On the Robustness of MIMO LMMSE Channel Estimation
ABSTRACT The robustness of the linear minimum mean square error (LMMSE) channel estimator is studied with respect to the reliability of the estimated channel correlation matrix used for its implementation. The analysis is of interest in practical applications of multipleinput multipleoutput (MIMO) systems, where a perfect estimate of the channel correlation matrix is not available. The channel estimation mean square error (MSE) is analytically analyzed assuming a general structure for the estimated channel correlation matrix used to implement the LMMSE channel estimator. The obtained results are successively detailed to the case of channel correlation matrices derived by sample correlation estimation methods. It is observed that the use of a coarse estimate of the channel correlation matrix can lead to a severe degradation on the LMMSE channel estimator performance, whereas the simpler leastsquare (LS) channel estimator may provide comparatively better results. Nevertheless, it is shown that a robust approach, although suboptimal, relies on implementing the LMMSE channel estimator by assuming transmissions over uncorrelated channels, since, with such an assumption, the resulting estimation MSE is certainly smaller than for the LS channel estimator.

Conference Paper: Sum mutual information of blockfaded MIMO MAC with LMMSE channel estimation for packet transmission.
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ABSTRACT: The effect of a trainingbased linear minimum mean squared error (LMMSE) channel estimator on the sum mutual information of the multipleinput multipleoutput (MIMO) multiple access channel (MAC) is investigated. The contribution of the present work consists in relating informationtheoretic bounds on the sum mutual information with practical system parameters that impact on the trainingbased LMMSE MIMO channel estimator. The unboundness of the sum mutual information and conservation of the multiplexing gain are shown for a block fading channel model even in the presence of channel estimation errors. Then, as an application of the bounds, mutual informationmaximizing power allocation strategies under total energy constraints are considered.14th International Symposium on Wireless Personal Multimedia Communications, WPMC 2011, Brest, France, October 37, 2011; 01/2011
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On the Robustness of MIMO LMMSE
Channel Estimation
Antonio Assalini, Member, IEEE, Emiliano Dall’Anese, Student Member, IEEE,
and Silvano Pupolin Senior Member, IEEE
Abstract—The robustness of the linear minimum mean square
error (LMMSE) channel estimator is studied with respect to
the reliability of the estimated channel correlation matrix used
for its implementation. The analysis is of interest in practical
applications of multipleinput multipleoutput (MIMO) systems,
where a perfect estimate of the channel correlation matrix is
not available. The channel estimation mean square error (MSE)
is analytically analyzed assuming a general structure for the
estimated channel correlation matrix used to implement the
LMMSE channel estimator. The obtained results are successively
detailed to the case of channel correlation matrices derived by
sample correlation estimation methods. It is observed that the
use of a coarse estimate of the channel correlation matrix can
lead to a severe degradation on the LMMSE channel estimator
performance, whereas the simpler leastsquare (LS) channel
estimator may provide comparatively better results. Nevertheless,
it is shown that a robust approach, although suboptimal, relies
on implementing the LMMSE channel estimator by assuming
transmissions over uncorrelated channels, since, with such an
assumption, the resulting estimation MSE is certainly smaller
than for the LS channel estimator.
Index Terms—Channel estimation, MIMO systems, least mean
square methods, correlation matrix, performance analysis.
I. INTRODUCTION
The LMMSE channel estimator [1] is frequently adopted for
practical implementation and theoretical analysis of MIMO
systems, since it presents good estimation performance for
any value of the signaltonoise ratio (SNR) [2]. Moreover, it
provides an estimation error that is statistically orthogonal to
the estimate itself, which makes tractable and meaningful the
analysis of the system performance limits [3], [4]. However,
the optimal implementation of the LMMSE channel estimator
requires perfect knowledge of the channel correlation matrix.
In the literature, such information is commonly assumed
available at the receiver, whereas the problem of estimating
the channel correlation matrix is faced separately from the
channel estimation task.
The statistical issue of estimating the correlation matrix of
a random process, on the basis of a sample of observations,
is a relevant topic that attracts interest and effort; in fact
many applications rely on an accurate estimation of correlation
matrices [5]–[12]. Nevertheless, it may be difficult to obtain
Copyright c ?2010 IEEE. Personal use of this material is permitted. How
ever, permission to use this material for any other purposes must be obtained
from the IEEE by sending a request to pubspermissions@ieee.org.
