100 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 2, FEBRUARY 2008

On the Training of MIMO-OFDM Channels withLeast Square Channel Estimation and Linear Interpolation

Seung Joon Lee, Senior Member, IEEE

Abstract— The effect of pilot arrangement is investigated onthe performance of trained MIMO-OFDM channels where fadingis time-flat in a block of coherence-time duration but frequency-selective. The MIMO-OFDM channel is considered to be trainedby using least square channel estimation on pilot subcarriersand by applying linear interpolation on data subcarriers. Theachievable rate of such a system is derived employing thegeneralized mutual information approach. Numerical examplesare presented examining the optimal density of pilots in the two-dimensional time-frequency domain.

Index Terms— Generalized mutual information, least squarechannel estimation, linear interpolation, multiple input multipleoutput, OFDM.

I. INTRODUCTION

MULTIPLE-INPUT multiple-output wireless communi-cation systems combined with orthogonal frequency

division multiplexing technologies (MIMO-OFDM) are at-tracting huge attention and being intensively studied anddeveloped [1]–[3]. A simple but popularly used scheme fortraining OFDM channels, especially in practice, is to obtainleast square (LS) channel estimates on pilot subcarriers andthen to linearly interpolate channel gains on data subcarriersusing the LS channel estimates [4], [5].

There are two kinds of information theoretical approachesin analyzing the achievable rate or the capacity of MIMOfading channels with channel estimation errors. One is touse the lower bound on the capacity [6]–[9] and the otheris to use the generalized mutual information (GMI) [10],[11]. The former considers the worst case equivalent noisewithout any assumption of decoding structure and the latterassumes Gaussian encoding and a nearest neighbor decoderwhile leaving the equivalent noise as it is. Both approacheshave been applied only to the case when channel estimatesare uncorrelated with channel estimation errors as in linearminimum mean square error (LMMSE) estimators. Recently, itwas reported in [12] that the GMI approach can be also appliedto least square (LS) channel estimation where estimates andestimation errors are correlated.

Linear interpolation has different statistical properties fromthose of LMMSE and LS channel estimations in that estima-tion errors are correlated with both estimates and channels,while they are uncorrelated with estimates in the LMMSEestimation and uncorrelated with channels in the LS estima-tion. The correlation between estimation errors and estimatesmakes it difficult to apply the approach of the lower bound

Manuscript received September 23, 2007. The associate editor coordinatingthe review of this letter and approving it for publication was E. Baccarelli.

S. J. Lee is with the Electronics & Telecommunications Research Institute,Daejeon 305-700, Korea (e-mail: s.j.lee@ieee.org).

Digital Object Identifier 10.1109/LCOMM.2008.071566.

on the capacity. However, the correlation of estimation errorswith estimates or channels does not impose restriction onapplicability of the GMI approach, which will be confirmedin this letter.

In this letter, assuming that channel experiences time-flatfading in a coherence-time duration and frequency-selectivefading, we analytically derive the average GMI of the MIMO-OFDM channels with LS channel estimation and linear inter-polation and then, based on the result, we obtain the optimalpilot arrangement maximizing the overall system performance.The remainder of this letter is organized as follows. In SectionII, a system model for training the MIMO-OFDM channel isdescribed. In Section III, the average GMI is derived for thetrained MIMO-OFDM fading channels. In Section IV, numer-ical examples are presented to show how the time-frequency-domain density of pilots affects the spectral efficiency (averageGMI per subcarrier). We draw conclusions in Section V.

II. SYSTEM MODEL

A MIMO-OFDM fading channel with M transmit antennasand N receive antennas at each time on each frequency isrepresented as

y = Hx + z (1)

where y is an N -dimensional output vector, x is an M -dimensional input vector, H is an N ×M channel matrix, andz is an N -dimensional additive noise vector. In (1), indexesdifferentiating time and frequency are omitted for the sake ofnotational simplicity. We assume that E

[xx†] = ρd

M IM fordata subcarriers, where ρd denotes the average signal-to-noiseratio (SNR) at each receive antenna for a data subcarrier andIM denotes the M ×M identity matrix. We also assume thatE

