IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008

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Resolving Multidimensional Ambiguity in Blind Channel Estimation of MIMO-FIR Systems via Block Precoding Feifei Gao, Student Member, IEEE, and Arumugam Nallanathan, Senior Member, IEEE

Abstract—In this paper, we consider the identification problem of multiinput–multioutput (MIMO) finite-impulse-response systems via second-order statistics only. By assigning different block precoders to different transmitters, we develop a new technique that allows blind MIMO channel identification up to a scalar ambiguity for each transmitter. We provide sufficient conditions for removal of the matrix ambiguity for a specific set of precoding matrices and derive a general theorem for other kinds of precoding matrices based on a reasonable conjecture. This theorem is firmly tested via numerical examples. Two potential precoding schemes are proposed, considering different ways of eliminating interblock interference. Finally, numerical results are provided to verify our analysis. Index Terms—Ambiguity, blind channel estimation, block transmissions, multiinput multioutput (MIMO), precoding, subspace (SS) method.

I. I NTRODUCTION

M

ULTIANTENNA transmission over multiinput– multioutput (MIMO) channels has recently been proven effective in combating fading, as well as enhancing data rates [1], [2]. Since coherent detection requires accurate channel state information, channel estimation has become a critical component in a variety of modern wireless communication systems. Many communication systems identify the channel coefficients by transmitting pilot symbols that are known to both transmitters and receivers. These pilot-aided schemes, however, reduce the transmission-bandwidth efficiency [3]. Therefore, blind-channel-estimation algorithm has received considerable attention during the past few decades. The subspace (SS)-based blind-channel-estimation algorithm has been developed in [4]–[7] for either single-input– single-output systems (SISO) or single-input–multioutput systems, where the channel coefficients between the transmitter, and all the receivers could be identified within a complex scalar ambiguity. This scalar ambiguity could easily be removed by further transmission of one or a few amount of pilot symbols. However, when the SS method is directly applied to the MIMO system [8]–[10], the channels could only be estimated Manuscript received March 16, 2005; revised November 10, 2005, July 20, 2006, and November 24, 2006; accepted April 5, 2007. The review of this paper was coordinated by Prof. R. Heath. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260 (e-mail: feifeigao@ nus.edu.sg; [email protected]). Digital Object Identifier 10.1109/TVT.2007.904542

up to an unknown matrix ambiguity. This matrix ambiguity is a multidimensional problem and is not acceptable in most applications. In [11], this matrix ambiguity is resolved under the assumption that sources from different transmitters are nonGaussian and statistically independent. The property of cyclostationary is exploited in [12] and [13]. Other works discussing linear prefiltering via z-domain polynomial analysis can be found in [14]–[16] and the references therein. In a recent work [17], a linear-precoding technique is applied at the inputs for MIMO orthogonal frequency-division multiplexing (OFDM)modulation transmissions. This method needs the transmitted symbols to be strictly white and is mainly proposed for blind channel estimation in a multiinput–single-output (MISO) scenario. In this paper, we propose a new way of resisting the matrix ambiguity when the SS algorithm is applied to the MIMO systems. We find that by dividing the data sequences into blocks and assigning different block precoders to different transmitters, it is possible to reduce the matrix ambiguity to a scalar ambiguity for each transmitter. Note that block precoding is a well-studied topic through the literatures [18], [19] to improve the performance of the detection. However, few works on block precoding have been proposed regarding the resistance of the channel-estimation ambiguity in the MIMO frequencyselective channels. We provide strict conditions on removing the matrix ambiguity for a specific set of precoding matrices, namely, zero-padding matrix. Then, a more general conclusion on resisting the matrix ambiguity is derived based on a reasonable conjecture. Various numerical examples are provided to demonstrate the effectiveness of our proposed algorithms. This paper is organized as follows. Section II presents the system model of MIMO transmissions. Section III presents our proposed precoding schemes. Section IV discusses the ambiguity and the identifiability issues. Numerical examples are provided in Section V to exhibit the effectiveness of our algorithms. Finally, conclusion is presented in Section VI, and the proof for the theorem is given in the Appendix. The following notations are used in this paper. Transpose, complex conjugate, Hermitian, inverse, and pseudoinverse of matrix A are denoted by AT , A∗ , AH , A−1 , and A† , respectively. [A]ij stands for the (i, j)th entry of the matrix A, tr(A) denotes the trace operation, ⊗ represents the Kronecker product, I is the identity matrix, and E{·} is the statistical expectation. The MATLAB notations for rows and columns are used in this paper. For example, A(:, p) represents the

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pth column of the matrix A, and A(:, p1 : p2 ) represents the submatrix obtained by extracting columns p1 to p2 from the matrix A, respectively. II. S YSTEM M ODEL Let us consider an Nt -input Nr -output linear finite-impulseresponse (FIR) MIMO system. The data streams from transmit antennas are denoted as {biτ }, i = 1, . . . , Nt , and τ = 0, 1, . . . , M − 1, where i and τ are the transmitter and time indexes, respectively. The data sequences are then divided into consecutive blocks with the length of K, namely T  si (m) = bi(mK) , bi(mK+1) , . . . , bi(mK+K−1) m = 0, 1, . . . , M/K − 1 (1) where m is the block index. Without loss of generality, M is taken as an integer multiple of K. Assume that perfect synchronization is achieved at the receivers. The data stream obtained from the jth receive antenna is denoted as {djτ }, j = 1, . . . , Nr , which could be divided into blocks as T  rj(m) = dj(mK) , dj(mK+1) , . . . , dj(mK+K−1) . (2) For convenience, we assume that all channel responses between different pairs of transmitters and receivers have the same channel order L. Let hij = [hij,0 , hij,1 , . . . , hij,L ]T  hij,L · · · hij,0 · · ·  .. .. .. .. MK (hij ) =  . . . . 0

···

hij,L

···



0 ..  . 

