Asymptotic Ergodic Capacity of Wideband MIMOChannels with Separately-Correlated Rician Fading∗

Giorgio TariccoPolitecnico di Torino – Dipartimento di Elettronica

corso Duca degli Abruzzi 24 – 10129, Torino, ItalyE-mail: taricco@polito.it

Erwin RieglerForschungszentrum Telekommunikation WienDonau-City-Strasse 1 – 1220 Vienna, Austria

E-mail: riegler@ftw.at

Abstract—The mutual information of a wideband Rician fadingcorrelated MIMO channel is approximated by an analytic asymp-totic method. The method is applied to derive the optimum inputsignal covariance matrix and the corresponding ergodic capacity.The numerical accuracy of the method is investigated and theeffectiveness of covariance optimization is assessed with respectto the level of spatial correlation and the Rice factor.

I. INTRODUCTION

Many research studies in multiple-input multiple-output(MIMO) communications address the problem of derivingthe channel capacity. Probably one of the most successfulresults in this area is due to Telatar, who found closed-form expressions for the capacity of the independent Rayleighfading MIMO channel [17]. Subsequent studies showed thatthis channel model is not always appropriate because it doesnot take into account spatial correlation and the presenceof a line-of-sight (LOS) component. Many experimental andtheoretical works addressed the issue of channel modeling [5].A good trade-off between model complexity and accuracy isachieved by the so-called separately-correlated channel [7].

Another active research area in MIMO communicationstargeted the extension to the wideband channel, in most casesbased on an OFDM approach [1], [8], [11]. Recently, Mous-takas and Simon studied the asymptotic mutual informationof the wideband Rayleigh fading correlated MIMO channelin [10], extending one of their own earlier studies, relevant tothe narrowband channel case [9]. The analytic approach usedin these works is based on the so-called replica method, widelyadopted in theoretical physics because of its effectiveness inaddressing complex system scenarios where other methods,such as the one based on Stieltjes transform proposed in [4],provide equivalent results with a greater mathematical effort.

In this work we address the case of a wideband Ricianfading MIMO channel with separate correlation (see the Sys-tem Model section for the precise assumptions). We derive anexpression of the average mutual information in the case ofasymptotically large number of antennas. This result representsa nontrivial extension of earlier results [16] addressing thenarrowband channel case. By using this result, we derivethe ergodic capacity of the channel and the corresponding

∗This work is supported by the STREP project No. IST-026905 (MASCOT)within the sixth framework programme of the European Commission.

(optimum) input signal covariance matrix. Then, we study thenumerical accuracy of the asymptotic approximation proposedand analyze the effectiveness of covariance optimization withrespect to the spatial correlation and the Rice factor.

II. SYSTEM MODEL

We consider a wideband frequency-selective separately-correlated Rician fading MIMO channel with nT transmit andnR receive antennas. We assume that the channel bandwidth isB and the delay spread is τ . Then, the maximum number ofresolvable paths of the channel is L = �Bτ�, and the channelequation can be written as follows:

y[n] =∑�∈L

H[�]x[n − �] + z[n], n ∈ N . (1)

Here, we defined: the frame length N ; the sets L �{0, . . . , L − 1} and N � {0, . . . , N − 1}; the transmittedsignal vector x[n] ∈ C

nT×1 at time n; the impulse responsechannel matrix H[�] ∈ C

nR×nT , sampled at delay time �; theadditive noise vector z[n] ∈ C

nR×1 at time n; and the receivedsignal vector y[n] ∈ C

nR×1 at time n.We assume that the additive noise is independent over time

and uncorrelated: z[n] ∼ Nc(0, InR)1.

