Optimal superimposed training design for

() 3206 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 Optimal Superimposed Training Design for Spatially Correlated Fading MIMO Channels Vu Nguyen, Hoang D Tuan, Member, IEEE,[.]

Optimal Superimposed Training Design for

Spatially Correlated Fading MIMO Channels

Vu Nguyen, Hoang D Tuan, Member, IEEE, Ha H Nguyen, Senior Member, IEEE

and Nguyen N Tran, Student Member, IEEE

Abstract—The problem of channel estimation for spatially

cor-related fading multiple-input multiple-output (MIMO) systems

is considered Based on the channel’s second order statistic, the

minimum mean-square error (MMSE) channel estimator that

works with the superimposed training signal is first developed.

The problem of designing the optimal superimposed signal

is then addressed and solved with an iterative optimization

algorithm Results show that under the constraint of equal

training power and bandwidth efficiency, our optimal design of

the superimposed training signal leads to a significant reduction

in channel estimation error when compared to the conventional

design of multiplexing training, especially for slowly

time-varying channels with a large coherence time The issue of power

allocation between the information-bearing and training signals

for detection enhancement is also investigated Simulation results

demonstrate excellent bit-error-rate performance of orthogonal

space-time block codes with our proposed channel estimation.

Index Terms—MIMO channel, spatial correlation, channel

estimation, MMSE estimation, training signal, training design,

time-multiplexing training, superimposed training.

I INTRODUCTION

THE use of multiple antennas at both the transmitter

and the receiver to create the so-called multiple-input

multiple-output (MIMO) communication systems has been

shown to greatly increase the data rate of the wireless

trans-mission medium [21], [30] This is especially true when the

channel fades among the transmitter-receiver pairs are

inde-pendently Rayleigh distributed [5], [30], [38] In particular, it

is shown in [30] that the capacity of a MIMO wireless channel

increases linearly with the number of antennas

The assumption of independent fades requires that the

antennas be placed sufficiently far apart, both at the transmitter

and the receiver In many practical applications, meeting such

requirements might be very expensive and impractical (such

as for the antennas in hand-held mobile units) It is therefore

more practical and useful to consider spatial correlations

Manuscript received March 2, 2007; revised August 1, 2007; accepted

October 1, 2007 The associate editor coordinating the review of this paper

and approving it for publication is D Dardari This work is supported by the

Australian Research Council under grant ARC Discovery Project 0556174 A

part of this work was presented at the IEEE Second International Workshop on

Computational Advances in Multi-Sensor Adaptive Processing, St Thomas,

U.S Virgin Islands, USA, 12-14 December 2007.

Vu Nguyen, Hoang D Tuan, and Nguyen N Tran are with the

School of Electrical Engineering and Telecommunications, the University

of New South Wales, Sydney, NSW 2052, Australia (e-mail: {q.nguyen,

nam.nguyen}@student.unsw.edu.au, h.d.tuan@unsw.edu.au).

Ha H Nguyen is with the Department of Electrical and Computer

Engi-neering, University of Saskatchewan, 57 Campus Dr., Saskatoon, SK, Canada

S7N 5A9 (e-mail: ha.nguyen@usask.ca).

Digital Object Identifier 10.1109/TWC.2008.070250.

among different sub-channels of the MIMO channel matrix [5], [13], [28] Compared to an independent fading MIMO channel, the results in [3], [10]–[12], [28] show that the capacity of a spatially-correlated fading MIMO channel is substantially reduced

Capacity reduction due to spatially-correlated fading can be partially alleviated by precoding the transmitted signal [20] This technique however requires the knowledge of the channel state information at the transmitter, which is not always available Furthermore, the MIMO channel capacity can be further reduced if inaccurate channel state information is obtained at the receiver [30] In other words, accurate channel estimation is very important to fully exploit the advantages

of MIMO wireless communications The correlated fading channel is often estimated by a training sequence, which can be either time-multiplexing (TM) training (see e.g., [26], [32] for single-input multiple-output (MISO) channels and [7], [19] for MIMO channels), frequency-multiplexing [14], [18] or superimposed (SP) training (see e.g., [17], [36] for single-input single-output (SISO) channels and [33] for MISO channels) In superimposed traning, the training symbols are superimposed on the precoded data for transmission In fact, superimposed traning includes both time-multiplexing and frequency-multiplexing as special cases, which correspond

to sending the non-zero training symbols when the data symbols are zero or sending the non-zero training symbols over subcarriers that are not occupied by the data symbols (i.e., pilot subcarriers) Because superimposed training is a general and powerful framework, it has recently received a growing interest in the research community [14], [15], [18], [33]

In SP training, since the received signal is a superposition

of the data-bearing signal, training signal and noise, a popular design approach is to decouple channel and symbol estimation [14], [18], [22], [37] This can be done by designing the precoding and training matrices so that the data-bearing and training signals belong to complementary signal subspaces Then, the data-bearing signal, which is considered as the unwanted noise in channel estimation, can be completely removed and channel estimation is carried out based on the training symbols An alternative approach is to perform joint channel/symbol estimation at the receiver and design the training signal accordingly Due to its more complicated processing and marginal advantage [37], joint channel/symbol estimation is less preferred than decoupled channel and sym-bol estimation This paper, therefore, is only concerned with

1536-1276/08$25.00 c  2008 IEEE

decoupled channel and symbol estimation approach.