The authors are with the University of Padua, Department of Information
Engineering (DEI), Via Gradenigo 6/B, 35131, Padova, Italy (email: {assa,
edallane, pupolin}@dei.unipd.it).
a reliable estimate of the channel correlation matrix since
we would need to observe a large sample set over a long
observation interval, in order to capture the secondorder
statical properties of the channel. In fact, in many cases,
channels are slow timevarying and, moreover, the fraction
of time dedicated to the transmission of training data has
to be small in order to keep the overall system spectral
efficiency. Furthermore, in some propagation scenarios, e.g.,
vehicular transmissions, the channel is notstationary and then
the channel correlation matrix may change with time [13],
[14]. Consequently, most practically, an inexact estimate of
the channel correlation matrix is likely to be employed to
implement the LMMSE channel estimator.
Therefore, we study the effect of the use of a not perfectly
estimated channel correlation matrix in the implementation of
the LMMSE channel estimator. The most closely related work
on this subject is due to Czink et al. in [5], where a novel
estimator of the channel correlation matrix was derived and
employed to implement the LMMSE channel estimator. The
analysis of the resulting performance was studied by means
of computer simulations considering different transmission
configurations. In this letter, we propose an analytical study of
the problem by assuming a general structure of the estimated
channel correlation matrix.1In particular, we compute the
channel estimation MSE that corresponds to the LMMSE
channel estimator implemented with a given but imperfect
estimate of the channel correlation matrix. It is shown that
for asymptotically high and low values of SNR, coarsely
estimated channel correlation matrices do not affect estimation
performance. On the contrary, for finite values of SNR, the per
formance degradation of the LMMSE channel estimator can be
so severe that better results can be provided by the LS channel
estimator, which does not require any statistical knowledge
about channel and SNR. Nevertheless, it is proved that if
the LMMSE channel estimator is implemented by assuming
spatially independent radio links, then we actually achieve the
same channel estimation MSE as it would be obtained by
estimating really independent links, no matter whatever the
correlation values of the real channel are. Moreover, despite
the suboptimal setting, the resulting MSE performance is
certainly improved with respect to the LS channel estimator.
The obtained results also suggest an enhancement to the
commonly used sample correlation matrix estimation method.
The outline of this letter is as follows. In Section II we
1Preliminary results were presented in [15], where we reported an analysis
based on bounding methods.
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introduce the system model. In Section III we briefly recall the
necessary elements of LMMSE channel estimation. Section IV
deals with the performance analysis of the LMMSE channel
estimator provided with an imperfect knowledge of the channel
correlation matrix. Practical estimators for the channel corre
lation matrix are considered in Section V, while numerical
results are reported in Section VI. Concluding remarks are
drawn in Section VII. In the Appendix we report details of
derivation and some further insights.
II. SYSTEM MODEL
It is considered a pointtopoint openloop narrowband
MIMO system with Nttransmit and Nrreceive antennas. The
Nt×1 vector xnand the Nr×1 vector yndenote, respectively,
the transmitted symbols and the received signal samples at
time index n. We assume that a known fixed training preamble
of L consecutive symbol vectors Xτ = [x1,x2,...,xL]T,
where the superscript (·)Tstands for transposition, is ap
pended at the beginning of each transmitted data frame for
synchronization and channel estimation purposes. It is known
that, over spatially uncorrelated channels, LMMSE channel
estimation performance is optimized by using orthogonal train
ing sequences, verifying X∗
symbol power Pτ [3]; the symbol Ik designates the identity
matrix of dimension k×k while the superscript (·)∗stands for
Hermitian transposition. The same policy can be adopted when
no channel state information is available at the transmitter.2In
particular, by setting L = Nt, the equality X∗
holds and, furthermore, the number of corresponding receiver
measurements is equal to the number NtNrof channel links
to estimate. This choice is optimal in many cases of interest
[3], although longer training sequences provide smaller esti
mation MSE [5], but at the cost of a reduced overall spectral
efficiency [3].
