IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 9, SEPTEMBER 2006

M

M

[5] P. P. Vaidyanathan, “Theory and design of -channel maximally decimated quadrature mirror filters with arbitrary , having the perfect-reconstruction property,” IEEE Trans. Acoust., Speech. Signal Process., vol. ASSP-35, no. 4, pp. 476–492, Apr. 1987. [6] W. W. Jones and K. R. Jones, “Narrowband interference suppression using filter bank analysis/synthesis techniques,” in Proc. Military Communications Conf. (MILCOM), Oct. 1992, pp. 898–902. [7] A. Ranheim, “Narrowband interference rejection in direct-sequence spread-spectrum system using time-frequency decomposition,” IEEE Proc. Communications, vol. 142, pp. 393–400, Dec. 1995. [8] S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 7, pp. 674–693, Jul. 1989. [9] M. J. Medley, G. J. Saulnier, and P. K. Das, “Applications of the wavelet transform in spread-spectrum communications systems,” in SPIE Proc. —Wavelet Applications, Apr. 1994, vol. 2242, pp. 54–68. [10] M. V. Tazebay and A. N. Akansu, “Adaptative subband transforms in time-frequency excisers for DS-SS communications systems,” IEEE Trans. Signal Process., vol. 43, no. 11, pp. 2776–2782, Nov. 1995. [11] J. J. Pérez, M. A. Rodríguez, and S. Felici, “Interference excision algorithm for frequency hopping spread spectrum based on undecimated wavelet packet transform,” Electron. Lett., vol. 38, no. 16, pp. 914–915, Aug. 2002. [12] E. Pardo, J. J. Pérez, and M. A. Rodríguez, “Interference excision in DSSS based on undecimated wavelet packet transform,” Electron. Lett., vol. 39, no. 21, pp. 1543–1544, Oct. 2003. [13] M. Shensa, “The discrete wavelet transform: Wedding the à Trous and Mallat algorithms,” IEEE Trans. Signal Process., vol. 40, no. 10, pp. 2464–2482, Oct. 1992. [14] M. K. Simon, Spread Spectrum Communications Handbook. New York: McGraw-Hill, 1994.

Identifiability of Data-Aided Carrier-Frequency Offset Estimation Over Frequency Selective Channels Feifei Gao and A. Nallanathan, Senior Member, IEEE

Abstract—Carrier-frequency offset (CFO) must be compensated before channel estimation and coherent detection. Several data-aided CFO estimation algorithms have been proposed recently. However, an improper selection of training sequences may cause the identifiability problem which results in failure of CFO estimation. In this correspondence, we present a detailed study on identifiability issue and derive two new theorems for data-aided CFO estimation. The first theorem is suitable for all training sequences. The second one mainly deals with a popular set of training sequences that is deemed as optimal for channel estimation. Simulation results are provided to validate the proposed study. Index Terms—Carrier-frequency offset (CFO), channel estimation, identifiability, orthogonal frequency-division multiplexing (OFDM), preamble.

3653

for coherent detection at receivers [3]–[5]. Meanwhile, the preamble is also used to estimate carrier-frequency offset (CFO) arising from transceiver oscillator mismatches and/or Doppler shifts [6]–[9]. In [6], the channel is considered to be deterministic, and a maximumlikelihood (ML) CFO estimator is proposed based on fast Fourier transform (FFT). A parallel algorithm for CFO estimation in a statistical channel is developed in [7]. In [8], an adaptive ML approach for joint CFO and CIR estimation is addressed. All these pilot-based algorithms work well as can be seen in their simulations. Although the identifiability of blind CFO estimation has been addressed in [10], no study regarding the identifiability issue of data-aided CFO estimation has yet been reported, which, if ignored, can cause failure in the estimation of CFO and, consequently, failure in CIR estimation. Before proceeding, we wish to define the term “identifiability” that will be used in this correspondence. In [10], since more than one CFO may be detected simultaneously, the issue is more precisely called ambiguity problem. However, in this correspondence, the CFO is considered not identifiable when no CFO can be found through the estimator. In the following discussion, we focus on the CFO identifiability problem. The consideration for ambiguity issue is out of the scope of this correspondence. This correspondence is organized as follows. Section II describes the basic model for frequency-selective channel, as well as the dataaided ML CFO estimation algorithm. Section III presents the study of the identifiability problem for data-aided CFO estimation. Two useful theorems are provided in this section. Section IV shows the simulation results validating the proposed study. Finally, conclusions are made in Section V, and the proof for Theorem 2 is given in Appendix. II. PROBLEM FORMULATION Similar to [6], the frequency-selective channel is considered to be quasi-static, i.e., the CIR and CFO are supposed to remain constant during one data frame but can vary from frame to frame. Assume that Ts , where =Ts denotes the channel length is upper bounded by L the data sampling rate. The corresponding discrete channel response is h0 ; ; hL T . In then represented by the L 2 vector ; s01 ; s0 ; ; sM 01 g with front of each frame, the preamble fs0L ; length M L is inserted. The received samples, after dropping the first L symbols, is free from interframe interference and can be expressed as

