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670 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007

Blind Channel Estimation forMIMO-OFDM Systems

Changyong Shin, Member, IEEE, Robert W. Heath, Jr., Senior Member, IEEE, and Edward J. Powers, Fellow, IEEE

Abstract—By combining multiple-input multiple-output(MIMO) communication with the orthogonal frequency divisionmultiplexing (OFDM) modulation scheme, MIMO-OFDM sys-tems can achieve high data rates over broadband wireless chan-nels. In this paper, to provide a bandwidth-efficient solution forMIMO-OFDM channel estimation, we establish conditions forchannel identifiability and present a blind channel estimationtechnique based on a subspace approach. The proposed methodunifies and generalizes the existing subspace-based methods forblind channel estimation in single-input single-output OFDMsystems to blind channel estimation for two differentMIMO-OFDM systems distinguished according to the numberof transmit and receive antennas. In particular, the proposedmethod obtains accurate channel estimation and fast convergencewith insensitivity to overestimates of the true channel order. Ifvirtual carriers (VCs) are available, the proposed method canwork with no or insufficient cyclic prefix (CP), thereby potentiallyincreasing channel utilization. Furthermore, it is shown underspecific system conditions that the proposed method can beapplied to MIMO-OFDM systems without CPs, regardless of thepresence of VCs, and obtains an accurate channel estimate with asmall number of OFDM symbols. Thus, this method improves thetransmission bandwidth efficiency. Simulation results illustratethe mean-square error performance of the proposed method vianumerical experiments.

Index Terms—Blind channel estimation, cyclic prefix, multiple-input multiple-output (MIMO) system, orthogonal frequencydivision multiplexing (OFDM), subspace method, virtual carrier.

I. INTRODUCTION

R ECENTLY, increasing interest has been concentratedon modulation techniques that provide high data

rates over broadband wireless channels for applications, in-cluding wireless multimedia, wireless Internet access, andfuture-generation mobile communication systems. Orthogonalfrequency division multiplexing (OFDM) is a promising digitalmodulation scheme to simplify the equalization in frequency-selective channels [1], [2]. The main benefit is that it simplifiesimplementation, and it is robust against the frequency-selectivefading channels. Multiple-input multiple-output (MIMO) com-munication, enabled by multiple transmit and receive antennas,

Manuscript received April 15, 2005; revised April 19, 2006 and April 25,2006. This paper was presented in part at the IEEE Global Telecommunica-tions Conference, Dallas, TX, November 2004. The review of this paper wascoordinated by Prof. Z. Wang.

C. Shin is with the Communications and Network Laboratory, SamsungAdvanced Institute of Technology (SAIT), 446-712 Korea (e-mail: [email protected]).

R. W. Heath, Jr. and E. J. Powers are with the Wireless Networking andCommunications Group (WNCG), Department of Electrical and ComputerEngineering, The University of Texas at Austin, Austin, TX 78712-0240 USA(e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TVT.2007.891429

can increase significantly the channel capacity (see, e.g., [3]–[6]and references therein). Thus, MIMO-OFDM systems, whichcombine the OFDM with MIMO communication, can providea high-performance transmission (see, e.g., [3], [4], and [7]–[9]and references therein).

In a MIMO-OFDM system, a coherent signal detection re-quires a reliable estimate of the channel impulse responsesbetween the transmit and receive antennas. These channelscan be estimated by sending training sequences. The trainingrequirements, however, are significant [10]–[18]. Furthermore,transmitting the training sequences is undesirable for certaincommunication systems [19], [20]. Thus, blind channel esti-mation for MIMO-OFDM systems has been an active area ofresearch in recent years. Zhou et al. [21] proposed a subspace-based blind channel estimation method for space-time codedMIMO-OFDM systems using properly designed redundantlinear precoding and the noise subspace method [22]–[24].Bölcskei et al. [25] proposed an algorithm for blind channelestimation and equalization for MIMO-OFDM systems usingsecond-order cyclostationary statistics induced by employinga periodic nonconstant-modulus antenna precoding. Yatawattaand Petropulu [26] presented a blind channel estimation methodbased on a nonredundant linear block precoding and cross-correlation operations. Zeng and Ng [27] proposed a subspacetechnique based on the noise subspace method for estimatingthe MIMO channels in the uplink of multiuser multiantennazero-padding OFDM systems [28], [29].

A variety of second-order statistics (SOS)-based blind es-timators (see, e.g., [22]–[24], and [30]–[51] and referencestherein) have been presented since Tong et al. [52] introduced aSOS-based technique for the blind identification of single-inputmultiple-output systems. Among those methods, the noise sub-space method is believed to be one of the most promising due toits simple structure and good performance. Thus, by exploitingthe fundamental structure of the noise subspace method, sub-space methods [53]–[55] for single-input single-output (SISO)OFDM systems have been proposed and achieved good esti-mation performance. Muquet et al. [53] developed a subspacemethod for SISO-OFDM systems by utilizing the redundancyintroduced by the cyclic prefix (CP) insertion, and derived acondition for channel identifiability. For shaping of the transmitspectrum, practical OFDM systems are not fully loaded [56].The subcarriers that are set to zero without any information arereferred to as virtual carriers (VCs) [57]. Other than the CP,the presence of VCs provides another useful resource that canbe used for channel estimation. Li and Roy [54] proposed asubspace blind channel estimator for SISO-OFDM systems byconsidering the existence of VCs and provided a condition for

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SHIN et al.: BLIND CHANNEL ESTIMATION FOR MIMO-OFDM SYSTEMS 671

channel identifiability. They showed that their estimator basedon the noise subspace method can achieve better estimationperformance than other blind estimation techniques [55], [58].In particular, they demonstrated that the CPs are more usefulfor their estimator than VCs. In [54], however, the reason whythe utilization of CPs rather than VCs increases the accuracyof channel estimation was not discussed. In addition, it wasnot considered how the channel estimation performance can beimproved in cases with no or insufficient CPs.

In this paper, we unify and generalize the SISO-OFDMsubspace methods in [53] and [54] to the case of blind channelestimation for spatial multiplexing MIMO-OFDM systems withany number of transmit and receive antennas. We present a newblind channel estimator based on the noise subspace methodand establish conditions for blind channel identification inspatially multiplexed MIMO-OFDM systems. The proposedmethod works regardless of the presence of VCs, can use aslittle as one received OFDM symbol for a filtering matrix,and operates with any number of transmit or receive antennas.Considering the presence of VCs, the proposed method can beapplied to MIMO-OFDM systems without CPs, where blind es-timation techniques based on CPs cannot be employed, therebyproviding the systems with the potential to achieve a higherchannel utilization. For MIMO-OFDM systems with CPs, theproposed method can provide an additional performance gainwith respect to the existing blind channel estimation methods.We provide numerical results that illustrate tradeoffs in mean-square error as a function of the CP length, number of VCs, andnumber of OFDM symbols used in the estimate.

Compared with [25] and [26], we do not use a transmitprecoding to aid our blind channel estimator. Furthermore,by virtue of our subspace approach, we need fewer OFDMsymbols to obtain a reliable estimate. Our estimator, however,has an additional full rank requirement that is not present in theprecoding-based methods. The combination of our estimatorand the precoding-based methods is an interesting topic forfuture work. Our approach generalizes the studies in [53] and[54] to operate with multiple transmit and multiple receiveantennas under the assumption that spatial multiplexing isused at the transmitter. With one transmit and receive antenna,our approach and identifiability conditions are simplified tothose presented in [53] and [54]. A subspace-based method forMIMO-OFDM systems with spatial multiplexing was proposedin [59]. Compared with [59], we also consider the case of excesstransmit antennas. Furthermore, the identifiability conditionsprovided in [59] are not complete. Specifically, their full rankrequirement does not appear to be sufficient, and the channelambiguity condition does not appear to be comprehensive.

The rest of this paper is organized as follows. In Section II,we briefly describe a MIMO-OFDM system model. InSection III, we establish conditions for the MIMO-OFDMchannel identifiability by generalizing the conditions presentedin [54] and develop a blind channel estimation scheme basedon the noise subspace method. Section IV contains simulationresults demonstrating the performance of the proposed method.Finally, a conclusion is provided in Section V.

The notations used in this paper are as follows. Matricesand vectors are denoted by symbols in boldface, and (·)∗,

Fig. 1. MIMO-OFDM system model with Mt transmit and Mr receiveantennas.

Fig. 2. Baseband OFDM system with virtual carriers for the jth transmitantenna and the ith receive antenna.

