Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2007, Article ID 29749, 9 pagesdoi:10.1155/2007/29749

Research ArticleEigenstructures of MIMO Fading Channel CorrelationMatrices and Optimum Linear Precoding Designs forMaximum Ergodic Capacity

Hamid Reza Bahrami and Tho Le-Ngoc

Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montreal, QC, Canada H3A 2A7

Received 27 October 2006; Revised 10 February 2007; Accepted 25 March 2007

Recommended by Nicola Mastronardi

The ergodic capacity of MIMO frequency-flat and -selective channels depends greatly on the eigenvalue distribution of spatial cor-relation matrices. Knowing the eigenstructure of correlation matrices at the transmitter is very important to enhance the capacityof the system. This fact becomes of great importance in MIMO wireless systems where because of the fast changing nature of theunderlying channel, full channel knowledge is difficult to obtain at the transmitter. In this paper, we first investigate the effect ofeigenvalues distribution of spatial correlation matrices on the capacity of frequency-flat and -selective channels. Next, we introducea practical scheme known as linear precoding that can enhance the ergodic capacity of the channel by changing the eigenstructureof the channel by applying a linear transformation. We derive the structures of precoders using eigenvalue decomposition andlinear algebra techniques in both cases and show their similarities from an algebraic point of view. Simulations show the ability ofthis technique to change the eigenstructure of the channel, and hence enhance the ergodic capacity considerably.

Copyright © 2007 H. R. Bahrami and T. Le-Ngoc. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

1. INTRODUCTION

It has been shown that the capacity of a MIMO system isgreatly reduced by spatial correlation in the underlying chan-nel [1, 2]. Spatial correlation can reduce the rank of the chan-nel matrix, and hence greatly surpasses the multiplexing gainof a MIMO system. Various techniques that have been pro-posed in the literature to reduce the correlation effects arebased on two main approaches. One aims to avoid corre-lation in the channel by antenna beamforming [3, 4]. Theother tries to cancel the existing channel correlation by suit-able methods at the transmitter or receiver. In this paper,our focus is on the linear precoding technique based on theknowledge of correlation at the transmitter, aiming to in-crease the ergodic capacity of fading channels by modifyingthe eigenvalue spread of the channel correlation matrices.

Linear precoder design in MIMO systems is a relativelysimple (in term of implementation and design complexity)strategy that tries to improve the transmission quality andrate by optimal allocation of resources such as power andbits over multiple antennas, based on the channel properties.Design of the precoders based on full channel knowledge forMIMO systems in frequency-flat and -selective channels has

been investigated by many works. For a detailed overview onthe designs for frequency-flat channels, see [5, 6]. While wesee a number of different precoder structures for frequency-flat fading channel proposed in the literature, there are fewerpapers addressing MIMO precoding designs in a frequency-selective fading environment [7]. In designs based on fullchannel knowledge, it is assumed that the transmitter hasthe instantaneous channel information and based on thisinformation, a metric related to performance, such as pair-wise error probability (PEP) or minimum mean-square error(MMSE) or rate (ergodic capacity or probability of outage),is defined and optimized by selection of proper linear pre-coder.

In a fast fading environment, however, the assumption offull channel knowledge at the transmitter is no longer real-istic due to the finite delay in channel response estimationand reporting. Hence, it is more reasonable to assume thatthe transmitter knows only partial channel knowledge suchas spatial correlation information, that is, transmit and re-ceive correlation matrices.

Optimal precoding designs using PEP criterion based ontransmit and receive correlation matrices were presented in[8, 9], respectively. In [10], optimal precoding designs based

2 EURASIP Journal on Advances in Signal Processing

on both transmit and receive correlation matrices were de-veloped for three different criteria, that is, PEP, MMSE, andergodic capacity. The results indicate that the optimal pre-coder structures for these criteria are very similar. All of theabove designs are for flat fading channels.

In this paper, we investigate the channel correlationeffects on the capacity of frequency-flat and frequency-selective fading channels from an algebraic viewpoint anddevelop the corresponding linear precoding structures tomaximize their ergodic capacity. We show that the eigen-values of the correlation matrices play a key role in the er-godic capacity of fading channels. In particular, the effectof correlation on the capacity of the system becomes morepronounced with increase in the eigenvalue spread of thespatial correlation matrices. Therefore, in general, our focusis to find how we can modify the eigenvalues of the chan-nel correlation matrices to enhance the capacity. Based onlinear algebraic structures of frequency-flat and frequency-selective fading MIMO channels, we construct suitable ana-lytical models to include channel spatial correlations in bothcases, and derive the precoding matrix structures that canmaximize their ergodic channel capacity. In both cases, theprecoding matrices are closely related to the eigenstructures(eigenvalues and eigenvectors) of spatial correlation matri-ces. We further show that the structure of the precoder in thefrequency-flat case is an eigenbeamformer with beams point-ing to the eigenmodes of the transmit correlation matrix. Fora frequency-selective fading channel with L independent ef-fective paths, the precoder can be constructed as a number ofparallel precoders for frequency-flat fading channels. In thissense, there is a kind of duality between precoder design forfrequency-flat and -selective channels.