We assume uncorrelated scattering, so that all matrices H[�]are independent. We consider a separately correlated Ricianfading model with common spatial correlation among the usersat the receiver. Thus, we can write

H[�] = E[H[�]] + R[�]1/2Hw[�]T[�]1/2. (2)

Here, the matrices Hw[�] have all iid entries distributedas Nc(0, 1) (in compliance with the uncorrelated scatteringassumption), R[�] is the receive spatial correlation matrix,and T[�] is the transmit spatial correlation matrix. Since anonzero average channel matrix derives from a line-of-sightcomponent, which typically affects only the signal path withthe lowest delay, it will be assumed that

E[H[�]] ={

H � = 00 � > 0 . (3)

1In the presence of spatial correlation, z[n] ∼ Nc(0,ΣZ), we can reduceto the uncorrelated case pre-multiplying y[n] by Σ

−1/2Z . The resulting noise

vector will be uncorrelated with identity covariance matrix and the channelmatrices will be all pre-multiplied by the same constant matrix Σ

−1/2Z and

the resulting problem will be still tractable under our approach.

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The Rician factor (defined as the ratio between the LOS andthe scattered received power when the input signals are iidwith power P/nT) and the signal-to-noise ratio (SNR) aregiven by:

K � ‖H‖2∑�∈L Tr(R[�]) Tr(T[�])

(4)

and

ρ �(K + 1)P

∑�∈L Tr(R[�]) Tr(T[�])nRnT

, (5)

respectively.

A. OFDM signaling

We consider a communication system based on orthogonalfrequency-division multiplexing (OFDM). It is well knownthat OFDM allows to convert the frequency-selective fadingchannel into a set of parallel frequency-flat channels. The keyassumption of OFDM is the insertion of a cyclic prefix (CP)in all frames, whose length is equal to the delay spread. TheCP assumption implies the following set of conditions:

x[n] = x[n + N ] n = −L + 1, . . . ,−1. (6)

It must be noted that the presence of a CP reduces the availablethroughput by a factor κ � N

N+L−1 .Next, we calculate the discrete Fourier transform (DFT)

of the channel equation. Let ω � exp(−j 2π/N) so that, inconsideration of the CP condition (6), we can transform thechannel equation (1) into

y[m] �∑n∈N

y[n]ωmn

=∑n∈N

∑�∈L

H[�]x[n − �]ωmn +∑n∈N

z[n]ωmn

=∑�∈L

H[�]ωm�∑n∈N

x[n]ωmn +∑n∈N

z[n]ωmn

= H[m]x[m] + z[m], (m ∈ N ). (7)

Here, we defined the following vector/matrix DFTs: H[m] �∑�∈L H[�]ωm�, x[m] �

∑n∈N x[n]ωmn, and z[m] �∑

n∈N z[n]ωmn.

B. Frequency-domain equivalent channel

First, we notice that the frequency-domain noise vectors areiid since:

E[z[m]z[m′]H] =∑n∈N

E[z[n]z[n]H]ω(m−m′)n

={

0 m �= m′

NInRm = m′ . (8)

However, the frequency-domain channel matrices H[m] maybe statistically dependent, even though the H[�] matricesare independent. This complicates the analysis of the mutualinformation because the resulting channel would be frequency-varying. However, under assumption (3), the frequency-domain channel matrices are marginally identically distributed.In fact, it is easy to see that E[H[m]] = H for all m ∈ N .

Moreover, setting h[�] � vec(H[�]), h[m] � vec(H[m]),h[�] � h[�]−E[h[�]], for � ∈ L and m ∈ N , and h � vec(H),we have:

E

[h[m]h[m′]H

]= E

[h[m]

]E

[h[m′]

]H+∑�∈L

E

[h[�]h[�]H

]ω(m−m′)�

= hhH +∑�∈L

TT[�] ⊗ R[�]ω(m−m′)�, (9)

which is independent of m when m = m′, as required.Then, we follow the arguments reported in [1]. Assuming

every OFDM symbol corresponds to independent realizationsof the channel impulse response matrices, the mutual infor-mation will be given (in nat) by

I(Q[0], . . . , Q[N − 1]) =1

N + L − 1·∑

m∈Nln det(InR

+ H[m]Q[m]H[m]H), (10)

where Q[m] = E[x[m]x[m]H].Typically, the ergodic channel capacity is calculated under

an average time-domain power constraint:∑n∈N

E[Tr(x[n]x[n]H)] ≤ P. (11)

