Int. J. Communications, Network and System Sciences, 2010, 3, 962-971
doi:10.4236/ijcns.2010.312131 Published Online December 2010 (http://www.SciRP.org/journal/ijcns)
Performance Improvement in Estimation of Spatially
Correlated Rician Fading MIMO Channels Using a New
LMMSE Estimator
Hamid Nooralizadeh1, Shahriar Shirvani Moghaddam2
1Electrical Engineering Department, Islamshahr Branch, Islamic Azad University, Islamshahr, Iran
2DCSP Research Laboratory, Department of Electrical and Computer Engineering,
Shahid Rajaee Teacher Training University, Tehran, Iran
Received August 18, 2010; revised September 23, 2010; accepted November 2, 2010
Abstract
In most of the previous researches on the multiple-input multiple-output (MIMO) channel estimation, the
fading model has been assumed to be Rayleigh distributed. However, the Rician fading model is suitable for
microcellular mobile systems or line of sight mode of WiMAX. In this paper, the training based channel es-
timation (TBCE) scheme in the spatially correlated Rician flat fading MIMO channels is investigated. First,
the least squares (LS) channel estimator is probed. Simulation results show that the Rice factor has no effect
on the performance of this estimator. Then, a new linear minimum mean square error (LMMSE) technique,
appropriate for Rician fading channels, is proposed. The optimal choice of training sequences with mean
square error (MSE) criteria is investigated for these estimators. Analytical and numerical results show that
the performance of proposed estimator in the Rician channel model compared with Rayleigh one is much
better. It is illustrated that when the channel Rice factor and/or the correlation coefficient increase, the per-
formance of the proposed estimator significantly improves.
Keywords: Channel Estimation, MIMO, Spatially Correlated Rician Fading, Optimal Training Sequences,
LS, Generalized LMMSE
1. Introduction
Due to high capacity and diversity gain, multiple input
multiple output (MIMO) systems have received consid-
erable attention in wireless communications. It has been
demonstrated that when the fades connecting pairs of
transmit and receive antenna elements are independent,
identically distributed (i.i.d.), the capacity of a Rayleigh
distributed flat fading channel increases almost linearly
with the minimum number of transmitter and receiver
antennas [1-3]. Moreover, in [3] it is indicated that Ri-
cian fading can improve the capacity of a multiple an-
tenna system, especially if the transmitter knows the
value of the Rice factor.
In order to attain the advantages of MIMO systems, it
is necessary that the receiver and/or transmitter have
access channel state information (CSI). One of the most
usual approaches to identify MIMO CSI is training based
channel estimation (TBCE). This class of estimation is
attractive especially when it decouples symbol detection
from channel estimation and thus simplifies the receiver
implementation and relaxes the required identification
conditions.
The optimal choice of training signals is usually inves-
tigated by minimizing mean square error (MSE) of the
linear MIMO channel estimator. In the literature, it is
perceived that optimal design of training sequences for
MIMO channel estimation is connected with the channel
statistical characteristics, e.g., fading model and the
channel noise model. For example, in [4], a sub-matrix
of the discrete Fourier transform (DFT) matrix has been
used to identify the Rayleigh distributed flat fading
MIMO channel. In [5-7], in order to estimate MIMO
inter symbol interference (ISI) channel, the delta se-
quence is used as optimal training. Further studies are
reported in [8-13] using optimal training and considering