The rest of this paper is organized as follows. In Section 2,we consider the case of frequency-flat fading channels. Theso-called Kronecker model is introduced to represent thespatial correlation effects in MIMO system. Based on thischannel model, we investigate the effects of eigenvalues ofspatial correlation matrix and their spread on the ergodicchannel capacity and develop the corresponding precodingstructure based on the eigenstructure of the channel spatialcorrelation matrix in order to maximize the ergodic capacity.In Section 3, we consider the case of frequency-selective fad-ing channels. We develop a comprehensive linear algebraicmodel of a frequency-selective fading channel with L effectivepaths in terms of channel correlation matrices. Furthermore,we analyze the effects of the eigenstructures of the channelcorrelation matrices on the ergodic capacity of a frequency-selective fading channel and develop the optimum precoderbased on the eigenstructures of spatial correlation matricesof L effective channel paths for maximum ergodic capacity.It is shown that the structure includes L parallel precoders,each for one frequency-flat fading channel path, with specificpower loadings. Section 4 presents illustrative examples withnumerical results and plots. The effects of eigenvalues of spa-tial correlation matrices on the capacity of both frequency-flat and frequency-selective fading channels in various con-ditions are presented. The effects of precoding on the eigen-value spreads of the channel correlation matrices are also

shown. Furthermore, performance in terms of achievable er-godic capacity versus SNR of the proposed precoders is eval-uated and compared with that of systems using no precodingin various scenarios by means of simulation. It is shown thatthe precoders perform well in changing the eigenstructure(mainly eigenvalue spread) of the channels in favor of chan-nel capacity. In other words, the precoders are capable of pro-viding a considerable capacity gain in different propagationscenarios by changing the characteristics of the channel cor-relation matrices. Finally, Section 5 includes with concludingremarks.

2. FREQUENCY-FLAT FADING CHANNEL

2.1. System model

Consider a MIMO transmission system over a frequency-flatfading channel, using transmitter and receiver equipped withM and N antennas, respectively. The discrete-time wirelessMIMO fading channel impulse response can be assumed tobe an N ×M matrix H and the system model (input-outputrelationship) at the kth time instant can be written as

y(k) = H(k) · s(k) + n(k), (1)

where s[k] is the transmitted M × 1 data vector with statisti-cally independent entries and y[k] and n[k] denote the N×1received and noise vectors, respectively. We assume that theelements of H and n are complex Gaussian random variableswith 1/2 variance per dimension, and E{nnH} = σ2

nIN , whereσ2n is the noise variance and IN is the identity matrix of sizeN . Besides, E{·} denotes expectation and superscript H isHermitian (conjugate transpose) operator.

For simplicity, we assume the receiver (e.g., mobile unit)to be surrounded by local scatterers so that fading at the mo-bile unit is spatially uncorrelated while transmitter (e.g., basestation) is located in a high altitude, and therefore the fadingis correlated at base station. However, it is straightforward togeneralize the model to the case when both transmitter andreceiver are spatially correlated.1 Due to the assumed spatialcorrelation at transmit side, the elements of each row of Hare correlated and for each row, we can write

RT = E{

hHi hi

}, i = 0 · · ·N − 1, (2)

where hi is the ith row of H. RT is the M ×M transmit non-negative, semidefinite correlation matrix, and hence can berepresented as RT = R1/2

T RH/2T (Choleski factorization). Sub-

sequently, the channel matrix H can be represented as

H = GR1/2T , (3)

where G is an uncorrelated N × M matrix with i.i.d. zero-mean normalized Gaussian distributed entries, that is, G ∼

CN(0, 1). The model proposed here is usually known as Kro-necker model in the related literature [11].

1 For a more detailed analysis including receive correlation, see [10].

H. R. Bahrami and T. Le-Ngoc 3

2.2. Effect of correlation

We start by defining the mutual information in each channeluse. We assume an independent and invariant realization ofthe channel matrix in each channel. Using (1), the mutualinformation of such a system is defined as [12, 13]

I(s; y) = log[

det(

IN +1σ2n

HΣHH)]

, (4)

where Σ is the M ×M covariance matrix of Gaussian inputx with a maximum power limit due to the total power limi-tation at transmitter, that is, tr(Σ) ≤ P. Note that the instan-taneous capacity of the system is defined as the maximum ofmutual information over all covariance matrices Σ that sat-isfy the power constraint, and the ergodic capacity is the en-semble average over instantaneous capacity.

In the following analysis, for simplicity we assume anequal power allocation Σ = (P/N)IM . Based on this assump-tion, the mutual information in (4) can be written as

I(s; y) = log[

det(

IN +P

Nσ2n

HHH)]

. (5)

This is in fact the instantaneous channel capacity when trans-mitter has no knowledge about the channel. Our objective isto understand the effect of transmit correlation matrix (morespecifically its eigenvalues) on the ergodic capacity of the sys-tem.

Lemma 1. Instantaneous mutual information has the follow-ing distribution:

I ∼ log det(

IM +P

Nσ2nΔGHG

), (6)

where Δ = diag{δi(RT)}M−1i=0 , that is, δi’s are the eigenvalues of

transmit correlation matrix.

Proof. Substituting H from (3) into (5) will result in

I = log[

det(

IN +P

Nσ2n

GRTGH)]

. (7)

By applying the eigenvalue decomposition of RT = ΦΔΦH

and the fact that GΦ ∼ G (and hence ΦHGH∼ GH), (7) can

be rewritten as

I ∼ log det(

IN +P

Nσ2n

GΔGH). (8)

Using the matrix equality det(I + AB) = det(I + BA) willcomplete the proof.