The noisy samples received in correspondence to the train
ing symbols for the mth frame are given by Yτ,m
[y1,m,y2,m,...,yL,m]T= XτHm+ Nτ,m, where Hm is
the Nt× Nr channel matrix with zeromean unitarypower
circularly symmetric complex Gaussian entries. The chan
nel is stationary with associated correlation matrix R =
E[vec(Hm)vec(Hm)∗], with vec(·) denoting the vectorization
operator, which creates a column vector by stacking the
columns of a matrix below one other. The matrix R is assumed
positive definite. The propagation channel is static during
the transmission of a training sequence. The additive white
Gaussian noise (AWGN) components in Nτ,mare independent
with zeromean and variance σ2
In the following, we do not introduce any further constraint
on R since we propose a study that holds for any model
adopted for the channel correlation matrix. A structured ex
ample for R shall be considered in Section VI for numerical
analysis validation.
τXτ = PτLINtfor a fixed per
τXτ = XτX∗
τ
=
n.
2Training sequences and antenna power loading could be further optimized
if the channel covariance matrix would be available at the transmitter [7],
[16]. In [16] it was shown that orthogonal training sequences are still optimal
when no transmitter correlation is present, regardless of the correlation at the
receiver.
III. PRELIMINARY ON LMMSE CHANNEL ESTIMATION
In this section we review some basic results on LMMSE
channel estimation with particular emphasis on the estimation
error correlation matrix and the resulting estimation MSE.
We first exploit the vec(·) operator properties to arrange the
received samples as
vec(Yτ,m) =¯Xτvec(Hm) + vec(Nτ,m) ,
where¯Xτ = INr⊗ Xτ, with ⊗ denoting the Kronecker
product. It results that¯X∗
INr⊗¯X∗
PτL. Therefore, the LMMSE channel estimator, based on
Yτ,mand Xτ, is obtained as [1]
vec(?Hm) =?R−1σ2
The correspondent channel estimation error matrix EH,m=
Hm−?Hm has Gaussian distributed entries and associated
RE= E[vec(EH,m)vec(EH,m)∗]
=?R−1+ β INtNr
where we defined β = ¯Pτ/σ2
positive definite with eigenvalues λi> 0, i = 1,...,NtNr,
the inverse R−1is positive definite, with eigenvalues 1/λi,
i = 1,...,NtNr [17]. Therefore, the estimation MSE for a
fixed R and β is equal to
?1
(1)
τ¯Xτ = (INr⊗¯X∗
τ)(INr⊗¯Xτ) =
τ¯Xτ = INr⊗ (PτLINt) =¯PτINtNr, with¯Pτ =
n+¯X∗
n+¯PτINtNr
τ¯Xτ
?−1¯X∗
τvec(Yτ,m)
?−1¯X∗
=?R−1σ2
τvec(Yτ,m) .
(2)
correlation matrix given by [1]
?−1,
(3)
n. As we assume that R is
MSE(R,β)=Tr(RE)=
NtNr
?
i=1
λi
+ β
?−1
=
NtNr
?
i=1
λi
1 + β λi
.
(4)
Note that, since Tr(R) = NtNr is fixed for construction,
the spatially uncorrelated channel, i.e., R = INtNr(λi= 1
for all i), leads to the highest estimation MSE:
MSE(INtNr,β) =NtNr
1 + β≥ MSE(R,β) ,
(5)
and, conversely, we can generally expect that correlated chan
nels can be estimated more accurately, provided that the
secondorder statistical description of the channel is perfectly
known.
IV. IMPERFECT CHANNEL CORRELATION INFORMATION
In practice the channel correlation matrix R is firstly
estimated at the receiver, and the resulting estimate?R is then
In Appendix it is shown that the resulting correlation matrix of
the channel estimation error,3for a LMMSE channel estimator
implemented with a given estimate?R, becomes
βWW∗+ (W − INtNr)R(W − INtNr)∗,
used, in place of R, in (2) to estimate the channel coefficients.
RE=1
(6)
3Concisely, note that with?R in place of R, (2) is not the LMMSE channel
estimator.
estimator anymore, but it is an imperfectly implemented LMMSE channel
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where W = (1
given β and?R, the correspondent channel estimation MSE
MSE(?R,R,β)=Tr(RE)=1
where we used the equality Tr(AA∗) = ?A?2, which holds
for any given matrix A, with ?·? denoting the Frobenius norm.