However, applying the inverse DFT (IDFT)

x[n] =1N

∑m∈N

x[m]ω−mn,

we can obtain the following equivalent average frequency-domain power constraint:

P ≥∑n∈N

E[Tr(x[n]x[n]H)]

=∑n∈N

∑m∈N

∑m′∈N

E[Tr(x[m]x[m′]H)]N2

ω−(m−m′)n

=1N

∑m∈N

E[Tr(x[m]x[m]H)]. (12)

Therefore, the MIMO-OFDM ergodic capacity is defined asthe maximum of E[I(Q)] under the constraint∑

m∈NTr(Q[m]) ≤ NP. (13)

Furthermore, in accordance with [10, Sec. III-A], we can showthat the ergodic capacity achieving matrices are independentof m. In fact, let use define:

φ(Q) � E[ln det(InR+ H[m]QH[m]H)].

This is a concave function of Q from [3, Th. 17.9.1]. Hence,we can apply Jensen’s inequality and obtain:

E[I(Q[0], . . . , Q[N − 1])] =1

N + L − 1

∑m∈N

φ(Q[m])

≤ κφ

(1N

∑m∈N

Q[m])

.

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Since the last matrix argument is independent of m, weconclude that the ergodic capacity is achieved by a covariancematrix that is constant for all values of m. Therefore, werestrict to consider the case of equal covariance matricesQ[m] = Q with the power constraint

Tr(Q) ≤ P.

III. ASYMPTOTIC ANALYSIS

In this section we provide an analytic method to calculatethe asymptotic ergodic mutual information of the OFDM-MIMO fading channel described above. The method proposedis a generalization of the one presented in [14], [15] relevant tothe narrowband channel. Recalling that each frequency-domainchannel matrix H[m] has the same probability distribution, ourgoal is to calculate the mean

E[ln det(InR+ HQHH)],

where

vec(H) ∼ Nc

(h,∑�∈L

TT[�] ⊗ R[�])

,

according to (9). The ergodic mutual information of ourOFDM-MIMO fading channel model is obtained as

E[I(Q)] = κE[ln det(InR+ HQHH)].

A. Asymptotic CGF

Repeating the main steps of the asymptotic analysis devel-oped in [14], [15], we write the cumulant generating function(CGF) of the mutual information as follows:

G(ν) � E

[det(InR

+ HQHH)−ν]

= E

[ ∫CnR×ν

dU∫

CnT×ν

etr{− π(UHU + VHV)

}etr{− π(UHHQ1/2V − VHQ1/2HHU)

}dV].

Using the following result (see, e.g., [14], [15]):

E[etr(MW − WHMH)] = exp(−‖M‖2),

which holds for any random matrix W with iid Nc(0, 1)entries, we obtain:

G(ν) =∫

CnR×ν

dU∫

CnT×ν

dVetr[−π(UHU + VHV)]

etr{− π(UHHQ1/2V − VHQ1/2HHU)

−π2∑�∈L

UHR[�]UVHT[�]V}

=∫

CnR×ν

dU∫

CnT×ν

dVetr[−π(UHU + VHV)]

etr[−π(UHHQ1/2V − VHQ1/2HHU)]∏�∈L

∫dμ(W�,Z�)etr

{W�Z� − πW�UHR[�]U

}etr{− πZ�VHT[�]V

}.

Here, integration over W� is carried out on Rν×ν + W�,0

for some W�,0 ∈ Cν×ν , and the integration over Z� is

carried out on (j R)ν×ν + Z�,0 for some Z�,0 ∈ Cν×ν .

Moreover, dμ(W�,Z�) = dW�d(Z�/(j 2π)). Applying theidentity Tr(MXHNY) = vec(X)H(MT ⊗ N)vec(Y) [6] inthe last integral, we obtain the following result:

G(ν) =∫ {∏

�∈Letr(W�Z�)dμ(W�,Z�)

}× (14)

det

(IνnR

+∑

�∈L W� ⊗ R[�] Iν ⊗ HQ1/2

−Iν ⊗ Q1/2HH IνnT+∑

�∈L Z� ⊗ T[�]

)−1

Here and in the following we define T[�] � Q1/2T[�]Q1/2.

