IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 9, NOVEMBER 2012

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Reduced Complexity Power Allocation Strategies for MIMO Systems With Singular Value Decomposition Alberto Zanella, Senior Member, IEEE, and Marco Chiani, Fellow, IEEE

Abstract—We consider wireless multiple-input–multiple-output (MIMO) systems in fading environments, with frequency flat fading, channel state information at both transmitter and receiver sides, and linear precoding based on singular value decomposition (SVD). The optimal solution for these MIMO SVD systems, in terms of achievable rate, requires water filling to optimally allocate power to the different channel eigenmodes. Alternatively, reduced complexity power allocation methods can be employed. We propose two power allocation techniques that only require statistical knowledge of the channel matrix coefficients and do not need knowledge of the instantaneous values of the channel state. To study these power allocation methods, we introduce a new expression for the exact distribution of the eigenvalues of Wishart matrices, where the probability density function of the !th largest eigenvalue is given as a sum of terms of the form xβ e−xδ . The expression is here used, in the context of MIMO SVD systems, to obtain the achievable rate for both zero-outage and nonzero-outage strategies. We show that low-complexity methods have performance very similar to water-filling methods. Index Terms—Multiple-input–multiple-output (MIMO) systems, power allocation, singular value decomposition (SVD), Wishart matrices.

I. I NTRODUCTION

T

HE PIONEERING contributions of [1] and [2] have shown that multiple-input–multiple-output (MIMO) systems can provide very large spectral efficiency, making them suitable for high throughput wireless systems. Furthermore, in the presence of frequency-selective channels, the combined use of orthogonal frequency-division multiplexing (OFDM) and MIMO can effectively counteract the effect of intersymbol interference by transforming the frequency-selective channel into a number of frequency-flat MIMO channels [3]. Closedform expressions for the ergodic and outage capacity in the presence of frequency-flat fading environments and perfect channel state information at the receiver (CSIR) have been obtained, over the past few years, for the cases of uncorrelated [4], semicorrelated [5], [6], and fully correlated MIMO channels [7]. The performance of minimum mean-square errorManuscript received December 16, 2011; revised May 7, 2012 and July 17, 2012; accepted July 18, 2012. Date of publication July 27, 2012; date of current version November 6, 2012. This work was supported in part by the EC under the ICT Seventh Framework Programme, Grant 288502 (CONCERTO) and in part by the project PEGASUS (MS01-00024), funded by the Italian Ministry for Economic Development. The review of this paper was coordinated by Prof. Y. Gong. The authors are with the IEIIT/CNR, DEI, University of Bologna, 40136 Bologna, Italy (e-mail: [email protected], [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2012.2210575

based MIMO schemes, in terms of symbol error probability (SEP), has been studied in [8]. When the channel is known at both the transmitter and the receiver, the capacity is obtained with MIMO transmission based on singular value decomposition (SVD) with water-filling power allocation [3], [4]. The ergodic capacity of a MIMO SVD system has been investigated in [9] and [10] in the presence of uncorrelated Rayleigh and Rician fading, respectively. The performance of a MIMO SVD system has been investigated in [11]–[13] in the context of uncoded transmission. More precisely, closed-form expressions for the symbol error rate, averaged over all SVD subchannels (also called spatial channels or eigenmodes), have been derived in [11] for Rayleigh fading. The SEP for each SVD subchannel has been analyzed in [12] for large signal-to-noise ratio (SNR) values. The SEP of each individual SVD subchannel has been investigated in [13] in the presence of Rician fading. The performance of power allocation strategies for MIMO SVD systems with adaptive modulation has been studied in [14]. Power allocation strategies for MIMO SVD have also been investigated in [15] and [16] in the presence of delay constraints. In [15], some long- and short-term energy constraints have been considered, and the optimal power allocation has been derived. In [16], suboptimal power allocation strategies have been evaluated in the context of MIMO OFDM systems. The advantage of using the channel state information (CSI) of all users at the base station is exploited in [17], where an SVDbased joint multiuser transmitter and multiuser detector-aided MIMO system is proposed and investigated. It is well known that the performance of MIMO SVD systems depends on the distribution of the eigenvalues of HH† , where H denotes the channel gain matrix.1 More specifically, the performance analysis of MIMO SVD systems requires the evaluation of expressions like E{ς(λ! )}, where λ! is the #th largest eigenvalue of HH† , and ς(·) is a suitable function. When the elements of the channel matrix can be modeled as complex Gaussian random variables, i.e., when the propagation environment is characterized by Rayleigh or Rician fading, HH† becomes a Wishart (or pseudo-Wishart) matrix [18], [19]. Expressions for the cumulative distribution function (cdf) of the largest, smallest, and #th largest eigenvalue of a Wishart matrix have been obtained in previous works (see, for instance, [20] and [21]); however, direct computation of the corresponding probability density function (pdf) from the cdf is quite complex, and the methodology used to derive the cdfs was specific to the 1 The

superscript † denotes conjugation and transposition.

0018-9545/$31.00 © 2012 IEEE

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 9, NOVEMBER 2012

particular Wishart matrix considered (correlated, uncorrelated, central, and noncentral). Apart from [20] and [21], several other papers derived exact expressions for the largest and the most general #th largest eigenvalue. However, these results, which will be briefly summarized in Section IV, do not usually lead to closed-form expressions for E{ς(λ! )} for many cases of interest. Motivated by this fact, we propose a methodology that allows the derivation of the pdf of the #th eigenvalue of both central Wishart and pseudo-Wishart matrices or quadratic forms with arbitrary one-sided correlation matrix, expressed by sums of terms in the form xβ e−xδ , leading to the direct and straightforward evaluation of several statistical expectations, which are of interest for the performance analysis of MIMO systems. These results are used in this paper to investigate the performance of some power allocation methods for MIMO SVD systems. In particular, we propose two novel low-complexity power allocation methods and derive closed-form expressions for the achievable rate for both zero-outage and nonzero-outage strategies. We also show that the performance of our low-complexity methods is comparable with that of conventional water-filling methods in propagation environments with both uncorrelated and semicorrelated (that is, with correlation among either transmit or receive antennas) Rayleigh fading. Our results can be summarized as follows: 1) We propose a reduced complexity power allocation algorithm, denoted as fixed energy (FE), where the amount of energy allocated to each spatial channel is computed a priori based on the channel statistics. With such a method, power allocation is evaluated only once and remains fixed over the different realizations of fading. Since FE requires solving an optimization problem, we also obtain a simple approximate solution called the approximate FE (AFE) method. 2) We propose a simplified FE algorithm called uniform allocation (UA), where the total energy is uniformly assigned to a subset of the spatial channels based on the channel statistics and SNR conditions. 3) We propose an expression for the pdf of the #th eigenvalue of central Wishart and pseudo-Wishart matrices. This result, which can applied to the performance analysis of several MIMO SVD systems, is used to obtain closedform expressions for the rate obtained using the power allocation schemes AFE and UA. 4) As an example of the application of the pdf expression discussed in the previous point, we consider the algorithm proposed in [16], denoted as fixed rate (FR), and evaluate a closed-form expression for the mean (over fast fading fluctuation) energy necessary to obtain the target rate. 5) We show that the performance of UA and AFE is similar to that of the more complex water-filling approach. We also show that AFE is quite robust in the presence of imprecise knowledge of the channel state statistics. This paper is organized as follows. In Section II, we discuss the various energy allocation methods in MIMO SVD systems. The proposed energy allocation methods are presented in Section III. Closed-form expressions for the pdf of λ! and for some expectations of functions of λ! are calculated in

