Convex Optimization Theory Applied to Joint ... - CiteSeerX

Convex Optimization Theory Applied to Joint Beamforming Design in Multicarrier MIMO Channels Daniel P´erez Palomar1, John M. Cioffi2 , Miguel Angel Lagunas 1,3, and Antonio Pascual Iserte 1 [email protected], [email protected], [email protected], [email protected] 1

Universitat Polit`ecnica de Catalunya (UPC) Dept. of Signal Theory and Comm. c/ Jordi Girona, 1-3, M`odul D5, Campus Nord 08034 Barcelona (Spain)

2 Stanford University Packard Elect. Eng. Building 350 Serra Mall, Room 363, Stanford, CA 94305-9515, USA

Abstract— This paper addresses the joint design of transmit and receive beamvectors for a multicarrier MIMO channel within the general and powerful framework of convex optimization theory. From this perspective, a great span of design criteria can be easily accommodated and efficiently solved even though closedform expressions may not be available. Among other criteria, we consider the minimization of the average bit error rate (BER) and also of the maximum BER among all carriers for a given signal constellation. We show how to include additional constraints to control the Peak-to-Average Ratio (PAR) in the system design.

I. I NTRODUCTION Recently, multi-input multi-output (MIMO) channels arising from the use of multiple antennas both at the transmitter and at the receiver have drawn considerable attention because they provide a significant improvement not only in terms of spectral efficiency but also in terms of link reliability. A multicarrier approach is suitable for transmission over frequency-selective channels since it is a capacity-lossless scheme [1]. In addition, each carrier can be treated as a transmission over a flat channel, which greatly simplifies the system design. A capacity-achieving design implies that the channel matrix is diagonalized at each carrier and then a water-filling solution is used on the spatial subchannels (or channel eigenmodes) of all carriers [1]. However, this solution has the implication that a different signal constellation and coding scheme have to be used for each eigenmode and carrier according to its allocated power. To reduce the complexity of the optimal solution, for example, the system can be constrained to use a single spatial eigenmode per carrier and the same constellation on all carriers (or just on the carriers with high gain) such as QPSK or 64QAM (this applies to the European standard HIPERLAN/2 [2] and to the US standard IEEE 802.11 for Wireless Local Area Networks (WLAN)). In such a case (equal-rate transmission), the spatial diversity can still be used as a means to improve the link reliability, i.e., to reduce the bit error rate (BER). In this paper, we consider a multicarrier transmission system over a frequency-selective MIMO channel with perfect channel knowledge at both sides of the communication link 1 . We assume that the constellations on all carriers are given, either all equal or approximating a water-filling solution, and consider

3

Centre Tecnol`ogic de Telecom. de Catalunya (CTTC) c/ Gran Capit 2-4, Edif. Nexus I, Pta 2 08034 Barcelona (Spain)

x1

bk

xk

n

T

xN

Fig. 1.

x1

sk n

T

Hk n

R

yk

x^k

ak

H

n

R

n

R

1xn

xn

T

R

Transmit-receive beamforming scheme at the kth carrier.

x^N

that beamforming is used both for transmission and reception 2. The problem of jointly designing the transmit and receive beamvectors for each carrier is cast under the general and powerful framework of convex optimization theory. From this perspective, a great number of interesting design criteria can be easily accommodated and efficiently solved even though closed-form expressions may not exist. To be more specific, we use different design criteria based on optimizing some function of the mean squared error (MSE), the signal to interferenceplus-noise ratio (SINR), and the BER at each carrier. Furthermore, we show how to include additional constraints to control the Peak-to-Average Ratio (PAR) in the system design. II. S IGNAL M ODEL The communication over a frequency-selective MIMO channel with nT transmit and nR receive antennas can be represented in a multicarrier fashion (see Fig. 1) as yk = Hk sk + nk

1≤k≤N

(1)

where k denotes the carrier index, N is the number of carriers, I nT ×1 is the transmitted vector, H k ∈ C I nR ×nT is sk ∈ C I nR ×1 is the received vector, and the channel matrix, y k ∈ C I nR ×1 is a zero-mean circularly symmetric complex nk ∈ C Gaussian noise vector with arbitrary covariance matrix R k . Since transmit beamforming is used at each carrier, the transmitted signal is sk = bk xk

1≤k≤N

(2)