H. NOORALIZADEH ET AL.
963
a few aspects, e.g., the peak to average power ratio
(PAPR) constraint on training sequences.
In [4,14-19] the spatially correlated fading MIMO
channel is considered. In [14], the frequency offset and
channel gain estimation is considered for MIMO ISI
correlated fading channels. In [15,16], it is investigated
that the impacts of spatial correlation are helpful not only
to improve channel estimation but also to decrease the
training length. It is noteworthy that the spatial correla-
tion harms channel capacity [20].
In [4], the performance of the least squares (LS),
scaled LS (SLS), minimum mean square error (MMSE),
and relaxed MMSE (RMMSE) estimators is studied in
the Rayleigh fading MIMO channel. The MMSE esti-
mator has the best performance among the estimators,
because it can employ more a priori knowledge about the
channel.
In most previous works on the MIMO channel estima-
tion, the channel fading is assumed to be Rayleigh dis-
tributed. Of course, the Rayleigh fading model is known
to be a reasonable assumption for fading encountered in
many wireless communications systems. However, Ri-
cian fading model is suitable for suburban areas where a
line of sight (LOS) path often exists. This may also be
true for microcellular or picocellular systems with cells
of less than several hundred meters in radius.
In [21], the TBCE scheme is investigated in MIMO
Rician flat fading channels. By the new method of
shifted scaled least squares (SSLS), it is shown that in-
creasing the channel Rice factor improves the perform-
ance of channel estimation. However, the SSLS channel
estimator is only appropriate for uncorrelated Rician
channels because this estimator cannot exploit the
knowledge of spatial correlation of the MIMO channels.
In uncorrelated fading, it is assumed that antenna ele-
ments are placed sufficiently apart. However, it is not
always realized in practice due to insufficient antenna
spacing when the channel estimation is used in compact
terminals. The linear MMSE (LMMSE) channel estima-
tor of [4] is appropriate for spatially correlated channels.
Nevertheless, this estimator cannot benefit from the Rice
factor of the Rician fading channels.
In this paper, a general form of the LMMSE channel
estimator is proposed that is appropriate for spatially
correlated Rician fading MIMO channels. We extend the
results of [4] in the Rayleigh fading model to the more
general Rician fading case. It is shown that this estimator
can exploit the knowledge of both spatial correlation and
Rician fading of the MIMO channels.
First, the traditional LS method is examined. It is dem-
onstrated that the performance of this estimator is inde-
pendent of the Rice factor. Then, the proposed MIMO
channel estimator is introduced; we refer to it as gener-
alized linear minimum mean square error (GLMMSE)
estimator. It is shown that in the spatially correlated Ri-
cian fading MIMO channel when the Rice factor and/or
the correlation coefficient increase, the accuracy of the
GLMMSE estimator improves. Note that the perform-
ance of the proposed estimator in the Rician fading
channels improves because:
1) We consider the effect of Rice factor and suggest a
new formulation comparing with other references as [4]
and [22].
2) We design the optimal training sequence appropri-
ate for spatially correlated Rician channel model.
It means that the optimal training sequence and the
LMMSE formulation in the Rayleigh channel estimation
[4] are not suitable for Rician channel estimation.
The rest of this paper is organized as follows. Section
2 introduces the channel model. The performance of the
LS and GLMMSE channel estimators and the optimal
training sequence design are investigated in Sections 3
and 4. Simulation results are presented in Section 5. Fi-
nally, Section 6 concludes this paper.
2. Channel Model
For flat MIMO channels, the block fading model is as-
sumed. It means that the channel response is fixed within
one block and changes from one block to another ran-
domly. The transmitter and receiver are equipped with
NT and NR antennas, respectively. During the training
period, the received signal in such a system can be writ-
ten in the matrix form as

YHXV (1)
where Y, X and V are the complex NR-vector of received
signals on the NR receive antennas, the possibly complex
NT-vector of transmitted signals on the NT transmit an-
tennas, and the complex NR-vector of additive receiver
noise, respectively. The elements of noise matrix are i.i.d.
complex Gaussian random variables with zero-mean and
2
n

variance, and the correlation matrix of V is then
given by


2
VP nRN
=E N

 RVV I (2)
where NP is the number of transmitted training symbols
by each transmitter antenna, P I
N
is the NP × NP identity
matrix, (٠)H denotes the matrix Hermitian, and E{٠} is
the mathematical expectation.
The channel matrix H in the model (1) is the NR × NT
matrix of complex fading coefficients. The (r, t)th ele-
ment of the matrix H denoted by hr,t represents the fading
coefficient value between the rth receiver antenna and the
tth transmitter antenna. The elements of H are Gaussian
with independent real and imaginary parts each distrib-

H. NOORALIZADEH ET AL.
964
uted as N

2
2,



. So, the elements hr,t of H are iden-
tically distributed complex Gaussian random variables
hr,t ~ CN