Section IV. Numerical results are shown in Section V. Conclusions are given in Section VI. Throughout this paper, we will use fX (x) to denote the pdf of the random variable (r.v.) X, and E{·} denotes the expectation operator. In case of multivariate r.v.s X1 , . . . , Xq , with X1 ≥ X2 ≥ · · · ≥ Xq , we will use f!:q (x) to denote the marginal pdf of the #th largest r.v X! . We will use bold for vectors and matrices, so that x ∈ Cm×1 denotes a vector of m complex elements, and A ∈ Cm×n denotes an (m × n) matrix with complex elements ai,j , with aj denoting the jth column vector of A. We will use |A| to denote the determinant of A ∈ Cm×m and the superscript † for conjugation and transposition. II. E NERGY A LLOCATION IN M ULTIPLE -I NPUT-M ULTIPLE -O UTPUT S INGULAR VALUE D ECOMPOSITION A. System Model The discrete-time baseband received vector y ∈ CNR ×1 for a MIMO system with NT transmit and NR receive antennas is given by [22], [23] y = Hz + ν

(1)

where z ∈ CNT ×1 is the transmitted signal vector, H is the channel matrix, and ν ∈ CNR ×1 is the additive circularly symmetric complex Gaussian (CSCG) noise vector with E{ν} = 0 and E{νν † } = N0 I. The elements of H, hi,j , are modeled as CSCG, with E{|hi,j |2 } = 1. In the presence of CSI at the transmitter, an SVD of H = UDV† can be performed, where U ∈ CNR ×NR , V ∈ CNT ×NT are unitary matrices, and √ D∈ CNR ×NT is a diagonal matrix whose nonzero entries λi are the square roots of the nonzero eigenvalues of HH† . To exploit ˆ is precoded at the properties of the SVD, the symbol vector z the transmitter to obtain z = Vˆ z , and at the receiver, the vector ˆ = U† y. Thus, the decoder y is multiplied by U† to obtain y output can be written as ˆ = Dˆ ˆ y z+ν

(2)

ˆ and ν have the same ˆ = U† ν. Since U is unitary, ν where ν ˆ can be written as distribution. The ith component of y ! yˆi = λi zˆi + νˆi i = 1, . . . , Nmin (3) ∆

where Nmin = min{NR , NT }. Equation (3) shows that, using SVD, the MIMO channel has been transformed into Nmin parallel (spatial) channels (also called eigenmodes or eigenchannels), with channel gains (in power) λ = (λ1 , . . . , λNmin ). B. Energy Allocation Techniques for MIMO SVD Systems Once the MIMO channel has been decomposed into Nmin parallel channels by SVD, we can decide how to use these channels and, in particular, how much energy to allocate to each eigenmode. The simplest strategy of the first type is the uniform power allocation (UPA), where the total energy E is uniformly subdivided over the eigenmodes. If all modes are used, UPA gives the same available rate as MIMO with CSIR only, but with the advantage that only linear processing is needed at the

ZANELLA AND CHIANI: REDUCED COMPLEXITY POWER ALLOCATION STRATEGIES FOR MIMO SYSTEMS WITH SVD

transmitter and receiver sides, instead of the more complicated maximum likelihood or other suboptimal decoders [22], [23]. In general, energy adaptation strategies can be divided into the following two categories. 1) Constant Instantaneous Total Energy (CITE): With CITE, the total available energy E is constant in time and is allocated over the eigenmodes, depending on the instantaneous channel gains λ! to maximize the total rate. Given the total instantaneous energy E, these methods imply that the total available rate R = R(λ, E) is time varying (depending on the channel realizations). Therefore, if the source has a fixed rate Rsource , we will have Rsource < R for some channel realizations, meaning that it is possible to transmit with an arbitrary low error probability, and Rsource > R for some others, resulting in an outage. The mean over the channel ensemble gives the average available rate, which depends on E. The optimal allocation method is obtained by applying the well-known water-filling principle, which requires that elements zˆi have to be independent, zero mean, and Gaussian [3], [4], [24]. The achievable rate of CITE with water filling, here denoted as WE, is given by [4] RWE = RWE (λ, E) =

k WE "

log2 (µWE (kWE ) λi )

(4)

i=1

The power allocation strategy for CITR consists of minimizing the transmitted energy under the constraint that the total instantaneous rate is equal to a fixed value R. The optimal solution, here denoted as WR, requires water filling again. The values of energy Ei has to be chosen as in (6), but now µWE (k) must be replaced by [15] 2R/k µWR (k) = ) +1/k *k λ i i=1

Ei = µWE (k) =

1 k

k

" E + λ−1 N0 i=1 i

$

(5)

and kWE = {max k : µWE (k) ≥ λ−1 k }. % min The choice of Ei = E{|ˆ zi |2 }, with N I=1 Ei ≤ E, achieving the rate RWE is & ' ( N µ (k )−λ−1 i = 1, . . . , kWE i Ei = 0 WE WE (6) 0 i = kWE +1, . . . , Nmin . Note that, while E is constant, the allocated energies Ei = Ei (λ, E) and the resulting rate RWE = RWE (λ, E) change with the channel eigenvalues. Note also that the foregoing criterion is usually denoted as space water filling owing to the presence of a constraint on the instantaneous energy. Ergodic capacity with water filling can be obtained by imposing a constraint of the mean (over fast fading fluctuations) energy (giving the so called space–time water filling) [9], [10], [25]. 2) Constant Instantaneous Total Rate (CITR): With CITR, the total rate (the sum of the eigenmodes rates) is kept constant over all channel realizations by properly allocating the necessary energy to the eigenmodes. Given the target total instantaneous rate R, these methods imply that the transmitted total energy E = E(λ, R) is time varying (depending on the channel realizations). The mean over the channel ensemble gives the ¯ = Eλ {E}, which depends on average transmitted energy E R. In this case, a source transmitting at a constant rate not exceeding R will never experience outage. These are therefore called zero-outage power allocation strategies.

