Mobile WiMAX Impact of Channel Estimation Error on the Performance of Limited Feedback Linear...
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Abstract — The mobile WiMAX standard (802.16e) uses
multiple-input multiple-output (MIMO) limited feedback linear
precoding to exploit the channel state information at the
transmitter. Although the performance of limited feedback
linear precoding in relation to traditional open-loop MIMO-
OFDM has been extensively studied in the literature, these
studies commonly assume perfect channel estimation at the
receiver. In a practical OFDM-based system, the estimated
channel matrix often differs from the actual channel matrix due
to errors incurred in the channel estimation process. This
results in degraded performance relative to the case with
perfect channel estimation. To date, few researchers have
studied the impact of channel estimation error on the
performance of an OFDM limited feedback linear precoding
system. This paper investigates the channel estimation errorusing 1) an MMSE channel estimator that takes into account
the subcarrier correlation when estimating the channel, 2) a
Low Rank (LR) channel estimator that relaxes the requirement
for a perfect channel covariance matrix in the MMSE receiver,
and 3) a ZF estimator where this correlation information is
ignored. Simulation results show that with the MMSE
estimator the system suffers very little array gain loss with a
performance degradation of 0.2dB SNR. Compared to the
MMSE estimator, the LR estimator incurs a small performance
loss of around 0.5dB. Finally, when the ZF estimator is
implemented, a significant performance degradation is
observed with approximately 4-5dB loss in array gain loss.
Index Terms —802.16e, WiMAX, MIMO, linear precoding,
limited feedback.
I. I NTRODUCTION
The first WiMAX systems were based on the IEEE
802.16-2004 standard [1]. This targeted fixed broadband
wireless applications via the installation of Customer
Premises Equipment (CPE). In December 2005 the IEEE
completed the 802.16e-2005 [2] amendment, which added
new features to support mobile applications.
Mobile WiMAX now supports both open-loop and
closed-loop multiple-input multiple-output (MIMO)
techniques. Open-loop techniques, such as space time block
coding (STBC) and spatial multiplexing (SM), can be used
to increase diversity gain or system throughput without the
need for channel state information (CSI) at the transmitter.However, recent work [3, 4] has reported further increases in
system performance (both diversity and array gain) and
throughput by applying linear precoding closed-loop
techniques at the transmitter that exploit knowledge of the
CSI.
The key idea behind linear precoding is to customize the
transmit signal by pre-multiplication with a precoding
matrix. It is well-known that singular value decomposition
(SVD) linear precoding provides the highest achievable
performance [4]. However, the SVD approach requires
perfect CSI at the transmitter, which cannot be achieved in a
MIMO Mobile WiMAX system with numerous antennas,
subcarriers, and a rapidly changing channel. The need to
reduce the amount of CSI feedback information motivates
the use of a codebook based linear precoding technique [5,
6]. Here, the mobile station (MS) calculates the optimal
precoding matrix for each subcarrier and feeds back the
matrix, rather than the CSI, to the base station (BS).
Specifically, the optimal precoding matrix is constrained to
one of N distinct matrices, which are referred to as codebook
entries, designed offline and known to both the MS and BS.
The MS identifies the optimal precoding matrix based on the
current CSI. Since the codebook is known at the BS, the MS
only needs to feedback a binary index of the optimal precoding matrix, rather than the entire precoding matrix
itself. For each combination of the number of transmit ( T N )
and receive ( R N ) antennas, the 802.16e standard defines
two codebooks: one with 8 entries and the other with 64
entries [7]. These correspond to 3-bit and 6-bit codebook
indices for each precoding matrix respectively.
The performance improvement of codebook based linear
precoding MIMO-OFDM systems has been previously
reported in the literature [8]. However, results are often
based on the assumption that the channel is perfectly
estimated at the receiver. In practice, the OFDM channel
estimator at the receiver always leads to imperfect channel
estimation. If the channel is not perfectly known at the
receiver, the performance of the linear precoding technique
will degrade due to two reasons: 1) the receiver will select
the optimal precoding matrix based on the errored channel
estimation H , and not the true channel H, and 2) the use of
errored channel information at the receiver (e.g., H is used
instead of H in the MMSE receiver of the SM system). This
paper investigates the impact of channel estimation error on
the performance of a linear precoding mobile WiMAX
system using the distributed PUSC subcarrier permutation
scheme.