For the estimation of independent Rayleigh fading channels,

the works in [14], [18], [37] conventionally treat the received

signal along time, i.e., as a concatenation of the columns of

the received signal matrix The subspaces of the data-bearing

and training signals thus depend on the unknown channel

matrix Consequently, some special structures on both the

precoding and training matrices are imposed This together

with several complex arrangements make these matrices

com-mutative with the channel matrix and hence, decouple the

two subspaces of the data-bearing and training signals A

novel approach has also been recently developed in [22], [25],

[31], where the received signal is viewed along space, i.e.,

as a concatenation of the rows of the received signal matrix

The most important consequence of this view is that the two

subspaces of interest are independent of the channel matrix

This means that, if designed properly, the orthogonality of the

precoding and training matrices already guarantees subspace

complementarity Therefore there is much more freedom in the

optimal design of these matrices Indeed, the results in [22],

[31] demonstrate that this design approach is superior than the

designs in [14], [18], [37] in terms of estimation performance,

symbol detectability and computational complexity

This paper adopts the approach in [22], [31] to design

the optimal superimposed training signal for the channel

estimation of correlated block-fading MIMO channels For

uncorrelated block-fading MIMO channels, precoding is well

known to be useless [4], [6], [16], [18] For these MIMO

channels, the optimal TM training and superimposed training

can be easily seen to be the same with a scaled identity matrix

as the optimal training matrix [24], [25] However, as pointed

out in [9], the design of training signals for correlated fading

MIMO channels is quite challenging in general This is due

to the large number of channel parameters involved and the

complex nature of the correlated channel coefficients In fact,

while the design of TM training signal is quite straightforward

for independent fading channels [8], it is still a difficult

problem for the correlated fading channels [9] In particular,

sub-optimal TM training signals for spatially-correlated fading

channels were proposed in [9] for only two extreme cases

of low and high signal-to-noise ratios (SNRs) It is still not

clear what is the optimal TM training signal for a given SNR

level Similarly, while the design of the optimal superimposed

training signal is well understood for independent fading

chan-nels [14], [22], [31], [37], the design for spatially-correlated

fading channels at any SNR level has not been addressed This

paper solves this challenging design problem by developing an

efficient iterative optimization algorithm to find the solution

Results show that the proposed design of the superimposed

training signal performs much better than the TM

training-based estimation considered in [9]

The remaining of this paper is organized as follows Section

II introduces the system model and describes the design

prob-lem The optimal superimposed training design is presented

in Section III with an iterative optimization algorithm The

issue of optimal power allocation for the training signal is

addressed in Section IV Section V provides numerical results

to illustrate the advantages of the proposed design over the

existing ones Section VI concludes the paper

Notation: Boldface upper (lower) letters denote matrices (column vectors) The operation vec(·) means matrix vector-ization which forms a column vector by vertically stacking the columns of a matrix For a matrix A, its transposition, Hermitian adjoint and trace are denoted by AT, AH and trace(A), respectively IN is the identity matrix of sizeN ×N ,

0N ×M is theN × M zero matrix and (∗)N ×M stands for any matrix of sizeN × M UN andDN denote the sets ofN × N unitary matrices and diagonal matrices, respectively For two Hermitian matrices X and Y, X≤ Y means that Y − X is positive semi-definite Similarly, X < Y implies that Y−X is positive definite The symbol ⊗ is used for Kronecker matrix product, while E(A) is the expectation of random matrix A For any x, define (x)+= max(x, 0)

Furthermore, some properties of Kronecker product trans-formations and positive definite matrices used in this paper are as follows:

(P2) (A ⊗ B)H = AH⊗ BH

(P3) (A ⊗ B)−1= A−1⊗ B−1

(P4) trace(A ⊗ B) = trace(A)trace(B).

(P6) If UN ∈ UN and UM ∈ UM, then UM ⊗ UN ∈ UN M

(P7) If 0 < X < Y, then trace(X) < trace(Y) and trace(X−1) > trace(Y−1)

(P8) If X≥ 0, then X ⊗ IN ≥ 0, ∀N

(P9) If 0 < X ∈ CN ×N, then trace(X−1) ≥

N



i=1

1/X(i, i)

II SYSTEMMODEL

Consider a narrowband frequency-flat block-fading MIMO channel withN transmit and M receive antennas (see Fig 1) The information-bearing symbols are grouped into blocks of size Ns, namely s(k) = [s(kNs), s(kNs+ 1), , s(kNs+

Ns− 1)]T, wherek denotes the block index Then each block s(k) is encoded and/or multiplexed in space and time, which

is generally represented by a block labeled with space-time coding (STC) in Fig 1 Thus the system under consideration can accommodate any specific space-time schemes such as the Alamouti’s orthogonal space-time block codes or the BLAST-type schemes [27] The output of the space-time encoder consists ofN vectors, xi∈ CK×1, i = 1, , N , each having lengthK (K ≥ N ) symbols The information-bearing signal can therefore be represented by the following matrix:

X= [x1, x2, , xN]T ∈ CN ×K (1) Before directed to the transmit antennas, the signal matrix X

is first precoded by post-multiplying with a precoding matrix

P = [p1, p2, , pK]T ∈ CK×(K+L), where L ≥ N , to produce the following precoded signal matrix:

D:=

⎡

⎢

dT 1

dTN

⎤

⎥

⎦ = XP =

⎡

⎢

xT

1P

xTNP

⎤

⎥

⎦ ∈ CN ×(K+L) (2)

Here, L represents the number of redundant vectors resulted

by precoding the transmitted signal In general, it is desir-able to have L as small as possible in order to improve

Space-time coding

(STC)