B. Ergodic mutual information

We assume that the saddlepoint of the integrand in (14)corresponds to the replica symmetric point W� = w�Iν ,Z� =z�Iν , for � ∈ L. Under this assumption, we can get the valuesof w�, z� by solving the following set of fixed-point equations:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

w� = Tr{[

InT+∑

�∈L z�T[�] + Q1/2HH

×(InR

+∑

�∈L w�R[�])−1

HQ1/2]−1

T[�]}

z� = Tr{[

InR+∑

�∈L w�R[�] + HQ1/2

×(InT

+∑

�∈L z�T[�])−1

Q1/2HH]−1

R[�]} (15)

for all � ∈ L. The resulting asymptotic approximation of theergodic capacity is given by:

μI(Q) = ln det{InR

+∑�∈L

w�R[�]}

+ ln det{InT

+[∑

�∈Lz�T[�] + HH

(InR

+∑�∈L

w�R[�])−1

H]Q}

−∑�∈L

w�z� (16)

C. Ergodic capacity

To maximize the ergodic mutual information (16) withrespect to the input covariance matrix Q we have to solvethe following optimization problem:

maximize μI(Q)

subject to Q ≥ 0, Tr(Q) ≤ P.

Paralleling the approach of [19], we can see that the objectivefunction is concave with respect to Q and the constraintssatisfy Slaters qualification condition [2]. Thus, the Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficientfor optimality [2]. Then, to derive the KKT conditions, weform the Lagrangian

L(Q, λ,Ψ) = μI(Q) − λ[Tr(Q) − P ] + Tr(ΨQ) (17)

for λ ≥ 0,Ψ ≥ 0. After nulling the variation of L(Q, λ,Ψ)with respect to Q, we obtain the KKT conditions:⎧⎪⎨⎪⎩

(InT+ ΞQ)−1Ξ = λInT

− ΨTr(Q) = P

Tr(ΨQ) = 0(18)

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where we defined the matrix

Ξ �∑�∈L

z�T[�] + HH

(InR

+∑�∈L

w�R[�])−1

H.

The solution of eqs. (18) depends on the rank of the matrixΞ. Let us assume that Ξ = UH

xΛxUx is the orthogonaldecomposition of Ξ with Λx = diag(Λx,0) and unitaryUH

x, where Λx is a positive definite diagonal matrix. Then,if Px =rank(Ξ) ≤ nT, we can see that

ln det(InT+ ΞQ) = ln det(Iρx

+ ΛxQ1),

where Q1 is the ρx × ρx upper left submatrix of UxQUHx.

Finally, the ergodic capacity achieving covariance matrix isgiven by

Q = Uxdiag((λ−1Iρx

− Λ−1x )+,0

)UH

x (19)

where (·)+ denotes the elementwise positive part of theelements of the matrix argument, i.e., (x)+ = x for x ≥ 0and 0 otherwise. The Lagrange multiplier λ is determined bysolving the constraint equation Tr(Q) = P . It can be noticedthat eq. (19) can be written as

Q = Ux(λ−1InT− Λ−1

x )+UHx

by assuming that matrix inversion maps the null entries on themain diagonal of Λx to positive infinity.

Remark III.1 It is worth noting that the ergodic capacityachieving covariance matrix found has the typical waterfillingform arising in many information theoretic optimization prob-lems (see [3], [19]). The ergodic channel capacity is foundby solving the joint set of equations given by (15) and (19).This can be achieved by an iterative algorithm that starts froman initial value of the variables (w�, z�, Q) and proceeds byapplying (15) and (19) alternately until it converges.

Remark III.2 In the special case when R[�] = R, constantfor all � ∈ L, we have a substantial simplification of theasymptotic mutual information equations. In fact, eqs. (15)and (16) become{

w = Tr{[InT+ zT + Q1/2HH(InR

+ wR)−1HQ1/2]−1T}z = Tr{[InR

+ wR + HQ1/2(InT+ zT)−1Q1/2HH]−1R}

(20)and

μI(Q) = ln det(InR+ wR)

+ ln det{InT+ [zT + HH(InR

+ wR)−1H]Q} − wz,

(21)

respectively. In those equations we set T �∑

�∈L T[�] andT � Q1/2TQ1/2.