As previously mentioned, the ergodic capacity is definedas C = E{I}. The importance of C comes from the fact thatat transmission rates lower than C, the error probability ofa good code decays exponentially with the transmission rate.Here, our objective is to investigate the effects of the eigen-values of transmit correlation matrix on C. We show the im-portance of these eigenvalues in two ways. First, following the

same approach as in [14], we consider the asymptotic case oflarge number of receive antennas, based on the law of largenumbers, when the number of receive antennas (N) is large(1/N)GHG → IM , and is hence in the limit

C = E{

log det(

IM +P

Nσ2nΔGHG

)}

= log det(

IM +P

σ2nΔ)

=M∑

i=1

log(

1 +P

σ2nδi

),

(9)

where δi’s are the diagonal entries of Δ, the eigenvalue matrixcheck of transmit correlation matrix RT . This clearly showsthe effect of the eigenvalues of correlation matrix on the er-godic capacity of a frequency-flat MIMO channel. It is alsopossible to derive the same result by applying Jensen’s in-equality [15] to the first equation in (9) to compute an upperbound on ergodic capacity. Using Jensen’s inequality,

C ≤ CUB = log detE{(

IM +P

Nσ2nΔGHG

)}. (10)

Since the entries of G are Gaussian with zero mean and 1/2variance per dimension, it follows that

C ≤ CUB = log detE{(

IM +P

σ2nΔ)}

=M∑

i=1

log(

1 +P

σ2nδi

).

(11)

This bound gets tighter by increasing N , the number of thereceive antennas, and the inequality in (11) becomes equalityin the limit of large N .

The following lemma specifies the optimal case for eigen-values in order to maximize the ergodic capacity. We borrowthis lemma from [14].

Lemma 2. For tr(RT) = 1, the ergodic capacity is maximizedwhen δi = 1/N , i = 1 · · ·N .

The proof is straightforward and can be obtained by sym-metry argument.

Lemma 2 shows that the best case is when the channel isindeed uncorrelated, that is, RT = (1/N)IN . Now the ques-tion is what transmit strategy can be used when the channelis correlated. Our focus here is to devise a linear transmissionstrategy to maximize the ergodic capacity when the transmitcorrelation matrix is not identity.

In wireless communications, this question is also appeal-ing from another point of view. Here, we assume that we justknow the transmit correlation matrix RT at the transmitterand do not have information of the channel matrix H. Thisassumption becomes more important in wireless channelswhere the channel changes very fast (i.e., fast fading chan-nels), since it is difficult, or sometimes impossible to acquireinstantaneous channel response, H, at the transmitter. Onthe other hand, transmit correlation matrix (or any other

4 EURASIP Journal on Advances in Signal Processing

second-order statistics) of the channel changes much slowlycompared to instantaneous channel response, H. Therefore,it is possible in fast fading environment to obtain an accu-rate transmit correlation matrix at the transmitter. Some-times in the related literature, such information is calledpartial channel information. In state-of-the-art communi-cations systems, these types of channel information becomemore and more important as we are interested in transmit-ting information to high-speed mobile units.

Our objective through this paper is to apply an M ×Mlinear transformation (precoding) W over information sym-bols s to get anM×1 transmit vector x, that is, x = Ws, underthe power constraint. The precoding matrix is selected suchthat a performance metric (e.g., the ergodic capacity) is op-timized. We assume that the transmitter is just informed ofthe transmit correlation matrix RT . We treat the flat-fadingchannel in this section and the frequency-selective fadingcases in the next section.

2.3. Precoder design

We assume that the receiver has the perfect channel informa-tion but the transmitter knows only spatial and path correla-tion matrices. Our objective is to design the precoding matrixW to maximize the ergodic capacity for a given total trans-mit power. Applying precoding matrix, the ergodic capacityof the MIMO system in a frequency-flat fading channel cannow be written as

C = E[

log2

(det

(IM +

1Nσ2

nWHHHH W

))]. (12)

Note that the power constant P in (7) is now considered inthe elements of precoder matrix, and hence a power con-straint is applied to its entries, that is, tr{WWH} ≤ P. Get-ting expectation from the log-function in (12) is very hard (ifnot impossible). By applying Jensen’s inequality [15] to log-det function, that is, E{log[det(A)]} ≤ log[det(E{A})], wecan derive an upper bound on the ergodic capacity as

C ≤ CUB = log2

(detE

(IM +

1Nσ2

nWHRH/2

T GHGR1/2T W

)),

(13)

where H has been substituted from (3).

Lemma 3. The optimum precoding matrix for frequency-flatfading channel is directly related to the eigenvector matrix oftransmit correlation matrix RT and can be written as W =ΦΣ1/2Γ, where Φ is the eigenvector matrix of RT , Σ is a diago-nal matrix called power loading matrix whose entries should becomputed for optimality, and Γ is an arbitrary unitary matrix.

Proof. By taking the expectation in (13) and eigendecompo-sitions WWH = ΨΣΨH and RT = ΦΔΦH , we obtain

C ≤ CUB = log2

(det

(IM +

κ

σ2nΨΣΨHΦΔΦH

)), (14)

where κ is a constant that can be calculated by taking the ex-pectation of the components of G. Our aim is to find W thatmaximizes (14) under the power constraint, that is,

max log2

(det

(IM +

κ

σ2nΨΣΨHΦΔΦH

))s.t. tr(Σ) ≤ P.

(15)

Note that tr{WWH} ≤ P will directly result in tr{Σ} ≤ P. Us-ing Hadamard’s inequality [16], the above optimization canbe achieved when the argument of the determinant is a di-agonal matrix. To this end, we should have Ψ = Φ. In otherwords, the singular matrix of the precoder matrix should bethe same as the singular matrix of transmit correlation ma-trix. Therefore, the precoder structure can be written as

W = ΦΣ1/2Γ, (16)

where Γ is an arbitrary unitary matrix that has no effect onthe system performance, and therefore can be set to iden-tity for simplicity and Σ (the eigenvalue matrix of W) is thepower loading matrix that should be optimized.