[zz†

]= IN , and that the elements of H are independent

and identically distributed (i.i.d.) complex Gaussian with unitvariance. Pilot and data subcarriers are arranged as in Fig.1, referring to examples of practical systems [13], [14]. Thearrangement shown in Fig. 1 is different from the comb-typepilot arrangement introduced in [4], [5] since all subcarriersare not used as pilots on specific frequencies, while so arethey in the comb-type arrangement. There are T symbols inthe coherence-time duration, over which fading is time-flat.At frequency k · L (k = · · · ,−1, 0, 1, · · · ), Tt subcarriersare used for pilots and T − Tt subcarriers are used for data,where integer L denotes the frequency-domain interval ofpilot subcarriers. At frequency l �= kL, all T subcarriersare used for data. Least square (LS) channel estimates areattained at frequency kL and linear interpolation is employedat frequency l �= kL using the adjacent LS estimates. Weconsider that Tt is multiples of M so that the same numberof pilots are used for each transmit antenna.

1089-7798/08$25.00 c© 2008 IEEE

LEE: ON THE TRAINING OF MIMO-OFDM CHANNELS WITH LEAST SQUARE CHANNEL ESTIMATION AND LINEAR INTERPOLATION 101

TimeFreq

uenc

y

TtT

L

: pilot subcarrier : data subcarrier

l=0l=1

l=L

l=L 1

Fig. 1. The arrangement of pilot and data subcarriers.

III. ACHIEVABLE RATE OF MIMO-OFDM CHANNELS

TRAINED USING LS ESTIMATION AND LINEAR

INTERPOLATION

We let fd denote the subcarrier spacing and hmn[l] =hmn(fdl) denote the channel gain from transmit antenna m toreceive antenna n at frequency l. We hereafter drop antennaindexes m and n for the sake of notational simplicity. Thefrequency-domain correlation of h[l] is defined as γ[s] �E {h[l] · h∗[l + s]} where ∗ denotes complex conjugate. Linearinterpolation at frequency l is given by

h[l] =lh(LS)[L] + (L − l)h(LS)[0]

L, 0 ≤ l < L (2)

where h(LS)[kL] (k = 0, 1) denote the LS channel estimatesobtained on pilot subcarriers. We see h[0] = h(LS)[0].

Defining the estimation error as he[l] � h[l]−h[l], we obtainthe variances of h[l] and he[l] and the correlation1 between h[l]and he[l] as

σ2h[l]

=l2 + (L − l)2

L2

(γ[0] + σ2

he[0]

)+ 2

l(L − l)L2

�{γ[L]}(3)

σ2he[l] = σ2

h[l]− 2

l

L�{γ[L− l]}− 2

L − l

L�{γ[l]}+ γ[0] (4)

and

ζ[l] � E {h[l]h∗e[l]} =

lγ[L − l] + (L − l)γ[−l]L

− γ[0], (5)

respectively, where σ2he[0] is the variance of LS estima-

tion errors and �{·} denotes the real part of a com-plex value. The random variable h[l] conditioned on he[l]is Gaussian with mean (1 + ζ[l]/σ2

he[l])he[l] and variance

1 − |ζ[l]|2/σ2he[l]. Using the Bayes’ rule Pr(he[l]|h[l]) =

1This correlation vanished for the LS channel estimation in [12]. However,it does not restrict applicability of the GMI approach, as shown in this section.

Pr(h[l]|he[l]) Pr(he[l])/Pr(h[l]), we have that

E{H[l]H†[l]∣∣H[l]} =

(1 − 2µ1[l])H[l]H†[l] + E{He[l]H†e[l]

∣∣H[l]} (6a)

E{He[l]H†e[l]

∣∣H[l]} = |µ2[l]|2H[l]H†[l] + Mµ3[l]IN (6b)

whereµ1[l] � (σ2

he[l] + �{ζ[l]})/σ2h[l]

(7a)

µ2[l] � (σ2he[l] + ζ∗[l])/σ2

h[l](7b)

µ3[l] � (σ2he[l] − |ζ[l]|2)/σ2

h[l](7c)

and frequency index l appears on matrices for clarity. It isnoticed that if ζ[l] = 0, then µ1[l] = µ2[l] = µ3[l] and (6)looks similar to the case of LS channel estimation.

