +1 representing a binary one and cl = −1 representing a binary zero, we define LA (cl )

4655 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

from outer channel decoder as LA (cl ) ≡ log

P [cl = +1] P [cl = −1]

(5)

where l = 1, . . . , nt Mc . Moreover, LA (cl ) can also be viewed as the extrinsic information learned at the outer channel decoder. The a posteriori LLR LD (cl |y(k)) for bit cl , conditioned on received vector y(k) is similarly defined as LD (cl |y(k)) ≡ log

P [cl = +1|y(k)] P [cl = −1|y(k)]

(6)

where P [cl = m|y(k)], m = ±1, is the a posteriori probability (APP) of bit cl . Using Bayes’ theorem, (6) can be rewritten as P [cl = +1] P [y(k)|cl = +1] + log LD (cl |y(k)) = log P [y(k)|cl = −1] P [cl = −1] (7) = LE (cl ) + LA (cl ) where the first term in (7), denoted as LE (cl ), is the extrinsic information delivered by MIMO detector, based on the received vector y(k) and prior information about the coded bits LA (cl ). “New” (extrinsic) information learned at the detection stage can easily be separated from a posteriori LLR LD (cl ) by subtracting off the a priori LLR LA (cl ). That is, LE (cl ) = LD (cl |y(k)) − LA (cl ).

(8)

In view of (8), extrinsic information LE (cl ) is then fed into outer channel decoder as a priori information on the coded bit cl . III. SIC-AMMSE D ETECTOR As a priori LLR becomes available, we form symbol mean x ¯i (k), i = 1, 2, . . . , nt , as  xP [xi (k) = x], (9) x ¯i (k) =

The LMMSE filter wi (k) is chosen to minimize the meansquare error (MSE) between the transmit symbol xi (k) and the filter output x ˆi (k). Equivalently, LMMSE filtering is precisely stated in the following optimization problem: minimize E|xi (k) − x ˆi (k)|2 subject to x ˆi (k) = wi (k)† yi (k),

where (·)† denotes conjugate-transpose. Hence, the optimal LMMSE filter coefficient wi (k) is obtained by solving (12). It can be shown that the optimal solution [6], [8]–[10] is given by,

−1 N0 Inr + H(k)∆i (k)H(k)† hi (k), (13) wi (k) = Es where the covariance matrix ∆i (k) is ∆i (k) = diag

P [xi (k) = x] =

Mc  ˜ l=1

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e−xl˜LA (cl˜)

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(10)

For ith transmit antenna, interference from rest of nt − 1 antennas is “parallel” cancelled to obtain

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and σx2n (k) , n = 1, 2, . . . , nt with n = i, is the transmit symbol variance and generally can be computed as,  σx2i (k) = |x − x ¯i (k)|2 P [xi (k) = x]. (15) x∈X

In view of (13), the LMMSE filter adapts its filter coefficients according to the quality of soft interference cancelled symbols through covariance matrix ∆i (k). The optimization problem in (12) for the LMMSE filter coefficient can be generalized. Realizing the fact that a priori information about the current detection symbol x ˆi (k) becomes available after first turbo iteration, we then seek an affine estimator of the following form: ˆi (k)|2 minimize E|xi (k) − x subject to x ˆi (k) = wi (k)† yi (k) + mi (k),

(16)

where {wi (k), mi (k)} are to be determined and yi (k) is obtained from (11). To find wi (k) and mi (k), we rely on the following two observations: 1) the affine estimator should be unbiased, and 2) the filter coefficient wi (k) should be chosen optimally to minimize MSE in (16). ¯i (k) For the unbiased estimator, we must have Eˆ xi (k) = x where x ¯i (k) is defined in (9). Taking expectation on both side of the constraint equation of (16) shows that mi (k) must satisfy (17) x ¯i (k) = wi (k)† Eyi (k) + mi (k), ¯ i (k) = x where Eyi (k) = y ¯i (k)hi (k). To minimize MSE, we uses expression (17) for mi (k) and eliminate it from the constraint equation in (16), which becomes,   ¯ i (k) . ¯i (k) − wi (k)† y (18) x ˆi (k) = wi (k)† yi (k) + x

yi (k) nt 



(14)

x∈X

where X is the complex constellation set and P [xi (k) = x] refers to a priori symbol probability. Assuming bits within symbol x are statistical independent and let x˜l represents the ˜lth bit value of symbol x (i.e. x˜ = +1 means ˜lth bit of symbol l x is binary one), where ˜l = 1, 2, . . . , Mc , then P [xi (k) = x] can be computed as