2
12,2j

  for r = 1, 2,, NR and t =
1, 2,, NT. The magnitude of the elements of H has the
Rician distribution
1
mn
mn R
rN




 (7)
 



2
1
212 1
a
A
f
aaeI


 
 

Therefore, the correlation matrix of the spatially cor-
related Rician fading MIMO channel can be expressed in
Equation (8).
a  (3) Note that when ρ = 0, (8) reduces to the special case of
(12) in [3] and when
κ = 0, it reduces to the spatially
correlated Rayleigh fading channel introduced in [4].
Using (5), the covariance matrix of the described Rician
fading model can be written as (9).
where I◦ is the modified Bessel function of first kind, of
order zero, and the Rice factor,
κ , can be defined as
2
2
2



If the elements hr,t of H are uncorrelated (ρ = 0), we
can write the result as
(4)
1T
R
H
N
N

 CI

(10)
For notational convenience, we have also presented
the normalization μ2 + 2σ2 = 1. Note that (3) reduces to
the Rayleigh probability density function (pdf) when
κ =
0. If elements of H are distributed as described above, H
will be a complex normally distributed matrix, denoted
as H ~ CN (M, CH) where CH and M are the Hermitian
covariance matrix and the mathematical expectation ma-
trix of the H, respectively. The matrix M can be written
as follows:
The elements of H and noise matrix are independent
of each other. In order to estimate the channel matrix, it
is required that NP ≥ NT training symbols are transmitted
by each transmitter antenna. The function of a channel
estimation algorithm is to recover the channel matrix H
based on the knowledge of Y and X.
3. LS Channel Estimator

1
2
R
T
N
N
j


M1 (5)
Consider that H is an unknown deterministic matrix.
To identify it from (1), the LS approach minimizes



H
tr YHXYHX which results in
Here,
N
R
T
N is an NR × NT matrix whose entries are
all 1. We assume that the elements hr,t of H are correlated.
Suppose that
1

0
mn



1 is the correlation coef-
ficient of elements in the columns mth and nth of the H.
Therefore, the correlation of any two elements from mth
and nth columns of H is expressed in the following form:

1
ˆHH
LS

HYXXX (11)
where tr {٠} denotes the trace of a matrix, (٠)–1 denotes
the matrix inverse. The LS error criterion (MSE) is de-
fined by

*22
2
11 1
mn
im in
mn mn
Eh h  




 





 


2
ˆ
LS LS
F
JEHH (12)

(6)
where 2
F

denotes the Frobenius norm. Let us write
from (1) and (11)
where m, n = 1, 2,, NT, i = 1, 2 ,, NR, and (٠)*
denotes the complex conjugate. Then, the (m, n)th ele-
ment of the channel correlation matrix can be written
as



1
1
ˆHH
LS
HH


 

HHHHXVX XX
VX XX
(13)
1 2
2
123
1
1
1
1
T
T
TT T
N
N
R
H
NN N
N
 



 






 



 


R


 




(8)
1 2
2
123
1
1
1
1
T
T
TT T
N
N
HR
HH
NN N
N
 





 







 









CRMM




(9)

H. NOORALIZADEH ET AL.
965
Using (2) and (13), the MSE (12) can be rewritten as
 

2
1
2HH H
LSn R
F
JE Ntr 


 


VX XXXX

1
H
p



(14)
Let us find X which minimizes (14) subject to a trans-
mitted power constraint. This is equivalent to the fol-
lowing optimization problem:



1
X
min .
H
trS Ttrp
XXXX (15)
where p is a given constant value considered as the total
power of training matrix X. To solve (15), the Lagrange
multiplier method is used. The problem can be written as



1
,
HH H
Ltr tr 



XXXXXX (16)
where η is the Lagrange multiplier. By differentiating (16)
with respect to XXH and setting the result equal to zero,
it is obtained that the optimal training matrix should sat-
isfy the Equation (17)
1
T
H
N

XXI (17)
Equation (17) can be expressed in the following form
using the constraint