(7)

where kWR = {max k : µWR (k) ≥ λ−1 k }. As already observed, this allocation is zero-outage as the value of rate R is guaranteed for any instantaneous state of the channel. Note that, although the available instantaneous rate R is constant, the allocated energies Ei = Ei (λ, R) and the total energy E = E(λ, R) change with the channel eigenvalues. A suboptimal, but less complex, power allocation algorithm has been proposed in [16] in the context of MIMO OFDM systems. Starting from the available rate R and regardless of the instantaneous values λi may take, the FR algorithm fixes the rate ri to be allocated to the ith spatial channel [16]. The rates are specified based on the knowledge of some moments of the spatial channels. Once the rate is assigned (offline), the power allocation can be easily obtained as

where #

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N0 (2ri − 1) λi

i = 1, . . . , Nmin .

(8)

Note that, although the available rate R is constant, the allocated energies Ei = Ei (λ, R) and the total instantaneous energy E = E(λ, R) change with the channel eigenvalues. The optimal choice of ri that minimizes the total instantaneous energy E is [16] , ) + µFR (kFR ) log i = 1, . . . , kFR 2 ϑ i (9) ri = 0 i = kFR + 1, . . . , Nmin ∆

where ϑi = E{1/λi }, kFR = {max k : µFR (k) ≥ ϑk }, and µFR (k) = 2R/k

#

k -

i=1

ϑi

$1/k

(10)

.

As shown in (9) and (10), the rate is assigned a priori by using the knowledge of ϑi = E{1/λi }, leading to a zero-outage strategy. Starting from (8), the mean (over fast fading fluctuations) value of the total transmitted energy can be written as ¯ = Eλ {E} = N0 E = N0

#

kFR " i=1

(2ri − 1)ϑi

kFR µFR (kFR ) −

kFR " i=1

ϑi

$

.

(11)

Numerical results in [16] indicate that the performance of FR (in terms of overall amount of energy spent to obtain a given rate) is comparable with the most complex WR.

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 9, NOVEMBER 2012

Therefore, FR might be preferred since it does not require water filling for each channel realization. As shown in (8), only the evaluation of Ei is necessary once the channel is known at the transmitter. The energy adaptation strategy of FR needs to have a reliable estimate of ϑi . In the case of erroneous estimate of the true expectation (i.e. ϑ˜i '= ϑi ), the mean energy becomes / kest . " µ ˜FR (kest ) ¯ Eest = N0 − 1 ϑi (12) ϑ˜i i=1

* where µ ˜FR (k) = 2R/k ( ki=1 ϑ˜i )1/k , and kest = {max k : µ ˜FR (k) ≥ ϑ˜k }. The performance of FR has been investigated in [16] via simulations. The results in Section IV on the moments of functions of ordered eigenvalues of HH† allow us to obtain closed-form expressions for the average (with respect to fading) energy necessary by FR to obtain a given rate. III. P ROPOSED S IMPLIFIED C ONSTANT I NSTANTANEOUS T OTAL E NERGY S TRATEGIES

Motivated by the idea of FR of using the statistical knowledge of channel statistics to reduce the complexity of the power allocation algorithm, we propose some novel allocation techniques that are suitable for CITE strategies. Compared with the conventional water filling approach, the proposed solutions calculate the amount of energy allocated to each spatial channel by exploiting the statistical knowledge of the channel conditions. A. FE and AFE Power Allocation Methods The first allocation scheme is denoted as FE and can be seen as a low-complexity alternative to the conventional water-filling method WE. The main characteristic of FE is that the fraction of energy allocated to each spatial channel is evaluated a priori based on channel statistics [26]. More specifically, the value of energy assigned to each spatial channel is chosen to maximize the average (over fading realizations) obtainable rate. The FE problem can be formulated as follows. Find the ˘i , i = . . . Nmin , that maximize (fixed) values E ,N # $0 min " ˘i E E log2 1 + λi N0 i=1 % min ˘ subject to the constraint N i=1 Ei ≤ E. Using the method of Lagrange multipliers, E˘i can be found by solving the following system:  %N min ˘   i=1 Ei = E ˘i ) dhi (E (13) + η = 0, i = 1, . . . , Nmin ˘i dE   ˘i ≥ 0, E i = 1, . . . , Nmin ∆

where η is the Lagrange multiplier, and hi (x) = E{ln(1 + λi x/N0 )}. Since & 5 ˘i ) λi dhi (E =E (14) ˘i ˘i dE N0 + λ i E

we see that, once we are able to compute the expectation in (14), system (13) can be solved by using numerical optimization techniques. To obtain closed-form, although approximate, solutions for (13), we apply Jensen’s inequality to hi (x) and obtain hi (x) ≤ ln(1 + ωi x/N0 ), where ωi = E{λi }. Now, system (13) can be approximated as  %Nmin ˘i = E  i=1 E ωi + η = 0,  N0 +ωi E˘i ˘ Ei ≥ 0,

i = 1, . . . , Nmin

(15)

i = 1, . . . , Nmin .

The power allocation method corresponding to the optimization problem (15) is denoted as AFE, whose solution is & ' ( −1 ˘i = N0 µAFE (kAFE ) − ωi , i ≤ kAFE E 0, kAFE < i ≤ Nmin (16) % k −1 with µAFE (k) = k −1 (E/N0 + i=1 ωi ), and kAFE = {max k : µAFE (k) ≥ ωk−1 }. The average rate of AFE can be written as R=

N min " i=1

) +7 6 ˘i /N0 E log2 1 + λi E

(17)

which can be calculated in closed-form once the distribution of the ith eigenvalue is known. Note that, from a computational point of view, the complexity of the evaluation of µWE (k) and µAFE (k) is the same. The difference is that µWE (k) has to be evaluated for each channel realization H, whereas µAFE (k) is evaluated a priori (based on the channel statistic) and needs to be computed only once. Numerical results will show that the performance of the suboptimal AFE method is comparable with that of the more complex water-filling WE. B. UA Over a Selected Subset of the Eigenmodes With UA, the energy is uniformly allocated over the kUA strongest modes, where kUA has to be properly designed. This subset selection with uniform energy allocation (UA) is thus a CITE method like WE, FE, and AFE. The mean rate achievable by UA can be written as R=

k UA " i=1

&

E log2

.