(·)T, and (·)H represent complex conjugate, transpose, andHermitian, respectively. rank(X) and span(X) mean the rankof a matrix X and the subspace spanned by the column vectorsof a matrix X, respectively. ∗ and ⊗ stand for the convolutionand the Kronecker product, respectively. wN is equal to ej2π/N .Im denotes the m×m identity matrix, and 0 stands for theall-zeros matrix of appropriate dimensions. diag(x) denotes adiagonal matrix with x on its main diagonal. E· denotesstatistical expectation. x[1 : k] denotes the first k elements ofa vector x. X[i1 : i2, j1 : j2] denotes a submatrix obtained byextracting rows i1 through i2 and columns j1 through j2 froma matrix X. If no specific range appears at the row or columnposition in notation, then all rows or columns are considered toconstitute the submatrix. [X]i,j denotes the (i, j)th element ofa matrix X. x is the largest integer less than or equal to x.Also, x indicates the nearest integer that is not smaller thanx. 〈x〉y means the integer remainder after x is divided by y.det(X) denotes the determinant of a square matrix X. ‖ · ‖2

and ‖ · ‖F mean the l2-norm and the Frobenius matrix norm,respectively. CN (0, σ2) denotes a circular symmetric complexGaussian distribution with zero mean and variance σ2.

II. MIMO-OFDM SYSTEM MODEL

In this section, we describe the MIMO-OFDM system modelwith Mt transmit and Mr receive antennas considered in thispaper, as illustrated in Fig. 1. Fig. 2 shows the basebandmodel of an OFDM system for the jth transmit and ith receiveantennas in Fig. 1, which has N subcarriers and uses thesubcarriers numbered k0 to k0 + D − 1 for information data.The remaining N −D unmodulated carriers are referred to asVCs that are needed for the input signal pulse shaping by atransmit filter such as the raised cosine filter with a roll-off


factor [57], [60]. If we set k0 to 0 and D to N , the systemno longer has VCs. Thus, our system model can be appliedto both systems with and without VCs. Let the nth block ofthe frequency-domain information symbols in the jth transmitantenna be written as

dj(n) = [dj(n, k0), dj(n, k0 + 1), . . . , dj(n, k0 + D − 1)]T

(1)

where the subscript j is the transmit antenna index with 1 ≤j ≤ Mt. Assuming that the length of the CP is P , each OFDMmodulator adds N −D zeros for VCs to the data block in (1),applies an N -point inverse fast Fourier transform (IFFT) to thisblock, and inserts the CP in front of the IFFT output vector,which is a copy of the last P samples of the IFFT output. Thisresults in the time-domain sample vector of the nth OFDMsymbol written as

sj(n) = [sj(n,N − P ), . . . , sj(n,N − 1),

sj(n, 0), . . . , sj(n,N − 1)]T . (2)

To generate the continuous-time signal to be sent on the chan-nel, each element in the vector sj(n) is pulse-shaped by atransmit filter gtx[t]

sj [t] =∞∑

n=−∞

Q−1∑k=0

sj (n, 〈N − P + k〉N ) gtx [t− (k + nQ)T ]

(3)

where Q = N + P , and T is the sample duration in the timedomain. By denoting sj(n, 〈N − P + k〉N ), meaning the kthsample of the nth OFDM symbol in the time domain, assj((k + nQ)T ), and k + nQ as a, the transmitted signal sj [t]can be concisely expressed as

sj [t] =∞∑

a=−∞sj(aT )gtx[t− aT ]. (4)

Then, Mt transmit antennas simultaneously transmit the signalss1[t], s2[t], . . . , sMt

[t].During the transmission, the transmitted signal sj [t] from

the jth transmit antenna passes through a dispersive channelwith an impulse response cij [t], it gets corrupted by a spatiallyuncorrelated additive white Gaussian noise vi[t], and it finallyenters into a front-end receive filter grx[t] at the ith receiveantenna. When we denote the composite channel impulse re-sponse between the jth transmit antenna and the ith receiveantenna as hij [t] = gtx[t] ∗ cij [t] ∗ grx[t], and the filtered noiseat the ith receive antenna as ηi[t] = vi[t] ∗ grx[t], the receivedsignal ri[t] at the ith receive antenna is expressed as

ri[t] =Mt∑j=1

∞∑a=−∞

sj(aT )hij [t− aT ] + ηi[t]. (5)

We suppose that the composite channel impulse responseshij [t] have the finite support [0, (L + 1)T ) with L ≤ P , whichguarantee that ri[t] is not contaminated by previous OFDMsymbols. By sampling ri[t] at a rate 1/Ts = q/T with a positive

integer q, the sampled received signal ri[εi + mTs] = ri[εi +(mT/q)] at the ith receive antenna is given as

ri

[εi+

mT

q

]=

Mt∑j=1

m/q∑a=m/q−L

sj(aT )

× hij

[εi+

mT

q− aT

]+ηi

[εi+

mT

q

]

=Mt∑j=1

L∑l=0

sj

(⌊m

q

⌋T − lT

)

× hij

[εi+

〈m〉qTq

+lT

]+ηi

[εi+

mT

q

](6)

where εi ∈ [0, Ts) is the sample timing error at the ith receiveantenna.

III. SUBSPACE-BASED BLIND CHANNEL ESTIMATION

In this section, we establish conditions for the channelsto be identifiable and present a subspace method for blindchannel estimation for two MIMO-OFDM system structures.We distinguish between two different MIMO-OFDM systemstructures: one with Mt ≤ Mr and the other with Mt > Mr.

A. MIMO-OFDM System With Mt ≤ Mr

Based on the systems illustrated in Figs. 1 and 2, we denotethe information symbols before an OFDM modulation as

d(n, k)= [d1(n, k), d2(n, k), . . . , dMt(n, k)]T (7)

dn =[d(n, k0)T, d(n, k0+1)T, . . . ,d(n, k0+D − 1)T

]T(8)

where dj(n, k) is an information symbol loaded on the kthsubcarrier in the nth OFDM symbol to be transmitted fromthe jth transmit antenna. By collecting J consecutive OFDMsymbols from Mt transmit antennas, the information symbolvector d(n) is constructed as given in

d(n) =[dT

n ,dTn−1, . . . ,d

Tn−J+1

]T. (9)

When we define the matrices W(i), W, and W associated withthe IFFT as, respectively

W(i) ∆=1√N

[wik0

N , wi(k0+1)N , . . . , w

i(k0+D−1)N

](10)

W ∆=[W(N−1)T, . . . ,W(0)T,

W(N−1)T, . . . ,W(N−P )T]T

(11)

W ∆= IJ ⊗ W ⊗ IMt(12)


and denote the time-domain signal vector s(n) to be transmittedafter the OFDM modulation as

s(n, k) = [s1(n, k), s2(n, k), . . . , sMt(n, k)]T (13)

sn =[s(n,N−1)T, . . . , s(n, 0)T,

s(n,N−1)T, . . . , s(n,N−P )T]T

(14)

s(n) =[sTn , s

Tn−1, . . . , s

Tn−J+1

]T(15)

we obtain the relationship given as

s(n) = Wd(n). (16)

In (13), sj(n, k) means the (k + P + 1)th element of the vectorsj(n) in (2).

By sampling a received signal at each receive antenna witha rate 1/T , meaning q = 1 in (6), we can consider the dis-crete composite channels as given in (6) instead of continuouschannels at Mr receive antennas. We assume that the discretecomposite channels between Mt transmit antennas and Mr

receive antennas are modeled as an Mr ×Mt finite impulseresponse filter with L as the upper bound on the orders of thesechannels. When we denote hij [εi + lT ] in (6) as hij(l), and thelth lag of the MIMO channel as

h(l) =

h11(l) h12(l) · · · h1Mt

(l)h21(l) h22(l) · · · h2Mt

(l)...

......

...hMr1(l) hMr2(l) · · · hMrMt

(l)

(17)

respectively, the matrix transfer function H(z) is given as

H(z) =L∑

l=0

h(l)z−l. (18)

Denoting ri[εi + mT ] in (6) as ri(m), and rearranging ri(m)according to ri(n, k) = ri(k + nQ), we express the receivedsignal at Mr receive antennas as

r(n, k) = [r1(n, k), r2(n, k), . . . , rMr(n, k)]T (19)

rn =[r(n,Q− 1)T, r(n,Q− 2)T, . . . , r(n, 0)T

]T.

(20)

By collecting J consecutively received OFDM symbols, thereceived signal vector r(n) is given as

r(n) =[rT

n , rTn−1, . . . , rn−J+1 [1 : (Q− L)Mr]

T]T

. (21)

Similarly, denoting ηi[εi + mT ] in (6) as ηi(m), and rearrang-ing ηi(m) according to ηi(n, k) = ηi(k + nQ), we write theadditive noise at Mr receive antennas as

η(n, k) = [η1(n, k), η2(n, k), . . . , ηMr(n, k)]T (22)

ηn =[η(n,Q− 1)T,η(n,Q− 2)T, . . . ,η(n, 0)T

]T(23)

η(n) =[ηT

n ,ηTn−1, . . . ,ηn−J+1[1 : (Q− L)Mr]T

]T.