By substituting (16) into (13), the optimization problemcan be rewritten as

maxΣ

log2

(det

(I +

κ

σ2nΣΔ))

s.t. tr(Σ) ≤ P, (17)

with the following solution for elements of Σ:

σi =[v − σ2

n

κδi

]+

, i = 0 : M − 1, (18)

where [x]+ = max[0, x] for a scalar x, σi and δi are the di-agonal entries of Σ and Δ, respectively and v is the constantdetermined by the power constraint. At the optimum point,the power inequality tr{Σ} ≤ P becomes equality.

In fact, the precoder changes the eigenvalues of the chan-nel to optimize the ergodic capacity. The new eigenvaluesof the product of the channel matrix and precoder matrixare σiδi, i = 0 · · ·M − 1 (instead of δi). Precoder tends toincrease the larger eigenvalues compared to small eigenval-ues and increase the eigenvalue spread of the product matrixHW. Therefore, δi > δj results in σi > σj . This power alloca-tion process is known as waterpouring in which the precoderpours more power to stronger eigenvalues (or eigenmodes)and allocates less to weaker ones.

3. FREQUENCY-SELECTIVE FADING CHANNEL

3.1. System model

We consider a transmission system with M transmit and Nreceive antennas in a frequency-selective fading channel. Be-cause of the delay spread in the frequency-selective fadingchannel, the received signal is a function of the input signal atdifferent time instants. The frequency-selective fading chan-nel can be modeled as an L-tap FIR filter shown in Figure 1,and each tap denotes a resolvable channel path representedby an N ×M matrix Hl, l = 0, . . . , (L− 1).

H. R. Bahrami and T. Le-Ngoc 5

· · ·

· · ·

· · ·

HL−1

Δ

H0 H1

ΔΔs

y + +

Figure 1: Frequency-selective MIMO channel model.

Consider a transmitted block of K +L vectors of size M×1, organized as a long (K + L)M × 1 vector, where K is anarbitrary value, s(k) = [s(k(K +L)), . . . , s(k(K +L) +K +L−1)]T . At the receiver, we eliminated the first L vectors of sizeN × 1 to remove the interblock interference (IBI), and stackK remaining received vectors of size M × 1 to form a longKM×1 vector, y(k) = [y(k(K +L) +L), . . . , y(k(K +L) +K +L− 1)]T ,

y(k) = H · s(k) + n(k), (19)

where n(k) = [n(k(K+L)+L), . . . , n(k(K+L)+K+L−1)]T isthe long KM× 1 vector of K subsequent noise vectors of sizeM × 1, and H is the NK ×M(K + L) block-Toeplitz channelmatrix:

H =

⎡

⎢⎢⎢⎢⎢⎢⎣

HL−1 HL−2 · · · H0 0 · · · 00 HL−1 HL−2 · · · H0 · · · 0...

......

......

......

0 · · · HL−1 HL−2 · · · H0 00 · · · 0 HL−1 HL−2 · · · H0

⎤

⎥⎥⎥⎥⎥⎥⎦

. (20)

The N ×M matrix Hl(k) represents the spatial response cor-responding to the resolvable channel path l, l = 0, 1, . . . ,L−1,at the instant k. Its entry, hnm(k) is the complex-valued ran-dom gain from the mth transmit to nth receive antennas overthe effective path l at the instant k, assumed to be unchangedduring a frame transmission. Assuming that the receive cor-relation matrix is identity for all paths, the channel path ma-trix Hl(k) can be written as

Hl(k) = Gl(k)R1/2T ,l , l = 0 : L− 1. (21)

Note that the spatial correlation matrix is a function of trans-mit antennas (such as antenna spacing and antenna pat-tern) and channel physical characteristics (angular spreadand power angular spread). The former parameter is thesame for all paths while the latter is different from one pathto another. This results in different channel path transmitcorrelation matrices RT ,l, l = 0 · · ·L − 1. We assume thatthe power of the lth path has been considered in the diago-nal entries of its spatial correlation matrix RT ,l. Due to thedifferent delays between L effective paths, (20) can providea frequency-selective fading MIMO channel model, whileeach individual Hl(k) just represents a frequency-flat fadingMIMO channel.

The block-Toeplitz channel matrix in (20) can be writtenas

H =

⎡

⎢⎢⎢⎢⎢⎢⎣

HHEHE2

...HEK−1

⎤

⎥⎥⎥⎥⎥⎥⎦

= (IK ⊗H) ·

⎡

⎢⎢⎢⎢⎢⎢⎣

IM(K+L)

EE2

...EK−1

⎤

⎥⎥⎥⎥⎥⎥⎦

= (IK ⊗H) · E,

(22)

where ⊗ stands for Kronecker product, the N × M(K + L)matrix H = [HL−1, HL−2, . . . , H0, 0, . . . , 0] is the first row ofH in (20), and IK is the K × K identity matrix. The M(K +L)×M(K +L) matrix E is a column switching matrix and hasthe following structure:

E =[

0M(K+L−1)×M IM(K+L−1)

IM 0M×M(K+L−1)

]

, (23)

where 0M(K+L−1)×M and 0M×M(K+L−1) are the M(K+L−1)×Mand M ×M(K + L − 1) zero-matrices, respectively. We canverify the following properties of Ei, i = 0, 1, . . . ,K − 1.

(i) Ei can be obtained by applying column switching in anidentity matrix, and Ei(Ei)T = I. Therefore, the eigen-values of I and Ei have the same absolute values, anddet(Ei) = ±1.