E 1 + λi kUA N0

/5

(18)

which can be calculated in closed-form using the distribution of the kUA strongest eigenvalues of HH† . Note that the allocation method (UA) belongs to the class of partial (CSI) techniques. Feedback information required by UA is limited to the kUA strongest eigenvectors, and the remaining Nmin − kUA eigenvectors are not fed back to the transmitter. Numerical results will show that this simple selection method, with the proper choice of kUA , gives a rate that is comparable with those of AFE and WE.

ZANELLA AND CHIANI: REDUCED COMPLEXITY POWER ALLOCATION STRATEGIES FOR MIMO SYSTEMS WITH SVD

IV. M OMENTS OF F UNCTIONS OF THE O RDERED E IGENVALUES OF HH†

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The constant K is given by

The analysis of the previously described allocation methods requires the evaluation of statistical averages of functions of the eigenvalues of HH† . A friendly expression for the largest eigenvalue for the uncorrelated central Wishart case (which corresponds to the uncorrelated Rayleigh fading case) was proposed in [27]. This expression is composed by a sum of terms of the form di,k xk e−ix , which makes it extremely useful for the evaluation of expressions such as E{ς(λmax )}. However, the evaluation of the coefficients di,k requires symbolic computation. A more efficient algorithm to obtain coefficients di,k was proposed in [28], still requiring symbolic computation. A closed-form expression for the moment generating function (MGF) of the largest eigenvalue for both uncorrelated and correlated central Wishart cases can be found in [29]. This expression can be used to obtain closed-form expressions for the ergodic capacity and for the SEP of the MIMO maximal ratio combiner. Although the expression for the MGF is written as the sum of terms xβ e−xδ , its evaluation is not straightforward. Expressions for the pdf of the #th largest eigenvalue for uncorrelated central, correlated central, and uncorrelated noncentral Wishart cases were proposed in [30]–[32]. The cdf and pdf for the case of uncorrelated central Wishart were also proposed in [33]. An expression for the cdf and a first-order expansion for the pdf of λ! in the uncorrelated noncentral case was given in [34]. Unfortunately, except obviously the firstorder expansion in [34], all these expressions do not lead to closed-form expressions after further operations of integration. Motivated by these results, we here propose an expression for an arbitrary eigenvalue of a Wishart matrix in % ordered βn −xδn x e , where βn and δn are suitable the form of n constants [35].

*L mi p (−1)p(n−nmin ) i=1 φ(i) K= (m i m j *L * ' Γ(nmin ) (p) i=1 Γ(mi ) (mi ) i φ(2) . . . > φ(L) −1 are the L distinct eigenvalues of %LΦ , with associated multiplicities m1 , . . . , mL such that i=1 mi = n. The (n × n) matrix G(x, φ) has elements , gi (xj ) = (−xj )d(i) e−φ(e(i)) xj , j = 1, . . . , nmin gi,j = n−j−d(i) g¯i,j = [n − j]d(i) φ(e(i)) , j = nmin + 1, . . . , n (21) ∆ where [a]n → a(a − 1) · · · (a − n + 1), and e(i) denotes the unique integer such that m1 + · · · + me(i)−1 < i ≤ m1 + · · · + me(i) d(i) =

e(i) "

k=1

mk − i.

Proof: The proof is given in [36]. ! Note that Lemma 1 gives, in a compact form, the general joint distribution for the eigenvalues of a central Wishart (p ≥ n) and central pseudo-Wishart (n ≥ p), with arbitrary onesided correlation matrix, even with nondistinct eigenvalues. To obtain a friendly expression for the pdf of the largest eigenvalue of HH† , we introduce Lemma 2 and Theorem 1. The theorem represents one of the main results of this paper. Lemma 2: Letting X1 , X2 , . . . , Xnmin be i.i.d. or exchangeable r.v.s, then the marginal pdf of the #th largest r.v. can be written as [37, p. 99, eq. (5.3.1)] . /. / n min " s−1 nmin f!:nmin (x) = (−1)s−! fmin:s:nmin (x) (22) #−1 s s=!

†

A. Derivation of the pdf of the #th Largest Eigenvalue of HH

When the elements of the channel matrix H are modeled as zero-mean complex Gaussian random variables, i.e., when the propagation environment is characterized by Rayleigh fading, HH† becomes either a central Wishart or a central pseudoWishart matrix [19]. The joint pdf of the eigenvalues for Gaussian quadratic forms and central Wishart matrices with arbitrary one-sided correlation matrix can be expressed in a concise way by using the following lemma. Lemma 1: Denoting by A a complex Gaussian (p × n) random matrix with zero mean, unit variance, independent identically distributed (i.i.d.) entries and by Φ an (n × n) positive definite matrix, the joint pdf of the (real) nonzero ordered eigenvalues λ1 ≥ λ2 ≥ . . . ≥ λnmin of the (p × p) quadratic form W = AΦA† is fλ (x1 , . . . , xnmin ) = K |V(x)| · |G(x, φ)|

nmin

ξ(xi )

(19)

i=1

where nmin = min(n, p), ξ(x) = xp−nmin , and V(x) is the (nmin × nmin ) Vandermonde matrix with elements vi,j = xi−1 j .

where fmin:s:nmin (x) denotes the pdf of the smallest r.v. considered in any arbitrary subset of s r.v.s. over the set of nmin r.v.s. X1 , . . . , Xnmin .2 Theorem 1: Let λ1 , λ2 , . . . , λnmin be the nonzero ordered eigenvalues of a (p × p) quadratic form W, whose joint pdf is described by (19)–(21). Then, the pdf of the #th largest eigenvalue λ! is f!:nmin (x) . /. / n min " s − 1 nmin sK (−1)s−! = #−1 s nmin ! s=! %s "" × sgn(α)sgn(σ)(−1) l=1 d(σl ) As,nmin (α, σ) α σ " α +d(σ )+p−n −1+%s−1 t −x %s φ s min l l=1 (e(σl )) l=1 × x s e ×

s−1 l=1

t

l φt(e(σ l ))

(αl + d(σl ) + p − nmin + 1)! tl !

(23)

2 X and X nmin in [37] denote the largest and the smallest r.v., respectively. 1 Furthermore, the lemma in [37] is expressed in terms of cumulative density functions. The extension to pdfs is straightforward.