(12)

(xn (k) − x ¯n (k))hn (k) + n(k).

Equivalently,

n=1,n=i

(11)

¯ i (k)) . x ˆi (k) − x ¯i (k) = wi (k)† (yi (k) − y

4656 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

(19)

(19) shows that the desired optimal filter coefficients wi (k) ¯ i (k)) to should map the now zero-mean variable (yi (k) − y ¯i (k)). In other words, another zero-mean variable (ˆ xi (k) − x optimization problem in (16) is reduced to, minimize E|xi (k) − x ˆi (k)|2 ¯ i (k)) . subject to x ˆi (k) − x ¯i (k) = wi (k)† (yi (k) − y (20) In view of (20), it should be obvious that LMMSE formulation is really a sub-class of a more general AMMSE formula¯ i (k) are zero throughout tion. If we assume both x ¯i (k) and y the iterative detection and decoding process, then (20) has exactly the same formulation as (12). More interestingly, (20) can also be derived from the original observation y(k) in (3) instead from yi (k) in (11). To see this, we just substitute (11) into the constraint equation of (20): minimize E|xi (k) − x ˆi (k)|2 ¯ (k)) + x subject to x ˆi (k) = wi (k)† (y(k) − y ¯i (k). It can be shown that the optimal filter coefficient for (20) is given by, −1 2  wi (k) = N0 Inr + H(k)∆(k)H(k)† σxi (k) hi (k) (21) where the covariance matrix ∆(k) can be expressed as 

(22) ∆(k) = diag σx21 (k) , · · · , σx2i (k) , · · · , σx2n (k) t

σx2i (k) ,

and i = 1, 2, . . . , nt , is transmit symbol variance defined in (15). Therefore, the optimal solution to (16) is given by ¯ i (k)) + x x ˆi (k) = wi (k)† (yi (k) − y ¯i (k) (23)

Examining (21), (23), (26) and (27) shows this generalized formulation simplifies the calculation of detection symbol estimate greatly. The optimal AMMSE filter coefficient wi (k) obtained in (21) is clearly a function of estimated symbol variance σx2i (k) . At the beginning of iterative process (i.e. first turbo iteration), no a priori information is available. Then, σx2i (k) equals its maximum value Es and (21) reduces to (13) which is further simplified to

−1 N0 Inr + H(k)H† (k) hi (k). (28) wi (k) = Es Meanwhile, the estimated symbol mean x ¯i (k) which is calculated from a priori information equals zero. Hence, (25) has the same conditional mean and variance as SIC-LMMSE detector in [6], [8]–[10]. As number of turbo iteration increases, estimated symbol x ¯i (k) approaches the true transmit symbol xi (k) while the estimated symbol variance σx2i (k) approaches zero because availability of a priori information. As a result, the output of AMMSE filter becomes the estimated symbol mean x ¯i (k) since wi (k) also approaches zero as clearly indicated by (21). In general, output of AMMSE filter is a combination of filter suppression and estimated symbol mean depending on the level of a priori LLR. The likelihood function for the detection symbol estimate can also be simplified and calculated via quantization due to the AMMSE formulation. Notice that the a priori symbol probability in (10) can be reexpressed in the log-domain as,   Mc  1 . (29) P [xi (k) = x] = log  −xl˜LA (cl˜) 1 + e ˜ l=1

where wi (k) is obtained from (21). In order to calculate the a posteriori LLR from the output of AMMSE estimator in (23), we again rely on the Gaussian approximation of the ISI-plusnoise term as in [15]. Upon rewriting (23) shows that

Realizing that log of products in (29), it can further simplify to Mc 
 (30) P [xi (k) = x] = −log 1 + e−xl˜LA (cl˜) .