( +1) 1 h = [ ... ( + 1) 1 ~ ... ~ ~ ... ~

+

x = 8Ah + n (1) where A is the M 2 (L +1) Toepliz matrix with its (p; q )th entry given by

I. INTRODUCTION Many practical communication systems over frequency-selective channels transmit frames of data that are preceded by a preamble of known symbols [1], [2]. The preamble usually serves as the training sequence for channel impulse response (CIR) estimation that is critical

n

1

[A]pq = s~p0q

(2)

and is the M 2 vector representing the white Gaussian noise with zero mean and variance n2 at each sampling time. Matrix is an M 2 M diagonal matrix with the form

8

8 = diag 1; ej  ; . . . ; ej  M 0 2

Manuscript received September 16, 2005; revised November 8, 2005. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ananthram Swami. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, 119260 (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TSP.2006.879277

]

=1

2 (

1)

(3)

and  fTs is the normalized frequency offset. The probability density function of , given  and , is

f (xj; h) =

1053-587X/$20.00 © 2006 IEEE

x

h

exp 0 kx 0 8Ahk (n ) n 1

2

M

2

2

:

(4)

3654

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 9, SEPTEMBER 2006

The joint ML estimates of ,

h can be obtained from

P

[^; h^ ] = arg max f (xj; h) ;h

(5)

kx 0 8Ahk2 : [^; h^ ] = arg min ;h

(6)

or equivalently

A

For joint estimation of CFO and CIR, should be designed as a fullcolumn-rank matrix such that is identifiable. Then, the ML estimate of , given , is obtained from (6) as

h

h

h^ = (88A)y x = (AH A)0 AH 8H x 1

(7)

where y denotes the pseudoinverse operation. Substituting (7) into (6), the ML estimate of  can be found by

^ = arg min 

x 0 8 A (A H A )0 A H 8 H x 1

= arg max g ( ) 

supposed to be L + 1, there will be exactly L + 1 ones on the diagonal of A . Theorem 1 is a general statement for all those training sequences making g () independent of . Unfortunately, there is no apparent way to relate Theorem 1 to those training sequences that could cause the identifiability problem. Therefore, we next restrict our attention to a special set of training sequences, and discuss its related identifiability issue. Note that among all kinds of training sequences, those satisfying

2

(8)

AH A = P I L

( +1)

2(L+1)

(11)

have received wide attention, where P is the total power assigned for training. One reason is that training sequences satisfying (11) are considered optimal purely for channel estimation over frequency-selective channels [3]–[5]. Second, in [9], training sequences from (11) are considered as optimal for jointly minimizing the asymptotical CRB of both CFO and CIR, based on a min-max approach. Therefore, it is necessary to make a specific consideration for this group of training sequences. From Theorem 1, we know that if M = L + 1, A would be an identity matrix for arbitrary training sequences from (11). Therefore, M > L +1 is always assumed in the remaining part of this correspondence. Let

P

where

~s = [~s0 ; s~2 ; . . . ; s~M 01 ]T g() = xH 8PA8H x

P

(9)

AA A A A h

denotes the cost function of , and A = ( H )01 H represents the projection matrix onto the subspace spanned by . Equations (8) and (7) are the ML estimator for  and that have been proposed in [6] and [8].