H
tr p

XX ,
T
H
N
T
p
N
XXI (18)
Therefore, any training matrix with orthogonal rows of
the same normT is optimal. Let us dictate PAPR
constraint on X that is considered in [4,11,12]. To satisfy
this constraint, a properly normalized sub-matrix of the
DFT matrix can be used
pN

 
1
11
11 1
1
1
P
PP
TT
PP
N
NN
PT
NNN
NN
WW
p
NN
WW



1
P







X


 


(19)
where

exp 2
k
Wj  

k . Substituting (18) back into
(14), the channel estimation error under optimal training
is given by

22
min
nTR
LS
NN
Jp

 (20)
In a particular case that NT = NR =1, single-input sin-
gle-output (SISO) channel, the MSE of (20) is minimum.
Then, increasing the number of antennas results in higher
MSE. On the other hand, the capacity of an MIMO
channel increases when the number of antennas is in-
creased. Note that the error in (20) is proportional to the
square of NT. This causes a certain restriction in the
number of transmit antennas as compared with the num-
ber of receive antennas used. For optimal training which
satisfies (18), the LS channel estimator (11) yields
ˆ
H
T
LS
N
p
HYX (21)
These results are the same as [4], because the LS esti-
mator cannot exploit any statistical knowledge about the
Rayleigh or Rician fading channels. In the next section,
we derive new results in the Rician channel model by the
new GLMMSE estimator.
4. Proposed GLMMSE Channel Estimator
For linear model (1), the MMSE and LMMSE estimators
are identical [23]. So, let us obtain a general form of lin-
ear estimator, appropriate for Rician fading channels,
that minimizes the estimation MSE of H. It can be ex-
pressed in the following form:



 
ˆY
GLMMSE EEHHYAMYM 
XA
(22)
Here, has to be obtained so that the following
MSE is minimized:
A


2
ˆ
GLMMSEGLMMSE F
JEHH (23)
The optimal can be found from ∂JGLMMSE /∂=
0 and it is given by
AA


1
2
P
HH
H
nRN H
N 

AXCXI XC
 (24)
Proof: See the Appendix.
Substituting back into (22), the GLMMSE chan-
nel estimator of H can be rewritten as
A


1
2
ˆ
P
HH
GLMMSEHnR NH
N 

 HMYMXXCXIXC
(25)
Note that in the Rayleigh fading channel, M = 0, CH =
RH. This estimator not only utilizes received and trans-
mitted signals but also takes the advantages of the chan-
nel first and second-order statistics. The required
knowledge of the channel statistics can be estimated by
some methods. For instance, the problem of estimating
the MIMO channel covariance, based on limited amounts
of training sequences, is treated in [24]. Moreover, in
[25], estimation of the channel autocorrelation matrix is
performed by an instantaneous autocorrelation estimator
where only one channel estimate (obtained by a very low
complexity channel estimator) has been used as input.
The performance of the GLMMSE channel estimator
is measured by the error matrix ε = H – ĤGLMMSE, whose
pdf is Gaussian with zero mean and the following co-

966 H. NOORALIZADEH ET AL.
variance matrix:

1
1
2
1-
H-H
H
nR
EN



 


CR εε CXX



(26)
Therefore, the estimation error can be computed as

 

2
1
1
2
ˆ
1
H
GLMMSEGLMMSE F
H
H
nR
JE Etr
tr trN




 



 
 
 

HH εε
CC XX


(27)
Let us find X which minimizes the channel estimation
error subject to a transmitted power constraint. Thus, we
seek the matrix X that is the solution to the optimization
problem (28)

1
1
2
1
min .
HH
H
nR
trS T trp
N 






 
 

XCXX XX (28)
To solve (28), the Lagrange multiplier method is ap-
plied. The problem can be written as


1
1
2
1
,
H
H
nR
H
LtrN
tr p








 






XXC XX
XX
H





(29)
By differentiating (29) with respect to XXH and equat-
ing to zero, we have
2
2
T
HnR 1
N
nRH
NN 


XXIC (30)
Using the constraint , (30) can be ex-
pressed as

H
tr pXX

21
2
T
nR H
H1
N
nRH
T
pNtr N
N





C
XXIC (31)
By applying (31) in (27),

12
1
H
HnR
N 
CXX will
be a diagonal matrix. Therefore, according to the lemma
1 in [4], we obtain that the MSE (27) will be minimized
as