(3) (4)

hij,0

represent the (L + 1) × 1 equivalent discrete channel vector and the K × (K + L) channel matrix from the ith transmitter to the jth receiver, respectively. The subscript K denotes the number of rows in MK (hij ). The combined channel matrix from the ith transmitter to all the receivers can be represented by the KNr × (K + L) matrix T  T K (hi ) = MK (hi1 )T , MK (hi2 )T , . . . , MK (hiNr )T (5) where T  T T hi = hT (6) i1 , hi2 , . . . , hiNr is the combination of channel vectors from the ith transmitter to all the receivers. The overall KNr × (K + L)Nt channel matrix for the MIMO system is ΠK (hij ) = [T K (h1 ), T K (h2 ), . . . , T K (hNt )] .

(7)

The overall received signal block is then modeled as T  r(m) = r1 (m)T , . . . , rNr (m)T Nt = T K (hi )¯si (m) + n(m)

T  ¯si (m) = siL (m − 1)T , si (m)T T  ˜si (m) = ¯s1 (m)T , . . . , ¯sNt (m)T

(9) (10)

where n(m) is the KNr × 1 vector whose entry represents the independent identically distributed white Gaussian noise with variance σ 2 . siL (m − 1) denotes the last L entries of the vector si (m − 1), ¯si (m) is the (K + L) × 1 vector denoting the combination of siL (m − 1) and si (m) due to the interblock interference (IBI), and ˜s(m) is the (K + L)Nt × 1 vector representing the effective transmitted symbol vector for the mth received block r(m). An alternative representation of the channel matrix is also provided for later use. Let Hl denote the Nr × Nt matrix with its (j, i)th entry given by hij,l . After properly permutating the rows and columns of ΠK (hij ), the channel matrix can be expressed as  L  H HL−1 · · · H0 0 ··· 0 L L−1 0  0 H ··· H ··· 0  H . . ΓK (Hl ) =  .. .. .. .. ..  ... . . . . . ..  0

···

0

HL HL−1 · · · H0

(11)

Remark 1: The algorithms in this paper are presented based on the channel model ΠK (hij ). However, theorems or lemmas are provided based on the second channel model ΓK (Hi ). One can easily establish the equivalence between ΠK (hij ) and ΓK (Hl ). Lemma 1 [20], [21]: The channel identifiability up to an ambiguity matrix by using only the second-order statistics (SOS) of the received signals is ensured if the following conditions are satisfied. 1) ΠK (hij ) and ΓK (Hl ) are tall matrices. 

l −l 2) H(z) = L is irreducible and column-reduced. l=0 H z L 3) H is of full column rank. Since different channel vectors are random vectors, these conditions can be satisfied for most of the practical MIMO systems [16]. Therefore, the MIMO channel discussed in this paper is always considered to satisfy the conditions previously listed, which also indicates that ΠK (hij ) and ΓK (Hl ) are fullcolumn-rank matrices. III. P ROPOSED P RECODING T ECHNIQUES We found that, by applying different precoders to different transmitters, it is possible to reduce the matrix ambiguity to one scalar ambiguity for each transmitter. The intuitive explanation is that, since we artificially introduce certain “differences” among different transmitters, the channel can be discriminated from each other by exploiting these “differences.” A. Normal Precoding

i=1

= ΠK (hij )˜s(m) + n(m)

with

(8)

An illustration of normal precoding scheme at the ith transmitter is shown in Fig. 1, where the original data sequence is

GAO AND NALLANATHAN: RESOLVING MULTIDIMENSIONAL AMBIGUITY IN BLIND CHANNEL ESTIMATION

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signal vector r(m) can be reexpressed as r(m) =

Nt i=1

=

Nt



FLL i T K+1 (hi ) 0

0 Fi

siL (m − 1) + n(m) si (m)

¯ i¯si (m) + n(m) T K+1 (hi )F

(17)

i=1

¯ i is the corresponding (K + L + 1) × (K + L) where F matrix. Define ¯ 2, . . . , F ¯N } ˜ = diag{F ¯ 1, F F t Fig. 1.

Normal precoding.

first divided into continuous block si (m) and is then precoded by a (K + 1) × K matrix Fi , resulting in a new data block ui (m) with the length K + 1, namely ui (m) = Fi si (m).

