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A general framework for MIMO transceiver design with imperfect CSI and transmit correlation

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A general framework for MIMO transceiver design with imperfect CSI and transmit correlation
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  A General Framework for MIMO TransceiverDesign with Imperfect CSI and Transmit Correlation Minhua Ding † , Steven D. Blostein ‡ , Wai Ho Mow † , Constantin Siriteanu # † Hong Kong University of Science and Technology, Hong Kong, P. R. China ‡ Queen’s University, Kingston, Ontario, Canada  # Seoul National University, Seoul, KoreaEmail:  † { eemding, eewhmow } @ust.hk,  ‡ steven.blostein@queensu.ca,  # costi@cse.snu.ac.kr  Abstract —Assuming perfect channel state information (CSI),linear precoding/decoding for multiple-input multiple-output(MIMO) systems has been considered in the literature undervarious performance criteria, such as minimum total mean-square error (MSE), maximum mutual information, and min-imum average bit error rate (BER). It has been shown thatthese criteria belong to a set of reasonable Schur-concave orSchur-convex objective functions of the diagonal entries of thesystem mean-square error (MSE) matrix. In this paper, assumingonly the knowledge of   channel mean and transmit correlation at both ends, a general theoretical framework is presented toderive the optimum precoder and decoder for MIMO systemsusing these objective functions. It is shown that for all theseobjective functions the optimum transceivers share a similarstructure. Compared to the case with perfect CSI, a linear filteris added to both ends to balance the suppression of channel noiseand the additional noise induced from channel estimation error.Simulation results are provided. 1 I. I NTRODUCTION The performance of multiple-input multiple-output (MIMO)systems depends on the availability of channel state informa-tion (CSI) at the transmitter (CSIT) and/or at the receiver(CSIR) [1]. Previously, optimum precoding or joint precod-ing/decoding for MIMO spatial multiplexing systems hasbeen obtained using mean-square error (MSE)-related designcriteria under different CSI assumptions [2]-[13]. Assumingperfect CSI at both ends, optimum transceivers are derived forminimizing total MSE or for maximizing capacity [2][3][4].In [5], assuming perfect CSI, the optimum transceivers areobtained for a set of MSE, signal-to-interference-plus-noise(SINR), or bit error rate (BER)-related design criteria, whichare Schur-convex or Schur-concave functions of the diagonalentries of the MIMO system MSE matrix and include theminimum total MSE and maximum capacity design criteriaas special cases.CSI is imperfect in practice, and there have been robust de-signs which take this fact into account. Transceiver optimiza-tion has been considered assuming  perfect CSIR  and imperfectCSIT (channel mean and/or channel covariance information)(see [6, Sec. VII], and references therein). In [7][8], the sameimperfect CSI is assumed at both ends of a MIMO link without explicit consideration of channel correlation. In [9][6,Sec. VI], transceiver designs have been studied assuming, 1 The work in this paper was supported by the Hong Kong Research GrantsCouncils under project number 617087. at both ends, the imperfect CSI composed of   channel meanand receive correlation information . In [10][11][13], optimumsignaling for a capacity lower-bound (i.e., minimizing thedeterminant of the system MSE matrix [13]) has been studiedassuming imperfect CSI at both ends with  channel meanand transmit correlation information , where the closed-formtransmit covariance matrix has been found