I nT ×1 is the transmit beamvector and x k is the where bk ∈ C transmitted symbol at the kth carrier. The receiver also uses beamforming (which is optimal given that the transmitter is using beamforming):

x ˆk = aH k yk This work was partially supported by the Spanish Government (CICYT) TIC2000-1025, TIC2001-2356, TIC2002-04594, FIT-070000-2000-649 (MEDEA + A105 UniLAN); the Catalan Government (DURSI) 1999FI 00588, 2001SGR 00268; and Samsung Advanced Institute of Technology. 1 It is possible to take into account channel estimation errors in the design but this is out of the scope of the present paper.

x^1

nk

1≤k≤N

(3)

2 Beamforming in a MIMO channel by using transmit and receive beamvectors is in principle suboptimal since it implies a rank-one transmit covariance matrix, although for scenarios with high spatial correlation it is almost optimal. In [3], the more general case of multiple beamforming (transmit and receive beam-matrices instead of beamvectors) is considered.

I nR ×1 is the receive beamvector and x where ak ∈ C ˆ k the estimated symbol at the kth carrier. The transmitter is constrained in its average transmit power as N


E [||sk ||2 ] =

k=1

N


2

bk  ≤ PT

(4)

k=1

where unit-energy transmitted symbols E [|x k |2 ] = 1 are assumed and PT is in units of energy per transmission of N symbols. III. C ONVEX O PTIMIZATION P ROBLEMS A general convex optimization problem (with no equality constraints) is of the form: min f0 (x) x (5) s.t. fi (x) ≤ 0 1≤i≤m n×1

where x ∈ IR is the optimization variable, f 0 (x) is the objective function, and f 0 (x), · · · , fm (x) are all convex functions. In some cases, convex optimization problems can be analytically solved using duality theory [4] and closed-form expressions can be obtained. The underlying idea is to take the constraints into account by augmenting the objective function with a weighted sum of the constraint functions, obtaining the Lagrangian m
L(x,λ) = f0 (x) + λi fi (x) (6) i=1

where the λi ’s are the Lagrangian multipliers. A central result in duality theory is that a solution of (5) is optimal if and only if it satisfies the Karush-Kuhn-Tucker (KKT) conditions which are obtained from the gradient of the Lagrangian and the constraints (the reader is referred to [4] for details). Many design problems arising in engineering can be cast (or recast) in the form of a convex optimization problem. In general, some manipulations are required to convert the problem into a convex one (unfortunately, this is not always possible). The interest of expressing a problem in convex form is that, although analytical closed-form solutions may not exist, it can still be solved (numerically) very efficiently, both in theory and practice. Recently developed interior-point methods (which basically deal with the constrained problem by solving a sequence of unconstrained problems for which a Newton method can be efficiently used) can be used to iteratively solve convex problems very efficiently (see [4] for practical implementation details). To be more specific, global solutions are found with a computational time that is always small and grows gracefully with problem size [4]. Another interesting feature of expressing a problem in convex form is that additional constraints can be straightforwardly added as long as they are convex (see Section V). Convex optimization theory has been used in related areas such as FIR filter design [4], antenna array pattern synthesis [4], power control for interference limited wireless networks, and beamforming design in multiuser scenarios with multiantenna base stations [5]. IV. J OINT T X -R X B EAMFORMING D ESIGN This section deals with the joint transmit-receive beamforming design according to a variety of criteria by casting the problem in convex form (c.f. [3]).

A. MSE-based criteria The MSE at the kth carrier is 2 H xk − xk |2 ] = |aH MSEk = E [|ˆ k Hk bk − 1| + ak Rk ak (7)

which is a non-convex function in a k and bk . However, for a fixed ak , it is convex in bk and vice-versa. Given b k , we can independently minimize each MSE k with respect to each ak with solution:  −1 H ak = Hk bk bH Hk bk . (8) k Hk + Rk The resulting MSE is then MSEk =