The paper is organized as follows: Section II describes
important parameters used in the mobile WiMAX simulator.
An overview of precoded spatial multiplexing and dominant
eigenbeamforming for mobile WiMAX systems is described
in Section III. Training-based channel estimation techniques
supported in the mobile WiMAX standard are described in
Section IV. Section V investigates the impact of channel
estimation error on the performance of a precoded mobile
WiMAX system. Finally, conclusions are presented in
Section VI.
II. LINK LEVEL MOBILE WIMAX SIMULATOR
A detailed downlink Mobile WiMAX link-level simulator
Mobile WiMAX: Impact of channel estimation error on
the Performance of Limited Feedback Linear PrecodingMai Tran, Andrew Nix, and Angela Doufexi
Centre for Communications Research, Merchant Venturers Building,
University of Bristol, Bristol BS8 1UB, UK
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[9] using the PUSC subcarrier permutation and
convolutional coding with soft Viterbi decoding has been
implemented by the authors based on the 802.16e-2005
standard [2]. The simulator models a cell with an omni-
directional basestation (BS) and three mobile stations (MS)
randomly situated in the cell. In the downlink, each MS is
randomly allocated 5 out of a total of 15 subchannels. The
BS transmits data simultaneously to 3 MS, with each sharing
a common OFDMA symbol. Table I summarises the
OFDMA parameters used in the Mobile WiMAX simulator.
A detailed description of the simulator can be found in [9].
TABLE I: OFDMA PARAMETERS
Parameter Value
Carrier frequency (GHz) 2.3
FFT size 512
Channel bandwidth (MHz) 5
Sampling frequency F s (MHz) 5.6
Sampling period 1/ F s ( s) 0.18
Subcarrier frequency spacing ∆ f =F s /N FFT (kHz) 10.94
Useful symbol period T b = 1/ ∆ f (µs) 91.4
Guard Time T g = T b/8 (µs) 11.4
OFDMA symbol duration T s =T g +T b (µs) 102.9
Number of used subcarriers ( N used ) 421
Number of pilot subcarriers 60
Number of data subcarriers 360
Number of data subcarriers in each subchannel 24
Number of subchannels 15
Number of users ( N users) 3
Number of subchannels allocated to each user ( N ALSU ) 5
Based on the ETSI 3GPP2 spatial channel model (SCM)
[10, 11], urban micro and urban macro tapped delay line
(TDL) channels were generated for use in this analysis. The
TDL comprises 6 taps with non-uniform delays. Each tap
experiences Rayleigh fading based on an MS velocity andthe traditional Jake’s Power Doppler Spectrum [12]. The
antenna element separation is 10 λ at the BS and 0.5 λ at
the MS, where λ represents the carrier wavelength.
III. LINEAR PRECODING
This section summarizes two different linear precoding
systems, namely linear precoding spatial multiplexing (SM
PRE) and dominant eigenbeamforming (DE), both of which
are implemented in the mobile WiMAX simulator.
A. Linear precoding spatial multiplexing (SM PRE)
For purposes of simplicity, a generic linear precoding
spatial multiplexing system for a single subcarrier is
illustrated in Fig. 1.In the case of an OFDM mobile WiMAX system, the k -th
subcarrier is allocated a precoding matrix k F , and the
1 R N × receive symbol vector k y is given by
sk k k k k
E
M = +y H F s n (1)
where k is the subcarrier index, s E is the total transmit
power for the k -th subcarrier, M is the number of spatial
streams ( T M N < ), k H is the R T N N × normalised channel
matrix, k s is an 1M × transmit data symbol vector (which is
spread over T N transmit antennas by multiplying by an
T N M × precoding matrix k F ), and k n is an 1 R N × noise
vector whose entries are complex, independent and
identically distributed (i.i.d) additive white Gaussian noise
(AWGN) samples with zero mean and variance 2σ .