IV. NUMERICAL RESULTS

We present some numerical results to illustrate the accuracyof the asymptotic analytic method proposed to determinethe capacity of the wideband MIMO communication channelconsidered. Here we disregard the rate reduction due to CPinsertion (equivalently, we assume N → ∞). Specifically,we study the effects of spatial correlation with a strongLOS component corresponding to a Rice factor K = 0 dB.We consider the following parameters of the communicationscenarios:

• N. of antennas: nT = nR = 4, n. of paths: L = 3.• Mean channel matrix: (H)ab = h0 (constant).• Correlation: (R[�])ab = α

|a−b|� and (T[�])ab = ρ�β

|a−b|� ,

with ρ� = 0.9�.The effects of spatial correlation are assessed by comparingthe ergodic capacity and the iid mutual information (cor-responding to Q = qInT

) in three different scenarios: i)uncorrelated (Fig. 1); ii) partly correlated (i.e., correlated onlyover the paths without a LOS component, Fig. 2); and iii)completely correlated (i.e., correlated over all paths, Fig. 3).In all figures, solid and dashed lines with markers correspondto the ergodic capacity and to the iid mutual information,respectively, obtained in both cases via the asymptotic analyticapproach developed in this work. Square markers representthe corresponding Monte-Carlo simulation results. We can seethat, in all cases considered, simulation results are in closeagreement with the analytic ones. The figures also plot theoptimum covariance matrix eigenvalue distribution (solid lineswithout markers, with sum normalized to 1). These curvesconfirm the well known fact that beamforming (correspondingto a single dominant eigenvalue, only one curve stuck to1) is optimal below a certain SNR. These results show thatthe threshold increases as the correlation increases, too. Theoptimization of the covariance matrix is more effective as thelevel of spatial correlation increases. The results show thatcovariance optimization can compensate almost completely theiid mutual information degradation due to increased correla-tion. Similar results, not reported for space limitation, showthat the effectiveness of covariance optimization increases asthe Rice factor increases. On the contrary, with Rayleighfading, the advantage is more limited.

V. CONCLUSIONS

We provided an analytic asymptotic (in the number ofantennas) method to approximate the mutual information of awideband Rician fading separately correlated MIMO channel.By using this method, we obtain the asymptotically optimuminput signal covariance matrix, yielding the ergodic capacityof the channel considered. As noticed in [4], the asymptoticmethod is substantially more efficient than numerical opti-mization methods such as the one proposed in [18] (interiorpoint with barrier). We show by numerical results that theasymptotic approximation is accurate even when the numberof antennas is small. Finally, using our proposed method, weanalyze the effectiveness of covariance optimization related tothe level of spatial correlation and the Rice factor.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.978-1-4244-2324-8/08/$25.00 © 2008 IEEE. 4

−20 −15 −10 −5 0 5 100

1

2

3

4

5

6

SNR (dB)

Mut

ual i

nfor

mat

ion

(bit/

s/H

z)4x4 MIMO,L=3,α

l=(0.0 0.0 0.0 ),β

l=(0.0 0.0 0.0 ),ρ

l=(1.0 0.9 0.8 ),K=0 dB

iidoptimum

Fig. 1. System capacity and iid mutual information versus the SNR for theuncorrelated scenario. Optimum covariance normalized eigenvalues are alsoreported.

−20 −15 −10 −5 0 5 100

1

2

3

4

5

6

SNR (dB)

Mut

ual i

nfor

mat

ion

(bit/

s/H

z)

4x4 MIMO,L=3,αl=(0.0 0.9 0.9 ),β

l=(0.0 0.9 0.9 ),ρ

l=(1.0 0.9 0.8 ),K=0 dB

iidoptimum

Fig. 2. Same as Fig. 1 for the partly correlated scenario.

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