¯i (k)) + x ¯i (k) x ˆi (k) = wi (k)† hi (k)(xi (k) − x n t  wi (k)† hn (k)(xn (k) − x ¯n (k)) +

With (30) and Max-log approximation [16], a posteriori LLR LD (c˜l |ˆ xi (k)) becomes (31). With these approximation, comxi (k)) only needs search over 2Mc hypotheses. puting LD (c˜l |ˆ The term − η21(k) |ˆ xi (k) − αi (k)|2 in (31) can be viewed i as Euclidean distance between detection symbol x ˆi (k) and “scaled version” of the actual constellation symbol in X . Then, it is obvious that hypothesis xmin , xmin ∈ X , which is closest to x ˆi (k) in Euclidean distance maximizes the term xi (k) − αi (k)|2 . As number of turbo iterations in− η21(k) |ˆ i creases, conditional variance ηi2 (k) approaches zero indicating also the conditional mean x ¯i (k) is most likely be one of actual constellation symbol in X . This observation leads us to the following quantization  1 A x ˆi (k) = xmin , xi (k) − αi (k)|2 = − 2 |ˆ (32) 0 x ˆi (k) = xmin , ηi (k)

+

n=1,n=i wi (k)† ni (k)

(24)

where the last two terms in (24) are viewed as ISI-plus-noise term. Given the knowledge of xi (k), the output of AMMSE filter is approximated as complex Gaussian distributed. More specifically, P [ˆ xi (k)|xi (k) = x] ∼ Nc (αi (k), ηi2 (k)) − 21 |ˆ xi (k)−αi (k)|2 1 = e ηi (k) 2 πηi (k)

(25)

where αi (k) is the conditional mean ¯i (k)) + x ¯i (k) αi (k) = wi (k)† hi (k)(x − x

(26)

and ηi2 (k) is the conditional variance and can be computed as ηi2 (k) = σx2i (k) wi (k)† hi (k)(1 − hi (k)† wi (k)).

˜ l=1

where A is the quantization value which refers to maximum value LLR may take. Then, SIC-AMMSE detector forms “new” (extrinsic) LLR LE (c˜l ) as

(27)

xi (k)) − LA (c˜l ). LE (c˜l ) = LD (c˜l |ˆ

4657 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

(33)

  Mc  
 1 xi (k) − αi (k)|2 + LD (c˜l |ˆ xi (k)) ≈ max − 2 |ˆ −log 1 + e−xl˜LA (cl˜)  x∈X˜+1  ηi (k) ˜ l l=1   Mc   
 1 2 −xl˜LA (cl˜) − max |ˆ x (k) − α (k)| + −log 1 + e − i i 2  x∈X˜−1  ηi (k)

(31)

˜ l=1

l

TABLE I 0

10

SIC-AMMSE DETECTION ALGORITHM for i = 1, 2, . . . , nt 1. Find a priori symbol probability P [xi (k) = x] as in (10). 2. Find symbol mean x ¯i (k) and variance σx2 (k) in (9) and (15). i end   −1 via RUA in [13]. 3. Construct N0 Inr + H(k)∆(k)H(k)† for i = 1, 2, . . . , nt 4. Perform soft interference cancellation as in (11). 5. Obtain wi (k) as in (21). ¯ i (k)) + x 6. Form AMMSE estimate x ˆi (k) = wi (k)† (yi (k) − y ¯i (k). 7. Compute αi (k) and ηi2 (k) as in (26) and (27). 8. Form a posteriori LLR LD (c˜l |ˆ xi (k)) as in (31). 9. Extract extrinsic information LE (c˜l ) in (33). end