Applying (6) and (7) to [12, eq. (5)] and letting{λ1, · · · , λLm

} denote the eigenvalues of H[l]H†[l] whereLm = min(M,N), we obtain the generalized mutual infor-mation (GMI) as

GMI ({λi}) = supθ>0

g (θ, {λi}) (8a)

g (θ, {λi}) �Lm∑i=1

{ln

(1 + θ

ρd

Mλi

)−

θ2(1 + ρdµ3[l] + |µ2[l]|2 ρd

M λi

) − θ(1 − 2µ1[l])1 + θ ρd

M λi

ρd

Mλi

}.

(8b)

Then the average GMI at frequency l, GMI[l], can be calcu-lated in the same way as in [12, eq. (8)] with (1+σ2

he) in [12,

eq. (8a)] replaced by σ2h[l]

of (3). Since there is no closed formsolution for the optimal value θ∗ of θ maximizing g(θ, {λi})for Lm > 1, we use the following high rate and low rateapproximations:

θ∗(high rate)({λi}) =Lm

(1 + ρdµ3[l]) Lm + |µ2[l]|2 ρd

M

∑Lm

i=1 λi

(9)and

θ∗(low rate)({λi}) =(1 − µ1[l])

∑Lm

i=1 λi

(1 + ρdµ3)∑Lm

i=1 λi + |µ2[l]|2 ρd

M

∑Lm

i=1 λ2i

.

(10)

IV. NUMERICAL EXAMPLES

In order to examine the effect of pilot arrangement, Tt

and L, on the overall performance, we consider the spectralefficiency defined as

Spectral Efficiency =(T − Tt)GMI[0] + T

∑L−1l=1 GMI[l]

TL(11)

which means the sum of average GMIs divided by thenumber of pilot and data subcarriers. We assume thecarrier spacing fd = 10.9375 kHz [15] and the SNRof pilot subcarrier equal to that of data subcarrier ρd,which results in the variance of LS estimation errorsσ2

he[0] = M/(ρdTt). We use a highly frequency-selectivefading channel model of ‘Pedestrian-B’ [16] represented as

102 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 2, FEBRUARY 2008

0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

6

TL

Spe

ctra

l Effi

cien

ty [b

its/c

hann

el u

se]

t

SNR=10 dBSNR=20 dB

Fig. 2. Spectral efficiency as a function of Tt and L when M = N = 1and T = 12.

h[l] = (1/A)∑5

i=0 ai exp(−j2πfdlτi) where (τ0, · · · , τ5) =(0, 0.2, 0.8, 1.2, 2.3, 3.7) [µsec], (E{|a0|2}, · · · , E{|a5|2}) =(0,−0.9,−4.9,−8,−7.8,−23.9) [dB], and A is a normalizingfactor. When Lm is 2 or 4, we employ θ∗(low rate)(·) for SNR=0dB and θ∗(high rate)(·) for SNR=10 or 20 dB.

Fig. 2 shows the spectral efficiency as a function of Tt andL when M = N = 1 and T = 12, where it is seen that thesensitivity of the spectral efficiency to (Tt, L) increases whenthe SNR is larger and it is not much high around the optimalpair of (Tt, L). The optimal pairs of (Tt, L) maximizing thespectral efficiency are tabulated in Table I, considering variousvalues of SNR, M(= N), and T , where the entries within eachparenthesis denote the optimal Tt, L, and spectral efficiency inorder. The followings are observed. When the SNR increases,lower time-domain density (smaller Tt) and higher frequency-domain density (smaller L) of pilots are required. Since for afixed Tt the SNR of pilot subcarrier per each transmit antennadecreases as M increases, larger Tt and L are necessaryfor increasing M(= N). If the coherence-time T is larger(from 12 to 24), the optimal Tt increases and the optimal Ldecreases. The spectral efficiency gain by increasing T is alittle enlarged when M(= N) increases.

V. CONCLUSION

The generalized mutual information (GMI) of MIMO-OFDM fading channels was derived considering that channelestimation was performed using least square channel esti-mation on pilot subcarriers and linear interpolation on datasubcarriers. The result from the case of LS channel estimationwas easily extended to obtain the GMI with linear interpo-lation. The GMI with linear interpolation was derived as afunction of the frequency-domain autocorrelation of channel.Numerical examples were given to show how sensitive thespectral efficiency is to the time-frequency-domain densityof pilots and to present the optimal density of pilots infrequency-selective and time-flat fading for a given coherence-time duration.