The notation MSE(?R,R,β) emphasizes that (7) represents
estimating a channel having correlation matrix R, but by using
the estimated correlation matrix?R in the implementation of
to obtain a prompt analytical test of the accuracy of an
estimator of the channel correlation matrix, considering its
effect on the LMMSE channel estimator. We note that, for
the channel estimation issue, the performance measure (7)
is more meaningful than the MSE of the estimation of the
channel correlation matrix, i.e, E[?R−?R?2], since it directly
of the LMMSE channel estimator, rather than considering the
sole accuracy of the estimate R.
Based on (6) and (7), we can now study the performance of
the LMMSE channel estimator for low and high values of β.
β?R−1+ INtNr)−1. Therefore, by (6), for a
can be numerically evaluated as
β?W?2+?(W−INtNr)R1/2?2,
(7)
the channel estimation MSE, obtained for a fixed β, by
the LMMSE channel estimator. The form (7) can be used
emphasizes the impact of an estimate?R on the performance
A. Impact of β and relation with the LS channel estimator
The value of β =¯Pτ/σ2
on the system SNR and on the processing gain achievable by
using longer training sequences. Indeed, the value of β can be
increased by increasing either the power Pτduring the training
phase or the length L of the training sequences.
For low values of β, it is readily seen from (6) that
the estimation error is high, although it is bounded, since
RE
−→ R, and then MSE(?R,R,β → 0) = Tr(R) = NtNr,
still find that the performance does not depend on?R, but in
MSE decreases with β as MSE(?R,R,β → +∞) = NtNr/β .
found by fixing the estimate?R. For a specific estimator of
relates to β (see Section V and the Appendix for an example).
The obtained limit result for high values of β also points out
the asymptotic equivalence between LMMSE and LS channel
estimators. Actually, this is a wellknown equivalence for the
case of perfect knowledge of R, but it still holds when just an
imperfect estimate?R is employed by the estimator. The LS
vec(?HLS
Therefore, let ELS
E[vec(ELS
Tr(RLS
n= PτL/σ2
mconveys information
β→0
independently of the estimate?R. For high values of β, we
this case we obtain RE
−→ (1/β)INtNrand the estimation
β→+∞
We remark that the obtained asymptotic results have been
R, we may also consider how the accuracy of the estimate?R
estimate is obtained as
m)=1
¯Pτ
¯X∗
τvec(Yτ,m)=vec(Hm)+1
¯Pτ
¯X∗
τvec(Nτ,m).
(8)
=
H,m= Hm−?HLS
m, we find that RLS
E
H,m)vec(ELS
E) = NtNr/β, as aforesaid.
H,m)∗] = (1/β)INtNr and MSELS =
B. Assuming uncorrelated channels
We have seen that, for not too small or high values of β,
the channel estimation error has a secondorder statistical de
scription dependent on both the real and the estimated channel
correlation matrices. At the beginning of a communication,
when the channel has been observed for a not sufficiently long
time, the achievable estimate?R can be very inaccurate, and the
on the estimator performance. In this case, it is advisable to
find a suitable suboptimal setting for the LMMSE channel
estimator, such that the estimator itself would be capable of
providing pretty good performance, regardless of the unknown
real channel correlation conditions. Therefore, to this aim, we
now investigate the LMMSE channel estimator performance
assuming?R = INtNr, i.e., supposing transmissions over
we obtain W = (?R−1/β+INtNr)−1= β/(1+β)INtNr, and
RE=
1 + β
We observe that (9) and (3) have a common eigenvector basis,
common with R, but, however, they have different eigenval
ues. Nevertheless, by the trace of REin (9), the resulting esti
mation MSE is equal to MSE(?R = INtNr,R,β) = Tr(RE) =
is exactly the same as the estimation MSE given in (5),
which was obtained for transmissions over really uncorrelated
channels. Therefore, we have found that if we assume the
channel to be uncorrelated, then the resulting channel estima
tion MSE is independent of R and it is equal to the estimation
MSE that would be obtained for transmissions over really
uncorrelated channels. It appears that setting?R = INtNrcan
channel estimator when no reliable information about R is
available at the receiver.
Moreover, we note that MSE(INtNr,R,β) = NtNr/(1 +
β) < MSELS = NtNr/β, and then, despite the suboptimal
setting, the LMMSE channel estimator implemented with?R =
estimator.