1

x

2

x

N

x

Precoding by post-multiplying with matrix

P

1

d

1

c

2

d

2

c

N

d

N

c

Superimposed

Ant-2

Ant-N

P/S

P/S

P/S

Space-Time Decoding Ant-1

Ant-M

S/P

Ant-2

S/P

S/P

Channel Estimator

Hˆ

(a) Transmitter

(b) Receiver

Information

symbols

Decoded symbols

Decoupling by post-multiplying

with matrix Q

1

y

2

y

M

y

Fig 1 An equivalent discrete-time baseband MIMO system.

the transmission efficiency Next, a training matrix C =

[c1, c2, , cN]T ∈ CN ×(K+L)is added (i.e., superimposed)

to the precoded matrix D Finally, the nth row of D + C,

namely dT + cT is serially transmitted over thenth transmit

antenna, n = 1, , N It should be pointed out that the

above superimposed training is performed after the

space-time encoder Therefore our training design is flexible and

can accommodate any space-time code

Assume that the fading channel remains constant during

every block of(K + L) symbols, but changes independently

from block to block Typically, this assumption implies that the

exact statistical behavior of the correlation in time is neither

available nor exploited, but only the coherence time is used

to determine the block length In practical systems, the speed

of mobility and the transmission rate determine the coherence

time (in units of information symbols) The coherence time in

turn dictates the block length [27]1 With this assumption and

since transmission and reception are conducted on a

block-by-block basis, the time index is omitted for convenience

Let H be theM × N MIMO channel matrix in an arbitrary

transmission block To reflect spatially-correlated fading, the

channel matrix is represented as follows [9]:

where Σr and ΣtareM × M and N × N known covariance

matrices that capture the correlations of the transmit and

1 For example, in a typical cellular system operating at a carrier frequency

of f c = 1.9 GHz, a mobile speed of v = 36 km/h translates to a coherence

time of about T c = c/(8f c v) ≈ 1.97 ms, where c = 3 × 10 8 m/s is the

speed of light If the data rate is 250 kbps and a 16-QAM constellation is

used, then the block length would be about 123 QAM symbols.

receive antenna arrays, respectively The matrix Hw is an

M × N matrix whose entries are independent and identically-distributed (i.i.d.) circularly symmetric complex Gaussian ran-dom variables of unit variance, i.e., CN (0, 1) In particular E[vec(Hw)vecH(Hw)] = IMN The known matrices Σr and

Σthave the following forms:

Σr=

⎡

⎢

⎢

⎣

r∗

r∗ 1M r∗

⎤

⎥

⎥

⎦,

(4)

Σt=

⎡

⎢

⎢

⎣

1 t12 · · · t1M

t∗

. .

t∗ 1M t∗

⎤

⎥

⎥

⎦,

(5)

wheretij (rnm, resp.) withi = j (n = m, resp.) reflects the correlated fading between the ith and the jth (nth and mth, resp.) elements of the transmit (receive, resp.) antenna array The elements of Σrand Σtcan be specified, for example, by using the one-ring model in [28] The covariance matrix of the overall channel matrix H can be easily shown to be

R= E[vec(H)vecH(H)] = Σt⊗ Σr

At the receiver, the received signal matrix is given as:

where N ∈ CM×(K+L) is the matrix of additive white Gaussian noise (AWGN) samples Furthermore, the following assumptions are made for the input/output channel model in (6):

(A1) The information-bearing symbols are independent,

zero-mean and with variance σ2

x, i.e., E(XXH) = Kσ2

xIN Note that this assumption is valid if X is obtained

by simply multiplexing the information symbol block s(k) in space and time When X is obtained by space-time encoding the information block s(k), the correlation matrix E(XXH) generally admits a different form Nev-ertheless, the technique presented in this paper can be easily extended to cover any other form of E(XXH)

(A2) The AWGN samples are also independent, zero-mean and

with variance σ2

n, i.e., E(NNH) = (K + L)σ2

nIM

(A3) The average transmitted power, including the powers of

the information-bearing and training signals, is normal-ized as σ2

x+ σ2

c = 1, where σ2

c = trace(CC

H

)

N (K+L) is the average power of the training signal

Moreover, the precoding matrix P is full rank and satisfies

The above constraint is to ensure that the average transmitted power of the information-bearing signal is unchanged after precoding Mathematically, this is verified as

σd2=trace

E(DDH)

N σ2

xtrace(PPH)

2

x

In [14], [18], [37], the signal matrix X is precoded as PX,

or equivalently, each column xi of X is precoded by Pxi

Then the information-bearing signal HPxi at the receiver

side belongs to a subspace governed by the unknown channel

matrix H Similarly, the training signal Hci, whereci is the

ith column of the training matrix C, belongs to a subspace that

can only be determined by knowing H It follows that it is not

easy to decouple these two unknown subspaces [14], [18], [37]

for convenient and effective channel estimation Consequently,

the optimal training matrix C cannot be readily derived

On the contrary, it can be seen that our precoded signals,

xTiP, i = 1, , N , belong to the subspace ΥP ⊂ CK+L

spanned by the rows pT

i, i = 1, , K, of the precoding matrix P Then the rows of the information-bearing part HXP

in the received signal matrix Y also belong to ΥP, which

is independent of the unknown channel matrix H Moreover,

the rows of the training part HC belongs to the subspace

ΥC ⊂ CN +L spanned by the rows of the training matrix

C, which is also independent of H Therefore, in order to

estimate the channel matrix H in an effective way, the received

signal matrix Y is post-multiplied with the decoupling matrix

Q = [q1, q2, , qK+L]T ∈ R(K+L)×N, which is chosen

such that

The matrix Q for channel estimation is also full rank and

satisfies QHQ = IN, which means that the noise is not

enhanced by the decoupling operation

Thus, the decoupled signal matrix for channel estimation is

expressed as

which is free of the unwanted component HX, and hence H

can be efficiently estimated

Although (8) is the most important relationship between the

precoding matrix P and the decoupling matrix Q for efficient

channel estimation, the precoding matrix P can be further

designed to improve the performance of symbol detection as

follows [31] First, choose the decoupling matrix for symbol

detectionas QD = PH

PPH

−1

, where P is also chosen such that CPH = 0 Then, by post-multiplying the received

