862 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 11, NOVEMBER 2007

Effect of Least Square Channel Estimation Errors onAchievable Rate in MIMO Fading Channels

Seung Joon Lee, Senior Member, IEEE

Abstract— The achievable rate of a multiple input multipleoutput (MIMO) fading channel is derived in the form ofgeneralized mutual information (GMI) when least square (LS)channel estimation is employed to gain channel state informationat receivers. The derived average GMI of a channel with LSchannel estimation is compared with the known average GMI(equivalently, the lower bound on the ergodic capacity) ofa channel with linear minimum mean square error channelestimation. An effective approximation method is also proposedto ease the numerical calculation of the average GMI of a MIMOchannel with LS channel estimation.

Index Terms— Generalized mutual information, least squarechannel estimation, multiple input multiple output.

I. INTRODUCTION

RECENTLY, the effect of channel estimation errors onthe capacity or achievable rate of multiple input multiple

output (MIMO) as well as single input single output (SISO)channels has been investigated [1]–[7]. Works [1]–[5] derivedthe lower bound on the capacity without any assumptionof decoding structure, while works [6], [7] obtained theachievable rate or the generalized mutual information (GMI)assuming a (weighted) nearest neighbor decoder. The GMIderived in [7] is found to be the same as the lower bound onthe capacity derived in [1].1

All the works except [5] assumed linear minimum meansquare error (LMMSE) channel estimation. LMMSE channelestimation has the useful feature that channel estimates areuncorrelated with channel estimation errors, which makes iteasy to analyze the effect of channel estimation errors onthe capacity. Least square (LS) channel estimation is alsopopularly used in practical systems due to its simplicity (e.g.,in [8], [9]). In the LS channel estimation, channel estimatesare correlated with channel estimation errors. This correlationmakes it difficult to analytically investigate the relationshipbetween the LS channel estimation errors and the capacity.Although [5] proposed a formula for the lower bound on thecapacity of MIMO channels with LS channel estimation, thederivation procedure missed the above-mentioned correlation,and the results are misleading, as will be shown in this letter.

In this letter, when the LS channel estimation is used, theGMI (implying the achievable rate with Gaussian coding) isderived by following a procedure similar to that in [7]. The

Manuscript received July 10, 2007. The associate editor coordinating thereview of this letter and approving it for publication was Shuguang (Robert)Cui.

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

Digital Object Identifier 10.1109/LCOMM.2007.071150.1Eq. (23) in [7] can be verified to be the same as eq. (17) in [1]. It is claimed

in [7] that different approaches of [7] and [1] suggest different methods ofoptimizing the training sequence, which is beyond the discussion of this letter.

remainder of this letter is organized as follows. In SectionII, a system model is described including a debate aboutthe applicability of the lower bound on the capacity to thesystem under consideration. In Section III, the GMI is derivedfor MIMO channels with LS channel estimation. In SectionIV, it is examined how to effectively perform the numericalcalculation of the average GMI, and then numerical examplesare presented to compare the average GMIs of MIMO channelswith different (perfect, LMMSE, and LS) channel estimations.We draw conclusions in Section V.

II. SYSTEM MODEL

A channel with M transmit antennas and N receive anten-nas at time k is represented as

yk = Hxk + zk (1)

where yk is an N -dimensional output vector, xk is an M -dimensional input vector, H is an N × M channel matrix,and zk is an N -dimensional additive complex Gaussian noisevector. We assume that E

[xkx

†k

]= ρd

M IM , E[zkz

†k

]=

IN , and the elements of H are independent and identicallydistributed (i.i.d.) complex Gaussian with unit variance, whereIM denotes the M × M identity matrix and ρd denotes theaverage signal-to-noise ratio (SNR) at each receive antennafor a data symbol. The least square (LS) channel estimationmodel is described as

H = H + He (2)

where H is an estimate of H and He is an estimation errormatrix independent of H with E

[HeH†

e

]= Mσ2

heIN . Then

(1) is rewritten as

yk = Hxk + zk − Hexk. (3)

If the two terms, Hxk and vk � zk − Hexk, onthe right side of (3) are uncorrelated, then the capac-ity C of the system (3) is lower-bounded as C ≥log2

∣∣∣∣IM×M + ρd

M HH†(

E[vkv

†k

∣∣∣H])−1∣∣∣∣ [bits/channel use]

where | · | denotes the determinant of a matrix, † denotes com-

plex conjugate transpose, and E[· ∣∣H]

denotes conditional

expectation (conditioned on H) [1]. However, if those twoterms are correlated as in the case of LS channel estimation,this lower bound on the capacity is not valid.2 Alternatively,we rely on the generalized mutual information (GMI), whichmeans the maximum communication rate at which the averageprobability of error vanishes as the Gaussian codeword lengthincreases [6], [7].