2
min 2
T
GLMMSE
Rn H
N
JpN tr


C1 (32)
In a particular case that the elements hr,t of H are un-
correlated (ρ = 0), the covariance matrix of H is diagonal
and from (10)

11
H
T
R
tr N
N


C (33)
Using (33), it is observed that (31) is the same as (18).
It means that in the case of ρ = 0, both the LS and
GLMMSE approaches have the same condition on the
optimal training matrices.
Using (10) and (18), the GLMMSE channel estimator
(25) reduces to


ˆ1
H
GLMMSE
  HMYX (34)
where
 
22
,
11
PP
nP TnP
NpN
pNpN N

 

 T N
(35)
Substituting (33) back into (32), MSE in the particular
case of ρ = 0 is given by (36)



2
min 21
RT
GLMMSE
nT
NN
JpN

 (36)
Equation (36) shows that when the Rice factor, κ ,
increases, the MSE considerably decreases. In other
words, in the Rician fading channel model compared
with Rayleigh one, the obtained MSE improves.
Increasing the channel Rice factor causes decreasing the
MSE, and for higher values of
κ , the MSE is
proportional to 1/
κ . When κ = 0, (36) is identical to the
acquired result in [4] for RMMSE channel estimator.
In general, the covariance matrix of the H is given by
(9) and the condition for the optimal training matrix of
the GLMMSE channel estimator is different from that of
the LS estimator. We seek the matrix X that is the solu-
tion to (31). Thus, the eigen-value decomposition (EVD)
of CH in the form of CH = QΛQH is used, where Λ is a
diagonal matrix containing the nonnegative eigenvalues
of the CH as its diagonal elements and Q is a unitary ma-
trix containing the eigenvectors of the CH in its columns.
Using this notation, (27) can be rewritten as
1
1
2
1
1
2
1
1
HH
GLMMSE
nR
HH
nR
Jtr N
tr N
































QΛQXX
ΛQXXQ
(37)
Equation (37) can be reduced by replacing


2
1H
nR
N  QX
by X




1
1H
GLMMSE
Jtr 

ΛXX
 (38)
Also, the total transmitted training power constraint in
(28) can be rewritten in the following form:
 
22
1
HHH
nR nR
p
tr tr
NN 
XXQXX Q
 (39)
Using lemma 1 of [4], the minimum of (38) will be
obtained if X̃X̃H has the following diagonal structure:

H. NOORALIZADEH ET AL.
967

2
2
2
12
,,,
T
H
N
diag xxxXX
  
 (40)
Then, the optimal training matrix for the GLMMSE
channel estimation method can be found by solving the
following constrained optimization problem:



1
1
2
min .
HH
nR
p
trS T trN



X
ΛXX XX

  (41)
Using Lagrange multiplier method and taking into ac-
count (40), the optimal training matrix of the GLMMSE
method can be found by minimizing the following func-
tion:
 





1
1
2
1
2
1
1
22
1
,
T
T
HH
H
nR
N
ii
i
N
inR
i
Ltr
trp N
x
xpNN


















XX ΛXX
XX
 


T


(42)
where λi for i = 1, 2,, NT are the nonnegative eigen-
values of the CH. Differentiating (42) with respect to |xi|2
for i = 1, 2,, NT and setting the results equal to zero
yields

2
2
1
1,1,2,,
T
ii
i
x







N
(43)
The water-filling-type solution of this problem is
11
1
,
0,
i
i
i
if
x
if
i













 (44)
The constant η◦ = η–0.5 should be adjusted so that the
transmitted power constraint (39) is satisfied. If NP = NT,
then the optimal X̃ can be written in the following matrix
form:

1/2
1
T
N






XIΛ

 (45)
where the operator [٠]+ is interpreted as meaning that all
negative entries of a real matrix are replaced by zeros
and all nonnegative entries are leaved unchanged. Finally,
the optimal training matrix can be written as