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where "

t

Starting from (27), fmin:s:nmin (x) can easily be written as "

α1 +d(σ1 )+p−nmin −1

"

t1 =0

αs−1 +d(σs−1 )+p−nmin −1

"

···

(24)

ts−1 =0

α and σ are permutations of the integers 1, 2, . . . , nmin and 1, 2, . . . , n, respectively. The sgn (π) denotes the sign of permutation π, and As,nmin (α, σ) is defined as follows: n -

n−l−d(σ ) l]d(σl ) φ(e(σl )) l

[n − l=nmin +1 nmin (αl + d (σl ) + p − nmin 1)! × (−1)d(σl ) . αl +d(σl )+p−nmin φ(e(σ l=s+1 l ))

As,nmin (α, σ) =

(25)

Proof: We start the proof by applying Lemma 2 to the nonzero ordered eigenvalues λ1 , λ2 , . . . , λnmin of W. Since the eigenvalues λ1 , . . . , λnmin are not i.i.d., the application of Lemma 2 requires that they are exchangeable, which is fortunately the case for the joint pdf in (19). Note that the results of Lemma 2 can also be applied to the case of noncentral uncorrelated Wishart matrices, whose pdf can be written as [31, eq. (1), Table I]. Lemma 2 was also used in [33] for the evaluation of the cdf and pdf of λ! . Unfortunately, the expression obtained in [33] for the pdf cannot be used to obtain closed-form expressions for E{ς(λ! )} in many cases of interest. ∆ To obtain fmin:s:nmin (x) in (22), we evaluate fλ:nmin (x) = fλ1 ,...,λs :nmin (x1 , . . . , xs ), which denotes the joint pdf of s unordered eigenvalues taken over the original ensemble of nmin . The distribution fλ:nmin (x) can be calculated as 8 ∞ 8 ∞ K fλ:nmin (x) = ··· |V(x)| · |G(x, φ)| nmin ! 0 0 nmin ξ(xl )dxs+1 . . . dxnmin (26) × l=1

where we have integrated out nmin − s variables from the joint pdf in (19). By using the approach followed in [5] for the proof of Corollary 2, we get 8 ∞ 8 ∞ K "" fλ:nmin (x) = ··· sgn(α)sgn(σ) nmin ! α σ 0 0 nn min vαl (xl )ξ(xl ) gσl ,l dxs+1 · · · dxnmin × l=1

=

K

l=1

""

sgn(α)sgn(σ)Bs,nmin (α, σ) nmin ! α σ s × vαl (xl )gσl (xl )ξ(xl ) (27) l=1

where Bs,nmin (α, σ) "

#

n -

g¯σl ,l

l=nmin +1

×

nmin

l=s+1

8

vαl (x)gσl (x)ξ(x)dx.

sK " " sgn(α)sgn(σ)Bs,nmin (α, σ) nmin ! α σ

×vαs (xs )gσs (xs )ξ(xs )

s−1 -8 ∞ l=1

xs

vαl (x)gσl (x)ξ(x)dx

(29)

where we have integrated the ordered joint pdf s!fλ:nmin (x) over the variables x1 , . . . , xs−1 . By replacing vαl and ξ(x) with xαl −1 and xp−nmin , respectively, and substituting (21) in (29), we obtain fmin:s:nmin (x) sK " " sgn(α)sgn(σ)(−1)d(σs ) = nmin ! α σ

s +p+d(σs )−nmin −1 −xs φ(e(σs )) × As,nmin (α, σ)xα e s s−1 -8 ∞ −xφ × (−1)d(σl ) xαl +p+d(σl )−nmin −1 e (e(σl )) dx

xs

l=1

(30)

where As,nmin (α, σ) is given by (25). The integral in (30) can further be simplified by using the identity 8 ∞ −xφ xαl +p+d(σl )−nmin −1 e (e(σl )) dx xs

= (αl + d(σl ) + p − nmin − 1)! ×e

−xs φ(e(σ )) l

αl +d(σl )+p−nmin −1

"

tl =0

(xs φ(e(σl )) )tl . tl ! (31)

By substituting (31) in (30) and then (30) in (22), we obtain (23). ! B. Considerations About Theorem 1 Theorem 1 leads to the following remarks. Remark 1: Equation (23) is expressed as a sum of terms of the form xβ e−xδ . As a consequence, the pdf of λ! is given by a weighted sum of Gamma distributions and can be referred to as a Gamma mixture distribution [38]. Remark 2: The pdf of λ! can be written, in a more compact formulation, as " f!:nmin (x) = .(n)xβ(n) e−xδ(n) (32) n

where

$

∞ 0

fmin:s:nmin (x) =

" n

(28)

"

"""" s

α

σ

t1

···

"

(33)

ts−1

and the parameters .(n), β(n), and δ(n) can easily be obtained from (23).

ZANELLA AND CHIANI: REDUCED COMPLEXITY POWER ALLOCATION STRATEGIES FOR MIMO SYSTEMS WITH SVD

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TABLE I COEFFICIENTS Ωs,θ FOR THE UNCORRELATED WISHART CASE (nmin = 2, 3)

TABLE II COEFFICIENTS Ωs,θ FOR THE UNCORRELATED WISHART CASE (nmin = 4, 5 AND nmax = nmin )

Remark 3: For the case of correlated central Wishart matrix, the parameter .(n) is given as . /. / %s nmin sK s−!+ d(σl ) s − 1 (CC) l=1 .(n) = (−1) #−1 s nmin ! × sgn(α)sgn(σ)As,nmin (α, σ)

s−1 -

l φt(e(σ l ))

l=1

(αl + d(σl ) + p − nmin + 1)! × tl !

(34)

δ(n)(CC) =

s "

s−1 "

tl

(35)

l=1

(36)

φ(e(σl )) .

l=1

Remark 4: For the case of uncorrelated central Wishart matrix, .(n), β(n), and δ(n) can be written as (UC)

.(n)(UC) =

×

.

s−1 #−1

s(−1)s−! sgn(α)sgn(σ)As,nmin (α, σ) Γ(nmin ) (nmin )Γ(nmin ) (nmax )nmin !

/.

nmin s

/ s−1 l=1

β(n)

(nmax − nmin + αl + σl − 2)! tl !

= nmax − nmin + αs + σs − 2 +

A(UC) s,nmin (α, σ) =

nmin

l=s+1

t

=

α1 +σ1 +n" max −nmin −2 t1 =0

···

αs−1 +σs−1 +nmax −nmin −2

"

.

ts−1 =0

(40) Remark 5: In the case of uncorrelated central Wishart matrix, fmin:s:nmin (x) in (22) can be simplified as fmin:s:nmin (x) = e

−sx

s(nmax +nmin −s−1)

"

Ωs,θ xθ

(41)

s−1 "

which shows that the number of nonzero coefficients Ωs,θ cannot exceed (s − 1)nmax + (s + 1)(nmin − s) + 1. The lowest index is justified from (38) by considering the cases αs = σs = 2 and tl = 0 for l = 1, . . . , s − 1. The value of s(nmax + nmin − s − 1) is obtained by assuming tl = nmax − nmin + αl + σl − 2 for l = 1, . . . , s −%1. In such a case, (38) be− nmin − 2) + sl=1 (αl + σl ), and the maxicomes s(nmax% mum value of sl=1 (αl + σl ) is 2s nmin − s(s − 1). To give some examples, the coefficients Ωs,θ are given in Tables I–III for some cases of interest. The coefficients are given as (Ωs,0 , Ωs,1 , . . .). C. Expectations of Functions of Eigenvalues

(37)

where nmax = max{p, n} (UC)

"

θ=nmax −nmin

whereas β(n) and δ(n) can be written as β(n)(CC) = αs + d(σs ) + p − nmin − 1 +

δ(n)(UC) = s, and

tl (38)

l=1

(αl + σl − 2 + nmax − nmin )!