(12)

At the receiver, the received data stream {djτ }, τ = 0, . . . , M (1 + 1/K) − 1 is divided into blocks of length K + 1, namely T  rj (m) = dj(m(K+1)) , . . . , dj(m(K+1)+K) m = 0, . . . , M/K − 1 (13) as shown in Fig. 1. The overall received signal vector r(m) can be written as r(m) =

Nt

T K+1 (hi )¯ ui (m) + n(m)

(14)

i=1 T T ¯ i (m) = [uT where u iL (m − 1), ui (m)] , and uiL (m − 1) denotes the last L entries of the block vector ui (m − 1). Note that T K+1 (hi ) is of dimension Nr (K + 1) × (K + L + 1) but possesses a similar structure as (5). By denoting FiL = F(K + 2 − L : K + 1, :) as the last L rows of the matrix Fi , the received signals can be rewritten as

r(m) =

Nt i=1

=

Nt

T K+1 (hi )

FiL 0

si (m − 1) 0 + n(m) Fi si (m)

˘ i˘si (m) + n(m) T K+1 (hi )F

(15)

i=1

˘ i is the corresponding (K + 1 + L) × 2K matrix, where F and ˘si(m) is constructed by two consecutive blocks of source ˘ i has vectors. In order to guarantee the recoverability of ˘si(m) , F to be either a full-rank tall or a square matrix. We then design matrix Fi such that all the first K − L columns of FiL are zero, namely FiL = [ 0 FLL i ]

(16)

where FLL = F(K + 2 − L : K + 1, K + 1 − L : K) is coni structed by the last L rows and the last L columns of Fi . The

(18)

as the (K + 1)Nt × KNt block diagonal matrix representing the overall precoding matrix. The received signal covariance matrix is given by 

R = E r(m)rH (m) ˜ sF ˜ H ΠK+1 (hij )H + σ 2 IN (K+1) = ΠK+1 (hij )FR r

(19)

where Rs = E{˜s(m)˜sH (m)} is the (K + L) × (K + L) source covariance matrix. Since, normally, no two sources are fully correlated with each other,1 Rs can always be considered ˜ is a full-columnas a nonsingular matrix. Thus, ΠK+1 (hij )F rank matrix, and the covariance matrix R can be eigendecomposed as R = EΛEH + σ 2 GGH

(20)

where the (K + L)Nt × (K + L)Nt diagonal matrix Λ contains the signal-SS eigenvalues of R. In turn, the columns of the (K + 1)Nr × (K + L)Nt matrix E contain the signal-SS eigenvectors of R, whereas the (K + 1)Nr × ((K + 1)Nr − (K + L)Nt ) matrix G is composed of the noise-SS eigenvectors of R. From the SS detection theory [4], we know that the noise SS spanned by G is the orthogonal complement space spanned ˜ Hence, for different T K+1 (hi ), the following by ΠK (hij )F. equation holds: ¯ i (:, p) = 0, GH T K+1 (hi )F ⇒

(G ip )H hi = 0



H hH i G ip (G ip ) hi = 0

p = 1, . . . , K + L

(21)

where G ip can straightforwardly be calculated from GH and ¯ i (:, p). F Let Φi =

K+L

G ip (G ip )H .

(22)

p=1

ˆ i , can be obtained The estimate of hi , which is denoted as h from the eigenvector of Φi that corresponds to the smallest eigenvalue; therefore, different hi can be determined from its corresponding Φi . 1 Partly

correlated sources do not affect the rank of Rs .

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where G y,ip is obtained from (G y,ip )H hi = GH y TK+1−L (hi )Fi (:, p),

Fig. 2. Simplified precoding.

Remark 2: To resolve the matrix ambiguity, it is necessary that Φi has only one zero eigenvalue for each i. The related discussion is provided in Section IV. A necessary condition is ¯ i should span different quoted here, which states that different F SSs from each other. B. Simplified Precoding Due to the existence of IBI, the proposed algorithm forces an L × (K − L) submatrix of Fi to be zero, thus reducing the flexibility of precoder design. An alternative way of eliminating the IBI can simply be implemented by deleting the first L elements in each received block rj (m). The remaining symbol vector at the jth receive antenna can be expressed as T  yj (m) = dj(m(K+1)+L) , . . . , dj(m(K+1)+K) .

(23)

Then, the additional constraint (16) is removed. The transmission scheme is shown in Fig. 2, and the overall signal vector y(m) can be written as T  y(m) = y1 (m)T , y2 (m)T , . . . , yNr (m)T ˜ s(m) + n(m) = ΠK+1−L (hij )F˜

˜ to represent the overall precoding matrix where we still use F and ˜s(m) to represent the overall source vectors, namely ˜ = diag{F1 , F2 , . . . , FN } F t  T  T ˜s(m) = s1 (m), . . . , sT . Nt (m)

(25) (26)

The new noise SS matrix Gy is obtained from the eigen decomposition of

 Ry = E y(m)yH (m)

(27)

and is of dimension (K + 1 − L)Nr × ((K + 1 − L)Nr − KNt ). Define Φiy =

K p=1

G y,ip (G y,ip )H

Similar to normal precoding, the channel vector hi can be estimated from the eigenvector of Φiy that corresponds to the smallest eigenvalue. Remark 3: Note that the way of generating precoders now becomes much easier. We can randomly generate a full-rank (K + 1) × (K + 1) matrix F and then select K different columns to form Fi , by which means we can guarantee that different Fi is a full rank and spans different SS from each other. However, since the simplified algorithm does not consider the contribution of the first L symbols in the received block for each receiver, it cannot make full use of the received symbol blocks. Remark 4: Let us take a look at the maximum value of Nt . If the precoders of dimension (K + 1) × K are used, as previously assumed, the maximum value of Nt is K + 1 since we can at most construct the K + 1 precoders that span different SSs from each other. Moreover, we can also use a taller precoding matrix. If the precoders of dimension (K + q) × K q , are used, the maximum value of Nt can be calculated as Ck+1 where C is the notation of combination. IV. A MBIGUITY R ESISTANCE AND C HANNEL I DENTIFIABILITY It is known that the channel matrices Πζ (hij ) and Γζ (Hl ) that satisfy Lemma 1 are full-rank-tall matrices, where the value of ζ takes K + 1 for normal precoding or K + 1 − L ¯ i are full ranks for all for simplified precoding. If Fi and F ˜ i’s, then Πζ (hij )F is also a full column rank. Therefore, the SS-based detection [4] could be applied. Denote the estimates ˆ ij ) and Γζ (H ˆ l ), respectively. of channel matrices as Πζ (h ˆ ij )F ˜∈ From the SS detection theory, we know that span(Πζ (h ˜ span(Πζ (hij )F). Therefore ˆ ij )F ˜ = Πζ (hij )FA ˜ Πζ (h