represent the last M symbols in the preamble, and define the M as normalized DFT matrix

W

W = [ w ; . . . ; wM ] whose (p; q )th entry is given by

A

P

h

P

[W]pq = p1 e0j2(p01)(q01)=M :

(14)

s = W~s = [s ; s ; . . . sM 0 ]T

(15)

M

Then, the vector

A

h

2M (13)

1

III. IDENTIFIABILITY ON CFO ESTIMATION It was pointed out in [6] that if matrix is square and nonsingular, then A is an identity matrix, and g () is independent from . Consequently, maximizing g () becomes meaningless, and  cannot be estimated. Furthermore, (7) cannot give a correct estimation of channel vector . Intuitively, when is square and nonsingular, M is equal to L + 1. Then, there are L + 1 equations, no matter independent or not, but L + 2 unknown parameters to estimate. These L + 2 parameters correspond to L + 1 channel taps and the frequency offset. Then, no solution is possible since there are more unknowns. Therefore, a necessary condition for a joint estimation of  and is M > L + 1. However, even if M > L + 1, the identifiability problem may still exist. This issue has not been presented in [6]–[9] and all other CFO related literatures. Theorem 1: The cost function g () is independent of  if and only if the (p; q )th entry of A takes the value as

(12)

0

2

1

is considered as the virtual training sequence on M subchannels. Training sequences fs~0L ; . . . ; s~M 01 g that satisfy (11) are concluded as follows. Criterion 1 [3]: Training sequences satisfy (11) if and only if • fs~0L ; . . . ; s~01 g is zero sequence; • fs0 ; . . . ; sM 01 g satisfies M 01 s e0j 2(k(m01)=M ) = 0 m=0 m M 01 s 2 e0j 2(k(m01)=M ) = 0 m=0 m

j j

; for k = 1; 2; . . . ; L: (16)

= q = ki ; i = 1; . . . ; L + 1 (10) [PA ]pq = 10; potherwise Criterion 1 is claimed to be optimal since it does not assign power to the first L and last L symbols in the preamble; namely, +1 i gLi=1 where the index set K1 = fk contains the position of ones on fs~0L ; . . . ; s~01 g, and fs~M 0L ; . . . ; s~M 01 g are all zeros. Therefore,

P

the diagonal of A . Proof: The sufficient part of the proof is obvious. The necessary part can be proved as follows. Since 8 is a diagonal matrix, it is easily known that the only case for g () to be independent of  is when A becomes a diagonal matrix. Recalling that A is a projection matrix, the diagonal element of A should be either 1 or 0. Since rank( ) is

P

P

P A

for a fixed power P , it can achieve the best estimation result. Note, however, M 0 L > L + 1 is required for a joint estimation of CFO and CIR. A suboptimal choice arises from the popular OFDM based transmissions, where the first L symbols in the preamble are used as cyclic prefix (CP) whose power is generally not considered for optimal

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 9, SEPTEMBER 2006

3655

training design [4] and is discarded at the receiver. In this case, P only denotes the power of ~ or, equivalently, the power of fs~0 ; . . . ; s~M 01 g. So, the optimal training design criterion should be modified and is cited as follows. Criterion 2 [4], [11]: For CP-based transmissions, training sequences satisfy (11) if and only if

s

M 01 m=0

jsm j e0j  k m0 2

2 ( (

1)=M )

= 0;

= 1; 2; . . . ; L:

for k

(17)

s

Theorem 2: For all training sequences that satisfy (11), the cost function g () is independent of  if and only if is a scalar multiple . of one column of Proof: See Appendix. Several remarks are made for Theorem 2 as follows. Remark 1: It can be readily checked that the first M 0 L columns of are included in criterion 1, and all columns of are included in criterion 2. Therefore, they may be used as the optimal preamble for channel estimation under different transmission modes. However, when CFO also need to be estimated, as is usually the case in most of practical communication systems, this set of training sequences should be avoided. Remark 2: If is taken from the mth column of , the time-domain training sequence is all zero except at the position s~m01 . In this case, the received signal becomes

W

W

W

W

s

x = pP8[0; . . . ; 0; h ; . . . ; h ; 0; . . . ; 0] : (18) Obviously, there is no way to separate either  from h or h from . Practically, one may easily realize that this set of training sequences T