1

1+

. H −1 bH k Hk Rk Hk bk

(9)

which is still non-convex in b k . The MSE at each carrier can be lower-bounded as 1 1 ≥ (10) −1 H H 1 + λ 1 + bk Hk Rk Hk bk max,k zk −1 where λmax,k is the maximum eigenvalue of (H H k Rk Hk ) 3 H (which is real and positive ) and zk
bk bk is the power allocated to the kth carrier. The lower bound is achieved when bk lies along the direction of the eigenvector associated to λmax,k . Therefore, we can now focus on the minimization of the minimum value of each MSE k , which is a convex function of zk , by properly allocating the available transmit power over the carriers. In the following, we consider three heuristic criteria. 1) WSUM-MSE criterion: The minimization of the sum of the MSE’s was considered in [6]. We deal with the minimization of the weighted sum of the MSE’s (WSUM MSE) w MSEk (equivalently, the weighted arithmetic k k mean) as was extended in [7]. The problem in convex form (the objective is convex and the constraints linear) is  1 min k wk 1+λmax,k zk {zk }  (11) s.t. k zk ≤ PT , zk ≥ 0 1 ≤ k ≤ N.

where the optimization variables are the z k ’s. From the KKT conditions (see Section III), the following water-filling solution is obtained: +  1/2 −1/2 zk = µ−1/2 wk λmax,k − λ−1 (12) max,k +

where (x)
max (0, x) and µ−1/2 is the water-level chosen to satisfy the power constraint with equality. 2) WPROD-MSE criterion: The minimization of the weighted product of the MSE’s (WPROD-MSE)  wk (equivalently, the weighted geometric mean) k (MSEk ) expressed in convex form (since the objective function is log-convex, it is also convex [4]) is wk   1 min k 1+λmax,k zk {zk }  (13) s.t. k zk ≤ PT , 1 ≤ k ≤ N. zk ≥ 0 3 Strictly speaking, λ max,k can be zero (for Hk = 0), but in practice this happens with probability zero.

This problem, however, is better solved if the logarithm of the objective is used as objective. The obtained solution from the KKT conditions is +  (14) zk = µ−1 wk − λ−1 max,k where µ−1 is the water-level chosen to satisfy the power constraint with equality. Note that for the unweighted case (w k = 1) the power allocation (14) coincides with the capacityachieving solution, i.e., the classical water-filling solution [1]. 3) WMAX-MSE criterion: In general, the performance is limited by the carrier with highest MSE. It makes sense then to minimize the maximum of the MSE’s. In case that different signal constellations are used, it may be interesting to minimize the maximum of the weighted MSE’s (WMAXMSE) maxk {wk MSEk }. The problem in convex form (the objective is linear and the constraints convex) is min

t,{zk }

s.t.

t 1 t ≥ wk 1+λmax,k zk  z ≤ P , k T k zk ≥ 0.

1 ≤ k ≤ N,

(15)

where the optimization variables are t and the z k ’s. From the KKT conditions, the following multi-level water-filling solution is obtained: +  1/2 1/2 −1/2 zk = µ ¯k wk λmax,k − λ−1 (16) max,k 1/2

where the water-levels {¯ µ k } are chosen to satisfy the power 1 constraint with equality and also t = w k 1+λmax,k zk ∀k : t ≤ wk . For the unweighted case w k = 1, all the powers are strictly positive (otherwise the trivial solution t = 1, z k = 0 ∀k follows). The power allocation in (16) simplifies then to zk = λ−1 max,k 

PT −1 l λmax,l

(17)

−1 where λmax,k is the maximum eigenvalue of (H H k Rk Hk ) H and zk
bk bk . The upper bound can be achieved by choosing bk to lie along the direction of the eigenvector associated to λmax,k . Therefore, we can now focus on the maximization of the maximum value of each SINR k , which is a concave (linear) function of z k , by properly allocating the available transmit power over the carriers. In the following, we consider three heuristic criteria. 1) WSUM-SINR criterion: The maximization of the  weighted sum of the SINR’s (WSUM-SINR) k wk SINRk in convex form (the objective and the constraints are all linear) is  max k wk λmax,k zk {zk }  (22) s.t. k zk ≤ PT , zk ≥ 0 1 ≤ k ≤ N.