signal matrix in (6) with QD, the decoupled signal matrix for

symbol detection can be expressed as

Letxi and qibe theith columns of the estimated data matrix



X and QD, respectively Under the minimum mean-square

error (MMSE) criterion for symbol detection, theith column

of X is recovered based on the channel estimate H and the

matrix YQD as follows [8]:



xi=



1

σ2

x

σ2

n||qi||2HHH

−1

1

σ2

n||qi||2HHYqi (11) The total mean-square error (MSE) of symbol detection can

be shown to be

εX(QD) =

K



tr



1

σ2 x

σ2

n||qi||2HHH

−1 (12)

Since QHQD = (PPH)−1 and due to the the power con-straint in (7), the problem of precoding matrix design can be stated as

min

tr{[Q H

Q D ] − 1 }=K+L

K



i=1

tr

σ2 x

σ2

n||qi||2HHH

−1 (13) The closed-form solution to the above optimization problem has been shown in [31] to have the following structure:

PPH = (QHDQD)−1= K + L

Similar to [31], based on (8) and (14), the matrices P and

Qare designed as follows:

K+L

K O(1 : K, :) ∈ CK×(K+L)

(K + 1) : (K + N ), : ∈ C(K+L)×N, (15) where O(1 : K, :) and O

(K + 1) : (K + N ), : keep only rows 1 toK and rows (K + 1) to (K + N ) of an orthogonal

matrix O∈ C(K+N )×(K+L) As an example, O can be formed

by keeping the first (K + N ) rows of an unitary UK+L ∈

UK+L, i.e.,

Now, the key issue is how to design the superimposed training matrix C that results in the best estimation of the channel matrix H based on the input/output model in (9) When H is uncorrelated, this design is quite straightforward and a closed-form solution for the optimal C can be easily derived [31] In contrast, due to the spatial correlations among channels of different transmit-receive antenna pairs, the design under consideration is quite complicated Though a closed-form solution is not yet available, the next section proposes an iterative optimization algorithm to effectively find the optimal solution

III OPTIMALDESIGN OF THESUPERIMPOSEDTRAINING

SIGNAL

Let PT = N (K + L)σ2

c According to (A3), the design

of the training matrix C is subject to the following power constraint:

Rewrite (9) as y= Ch+ n, where y = vec(YQ) ∈ CMN,

n= vec(NQ) = (QT⊗ IM)vec(N) ∈ CMN, C= (CQ)T⊗

IM ∈ CMN ×MN and h= vec(H) ∈ CMN Since QHQ=

IN, it follows that E[nnH] = (QHQ)T ⊗ σ2

nIM = σ2

nIMN Based on the received signal vector y, the linear minimum mean-square error (MMSE) estimation of the channel vector

his:

ˆ

h= R CH( CR CH+ σn2IMN)−1y, (18) where, recall that, R= Σt⊗ Σr Furthermore, the covariance matrix of the estimation error vector is

E: = E[(h − ˆh)(h − ˆh)H]

=

R−1+ CH(σ2

nIM N)−1C −

1

= (Σt⊗ Σr)−1+ 1

σ2CT H(QQH)TCT⊗ IM

−1

.(19)

The objective is to design C to further minimize (19) subject

the the power constraint in (17)

From (15), QQH admits the following singular value

decomposition (SVD):



IN 0N ×(K+L−N )

0(K+L−N )×N 0K+L−N

 When O is specified as in (16), UQcan be obtained by simply permuting

the rows of UK+L

Now, perform the following transformation:

¯

C= UT

One has trace( ¯C ¯CH) = trace(UT

QCTCT HCT H) = trace((CHC)T) = trace(CCH) Thus, under the power

constraint in (17), the optimal design of C ∈ CN ×(K+L),

that aims at minimizingE, can be formulated as the following

constrained optimization problem in ¯C∈ C(K+L)×N:

min

¯

C trace

(Σt⊗ Σr)−1+ 1

σ2 n

¯

CHΛ ¯C⊗ IM

−1

subject to trace( ¯C ¯CH) ≤ PT (22)

In what follows, the approach of matrix partition for

matrix inequalities [23] is employed to derive the

solu-tion to the above optimizasolu-tion problem Suppose that ¯C

is the optimal solution of Problem (22) Partition ¯C =



CU

CL



, CU ∈ CN ×N, CL∈ C(K+L−N )×N and define ¯C0=



CU

0(K+L−N )×N



It can be verified that ¯CHΛ ¯C= CH

UCU =

¯

CH0Λ ¯C0 On the other hand, whenever CL = 0, one has

PT = trace( ¯C ¯CH) = trace(CH

UCU) + trace(CH

LCL) >

trace(CH

UCU) = trace( ¯C0C¯H0 ) It then follows that,

when-ever CL = 0 there exists λ > 1 such that PT =

trace(λ2C¯

0C¯H

0 ) Now, applying properties (P7) and (P8)

yields

trace

(Σt⊗ Σr)−1+ 1

σ2 n

¯

CHoptΛ ¯Copt⊗ IM

−1

>

trace

(Σt⊗ Σr)−1+ 1

σ2 n

(λ ¯C0)HΛ(λ ¯C0) ⊗ IM

−1

(23) which contradicts with the assumption that ¯C is the optimal

solution of Problem (22)