in [13]. Under thesame CSI assumption, optimum transceivers to minimize thetotal MSE (trace of the system MSE matrix) have been foundin [12, Sec. III][13]. It is worth pointing out that, with  channelmean and transmit correlation information at both ends , thetransceiver optimization problem is nontrivial compared to theperfect CSI case.In this paper, we consider the MIMO transceiver designwith channel mean and transmit correlation information at bothends as in [12][13]. This scenario is particularly interesting inpractical downlink transmissions, where the channels arisingfrom base station antennas are correlated. With this assumptionof CSI, the optimum precoder-decoder pairs for the minimumtotal MSE design and the maximum capacity lower-bounddesign have been derived based on the associated optimalityconditions [12][13]. However, this approach involves matrixdifferentiation and has to be applied individually for dif-ferent objective functions. On the other hand, the optimumtransceivers derived share the same structure, which impliesthat a unified approach might be possible.In light of the results from [5], here we present  a generaltheoretical framework   to derive the optimum transceivers forvarious practical designs (as summarized in [5], includingthose in [12][13] as special cases) under the same imperfectCSI. The approach taken here is to equivalently reformulatethe srcinal design problem using the notion of “reasonablefunctions”, and then apply majorization theory [5][17]. Weobtain the optimum transceiver for the whole set of designcriteria which are Schur-convex or Schur-concave functionsof the diagonal entries of the MIMO system MSE matrix.Assuming imperfect CSI, the analysis can also be extendedfor transceiver optimization for MIMO-OFDM systems usingcyclic prefix (CP) and without subcarrier cooperation.Notation:  E {·}  stands for statistical expectation, tr ( · )  fortrace, and  det( · )  for determinant.  ( · ) H  means complex con- jugate transpose (Hermitian).  A  ≻  B  means that  ( A − B ) is positive definite.  ( b ) +  = max( b, 0) .  N  c ( · , · )  denotes thecomplex Gaussian distribution.  I  is the identity matrix and  1 978-1-4244-5213-4/09/ $26.00 ©2009 IEEE182 Authorized licensed use limited to: Queens University. Downloaded on July 05,2010 at 17:19:19 UTC from IEEE Xplore. Restrictions apply.  is reserved for the all-one vector. diag ( A )  and eig ( A )  denotevectors whose entries are the diagonal entries and eigenvaluesof a positive semidefinite matrix  A , respectively. For square B ,  [ B ] ii  denotes the  i -th diagonal entry of   B .II. S YSTEM MODEL AND PROBLEM FORMULATION  A. System model It is assumed that  n T   ( n R ) antennas are used at thetransmitter (receiver). The information streams to be sent aredenoted by a  B × 1  vector s , where the number of data streams, B  ( ≤ n T  ), is chosen and fixed. A  n T   × B  precoder, denotedby  F , is employed at the transmitter, taking the available CSIinto account. After precoding, the data vector is transmittedacross a slowly-varying flat-fading MIMO channel, describedby the  n R × n T   matrix  H . The  n R × 1  received signal vectorat the receive antennas is y = HFs + n ,  (1)where n is the AWGN with distribution  N  c (0 ,σ 2 n · I ) . The inputsignal s is assumed to be zero-mean and white ( R ss  = I ), andindependent of channel realizations. In the receiver, a lineardecoder, described by the  B × n R  matrix  G , is employed torecover the srcinal information. After decoding, the signalvector  r  is given by  r = Gy = G ( HFs + n ) .The MIMO channel is modeled as in [14]:  H = H w R 1 / 2 T   ,where H w  is a matrix whose entries are independent and iden-tically distributed (i.i.d.)  