The optimal solution allocates all the available power to the carrier with maximum value of w k λmax,k . Although the solution maximizes indeed the weighted sum of the SINR’s, it is unacceptable due to the extremely poor spectral efficiency (only one carrier would be conveying information). 2) WPROD-SINR criterion: The maximization of the product of the SINR’s (WPROD-SINR)  weighted wk (SINR ) expressed in convex form (the geometric k k mean is a concave function [4]) is  w ¯k max k (λmax,k zk ) {zk }  (23) s.t. k zk ≤ PT , 1 ≤ k ≤ N. zk ≥ 0  where w ¯k = wk / ( l wl ). From the KKT conditions, the optimal solution is obtained as PT zk = wk  . (24) l wl For the unweighted case w k = 1, the solution in (24) simplifies to a uniform power allocation (as obtained in [8]) zk = PT /N.

which coincides with the solution obtained in [8]. B. SINR-based criteria The SINR at the kth carrier is: SINRk =

2 |aH k Hk bk | H ak Rk ak

(18)

which is non-concave in a k and bk (note that we are now interested in obtaining concave functions since we want to maximize the SINR’s). Given b k , we can independently maximize each SINRk with respect to each ak . The solution is given by the eigenvector corresponding to the maximum gen H eralized eigenvalue of the matrix pencil Hk bk bH k Hk , Rk ak =

αk R−1 k Hk bk .

(19)

The SINR is then SINRk =

H −1 bH k Hk Rk Hk bk .

(20)

which is still non-concave in b k . The SINR at each carrier can be upper-bounded as H −1 bH k Hk Rk Hk bk ≤ λmax,k zk

(21)

(25)

Note that for this simple unweighted case, it is not necessary to resort to the KKT conditions since the result in (25) can be easily  obtained from  the arithmetic-geometric mean inequality ( k zk )1/N ≤ N1 k zk with equality if and only if z k = zl ∀k, l. 3) WMIN-SINR criterion: The maximization of the minimum weighted SINR’s (WMIN-SINR) min k {wk SINRk } in convex form (the objective function and the constraints are all linear) is max t t,{zk }

s.t.

t≤ wk λmax,k zk k zk ≤ PT , zk ≥ 0.

1 ≤ k ≤ N,

(26)

with solution readily found (by noting that the constraint on t must be satisfied with equality for all k) as PT zk = wk−1 λ−1 . (27) max,k  −1 −1 w l l λmax,l Note that, for the unweighted case w k = 1, (27) coincides with (17).

C. BER-based criteria In this subsection, we deal directly with the minimization of the average uncoded BER over all carriers, which can be considered as the best criterion 4 . Under the Gaussian assumption, the symbol error probability P e can be analytically expressed as a function of the SINR:   Pe = α Q β SINR (28) where α and β are constants that depend on the signal constellation and Q is the Q-function defined as Q (x) = ∞ −λ2 /2 √1 e dλ. The BER can be approximated as BER ≈ 2π x Pe /k where k is the number of bits per symbol (assuming that a Gray encoding is used to map the bits into the constellation points). √  1) SUM-BER: Since Q β x is convex (its second derivative is non-negative), the minimization of the sum of the BER’s in convex form is 5    min βk λmax,k zk k αk Q {zk }  (29) s.t. k zk ≤ PT , zk ≥ 0. Unfortunately, (29) does not seem to have a simple closedform solution and one has to resort to interior-point methods as explained in Section III. For completeness and implementation purposes, we give the first and

second derivatives √  β −βx/2 −1/2 ∂ of the Q-function: ∂x Q βx = − 8π e x and

    √ β −βx/2 −1/2 ∂2 1 −1 x +β . βx = 2 8π e x ∂x2 Q V. I NTRODUCING A DDITIONAL C ONSTRAINTS : PAR One of the nice properties of expressing a problem in convex form is that additional constraints can be added as long as they are convex without affecting the solvability of the problem. Of course, with additional constraints, the closed-form solutions previously obtained (i.e., (12), (14), (16), (17), (24), (25), and (27)) are not valid. One of the main practical problems that multicarrier systems face is the Peak to Average Ratio (PAR). Indeed, multicarrier signals exhibit Gaussian-like time-domain waveforms with relatively high PAR, i.e., they exhibit large amplitude spikes when several frequency components add in-phase. These spikes may have a serious impact on the design complexity and feasibility of the transceiver’s analog front-end (i.e., high resolution of D/A-A/D converters and power amplifiers with a linear behavior over a large dynamical range). In practice, the transmitted signal has to be clipped when it exceeds a certain threshold. A variety of techniques have been devised to deal with the PAR [9]. In this section we show how the PAR can be taken into account into the design of the beamvectors using a convex optimization framework. Note that the already existing techniques to cope with the PAR and the proposed approach are not exclusive and can be simultaneously used. 4 In practice, multicarrier communication systems use some type of coding over the carriers and/or over different transmissions to reduce the BER (usually some orders of magnitude). The minimization of the average uncoded BER is in any case a meaningful design criterion since the final coded BER depends strongly on the uncoded BER (if the decoding is based on hard decisions, both quantities are strictly related). 5 Note that the extension to the minimization of the weighted sum of the BER’s is straightforward simply by using wk αk instead of αk .