(24)

When we define a (JQ− L)Mr × JQMt channel matrix H as

H ∆=

h(0) · · · h(L) 0 · · · 00 h(0) · · · h(L) · · · 0...

. . .. . .

0 · · · 0 h(0) · · · h(L)

(25)

the received signal vector r(n) in (21) can be written in a matrixform as

r(n) = Hs(n)+η(n) = HWd(n)+η(n) ∆= Ξd(n)+η(n).(26)

By assuming that the Nyquist pulse shaping is employed,η(n) is considered as a spatially and temporally uncorrelatedcomplex Gaussian noise vector with the zero mean vector andthe covariance matrix σ2

ηI(JQ−L)Mr.

For the MIMO channel to be identified by the noise subspacemethod [22], the matrix Ξ in (26) should have a full columnrank. The following Theorem 1 gives a necessary and sufficientcondition for the full column rank requirement.Theorem 1: In the case of Mt ≤ Mr and L ≤ (Q−D),

the matrix Ξ in (26) has a full column rank, if and only ifrank(H(wi

N )) = Mt for all i ∈ kk0+D−1k=k0

.Proof: Refer to Appendix I.

This theorem generalizes the identifiability condition forSISO-OFDM systems in [54] to the condition for MIMO-OFDM systems. As we can see from the above theorem,the identification of the MIMO-OFDM channel based on thenoise subspace method requires the frequency-domain MIMOchannel matrices at the subcarriers, where information symbolsare loaded, to have a full column rank. In addition, the identifi-ability condition needs an upper bound for the MIMO channelorder rather than accurate knowledge of the MIMO channelorder. Since the length of the CP is usually set to be greaterthan the channel delay spread in the practical MIMO-OFDMsystems, we can consider the CP length as an upper bound ofthe MIMO channel order.

In our derivation, we consider the MIMO channels satisfyingthe condition of rank(H(wi


as stated in Theorem 1. In addition, we suppose that theadditive noise is uncorrelated with the transmitted signal, andthe autocorrelation matrix Rdd = Ed(n)d(n)H of the in-formation symbol vector d(n) in (9) has full rank. When theautocorrelation matrix Rrr = Er(n)r(n)H of the receivedsignal vector r(n) in (21) is diagonalized through the eigen-value decomposition, we can partition the eigenvectors U intothe vectors Us spanning a signal subspace span(Us) and thevectors Un spanning a noise subspace span(Un) [22] as

U=[Us|Un]=[u1, . . . ,uJDMt

|uJDMt+1, . . . ,u(JQ−L)Mr

].

(27)

Since span(Ξ) and span(Us) share the same JDMt-dimensional space and are orthogonal to span(Un), we havethe following orthogonal relationship [22]:

uHk Ξ = 0 for all k ∈ n(JQ−L)Mr

n=JDMt+1 . (28)


Let us define the (L + 1)Mr × 1 channel response vector hi

associated with the channel impulse responses between the ithtransmit antenna and Mr receive antennas, and the channelcoefficient matrix H consisting of hi as, respectively

hi∆=[h(0)[:, i]T, h(1)[:, i]T, . . . ,h(L)[:, i]T

]T,

1 ≤ i ≤ Mt (29)

H ∆= [h1,h2, . . . ,hMt] =

[h(0)T,h(1)T, . . . ,h(L)T

]T.

(30)

Under the appropriate conditions detailed in Theorem 2 andLemma 1 to be given below, the noise subspace can determinethe channel coefficient matrix H up to an Mt ×Mt multiplica-tive matrix associated with the number of transmit antennas.

Let H′ be a matrix that has the same dimension as thatof H. Let H′ be a nonzero matrix constructed from H′ inthe same manner as the matrix H is constructed from H. Inaddition, we denote H′W as Ξ′, and

∑Ll=0 h′(l)z−l as H′(z).

By using these notations, we state Theorem 2 and Lemma 1 as-sociated with the ambiguity of an estimated MIMO channel asfollows.Theorem 2: Assume that the matrix Ξ in (26) has a full

column rank with J ≥ 2, Mr ≥ Mt, and (Q−D) ≥ L. Then,H′ is equal to HΩ with an Mt ×Mt invertible matrix Ω, if andonly if span(Ξ′) is equal to span(Ξ).

Proof: Refer to Appendix II. By Theorem 2, a scalar channel ambiguity for SISO-OFDM

systems in [54] is extended to a matrix channel ambiguity forMIMO-OFDM systems. The channel ambiguity is inherent toblind estimation schemes and can be resolved by exploiting thetechniques based on independent component analysis [61] (andreferences therein) and/or a small number of pilot symbols asgiven in [62] and [27]. Furthermore, since practical MIMO-OFDM systems provide pilot symbols for the purpose of syn-chronization, these pilot symbols can be used to resolve theambiguity matrix.

In particular, when a MIMO-OFDM system structure pos-sesses the specific conditions given in the following Lemma 1,we can estimate a MIMO channel with J ≤ 2.

Lemma 1: In the case of Mt < Mr and L ≤ (JD − 1)/(Mt + 1) with J ≤ 2, assume that h(0), h(L), and H(z) havea full column rank for all z. Then, H′ is equal to HΩ withan Mt ×Mt invertible matrix Ω, if and only if H′ has a fullcolumn rank and span(Ξ′) is equal to span(Ξ).

Proof: Refer to Appendix III. Since Lemma 1 allows a MIMO channel to be estimated with

J ≤ 2, we can obtain a MIMO channel estimate by exploitinga small number of OFDM symbols with J = 1. Furthermore,although the channel conditions required by Lemma 1 arestricter than those in Theorem 2, we note that the conditionsin Lemma 1 do not impose any constraints on the numberof CPs associated with the number of VCs. Thus, we canestimate a MIMO channel without CPs as long as the conditionsare satisfied, thereby increasing the transmission bandwidthefficiency. Obviously, the above theorems and lemma are stillvalid for a MIMO-OFDM system with no VCs by setting k0 to0 and D to N .

To find the signal and noise subspaces, the true Rrr isrequired. In practice, Rrr is estimated over Nb blocks by

Rrr =1Nb

Nb−1∑n=0

r(n)r(n)H. (31)

Thus, when a MIMO channel is estimated by the orthogonal re-lationship in (28), only estimates Un of the eigenvectors span-ning the noise subspace, which are obtained by the eigenvaluedecomposition of Rrr, are available in practice. In this case,we can obtain the channel matrix estimate H by minimizing aquadratic cost function C(H) given as

C(H) =(JQ−L)Mr∑k=JDMt+1

∥∥uHk Ξ∥∥2

2=

(JQ−L)Mr∑k=JDMt+1

∥∥uHk HW∥∥2

2.

(32)

Partitioning the eigenvector estimate uk with dimension(JQ− L)Mr into JQ− L equal segments as given in

uk =

v(k)

1

v(k)2...

v(k)JQ−L

(33)

constructing the (L + 1)Mr × JQ matrix Vk as

Vk =

v(k)

1 v(k)2 · · · v(k)

JQ−L 0 · · · 0

0 v(k)1 v(k)

2 · · · v(k)JQ−L · · · 0

.... . .

. . .. . .

. . .0 · · · 0 v(k)

1 v(k)2 · · · v(k)

JQ−L

(34)

and defining the matrix Ψ as

Ψ ∆=(JQ−L)Mr∑k=JDMt+1

Vk(IJ ⊗ W∗WT)VH

k (35)

we can write a cost function∑Mt

i=1 hHi Ψhi equivalent to C(H).

By imposing the constraints such as ‖hi‖2 = 1 for 1 ≤ i ≤Mt to avoid trivial solutions, the estimate H of the channelcoefficient matrix H in (30) is obtained by

H =[h1, h2, . . . , hMt

]= arg min

‖hi‖2=1

(Mt∑i=1

hiHΨhi

).