N  c (0 , 1) . The matrix  R T   representsnormalized transmit correlation with diagonal entries all equalto one. We assume that  R T   is invertible.  B. Description of the CSI  As in [10][12], MMSE estimation of   H w  is performed atthe receiver, which yields H w  = ˆ H w + E w , with  ˆ H w  being theestimate of  H w  and E w  being the error matrix.  ˆ H w  and E w  aremutually uncorrelated, and are both spatially white with entries  N  c (0 , 1 − σ 2 E  )  and  N  c (0 ,σ 2 E  ) , respectively. Variance  σ 2 E   = E {| H wji | 2 }− E {| ˆ H wji | 2 } . The CSI model is thus described by H = ( ˆ H w  + E w ) R 1 / 2 T   = ˆ H + E , where  ˆ H = ˆ H w R 1 / 2 T   is theestimated channel matrix (channel mean) and  E  =  E w R 1 / 2 T   .Below we assume that  ˆ H , R T  ,  σ 2 E   and  σ 2 n  are known to bothends of the link, which is also referred to as  channel meanand transmit correlation information.  It is assumed that CSITis obtained by perfect feedback of CSIR via a dedicated link.With the above CSI model, the received signal vector  y  isgiven by y = ˆ HFs + EFs + n =  ˆ HFs + E w R 1 / 2 T   Fs + n , and r = Gy . The system MSE matrix is calculated asMSE ( F , G )  def  =  E  ( r − s )( r − s ) H   = G ˆ HFF H   ˆ H H  G H  − G ˆ HF − F H   ˆ H H  G H  + I B  + [ σ 2 n  +  σ 2 E  · tr ( R T  FF H  )] GG H  .  (2)Note that  E  E w AE H w   =  σ 2 E   ·  tr ( A )  · I , if the entries of matrix  E w  are i.i.d.  N  c (0 ,σ 2 E  ) . The optimum linear MMSEdata estimator [15] is used at the receiver, i.e., G opt  = F H   ˆ H H  { ˆ HFF H   ˆ H H  + [ σ 2 n  +  σ 2 E  tr ( R T  FF H  )] I } − 1 . (3)Substituting (3) into (2), we obtain the MSE matrix in termsof   F  alone:MSE ( F ) =  I B  +  F H   ˆ H H   ˆ HF σ 2 n  +  σ 2 E  · tr ( R T  FF H  )  − 1 .  (4) C. Problem formulation Our goal here is to find the optimum  F  which minimizes aset of reasonable 2 Schur-convex or Schur-concave objectivefunctions [denoted as  g ( · ) ] [5] of the diagonal entries of MSE ( F )  subject to a total power constraint: min F g ( diag [ MSE ( F )]) ,  subject to tr ( FF H  ) ≤ P  T  .  (5)It can be shown that a global minimum exists for continuous  g functions, since the feasible set is a finite-dimension Frobeniusnorm ball [16]. Based on the optimized  F , we can evaluatethe performance of different designs with imperfect CSI. When σ 2 E   = 0 , the problem formulation in (5) reduces to that in [5],or those in [2][3][4] when the objective function is the trace ordeterminant of MSE ( F ) . Furthermore, when  σ 2 E    = 0  and theobjective function is the trace or determinant of MSE ( F ) , theoptimum F has also been determined in [12][13]. However, themethodology used in [12][13] depends on the differentiation of the objective function with respect to the precoder and decodermatrices, and has to be applied to each objective functionindividually. Here we will provide a general framework tofind the optimum  F  for a set of objective functions (different g ’s) without matrix differentiation.III. G ENERAL RESULTS For convenience, define T = [ σ 2 n · I n T   +  σ 2 E  · P  T   · R T  ] .  (6)Below we assume that the number of data stream,  B , is equalto  r , the rank of the estimated channel  ˆ H .  A. General results Proposition 1 : Assume that  g  :  R B +  → R  is reasonable(i.e., it is an increasing function in each of its arguments). •  If   g  is Schur-concave, then the optimum F for (5) is givenby: F = [ σ 2 n · I n T   +  σ 2 E  · P  T   · R T  ] − 12 VΦ F  1 ,  (7)where  Φ F  1  is a diagonal matrix satisfying the powerconstraint with equality, and  V  is obtained from thefollowing eigen-value decomposition: T − 12  ˆ H H   ˆ HT − 12 = [ V  ˜ V ]  Λ  00 ˜ Λ  [ V  ˜ V ] H  .  (8)In (8), Λ is a diagonal matrix whose diagonal entries arethe non-zero eigenvalues arranged in decreasing order.  ˜ Λ is a zero matrix, and  ˜ V  consists of basis vectors of the 2 A function  g  :  R B +  → R  is reasonable if it is increasing in each of itsarguments [5]. This definition fits in the context of linear precoding/decodingdesign for MIMO systems. 183 Authorized licensed use limited to: Queens University. Downloaded on July 05,2010 at 17:19:19 UTC from IEEE Xplore. Restrictions apply.  null space. V is composed of eigenvectors correspondingto the nonzero eigenvalues. •  If   g  is Schur-convex, then the optimum  F  for (5) is of the form: F = [ σ 2 n · I n T   +  σ 2 E  · P  T   · R T  ] − 12 VΦ F  2 U ,  (9)where Φ F  2  is diagonal, and U is a unitary matrix chosento make the diagonal entries of the resulting MSE ( F ) equal. Proof  : First, we show in Appendix A that if   g  is reasonable,then the minimum of (5) is achieved when the constraint issatisfied with equality, i.e., tr ( FF H  ) =  P  T  . Then (5) can beequivalently formulated as min F g  diag  I +  P  T   · F H   ˆ H H   ˆ HF tr { F H  [ σ 2 n I n T   +  σ 2 E  P  T  R T  ] F }  − 1  subject to tr ( FF H  ) =  P  T  .  (10)Without loss of generality,  F  can be expressed as F  =  T − 12 [ V  ˜ V ][ Φ H F   ˜ Φ H F   ] H  =  T − 12 [ VΦ F   + ˜ V ˜ Φ F  ] ,  (11)where  V  and  ˜ V  are both from (8), and  Φ F   and  ˜ Φ F   are  arbitrary  r × r  and  ( n T  − r ) × r  matrices, respectively. Define F   =  VΦ F   and  F ⊥  = ˜ V ˜ Φ F  . It is shown in Appendix Bthat, to achieve the minimum, F ⊥  = 0 , i.e., F = T − 1 / 2 VΦ F  .Substituting this into (10), after some algebra, we can showthat (10) is equivalent to min Φ F  g  diag  I +  P  T  Φ H F  ΛΦ F  tr { Φ H F  Φ F  }  − 1  subject to tr ( Φ H F  V H  T − 1 VΦ F  ) =  P  T  ,  (12)where  V  and  Λ  are from (8).From (12), it is the structure of   Φ F   that determines thevalue of the objective function. The norm of   Φ F   does notaffect it.To proceed, we need the following results from [5][17]. Let M  be a  n × n  positive semidefinite matrix. Let the entriesof diag ( M )  and eig ( M )  be arranged in decreasing order,respectively. Then diag ( M )  is majorized by eig ( M ) , and sois  tr ( M ) n  1  by diag ( M ) . If   f   :  R n → R  is Schur-concave,then  f  [ eig ( M )]  ≤  f  [ diag ( M )] . If   f   is Schur-convex, then f  [ tr ( M ) n  1 ] ≤ f  [ diag ( M )] .Thus, for a Schur-concave  g , the minimum of (12) isachieved when  Φ F   is diagonal with its diagonal entriesproperly arranged, denoted as  Φ F  1 . This gives us (7) 3 .On the other hand, for a Schur-convex  g , the minimum isachieved when all the diagonal entries are made equal and 3 It can be shown that if a non-diagonal matrix  Φ F   (satisfying the powerconstraint) achieves a certain value of the Schur-concave objective functionin (12), then there exists a diagonal matrix (satisfying the power constraint)that achieves a value of the objective function no greater than that achievedby the non-diagonal matrix. Therefore, the optimum  Φ F   for (12) must bediagonal. Due to space limitations, we do not elaborate on this here. their sum [trace of the matrix appeared in the objective of (12) as the argument of the diag function] is minimized. Sincethe trace function is itself a Schur-concave objective functionof the diagonal entries of the MSE matrix, the optimum  Φ F  for (12) has the form  Φ F  2 U , where  Φ F  2  is diagonal and  U is a unitary matrix which renders the diagonal entries of theresulting MSE matrix equal. Therefore, (9) is proved.   