T N x
R N y
1 x
1 y
Fig. 1: Linear precoding spatial multiplexing system block diagram
In this paper the received symbol vector k y is decoded
using an MMSE linear decoder k G , given by
-12* * * *n
k k k k k M k k
s
M
E
σ
G = F H H F + I F H . (2)
The optimal precoding matrix opt F is determined for each
subcarrier using the minimum mean square error (MSE)
criterion [5] as
( )1
* * * *
2
s sk M k k k k
n
E E MSE
M M σ
−
= +
F I F H H F (3)
where
( )( )argminik
i
opt k Q
trace MSE ∈
=F
F F (4)
and Q is the codebook (which is known to both the BS and
MS). Q is constructed using the methods described in
section 8.4.5.4.10.15 of [2].
B. Dominant Eigenbeamforming (DE)
The second linear precoding system considered in this
paper is dominant eigenbeamforming (DE), as illustrated in
Fig. 2 for the k -th subcarrier. Here the BS transmits a single
spatial stream across the T N transmit antennas.
T N x
R N
1 x
1 y
Fig. 2: Dominant eigenbeamforming system block diagram
In a precoded mobile WiMAX system, the k -th subcarrier
is assigned a 1T N × precoding vector k f . The receive
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symbol vector k y for a DE system can be expressed as
k s k k k k E s= +y H f n . (5)
In this paper k y is decoded using a traditional maximum
ratio combiner g [6].
2/k k k k k =g H f H f (6)
The optimal precoding vector opt f is determined from (7)
using the criterion defined in [6]. 2
2argmax
ik
i
opt k k Q∈
=f
f H f (7)
IV. CHANNEL ESTIMATION IN MOBILE WIMAX
In order to perform channel estimation, the mobile
WiMAX standard supports a training-based technique [2,
13] where known symbols are transmitted to aid the
receiver’s channel estimation algorithm. There are two ways
to transmit training symbols: 1) transmitting preamble-based
symbols in which known preambles are sent at the beginning
of each frame, and 2) transmitting pilot-based symbols
where several known pilots are inserted into each OFDM
symbol within a frame in order to track the changing channel
between OFDM symbols.Our mobile WiMAX simulator assumes a block fading
channel where the channel remains constant over a WiMAX
transmission frame, but changes between frames. Therefore
only the preamble is needed in our simulator to estimate the
channel. With a frame duration of 5ms, as defined in the
standard, this block fading assumption is valid for mobile
applications with velocities up to 80 km/h (i.e., a coherence
time of 6 ms). A preamble-based OFDM channel estimation
system with N subcarriers is often modelled as
=y Xh + n (8)
where X is an N N × diagonal matrix whose diagonal
elements are the pilot symbols in the frequency domain, h is
an 1 N ×
complex channel vector whose entries are thefrequency response of N subcarriers, and n is an 1 N × noise
vector of independent and identically distributed complex,
zero-mean Gaussian noise variables with variance 2σ .
Without loss of generality, we assume that the channel is
normalised such that { }21k E h = and { }2
, 1k k E X = . The
channel estimate h can be obtained using the zero forcing
(ZF) or the minimum mean-square error (MMSE) channel
estimator [14-16]. For example, by using the ZF channel
estimator, the channel estimate vector h is given by1ˆ
ZF −=h X y . (9)
The ZF channel estimator is implemented with very low
complexity but fails to consider the potentially significantcorrelation between subcarriers, and therefore suffers from a
high mean-square error [14]. In order to improve the quality
of the channel estimate, an MMSE based channel estimator
[14, 16] that minimizes the mean-square error by leveraging
the subcarrier correlation, can be used. The MMSE channel
estimate ˆMMSE h in the frequency domain is given by [14]
( )1
2ˆ ˆMSE hh hh ZF σ
−= +h R R I h (10)
where R hh= E {hh*} denotes the auto-covariance matrix of
the channel vector h and I denotes the N N × identity
matrix.