(24)

(28)

p = 1, . . . , K. (29)

(30)

where A is an unknown Nt (ζ + L − 1) × Nt (ζ + L − 1) matrix. A. Ambiguity Resistance Theorem 1: Suppose that the estimate of H(z) is achieved ˆ within an ambiguity matrix, namely, H(z) = H(z)B, where B ¯ i (for normal precoding) or is an Nt × Nt unknown matrix. If F Fi (for simplified precoding) is a full rank and spans different spaces from each other, then B must be a diagonal matrix, which indicates a scalar ambiguity for each transmit antenna. Proof: Denote Hc = [h1 , h2 , . . . , hNt ]. The condition ˆ H(z) = H(z)B can equivalently be expressed as ˆ c = Hc B or H

ˆ l = Hl B H

(31)

ˆ 1, h ˆ 2, . . . , h ˆ N ] is the estimate of Hc , and H ˆ c = [h ˆ l is where H t l the estimate of H . We only need to prove that B is a diagonal matrix.

GAO AND NALLANATHAN: RESOLVING MULTIDIMENSIONAL AMBIGUITY IN BLIND CHANNEL ESTIMATION

ˆ ij ) can be rewritten as If (31) holds, Πζ (h ˆ ij ) = Πζ (hij )B Πζ (h

(32)

where 

[B]11 Iζ+L  .. B= . [B]Nt 1 Iζ+L

··· .. . ···

 [B]1Nt Iζ+L  ..  . [B]Nt Nt Iζ+L

(33)

where [B]pq is the (p, q)th entry of B. By substituting (32) into (30), we obtain ˜ = Πζ (hij )FA. ˜ Πζ (hij )BF

(34)

From Lemma 1, we know that Πζ (hij ) is a full-rank-tall matrix. Then, (34) indicates that ˜ = FA. ˜ BF

(35)

Divide A into blocks as 

A11  A =  ... ANt 1

··· .. . ···

 A1Nt ..  . .  ANt Nt

(36)

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resorts to the z-domain polynomial analysis [20]. However, since the convolution property of the system is broken by block precoding, it is quite hard to implement the polynomial analysis here. Conjecture 1: Suppose that the conditions listed in Lemma 1 ¯ i (for normal are satisfied. If Fi (for simplified precoding) or F precoding) is a full-rank matrix, then the channel identifiability ˆ H(z) = H(z)B for the proposed algorithms can be guaranteed. An intuition explanation is obtained from [5], [22]–[24], where the redundant precoding is applied to guarantee the channel identifiability, even if the overall channel matrix may not be a full rank. Therefore, it makes no sense that the application of full-rank redundant precoders may destroy the identification of the system whose identifiability is originally guaranteed (under Lemma 1). We now provide a firm study on channel-estimation identifiability for a specific set of precoders, namely, zero-padding precoders introduced in [25, ch. 10]. Discussion on more general precoders is still an open problem and is currently under investigation. Theorem 2: For simplified precoding algorithm, if Fi is obtained by deleting one column from IK+1 (zero-padding precoders), then sufficient conditions for (31) to hold are the following. 1) (K − 1)/2 > L + 1, where · denotes the largest integer that is no bigger than the specified value. 2) Nr > (Nt ((K −1)/2−1)/((K −1)/2−L−1)).

From (35) and (36), we obtain 

··· .. . ···

[B]11 F1  ..  . [B]Nt 1 F1

 [B]1Nt FNt  ..  . [B]Nt Nt FNt  ··· F1 A11  . .. .. = . FNt ANt 1 · · ·

 F1 A1Nt  ..  . FNt ANt Nt

(37)

¯ i for normal precoding. where Fi should be changed to F Therefore [B]ij Fj = Fi Aij

(38)

for all pairs of (i, j). Since Fi is a full rank and spans different SSs from each other, it can easily be known that 

Aii = [B]ii Iζ+L−1 , [B]ij = 0, Aij = 0,

for i = j . for i = j

Proof: See the Appendix. If Nr and K are sufficiently large, then the conditions listed in Theorem 2 could always be satisfied. However, since Theorem 2 is proved based only on several columns in Γζ (H), the sufficient conditions obtained are rather loose. Normally, even for smaller values of Nr and K that do not satisfy the condition in Theorem 2, the identification could still be achieved. Remark 5: Although Theorem 2 only considers a simplified precoding algorithm, a similar proof can straightforwardly be applied for zero-padding precoders in the normal precoding ¯ i will vary if algorithm. Note that the size and the structure of F zeros are inserted into the different positions. The proof is rather tedious, and therefore, it is omitted here. However, similar conclusion can be made that, if Nr and K are sufficiently large or are larger than the certain values, the channel identifiability can be guaranteed.