L

0

could cause an identifiability problem. However, the merit of Theorem 2 lies in its necessary part; namely, it proves that only those optimal training sequences obtained from columns of could cause an identifiability problem, and all other optimal training sequences obtained from (11) can always be safely used. Remark 3: From [3] and [11], we know that a properly designed periodic training sequences fs~0L ; . . . ; s~01 ; s~0 ; . . . ; s~M 01 g are also optimal purely for channel estimation. In this case, several training symbols on virtual subchannels are zero; thus, the CFO is identifiable according to Theorem 2. However, as pointed out in [6], periodic training will reduce the effective CFO region by a factor of N , where N is the number of symbols in one periodic sequence. As a result, only 0 2 [00:5=N; 0:5=N ] can be uniquely estimated.

W

IV. SIMULATION RESULTS In this section, we present computer simulations to verify the proposed studies. In all numerical examples, a three-ray channel model with exponential power delay profile

E jh(l)j2

= exp(0l=10);

l = 0 ; 1; 2

(19)

is used [5]. The phase of each channel ray is uniformly distributed over

[0; 2). The parameter M is taken as 16, and the data transmitted are

modulated by 16 phase-shift keying (16PSK). All numerical results are averaged over M = 200 Monte Carlo runs. The normalized estimation mean-square error (NMSE) of CFO is defined as NMSE(CFO) =

1 M (^i 0 0 )2

Mi

=1

02

(20)

where the subscript i refers to the ith simulation run. A. Example 1

s

s

[3], we know that from the first and eighth columns is deemed as optimal for channel estimation in any transmission mode, and from all these three columns are optimal for OFDM transmissions [11]. However, according to Theorem 2, these three training sequences fail to give a correct estimation of CFO. We consider two ML data-aided CFO estimators. For the estimator in [6], a 1024-point FFT is used in order to achieve a CFO resolution upper bounded by 0.001. For the adaptive method in [8], the initial point is taken as 0.33, which is quite close to the true CFO value. The NMSEs of CFO estimation versus signal-to-noise ratio (SNR) results for both algorithms are shown in Fig. 1. One can see from the simulation results that both these two algorithms fail to yield correct estimation for all three , which is compatible with Theorem 2. However, it seems that the adaptive method gives much better estimation than the FFT-based method. The reason could be explained as follows. Since the initial point of adaptive method is already taken closer to the true CFO value, even if the algorithm fails, it still works better than the FFT-based method. However, the performance of the adaptive algorithm does not change with SNR, which clearly shows the failure of the algorithm.

s

B. Example 2 In the second example, we consider a partially loaded data on virtual subchannels. Four symbols with the same value are loaded at the virtual subchannel {1,5,9,13}, respectively. It can be readily checked that this set of training data satisfy criterion 1, thus is considered optimal purely for channel estimation. According to Theorem 2, this selection of training sequence can identify CFO, but the effective estimation region is reduced by a factor of four. Therefore, the normalized CFO in this example is taken as small as  = 0:1. Again, a 1024-point FFT is used. For the adaptive method, the initial point is taken as 0.13 and 0.16, respectively. The performance NMSEs of CFO versus SNR are shown in Fig. 2, respectively. The joint estimation Cramér–Rao bounds (CRBs) [9] are also displayed in these two figures. It can be seen that the FFT-based method gives satisfactory results. However, for the adaptive method, the initial point is so crucial that even 0.03 difference in the starting point could totally fail the algorithm. V. CONCLUSION

s

In the first example, the normalized CFO is taken as  = 0:3, and from the first, eighth and fifteenth columns of are examined. From

W

Fig. 1. CFO failure due to the wrong selection of training sequence.