(ii) For an arbitrary M(K + L) ×M(K + L) matrix A thatcan be eigendecomposed as A = UΛUH and AEi =U1Λ1UH

1 , it follows that Λ1 = Λ and U1 = UEi.

From the above properties, (21), and (22), the channel modelcan be written as

H = (IK ⊗GR1/2T

)E = (IK ⊗G

)(IK ⊗ R1/2

T

)E, (24)

where G = [GL−1, GL−2, . . . , G0, 0, . . . , 0] is an N ×M(K + L)matrix whose elements Gl’s are N × M matrices with i.i.d.zero-mean complex Gaussian entries and 1/2 variance perdimension. The remaining entries are zero, that is, 0N×Mdenotes an N × M zero matrix. Furthermore, RT is theM(K + L) ×M(K + L) transmit correlation matrix with thefollowing structures:

RT =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

RT ,L−1

RT ,L−2 0. . .

RT ,0

0

0. . .

0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (25)

where RT ,l is the M ×M transmit correlation matrix associ-ated with the lth channel path as defined in (21).

3.2. Effect of correlation

The following lemma sheds some light on the effect oftransmit correlation matrices on the ergodic capacity offrequency-selective MIMO channel.

6 EURASIP Journal on Advances in Signal Processing

Lemma 4. The upper bound on ergodic capacity of a frequen-cy-selective channel is a function of a matrix representing thesum of the eigenvalue matrices of spatial correlation matricesof different paths, Λ =∑L−1

i=0 (Ei)T diag(Δl)Ei.Starting by the mutual information equation for frequency-

selective channel [17–19], write

I(s; y) = 1P

log[

det(

IM(K+L) +P

NKσ2n

HH

H)]

. (26)

Subsequently, for sufficiently large P, the ergodic capacity of afrequency-selective fading channel is

C = E{I(s; y)

} = 1PE{

log[

det(

IM(K+L) +P

NKσ2n

HH

H)]}

.

(27)

Using the eigenvalue decomposition RT = diag(ΦlΔlΦHl ), l =

0, . . . , (L− 1), and (24) for H, one obtains

HH

H = ET(

IK ⊗GH diag(Φl)

diag(Δl)

diag(ΦH

l

)G)

E

∼ ET(

IK ⊗GH diag(Δl)

G)

E,(28)

since diag(ΦHl )G ∼ G and GH diag(Φl) ∼ GH . The ergodic

capacity of a frequency-selective fading channel in (27) can nowbe rewritten as

C ≈ 1PE{

log[

det(

IM(K+L)

+P

NKσ2n

ET(

IK ⊗GH diag(Δl)

G)

E)]}

.

(29)

By using Jensen’s inequality and taking the expectation, derivean upper bound on (29) as

C ≤ CUB

= 1P

log2 detE{

IM(K+L)

+P

NKσ2n

ET(

IK ⊗GH diag(Δl)

G)

E}

= 1P

log2 det(

IM(K+L) +P

Kσ2n

ET(

IK ⊗ diag(Δl))

E).

(30)

The right-hand side matrices in (30) can be multiplied, andhence it can be written as the sum of K products:

C ≤ CUB= 1P

log2 det

(

IM(K+L) +P

Kσ2n

L−1∑

i=0

(Ei)T

diag(Δl)

Ei

)

,

(31)

where Ei (i = 0 · · ·L−1) denote the column-shifted versions ofE defined in (23). Therefore, the upper bound on ergodic capac-ity of a frequency-selective channel is a function of the sum ofeigenvalue matrices of spatial correlation matrices of differentpaths, that is, Λ =∑L−1

i=0 (Ei)T diag(Δl)Ei.

Lemma 4 shows the importance of the eigenvalues of thespatial correlation matrices (of different paths in a frequency-selective fading channel) in the upper bound on ergodic ca-pacity, and hence in ergodic capacity itself. The exact analysisof the effect of eigenvalue matrices on the ergodic capacityis however not easy. Nevertheless, generally, when the cor-relation matrices are such that matrix Λ is a scaled identitymatrix, the most convenient case is of course when there isno spatial correlation for different paths, that is, when all theeigenvalues are one (Δl = (1/M)IM , (l = 0 · · ·L − 1)), yetwe can also find other cases that correlation matrices are notidentity but the ergodic capacity of the channel is maximized.

3.3. Precoder design

Our objective in this subsection is to find the optimal pre-coder matrix W, to maximize the ergodic capacity in (27)for frequency-selective channel based on the partial chan-nel knowledge of only the spatial correlation matrices RT ,l

(l = 0 · · ·L − 1) available at the transmitter. Recall that theprecoding matrix at the transmitter is only needed to recom-pute over a long interval whenever the spatial correlation ma-trices are changed. This point makes this precoder suitablefor the channels with fast fading.

Lemma 5 specifies the structure of the precoder in thiscase.

Lemma 5. The M(K + L) ×M(K + L) linear precoding ma-trix W that maximizes the ergodic capacity of a frequency-selective fading channel of (24) is a block diagonal matrixW = diag(Wi), with (K + L) optimal M ×M matrices Wi =ΦiΣ

1/2i Γi, where Γi’s are M×M arbitrary unitary matrices, Σi’s

are diagonal matrices, and Φi’s are the M ×M unitary matri-ces resulting from eigendecomposition of transmit correlationmatrices RT ,l’s, l = 0, 1, . . . , (L− 1).

Proof. Based on (27), the ergodic capacity of the system usingthe precoder can be written as

C = 1PE{

log2 det(

IM(K+L) +P

NKσ2n

WHHH

HW)}

.