The PAR is defined as PAR
max

0≤t≤Ts

A2 (t) σ2

(30)

where Ts is the symbol period,  A(t) is the zero-mean transmitted signal, and σ 2 = E A2 (t) . Since the number of carriers is usually large (N ≥ 64), A(t) can be accurately modeled as a Gaussian random process (central-limit theorem) with zero mean and variance σ 2 [9]. Using this assumption, the probability that the PAR exceeds certain threshold or, equivalently, the probability that the instantaneous amplitude exceeds a clipping value A clip is

 Aclip Pr {|A(t)| > Aclip } = 2 Q . (31) σ The clipping probability of an OFDM symbol is then [9]

2N Aclip . (32) Pclip (σ) = 1 − 1 − 2 Q σ In other words, in order to have a clipping probability lower than P with respect to the maximum instantaneous amplitude Aclip , the average signal power must satisfy σ ≤ σclip (P ) =

Q−1



Aclip 1−(1−P )1/(2N ) 2

.

(33)

When using multiple antennas for transmission, the previous condition has to be satisfied for each transmit antenna. The resulting constraint is linear in the optimization variables z k ’s: N 2 1
 2 zk [umax,k ]i  ≤ σclip N

1 ≤ i ≤ nT

(34)

k=1

where umax,k is the eigenvector corresponding to the maxi−1 mum eigenvalue of (H H k Rk Hk ). Such a constraint has two effects in the solution: (i) the power distribution over the carriers changes with respect to the distribution without the constraint, and (ii) the total transmitted power drops when necessary. VI. S IMULATION R ESULTS For the numerical results, we have chosen the European standard for WLAN HIPERLAN/2 [2]. It is based on the multicarrier modulation OFDM (64 carriers are used in the simulations). We assume multiple antennas at both the transmitter and the receiver, obtaining therefore the multicarrier MIMO model used throughout the paper. We assume perfect channel knowledge at both sides of the communication link 6 . The frequency selectivity of the channel is modeled using the power delay profile type C for HIPERLAN/2 as specified in [10], which corresponds to a typical large open space indoor environment for NLOS conditions with 150ns average r.m.s. delay spread and 1050ns maximum delay (the sampling period is 50ns [2]). The spatial correlation of the MIMO channel is modeled according to the model Novi3 defined in [11], which corresponds to a reception hall. It provides a large open indoor 6 In practice, channel estimation errors exist and it is therefore necessary to either quantify the loss for each of the methods or directly modify the design to take into account possible channel estimation errors. This is however out of the scope of this paper.

Outage BER (QPSK) in a MIMO(2,2) channel

0

10

Prob. clipping (mu=4.0) in a MIMO(2,2) channel

0

10

PROD−MSE PROD−SINR SUM−MSE MAX−MSE SUM−BER

SUM−MSE SUM−MSE (PAR const.) −1

Prob. clipping

−1

10

−2

−2

10

BER

10

10

−3

−3

10

−4

10

10

−5

0

−2

−5

10

0

5 SNR (dB)

10

15

Fig. 2. BER (at an outage probability of 5%) vs. the SNR when using a QPSK constellation on a 2 × 2 MIMO channel.