(39)

From (32), we can easily evaluate closed-form expressions for the following cases of interest: 1) Mean value of 1/λ! & 5 " 1 (β(n) − 1)! E .(n) (42) = λ! δ(n)β(n) n which can be used for the performance evaluation of FR strategy. More specifically, (42) can be applied in (9) and (11) to evaluate the rate and the average energy allocated to the #th spatial channel, respectively.

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 9, NOVEMBER 2012

TABLE III COEFFICIENTS Ωs,θ FOR THE UNCORRELATED WISHART CASE (nmin = 4, 5 AND nmax = nmin + 1)

2) Mean value of λ! E {λ! } =

" n

.(n)

(β(n) + 1)! δ(n)β(n)+2

(43)

˘i for the AFE which can be used in (16) to obtain E strategy. Note that (43) is valid for arbitrary # except for the case # = q = p. ˘! ) 3) Mean value of λ! /(N0 + λ! E & 5 " β(n)+1 δ(n)N0 λ! N E .(n) 0β(n)+2 e E˘# = ˘! ˘ N0 + λ ! E E n ! . / δN0 × (β(n) + 1)!Γ −1 − β, (44) ˘! E where Γ(α, x) is the upper incomplete Gamma function [39, pp. 949, eq. (8.350.2)], which can be used to obtain a closed-form expression for (14). To derive (44), we used the following identity [39, eq. (3.383.10)]: 8

∞ 0

xa−1 e−dx dx = ca−1 ec d (a − 1)!Γ(1 − a, cd) x+c

(45)

valid for a, c, d ∈ * with a, c, d > 0. 4) Mean value of ln(1 + a! λ! ) E {ln (1 + a! λ! )} =

" n

.(n)

β(n)! β(n)+1 a!

eδ(n)/a#

9 /: /k . β(n)+1 . " a! δ(n) × Γ k − 1 − β(n), δ(n) a!

(46)

Fig. 1. Comparison between FR and WR for correlated Rayleigh fading with ρ = 0.7.

V. N UMERICAL E XAMPLES We now investigate MIMO SVD with several allocation methods for both uncorrelated and correlated (one-sided) MIMO channels. In the case of correlated fading, we assume correlation among receive antennas, and the (i, j)th element of the correlation matrix ΣR is taken as ρ|i−j| with ρ ∈ [0, 1) ¯ 0 and (exponential correlation case). SNR is defined as E/N E/N0 for CITR and CITE allocation strategies, respectively. For the allocation method AFE, power allocated is calculated using (16). The notation MIMO (t, r) refers to a MIMO system with t transmit and r receive antennas.

k=1

which can be used to evaluate the total rate for AFE (17) and UA (18) strategies. To derive (46), we used the following identity [40, eq. (78)]: 8 ∞ a " Γ(−a+k, d) ln(1+x)xa−1 e−dx dx = (a−1)! ed dk 0 k=1 (47) valid for a, d ∈ * with a, d > 0. Since Lemma 1 is valid for arbitrary values of p and n, the analysis is valid for the case of correlation among either transmit or receive antennas. More specifically, one should use NT = n, NR = p, and Φ = ΣT in the case of correlation (with covariance matrix ΣT ) among transmit antennas, and NT = p, NR = n, and Φ = ΣR in the case of correlation (with covariance matrix ΣR ) among receive antennas.

A. CITR Allocation Strategy Fig. 1 shows the comparison between WR and FR in the case of correlated fading with ρ = 0.7 for three configurations: 1) MIMO(2, 2); 2) MIMO(3, 3); and 3) MIMO(4, 4). Our new closed-form results for the FR algorithm have been also verified by simulation. Similar to the results in [16], we note that the difference between the two criteria tends to decrease for increasing numbers of antennas. Fig. 2 shows the effect of an erroneous estimate of ϑi on the performance of FR allocation. We assume that the propagation environment is characterized by correlated fading but that the computation of r1 [see (9)] is based on values of ϑi corresponding to uncorrelated Rayleigh fading. As shown by (12), the existence of a mismatch between ϑ˜i and ϑi increases the value of SNR necessary to obtain the assigned rate R. When

ZANELLA AND CHIANI: REDUCED COMPLEXITY POWER ALLOCATION STRATEGIES FOR MIMO SYSTEMS WITH SVD

Fig. 2. Performance of FR with imperfect estimation of ϑi and comparison with the case of perfect estimate. NT = NR = 4, correlated fading with ρ = 0.5 and 0.9.

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Fig. 4. Comparison between AFE and UA for correlated Rayleigh fading with ρ = 0.7 and different values of NT and NR .

Fig. 5. Impact of a mismatch in the choice of kAFE on the performance of AFE. Fig. 3. Comparison between AFE, WE, and CSIR for correlated Rayleigh fading with ρ = 0.8.

ρ = 0.5, the difference, in terms of SNR, between the perfect and erroneous estimate of ϑi is almost negligible. This behavior is justified by the fact that ϑ˜ ∼ = ϑ for ρ ≤ 0.5. The difference in terms of SNR becomes significant when channel correlation increases (ρ = 0.9). In particular, it is about 2 dB and almost constant for values of R ranging from 5 to 15 bits/s/Hz. B. CITE Allocation Strategy Fig. 3 shows the performance comparison between AFE allocation and WE allocation in the case of correlated fading case with ρ = 0.8. As a benchmark, the case of CSIR only is also added. The figure clearly shows that the performance of AFE is almost coincident with the more complex water filling approach (WE). We checked that this result practically holds independently on the number of antennas. This behavior is also confirmed in the case of uncorrelated Rayleigh fading case (for the sake of brevity, results are not shown here). Similar results are obtained in Fig. 4, which shows the same energy allocation algorithms (WE, AFE and CSIR) for values