Finally, the implementation of the LMMSE channel estima
tor with?R = INtNris computationally efficient. In fact, by
can be written as ([1], [18])
use of such an estimate is likely to have a detrimental effect
spatially uncorrelated channels. Hence, setting?R = INtNr,
then (6) becomes
?
1
?2
(R + β INtNr) .
(9)
1/(1 + β)2(Tr(R) + β Tr(INtNr)) = NtNr/(1 + β), that
be a robust choice for the implementation of the LMMSE
INtNris assured to provide a smaller MSE that the LS channel
(2) (with?R = INtNrin place of R) and (8), the estimator
vec(?Hm) =
Therefore, it has been proved the effectiveness of the
estimator (10) with respect to the LS channel estimator. Next,
we introduce practical estimators of R, before reporting a
numerical validation of our analysis considering an illustrative
example with a given model for R.4
β
1 + βvec(?HLS
m) .
(10)
4The analysis reported in this letter can also be applied to OFDM systems,
where LMMSE channel estimation is performed in the frequency domain
by exploiting pilot symbols transmitted by the OFDM subcarriers [19]. The
impact of other impairments, such as possible intercarrier interference [20],
may also be jointly taken into account.
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V. CHANNEL CORRELATION MATRIX ESTIMATION
The channel correlation matrix R can be estimated by
exploiting training sequences on Nmconsecutive data frames.
The most commonly used method realizes the sample corre
lation matrix by the LS channel estimations (8) as ([5], [6])
?RLS
Nm=
1
Nm
Nm
?
m=1
vec
??HLS
m
?
vec
??HLS
m
?∗
.
(11)
However, since we have shown that the MSE performance of
the estimator (10) are better than for the LS channel estimator,
an alternative estimator of R may be obtained by using (10) in
place of (8) in (11). Therefore, the following other estimator
can be adopted
?
In particular, over an ergodic channel, by using (8) in (11),
we find that in the limit of Nm going to infinity, (11)
and (12) approach?RLS
estimators are not consistent, although they are both asymptot
ically optimal with the increasing of β. Now, we can use the
estimates?RLS
large sample set limit, being known the expressions of?RLS
closedform directly by the real channel correlation matrix R
(this case of study is explicitly detailed in Appendix). On the
other hand, for finite values of Nm, we first resort to numerical
simulation for the computation of (11) and (12), and then we
use the obtained estimates into (7) to analytically compute
their correspondent channel estimation MSE.
It should be noted that, throughout this letter, perfect
knowledge of the ratio β has been assumed, since we focused
attention on the channel correlation matrix (in Appendix,
we study the effect of an imperfect estimate of β on the
estimator (10)). The AWGN is a fast timevarying random
process and then many independent, although not directly
observable, samples are available for the estimation of its
statistical power. On the contrary, since the channel changes
more slowly, the estimation of R needs several training phases,
i.e., Nm? 1, before collecting a sufficiently large sample set
to get an accurate estimate of the channel correlation matrix.
As shown in the next section, the above requirement can
be rather restrictive since it may take quite a few training
sequences before obtaining an effective estimate of R.
?RI
Nm=
β
1 + β
?2
?RLS
Nm.
(12)
∞= R + (1/β)INtNrand?RI
∞=
(β/(1 + β))2(R + (1/β)INtNr), respectively. Hence the two
Nmand?RI
∞, the channel estimation MSE can be computed in
Nmin (7) to find their correspondent
MSE of the LMMSE channel estimator. In the asymptotically
∞
and?RI
VI. NUMERICAL RESULTS
In this section, we provide numerical results that corroborate
and quantify the study reported in this letter. We consider a
MIMO system with Nt = Nr = 4 antennas. The channel
correlation matrix is assumed to meet a Kronecker product
model R = Rt⊗ Rr, where the elements of position (i,j) of
the transmitter and receiver correlation matrices are given by
Rt(i,j) = ρi−j
t
, i,j = 1,2,...,Nt, and Rr(i,j) = ρi−j
i,j = 1,2,...,Nr, respectively [21]. We also set ρt= ρr=
ρ, where ρ is a fixed correlation coefficient. The introduced
r
,
model is representative of a MIMO system with a uniform
linear antenna array (ULA) at either side of the radio link. The
value of β is assumed to be perfectly known at the receiver.