Remark 1 :  Proposition 1  reduces to the resultsin [2][3][4][5] when  σ 2 E   = 0 . Compared to the perfect CSIcase, a linear filter  T − 12 [see (6)] is added in the transceiver,which balances the suppression of channel noise and theadditional noise caused by channel estimation error. Theeffect of   σ 2 E   is coupled with transmit correlation  R T  . When Proposition 1  is applied to (5), it remains to determine theentries of   Φ F  1  or those of   Φ F  2 , and the original matrixoptimization problem (5) is now scalarized.  B. Applications We consider examples of   g  which satisfy the requirementsof   Proposition 1 . For brevity, define  [ V H  T − 1 V ] ii  =  β  i ,i  =1 ,...,r  =  B . Also recall that the entries of   Λ  [see (8)] arearranged  in decreasing order . i. Examples of Schur-concave functions Let  [ Φ F  1 ] ii  =  φ i ,i  = 1 ,...,r . [ Φ F  1  is defined in (7).]Define  x i  =  φ 2 i . (a) Minimization of weighted arithmetic mean of the MSEs Let  g 1 ( { [ MSE ( F )] ii } ri =1 ) =  ri =1 ( w i · [ MSE ( F )] ii ) , where { w i } ri =1  are positive weights. This objective function has beenconsidered in [4] assuming perfect CSI, which incudes theunweighted MMSE design and the maximum capacity designas special cases. By choosing different weights, one can alsodesign the transceiver to achieve different SNRs on differentsubchannels [4]. Clearly,  g 1  is reasonable. Per [5],  g 1  is Schur-concave. The problem in (5) is scalarized as min { x i } ri =1 r  i =1 w i ·  11 +  P  T  λ i x i  rm =1 x m subject to r  i =1 x i β  i  =  P  T  , x i  ≥ 0 , ∀ i.  (13)Solving this problem using the method of Lagrange multipli-ers, we obtain x i  =  w 12 i  λ − 12 i  P  T  ( P  T   +  a 1 ) − a 2 P  T  λ − 1 i ( P  T   +  a 1 ) a 3 − a 2 a 4  + .  (14)Let the integer  k  ( k  ≤  r ) denote the number of non-zero x ′ i s . Note that  k  can be readily determined using a procedureas in [12]. Then  a 1  =  ki =1  λ − 1 i  ,  a 2  =  ki =1  λ − 12 i  ,  a 3  =  ki =1  λ − 12 i  β  i  and  a 4  =   ki =1  λ − 1 i  β  i . Note that this resultcoincides with that obtained in [13, Sec. 3.6] using a differentapproach.After obtaining  x i  [as in (13)], we can obtain  φ i  = √  x i , ∀ i ,and thus  Φ F  1 . (b) Minimization of the geometric mean of the MSEs 184 Authorized licensed use limited to: Queens University. Downloaded on July 05,2010 at 17:19:19 UTC from IEEE Xplore. Restrictions apply.  Let  g 2 ( { [ MSE ( F )] ii } ri =1 ) = Π ri =1 [ MSE ( F )] w i ii  . This designproblem is shown to be related to the minimization of thedeterminant of the MSE matrix (or maximization of the mutualinformation) with perfect CSI [5]. Again,  g 2  is reasonable. Itis shown in [5] that  g 2  is Schur-concave. The problem in (5)is reduced to min { x i } ri =1 Π ri =1   11 +  P  T  λ i x i  rm =1 x m  w i subject to r  i =1 x i β  i  =  P  T  , x i  ≥ 0 , ∀ i.  (15)Solving this problem, we obtain x i  =  P  T  { w i ( P  T   +  b 3 ) − b 0 λ − 1 i  } ( P  T   +  b 3 ) b 1 − b 2 b 0  + .  (16)Let the integer  m  ( m ≤ r ) denote the number of non-zero  x ′ i s .Similar to  k  for (14),  m  can be readily determined (see [13]).Then  b 1  =  mi =1  β  i ,  b 2  =  mi =1  λ − 1 i  β  i ,  b 3  =  mi =1  λ − 1 i  , and b 0  =  mi =1  w i . In this case,  [ Φ F  1 ] ii  = √  x i .The optimum precoder obtained here is the sameas that used to maximize a mutual information  lower-bound   [13][10][11] when  w i  = 1 , ∀ i : max F tr { FF H }≤ P  T  log 2  det  I +  F H   ˆ H H   ˆ HF σ 2 n  +  σ 2 E  · tr ( R T  FF H  )  .  (17)The relationship between the minimization of the geometricmean and (17) is similar to that in the perfect CSI case, whereminimizing the geometric mean is equivalent to maximizingthe  exact   mutual information [5]. Remark 2 : [Relationship between minimization of theunweighted geometric mean of MSEs (maximization of the ca-pacity  lower-bound   in (17)) and minimization of the weightedarithmetic mean of MSEs] Let  w i  =  λ i , ∀ i  in (13) and let w i  = 1 , ∀ i  in (15), then (14) is equal to (16). ii. Schur-convex functions Schur-convex functions are involved in many system de-signs of interest, e.g., minimization of the maximum MSEfrom all data streams [5]. For all reasonable Schur-convexfunctions, the solutions to (5) are the same 4 . Further, theoptimum Φ F  2  in (9) is the same as the optimum Φ F  1  in (7) tominimize the trace of MSE ( F ) , which has been obtained whenminimizing the arithmetic mean of the MSEs with  w i  = 1 , ∀ i [see (13) (14) with  w i  = 1 , ∀ i ]. Remark 3 : For single-carrier MIMO, despite differentdesign criteria for particular applications, the minimizationsof the weighted arithmetic mean and geometric mean of the 4 It is important to note that our design (5) takes an averaging approach [see(2)]. In addition, we should be cautious when applying some design criteriawith imperfect CSI. For example, in the design to minimize the arithmeticmean of BERs from all data streams, by simulations, we have found that,with imperfect CSI, Q -function cannot be used to describe the BER of eachdata stream at high SNR. This is different from the perfect CSI case [5].Therefore, some Schur-convex (or Schur-concave) functions established inthe perfect CSI case have to be re-examined. MSEs [Sec. III-B–( i )] are the core of all designs. This is truefor both perfect and imperfect CSI cases. iii. Extension to CP-based MIMO-OFDM systems withimperfect channel estimation and transmit correlation Assuming imperfect channel estimation, it is straightforwardto extend our results in Sec. III-A to a CP-based MIMO-OFDM system [19] with individual processing 5 and powerconstraints on each subcarrier.On the other hand, assuming individual processing, whena sum power constraint is imposed on all subcarriers, weemploy a two-stage processing (primal decomposition). First,we initialize the power for each subcarrier, and apply ourresults on transceiver optimization to each subcarrier. Thenan outer power allocation is performed among all subcarriers.Iteration is performed until the globally optimum transceiversare obtained for all subcarriers.Note that the outer power optimization problem is nontrivialin the case of imperfect CSI, and remains to be solved.However, the structures of the optimum precoders can bereadily shown.IV. N UMERICAL  R ESULTS Per  Remark 3 , for single-carrier MIMO, all the designsdiscussed here are related to minimization of the arithmeticor geometric mean of the MSEs. Thus we refer the readersto [12][13], where numerical examples for the results inSec. III-B–( i ) of this paper can be found. For single-carrierMIMO, the corresponding BER results for all Schur-convexfunctions are the same. An example is given below (see Fig. 1).Let  n T   =  n R  = 4 , the number of data streams  B  = 3 . Thetransmit correlation model is given by:  ( R T  ) ij  =  ρ | i − j | for i,j  ∈{ 1 ,...,n T  } . Here  ρ  = 0 . 5 . The SNR in Fig. 1 is definedas  P  T  /σ 2 n . QPSK (4-QAM) is used for each data stream. Thesystem performance is shown in terms of the arithmetic meanof BERs (ABER  =  1 B  Bj =1  BER j ), and is obtained fromMonte Carlo simulations. For the imperfect CSI case, the errorvariance is modeled in the same way as in [12, Sec. II-B, Sec.III], and is set to be  σ 2 E   = 0 . 01478  for  ρ  = 0 . 