of NT ranging from 3 to 7. The number of receive antenna is NR = 5, and the correlation coefficient ρ = 0.7. Fig. 5 shows the effect of an imprecise choice of kAFE on the achievable rate. The figure considers a scenario with four transmit and receive antennas and three different values of SNR (0, 10, and 20 dB). We use the value of kAFE optimized for uncorrelated Rayleigh fading in an environment characterized by correlated Rayleigh fading, ranging from ρ = 0.1 to ρ = 0.95. The figure shows that the penalty in terms of achievable rate for this “mismatched” allocation is negligible for values of ρ smaller than 0.9. Therefore, the system is very robust against erroneous choices of the parameter kAFE . Subset selection with UPA (UA) is considered in Fig. 6, which considers a correlated fading environment with ρ = 0.8 and four transmit and receive antennas. Here, the optimal value of kUA is obtained by plotting the achievable rate for different values of kUA , ranging from 1 to 4. The optimal kUA for the different values of SNR is given by the envelope of the four curves. These curves also define the switching points of kUA , which represent the values of SNR corresponding to a change in the optimal value of kUA . For comparison, the curve corresponding to AFE is added. Results show that the

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 9, NOVEMBER 2012

presented two new approaches, called FE and subset selection with uniform energy allocation. For Rayleigh MIMO channels, the performance of these methods is related to statistical expectations of functions of the eigenvalues of Wishart matrices. We derived closed-form expressions for these expectations, which were then used to analytically compare the different allocation strategies. We have shown that the proposed suboptimal allocation methods perform very similarly to the optimal water-filling methods for both uncorrelated and correlated channels. R EFERENCES

Fig. 6. Comparison between AFE and UA for correlated Rayleigh fading with ρ = 0.8. TABLE IV SWITCHING POINT IN TERMS OF E/N0 (IN DECIBELS) OF THE OPTIMAL kUA AND kAFE . UNCORRELATED FADING

TABLE V SWITCHING POINT IN TERMS OF E/N0 (IN DECIBELS) OF THE OPTIMAL kUA AND kAFE . CORRELATED FADING WITH ρ = 0.8

performance of UA is almost coincident with that obtainable with AFE (and therefore very similar to the WE). Tables IV and V show kUA and kAFE for different values of SNR and channel conditions (uncorrelated Rayleigh fading and correlated Rayleigh fading with ρ = 0.8). More precisely, the values indicated in the tables refer to the switching point, in terms of SNR, from one value of the optimal kUA (or kAFE ) to another one. The tables show that the switching points of kAFE are usually smaller than those for kUA . This can be easily explained by observing that the energy allocated by ˘i ) can take any value in the AFE to each spatial channel (E continuous set [0, E], whereas the energy allocated by UA is limited to discrete values (E, E/2, E/3, . . . , E/Nmin ). The comparison between the results of Tables IV and V also shows that the switching points vary significantly in the presence of correlation. However, Fig. 5 has shown that the use of kAFE optimized for uncorrelated Rayleigh fading causes a limited loss in terms of achievable rate when used in the presence of correlated Rayleigh fading. VI. C ONCLUSION We studied the problem of energy/rate allocation in MIMOSVD. In addition to known water-filling and FR methods, we

[1] J. H. Winters, “On the capacity of radio communication systems with diversity in Rayleigh fading environment,” IEEE J. Sel. Areas Commun., vol. SAC-5, no. 5, pp. 871–878, Jun. 1987. [2] G. J. Foschini, “Layered space–time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., vol. 1, no. 2, pp. 41–59, Autumn 1996. [3] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, no. 3, pp. 357–366, Mar. 1998. [4] ˙I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [5] M. Chiani, M. Z. Win, and A. Zanella, “On the capacity of spatially correlated MIMO Rayleigh fading channels,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2363–2371, Oct. 2003. [6] P. J. Smith, S. Roy, and M. Shafi, “Capacity of MIMO systems with semicorrelated flat fading,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2781–2788, Oct. 2003. [7] H. Shin, M. Win, J. H. Lee, and M. Chiani, “On the capacity of doubly correlated MIMO channels,” IEEE Trans. Wireless Commun., vol. 5, no. 8, pp. 2253–2266, Aug. 2006. [8] A. Zanella, M. Chiani, and M. Z. Win, “MMSE reception and successive interference cancellation for MIMO systems with high spectral efficiency,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 1244–1253, May 2005. [9] S. K. Jayaweera and H. V. Poor, “Capacity of multiple-antenna systems with both receiver and transmitter channel state information,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2697–2709, Oct. 2003. [10] A. Maaref and S. Aissa, “Capacity of MIMO Rician fading channels with transmitter and receiver channel state information,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1687–1698, May 2008. [11] L. M. Garth, P. J. Smith, and M. Shafi, “Exact symbol error probabilities for SVD transmission of BPSK data over fading channels,” in Proc. IEEE Int. Conf. Commun., May 2005, pp. 2271–2276. [12] L. G. Ordonez, D. P. Palomar, A. P. Zamora, and J. R. Fonollosa, “Analytical BER performance in spatial multiplexing MIMO systems,” in Proc. IEEE Workshop SPAWC, Jun. 2005, pp. 460–464. [13] S. Jin, X. Gao, and M. McKay, “Ordered eigenvalues of complex noncentral Wishart matrices and performance analysis of SVD MIMO systems,” in Proc. IEEE Int. Symp. Inf. Theory, Seattle, WA, Jul. 2006, pp. 1564–1568. [14] Z. Zhou, B. Vucetic, M. Dohler, and Y. Li, “MIMO systems with adaptive modulation,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1828–1842, Sep. 2005. [15] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of blockfading channels with multiple antennas,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1273–1289, Apr. 2001. [16] J. H. Sung and J. R. Barry, “Approaching the zero-outage capacity of MIMO-OFDM without instantaneous water-filling,” IEEE Trans. Inf. Theory, vol. 54, no. 4, pp. 1423–1436, Apr. 2008. [17] W. Liu, L. Yang, and L. Hanzo, “SVD-assisted multiuser transmitter and multiuser detector design for MIMO systems,” IEEE Trans. Veh. Technol., vol. 58, no. 2, pp. 1016–1021, Feb. 2009. [18] R. A. Fisher, “Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population,” Biometrika, vol. 10, no. 4, pp. 507–521, May 1915. [19] A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Math. Stat., vol. 35, no. 2, pp. 475–501, 1964. [20] C. G. Khatri, “Distribution of the largest or the smallest characteristic root under null hyphotesis concerning complex multivariate normal populations,” Ann. Math. Stat., vol. 35, no. 4, pp. 1807–1810, Dec. 1964.

