664 IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 9, SEPTEMBER 2009

Comments on “Channel Estimation and Signal Detection forMIMO-OFDM with Time Varying Channels”

Fei F. Cao and Jiandong Li, Senior Member, IEEE

Abstract—It is shown in [1] that the proposed least squareschannel estimator for multiple input multiple output-orthogonalfrequency division multiplexing (MIMO-OFDM) systems willacquire improved performance as the polynomial order increases,which we find is not generally the case. Analyses and simulationresults are given to support our claim.

Index Terms—MIMO-OFDM, channel estimation, polynomialapproximation, fast fading channels.

I. INTRODUCTION

IN a recent letter [1], Song and Lim proposed a poly-nomial approximation based pilot-assisted least squares

(LS) channel estimator for multiple input multiple output-orthogonal frequency division multiplexing (MIMO-OFDM)systems in fast fading channels. It is claimed in [1] thatthe proposed LS channel estimator will obtain improvedperformance as the polynomial order increases, which we willshow is not generally the case. Particularly, we will showthat on the one hand, there is a restriction in the selectionof the maximum polynomial order; on the other hand, theLS estimator proposed in [1] will not necessarily acquireimproved performance as the polynomial order increases.In the following, we will use (·)T , (·)†, ⊗, and 〈·〉N todenote transpose, Moore-Penrose pseudo-inversion, Kroneckerproduct, and modulo-N operation, respectively. IM denotesthe M×M identity matrix and A(v, :) the matrix constructedwith the rows of A indicated by the index vector v. Weprescribe that the indices of row and column of a matrix beginfrom zero.

II. SYSTEM MODEL

We consider a MIMO-OFDM system with NT transmit an-tennas and NR receive antennas, the number of the subcarriersis N and we adopt a cyclic prefix (CP) with a length ofNCP that is larger than the maximum multipath spread of thechannel to avoid inter-block interference (IBI). We assumethat the number of the channel taps is L. The normalizedmaximum Doppler frequency is denoted as fdTB , where fd

and TB are maximum Doppler frequency and the duration ofone OFDM block excluding CP, respectively. We assume thatdifferent transmit-receive antenna pairs experience indepen-dent, identically distributed (i.i.d) fading channels. The time-domain signal transmitted by the tth (t = 1, . . . , NT ) transmit

Manuscript received April 7, 2009. The associate editor coordinating thereview of this letter and approving it for publication was J. Armstrong.

This work is supported by NSFC (60725105,60572146), 973 Program(2009CB320404), PCSIRT, 111 Project (B08038), and MOE (107103).

The authors are with the State Key Lab. of ISN, Xidian University,TaibaiNanlu No. 2, 710071, Xi’an, China (e-mail: f f cao@263.net).

Digital Object Identifier 10.1109/LCOMM.2009.090819

antenna is F HXt, where F is the unitary discrete Fouriertransform (DFT) matrix, Xt = [Xt(0), . . . , Xt(N − 1)]T isthe frequency-domain symbol vector from transmit antennat with the elements taken from a constellation set Ω ={a1, . . . , aQ}. The received time-domain signal at the rth (r =1, . . . , NR) receive antenna is yr = [yr(0), . . . , yr(N−1)]T =∑NT

t=1 Hr,tXt + wr, where wr = [wr(0), . . . , wr(N − 1)]T

is i.i.d. circularly symmetric complex Gaussian noise vectorwith zero-mean and covariance σ2

wIN ; Hr,t is the N ×Ntime-domain channel matrix with the (n, m)th entry being

Hr,t(n, m) =1√N

L−1∑l=0

[hr,t(n, l)exp(−j2πτlm/N)

×exp(j2πmn/N)],

n, m = 0, . . . , N − 1 (1)

where hr,t(n, l) is the channel gain of the lth tap at discretetime n from transmit antenna t to receive antenna r, and τl

the relative delay of the lth tap. At the rth (r = 1, . . . , NR)receive antenna, after DFT transform, the frequency-domainsignal at the mth (m = 0, . . . , N − 1) subcarrier is

Yr(m)=1√N

N−1∑n=0

yr(n)exp(−j2πmn/N)

=1N

NT∑t=1

N−1∑n=0

N−1∑k=0

L−1∑l=0

[Xt(k)exp(−j2πτlk/N)

×exp(j2πnk/N)exp(−j2πmn/N)hr,t(n, l)]+Wr(m)