(36)


When we find hi satisfying ∂(∑Mt

i=1(hHi Ψhi + λi(1−

‖hi‖22)))/∂h

Hi = 0 with a Lagrange multiplier λi, the

estimates hi of the channel response vectors hi, 1 ≤ i ≤ Mt

in (29) are the eigenvectors associated with the smallestMt eigenvalues of Ψ. Since the orthogonal relationshipuH

k Ξ = 0 in (28) can be rewritten as (IJ ⊗ WT)VHk hi = 0

for 1 ≤ i ≤ Mt, we should find hi closely orthogonal tocolumn vectors of Vk(IJ ⊗ W∗) with an estimate Vk of Vk.In addition, h1,h2, . . . ,hMt

should be linearly independentto satisfy the condition in Theorem 1. Thus, the solution of(36) satisfies these orthogonality and linear independenceconditions. Although the vectors h1,h2, . . . ,hMt

and thesolution of (36) span the same Mt-dimensional space in theideal case with the knowledge of true Rrr, we do not haveinformation about the direction and magnitude of each hi

in the space. This causes the channel ambiguity stated inTheorem 2 and Lemma 1. Thus, if the eigenvectors associatedwith the smallest Mt eigenvalues of Ψ are denoted ash′

1, h′2, . . . , h

′Mt

, respectively, we can express the estimatedchannel coefficient matrix H as

H =[h1, h2, . . . , hMt

]=[h′

1, h′2, . . . , h

′Mt

]Ω (37)

where Ω is an Mt ×Mt channel ambiguity matrix.In summary, insofar as the condition in Theorem 1 is sat-

isfied, the proposed subspace method can be applied to theblind channel estimation for a MIMO-OFDM system withMt ≤ Mr. In addition, we note that the condition requires anupper bound of a true MIMO channel order rather than theexact knowledge of the MIMO channel order. The estimatedMIMO channel has an ambiguity corresponding to an Mt ×Mt

invertible matrix given in Theorem 2. Furthermore, withoutrelying on the presence of VCs under the specific conditiongiven in Lemma 1, we can apply the proposed method witha smaller J to blind channel estimation for the MIMO-OFDMsystem without CPs. This increases the bandwidth efficiencyand makes it possible to obtain an accurate channel estimate byutilizing a small number of OFDM symbols.

Next, we extend the above results obtained in a MIMO-OFDM system with Mt ≤ Mr to the blind channel estimationfor a MIMO-OFDM system with Mt > Mr.

B. MIMO-OFDM System With Mt > Mr

To perform the blind channel estimation for a MIMO-OFDMsystem with Mt > Mr, we set the sampling rate at the receiverto q/T with q ≥ Mt/Mr in the system shown in Fig. 2. Byconsidering the discrete composite channel impulse responsebetween the jth transmit antenna and the ith receive antenna in(6), and defining h

(ξ)ij (l) with ξ = 〈m〉q as

h(ξ)ij (l) ∆= hij

[εi +

(ξ

q+ l

)T

](38)

we denote the lth lag of the oversampled MIMO channel as

h(l) =

h(0)11 (l) h

(0)12 (l) · · · h

(0)1Mt

(l)...

......

...h

(q−1)11 (l) h

(q−1)12 (l) · · · h

(q−1)1Mt

(l)

h(0)21 (l) h

(0)22 (l) · · · h

(0)2Mt

(l)...

......

...h

(q−1)21 (l) h

(q−1)22 (l) · · · h

(q−1)2Mt

(l)...

......

...h

(0)Mr1(l) h

(0)Mr2(l) · · · h

(0)MrMt

(l)...

......

...h

(q−1)Mr1 (l) h

(q−1)Mr2 (l) · · · h

(q−1)MrMt

(l)

. (39)

Assuming that the discrete composite MIMO channel has L asthe upper bound on the order of the channel, we construct the(JQ− L)qMr × JQMt channel matrix H, in the same way asgiven in (25), as

H =

h(0) · · · h(L) 0 · · · 00 h(0) · · · h(L) · · · 0...

. . .. . .

0 · · · 0 h(0) · · · h(L)

. (40)

Recalling m/q = m/q + (〈m〉q/q) and defining m′ and ξ

as m′ = m/q and ξ

= 〈m〉q , respectively, we denote ri[εi +

(mT/q)] in (6) as r(ξ)i (m′). Rearranging r

(ξ)i (m′) according

to r(ξ)i (n, k) = r

(ξ)i (k + nQ), we express the oversampled re-

ceived signal at Mr receive antennas as

r(n, k)=[r(0)1 (n, k), . . . , r(q−1)

1 (n, k), r(0)2 (n, k), . . . ,

r(q−1)2 (n, k), . . . , r(0)

Mr(n, k), . . . , r(q−1)

Mr(n, k)

]T(41)

rn =[r(n,Q− 1)T, r(n,Q− 2)T, . . . , r(n, 0)T

]T(42)

r(n)=[rT

n , rTn−1, . . . , rn−J+1[1 : (Q− L)qMr]T

]T. (43)

Similarly, denoting ηi[εi + (mT/q)] in (6) as η(ξ)i (m′), and

rearranging η(ξ)i (m′) according to η

(ξ)i (n, k) = η

(ξ)i (k + nQ),

we write the oversampled additive noise at Mr receiveantennas as

η(n, k)=[η(0)1 (n, k), . . . , η(q−1)

1 (n, k), η(0)2 (n, k), . . . ,

η(q−1)2 (n, k), . . . , η(0)

Mr(n, k), . . . , η(q−1)

Mr(n, k)

]T(44)

ηn =[η(n,Q− 1)T, η(n,Q−2)T, . . . , η(n, 0)T

]T(45)

η(n)=[ηT

n , ηTn−1, . . . , ηn−J+1[1 : (Q−L)qMr]T

]T. (46)


Fig. 3. Equivalent MIMO-OFDM system model with Mt transmit and qMr

receive antennas.

Then, the oversampled received signal vector r(n) in (43) canbe written in a matrix form as

r(n)=Hs(n) + η(n)=HWd(n) + η(n) ∆= Ξd(n) + η(n).(47)

When we consider (39) through (47), we can model a MIMO-OFDM system with Mt transmit and Mr receive antennas,where the received signals are oversampled by a factor of q, asan equivalent MIMO-OFDM system with Mt transmit and qMr

receive antennas. This equivalent system model is shown inFig. 3, where the received signal at each receive antenna is sam-pled at the rate 1/T . Since the equivalent system has qMr ≥Mt/MrMr ≥ Mt, we can apply Theorem 1 for the channelidentifiability of the system. Let us denote

∑Ll=0 h(l)z−l as

H(z). According to Theorem 1, if the condition in (48) issatisfied

rank(H(wi

N ))

= Mt for all i ∈ kk0+D−1k=k0

(48)

the matrix Ξ has a full column rank, which implies that theoversampled MIMO channel can be identified through the noisesubspace method.

In our derivation, we consider MIMO channels satisfying thecondition in (48). Furthermore, we note that since the samplingrate q/T is higher than the Nyquist rate, the oversampled noiseη(ξ)i (n, k) is not necessarily temporally uncorrelated. Although

it is possible to design a front-end receive filter grx[t] witha wider bandwidth to whiten the oversampled noise [63], wesuppose that the oversampled noise vector η (n) is generallycolored with the covariance matrix R

η ηthat has full rank. By

decomposing Rη η

as Rη η

= R1/2

η ηRH/2

η η, and whitening the

oversampled received signal vector r(n) in (47) by the inverseof R1/2

η η, denoted as R−1/2

η η, we obtain the whitened received

signal vector rw(n) as given in

rw(n) = R− 12

η ηr(n) = R− 1

2

η ηΞd(n) + R− 1

2

η ηη(n). (49)

We assume that the additive noise is uncorrelated with thetransmitted signal, and the autocorrelation matrix Rdd of the

information symbol vector d(n) has full rank. When the auto-correlation matrix R

rw rw= Erw(n)rw(n)H of the whitened

received signal vector rw(n) is diagonalized through the eigen-value decomposition, we can partition the eigenvectors U intothe vectors Us spanning a signal subspace span(Us) and thevectors Un spanning a noise subspace span(Un) as

U =[Us|Un

]=[u1, . . . , uJDMt

|uJDMt+1, . . . , u(JQ−L)qMr

]. (50)

Since span(R−1/2

η ηΞ) and span(Us) share the same JDMt-

dimensional space and are orthogonal to span(Un), we obtainthe orthogonal relationship given as

uHk R− 1

2

η ηΞ = 0 for all k ∈ n(JQ−L)qMr

n=JDMt+1. (51)

Defining the (L + 1)qMr × 1 channel response vector hi as-sociated with the channel impulse responses between the ithtransmit antenna and qMr receive antennas, and the channelcoefficient matrix H consisting of hi as, respectively

hi∆=[h(0)[:, i]T, h(1)[:, i]T, . . . , h(L)[:, i]T

]T, 1 ≤ i ≤ Mt

(52)

H ∆=[h1, h2, . . . , hMt

]=[h(0)T, h(1)T, . . . , h(L)T

]T(53)

and replacing Mr, uk, Ξ, H, hi, and H in (32) through(37) with qMr, R−H/2

η ηuk, Ξ, H, hi, and H, respectively, we

can construct a cost function in the same manner as givenin Section III-A. By minimizing this cost function, we canestimate the channel coefficient matrix H up to an Mt ×Mt

channel ambiguity matrix stated in Theorem 2 and Lemma 1.Furthermore, the channel ambiguity matrix can be resolved byusing the schemes given in [62] and [27].