First, we compare the performance of the two estimators
?RLS
ing sequences are generated independently of each other. In
the literature, this assumption is commonly introduced for the
study of estimators of correlation matrices. However, it should
be considered that it represents an optimistic assumption
since the sample correlation is calculated over independent
channel samples, that is the most favorable condition for the
convergence of the estimators. In practice, it means that two
consecutive training sequences should be sent at time instants
sufficiently exceeding the channel coherence time from each
other. Therefore, in a slow timevarying environment, it can
take a long time to achieve the same estimation performance as
for the temporally independent case with high values of Nm.
In Figs. 1 and 2 we report, respectively, the normalized
MSE of the channel correlation matrix estimators, i.e., E[?R−
?R?2]/?R?2, as a function of β and ρ for different values
while in Fig. 2 we set a fixed β = 10dB and we vary ρ.
In practice, in order to compute (11) and (12), for a fixed
β and R, we can directly use (8), since its noisy term is
AWGN with power 1/β per component. Incidentally, we also
confirmed the numerical results reported in this section by
using training sequences of length L = Nt and varying the
transmit power Pt to modify the value of β. Finally, note
that in the plots, and the same is used in Figs. 3(a) and
4(a), the style of lines, either solid or dashed, identifies the
estimator type, either?RLS
proposed estimator?RI
values of β, while the two estimators become equivalent with
the increasing of β. In Fig. 2 we note that with the increasing
of Nmand ρ, the performance gap between the two estimators
becomes smaller and the estimator?RLS
confirmed through direct computation of E[?R−?R?2]/?R?2
Appendix).
However, the differences between the two estimators almost
disappear once the performance measure of interest becomes
the resulting channel estimation MSE. In Figs. 3 and 4
we report the channel estimation MSE (7) normalized to
Tr(R) = NtNr, and averaged over different realizations of
the estimators?RLS
then, we use the resulting estimate in (7) to analytically
compute the channel estimation MSE. The process is repeated
and the results are averaged and normalized with respect to
Tr(R). Incidentally, we checked that exactly the same results
as for (7) are obtained through simulation of the LMMSE
channel estimator (3), adopting orthogonal training sequences
of length L = Nt.
Nmand?RI
Nm, given in (11) and (12), respectively. The
channel realizations Hmin correspondence to different train
of Nm. In Fig. 1 we set a fixed ρ = 0.5 and we vary β,
Nmor?RI
Nm, while the markers denote
different values of the sample set size Nm. We can see that the
Nmprovides improved performance in
almost any condition. The gain is more remarkable for small
Nmcan even outperform
the estimator?RI
with?R given by?RLS
Nm(this trend for high values of Nmcan be
∞and?RI
∞, respectively. See (A.4) in the
Nmand?RI
Nm. In effect, as before, we first
find an estimate of R by exploiting Nm training sequences,
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5
05 1015 202530
10−1
100
β [dB]
Normalized MSE – E[?R −?R?2]/?R?2
[dB]
Nm= 5
Nm= 10
Nm= 20
Nm= 50
Nm= 100
?RLS
Nm
?RI
Nm
Figure 1.
?RLS
Normalized MSE of the channel correlation matrix estimators
Nmand?RI
Nmas a function of β for different values of Nm. ρ = 0.5.
From Fig. 3(a), where R is fixed with ρ = 0.5, we note
that the channel estimation MSE is very similar for both
of the considered estimators of R. Contrary to Fig. 1, for
small values of β, the use of the estimator?RLS
These results do emphasize that the adoption of the metric
(7) is advisable in order to develop an effective estimator of
the channel correlation matrix to be employed for LMMSE
channel estimation.
Nmleads to
slightly better performance than using the estimator?RI
Nm.