The above result shows that the optimal solution of the

optimization problem in (22) must have the following form:

¯

Copt=



CU

0(K+L−N )×N



and the optimization problem in (22) is equivalent to the

following problem:

min

C U ∈C N ×N trace

(Σt⊗ Σr)−1+ 1

σ2 n

CHUCU⊗ IM

−1

Remark 1: The above optimization problem implies the

fol-lowing important consequence: As long as L ≥ N ,

perfor-mance of the channel estimation in terms of the mean-square

error does not depend on the actual value of L Thus, as far as channel estimation is concerned, choosing L = N

is optimal to maximize the system’s bandwidth efficiency It should be noted, however, that choosing L > N affects the precoding operation and might lead to a better performance with respect to a different criterion (such as the bit-error-rate (BER) performance, or the effective SNR considered in Section IV)

To find the solution to Problem (25), make the following SVDs:

Σt = UtΛtUHt , Ut∈ UN, Λt∈ DN,

Σr = UrΛrUHr , Ur∈ UM, Λr∈ DM Then, the objective in (25) can be evaluated as follows: trace

(UtΛ−1t UHt ) ⊗ (UrΛ−1r UHr) + 1

σ2 n

(CHUCH) ⊗ IM

−1

= trace

(Λt⊗ Λr)−1+ 1

σ2 n

Z⊗ IM

−1

where

Z= UHt CHUCUUt∈ CN ×N (27) The power constraint in (25) can also be expressed in terms

of the new variable Z as trace(CUCHH) = trace(Z) ≤ PT With the expressions of the objective and constraint in Z, the equivalent optimization problem is:

min

Z∈C N ×N trace

(Λt⊗ Λr)−1+ 1

σ2 n

Z⊗ IM

−1

subject to trace(Z) ≤ PT (28)

Using property (P9), we can easily see that

trace (Λt⊗ Λr)−1+ 1

σ2 n

Z⊗ IM

−1

≥

trace (Λt⊗ Λr)−1+ 1

σ2 n

diag[Z(i, i)]i=1, ,N⊗ IM

−1

and

trace(diag[Z(i, i)]i=1, ,N) = trace(Z)

Here diag[Z(i, i)]i=1, ,Nis the diagonal matrix with diagonal entries Z(i, i) This implies that the optimal solution of Prob-lem (28) must be diagonal Consequently, the optimization problem in (28) can be reformulated as follows:

min

0≤Λ C ∈D N

trace (Λt⊗ Λr)−1+ 1

σ2 n

ΛC⊗ IM

−1

subject to trace(ΛC) ≤ PT (29) The following theorem summarizes the optimal design problem, based on (21), (24), (27)

Theorem 1: The optimal solution ΛC,opt of Problem (29) provides the following optimal training signal:

Copt=

UHTt 

ΛC,opt 0N ×(K+L−N )

Remark 2: It is intuitively satisfying to observe that the precoding matrix P designed as in (15) and the optimal

training matrix Copt derived in (30) are orthogonal This can

be shown as follows:

CoptPH = 

UHTt 

ΛC,opt 0N ×(K+L−N )

UHQPH

UHTt 

ΛC,opt 0N ×(K+L−N )0N ×K

∗



The above implies that the components in the received signal

corresponding to the training signal (namely HCopt) and the

information-bearing signal (namely HXP) are guaranteed to

be orthogonal, regardless of the unknown channel matrix H

Unfortunately, a closed-form expression for the optimal

solution of Problem (29) is not available Nevertheless, the

following subsection provides an effective iterative algorithm

to solve Problem (29)

A Iterative Algorithm to Find the Optimal Solution of

Prob-lem (29)

For convenience, define

[δ1, δ2, , δMN]T

= [(Λt⊗ Λr)−1(1, 1), , (Λt⊗ Λr)−1(M N, M N )]T,

s = [s1, s2, , sN]T

Then Problem (29) can be equivalently re-expressed as

min

s∈R N

N



j=1

M



i=1

1

δ(j−1)N +i+σ12nsj

,

subject to

N



j=1

When there is only one receive antenna, i.e., M = 1,

the closed-form optimal solution of the above optimization

problem is well-known to have the water-filling structure (see

e.g [2]) The situation is quite different when M > 1 and

a closed-form solution is not expected Note that both the

objective and constraint functions of the above problem are

still convex in s In principle, the interior-point algorithms of

convex programming (see e.g [2]) can be applied However,

we shall exploit not only the convex structure of Problem

(33), but also its monotonic structure and provide an efficient

and fast computational algorithm to find its optimal solution

Undoubtedly, convexity and monotonicity are the most useful

properties in optimization [34], [35]

Specifically, for a decreasing function h(t) the nonlinear

scalar equation

can be solved online by the following iterative bisection

procedure (IBP):

• Ifh(t) < γ or h(¯t) > γ then there is no solution in [t, ¯t]

• Fort = (t + ¯t)/2 reset t = t if h(t) > γ and reset ¯t = t

ifh(t) < γ Repeat until h(t) = γ

Note that the objective functionf (s) in (33) is separable in

sj, i.e.,

f (s) =

N



j=1

fj(sj), where fj(sj) =

M



i=1

1

δ(j−1)N +i+σ1n2sj

Furthermore,fj(sj) is not only convex but also decreasing in

sj The Lagrangian of (33) is L(s, µ, µ1, , µN)

⎛

⎝

N



j=1

sj− PT

⎞

⎠ −

N



j=1

µjsj, µ ≥ 0, µj≥ 0

According to the Kuhn-Tucker condition for optimality of convex programming, the optimal solution of the optimization problem in (33) and the corresponding Lagrange multipliers must satisfy the following necessary and sufficient conditions:

∂L(s, µ, µ1, , µj)

∂sj + µ − µj= 0, whereµjsj = 0, j = 1, 2, , N

Therefore, the optimal solution of (33) can be expressed as

sj,opt= s+j(µ) := max{sj(µ), 0}, j = 1, , N, where:

• sj(µ) is the solution of the following nonlinear equation:

gj(sj) := −∂fj(sj)

∂sj

=

M



i=1

1

σ2 n

1 [δ(j−1)N +i+σ12nsj]2 = µ

(35) Thus for eachµ we can quickly locate s+j(µ) by the IBP described before Obviouslys+j(µ) is decreasing in µ

• The scalar Lagrange multiplierµ > 0 is such that g(µ) :=

N



j=1

sj,opt=

N



j=1

sj(µ) += PT (36)

The functiong(µ) is also decreasing in µ So again the IBP can be effectively used to locate the solution of (36)

To summarize, an effective procedure for locating the op-timal solution of the optimization problem in (33) is outlined below

• Computeµ and µ such that the solution of (36) belongs

to[µ, ¯µ]

• Apply IBP to locate the solution of Equation (36) The subroutines include (i) the computations of sj(µ) and ¯j(µ) such that the solution of (35) belongs to [sj(µ), ¯sj(µ)] and (ii) the application of IBP for locating the solutionssj(µ) of (35)

To make the above procedure completely realizable, the expressions of sj(µ), ¯sj(µ), µ and ¯µ are given next Define

δj,max= max

i=1,2, ,Mδ(j−1)N +i, (37)

δj,min= min

i=1,2, ,Mδ(j−1)N +i, j = 1, 2, , N (38)

It is obvious that

M

σ2

n(δj,max+ 1 sj)2 ≤ gj(sj) ≤ M

σ2

n(δj,min+ 1 sj)2 (39)

Hence, for a fixed µ, the solution sj(µ) of (35) belongs to

[sj(µ), ¯sj(µ)] with

sj(µ) := σ2n

M

µσ2 n

− δj,max

+

¯j(µ) := σ2n

M

µσ2 n

− δj,min

+

Also, it must be true thatµ ∈ [µ, ¯µ] for the solution of (36),

whereµ and ¯µ are the solutions of

N



j=1

and

N



j=1

respectively In other words, {sj(µ), µ} and {¯sj(¯µ), ¯µ} are

the water-filling structured optimal solutions and the

corre-sponding Lagrange multipliers of the following optimization

problems

min

s∈R N

N



j=1

M

δj,max+σ12nsj

:

N



j=1

sj≤ PT, sj≥ 0 (44) and

min

s∈R N

N



j=1

M

δj,min+ 1

σ 2nsj

:

N



j=1

sj≤ PT, sj≥ 0, (45)

respectively The technique to find these solutions is quite

standard and the details are omitted here

Until now, we have considered the problem of designing the

optimal training signal C under its fixed power constraint as

specified in (17) Under the total transmitted power constraint

stated in (A3), the following tradeoff arises The performance

of channel estimation can be improved by spending more

power for the training signal This, however, comes at the

expense of decreased transmitted power for the

information-bearing signal, leading to performance degradation of signal

detection This section considers this tradeoff problem and

proposes a sub-optimal power allocation that maximizes the

effective signal-to-noise ratio (SNR) The SNR is selected

because any increase in the effective SNR translates to an

increase in system capacity and/or quality of signal detection

Maximizing the effective SNR is also very helpful for

chan-nel decoding if error control coding is implemented in the

systems

With the optimal training matrix Copt derived in the

previ-ous section by (30), Equation (6) is rewritten as

Next, rewrite (10) to see the effect of channel estimation error

on data detection as follows:

where H is the estimated channel matrix and H = H − H

is the channel estimation error matrix Since H in (47)

is unknown and random and HX is uncorrelated to HX according to the orthogonality property [8], it is considered

as noise Thus, the effective SNR of the input/output model

in (47) is defined as

E

Here,

E

E HXXHHH

= Kσx2trace

E

HHH − E H HH

= Kσx2(M N − ǫ), whereǫ := traceE HHH , and

E

E  HXXHHH + trace

E

NQDQHDNH

xǫ + trace

E

NPH(PPH)−1(PPH)−HPNH

= Kσx2ǫ + M σn2 K

2

K + L.

It follows that SNReff =σ

2

x(M N − ǫ)

σ2

2 n

K

Note that ǫ is exactly the optimal value of the objective

function in Problem (29) As pointed out in Remark 1,ǫ does not depend on the actual value of L as long as L ≥ N On the other hand, it follows from (49) that the effective SNR increases withL, an intuitively satisfying result For simplicity and to maximize the system’s bandwidth efficiency, L = N

is chosen in the remaining of this paper

Furthermore, the following upper bound onǫ can be easily derived:

K + N

σ2 n

σ2 c

= βσ

2 n

σ2 c

This is because ΛC = PT

N IN = (K + N )σ2

cIN is a feasible solution of (29) Then, by replacing σ2

x with (1 − σ2

c), it is easy to see that SNReff in (49) is lower bounded as

SNReff(σc2) ≥(1 − σ

2

c)(M N σ2

c − βσ2

n)

βσ2

n− βσ2

nσ2

c+ γσ2 c

Instead of maximizing SNReff, we maximize its lower bound as given by the right-hand-side of (51) This maxi-mization leads to a sub-optimal power allocation as far as maximizing SNReff is concerned The sub-optimal solution of power allocation can be shown to be:

σ2 c,sub-opt =M N βσ

2

n−

M N γβσ2

n(−βσ2

n+ M N + γ)

M N (βσ2

(52) Since the total transmitted power is normalized to unity, the above expression essentially gives the fraction of the total power allocated to the training signal The above sub-optimal power allocation is used to obtain the simulation results presented in the next section

0 5 10 15 20

−25

−20

−15

−10

−5

0

SNR (dB)

PSPT ESPT TMT

K=10

K=60

Fig 2 Comparison of the mean-squared errors in channel estimation of

the 2 × 2 MIMO systems using different training signals: The proposed

superimposed training (PSPT), the equal-powered superimposed training

(ESPT) and the time-multiplexing training (TMT).