ZANELLA AND CHIANI: REDUCED COMPLEXITY POWER ALLOCATION STRATEGIES FOR MIMO SYSTEMS WITH SVD

[21] C. G. Khatri, “Non-central distribution of i-th largest characteristic roots of three matrices concerning complex multivariate multivariate normal populations,” J. Inst. Ann. Stat. Math., vol. 21, no. 1, pp. 23–32, Dec. 1969. [22] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005. [23] D. Tse and P. Viswanath, Fundamentals of Wireless Communications. New York: Cambridge Univ. Press, 2005. [24] T. A. Cover and J. A. Thomas, Elements of Information Theory, 1st ed. New York: Wiley, 1991. [25] A. Grant, “Capacity of ergodic MIMO channels with complete transmitter channel knowledge,” in Proc. 6th Australian Commun. Theory Workshop, Feb. 2005, pp. 125–129. [26] A. Zanella and M. Chiani, “Analytical comparison of power allocation methods in MIMO systems with singular value decomposition,” in Proc. IEEE Global Telecommun. Conf., Honululu, HI, Nov. 2009, pp. 1–7. [27] P. Dighe, R. Mallik, and S. Jamuar, “Analysis of transmit-receive diversity in Rayleigh fading,” IEEE Trans. Commun., vol. 51, no. 4, pp. 694–703, Apr. 2003. [28] A. Maaref and S. Aissa, “Closed-form expressions for the outage and ergodic Shannon capacity of MIMO MRC systems,” IEEE Trans. Commun., vol. 53, no. 7, pp. 1092–1095, Jul. 2005. [29] M. McKay, I. Collings, and P. Smith, “Capacity and SER analysis of MIMO beamforming with MRC,” in Proc. IEEE Int. Conf. Commun., Istanbul, Turkey, Jun. 2006, vol. 3, pp. 1326–1330. [30] M. Chiani and A. Zanella, “Joint distribution of an arbitrary subset of the ordered eigenvalues of Wishart matrices,” in Proc. IEEE Int. Symp. Pers., Indoor Mobile Radio Commun., Cannes, France, Sep. 2008, pp. 1–6. [31] A. Zanella, M. Chiani, and M. Z. Win, “On the marginal distribution of the Eigenvalues of Wishart matrices,” IEEE Trans. Commun., vol. 57, no. 4, pp. 1050–1060, Apr. 2009. [32] L. Ordoez, D. Palomar, and J. Fonollosa, “Ordered eigenvalues of a general class of Hermitian random matrices with application to the performance analysis of MIMO systems,” IEEE Trans. Signal Process., vol. 57, no. 2, pp. 672–689, Feb. 2009. [33] R. Kwan, C. Leung, and P. Ho, “Distribution of ordered eigenvalues of Wishart matrices,” Electron. Lett., vol. 43, no. 5, pp. 31–32, Mar. 2007. [34] S. Jin, M. McKay, X. Gao, and I. Collings, “MIMO multichannel beamforming: SER and outage using new eigenvalue distributions of complex noncentral Wishart matrices,” IEEE Trans. Commun., vol. 56, no. 3, pp. 424–434, Mar. 2008. [35] A. Zanella and M. Chiani, “The distribution of the #-th largest eigenvalue of central Wishart matrices and its application to the performance analysis of MIMO systems,” in Proc. IEEE Global Telecomm. Conf., New Orleans, LA, Dec. 2008, pp. 201–206. [36] M. Chiani, M. Z. Win, and H. Shin, “MIMO networks: The effects of interference,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 336–349, Jan. 2010. [37] H. David and H. Nagaraja, Order Statistics, 3rd ed. Hoboken, NJ: Wiley, 2003. [38] G. McLachlan and D. Peel, Finite Mixture Models. New York: Wiley, Sep. 2000. [39] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic, 1994. [40] M. S. Alouini and A. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165–1181, Jul. 1999.

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Alberto Zanella (S’99–M’00–SM’12) was born in Ferrara, Italy, in December 1971. He received the Dr. Ing. degree (with honors) in electronic engineering from the University of Ferrara, Ferrara, Italy, in 1996 and the Ph.D. degree in electronic engineering and computer science from the University of Bologna, Bologna, Italy, in 2000. He was a Researcher in 2001 and has been a Senior Researcher since 2006 with the CNR-CSITE (now a section of CNR-IEIIT). He was also an Adjunt Professor of electrical communication from 2001 to 2005, telecommunication systems in 2002, multimedia communication systems from 2006 to 2011, all with the University of Bologna. His research interests include MIMO, smart antennas, mobile radio systems, and ad hoc and sensor networks. Dr. Zanella currently serves as Editor for Wireless Systems, IEEE TRANSACTIONS ON COMMUNICATIONS. He either currently participates or has participated in several national and European projects. He was the Technical Co-Chair of the PHY track of the IEEE Wireless Communications and Networking Conference (WCNC) in 2009 and of the Wireless Communications Symposium (WCS) of IEEE Global Telecommunications Conference (Globecom) in 2009. He currently serves or has served on the Technical Program Committee of several international conferences, such as the International Conference on Communications (ICC), Globecom, WCNC, the International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), and the Vehicular Technology Conference (VTC). Marco Chiani (M’94–SM’02–F’11) was born in Rimini, Italy, in April 1964. He received the Dr. Ing. degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronic and computer science from the University of Bologna, Bologna, Italy, in 1989 and 1993, respectively. He is a Full Professor of telecommunications and the current Director of the Center for Industrial Research on ICT, University of Bologna. He is a frequent visitor with the Massachusetts Institute of Technology (MIT), Cambridge, where he presently holds a Research Affiliate appointment. He leads the research unit of the University of Bologna on cognitive radio and UWB (European project EUWB), on Joint Source and Channel Coding for wireless video (European projects Phoenix-FP6 and Optimix-FP7), and is a Consultant to the European Space Agency (ESA-ESOC) for the design and evaluation of error correcting codes based on LDPCC for space CCSDS applications. His research interests include wireless communication systems, MIMO systems, wireless multimedia, lowdensity parity-check codes (LDPCC), and UWB. Dr. Chiani is the past Chair (2002–2004) of the Radio Communications Committee of the IEEE Communication Society and past Editor of Wireless Communication (2000–2007) for the IEEE TRANSACTIONS ON COMMUNICATIONS. He received the Communication Theory Symposium Best Paper Award at the IEEE International Conference on Communications (ICC) in 2008, the Best Paper Award at the 2007 IST Mobile and Wireless Communication Summit, the Best Paper Award at the International Wireless Communication and Mobile Computing (IWCMC) in 2006, and the 2011 IEEE Communications Society Leonard G. Abraham Prize in the Field of Communications Systems.