(2)

where Wr(m) is frequency-domain noise and W r =[Wr(0), . . . , Wr(N − 1)]T = Fwr. Defining the subcarrierindex sets Λm = {m−C, m−C +1, . . . , m+C−1, m+C}and Λm = {k|k = 0, . . . , N − 1 andk /∈ Λm}, where C is aninteger generally larger than zero, then (2) can be expressedas

Yr(m) =NT∑t=1

L−1∑l=0

N−1∑n=0

G(m, n, l, t)hr,t(n, l) +

NT∑t=1

L−1∑l=0

N−1∑n=0

Ge(m, n, l, t)hr,t(n, l) + Wr(m) (3)

where G(m, n, l, t) =∑

k∈Λmg(m, k, n, l, t),

Ge(m, n, l, t) =∑

k∈Λmg(m, k, n, l, t), and g(m, k, n, l, t) =

( 1N )exp(−j2πτlk/N)exp(−j2πn(m − k)/N)Xt(k). Due to

the fact that for a given subcarrier m (0 ≤ m ≤ N − 1), themost significant ICI items are from its adjacent subcarriers,therefore we neglect the items that relate to Ge(m, n, l, t)

1089-7798/09$25.00 c© 2009 IEEE

CAO and LI: COMMENTS ON “CHANNEL ESTIMATION AND SIGNAL DETECTION FOR MIMO-OFDM WITH TIME VARYING CHANNELS” 665

and approximate (3) as

Y r ≈ Ghr + W r (4)

where Y r = [Yr(0), . . . , Yr(N − 1)]T ; G is an N ×NT LN matrix and its (m, q)th entry is G(m, n, l, t) witht = q/(LN) + 1, l = [q − (t − 1)LN ]/N, andn = 〈q〉N ; the NT LN × 1 vector hr is expressed as hr =[hr,1(0), . . . , hr,1(L − 1), . . . , hr,NT (0), . . . , hr,NT (L − 1)]T

where hr,t(l) = [hr,t(0, l), . . . , hr,t(N − 1, l)].

III. POLYNOMIAL APPROXIMATION BASED

LS ESTIMATOR

As in [1], pilot symbols are placed in NE equi-spacedclusters; each cluster consists of 2C + 1 consecutive pilotsubcarriers and the middle pilot subcarrier is extracted forchannel estimation (the number of the pilot subcarriers withinone OFDM block is therefore NE(2C +1)). For convenience,we define the 1×NE vector vE = [E(0), . . . , E(NE−1)] thatconsists of the indices of pilot subcarriers extracted for channelestimation, then the signal model for channel estimation at therth (r = 1, . . . , NR) receive antenna can be represented as

Y r(vE , :) ≈ G(vE , :)hr + W r(vE , :) (5)

With polynomial expansion [1], we have hr,t(n, l) =∑Dd=0 ar,t(d, l)nd + er,t(n, l), where D is the polynomial

order, ar,t(d, l) the polynomial coefficient, and er,t(n, l) thepolynomial approximation error (PAE). Defining hr = QDar

and neglecting the PAE, we can rewrite (5) as1

Y r(vE , :) ≈ G(vE , :)QDar + W r(vE , :) (6)

where QD is an NT LN ×NT L(D + 1) matrix and QD =INT L ⊗ P D, P D is an N × (D + 1) matrix with the(n, d)th entry being nd; ar = [ar,1(0), . . . , ar,1(L −1), . . . , ar,NT (0), . . . , ar,NT (L − 1)]T is an NT L(D + 1)×1vector where ar,t(l) = [ar,t(0, l), . . . , ar,t(D, l)]. It is shownin [1] that G(vE , :)QD is an NE×NT L(D + 1) matrix andif NE ≥ NT L(D + 1), G(vE , :)QD is full column rank andthe estimation of hr (r = 1, . . . , NR) can be obtained as

hr = QD[G(vE , :)QD]†Y r(vE , :) (7)

However, we indicate that ensuring NE ≥ NT L(D + 1) isonly one necessary condition for the feasibility of the abovepolynomial approximation based LS channel estimation, itdoes not ensure G(vE , :)QD is full column rank. Whetheror not G(vE , :)QD is full column rank is closely related tothe system parameters adopted, particularly, we have

Lemma 1: Besides the necessary condition that NE ≥NT L(D + 1), another necessary condition for the situationthat G(vE , :)QD is full column rank is D≤2C.