V ILLUSTRATIVERESULTS

This section provides simulation results to illustrate the

performance of the proposed optimal training design In all

simulations, the wireless channel model is assumed to be

quasi-static block Rayleigh fading and spatially correlated as

described in (3) The one-ring model in [28, E.q (6)] is used

to generate the elements of the covariance matrices Σr and

Σt Specifically,

Σt(n, m) ≈ J0



∆2π

λ dt|m − n|



Σr(i, j) ≈ J0

2π

λdr|i − j|



(54) where∆ is the angle spread in the one-ring model; dtanddr

are the spacings of the transmit and receive antenna arrays,

respectively; λ is the carrier wave-length and J0(·) is the

zeroth order Bessel function of the first kind Note that

the angle spread, ∆, and the antenna spacings, dt and dr,

determine how correlated the fading is at the transmit and

receive antenna arrays Unless stated otherwise, the values

of ∆ = 50, dt = 0.5λ and dr = 0.2λ are used in the

simulation to create highly correlated fading Since the average

transmitted power, including the training and data powers,

is normalized to unity as in assumption (A3), the received

SNR in dB is defined as SNR = −10log10σ2

n The power allocation for data and training signals in the proposed SP

training follows (52)

A Estimation Performance

Two different designs of superimposed training and one

conventional time-multiplexing training (TMT) design are

in-vestigated and compared The proposed superimposed training

(PSPT) signal is obtained with the iterative algorithm

de-scribed in Section III The equal-power superimposed training

(ESPT) signal is chosen as a scaled identity matrix with

power constraint P , which is the optimal training scheme

−14

−12

−10

−8

−6

−4

−2 0

SNR (dB)

PSPT ESPT TMT

K=10

K=60

Fig 3 Comparison of the mean-squared errors in channel estimation of the 4 × 4 MIMO systems using different training signals: The proposed superimposed training (PSPT), the equal-powered superimposed training (ESPT) and the time-multiplexing training (TMT).

for the uncorrelated MIMO channel The TMT design used

for comparison in this section is the improved version of the design proposed in [9] (which applies to the case of high or low SNR only [9, Subsection IV-C]) The improved design

is obtained by applying the iterative algorithm described in Section III to the optimization problem in [9, Equ (30)] As

in [9], the linear MMSE estimator is used for identifying the wireless channel Furthermore, the length of the TMT signal

is chosen to be the minimum length required for the channel estimation This minimum length is shown to be N symbols for a MIMO channel having N transmit antennas in [7] On the other hand, the use of precoding matrix P and decoupling matrix Q in this paper also introduces L = N redundant vectors per block Thus, the extra bandwidth consumption is the same for all the different training signals, namely the PSPT, ESPT and TMT signals Of course, the estimation performance

of different training designs is compared based on the same training power and additive Gaussian noise environment The normalized mean-square errors (normalized by E[||h||2]), expressed in dB, of the channel estimation provided

by the above three training signals are plotted versus SNR

in Figs 2 and 3 for the 2 × 2 and 4 × 4 MIMO channels, respectively In these two figures, two different lengths of the information vector X, namely K = 10 and K = 60, are considered It should be noted that the implementation of TMT

is independent of K as long as K ≥ N First, observe that

at almost any SNR level (except for SNR > 10 dB in the

2 × 2 system), the mean-square error is significantly reduced with the use of the PSPT signal instead of the ESPT A more important observation is that, compared to PSPT, using the TMT signal results in a larger MSE at any SNR level for both cases of MIMO channels and for both block lengths considered As expected, the advantage of the superimposed training (including PSPT and ESPT) over the TMT becomes more evident for the system with a larger block length In fact,

if the block length is not long enough, TMT can outperform ESPT as can be seen from Fig 3 for the 4 × 4 channel

0 5 10 15 20

−25

−20

−15

−10

−5

0

SNR (dB)

PSPT ESPT TMT

Fig 4 Comparison of the mean-squared errors in channel estimation of the

2 × 2 MIMO systems for different angle spreads (∆ = 15 0

and ∆ = 30 0

) and using different training signals (K = 60).