(32)

Following the same steps as in the previous case, we can de-rive the upper bound on ergodic capacity as

C ≤ CUB = 1P

log2 det(

IM(K+L) +P

Kσ2n

RTΨΣΨH)

, (33)

where RT is defined in (25), and RT =∑K−1

l=1 (El)TRTEl. Notethat RT is also a block diagonal matrix.

Considering that RT = diag(ΦiΔiΦHi ), i = 0, 1, . . . , (K +

L−1), and using det(I+AB) = det(I+BA), one can find Ψ =diag(Φi), i = 0, 1, . . . , (K + L − 1). Therefore, the precodingmatrix can be written as

W = diag(Φi)Σ1/2Γ

= diag(ΦiΣ1/2i Γi

), i = 0, 1, . . . , (K + L− 1),

(34)

H. R. Bahrami and T. Le-Ngoc 7

· · ·WK+L−1 WK+L−2

ΔΔ Δ

W0

x(k)

y(k)Stacking

Figure 2: Precoder structure for a frequency-selective fading chan-nel with L independent effective paths.

where Γi is an arbitrary unitary matrix that can be set to iden-tity for simplicity. Therefore, the transmit precoding matrixW is also a block diagonal matrix with (K+L) optimal M×Mmatrices Wi = ΦiΣ

1/2i Γi, where Φi is one diagonal block of

the eigenvector matrix of RT =∑K−1

l=1 (El)TRTEl.

Lemma 5 shows the structure of the optimal precodermatrix in this case. Applying this precoder matrix changesthe eigenvalues of the correlation matrices of the channelfrom (Ei)T diag(Δl)Ei, (i = 0 · · ·L−1) to Σi(Ei)T diag(Δl)Ei,(i = 0 · · ·L − 1). It remains to find the diagonal entries ofthe multiplier matrices Σi’s (i = 0 · · ·L − 1) to modify theeigenvalues in order to achieve the maximum upper boundon ergodic capacity in (33), that is,

maxΣ

log2 det(

IM(K+L) +P

Kσ2n

diag(Δi)Σ)

s.t. tr(Σ) : constant.(35)

Solving (35) results in the following relation:

σi =[

ν−(

P

Kσ2nδ(i mod M)

)−1]+

, i = 1, 2, . . . ,M(L + K),

(36)

where σi’s (called power loading coefficients) and δi’s are thediagonal entries of Σ and Δi, respectively, and v is the con-stant determined by the power constraint. The waterpouringequation in this case is a function of Δi the eigenvalue ma-

trices of RT = diag(ΦiΔiΦHi ), i = 0, 1, . . . , (K + L − 1). In

other words, these equations are not directly related to trans-mit correlation matrix RT defined in (25).

In other words, the precoding matrix for a frequency-selective fading channel with L independent effective pathsis block diagonal. Therefore, the corresponding structure canbe decoupled into (K+L)M×M precoders for frequency-flatfading channels as shown in Figure 2. Δ blocks in the pre-coder structure are just time delays. The construction of the(K + L) precoders requires to solve the eigendecompositionof an M(K + L) × M(K + L) matrix RT , or equivalently Ldifferent transmit correlation matrices of size M ×M.

4. NUMERICAL RESULTS

At first, we investigate the effect of eigenvalues of spatial cor-relation matrix on ergodic capacity of a frequency-selectivechannel. We consider a system with two receive antennas(N = 2) and different number of transmit antennas and

Cap

acit

y(b

ps/H

z)

SNR (dB)

Frequency-selective, no precodingFrequency-selective, with precodingFrequency-flat, no precodingFrequency-flat, with precodingFrequency-flat, uncorrelated

0

2

4

6

8

10

12

14

16

−15 −10 −5 0 5 10 15 20 25 30

Figure 3: Performance comparison in partially correlated channels.C

apac

ity

(bps

/Hz)

SNR (dB)

Frequency-selective, no precodingFrequency-selective, with precodingFrequency-flat, no precodingFrequency-flat, with precodingFrequency-flat, uncorrelated

0

2

4

6

8

10

12

14

16

−15 −10 −5 0 5 10 15 20 25 30

Figure 4: Performance comparison in fully correlated channels.

channel paths (i.e., M = 2, 4 and L = 2, 4). Figure 5 showsthe ergodic capacity of the system for different eigenvaluespreads (λmax/λmin): 1 (no correlation), 2 (partial correla-tion), and ∞ (full correlation). The results clearly indicatethat the capacity decreases with an increase in eigenvaluespread of the spatial correlation matrices.

Figure 6 compares the change in the eigenvalue spreadof specific channels after applying linear precoding fordifferent numbers of transmit antennas in frequency-flatand frequency-selective channels with two paths. Precoderincreases the eigenvalue spread in the sense that it increases

8 EURASIP Journal on Advances in Signal Processing

0 5 10 15 20 25 302

4

8

6

10

12

14

Cap

acit

y(b

ps/H

z)

SNR (dB)

No correlation, M = L = 2No correlation, M = 2, L = 4No correlation, M = L = 4Partial correlation, M = L = 2Partial correlation, M = 2, L = 4

Partial correlation, M = L = 4Full correlation, M = L = 2Full correlation, M = 2, L = 4Full correlation, M = L = 4

Figure 5: Ergodic capacity with different eigenvalue spreads andnumbers of transmit antennas and channel paths.

21

Eig

enva

lue

spre

ad

Frequency-flat, no precodingFrequency-selective, no precodingFrequency-flat, with precodingFrequency-selective, with precoding

1.5

2

2.5

3

3.5

3 4 5 7 8 9 106

Number of transmit antennas (M)

Figure 6: Eigenvalue spread before and after applying precoding.

large eigenvalues (magnifies the strong eigenmodes) while itdecreases small eigenvalues (weakens the weak eigenvalues)in order to improve ergodic capacity. The function of theprecoders in changing the eigenvalues of spatial correlationmatrices is also clear from (18) and (36).