2Nonetheless, this lower bound on the capacity was suggested in [5, eq.(22)], which will be shown to overestimate the performance in Section IV.

1089-7798/07$25.00 c© 2007 IEEE

LEE: EFFECT OF LEAST SQUARE CHANNEL ESTIMATION ERRORS ON ACHIEVABLE RATE IN MIMO FADING CHANNELS 863

III. GENERALIZED MUTUAL INFORMATION WITH LEAST

SQUARE CHANNEL ESTIMATION

As in [7], we first assume the following weighted nearestneighbor decoder

s = arg mins

1K

K∑k=1

∥∥∥W (yk − Hx(s)

k

)∥∥∥2

(4)

where W is the weighting matrix. In [7], matrix W wasdesigned to maximize the GMI, and it was found to dependon the statistics of channel estimation errors conditioned onthe channel estimates [7, eq. (22)]. However, we consider thatsuch statistics are not available at receivers in the LS channelestimation.3 So we set W = IN . Then we obtain the GMI as[7]4

GMI = supθ<0

E[θtr

{( ρd

MHeH†

e + zkz†k

)−

( ρd

MHH†+

zkz†k

)G−1

}+ ln |G|

∣∣∣H][nats/channel use]

(5a)

G = IN − θρd

MHH† (5b)

where tr(·) denotes a trace. Using E[HH†∣∣H]

=

E[HeH†

e

∣∣H]+(1−2µ)HH† and E

[HeH†

e

∣∣H]= MµIN +

µ2HH† for LS channel estimation where µ =σ2

he

1+σ2he

, (5) is

derived as (6) on the next page. If we let the eigenvalues ofHH† be λ1, λ2, · · · , λLm

where Lm � min(M,N), (6b) isrewritten as (7) on the next page. To obtain (7) from (6b), weuse that tr

{U · diag(λ1, λ2, · · · , λLm

) · U†} =∑Lm

i=1 λi forany unitary matrix U where diag({λi}) denotes the diagonalmatrix the entries of which are {λi}. The average GMI canbe calculated as

GMI � E [GMI({λi})]=

∫ ∞

0

· · ·∫ ∞

0

GMI({(

1 + σ2he

)ξi

})·f({ξi})dξ1 · · · dξLm

(8a)

where f({ξi}) is given by [10]

f (ξ1, ξ2, · · · , ξLm) =

1Lm!K

∏i

e−ξiξLM−Lmi

∏i<j

(ξi − ξj)2

(8b)and where LM = max(M,N) and K is a normalizing factor.

IV. NUMERICAL CALCULATION AND EXAMPLES

It is not difficult to obtain the value θ∗ of θ maximizingg (θ, {λi}) in (7) at least by numerical calculation for anynumber of antennas. However, it is found that a closed-formsolution for θ∗ as a function of {λi} is helpful for easilyevaluating the GMI of (8) by using common math tools (e.g.,Maple V). For Lm = 1 (that is, M = 1 or N = 1), θ∗ is

3If those statistics were available at receivers, then the receivers couldperform LMMSE channel estimation.

4Eq. (5) is the same as [7, eq. (17)] with the weighting matrix replaced bythe identity matrix. Eq. (17) of [7] is valid for LS channel estimation as wellas LMMSE channel estimation.

simply given by a root5 of a 2nd order polynomial equation.On the other hand, if Lm > 1, then there is no closed-formsolution for θ∗. Alternatively, we consider high rate and lowrate approximations for a closed-form of θ∗ as follows. Ifθ ρd

M λi � 1 (implying high achievable rate6), then the solutionof ∂g(θ,{λi})

∂θ = 0 is given by

θ∗(high rate)({λi}) =Lm

(1 + ρdµ) Lm + µ2 ρd

M

∑Lm

i=1 λi

. (9)

Also, if θ ρd

M λi → 0 (implying low achievable rate), then θ∗ isapproximated by

θ∗(low rate)({λi}) =(1 − µ)

∑Lm

i=1 λi

(1 + ρdµ)∑Lm

i=1 λi + µ2 ρd

M

∑Lm

i=1 λ2i

.

(10)Therefore, we consider that GMI is approximated and lowerbounded as

GMI ≥ max(E

[g

(θ∗(high rate)({λi}), {λi}

)],

E[g

(θ∗(low rate)({λi}), {λi}

)]). (11)

Assuming that ρt denotes the average SNR at each receiveantenna for a pilot symbol and Tt is the number of pilot sym-bols used for each channel estimation, the channel estimationerror is given by σ2

he= M

ρtTt. If the ratio of ρt to ρd is

fixed (e.g., ρt = ρd) with increasing ρd, we observe in (10)that limρd→∞ µ2ρd = 0, and then limρd→∞ θ∗(low rate)({λi}) =

11+ρdµ = limρd→∞ θ∗(high rate)({λi}). This means, interestingly,that θ∗(low rate)({λi}) of (10) can be also applied for high SNR.In the following numerical examples, it will be shown that(10) is a good approximation in the entire range of SNRs.