(39)

Therefore, B must be a diagonal matrix, and the multidimensional ambiguity is converted to a scalar ambiguity for each transmit antenna.  B. Identifiability Now, the question is whether (31) holds, which is normally categorized as the identifiability problem. Usually, the proof

V. S IMULATION R ESULTS In this section, we provide numerical examples to verify the theorems/algorithms developed in previous sections. As previously claimed, the sufficient conditions listed in Theorem 2 are quite loose; therefore, we will not restrict ourselves in testing only those big K and Nr . For simplicity, the parameters are chosen as K = 7, Nt = 2, Nr = 3, and L = 2. One can see through the simulations that channel identification can still be guaranteed within an unknown complex scalar for each

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Fig. 3. Detected constellation diagram for zero-padding precoders under noise-free case.

transmit antenna. The channel coefficients are randomly generated, which, in this simulation, are

h11 =[0.4608+1.8903i, 0.4574+0.2622i, 0.4507−0.8794i]T h12 =[0.4122−0.0678i, 0.9016−1.4460i, 0.0056+0.6126i]T h13 =[0.2974+1.3720i, 0.0492−0.4288i, 0.6932+0.1289i]T h21 =[0.6501−0.5702i, 0.9830+0.5521i, 0.5527−0.4084i]T h22 =[0.4001−1.1703i, 0.1988−0.5576i, 0.6252−0.2016i]T h23 =[0.7334−2.5679i, 0.3759−0.2724i, 0.0099−0.0160i]T

unless otherwise mentioned. The normalized-estimation meansquare-error (NMSE) of hi is defined as [21]

NMSE =

Nq ˆ i αi − hi 2 1 h Nq i=1 h 2i

(40)

ˆ i αi − hi 2 is minimized, where αi is chosen such that h and Nq = 100 is the number of Monte Carlo runs. The symbols from each transmitters are independently generated from 16-quadratic-amplitude modulation (16-QAM). The signal-tonoise ratio (SNR) is defined as the ratio between the symbol power at the transmitter and the noise power at the receiver, which does not take the channel effect into account.

A. Capability of Ambiguity Resistance We adopt the way in [16] to demonstrate the capability of the ambiguity resistance here. We first consider the zeropadding precoding matrix with F1 = I8×8 (:, 1 : 7) and F2 = I8×8 (:, 2 : 8). Totally, 30 blocks of signals are assumed to be sent from each transmit antenna, and the channel estimation is purely conducted based on the correspondingly received 30 blocks. For the noise-free case, the data patterns detected for each transmitter applying both algorithms are shown in Fig. 3. The constellations are drawn by placing the real parts of all the received 7 × 30 symbols onto the x-coordinate, whereas the corresponding imaginary parts are placed onto the y-coordinate. Clearly, the shape of the 16-QAM constellation is kept, but it is rotated and scaled compared with the standard 16-QAM constellation. Therefore, the matrix ambiguity reduces to one scalar ambiguity for each transmitter. Then, we consider the noisy case. Since we only wish to demonstrate the capability of ambiguity resistance of the proposed algorithms, the SNR is taken relatively higher as 25 dB. The detected data patterns are shown in Fig. 4. It can be seen that the proposed algorithms are still able to resolve the multidimensional ambiguity. It is also important to numerically test the capability of the ambiguity resistance for more general precoding matrices. We consider 103 simulation runs and assume the noiseless environment for test purposes only. In each simulation run, the precoding matrices are randomly generated.2 In this specific 2 The randomly generated matrix has a full rank with a probability of one. Since the SNR is infinite, we could even adopt an ill-conditioned precoding matrix without affecting the detection results.

GAO AND NALLANATHAN: RESOLVING MULTIDIMENSIONAL AMBIGUITY IN BLIND CHANNEL ESTIMATION

Fig. 4.

Detected constellation diagram for zero-padding precoders under noisy case SNR = 25 dB.

Fig. 5.

Detected constellation diagram for random precoders under noise-free case.

example, we also allow the channel to randomly change for different simulation runs. It is found that the detected symbol points exactly form the true 16-QAM constellation, except that the constellation patterns have different scaling factors and rotation angles for different simulation runs. Therefore, we only show the data pattern for the specific channel realization (40) in Fig. 5. By this example, we numerically prove that the matrix ambiguity could be resolved by using more general precoders.

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B. Performance of the Proposed Algorithms In this example, we demonstrate the performance comparison between the normal- and simplified-precoding techniques. The precoders for the two transmitters are taken as F1 = I8×8 (:, 1 : 7) and F2 = I8×8 (:, 2 : 8). Each precoder is then scaled to keep the power of the precoded signal unchanged. We choose to compare with the algorithm in [17], where the block

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Fig. 6. Channel-estimation nmses versus the SNR: Comparing two algorithms and the existing work [17].

Fig. 7.

¯ i. Performance comparison for different ways of generating F

Fig. 8.

Performance comparison for different linear precoders.

transmissions are also adopted for fair comparisons. In Fig. 6, the NMSEs of individual channel estimation versus the SNRs of these two estimators, as well as the algorithm in [17], are shown. The number of the transmitted symbol blocks is taken as 200 for all algorithms. From Fig. 6, we see that the performance of the normal blind algorithm is better than the simplified algorithm. The reason lies in that it exploits more information from the received signal. It is also seen that both the proposed algorithms outperform the one in [17] at relatively higher SNR region. The reason is that the latter method, although it is applied to block transmissions and possesses scalar ambiguity, is mainly targeted for blind channel estimation in MISO system.