The main drawback of the MMSE estimator is that it
requires a perfect channel covariance matrix at the receiver.
In practice the receiver does not often have this information
in advance, and hence this too needs to be estimated. The
work in [17] proposes a low-rank MMSE estimator (LR)
that uses the channel covariance matrix estimated from a
uniform power-delay profile (pdf). The estimated correlation between the m-th and the n-th subcarrier in this case is given
by
2
,
1, if
, if 1
m n j L
N m n
m n
r m ne
m n j L
N
π
π
−−
=
= ≠− −
2
(11)
where L is the number of samples in the guard interval, and
N is the number of subcarriers within an OFDM symbol. It
can be seen that this estimator only requires knowledge of
the guard interval length and the number of subcarriers in
the system. Results in [17] also show that the wrong channel
statistics (due to the use of a uniform pdf) only incurs asmall performance loss relative to the case with perfect
knowledge of the covariance matrix.
In a MIMO-OFDM system the received signal at each
receive antenna is the superposition of transmit signals from
T N transmit antennas. Therefore, in order to differentiate
the preamble signals transmitted from each antenna, an
independent pattern for transmitting preamble signals [18] is
implemented as illustrated in Fig. 3. The independent pattern
transmits preamble signals from each antenna at a time when
the other antennas keep silent. By doing this all the
preambles are received at the receiver without interfering
with one another.
Fig. 3: Independent pattern for transmitting preambles [18] for a MIMO
system with 2 transmit antennas
0 50 100 150 200 250 300 350 4000
0.2
0.4
0.6
0.8
1
1.2
1.4
Subcarrier index
C h a n n e l a m p l i t u d e
Perfect channel
Estimation at 0dB SNR
Estimation at 10dB SNR
Estimation at 20dB SNR
Fig. 4: MMSE channel estimation at various SNRs
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Fig. 4 and Fig. 5 compare the actual channel with the
channels estimated using the MMSE and ZF estimators,
correspondingly, for various SNR values. It can be seen that
the MMSE estimator achieves a very accurate channel
estimate. It can maintain a reasonable channel estimate
accuracy even at 0dB SNR. However when the ZF estimator
is used the channel estimate performance becomes very poor
at low SNRs. At an SNR of 0dB the ZF channel estimate is
unusable.
0 50 100 150 200 250 300 350 4000
0.5
1
1.5
2
2.5
3
Subcarrier index
C h a n n e l a m p l i t u d e
Perfect channel
Estimation at 0dB SNR
Estimation at 10dB SNR
Estimation at 20dB SNR
Fig. 5: ZF channel estimation at various SNRs
Fig. 6 compares the mean-squared error (MSE) between
the MMSE, LR and ZF estimators. As expected, the MSE of
the MMSE and LR estimators are much smaller than that of
the ZF estimator.
0 5 10 15 20 25 30 35 4010-6
10-4
10-2
100
SNR (dB)
M S E
LR uniform estimator
MMSE estimator
ZF estimator
Fig. 6: MSE of MMSE, LR and ZF estimators
V. SYSTEM PERFORMANCE ANALYSIS
This section studies the packet error rate (PER)
performance of the precoded MIMO mobile WiMAX
system with MMSE, LR and ZF channel estimation. Linear
precoding with channel estimation errors can be simulated
by assuming that the receiver selects the optimal precoder
matrix ( )ˆ f =F H using knowledge of the channel estimate
matrix H , and not the true channel H.
The ideal case with perfect channel knowledge at thereceiver (denoted as Perf H) is demonstrated as a benchmark
for PER comparison.
A. Dominant Eigenbeamforming
Fig. 7 shows the PER performance of the 2 2× QPSK
and 16QAM 3/4 rate DE systems using MMSE, LR and ZF
channel estimators. It can be seen that the performance for
the MMSE channel estimate is quite close to that of the ideal
channel system (degraded by approximately 0.2dB). The LR
estimator, although using inaccurate channel statistics, only
suffers an approximate 0.5dB loss relative to the MMSE
approach. This result agrees with the performance of the LR
estimator demonstrated in [17]. Finally, the ZF channel
estimation scheme suffers significant performance
degradation with a 4dB loss in array gain.