1/2
2
QT
nR N
N 





XII

1
(46)
The matrices Q and Λ are obtained from the EVD of
CH and the constant η◦ should be adjusted so that the
transmitted power constraint in (28) is satisfied.
5. Simulation Results
In this section, our goal is to compare the performance of
the LS and GLMMSE channel estimators in the Rayleigh
and Rician flat fading channels, numerically. Also, we
contrast the results with the LMMSE channel estimator
of [4] and SSLS channel estimator of [21]. For the sake
of simplicity and without loss of generality, we assume a
2 × 2 MIMO channel, i.e., NT = NR = 2. It is also sup-
posed that the spatially correlated Rician fading MIMO
channel has the covariance matrix (9). Hence, the ele-
ments of the covariance matrix of the channel can be
written in the following form:

,;0 1
1
Rkl
Hkl
N






C (47)
where k, l are the indexes of the array sensors. As a per-
formance measure, we consider the channel MSE, nor-
malized with the average channel energy as:



2
2
ˆ
F
F
E
NMSE
E


HH
H
(48)
The signal to noise ratio (SNR) is defined as:
2
n
p
SNR

 (49)
Figure 1 shows the normalized MSE (NMSE) of the
LS channel estimator with optimal training versus SNR
for various Rice factors of the channel. As it is expected,
this estimator cannot exploit the knowledge of the chan-
nel Rice factor; a phenomenon that is confirmed by this
figure. In [4] and [9], it is demonstrated that the LS esti-
mator does not require any knowledge about the channel.
Hence, it is also clear that the performance of this esti-
mator is independent of ρ and the type of channel fading.
The numerical and analytical results coincide when the
number of independent simulation runs reaches to 5000.
Using (20), the NMSE of the LS channel estimator is
plotted in Figure 1. As depicted in this figure, the ana-
lytical and numerical results are almost identical.
Figures 2 and 3 indicate the NMSE of the LS,
LMMSE of [4] and GLMMSE channel estimators with
orthogonal training of (19) versus SNR in the case of ρ =
0.1 and ρ = 0.8, respectively. It is observed that the pro-
posed GLMMSE estimator has the best performance
among the methods tested. Increasing the channel Rice
factor and/or the correlation coefficient of the array sen-
sor elements improves the performance of this estimator
especially at low SNRs compared with the fixed per-
formance of the LS estimator. Moreover, increasing ρ
improves the performance of the LMMSE channel esti-
mator. These results are due to the fact that the Bayesian

968 H. NOORALIZADEH ET AL.
Figure 1. NMSE of the LS channel estimator for various
Rice factors of the channel, NR = NT = 2 (Numerical and
analytical results).
Figure 2. NMSE of the LS, LMMSE [4] and GLMMSE ( κ =
1, 10) channel estimators in the case of orthogonal training
signals (NR = NT = 2, ρ = 0.1).
Figure 3. NMSE of the LS, LMMSE [4] and GLMMSE ( κ =
1, 10) channel estimators in the case of orthogonal training
signals (NR = NT = 2, ρ = 0.8).
estimators, e.g., the proposed GLMMSE channel esti-
mator can employ more a priori knowledge about the
channel. As depicted in Figures 2 and 3, at high SNRs,
the performances of the LS, LMMSE and GLMMSE
channel estimators are nearly identical, particularly for
low Rice factors and spatial correlations. However, at
higher
κ and ρ, the performance of the GLMMSE
estimator is still better than that of the LS estimator. Note
that in the special case,
κ = 0, the proposed estimator is
the same as the LMMSE estimator of [4]. However, in
the presence of LOS paths, it is obvious that the pro-
posed GLMMSE channel estimator outperforms the
LMMSE estimator of [4].
The NMSE of the LS, LMMSE and GLMMSE chan-
nel estimators with optimal training in the case of ρ = 0.1
and ρ = 0.8 is shown in Figures 4 and 5, respectively.
Figures 6 and 7 compare the NMSE of the LS and
Figure 4. NMSE of the LS, LMMSE [4] and GLMMSE ( κ =
1, 10) channel estimators in the case of optimal training
signals (NR = NT = 2, ρ = 0.1).
Figure 5. NMSE of the LS, LMMSE [4] and GLMMSE ( κ =
1, 10) channel estimators in the case of optimal training
signals (NR = NT = 2, ρ = 0.8).