In summary, when the received signal at each receive antennais oversampled by a factor of q ≥ Mt/Mr and the conditionin (48) according to Theorem 1 is satisfied, we can still applythe proposed method to the blind channel estimation for aMIMO-OFDM system with Mt > Mr. Again, the conditiondepends on an upper bound of a true MIMO channel orderrather than the exact knowledge of the MIMO channel order. AMIMO channel is estimated up to an Mt ×Mt ambiguity ma-trix given in Theorem 2. If the matrices h(0), h(L), and H(z)satisfy the same conditions as those required for h(0), h(L),and H(z) in Lemma 1, respectively, the proposed method withJ ≤ 2 is still applicable to the blind channel estimation for theMIMO-OFDM system without CPs, regardless of the existenceof VCs. This increases the bandwidth efficiency and enablesaccurate channel estimation by exploiting a small number ofOFDM symbols.


Fig. 4. Comparison of NRMSE performance when the sum of the number of virtual carriers (N − D) and the number of CPs (P ) is fixed to three. (a) NRMSEversus SNR for the number of OFDM symbols used for channel estimation Ns = 2000. (b) NRMSE versus the number of OFDM symbols used for channelestimation Ns for SNR = 25 dB.

IV. SIMULATION RESULTS

To evaluate the performance of the proposed method, weconsider a MIMO-OFDM system with two transmit antennas(Mt = 2) and two receive antennas (Mr = 2). The number ofsubcarriers N is set to 64. Information symbols di(n, k) areindependent and identically distributed (i.i.d.) 16-quadratureamplitude modulation symbols. Each channel tap hij(l) is i.i.d.and randomly generated from CN (0, σ2

h). The order of theMIMO channel is considered to be L = 3. We suppose thatthe channel is time-invariant during each channel estimation.For the fairness of performance comparison, the transmit powerper OFDM symbol is fixed to Es for all simulations, and theadditive noise at each receive antenna is a spatially uncorrelatedcomplex white Gaussian noise with zero mean and variance σ2

η

determined by the SNR defined as

SNR∆= 10 log10

Mt(L + 1)σ2hEs

(N + Po)σ2η

(dB) (54)

where Po is the maximum CP length used throughout thesimulations, which is set to three.

As a measure of performance, we consider the normalizedroot mse (NRMSE) given as

NRMSE =

√√√√√√ 1NmMtMr(L + 1)

Nm∑k=1

Mt∑i=1

∥∥∥h(k)i − h(k)

i

∥∥∥2

2∥∥∥h(k)i

∥∥∥2

2

(55)

where Nm is the number of Monte Carlo trials, the superscript krefers to the kth Monte Carlo trial, and h(k)

i and h(k)i represent

the true channel response vector and the estimated channel re-sponse vector after resolving a channel ambiguity, respectively.All the results are obtained by averaging Nm = 500 indepen-dent Monte Carlo trials. To isolate the impact of a scheme forresolving a channel ambiguity on channel estimation in com-puting NRMSE, we calculate the ambiguity Ω by minimizing‖[h(k)

1 ,h(k)2 , . . . ,h(k)

Mt] − [h′(k)

1 , h′(k)2 , . . . , h′(k)

Mt]Ω‖2

F as used

in [38] and [32]. [h′(k)1 , h′(k)

2 , . . . , h′(k)Mt

] is the estimated chan-nel coefficient matrix by the proposed method. By using this

approach, the NRMSE provides a measure of how well thetrue MIMO channel and the estimated MIMO channel by theproposed method span the same Mt-dimensional space.

In Fig. 4, the NRMSE performance of the proposed methodwith different combinations of the number of information sym-bols (D) and the number of CPs (P ) is compared with thatof the method in [25] that is marked with “Bölcskei.” In thecases for the proposed method, the redundancy (N −D + P )is fixed to three through various combinations of VCs andCPs. An observed OFDM symbol block J associated withthe dimensions of subspaces is fixed to two. To obtain theNRMSE performance as a function of SNR shown in Fig. 4(a),we use 2000 OFDM symbols. The NRMSE performance asa function of the number of OFDM symbols Ns used forchannel estimation in Fig. 4(b) is obtained by setting theSNR to 25 dB. For a fair comparison, the complex scalarambiguities αij from the method in [25] are also resolved

by minimizing ‖[h(k)ij (0), h(k)

ij (1), . . . , h(k)ij (L)]− αij [h

′(k)ij (0),

h′(k)ij (1), . . . , h′(k)

ij (L)]‖22. [h′(k)

ij (0), h′(k)ij (1), . . . , h′(k)

ij (L)] isthe estimated channel impulse response vector between the jthtransmit antenna and the ith receive antenna by the methodin [25].

As we can see from Fig. 4, the estimator errors of all thecases decrease with increasing SNR and OFDM symbol recordlength Ns. Furthermore, the proposed method demonstratesmuch better performance than the method in [25], which revealsthe fast convergence property of the noise subspace methodfor a small data record. In addition, there are performancegaps among the non-CP system (J = 2, D = 61, P = 0), theinsufficient CP systems (J = 2, D = 62, P = 1 and J = 2,D = 63, P = 2) and the CP-only system (J = 2, D = 64, P =3). This demonstrates that CPs are more useful for the noisesubspace-based estimation method than VCs. We discuss thisbenefit of CPs by referring to Fig. 5 that is obtained by using2000 OFDM symbols at the SNR of 25 dB and averaging 500independent trials. Fig. 5(a) shows the estimated eigenvalues ofthe autocorrelation matrix Rrr corresponding to the estimatedeigenvectors spanning the signal subspace in a descending orderof the eigenvalues. From Fig. 5(a), we note that as fewer CPs areused in the presence of VCs, the eigenvalues rapidly decrease.Although the boundary between the signal subspace and the


Fig. 5. Comparison of eigenvalue distributions when the sum of the number of virtual carriers (N − D) and the number of CPs (P ) is fixed to three, and 2000OFDM symbols are used for channel estimation at the SNR of 25 dB. (a) Distributions of estimated eigenvalues of the autocorrelation matrix Rrr correspondingto estimated eigenvectors spanning the signal subspace in a descending order of the eigenvalues. (b) Distributions of eigenvalues of the matrix Ψ in (35) in adescending order.

Fig. 6. Comparison of average Fubini–Study distances when the sum of the number of virtual carriers (N − D) and the number of CPs (P ) is fixed to three.(a) Average Fubini–Study distance versus SNR for the number of OFDM symbols used for channel estimation Ns = 2000. (b) Average Fubini–Study distanceversus the number of OFDM symbols used for channel estimation Ns for SNR = 25 dB.

noise subspace is theoretically given in (27), this boundary withthe rapidly decreasing eigenvalues is usually indistinguishablein the presence of additive noise. That is, it is difficult toaccurately differentiate eigenvectors spanning the signal sub-space and eigenvectors spanning the noise subspace from theestimated eigenvalues and eigenvectors in the presence of noise.This causes the performance degradation in the cases exploitingfewer CPs. In addition, the more rapid the eigenvalues decrease,the poorer the estimation performance becomes. Furthermore,since the eigenvectors corresponding to the seventh and eighthsmallest eigenvalues of the matrix Ψ in (35) provide the es-timated MIMO channel with an ambiguity for this simulationexample, a distinctive boundary between the sixth smallesteigenvalue and the seventh smallest eigenvalue is desirable. Asdemonstrated in Fig. 5(b), which shows the eigenvalues of Ψin a descending order, however, this boundary is not clear inthe cases with a reduced number of CPs. Also, CPs increasethe dimension of each eigenvector estimate spanning the noisesubspace, thereby imposing more constraints on the estimatesof the channel impulse responses. Thus, although the subspacedimension extended by larger CPs increases the computationalcomplexity of the proposed method, increasing CPs rather thanVCs can significantly improve the performance of the subspacemethod.

As another measure evaluating the closeness between twoMt-dimensional spaces spanned by the true MIMO channel and

the estimated MIMO channel by the proposed method, we canconsider the Fubini–Study distance [64] given as

dFS

(U(k)

h , H(k))

= arccos∣∣∣∣det

(U(k)

h

HH(k)

)∣∣∣∣ . (56)

In (56), U(k)h is a matrix consisting of eigenvectors associated

with nonzero eigenvalues of H(k)H(k)H, where H(k) is thekth realization of the true channel coefficient matrix and H(k)

is an estimate of H(k) obtained by the proposed method.By comparing the average Fubini–Study distances defined as(1/Nm)

∑Nm

k=1 dFS(U(k)h , H(k)) in Fig. 6, we evaluate the per-

formance of the proposed method for the systems consideredin Fig. 4. To compute the average Fubini–Study distances, weuse Nm = 500 independent trials. Fig. 6(a) shows the averageFubini–Study distance of the proposed method as a functionof SNR with the utilization of 2000 OFDM symbols for chan-nel estimation, whereas Fig. 6(b) demonstrates the averageFubini–Study distance of the proposed method as a functionof the number of OFDM symbols used for channel estimationwith the SNR fixed to 25 dB. As expected, the Fubini–Studydistances of all the cases still decrease with increasing SNRand OFDM symbol record length Ns. Furthermore, we can seethat the decreasing trends in the distances are similar to thosein the NRMSE performance in Fig. 4, which reconfirms that


Fig. 7. Comparison of NRMSE performance when an observed OFDM symbol block J increases. (a) NRMSE versus SNR for the number of OFDM symbolsused for channel estimation Ns = 2000. (b) NRMSE versus the number of OFDM symbols used for channel estimation Ns for SNR = 25 dB.