Next, we investigate the precoder performance in two dif-ferent cases of spatial correlation at the transmit and receivesides:

(i) partial spatial correlation, that is, with eigenvaluespread close to unity, and full-rank correlation matri-ces,

(ii) full spatial correlation, that is, with very large eigen-value spread, and rank-deficient spatial correlationmatrices.

As an illustrative example, we consider a MIMO systemwith 2 transmit and 2 receive antennas (M = N = 2) inboth frequency-flat and frequency-selective fading channels.The frequency-selective fading channel under considerationis represented by a 2-path model (L = 2). Furthermore, weassume that the channel paths are temporally uncorrelated.

Figures 3 and 4 illustrate the achievable capacity curves.For benchmark purpose, the capacity curve of an uncorre-lated frequency-flat fading channel is also included. In bothcases, precoders designed for frequency-flat and frequency-selective fading channels offer noticeable increases in the er-godic capacity of the system. In the case of partially cor-related channel, the curves are closer to the uncorrelatedfrequency-flat fading case. On the other hand, the precodersperform better when the channels are highly spatially corre-lated.

5. CONCLUDING REMARKS

We investigated the importance of eigenvalues of spatial cor-relation matrices on the ergodic capacity of frequency-flatand -selective MIMO channels. We showed that the ergodiccapacity depends greatly on the eigenvalue distribution ofspatial correlation matrices. In other words, knowing theeigenstructure of correlation matrices at the transmitter isvery important to enhance the capacity of the system. Basedon this fact, we first investigated the effect of eigenvaluesdistribution of spatial and path correlation matrices on thecapacity of frequency-flat and -selective channels. Next, weintroduced a linear scheme known as linear precoding thatcan enhance the ergodic capacity of the channel by chang-ing the eigenstructure of the channel by applying a lineartransformation. We derived the structures of precoders us-ing eigenvalue decomposition and linear algebra techniquesin both cases and show their similarities from an algebraicpoint of view. Simulations showed the ability of this tech-nique to change the eigenstructure of the channel, and henceto enhance the ergodic capacity considerably.

REFERENCES

[1] D. Chizhik, G. J. Foschini, M. J. Gans, and R. A. Valenzuela,“Keyholes, correlations, and capacities of multi-element trans-mit and receive antennas,” IEEE Transactions on Wireless Com-munications, vol. 1, no. 2, pp. 361–368, 2002.

[2] D. Gesbert, H. Bolcskei, D. A. Gore, and A. J. Paulraj, “Out-door MIMO wireless channels: models and performanceprediction,” IEEE Transactions on Communications, vol. 50,no. 12, pp. 1926–1934, 2002.

[3] J. Litva and T. K. Lo, Digital Beamforming in Wireless Commu-nications, Artech House, Boston, Mass, USA, 1996.

H. R. Bahrami and T. Le-Ngoc 9

[4] A. O. Boukalov and S. G. Haggman, “System aspects of smart-antenna technology in cellular wireless communications: anoverview,” IEEE Transactions on Microwave Theory and Tech-niques, vol. 48, no. 6, pp. 919–929, 2000.

[5] H. Sampath, P. Stoica, and A. J. Paulraj, “Generalized linearprecoder and decoder design for MIMO channels using theweighted MMSE criterion,” IEEE Transactions on Communi-cations, vol. 49, no. 12, pp. 2198–2206, 2001.

[6] H. Sampath, Linear precoding and decoding for multiple inputmultiple output (MIMO) wireless channels, Ph.D. thesis, Stan-ford University, Stanford, Calif, USA, May 2001.

[7] A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, andH. Sampath, “Optimal designs for space-time linear precodersand decoders,” IEEE Transactions on Signal Processing, vol. 50,no. 5, pp. 1051–1064, 2002.

[8] H. Sampath and A. J. Paulraj, “Linear precoding for space-time coded systems with known fading correlations,” IEEECommunications Letters, vol. 6, no. 6, pp. 239–241, 2002.

[9] S. Zhou and G. B. Giannakis, “Optimal transmitter eigen-beamforming and space-time block coding based on chan-nel correlations,” IEEE Transactions on Information Theory,vol. 49, no. 7, pp. 1673–1690, 2003.

[10] H. R. Bahrami and T. Le-Ngoc, “Precoder design based oncorrelation matrices for MIMO systems,” IEEE Transactions onWireless Communications, vol. 5, no. 12, pp. 3579–3587, 2006.

[11] D. Gesbert, H. Bolcskei, D. A. Gore, and A. J. Paulraj, “Out-door MIMO wireless channels: models and performanceprediction,” IEEE Transactions on Communications, vol. 50,no. 12, pp. 1926–1934, 2002.

[12] I. E. Telatar, “Capacity of multi-antenna Gaussian channel,”Tech. Rep., Bell Labs, Murray Hills, NJ, USA, 1995.

[13] G. J. Foschini and M. J. Gans, “On limits of wireless commu-nications in a fading environment when using multiple an-tennas,” Wireless Personal Communications, vol. 6, no. 3, pp.311–335, 1998.

[14] H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity ofOFDM-based spatial multiplexing systems,” IEEE Transactionson Communications, vol. 50, no. 2, pp. 225–234, 2002.

[15] T. M. Cover and J. A. Thomas, Elements of Information Theory,John Wiley & Sons, New York, NY, USA, 1991.