In Fig. 1, for single input and single output (SISO) chan-nels, the average GMI with LS channel estimation GMILS iscompared with the average GMI (= the ergodic capacity) withperfect channel estimation GMIideal, the average GMI (= thelower bound on the ergodic capacity) with LMMSE channelestimation GMILMMSE, and the result of [5, eq. (22)] intendedfor the lower bound on the ergodic capacity with LS channelestimation. The difference between GMILS and GMILMMSE isshown to be small. This is because, although the LS channelestimation error may be considerably larger than the LMMSEchannel estimation error for low SNR, the GMI is not sensitiveto channel estimation errors for low SNR. The differencebetween GMILS and GMILMMSE increases if the number ofpilot symbols Tt decreases. The result of [5, eq. (22)] isshown to misleadingly overestimate the performance. In Fig.2, for multiple input and multiple output (MIMO) channels,the approximate GMILS values are presented compared withGMIideal and GMILMMSE. Although the exact GMILS is notavailable, we expect that the proposed approximation (11)for GMILS (especially, E[g(θ∗(low rate)({λi}), {λi})]) has highaccuracy since it is shown to have only small difference fromGMILMMSE and the exact GMILS must be between GMILMMSE

and the proposed approximate (a lower bound on) GMILS.

5It is not shown here, since it is lengthy and straightforward.6For high SNR, the GMI is expected to be dominated by the first term

ln(1 + θ ρdM

λi) of the two terms within the summand in (7). Thus, largeθ ρd

Mλi implies high achievable rate.

864 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 11, NOVEMBER 2007

GMI = supθ<0

(ln |G| − θtr

{( ρd

ME

[HeH†

e

∣∣H]+ IN

)θρd

MHH†G−1 + (1 − 2µ)

ρd

MHH†G−1

})(6a)

= supθ>0

(ln

∣∣∣IN + θρd

MHH†

∣∣∣ − θtr

{((θρdµ + θ − 1 + 2µ)IN + θ

ρd

Mµ2HH†

) ρd

MHH†

(IN + θ

ρd

MHH†

)−1})

(6b)

GMI ({λi}) = supθ>0

Lm∑i=1

{ln

(1 + θ

ρd

Mλi

)− θ2

(1 + ρdµ + µ2 ρd

M λi

) − θ(1 − 2µ)1 + θ ρd

M λi

ρd

Mλi

}� sup

θ>0g (θ, {λi}) (7)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-10 -5 0 5 10 15

GMI ideal

GMILMMSE

GMILS

from [5, eqn. (22)]

GM

I [bi

ts/c

hann

el u

se]

T =1t

T =1t

T =2t

SNR per data symbol, dρ

Fig. 1. Comparison of average GMIs for SISO channels: (a) assuming nochannel estimation error (‘GMIideal’), (b) with LMMSE channel estimation(‘GMILMMSE’), and (c) with LS channel estimation (‘GMILS’). (The resultfrom [5, eq. (22)] is also included to check its validity.)

V. CONCLUSION

The effect of least square (LS) channel estimation wasinvestigated on the achievable rate of multiple input multipleoutput (MIMO) fading channels, relying on the generalizedmutual information (GMI) approach. To ease the numeri-cal calculation of the average GMI for MIMO channels, aheuristic but effective approximation method was proposed.The presented numerical results compared the derived averageGMI of channels with LS channel estimation and the knownaverage GMI of channels with linear minimum mean squarechannel estimation, and also verified the accuracy of theproposed approximation method.

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0

2

4

6

8

10

12

-10 -5 0 5 10 15SNR per data symbol, dρ

GM

I [bi

ts/c

hann

el u

se]

GMI ideal

GMILMMSE

GMILS

Low rate approx.

High rate approx.

GMILS

Max. of the above two

M=N=T =2

M=N=T =4t

t

Fig. 2. Comparison of average GMIs for MIMOchannels (‘High rate approx. GMILS’ corresponding toE[g(θ∗(high rate)({λi}), {λi})], ‘Low rate approx. GMILS’ corresponding toE[g(θ∗(low rate)({λi}), {λi})], and ‘Max. of the above two’ corresponding tomax(E[g(θ∗(high rate)({λi}), {λi})], E[g(θ∗(low rate)({λi}), {λi})]).

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