Fig. 8. Comparison of NRMSE and BER performances when the MIMO channel in (57) is estimated by using 2000 OFDM symbols. (a) NRMSE versus SNR.(b) BER versus SNR.

exploitation of CPs rather than VCs for the proposed methodimproves the closeness of the distance. Due to the similaritybetween the NRMSE and the Fubini–Study distance in oursimulations, we consider only the NRMSE performance in thesimulations hereafter.

In the cases having no or insufficient CPs, Fig. 7 showsthe NRMSE performance of the proposed method obtained byincreasing an observed OFDM symbol block J in two cases ofD = 61, P = 0 and D = 62, P = 1. We obtain the NRMSEperformance as a function of SNR in Fig. 7(a) by using 2000OFDM symbols. The NRMSE performance as a function ofthe number of OFDM symbols Ns in Fig. 7(b) is obtainedwith the SNR fixed to 25 dB. From Fig. 7, we notice thatthe channel estimation performance in both cases is improvedby increasing J which is associated with the dimensions ofsubspaces. In particular, increasing J from 2 to 3 significantlyimproves the estimation performance, whereas increasing Jfrom 3 to 4 results in trivial performance improvement. Thisillustrates that there might be an adequate dimension for thechannel estimation based on the noise subspace method andincreasing J more than the adequate dimension might notenhance remarkably the performance. Thus, even if the di-mension extended by a larger J increases the computationalcomplexity of the eigenstructure-based method, the proposedmethod with an adequate dimension is applicable to the MIMO-OFDM system with no or insufficient CPs and can achieve theimproved performance, thereby potentially leading to higherchannel utilization.

To demonstrate that the proposed method is insensitive to atrue channel order, we consider the following MIMO channelgiven in [25].

H(z)=[

0.4851 0.3200−0.3676 0.2182

]+[−0.4851 0.9387

0.8823 0.8729

]z−1

+[

0.7276 −0.12800.2941 −0.4364

]z−2. (57)

We assume that the upper bound of the channel order L isequal to three even if the true channel order is two, and thezero-forcing detection based on the estimated MIMO channel isused for symbol recovery. Fig. 8(a) and (b) shows the NRMSEand bit error rate (BER) performances as functions of SNRwhen the MIMO channel in (57) is estimated by using 2000OFDM symbols, respectively. As we can see from Fig. 8(a), theproposed method still achieves a good estimation performance,which demonstrates its insensitivity to a true channel order. Inaddition, the proposed method outperforms the method in [25].Even in this example, we observe that the utilization of CPsrather than VCs increases the accuracy of channel estimation.The BER performance in Fig. 8(b) reflects an influence ofthe channel estimation accuracy shown in Fig. 8(a) on symbolrecovery. In particular, we note that although the two casesof J = 2, D = 61, P = 0 and J = 2, D = 62, P = 1 usingthe proposed method achieve lower estimation errors than themethod in [25], they exhibit poorer BER performance than the


Fig. 9. Comparison of NRMSE performance for a MIMO-OFDM system without CPs when the observed OFDM symbol block J = 1 is used. The system hastwo transmit and three receive antennas, and the MIMO channel in (58) is considered. (a) NRMSE versus SNR for the number of OFDM symbols used for channelestimation Ns = 1000. (b) NRMSE versus the number of OFDM symbols used for channel estimation Ns for SNR = 25 dB.

method in [25]. This is due to the fact that the estimated channelmatrices in these cases, which are obtained by combiningblock Toeplitz matrices constructed from the estimated MIMOchannels with the IFFT matrices, tend to be ill-conditioned.In this case, the zero-forcing detection by the inversion of thechannel matrices may increase the detrimental effects of thechannel estimation error and the additive noise on the symbolrecovery. Thus, this ill-conditioning can result in poor BERperformance even with a small channel estimation error ora small amount of additive noise. On the other hand, as thelength of CPs increases, the BER performance of the proposedmethod is significantly improved and much better than that ofthe method in [25].

Finally, we consider a MIMO-OFDM system with two trans-mit antennas (Mt = 2) and three receive antennas (Mr = 3)and the MIMO channel in (58) to evaluate the performance ofthe proposed method with an observed OFDM symbol blockJ = 1

H(z)=

0.2200+j0.0850 0.0904−j0.23970.0397−j0.5745 0.0172−j0.1402−0.0096−j0.0873 0.4328+j0.2893

+

−0.1739−j0.7379 −0.1858 + j0.38280.7121+j0.0601 −0.0209−j0.1041−0.1419+j0.4276 −0.2524+j0.6386

z−1

+

0.1984−j0.4376 0.6829+j0.4328−0.1216+j0.2128 0.2530+j0.3916−0.0433−j0.6528 0.2312+j0.1217

z−2

+

0.3704−j0.0400 0.0855−j0.30390.2967−j0.0977 0.6466+j0.5773−0.2574−j0.5432 0.3431−j0.2673

z−3.

(58)

To achieve a high bandwidth efficiency, we do not insert a CPto each OFDM symbol to be transmitted. Fig. 9(a) shows theNRMSE performance as a function of SNR that is obtainedby using 1000 OFDM symbols. Fig. 9(b) demonstrates the

NRMSE performance as a function of the number of OFDMsymbols Ns that is obtained with the SNR fixed to 25 dB. Thechannel estimation errors of all the cases still decrease withincreasing SNR and OFDM symbol record length Ns, whichdemonstrates that the proposed method can achieve an accuratechannel estimation by using a smaller number of OFDM sym-bols with J = 1. In particular, the estimation performance in allthe cases is almost identical. This indicates that the dimensionsof the subspaces in all the cases reach an adequate dimensionfor the proposed subspace method.

Furthermore, by applying the proposed method to a MIMO-OFDM system with four transmit and two receive antennas,we also confirmed the similar results to those of the MIMO-OFDM system with two transmit and two receive antennasgiven above.

V. CONCLUSION

In this paper, we established the conditions for blind chan-nel identifiability in a MIMO-OFDM system and presenteda blind channel estimation scheme based on the noise sub-space method. The proposed method unifies and generalizesthe existing SISO-OFDM blind channel estimators to the caseof MIMO-OFDM with any number of transmit and receiveantennas. Furthermore, the proposed method achieves accuratechannel estimation and fast convergence. This method alsodemonstrates insensitivity to the exact knowledge of a trueMIMO channel order, which implies that it only requires anupper bound on the MIMO channel order. In terms of bothchannel estimation accuracy and convergence speed, increas-ing the length of CPs rather than the number of VCs forthe proposed method was found to significantly improve theperformance in the simulations. In addition, by increasingan observed OFDM symbol block to an adequate dimensionfor channel estimation, the proposed method can achieve anaccurate channel estimation in an MIMO-OFDM system withno or insufficient CPs, thereby potentially increasing channelutilization. Finally, when a system configuration is satisfiedwith the specific conditions given in Lemma 1, the proposedmethod can be applied to a MIMO-OFDM system without CPs,regardless of the presence of VCs, thereby achieving a higherbandwidth efficiency.


APPENDIX IPROOF OF THEOREM 1

First, we show that if rank(H(wiN )) = Mt for all i ∈

kk0+D−1k=k0

, the matrix Ξ has a full column rank. We defineλ(i) and Λ as

λ(i) ∆=w(k0+i−1)LN H

(w

(k0+i−1)N

)(59)

Λ ∆= diag (λ(1),λ(2), . . . , λ(D)) . (60)

By choosing the rows (j − 1)QMr + 1 to (j − 1)QMr +DMr for 1 ≤ j ≤ J from the matrix Ξ, we can generate aJDMr × JDMt submatrix Ξ that is a block diagonal ma-trix with (W ⊗ IMr

)[LMr + 1 : (L + D)Mr, :]Λs on its maindiagonal [54]. By using the structure of (W ⊗ IMr

)[LMr +1 : (L + D)Mr, :] and a Vandermonde matrix property [65],we can easily show that (W ⊗ IMr

)[LMr + 1 : (L + D)Mr, :]is a nonsingular matrix. In addition, if rank(H(wi

N )) = Mt

for all i ∈ kk0+D−1k=k0

, the matrix Λ has a full columnrank. Thus, rank((W ⊗ IMr

)[LMr + 1 : (L + D)Mr, :]Λ) =rank(Λ) = DMt, and consequently rank(Ξ) = JDMt. Sincethe submatrix Ξ is generated from the matrix Ξ by removingsome rows, rank(Ξ) becomes JDMt, which means that thematrix Ξ has a full column rank.