C. Different Precoder Design for Normal Precoding The precoder Fi can be generated by extracting different columns from an arbitrary full-rank square matrix. In order to apply both algorithms, Fi could be modified according to the following two ways. 1) Type 1: Force the first K entries of the last L rows in Fi to be zero, and then, scale to keep the power of signals unchanged, as indicated in (16). 2) Type 2: In addition to the operation in type 1), force the last L entries of the first K + 1 − L rows to be zero, and then, scale to keep the power of signals unchanged. In this case, Fi becomes a block diagonal matrix. Fig. 7 shows the NMSEs for the channel estimation versus the SNR of both algorithms. It can be seen that normal precoding, as well as simplified precoding by type 2) precoders, gives a satisfactory performance. However, both algorithms by type 1) precoders perform much worse than the other two cases. The reason is that forcing one corner in Fi to be zero only makes it “imbalanced” and ill-conditioned. Therefore, type 2) construction of linear precoders is preferred for the normal precoding algorithm.

D. Performance for Different Types of Precoders We compare the performance of different types of precoders in this example. The performances of the NMSEs versus the SNR for zero-padding precoders, the type 2) precoders extracted from the fast-Fourier-transformation (FFT) matrix, and the type 2) randomly generated precoders are shown in Fig. 8. For simplicity, only the curves for h1 are plotted. It is seen that the zero-padding precoders perform better than the other linear precoders. This phenomenon was also observed in [25], where the precoding is applied purely for SISO transmissions. Several possible reasons are provided in [25]. Intuitively, the zero-padding precoders do not introduce intersymbol interference, and the followed data detection undergoes less noise enhancement. VI. C ONCLUSION We have investigated a new way to identify blind channel estimation for MIMO-FIR systems based on the SOS.

GAO AND NALLANATHAN: RESOLVING MULTIDIMENSIONAL AMBIGUITY IN BLIND CHANNEL ESTIMATION

By assigning different precoders to different transmitters, the proposed algorithms can eliminate the higher dimensional ambiguity and are able to estimate the channel coefficient for different transmitters within a complex scalar ambiguity only. Strict conditions on identifiability are provided for zero-padding precoders, whereas identification for other precoding matrices is numerically tested. Other numerical examples on the performance of the proposed algorithms, as well as the comparison with the existing works, are also provided. A PPENDIX P ROOF OF T HEOREM 2 An equivalent expression for (30) is ˆ l )F = ΓK+1−L (Hl )F A ΓK+1−L (H

Case 1 v1 ≤ K/2 + 1, and v2 > K/2 + 1. The matrices ˆ l )F and ΓK+1−L (Hl )F can be partitioned as ΓK+1−L (H 

ˆ1 P l ˆ  ˆ2 ΓK+1−L (H )F = P 0  P1 ΓK+1−L (Hl )F =  P2 0

 0 ˆ3  P ˆ P4  0 P3  P4

0 ˆ H 0 0 H 0

The 2K/2th to the (2K/2 + 3)th column of (41) could be rewritten as 

  0 P1 ˆ  =  P2 H 0 0

ˆ = HA ˜ 2. H

 h0 1 h11

  . H=  ..  L h1 0

1

ˆ0 h 2 ˆ1 h 2 .. . ˆL h 2 0

0 h01 .. .

h02 h12 .. .

hL−1 1 hL 1

hL 2 0

 0 ˆ0  h 2  ..  .   L−1  ˆ h2 ˆL h



0 ˆ0  H  .  .  . H ˆ L−1 ˆL H

ˆ0   0 H ˆ 1   H0 H  ..    . .  =  .. ˆ L   HL−1 H HL 0

(43)

C2 . C4

(48)

0

From the last block row, we can get (49)

Since Hl is, by assumption, a full-column-rank matrix, then C 2 is a zero matrix. By substituting this result into (48), it can be further derived that

(44)

C 3 = 0.

(50)

Taking B = C 1 gives (31). Case 2 ˆ l )F v1 ≤ K/2 + 1, and v2 ≤ K/2 + 1. Matrices Γ(H and Γ(Hl )F can be divided as 



  .  L−1  h2 hL 2

 H0 1  H C1 ..   .  C 3 HL 

(42)

2

0 h02 .. .

(47)

HL C 2 = 0.

ˆ 1 and P1 are the Nr (K/2 − L) × (2K/2 − 1) where P ˆ 4 and P4 are the Nr ((K − 1)/2 − L) × matrices, whereas P (2(K − 1)/2 − 1) matrices. Define hli as the ith column ˆ and of Hl , namely, hli = [hi1,l , hi2,l , . . . , hiNr ,l ]T . Then, H H can be expressed as two Nr (L + 2) × 4 matrices with the form 0 ˆ0 h 1 .. . ˆ L−1 h 1 ˆL h

(46)

ˆ we obtain By properly rearranging the columns of H,

C1 = C4,

 ˆ0 h1 h ˆ1  1  . ˆ H =  ..  h ˆL 1 0

  ˜1 A 0 ˜2 . P3   A ˜3 P4 A

0 H 0

Note that P1 is obtained from the columns of Γ(K/2−L) (Hl ) and that P4 is obtained from the columns of Γ((K−1)/2−L) (Hl ). From Lemma 1, we know that P1 and P4 are of full column rank if Γ(K/2−L) (Hl ) and Γ((K−1)/2−L) (Hl ) are tall matrices, namely, Nr (K/2 − L) > 2K/2 and Nr ((K − 1)/2 − L) > 2(K − 1)/2. ˜ 3 can be calculated as zero matrices. Therefore ˜ 1 and A Then, A

(41)

˜ For convenience, where F is a row permutation matrix from F. we first provide the proof for Nt = 2. Then, we assume that Fi is obtained by deleting the vi th column from a (K + 1) × (K + 1) identity matrix, with v1 = v2 , and we divide the discussion into two cases.