[16] R. A. Horn and C. R. Johnson, Matrix Analysis, CambridgeUniversity Press, New York, NY, USA, 1985.

[17] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding forwireless communication,” IEEE Transactions on Communica-tions, vol. 46, no. 3, pp. 357–366, 1998.

[18] V. Kafedziski, “Capacity of frequency selective fading chan-nels with side information,” in Proceedings of the 32nd Asilo-mar Conference on Signals, Systems and Computers, vol. 2, pp.1758–1762, Pacific Grove, Calif, USA, November 1998.

[19] K. Liu, T. Kadous, and A. M. Sayeed, “Orthogonal time-frequency signaling over doubly dispersive channels,” IEEETransactions on Information Theory, vol. 50, no. 11, pp. 2583–2603, 2004.

Hamid Reza Bahrami received his B.S. andM.S. degrees both in electrical engineeringfrom Sharif University of Technology andUniversity of Tehran in 2001 and 2003, re-spectively. He is currently a Ph.D. Candidateat McGill University. His research interestis in the area of wireless communicationswith emphasis on transmission techniquesin MIMO systems.

Tho Le-Ngoc obtained his B.Eng. degree(with distinction) in electrical engineeringin 1976, his M.Eng. degree in micropro-cessor applications in 1978 from McGillUniversity, Montreal, and his Ph.D. degreein digital communications in 1983 fromthe University of Ottawa, Canada. During1977–1982, he was with Spar AerospaceLimited, involved in the development anddesign of satellite communications systems.During 1982–1985, he was an Engineering Manager of the RadioGroup in the Department of Development Engineering of SRT-elecom Inc., develop the new point-to-multipoint subscriber radiosystem SR500. During 1985–2000, he was a Professor at the Depart-ment of Electrical and Computer Engineering of Concordia Uni-versity. Since 2000, he has been with the Department of Electricaland Computer Engineering of McGill University. His research in-terest is in the area of broadband digital communications with aspecial emphasis on modulation, coding, and multiple-access tech-niques. He is a Senior Member of the Ordre des Ingnieur du Que-bec, a Fellow of the Institute of Electrical and Electronics Engineers(IEEE), a Fellow of the Engineering Institute of Canada (EIC), anda Fellow of the Canadian Academy of Engineering (CAE). He is therecipient of the 2004 Canadian Award in Telecommunications Re-search, and recipient of the IEEE Canada Fessenden Award 2005.

Photograph © Turisme de Barcelona / J. Trullàs

Preliminary call for papers

The 2011 European Signal Processing Conference (EUSIPCO 2011) is thenineteenth in a series of conferences promoted by the European Association forSignal Processing (EURASIP, www.eurasip.org). This year edition will take placein Barcelona, capital city of Catalonia (Spain), and will be jointly organized by theCentre Tecnològic de Telecomunicacions de Catalunya (CTTC) and theUniversitat Politècnica de Catalunya (UPC).EUSIPCO 2011 will focus on key aspects of signal processing theory and

li ti li t d b l A t f b i i ill b b d lit

Organizing Committee

Honorary ChairMiguel A. Lagunas (CTTC)

General ChairAna I. Pérez Neira (UPC)

General Vice ChairCarles Antón Haro (CTTC)

Technical Program ChairXavier Mestre (CTTC)

Technical Program Co Chairsapplications as listed below. Acceptance of submissions will be based on quality,relevance and originality. Accepted papers will be published in the EUSIPCOproceedings and presented during the conference. Paper submissions, proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.

Areas of Interest

• Audio and electro acoustics.• Design, implementation, and applications of signal processing systems.

l d l d d

Technical Program Co ChairsJavier Hernando (UPC)Montserrat Pardàs (UPC)

Plenary TalksFerran Marqués (UPC)Yonina Eldar (Technion)

Special SessionsIgnacio Santamaría (Unversidadde Cantabria)Mats Bengtsson (KTH)

FinancesMontserrat Nájar (UPC)• Multimedia signal processing and coding.

• Image and multidimensional signal processing.• Signal detection and estimation.• Sensor array and multi channel signal processing.• Sensor fusion in networked systems.• Signal processing for communications.• Medical imaging and image analysis.• Non stationary, non linear and non Gaussian signal processing.

Submissions

Montserrat Nájar (UPC)

TutorialsDaniel P. Palomar(Hong Kong UST)Beatrice Pesquet Popescu (ENST)

PublicityStephan Pfletschinger (CTTC)Mònica Navarro (CTTC)

PublicationsAntonio Pascual (UPC)Carles Fernández (CTTC)

I d i l Li i & E hibiSubmissions

Procedures to submit a paper and proposals for special sessions and tutorials willbe detailed at www.eusipco2011.org. Submitted papers must be camera ready, nomore than 5 pages long, and conforming to the standard specified on theEUSIPCO 2011 web site. First authors who are registered students can participatein the best student paper competition.

Important Deadlines:

P l f i l i 15 D 2010

Industrial Liaison & ExhibitsAngeliki Alexiou(University of Piraeus)Albert Sitjà (CTTC)

International LiaisonJu Liu (Shandong University China)Jinhong Yuan (UNSW Australia)Tamas Sziranyi (SZTAKI Hungary)Rich Stern (CMU USA)Ricardo L. de Queiroz (UNB Brazil)

Webpage: www.eusipco2011.org

Proposals for special sessions 15 Dec 2010Proposals for tutorials 18 Feb 2011Electronic submission of full papers 21 Feb 2011Notification of acceptance 23 May 2011Submission of camera ready papers 6 Jun 2011