Conversely, to prove that if the matrix Ξ has a full columnrank, then rank(H(wi


, we

show that if rank(H(wiN )) = Mt for some i ∈ kk0+D−1

k=k0,

then the matrix Ξ does not have a full column rank. Wecan construct a proper matrix Ξ column equivalent to Ξ byapplying elementary column operations [65] to the matrix Ξ,so that the submatrix Ξ[:, 1 : DMt] can be expressed as theproduct of a matrix having a full column rank of DMr andthe matrix Λ [54]. From the structure of this submatrix, itis observed that if any diagonal element of the block di-agonal matrix Λ does not have a full column rank, equiv-alently rank(H(wi

N )) = Mt for some i ∈ kk0+D−1k=k0

, thenrank(Ξ[:, 1 : DMt]) < DMt. Thus, the matrix Ξ does nothave a full column rank. Since rank(Ξ) = rank(Ξ) [65], thematrix Ξ does not have a full column rank if rank(H(wi

N )) =Mt for some i ∈ kk0+D−1

k=k0.

APPENDIX IIPROOF OF THEOREM 2

First, we prove that if span(Ξ′) is equal to span(Ξ), H′ =HΩ, where Ω is an Mt ×Mt invertible matrix. From thecondition of span(Ξ′) = span(Ξ), we know that there existsa nonsingular matrix A satisfying Ξ′ = ΞA. Let us partitionthe matrix A as

A [(iD + m)Mt + 1 : (iD + m + 1)Mt,

(jD + n)Mt + 1 : (jD + n + 1)Mt] = Ω(i, j)m, n . (61)

In (61), i, j, m, and n are integers satisfying 0 ≤ i, j ≤ J − 1and 0 ≤ m, n ≤ D − 1. Performing some mathematicalmanipulations to Ξ′ = ΞA and considering the submatrixΞ′ [iQMr + 1 : ((i + 1)Q− L)Mr, iDMt + 1 : (i + 1)DMt]for 0 ≤ i ≤ J − 1, we can obtain

(K1Θ ⊗ IMr)

H′(wk0N )Ω(i, i)

0, n

H′(w(k0+1)N )Ω(i, i)

1, n

...H′(w(k0+n−1)

N )Ω(i, i)(n−1), n

H′(w(k0+n+1)N )Ω(i, i)

(n+1), n

...H′(w(k0+D−1)

N )Ω(i, i)(D−1), n

= 0,

0 ≤ n ≤ D − 1 (62)

Θ = diag (θ(k0), θ(k0 + 1), . . . , θ(k0 + n− 1),

θ(k0 + n + 1), . . . , θ(k0 + D − 1)) (63)

θ(k) = w−kN − w−(k0+n)

N. (64)

In (62), the matrix K1 is generated by removing the(n + 1)th column from the matrix [W(N − 1)T,W(N −2)T, . . . ,W(L− P + 1)T]T. From the condition (Q−D) ≥L and the structure of K1 based on a Vandermonde matrix,we can show that the matrix K1 is a tall matrix with a fullcolumn rank. Since the matrix Ξ has a full column rank bythe assumption and span(Ξ′) = span(Ξ), Theorem 1 states thatthe matrix H′(wi

N ) for k0 ≤ i ≤ k0 + D − 1 has a full columnrank. Thus, we obtain Ω(i,i)

m,n = 0 for 0 ≤ i ≤ J − 1, 0 ≤ m,n ≤ D − 1, and m = n. By using this result, we can easilyshow that Ω(0,0)

m,m = Ω(1,1)m,m = · · · = Ω(J−1,J−1)

m,m for 0 ≤ m ≤D − 1 as well.

On the other hand, by considering the submatrix Ξ′[iQMr +1 : ((i + 1)Q− L)Mr, jDMt + 1 : (j + 1)DMt] for 0 ≤ i,j ≤ J − 1 and i = j, we obtain, for 0 ≤ n ≤ D − 1

([W(N − 1)T W(N − 2)T, . . . ,W(L− P )T

]T⊗ IMr

)

H′(wk0N )Ω(i, j)

0, n

H′(w(k0+1)N )Ω(i, j)

1, n

...H′(w(k0+D−1)

N )Ω(i, j)(D−1), n

= 0. (65)

Since the matrix [W(N − 1)T,W(N − 2)T, . . . ,W(L−P )T]T is a tall matrix with a full column rank and the matrixH′(wi

N ) for k0 ≤ i ≤ k0 + D − 1 has a full column rank,we obtain Ω(i,j)

m,n = 0 for 0 ≤ i, j ≤ J − 1, 0 ≤ m, n ≤ D −1, and i = j. Thus, the submatrix Ω(i,i)

m,m for 0 ≤ i ≤ J − 1and 0 ≤ m ≤ D − 1 is invertible. By using the above results


and rearranging Ξ′[(Q− L− 1)Mr + 1 : QMr, 1 : DMt] =(ΞA)[(Q− L− 1)Mr + 1 : QMr, 1 : DMt], we obtain

(K2 ⊗ IMr)(H′ − HΩ(0,0)

k−k0, k−k0)

=0, k0 ≤ k ≤ k0+D−1 (66)

K2

=

w

(N−Q+L)kN w

(N−Q+L−1)kN · · · w

(N−P+1)kN w

(N−P )kN

w(N−Q+L−1)kN w

(N−Q+L−2)kN · · · w

(N−P )kN 0

. ..

. ..

. .. ...

w(N−P )kN 0 · · · 0 0

(67)

thereby deriving H′ = HΩ(0,0)m,m for 0 ≤ m ≤ D − 1.

Since the matrix H(wiN ) for k0 ≤ i ≤ k0 + D − 1 has a

full column rank, rank(([1, w−iN , . . . , w−iL

N ] ⊗ IMr)H) =

rank(H(wiN )) = Mt ≤ rank(H) ≤ Mt [65], which means

that H has a full column rank. Thus, we have Ω(0,0)0,0 =

Ω(0,0)1,1 = · · · = Ω(0,0)

D−1,D−1. By denoting Ω(i,i)m,m for 0 ≤ i ≤

J − 1 and 0 ≤ m ≤ D − 1 as an invertible matrix Ω, weconclude H′ = HΩ.

Next, we show that if H′ = HΩ with an invertible matrixΩ, span(Ξ′) is equal to span(Ξ). By using H′ = HΩ, wecan write

Ξ′ = H′W

= H(IJ ⊗ IQ ⊗ Ω)(IJ ⊗ W ⊗ IMt)

= H(IJ ⊗ W ⊗ IMt)(IJ ⊗ ID ⊗ Ω)

= HW(IJ ⊗ ID ⊗ Ω)

=Ξ(IJ ⊗ ID ⊗ Ω). (68)

Since Ω is invertible, the matrix (IJ ⊗ ID ⊗ Ω) is a nonsingu-lar matrix. Thus, this means that span(Ξ′) = span(Ξ).

APPENDIX IIIPROOF OF LEMMA 1

First, we show that if rank(H′) = Mt and span(Ξ′) =span(Ξ), then H′ = HΩ with an Mt ×Mt invertible matrixΩ. Once we obtain a proof for the case with J = 2, the sameapproach can be applied to a proof for the case with J = 1,which is simpler than that of the case with J = 2. Thus, ourproof is focused on the case with J = 2 by setting J to two.

For brevity of notation, we define several matrices as follows:

B1∆=H [1 : (Q−D)Mr, 1 : (Q−D)Mt]

B2∆=H [1 : (Q−D)Mr, (Q−D)Mt + 1 : QMt]

B3∆=H [(Q−D)Mr + 1 : (Q + D − L)Mr,

(Q−D)Mt + 1 : QMt]

B4∆= H [(Q−D)Mr + 1 : (Q + D − L)Mr,

QMt + 1 : (Q + D)Mt]

B5∆= H [(Q + D − L)Mr + 1 : (2Q− L)Mr,

QMt + 1 : (Q + D)Mt]

B6∆= H [(Q + D − L)Mr + 1 : (2Q− L)Mr,

(Q + D)Mt + 1 : 2QMt]

W1∆= W [1 : (Q−D)Mt, 1 : DMt]

W2∆= W [(Q−D)Mt + 1 : QMt, 1 : DMt]

W3∆= W [QMt + 1 : (Q + D)Mt,DMt + 1 : 2DMt]

W4∆= W [(Q + D)Mt + 1 : 2QMt,DMt + 1 : 2DMt] .