H. NOORALIZADEH ET AL.
969
Figure 6. NMSE of the GLMMSE and LS channel estima-
tors for
κ = 5 with optimal and orthogonal training signals
(NR = NT = 2, ρ = 0.1).
Figure 7. NMSE of the GLMMSE and LS channel estima-
tors for
κ = 5 with optimal and orthogonal training signals
(NR = NT = 2, ρ = 0.8).
GLMMSE estimators with optimal and orthogonal train-
ing for κ = 5 in the case of ρ = 0.1 and ρ = 0.8, respec-
tively. It is observed that at high spatial correlation the
performance of the GLMMSE channel estimator with
optimal training is better than that of orthogonal training.
At low spatial correlation, the performance of the
GLMMSE channel estimator with optimal training and
orthogonal training is closely identical. In order to obtain
the advantage of the optimal training sequence design,
long-term statistics of the channel need to be estimated at
the receiver and fed back to the transmitter. Hence, when
the GLMMSE channel estimator is used to estimate
MIMO channel with low spatial correlation, the trans-
mitter has no need to the channel knowledge.
Figure 8. NMSE of the LS, SSLS [21] and GLMMSE chan-
nel estimators in the case of optimal training signals (NR =
NT = 2, ρ = 0.1, κ = 10).
Figure 9. NMSE of the LS, SSLS [21] and GLMMSE chan-
nel estimators in the case of optimal training signals (NR =
NT = 2, ρ = 0.8, κ = 10).
estimator is smaller than that of the SSLS channel esti-
mator, particularly at higher spatial correlations, because
the GLMMSE estimator can employ more a priori
knowledge about the channel than the SSLS estimator. It
is notable that the NMSE of the SSLS channel estimator
is independent of ρ.
6. Conclusions
We have proposed a new channel estimator (GLMMSE)
that is suitable for spatially correlated Rician fading
MIMO channel estimation. This estimator has better
performance than the SSLS estimator of [21] and
LMMSE estimator of [4]. Analytical and numerical re-
sults confirm the superiority of the GLMMSE estimator
in the mentioned channel model. It is demonstrated that
increasing κ and/or ρ decreases the NMSE of the of-
Finally, we compared the performance of the LS,
SSLS of [21] and GLMMSE channel estimators in Fig-
ures 8 and 9. Clearly, the NMSE of the proposed channel

970 H. NOORALIZADEH ET AL.
fered estimator. Hence, to obtain the given value of MSE,
the required SNR can be reduced in the Rician channel
estimation. Clearly, increasing the number of antennas in
MIMO systems leads to decreasing the performance of
estimators. In the Rician fading MIMO channel, the un-
favorable effect of increasing the number of antennas on
the performance of GLMMSE channel estimator can be
compensated. In other words, for the given values of
SNR and MSE, the number of antennas possibly in-
creases. Therefore, the Rician fading MIMO channels
result in a higher capacity than the Rayleigh fading
MIMO channels without increasing MSE. Moreover,
training length can be reduced in the presence of the spa-
tially correlated channel and/or Rician model to improve
the bandwidth efficiency without increasing MSE. It is
noteworthy that Rician fading is known as a more ap-
propriate model for wireless environments with a domi-
nant direct LOS path; and, in the microcellular mobile
systems, this model is better than the Rayleigh one.
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Appendix

 



2
TT
HH
GLMMSENHN
H
nR
Jtr
Ntr 
 

I AXCI XA
AA


(A-2)
Proof of Equation (24):
Using (22), the MSE (23) can be written as follows:







2
()
GLMMSE F
H
JE
Etr







HM YMXA
HM YMXA
HM YMXA



The optimal can be found from A


(A-1)
**2* 0 TT GLMMSE
HH nR
JN



 
XCXC XAA A

(A-3)
where (٠)T denotes the matrix transpose. Finally, we
have


1
2
P
HH
H
nRN H
N 

AXCXI XC
 (A-4)
With some calculations, the MSE (A-1) is given by