Data-aided CFO estimation has received much more attention during the past decade. Although the ML estimator of CFO has already been developed, another important study regarding the identifiability of the

3656

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 9, SEPTEMBER 2006

where JA is the M 2M matrix that permute the rows of A with indexes in K1 to the first L + 1 rows, and U is some (L + 1) 2 (L + 1) unitary matrix. Combining (21) and (23), we obtain

p

M WH SWL =

p

P JAT

UT ; 0(L+1)2(M 0L01)

T

:

(24)

The following equation can be derived from (24) H SW = 0 L W L (M 0L01)2(L+1)

(25)

where

 L = wk ; wk ; . . . ; wk W

(26)

contains columns of W whose column indexes fall in K0 . Suppose P  M elements in s with indexes in Ks = fi gP i=1 being nonzero. Define

Fig. 2. NMSE of CFO for partially loaded training.

wm = [[W] m ; [W] m ; . . . ; [W] m ]T B = [w 1 ; w 2 ; . . . ; w M ] s = [s ; s ; . . . ; s ]T :

algorithm has never been reported. In this correspondence, we address this issue and provide two theorems according to different groups of training sequences. The simulation results clearly validate the proposed analysis.

(27) (28) (29)

Equation (25) can be rewritten as APPENDIX PROOF: OF THEOREM 2 For training sequences obtained from criterion 1, since the first L and the last L symbols are zero, A is a column-wise circulant matrix. For training sequences obtained from criterion 2, due to the existence and the removal of CP, matrix A is also a column-wise circulant matrix. Therefore, A from both criteria can be rewritten in the following form:

A=

p

M WH SW

L

H SW = B L H W L L SBL H =B L [D1 s; D2 s; . . . ; DL+1 s] = 0(M 0L01)2(L+1)

where

S = diagfsg; Dl = diagfwl g; l = 1; . . . ; L + 1 BL = [w1 ; w2 ; . . . ; wL+1 ]

(21)

where WL contains the first L + 1 columns of W, and S is a diagonal matrix with the form S = diagfsg. Note that, if training sequences are not chosen from either criterion 1 or criterion 2, A is only a Toepliz matrix, and (21) does not hold.

 L = wk ; wk ; . . . ; wk B

From (11), we know that the rank of A is L + 1. Consequently, there exist exactly L +1 ones and M 0 L 0 1 zeros on the diagonal of PA . Let i gL+1 the indexes of these ones and zeros belong to the set K1 = fk i=1 M 0 L 0 1 and K0 = fki gi=1 with K1 K0 = f1; . . . ; M g. Note that K0 is guaranteed such that it is not an empty set because M > L + 1. Under the assumption of (11), PA can be rewritten as

PA =

1

P AA

H:

(22)

If (10) holds, then

JA A =

p

P

UT ; 0(L+1)2(M 0L01)

T

(23)

(31) (32) (33)

:

(34)

It can be readily checked that

H B L Dl = A. Necessary Condition

(30)

p1

M

w(k 0l+1)modM ; w(k 0l+1)modM ; . . . ; w[k

0l+1]modM

H

: (35)

H The term +1 resides in the index because for l = 1, columns in B L Dl does not change index numbers. Therefore, if s is a solution for (24), H it must lie in the null space spanned by all B L Dl or equivalently, the subspace spanned by the all vectors wk and their L circular shifts. To be compatible with column index set f1; . . . ; M g that does not include 0, the modular operation used here is a little bit different from the traditional one, where (cM mod M ) = M with c denoting an arbitrary integer. Case 1: ki are circularly continuous integers, and P = M . Without loss of generality, we assume that ki = (k1 + i 0 1)mod M . One can obtain from (30) and (35) that

wiHmodM s = 0

(36)

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 9, SEPTEMBER 2006

In summary, the only possible case for PA to be in the form of (10), assuming (11), is when s is obtained as the multiple of any column of W.

for

ic

= (k1

0 L)modM; . . . ; (k

1

+M

0 L 0 2)modM:

Since ic covers M 0 1 circularly continuous integers in f1; . . . ; M g, the only nonzero solution for (36) is that s = w(k +M 0L01)modM with pbeing any complex number. Considering the power constraint,

= P. p As k1 may vary from 1 to M , P wm , m = 1; . . . ; M are all possible solutions p for (30) or (25). One can further check that, if s is obtained as P wm , (11) and (24) hold. Case 2: ki are circularly continuous integers, and P < M . Similarly as (36), the following equation can be obtained from (30) and (35):

wH i modM s = 0;

3657

B. Sufficient Condition If s is the multiple of wm , m = 1; . . . ; M , then

p

s=

P wm :