Proof: We can express G(vE , :) as G(vE , :) = UB, Uis an NE×NENT L(2C + 1) matrix defined as

U =

⎡⎢⎣

XE(0) . . . 0...

. . ....

0 . . . XE(NE−1)

⎤⎥⎦ (8)

1It is clear that the signal model in this letter is essentially equivalent tothat in [1], i.e., all arguments presented in section III hold for the signal modelin [1]. We adopt different arrangements of the entries in G, QD , and ar (ascompared with those in [1]) for clarity of expression and convenience in theproof of Lemma 1.

where XE(i) = [XE(i),0,1, ..., XE(i),L−1,1, ..., XE(i),0,NT, ...,

XE(i),L−1,NT] (0 ≤ i ≤ NE − 1) is a 1×NT L(2C + 1)

vector, and XE(i),l,t = [Xt(E(i) − C), . . . , Xt(E(i) + C)].B is an NENT L(2C + 1)×NT LN matrix defined as B =[BT

E(0), . . . , BTE(NE−1)]T , BE(i) (0 ≤ i ≤ NE − 1) is an

NT L(2C + 1)×NT LN matrix

BE(i) =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

BE(i),0,1 · · · 0 · · · 0 · · · 0...

. . .... · · · ...

. . ....

0 · · · BE(i),L−1,1 · · · 0 · · · 0...

......

. . ....

......

0 · · · 0 · · · BE(i),0,NT· · · 0

.... . .

... · · · .... . .

...0 · · · 0 · · · 0 · · · BE(i),L−1,NT

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

where BE(i),l,t is a (2C+1)×N matrix with its (s, n)th entry(0≤s≤2C, 0≤n≤N − 1) calculated as

BE(i),l,t(s, n) =1N

exp[−j2πτl(E(i) − C + s)/N ]

×exp[−j2πn(C − s)/N ] (10)

Note that adopting the subscript l in XE(i),l,t is for clarityof expression and understanding, although XE(i),l,t is notrelated to l. It is straightforward that BE(i),l,t is full row rank,therefore Rank(BE(i)) = NT L(2C + 1). It is also clear thatthe mth (0≤m≤NT L(2C + 1) − 1) row of BE(i) and themth row of BE(j) (i �= j) are linearly correlated, thereforeRank(B) = NT L(2C + 1). Since U has rank NE , we haveRank[G(vE , :)]≤min[NE , NT L(2C+1)]. It is clear that QD

is full column rank, i.e., Rank(QD) = NT L(D+1), we haveRank[G(vE , :)QD]≤min[NE , NT L(2C + 1), NT L(D + 1)].Therefore, besides one necessary condition NE ≥NT L(D +1), another necessary condition for the situation that G(vE , :)QD is full column rank is NT L(2C + 1) ≥ NT L(D + 1),i.e., D≤2C.

It should be noted that the values of pilot symbols (theentries of U ) also play an important role in the rank ofG(vE , :)QD. However, it is evident that Lemma 1 holds eventhough optimum pilot symbols are adopted (design of theoptimum pilot symbols for the pilot pattern in [1] remainsan interesting topic and is beyond the scope of this letter).For random pilot symbols, our numerous simulation resultsshow that G(vE , :)QD is always full column rank so long asD≤2C.

Lemma 1 indicates that on the one hand, for a given C,selecting D >2C results in non-full column rank of G(vE , :)QD and therefore leads to unreliable channel estimation, thatis, there is a restriction in the selection of the maximumpolynomial order D when C is fixed. On the other hand,we find that even though it is ensured that G(vE , :)QD isfull column rank for D = 0, . . . , 2C, the LS estimation withthe largest D (i.e., 2C) does not necessarily obtain the bestperformance, which is explained as follows.

Let us draw a review at the signal model (6) based on whichthe LS estimation is performed, we see that the total interfer-ences is comprised of three items, i.e., the modeling errorintroduced by trivial ICI items, the interference introduced

666 IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 9, SEPTEMBER 2009

0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0

D=0D=1D=2D=3

(a)

(b)

MSE

(dB

)

SNR (dB )

0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0

D=0D=1D=2D=3

MSE

(dB

)

SNR (dB )

Fig. 1. Performances of the LS (in solid curves) and the LS-alter (in dashedcurves) estimators: (a) fdTB = 0.05, (b) fdTB = 0.2.

by PAE, and Gaussian noise. The magnitudes of the threeinterference items are closely related to the system parameters.Firstly, a large modeling error will be produced when fdTB

is large and the adopted C is relatively small. Secondly, whenD is not large enough, the PAE will become large. Thirdly,Gaussian noise will get relatively large when signal to noiseratio (SNR) decreases.