and when K = 10 This particular case clearly shows the

usefulness of our superimposed training design over the simple

ESPT

The difference in estimation performance between the PSPT

and the TMT depends mainly on their length ratio, which

is (K+N )N In general, the bigger the ratio (K+N )N is, the

larger the performance difference between PSPT and TMT

is, because the channel statistics is better incorporated for

estimation by superimposed training This can be clearly seen

from the results shown in Figs 2 and 3 for different values

ofK In practice, the value of K + N is determined by the

coherence time of the channel

In fact, when the coherence time is large, the estimation

performance of TMT can be improved by extending the

training length beyond the minimum required length of N

symbols [7], [9] However, a direct consequence of extending

the training length of TMT is a lower bandwidth efficiency

Therefore, taking both estimation performance and bandwidth

efficiency into account, PSPT is more attractive than TMT,

especially for a slowly time-varying wireless channel whose

coherence time can be very large

Figures 4 and 5 illustrate the impact of having larger

angle spreads, ∆ = 150 and ∆ = 300, on the estimation

performance of different training designs in both the 2 × 2

and4 × 4 systems Here K = 60 is considered Several

obser-vations can be made from these two figures First, all training

designs perform better when the angle spread increases This

is expected since a larger∆ makes the channel less correlated

Second, based on the performance difference between PSPT

and ESPT as well as the performance improvement when

going from ∆ = 150 to ∆ = 300, one concludes that the

2 × 2 MIMO channel can be considered spatially uncorrelated

for∆ ≥ 150, while∆ ≥ 300 makes the4 × 4 MIMO channel

uncorrelated Lastly, the PSPT scheme is seen to consistently

outperform the other two training schemes in these two figures

Next, the impact of antenna spacings on the estimation

performance is illustrated in Figs 6 and 7, where the

−12

−10

−8

−6

−4

−2 0 2 4

SNR (dB)

PSPT ESPT TMT

Fig 5 Comparison of the mean-squared errors in channel estimation of the

4 × 4 MIMO systems for different angle spreads (∆ = 15 0 and ∆ = 30 0 ) and using different training signals (K = 60).

−25

−20

−15

−10

−5

SNR (dB)

PSPT ESPT TMT

d

t=0.5λ, dr=0.2λ

d

t=0.2λ, dr=0.1λ

Fig 6 Comparison of the mean-squared errors in channel estimation of the 2 × 2 MIMO systems for different antenna spacings and using different training signals (K = 60).

squared errors of different training designs are plotted for two sets of antenna spacings, which are {dt = 0.5λ, dr = 0.2λ} and{dt= 0.2λ, dr= 0.1λ} These figures again confirm that all the training designs perform better in more correlated chan-nels as the consequence of having smaller antenna spacings And at any SNR level, the estimation performance of PSPT

is always the best for both MIMO channels

B Impact of Power Allocation

Fig 8 plots the average training power (as a fraction of the total power) computed as in (52), that maximizes the lower bound of SNReff when K = 10 and K = 60 It can be seen that a higher training power is needed for channel estimation

at a lower SNR level This is expected since the spatially correlated fading has a stronger effect on the quality of the channel estimation at the lower SNR level It is also evidenced from Fig 8 that a larger portion of the total power is spent

0 1 2 3 4 5 6 7 8

−16

−14

−12

−10

−8

−6

−4

−2

SNR (dB)

PSPT ESPT TMT

d

t=0.5λ, dr=0.2λ

d

t=0.2λ, dr=0.1λ

Fig 7 Comparison of the mean-squared errors in channel estimation of

the 4 × 4 MIMO systems for different antenna spacings and using different

training signals (K = 60).

0.1

0.15

0.2

0.25

0.3

0.35

0.4

SNR (dB)

4x4 MIMO, K=10 4x4 MIMO, K=60 2x2 MIMO, K=10 2x2 MIMO, K=60

Fig 8 Average training power that maximizes the lower bound of SNR eff

at different SNR levels.

for the training signal in the 4 × 4 MIMO system compared

to that in the2 × 2 MIMO system This is also expected since

there are more channel parameters to be estimated in the4 × 4

MIMO system than in the2 × 2 MIMO system Furthermore,

observe that the largerK is, the smaller the average training

power becomes This is also reasonable since with a largerK

the channel statistics is better incorporated for estimation by

superimposed training

The actual SNReff and its lower bound attained by the

proposed power allocation are plotted as functions of the

SNR in Fig 9 for the case of 2 × 2 MIMO system having

K = 60 Observe that the lower bound is very close to

the actual SNReff, which suggests the tightness of the lower

bound Moreover, shown in Fig 9 are plots of SNReff achieved

with several “ad-hoc” power allocation strategies It is obvious

that failing to allocate the training power as proposed in (52)

can significantly reduce SNReff

0 5 10 15 20

SNR (dB)

Proposed allocation Lower bound 40% training power 50% training power 60% training power

Fig 9 Plots of SNR eff and its lower bound for the MIMO systems with different power allocations (K = 60).

10−4

10−3

10−2

10−1

SNR (dB)

PSPT ESPT TMT Perfect

Fig 10 BER performance of the 2 × 2 MIMO system using full-rate Alamouti OSTBC and QPSK: Comparison of PSPT, ESPT, TMT and perfect channel estimation (K = 60).

C Bit-Error-Rate Performance

The final aspect to be investigated is the bit-error-rate (BER) performance of the MIMO systems that employ the proposed superimposed training design To this end, orthogonal space-time block codes (OSTBCs) together with the maximum likelihood (ML) decoding are incorporated in Fig 1 For the

2 × 2 system, the full-rate Alamouti code [1] is selected, whereas a half-rate OSTBC [29, E.q (5)] is applied for the

4 × 4 system Both systems use QPSK modulation with Gray mapping and the length of the information signal vector is set

toK = 60

The simulation results presented in Figs 10 and 11 were obtained with PSPT, ESPT, TMT and perfect channel estima-tion The BER performance with perfect channel estimation

is shown to serve as the performance benchmark Consistent with the relative comparison of estimation performance made before forK = 60, the BER performance with PSPT is better than that with ESPT and TMT in both the 2 × 2 and 4 × 4

training follows (52)

A Estimation Performance

Two different designs of superimposed training and one... using different training signals: The proposed superimposed training (PSPT), the equal-powered superimposed training (ESPT) and the time-multiplexing training (TMT).

for the uncorrelated... different training signals: The proposed

superimposed training (PSPT), the equal-powered superimposed training< /small>

(ESPT) and the time-multiplexing training