environment with two floors, which could easily illustrate a conference hall or a shopping galleria scenario. The matrix  channel generated was normalized so that n E [|Hij (n)|2 ] = 1. The SNR is defined as the transmitted power normalized with the noise variance. The results are given in terms of outage BER (averaged over the carriers), i.e., the BER that can be guaranteed with some probability or, equivalently, the BER that is not achieved with some small outage probability. The outage BER is a more realistic measure than the commonly used mean BER for typical systems with delay constraints. For the simulations, we have used the five unweighted design criteria: SUM-MSE, PROD-MSE, MAX-MSE, PRODSINR, and SUM-BER. Methods SUM-MSE and PROD-MSE are easily solved using the water-filling solutions of (12) and (14). Methods MAX-MSE and PROD-SINR also have closedform solutions given by (17) and (25). Method SUM-BER is solved using an interior-point method (see Section III). In Fig. 2, the BER is plotted vs. the SNR when using a QPSK constellation on a 2 × 2 MIMO channel. Clearly, the SUM-BER criterion has the lowest BER because it was designed for that. The SUM-MSE method performs very close to the SUM-BER for BER’s higher than 10 −2 . The MAX-MSE criterion performs extremely close to the SUM-BER for BERs lower than 10−2 . In overall, the MAX-MSE seems to be the second best criterion and the SUM-MSE the third. Regarding the PROD-MSE and the PROD-SINR criteria, they perform really bad in terms of BER. Now we consider the introduction of PAR constraints (interior-point methods are used to solve the problems). We parameterize  the clipping amplitude with respect to µ as Aclip = µ PT /nT . In Fig. 3, the BER is plotted vs. the SNR when using QPSK on a 2 × 2 MIMO channel for the SUM-MSE method with and without PAR constraints (µ = 4 and Pclip ≤ 10−2 ). With the additional PAR constraint, the BER is slightly higher. However, the system is guaranteed to have a clipping probability of at most 10 −2 unlike in the unconstrained case where nothing can be guaranteed. Recall that in a practical system, the final BER increases due to the clipping. VII. C ONCLUSIONS In this paper, we have formulated the joint design of transmit and receive beamvectors for a multicarrier MIMO channel

BER

−5

15

SUM−MSE SUM−MSE (PAR const.)

−1

10

10

10

Outage BER (QPSK) in a MIMO(2,2) channel

0

10

5 SNR (dB)

−3

10

−4

10

−5

10

−5

0

5 SNR (dB)

10

15

Fig. 3. Probability of clipping and BER (at an outage probability of 5%) as a function of the SNR for the SUM-MSE criterion with and without PAR constraint (µ = 4 and Pclip ≤ 10−2 ).

within the general and powerful framework of convex optimization theory. From this perspective, a great span of design criteria can be easily accommodated and efficiently solved even though closed-form expressions may not be available. In particular, we have considered some interesting design criteria based on the MSE, SINR, and BER, obtaining closed-form expressions in most cases. The simple criterion that minimizes the maximum MSE over all the carriers happens to have a really good performance with a low implementation complexity. We have also shown how to easily include additional constraints to control the PAR in the system design. R EFERENCES [1] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. on Communications, vol. 46, no. 3, pp. 357–366, March 1998. [2] ETSI, “Broadband radio access networks (BRAN); HIPERLAN type 2; physical (PHY) layer,” ETSI TS 101 475 V1.2.2, pp. 1–41, Feb. 2001. [3] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization,” to appear in IEEE Trans. on Signal Processing, submitted Feb. 2002 (revised Dec. 2002). [4] S. Boyd and L. Vandenberghe, Introduction to Convex Optimization with Engineering Applications, Course Notes (available at http://www.stanford.edu/class/ee364). Stanford University, 2000. [5] M. Bengtsson and B. Ottersten, “Optimal transmit beamforming using convex optimization,” submitted to IEEE Trans. on Communications, Oct. 1999. [6] J. Yang and S. Roy, “On joint transmitter and receiver optimization for multiple-input-multiple-output (MIMO) transmission systems,” IEEE Trans. on Communications, vol. 42, no. 12, pp. 3221–3231, Dec. 1994. [7] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion,” IEEE Trans. on Communications, vol. 49, no. 12, pp. 2198– 2206, Dec. 2001. [8] A. P. Iserte, A. I. Perez-Neira, D. P. Palomar, and M. A. Lagunas, “Power allocation techniques for joint beamforming in OFDM-MIMO channels,” in Proc. EUSIPCO 2002, Toulouse, France, Sept. 2002. [9] D. J. G. Mestdagh and P. M. P. Spruyt, “A method to reduce the probability of clipping in DMT-based transceivers,” IEEE Trans. on Communications, vol. 44, no. 10, pp. 1234–1238, Oct. 1996. [10] ETSI, “Channel models for HIPERLAN/2 in different indoor scenarios,” ETSI EP BRAN 3ERI085B, pp. 1–8, March 1998. [11] L. Schumacher, J. P. Kermoal, F. Frederiksen, K. I. Pedersen, A. Algans, and P. E. Mogensen, “MIMO channel characterisation,” Deliverable D2 V1.1 of IST-1999-11729 METRA project (available at http://www.istmetra.org), pp. 1–57, Feb. 2001.