We summarizes the above mentioned steps in Table I for SICAMMSE detector. Table I shows the SIC-AMMSE detection algorithm has obvious advantage in computational complexity. At the beginning of iterative process, no a priori information is available and SIC-AMMSE detector has the same computational complexity of O(n2t nr + n3r + nt Mc 2Mc ) as SIC-LMMSE detector [13]. For subsequent turbo iteration, detection estimate x ˆi (k) can be directly construct from x ¯i (k) depending on the level of a priori LLR LA (c˜l ). As number of turbo iterations increases, a priori information becomes more and more reliable while estimated symbol variance σx2i (k) approaches zero indicating the estimated symbol mean x ¯i (k) also approaches the true transmit symbol xi (k). Then, the dominant computation per transmit symbol involves only calculating a posteriori LLR xi (k)) in (31) via quantization in (32) with complexity LD (c˜l |ˆ of O(Mc 2Mc ). Therefore, SIC-AMMSE detector achieves an asymptotic complexity of O(nt Mc 2Mc ) per turbo iteration. IV. N UMERICAL R ESULTS In this section, we provide computer simulation results to show performance of the proposed front-end SIC-AMMSE detector in an iterative detection-decoding MIMO-OFDM system. We assume an equal number of transmit and receive antennas (i.e. nt × nt system). Most OFDM-PHY parameters such as: number of data sub-carriers, number of pilot sub-carriers and length of OFDM preamble are compatible with IEEE 802.11a standard [17]. The channel code which we adopts in this iterative detection-decoding MIMO-OFDM system is LDPC code with multiple rate compatibility [18]. The actual MIMO channel which we considered in simulation is taken from IEEE 802.11n channel model [19]. Specifically,

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we consider Channel Model D with 50ns RMS delay spread in the simulation. We compute the packet error rate (PER). Each packet consists of 1000 bytes of information bits. We further assume perfect timing synchronization, no frequency offset and perfect channel state information for the iterative receiver. Fig. 2 presents a PER performance comparison between SIC-LMMSE detector and SIC-AMMSE detector. For each packet transmission, we perform up to four turbo iterations, and 12 iterations within the LDPC decoder. Clearly, the performance gain by doing one extra turbo iteration is diminishing and converges at 4 turbo iterations. At 1% PER, both SIC-LMMSE detector and SIC-AMMSE detector provide a performance gain about 2 dB compared to single turbo iteration (i.e. MMSE suppression filter). We observe that our proposed SIC-AMMSE detection algorithm gives exactly the same performance compare to conventional full complexity SIC-LMMSE detection algorithm while being computational more efficient. Fig. 3 presents a PER comparison between HIC-LMMSE detector and SIC-AMMSE detector. At 1% with 4 turbo iterations, we observe that SIC-AMMSE detector outperforms HIC-LMMSE detector by 1 dB. Moreover, SIC-AMMSE detector also achieves a lower computational complexity than HIC-LMMSE detector. Fig. 4 and Fig. 5 present a PER of different spectral efficiency which ranges from 4 to 20 BPCU for 2 × 2 system and 4×4 system respectively. With different constellation map-

4658 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

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pings and code rates, the iterative receiver achieves variety of spectral efficiency while keeping linear asymptotic complexity. V. C ONCLUSION We have presented a computational more efficient frontend detection algorithm for iterative detection and decoding MIMO system, namely SIC-AMMSE detector. By generalizing the detection step in conventional LMMSE filtering process into AMMSE, this allows a more flexible allocation of computational power and more suitable for iterative processing receiver. Moreover, a complexity analysis demonstrates that the proposed system achieves a lower asymptotic complexity than HIC-LMMSE detector proposed in the past, but also has better PER performance. R EFERENCES [1] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun., vol. 44, no. 10, pp. 1261–1271, Oct. 1996. [2] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correction coding and decoding: Turbo codes,” in Proc. IEEE Int. Conf. Communications, Geneva, Switzerland, May 1993, pp. 1064–1070.

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Fig. 5. PER of different spectral efficiency for 4 × 4 SIC-AMMSE TurboLDPC L=1944.

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4659 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.