It is well known that, although LS estimation does not suf-fer from co-channel interference2 provided that G(vE , :)QD

is full column rank, it introduces noise enhancement. One mayadopt a larger D to reduce PAE, however, a larger D leads toa larger condition number (CN)3 of G(vE , :)QD, which maydeteriorate the estimation of ar and therefore increase channelestimation error (CEE) rather than decreasing it, especiallyat relatively low SNR range (this phenomenon is somewhatsimilar to the noisy polynomial fit problem, which can beverified by the ”polyfit” function in Matlab toolbox). Onlywhen the magnitude of the modeling error plus Gaussian noiseis very small can we expect that a larger D will lead to a betterperformance.

On the other hand, it is clear that we can not expectsatisfactory performance with a D that is too small. Ournumerous simulation results indicate that the value of Dshould be selected according to the system parameters, i.e.,the values of C, fdTB , and SNR.

IV. SIMULATION RESULTS

We consider a MIMO-OFDM system with N = 1028,NR = NT = 2, NCP = 256 and QPSK modulation,the system bandwidth is 6.4 MHz. The wireless channel isassumed to have L = 4 taps at relative delays of 0, 9.9, 17.1,and 36.9 μs with the relative powers of 0, −9.7, −19.2, and−22.8 dB, respectively. The inverse Fourier transform of theDoppler spectrum is the zeroth-order Bessel function of thefirst kind. For channel estimation, C = 1 is adopted as in [1],

2We call the entries of ar “co-channel signals” when estimating ar .3CN is a good evaluation of noise enhancement for matrices with full

column rank.

we adopt NE = 50 equi-spaced pilot clusters and each pilotcluster consists of 2C+1 = 3 consecutive pilot subcarriers, therandomly generated pilot symbols have the same modulationand power as those of data symbols. SNR is defined asNT (

∑L−1l=0 σ2

l )/σ2w where σ2

l is the variance of the lth tap.Mean square error (MSE) of channel estimation is defined as[1/(NT NR)]

∑NR

r=1

∑NT

t=1

∑L−1l=0 E[| hr,t(n, l) − hr,t(n, l) |2].

Fig. 1 shows the MSE performances of the LS estimatorproposed in [1] (in solid curves). We see that the LS estimatorobtains the worst performance for D = 3, which can beexplained by the non-full column rank of G(vE , :)QD asindicated by Lemma 1. We also see that adopting the maximumD (for C = 1, the maximum adoptable D is 2) does notlead to the best performance, and the estimator with D = 1underperforms that with D = 0 at relatively low SNR range,which can be explained by the large noise enhancementindicated in section III. The performance of the LS-alter4

estimator is also shown in Fig. 1 (in dashed curves). Wenote that compared with the LS estimator in [1], the LS-alter estimator suffers from less modeling error due to itsmaking use of all pilot symbols for calculating each elementof G(vE , :)5.

REFERENCES

[1] W. G. Song and J. T. Lim, “Channel estimation and signal detection forMIMO-OFDM with time varying channels,” IEEE Commun. Lett., vol.10, pp. 540-542, July 2006.

4The LS-alter estimator is the same as the LS estimator except that it adoptsan alternative strategy (suggested by one reviewer) to calculate G(vE , :),i.e., G(m, q) =

∑k∈VP

g(m, k, n, l, t), m ∈ VE , where VP is the set thatconsists of the indices of all pilot subcarriers and VE = {E(0), . . . , E(NE−1)}. In this case G(vE , :)QD is full column rank almost surely for D =0, 1, 2, 3, but is non-full column rank almost surely for D = 4, 5.

5It can be easily proven that the LS-alter and LS estimators obtain the sameperformance for D = 0. For D = 1, 2, our simulation results show that theLS-alter estimator can indeed obtain a marginal performance advantage overthe LS estimator, although their MSE curves almost overlap each other in Fig.1. Due to its suffering from large noise enhancement, the LS-alter estimatorwith D = 3 fails to outperform the case of D = 2.