5 . The twodesigns shown in Fig. 1 differ only in a unitary rotation, andthe comparison results are as they are designed to be.V. C ONCLUSIONS Assuming channel mean and transmit correlation informa-tion at both ends, optimum transceiver structures for MIMOsystems have been determined for a set of reasonable Schur-convex or Schur-concave objective functions of the diagonalentries of the system MSE matrix. Compared to the case withperfect CSI, a linear filter is added to both ends to balance thesuppression of channel noise and the additional noise inducedfrom channel estimation error. Results can also be applied tothe transceiver design assuming imperfect CSI for CP-basedMIMO-OFDM systems with noncooperative subcarriers. 5 This means that the subcarriers are not cooperating. Each subcarrier hasits own transceiver (precoder-decoder pair). 185 Authorized licensed use limited to: Queens University. Downloaded on July 05,2010 at 17:19:19 UTC from IEEE Xplore. Restrictions apply.  05101520253010 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR = P T  /  σ n2  (dB)    A   B   E   R  min arith−MSE, imperfect CSImin max−MSE, imperfect CSImin arith−MSE, perfect CSImin max−MSE, perfect CSI Fig. 1. ABER performance;  σ 2 E  = 0  (perfect CSI) or  σ 2 E  = 0 . 01478 (imperfect CSI).  n T   =  n R  = 4 ,  B  = 3 ,  ρ  = 0 . 5 . Here min arith-MSE(or max-MSE) denotes the design to minimize the arithmetic mean (or themaximum) of the MSEs from all data streams. Note that max-MSE is Schur-convex. A PPENDIX  AThe entries of   g  are the diagonal entries of MSE ( F )  andcan be represented as  e H i  [ MSE ( F )] e i , where  e i  is the  i -thcolumn of the identity matrix,  i  = 1 ,...,B . Suppose that F A is the optimum of (5) when the power constraint is  P  A  anddenote corresponding minimum of (5) as  v A . Let  P  B  > P  A and  ˘ F B  =   P  B P  A F A . Then, e H i  I +˘ F H B ˆ H H   ˆ H ˘ F B σ 2 n  +  σ 2 E  · tr ( R T   ˘ F B ˘ F H B )  − 1 e i = e H i  I +  F H A ˆ H H   ˆ HF AP  A P  B σ 2 n  +  σ 2 E  · tr ( R T  F A F H A )  − 1 e i < e H i  I +  F H A ˆ H H   ˆ HF A σ 2 n  +  σ 2 E  · tr ( R T  F A F H A )  − 1 e i ,  ∀ i.  (18)The last inequality follows from the fact that if   A ≻ B , then B − 1 ≻ A − 1 [18, p. 586, A.8, (vii)], and e H i  ( B − 1 − A − 1 ) e i  > 0 . Consequently, if   A  ≻  B ,  e H i  A − 1 e i  <  e H i  B − 1 e i . Denotethe value of the objective in (5) corresponding to  ˘ F B  as ˘ v B . Since  g  is increasing in each of its arguments, based on(18), we have  ˘ v B  < v A . Define the global minimum of (5)corresponding to  P  B  as  v B . Clearly,  v B  ≤  ˘ v B , since  v B  is theglobal minimum whereas  ˜ v B  is simply the cost of using onefeasible point. Therefore,  v B  < v A  for  P  B  > P  A . This showsthat the minimum of (5) must be achieved when the constraintis satisfied with equality.   A PPENDIX  BDue to space limitations, we can only present an outlinehere. Substituting (11) into (10), using the fact that F H  ⊥ F   = 0 , F H    F ⊥  = 0 ,  F H  ⊥ V  = 0 ,  F H    F   =  Φ H F  Φ F   and  F H  ⊥ F ⊥  =˜ Φ H F  ˜ Φ F  , after some calculations, we obtain the followingequivalent problem min Φ F  , ˜ Φ F  g  diag  I +  P  T   · Φ H F  ΛΦ F  / tr { Φ H F  Φ F  } 1 +  tr { ˜ Φ H F  ˜ Φ F  } / tr { Φ H F  Φ F  }  − 1  subject to tr  T − 1 ( F   + F ⊥ )( F   + F ⊥ ) H   =  P  T  .  (19)Using the same technique as in Appendix A, one can show thatthe objective in (19) is decreased if tr { ˜ Φ H F  ˜ Φ F  }  is decreased(assuming that  g  is reasonable, i.e., increasing in each of its arguments). Therefore, the objective is minimized whentr { ˜ Φ H F  ˜ Φ F  } = 0 , i.e.,  F ⊥  = 0 .   R EFERENCES[1] A. Paulraj, R. Nabar, D. Gore,  Introduction to Space-Time WirelessCommunications , Cambridge University Press, 2003.[2] J. Yang, S. 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