The comparison between Fig. 3(a) and Fig. 3(b) points out
that a large number Nmof sequences has to be sent in order
to obtain similar performance as for the perfect knowledge
of R case. Remarkably, the simple LS channel estimator can
even outperform the LMMSE channel estimator as long as the
estimate R is coarse and β is not too small. It is interesting
to note that very good results can be obtained with the setting
?R = INtNr, studied in Section IVB, for any value of β; the
knowledge of R. Nevertheless, for values of β > 10dB
also the simple LS channel estimator provides near optimal
performance. In Fig. 4, the different estimators are considered
by varying the value of the correlation coefficient ρ and then
changing the structure of R. Now, from the obtained results,
we can establish the following selection criterion, which is
based on the hypothetical knowledge of the range of values
of ρ and β: (i) For β ≤ 10dB and ρ < 0.5 use the setting
?R = INtNr; (ii) For β ≤ 10dB and ρ ≥ 0.5 use the setting
β > 10dB use the LS channel estimator.
achievable performance is very close to the case of perfect
?R = INtNruntil a reliable estimate of R is available; (iii) For
Therefore, although the LMMSE channel estimator is de
signed to minimize the estimation MSE, the performance
provided by its practical implementation, with a not well
estimated channel correlation matrix, can be severely compro
mised, so that other more robust and less complex approaches
can in practice provide better MSE performance.
00.1 0.20.30.40.50.6 0.70.80.9
10−2
10−1
100
Correlation coefficient – ρ
Normalized MSE – E[?R −?R?2]/?R?2
[dB]
Nm= 5
Nm= 10
Nm= 20
Nm= 50
Nm= 100
?RLS
Nm
˜RI
Nm
Figure 2.
?RLS
Normalized MSE of the channel correlation matrix estimators
Nmand?RI
0
Nm= 16
Nm= 20
Nm= 25
Nm= 50
?RLS
Nm
˜RI
Nm
Nmas a function of ρ for different values of Nm. β = 10dB.
010 20 30
−30
−25
−20
−15
−10
−5
β [dB]
Average Normalized Estimation Mean Square Error [dB]
(a)
0 1020 30
−30
−25
−20
−15
−10
−5
0
β [dB]
Average Normalized Estimation Mean Square Error [dB]
?R = R
?R = I
LS
(b)
Figure 3.
β for a LMMSE channel estimator implemented with an estimated channel
correlation matrix?R. ρ = 0.5.
VII. CONCLUSIONS
Average normalized channel estimation MSE as a function of
Although the LMMSE channel estimator theoretically
presents good estimation capabilities, it is sensitive to an
imperfect knowledge of the channel correlation matrix. Indeed,
the channel should be observed over many independent coher
ence intervals before getting a sufficiently accurate estimation
of the channel correlation matrix. This appears to be a very
stringent requirement for mobile communication systems, and
at the beginning of any wireless transmission. Nevertheless,
since it can be selfdefeating to make use of a not well
estimated channel correlation matrix in the implementation
of the LMMSE channel estimator, to suppose the MIMO
channel to be spatially uncorrelated is an effective choice,
at least until a reliable estimate of the channel correlation
matrix becomes available. With such an approach, the resulting
Page 6
6
00.1 0.20.30.40.50.60.7 0.80.9
−30
−25
−20
−15
−10
−5
0
Correlation coefficient – ρ
Average Normalized Estimation Mean Square Error [dB]
Nm= 16
Nm= 50
?RLS
Nm
˜RI
Nm
β = 10dB
β = 5dB
β = 20dB
(a)
0 0.10.20.3 0.40.50.6 0.70.8 0.9
−30
−25
−20
−15
−10
−5
0
Correlation coefficient – ρ
Average Normalized Estimation Mean Square Error [dB]
?R = R
?R = I
LS
β = 10dB
β = 20dB
β = 5dB
(b)
Figure 4. Average normalized channel estimation MSE as a function of ρ for a LMMSE channel estimator implemented with an estimated channel correlation
matrix?R. β = 10dB.
suboptimal LMMSE estimator outperforms the LS channel
estimator and it provides pretty good performance compared
with the optimally implemented LMMSE channel estimator.
APPENDIX
The first section of this Appendix reports the analytical steps
for the computation of the correlation matrix of the channel
estimation error given in (6). In the second section, some
of the results reported in the main body of this letter are
detailed considering an asymptotically large sample set for
the estimation of the channel correlation matrices. Finally, in
the last section we consider the impact of an estimated β on
the performance of the estimator (10).
A. Computation of RE
Using (1) into (2) with?R in place of R, and recalling that
vec(?Hm) = (?R−1σ2
= Wvec(Hm) +
¯X∗
τ¯Xτ=¯PτINtNr, we obtain
n+¯PτINtNr)−1× ...