(69)

By using the notations in (69), we can write the matrix Ξ withJ = 2 as

Ξ =

B1 B2 0 00 B3 B4 00 0 B5 B6

W1 0W2 00 W3

0 W4

=

Z1W2 0B3W2 B4W3

0 Z2W3

. (70)

In (70), the matrices Z1 and Z2 represent B1W1W−12 + B2

and B5 + B6W4W−13 , respectively. From the assumption of

rank(h(0)) = Mt and rank(H(z)) = Mt for all z, the polyno-mial matrix H(z) is irreducible [39], [66], [67]. In addition,H(z) is column reduced [39], [66], [67] by the assumptionof rank(h(L)) = Mt. Thus, the column polynomial vectors ofH(z) form a minimal polynomial basis [39], [66], [67] forspan(H(z)).1 Since h(L) has a full column rank, deg(H(z)[:,i]) = L2 for all i ∈ kMt

k=1. Therefore, by Theorem 1 in [39],the submatrix [B3, B4] has a full column rank with the condi-tions of Mt < Mr and L ≤ (2D − 1)/(Mt + 1). In addition,the submatrix W [(Q−D)Mt + 1 : (Q + D)Mt, :] is a blockdiagonal matrix with W2 and W3 on its main diagonal, andW2 and W3 are all nonsingular. Thus, the submatrix Ξ[(Q−D)Mr + 1 : (Q + D − L)Mr, :] has a full column rank, whichimplies that the matrix Ξ has a full column rank as well.

After we perform the eigenvalue decomposition of the auto-correlation matrix Rrr of the received signal vector r(n) in (26)and reorder its eigenvectors and the corresponding eigenvalues,we partition the unitary matrix U consisting of the eigenvectorsand the diagonal matrix D composed of the eigenvalues as inthe expressions in (71) and (72), respectively, shown at the topof the next page.

1span(H(z)) denotes the linear subspace over the field of scalar rationalfunctions spanned by the column vectors of H(z), i.e., the set of rational

function vectors written as∑Mt

i=1ci(z)H(z)[:, i], where ci(z) is a scalar

rational function (see [50] and references therein).2The degree of a Mr × 1 polynomial vector H(z)[:, i] =

[ [H(z)]1,i, [H(z)]2,i, . . . , [H(z)]Mr,i]T is defined as the greatest degree of

its components as given in deg(H(z)[:, i]) = max1≤k≤Mr deg([H(z)]k,i)(see [23], [39], [66], and [67] and references therein).


U =

U11 U12 U13 U14

U21 U22 U23 U24

U31 U32 U33 U34

(71)

D =

σ2

ηI(Q−D)Mr0 0 0

0 Σ + σ2ηI2DMt

0 00 0 σ2

ηI(2D−L)Mr−2DMt0

0 0 0 σ2ηI(Q−D)Mr

(72)

In (71), the submatrices Ui1, Ui2, Ui3, and Ui4 for 1 ≤i ≤ 3 have (Q−D)Mr, 2DMt, (2D − L)Mr − 2DMt, and(Q−D)Mr columns, respectively. The submatrices U1j , U2j ,and U3j for 1 ≤ j ≤ 4 have (Q−D)Mr, (2D − L)Mr, and(Q−D)Mr rows, respectively. The submatrix Σ in (72) isa 2DMt × 2DMt diagonal matrix with positive diagonal el-ements. From (26) and (70) through (72), we obtain the follow-ing relationship:

[B3, B4][W2 00 W3

]R

12dd = U22Σ

12 VH (73)

where Ed(n)d(n)H=Rdd =R1/2dd RH/2

dd , Σ=Σ1/2ΣH/2,and V is an unitary matrix. Assuming that Rdd has full rank,we obtain span([B3, B4]) = span(U22) from (73). Therefore,the matrix U22 has a full column rank. Defining a (2D −L)Mr × (2D − L)Mr − 2DMt matrix U⊥

22 whose columnvectors are linearly independent and span the subspace orthogo-nal to span(U22), we can find an invertible matrix G satisfyingthe following condition [65]:

Un =

U11 U13 U14

U21 U23 U24

U31 U33 U34

=

U11 0 U14

U21 U⊥22 U24

U31 0 U34

G

(74)

where Uij has the same dimension as that of Uij for 1 ≤i ≤ 3 and j = 1, 4. By defining the matrices B′

3 and B′4 as,

respectively

B′3

∆= H′ [(Q−D)Mr + 1 :

(Q + D − L)Mr, (Q−D)Mt + 1 : QMt]

B′4

∆= H′ [(Q−D)Mr + 1 :

(Q + D − L)Mr, QMt + 1 : (Q + D)Mt]

(75)

and using the condition of span(Ξ′) = span(Ξ) and the orthog-onal relationship in (28), we derive

UHnΞ′ =0 =⇒ (

U⊥22

)H[B′

3,B′4][W2 00 W3

]=0 ⇐⇒ (

U⊥22

)H[B′

3,B′4] = 0. (76)

Furthermore, we know that the column vectors of U⊥22 consti-

tute a basis for the left-hand nullspace of the matrix [B3, B4].By relying on Theorem 2 in [50] along with the relation-ship in (76) and the condition of rank(H′) = Mt, we obtainH′(z) = H(z)Ω with an Mt ×Mt invertible matrix Ω. That is,H′ = HΩ.

On the other hand, by following the same procedure as givenin (68), we can prove that if H′ = HΩ with an invertible matrixΩ, then span(Ξ′) = span(Ξ). In addition, since H′ = HΩ, wehave h′(0) = h(0)Ω. By the assumption of rank(h(0)) = Mt,we can conclude that H′ has a full column rank.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir constructive comments, which helped improve this paper.

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Changyong Shin (S’04–M’07) received the B.S. andM.S. degrees from Yonsei University, Seoul, Korea,in 1993 and 1995, respectively, and the Ph.D. degreefrom the University of Texas at Austin in 2006, all inelectrical engineering.

From 1995 to 2001, he was a Senior ResearchEngineer at LG Electronics Inc., Seoul, where heworked on digital video signal processing and verylarge scale integration design for digital signalprocessing. He is currently with Samsung AdvancedInstitute of Technology, Gyunggi-do, Korea. His re-

search interests cover multi-input–multi-output communications, multicarriermodulation, and signal processing for communications including channel esti-mation, signal detection, interference cancellation, space–time processing, andsynchronization.

Robert W. Heath, Jr. (S’96–M’01–SM’06) receivedthe B.S. and M.S. degrees from the University ofVirginia, Charlottesville, in 1996 and 1997, respec-tively, and the Ph.D. degree from Stanford Uni-versity, Stanford, CA, in 2002, all in electricalengineering.

From 1998 to 2001, he was a Senior Member ofthe Technical Staff and then a Senior Consultant withIospan Wireless Inc., San Jose, CA, where he workedon the design and implementation of the physical andlink layers of the first commercial MIMO-OFDM

communication system. In 2003, he founded MIMO Wireless Inc.: a con-sulting company dedicated to the advancement of MIMO technology. SinceJanuary 2002, he has been with the Department of Electrical and ComputerEngineering, University of Texas at Austin, where he is currently an AssistantProfessor and a member of the Wireless Networking and CommunicationsGroup. His research interests cover a broad range of MIMO communication,including limited feedback techniques, multihop networking, multiuser MIMO,antenna design, and scheduling algorithms, as well as 60-GHz communicationtechniques.

Dr. Heath serves as an Editor for the IEEE TRANSACTIONS ON

COMMUNICATIONS, and an Associate Editor for the IEEE TRANSACTIONS ON

VEHICULAR TECHNOLOGY, and he is a member of the Signal Processing forCommunications Technical Committee of the IEEE Signal Processing Society.

Edward J. Powers (F’83) received the B.S., M.S.,and Ph.D. degrees from Tufts University, Medford,MA, the Massachusetts Institute of Technology,Cambridge, and Stanford University, Stanford, CA,respectively, all in electrical engineering.

He is the Texas Atomic Energy Research Founda-tion Professor in engineering and Professor of elec-trical and computer engineering with the Universityof Texas at Austin. His primary professional interestslie in the innovative application of digital higherorder statistical signal processing in the analysis, in-

terpretation, and modeling of time series data characterizing nonlinear physicalphenomena, and the utilization of the wavelet transform and time–frequencyanalysis to detect and identify transient events in various physical systems.

Dr. Powers served as a chair of his department from 1981 to 1988. He is apast Editor of the IEEE TRANSACTIONS ON PLASMA SCIENCE.

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