19

ˆ1 Q ˆ˜ =  ˆ HF Q2 0  Q1 ˜ =  Q2 HF 0

0 ˆ H 0

 0 ˆ3 Q ˆ4 Q

(51)

0 H 0

 0 Q3  Q4

(52)

(45) ˆ 1 and Q1 are taken as the Nr (K/2 − L + 2) × where Q ˆ 4 and Q4 are taken as the 2K/2 matrices, and Q

20

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008

Nr ((K − 1)/2 − L − 1) × (2(K − 1)/2 − 2) matrices. ˆ and H are now Nr (L + 1) × 2 matrices, with the form H  ˆ0  H ˆ1  H ˆ =  . , H  ..  ˆL H



 H0 1 H   H=  ...  .

(53)

HL

The (2K/2 + 1)th and the (2K/2 + 2)th columns of (41) could be rewritten as 

  0 Q1 ˆ  =  Q2 H 0 0

0 H 0

  ¯1 A 0 ¯2 . Q3   A ¯3 Q4 A

(54)

Note that Q1 is obtained from the columns of Γ(K/2−L+2) (Hl ) and that Q4 is obtained from the columns of Γ((K−1)/2−L−1) (Hl ). From Lemma 1, we know that Q1 and Q4 are of full column rank if Nr (K/2 − L + 2) > 2K/2 + 4 and Nr ((K − 1)/2 − L − 1) > 2(K − 1)/2 − 2. Similar to case 1, we could arrive at (31). For cases {v1 ≤ K/2 + 1, v2 > K/2 + 1} and {v1 > K/2 + 1, v2 > K/2 + 1}, a similar discussion can be made. By combining all cases, Nr should satisfy  Nr >

 2 (K − 1)/2 − 2 . ((K − 1)/2 − L − 1)

(55)

For Nt > 2, basically, the proof can be divided into two cases. 1) All vi ’s are smaller than K/2 + 1, or all vi ’s are bigger than K/2 + 1. Then, similar equations as (52) and (53) can be obtained. 2) Otherwise, similar equations as (43) and (48) can be obtained. The remaining discussion is the same as that for Nt = 2. Note that P1 and P4 are still obtained from the columns of Γ(K/2−L) (Hl ) and Γ((K−1)/2−L) (Hl ) by keeping in mind that column selection of P1 and P4 from Γ(K/2−L) (Hl ) and Γ((K−1)/2−L) (Hl ) varies for different combinations of vi . Meanwhile, Q1 is still obtained from the columns of Γ(K/2−L+2) (Hl ), and Q4 is from the columns of Γ((K−1)/2−L−1) (Hl ). Therefore, Nr should satisfy  Nr >

 Nt ((K − 1)/2 − 1) . ((K − 1)/2 − L − 1)

(56)

A detailed discussion for Nt > 2 is omitted because it is quite straightforward. R EFERENCES [1] I. E. Telatar, Capacity of Multi-Antenna Gaussian Channels, 1995. Bell Labs Tech. Memo. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998.