TABLE I

OPTIMAL TIME-FREQUENCY DENSITY OF PILOTS,

(Tt, L, SPECTRAL EFFICIENCY)

M = N SNR T = 12 T = 24

10 dB (12,14,0.73) (16,13,0.76)

10 dB (5,9,2.6) (8,9,2.7)

20 dB (3,6,5.3) (4,5,5.4)

2

0 dB (12,14,1.3) (24,14,1.4)

10 dB (8,10,4.8) (10,9,5.0)

20 dB (4,6,9.8) (6,6,10)

4

0 dB (12,14,2.3) (24,14,2.7)

10 dB (12,11,9.0) (16,11,9.4)

20 dB (8,8,18) (12,7,19)

REFERENCES

[1] M. Jiang and L. Hanzo, “Multiuer MIMO-OFDM for next-generationwireless systems,” Proc. IEEE, vol. 95, no. 7, pp. 1430–1469, July 2007.

[2] A. Ghosh, D. R. Wolter, J. G. Andrews, and R. Chen, “Broadbandwireless access with WiMAX/802.16: Current performance benchmarksand future potential,” IEEE Commun. Mag., vol. 43, no. 2, pp. 129–136,Feb. 2005.

[3] H. Ekstrom, A. Furuskar, J. Karlsson, M. Meyer, S. Parkvall, J. Torsner,and M. Wahlqvist, “Technical solutions for the 3G long-term evolution,”IEEE Commun. Mag., vol. 44, no. 3, pp. 38–45, Mar. 2006.

[4] S. Coleri, M. Ergen, A. Puri, and A. Bahai, “Channel estimation tech-niques based on pilot arrangement in OFDM systems,” IEEE Trans.Broadcast., vol. 48, no. 3, pp. 223–229, Sept. 2002.

[5] O. Simeone, Y. Bar-Ness, and U. Spagnolini, “Pilot-based channelestimation for OFDM systems by tracking the delay-subspace,” IEEETrans. Wireless Commun., vol. 3, no. 1, pp. 315–325, Jan. 2004.

[6] B. Hassibi and B. M. Hochwald, “How much training is needed inmultiple-antenna wireless links?,” IEEE Trans. Inform. Theory, vol. 49,no. 4, pp. 951–963, Apr. 2003.

[7] J. Baltersee, G. Fock, and H. Meyr, “Achievable rate of MIMO channelswith data-aided channel estimation and perfect interleaving,” IEEE J.Select. Areas Commun., vol. 19, no. 12, pp. 2358–2368, Dec. 2001.

[8] T. Yoo and A. Goldsmith, “Capacity and power allocation for fadingMIMO channels with channel estimation error,” IEEE Trans Inform.Theory, vol. 52, no. 5, pp. 2203–2214, May 2006.

[9] M. Medard, “The effect upon channel capacity in wireless communica-tions of perfect and imperfect knowledge of the channel,” IEEE Trans.Inform. Theory, vol. 46, no. 3, pp. 933–946, May 2000.

[10] A. Lapidoth and S. Shamai, “Fading channels: how perfect need “perfectside information” be?” IEEE Trans. Inform. Theory, vol. 48, no. 5, pp.1118–1134, May 2002.

[11] H. Weingarten, Y. Steinberg, S. Shamai, “Gaussian codes and weightednearest neighbor decoding in fading multiple-antenna channels,” IEEETrans. Inform. Theory, vol. 50, no. 8, pp. 1665–1686, Aug. 2004.

[12] S. J. Lee, “Effect of least square channel estimation errors on achievablerate in MIMO fading channels,” IEEE Commun. Lett., vol. 11, no. 11,pp. 862–864, Nov. 2007.

[13] IEEE P802.16Rev2/D0b, Draft Standard for Local and metropolitan areanetworks, Part 16: Air Interface for Fixed and Mobile Broadband WirelessAccess Systems, June 2007.

[14] 3GPP TS 36.211, v1.2.0, Physical Channels and Modulation, June 2007.[15] T. Jiang, W. Xiang, H.-H. Chen, and Qiang Ni, “Multicast broadcast

services support in OFDMA-based WiMAX systems,” IEEE Commun.Mag., vol. 45, no. 8, pp. 78–86, Aug. 2007.

[16] ITU-R M.1225, Guidelines for evaluation of radio transmission tech-nologies for IMT-2000, 1997.