... ×¯X∗
1
¯PτW¯X∗
τ
?¯Xτvec(Hm) + vec(Nτ,m)?
τvec(Nτ,m) ,
(A.1)
where W = (1
of the channel estimation error correlation matrix, using (A.1)
and exploiting the statistical independence among channel,
data and noise, we write
??
= R + WRW∗+1
βWW∗− RW∗− WR ,
β?R−1+INtNr)−1. Therefore, by the definition
RE= Evec(Hm) − vec(?Hm)
??
vec(Hm) − vec(?Hm)
?∗?
(A.2)
that by collecting and reordering the terms gives (6).
B. Large sample set analysis
We find an analytical expression for the channel estimation
MSE using the correlation matrix estimators given in Section
V and assuming an asymptotically large sample set. Hence,
we write?RLS
decomposition of R, where U is a unitary matrix and Λ is the
diagonal matrix having the eigenvalues λi, i = 1,2,...,NtNr,
of R on its diagonal. Therefore, it can be shown that ([17])
W = (1
INtNr)−1U∗. Using this expression in (6), and computing the
trace of REas the sum of its eigenvalues, after some algebra,
we find
∞= R + (1/β)INtNrand?RI
∞= γ?RLS
∞with
γ = (β/(1 + β))2. Let R = UΛU∗be the eigenvalue
β(γ?RLS
∞)−1+ INtNr)−1= U(1
β
1
γ(Λ +1
βINtNr)−1+
MSE(γ?RLS
that provides a lower bound on the average channel esti
mation MSE when using the estimators (11) or (12) in (2).
In agrement with Section IVA, but directly by (A.3), for
small values of β, we obtain MSE(γ?RLS
MSE(γ?RLS
matrix estimators for an asymptotical high value of Nm:
∞,R,β) =
NtNr
?
i=1
β λi+ γ2(1 + β λi)2
β(1 + γ(1 + β λi))2,
(A.3)
∞,R,β → 0) = NtNr,
while for asymptotically high values of β the MSE results
∞,R,β → +∞) = NtNr/β.
We now compute the normalized MSE of the correlation
E[?R − γ?RLS
∞?2]
?R?2
=E[?(1 − γ)R + (1/β)INtNr)?2]
?R?2
= (1 − γ)2+2NtNr(1 − γ)/β
= (1 − γ)2+2NtNr(1 − γ)/β
≥E[?R −?RLS
=E[?R − γ(R + (1/β)INtNr)?2]
?R?2
?R?2
+NtNr(1/β)2
?R?2
+E[?R −?RLS
?R?2
.
∞?2]
?R?2
∞?2]
?R?2
(A.4)
Page 7
7
Therefore, for high values of Nm the estimator (11) has a
smaller MSE than the estimator (12).
C. Sensitivity of the estimator (10) to the estimation of β
Letˆβ be an estimated value for β. Hence, at the receiver,
the estimator (10) is implemented by usingˆβ in place of β as:
vec(?Hm) =ˆβ/(1 +ˆβ)vec(?HLS
vec(Hm) − vec(?Hm)
ing that X∗
dence between channel and noise, it can be readily shown
that the correlation matrix of the estimation error in this case
results
RE= E[vec(EH,m)vec(EH,m)∗]
?
that forˆβ = β it corresponds to (9). The channel estimation
MSE is then given by Tr(RE) = NtNr/(1 +ˆβ)2(1 +ˆβ2/β).
The numerical evaluation of the resulting MSE, which is not
reported herein for the sake of compactness, reveals that the
estimator is sensitive to an underestimation of the value of
β, while it is robust to an overestimation. On the other hand,
when the values of β is largely overestimated, the estimator
behaves like the LS channel estimator, in fact vec(?Hm) =
m). Therefore, by using (8), the
correspondent channel estimation error results: vec(EH,m) =
=(1 −ˆβ/(1 +ˆβ))vec(Hm) +
(ˆβ/(1 +ˆβ))(1/¯Pτ)¯X∗
τXτ= PτLINt, and by exploiting the indepen
τvec(Nτ,m). After some algebra, recall
=
1
1 +ˆβ
?2?
R +
ˆβ2
βINtNr
?
,
(A.5)
ˆβ/(1 +ˆβ)vec(?HLS
m) ? vec(?HLS
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