(41)

Consequently

PA = A(AH A)01 AH M H H H = P HW SWHL WL S W ~ LW ~ W; =W W L

(37)

(42)

where

for

ic

= ( k1

0 L)modM; . . . ; (k

1

+M

0 L 0 2)modM

~ L= W

M

P SWL= wm ; w m (

M ; . . . ; w(m+L)modM : (43)

+1)mod

Obviously, PA is a diagonal matrix with L)modM g.

or equivalently

1)modM; . . . ; (m +

GH s = 0

(39)

for

i = 1; . . . ; M

0 L 0 1;

l = 1; . . . ; L + 1:

Since the distance between all adjacent ki is smaller or equal to L, the set (ki 0 l + 1)modM will cover all integers in f1; . . . ; M g. It follows that

B H s = 0:

1

=

fm; (m

+

(38)

where G is obtained from B by deleting its [(k1 + M 0 L 0 1)mod M ]th column. Clearly, G is a P 2 (M 0 1) submatrix from W. From [12], we get the following lemmas. Lemma 1: Any (M 0 1) 2 (M 0 1) submatrix extracted from W is nonsingular. Lemma 2: Any P 2 (M 0 1) submatrix extracted from W with P  (M 0 1) is of full row rank. From Lemma 2, we know G is full row rank. Then, no trivial solution of s exists for (38). Case 3: ki are not circularly continuous. Without loss of generality, suppose k1 < k2 <; . . . ; < k(M 0L01) . In this case, the “distance” between adjacent ki and ki+1 , defined as the number of integers between ki and ki+1 (excluding ki and ki+1 ), must vary from 0 to L. We know from (30) and (35) that s will be orthogonal to L circular shifts of all wk , that is

wH (k 0l+1)modM s = 0

K

(40)

Since B is a P 2 M matrix obtained from W, it is obviously a full row rank matrix. Then, no trivial solution of s exists for (40).

REFERENCES [1] Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High Speed Physical Layer in the 5 GHZ Band, IEEE 802.11a, 1999. [2] J. Terry and J. Heiskala, OFDM Wireless LANs: A Theoretical and Practical Guide. Indianapolis, IN: Sams, 2001. [3] J. H. Manton, “Optimal training sequences and pilot tones for OFDM systems,” IEEE Commun. Lett., vol. 5, no. 4, pp. 151–153, Apr. 2001. [4] Z. Cheng and D. Dahlhaus, “Time versus frequency domain channel estimation for OFDM systems with antenna arrays,” in Proc. IEEE Int. Conf. Signal Processing (ICSP), Beijing, China, Aug. 2002, vol. 2, pp. 1340–1343. [5] M. Morelli and U. Mengali, “A comparison of pilot-aided channel estimation methods for OFDM systems,” IEEE Trans. Signal Process., vol. 49, no. 12, pp. 3065–3073, Dec. 2001. [6] ——, “Carrier-frequency estimation for transmissions over selective channels,” IEEE Trans. Commun., vol. 48, no. 9, pp. 1580–1589, Sep. 2000. [7] T. Cui and C. Tellambura, “Joint channel and frequency offset estimation and training sequence design for MIMO systems over frequency selective channels,” in Proc. IEEE GLOBECOM’04, Dallas, TX, Nov. 2004, vol. 4, pp. 2344–2348. [8] X. Ma, H. Kobayashi, and S. C. Schwartz, “Joint frequency offset and channel estimation for OFDM,” in Proc. GLOBECOM’03, San Fransisco, CA, Dec. 2003, vol. 1, pp. 15–19. [9] P. Stoica and O. Besson, “Training sequence design for frequency offset and frequency-selective channel estimation,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1910–1917, Nov. 2003. [10] X. Ma, C. Tepedelenlioglu, G. B. Giannakis, and S. Barbarossa, “Nondata-aided carrier offset estimators for OFDM with null subcarriers: Identifiability, algorithms, and performance,” IEEE. Trans. Commun., vol. 19, no. 12, pp. 2504–2515, Dec. 2001. [11] S. Ohno 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. [12] D. A. Harville, Matrix Algebra From a Statistician’s Perspective. New York: Springer-Verlag, 1997.