[3] L. Tong and S. Perreau, “Multichannel blind identification: From subspace to maximum likelihood methods,” Proc. IEEE, vol. 86, no. 10, pp. 1951– 1968, Oct. 1998. [4] E. Moulines, P. Duhamel, J. F. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Process., vol. 43, no. 2, pp. 516–525, Feb. 1995. [5] B. Muquet, Z. Wang, G. B. Giannakis, M. de Courville, and P. Duhamel, “Cyclic prefixing or zero padding for wireless multicarrier transmissions?” IEEE Trans. Commun., vol. 50, no. 12, pp. 2136–2148, Dec. 2002. [6] S. Roy and C. Li, “A subspace blind channel estimation method for OFDM systems without cyclic prefix,” IEEE Trans. Wireless Commun., vol. 1, no. 4, pp. 572–579, Oct. 2002. [7] K. Wei and B. Champagne, “Subspace-based blind channel estimation: Generalization and performance analysis,” IEEE Trans. Signal Process., vol. 53, no. 3, pp. 1151–1162, Mar. 2005. [8] D. Reynolds, X. Wang, and H. V. Poor, “Blind adaptive spacetime multiuser detection with multiple transmitter and receiver antennas,” IEEE Trans. Signal Process., vol. 50, no. 6, pp. 1261–1276, Jun. 2002. [9] Y. Hua and J. K. Tugnait, “Blind identifiability of FIR-MIMO systems with colored inputs using second order statistics,” IEEE Signal Process. Lett., vol. 7, no. 12, pp. 348–350, Dec. 2000. [10] H. Luo, R. Liu, X. T. Ling, and X. Li, “The autocorrelation matching method for distributed MIMO communications over unknown FIR channels,” in Proc. IEEE ICASSP, Salt Lake City, UT, May 2001, vol. 4, pp. 2161–2164. [11] S. Visuri and V. Koivunen, “Resolving ambiguities in subspace-based blind receiver for MIMO channels,” in Proc. 36th Asilomar Conf., Pacific Grove, CA, Nov. 2002, vol. 1, pp. 589–593. [12] I. Bradaric, A. P. Petropulu, and K. I. Diamantaras, “Blind MIMO FIR channel identification based on second order spectra correlations,” IEEE Trans. Signal Process., vol. 51, no. 6, pp. 1668–1674, Jun. 2003. [13] H. Bolcskei, R. W. Heath, Jr., and A. J. Paulraj, “Blind channel estimation in spatial multiplexing systems using nonredundant antenna precoding,” in Proc. 33th Asilomar Conf., Pacific Grove, CA, Oct. 1999, vol. 2, pp. 1127–1132. [14] A. Medles and D. T. M. Slock, “Linear precoding for spatial multiplexing MIMO systems: Blind channel estimation aspects,” in Proc. IEEE ICC, New York, Apr. 28–May 2, 2002, vol. 1, pp. 401–405. [15] H. Luo and R. W. Liu, “Blind equalization for MIMO for channels based only on second order statistics by use of pre-filters,” in Proc. IEEE SPAWC, Annapolis, MD, May 1999, pp. 106–109. [16] X. G. Xia, W. Su, and H. Liu, “Filterbank precoders for blind equalization: Polynomial ambiguity resistant precoders (PARP),” IEEE Trans. Circuits Syst. I, vol. 48, no. 2, pp. 19–209, Feb. 2001. [17] F. Gao and A. Nallanathan, “Blind channel estimation for MIMOOFDM systems via a nonredundant linear precoding,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 784–789, Jan. 2007. [18] Z.-Q. Luo, T. N. Davidson, G. B. Giannakis, and K. M. Wong, “Transceiver optimization for block-based multiple access through ISI channels,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 1037–1052, Apr. 2004. [19] D. J. Love and R. W. Heath, Jr., “Limited feedback unitary precoding for orthogonal space-time block codes,” IEEE Trans. Signal Process., vol. 53, no. 1, pp. 64–73, Jan. 2005. [20] Y. Li and K. J. R. Liu, “Blind identification and equalization for multipleinput/multiple-output channels,” in Proc. GLOBECOM, London, U.K., Nov. 1996, vol. 3, pp. 1789–1793. [21] A. Safavi and K. Abed-Meraim, “Orthogonal minimum noise subspace for multiple-input multiple-output system identification,” in Proc. IEEE SSP, Singapore, Aug. 2001, pp. 285–288. [22] S. Zhou, B. Muquet, and G. B. Giannakis, “Subspace-based (semi-) blind channel estimation for block precoded space-time OFDM,” IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1215–1228, May 2002. [23] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank precoders and equalizers. II. Blind channel estimation, synchronization, and direct equalization,” IEEE Trans. Signal Process., vol. 47, no. 7, pp. 2007–2022, Jul. 1999. [24] O. Shuichi and G. B. Giannakis, “Optimal training and redundant precoding for block transmissions with application to wireless OFDM,” IEEE Trans. Commun., vol. 50, no. 12, pp. 2113–2123, Dec. 2002. [25] G. B. Giannakis, Y. Hua, P. Stoica, and L. Tong, Signal Processing Advances in Wireless and Mobile Communications, vol. I. Englewood Cliffs, NJ: Prentice-Hall, Sep. 2000.

GAO AND NALLANATHAN: RESOLVING MULTIDIMENSIONAL AMBIGUITY IN BLIND CHANNEL ESTIMATION

Feifei Gao (S’05) received the B.Eng. degree in information engineering from Xi’an Jiaotong University, Xi’an, China, in 2002 and the M.Sc. degree from the McMaster University, Hamilton, ON, Canada, in 2004. He is currently working toward the Ph.D. degree with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests are in communication theory, broadband wireless communications, signal processing for communications, multiinput– multioutput systems, and array signal processing. Mr. Gao was a recipient of a President Scholarship from the National University of Singapore.

21

Arumugam Nallanathan (S’97–M’00–SM’05) received the B.Sc. degree in electrical engineering (with honors) from the University of Peradeniya, Peradeniya, Sri Lanka, in 1991, the Certificate of Postgraduate Study in electrical engineering from Cambridge University, Cambridge, U.K., in 1994, and the Ph.D. degree in electrical engineering from the University of Hong Kong, Kowloon, Hong Kong, in 2000. Since then, he has been an Assistant Professor with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests include orthogonal frequency-division-multiplexing systems, ultrawidebandwidth (UWB) communication and localization, multiinput–multioutput systems, and cooperative diversity techniques. He has published over 100 journal and conference papers in these areas. Dr. Nallanathan currently serves on the editorial board of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, Wireless Communications and Mobile Computing, and the EURASIP Journal of Wireless Communications and Networking as an Associate Editor. He also serves as a Guest Editor for the EURASIP Journal of Wireless Communications and Networking: Special Issue on UWB Communication Systems—Technology and Applications. He served as a Technical Program Cochair and as a Technical Program Committee Member for more than 25 IEEE international conferences. He currently serves as the General Track Chair for IEEE VTC 2008-Spring and a Cochair for the IEEE GLOBECOM 2008 Signal Processing for Communications Symposium and the IEEE ICC 2009 Wireless Communications Symposium. He is a corecipient of the Best Paper Award presented at the 2007 IEEE International Conference on Ultra-Wideband.

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