Fig. 8 illustrates the impact of MMSE, LR and ZF channel
estimations on the PER performance of a 2 2× QPSK and
16QAM 3/4 rate Alamouti system [19]. Comparing Fig. 7
with Fig. 8 it can be seen that different channel estimation
algorithms have a similar impact on the performance of
closed-loop and open-loop MIMO diversity systems. This
similar performance degradation initially seems counter-intuitive because the channel estimation in a closed-loop
MIMO system causes both an incorrect precoding matrix
ˆopt f at the transmitter and an errored channel H at the
receiver. This is expected to result in a higher performance
degradation than the open-loop MIMO system, where only
an errored channel H is experienced at the receiver. In fact,
a closed-loop MIMO system in this case is equivalent to an
open-loop MIMO system operating over a channel ˆopt Hf
with an errored channel estimate ˆˆopt Hf at the receiver. This
results in a similar performance loss to the open-loop
system.
-4 -2 0 2 4 6 8 10 12 14 16 1810
-3
10-2
10-1
100
SNR (dB)
P E R
QPSK 3/4 DE PRE 2x2 Perf H
16QAM 3/4 DE PRE 2x2 Perf H
QPSK 3/4 DE PRE 2x2 MMSE H
16QAM 3/4 DE PRE 2x2 MMSE H
QPSK 3/4 DE PRE 2x2 LR uni
16QAM 3/4 DE PRE 2x2 LR uni
QPSK 3/4 DE PRE 2x2 ZF H
16QAM 3/4 DE PRE 2x2 ZF H
Fig. 7: PER performance of 2x2 DE QPSK and 16QAM 3/4 rate systems
with MMSE, LR and ZF channel estimation
-2 0 2 4 6 8 10 12 14 16 18 2010
-3
10-2
10-1
100
SNR (dB)
P E R
QPSK 3/4 ST22 Perf H
16QAM 3/4 ST22 Perf H
QPSK 3/4 ST22 MMSE H
16QAM 3/4 ST22 MMSEH
QPSK 3/4 ST22 LR uni
16QAM 3/4 ST22 LR uni
QPSK 3/4 ST22 ZF H
16QAM 3/4 ST22 ZF H
Fig. 8: PER performance of 2x2 Alamouti QPSK and 16QAM 3/4 rate
systems with MMSE, LR and ZF channel estimation
-6 -4 -2 0 2 4 6 8 10 12 14 1610
-3
10-2
10-1
100
SNR (dB)
P E R
QPSK 3/4 DE PRE 4x2 Perf H
16QAM 3/4 DE PRE 4x2 Perf H
QPSK 3/4 DE PRE 4x2 MMSE H
16QA M 3/4 DEPRE 4x2 MMSE H
QPSK 3/4 DE PRE 4x2 LR uni
16QAM 3/4 DE PRE 4x2 LR uni
QPSK 3/4 DE PRE 4x2 ZF H
16QAM 3/4 DE PRE 4x2 ZF H
Fig. 9: PER performance of 4x2 DE QPSK and 16QAM 3/4 rate systems
with MMSE, LR and ZF channel estimation
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Fig. 9 illustrates the PER performance of a 4 2× QPSK
and a 16QAM 3/4 rate DE system using MMSE, LR and ZF
channel estimators. It demonstrates a performance
degradation of 0.2dB, 1dB, and 5dB for MMSE, LR and ZF
systems respectively.
B. Linear precoding spatial multiplexing
Fig. 10 studies the impact of MMSE, LR and ZF
estimators on the PER performance of 4 2× SM PRE QPSK
and 16QAM 3/4 rate systems. It presents the same performance degradation as observed for the DE system.
Compared to the ideal system with perfect channel
knowledge, the use of an MMSE estimator only incurs a
small array gain loss of 0.2dB. The LR estimator suffers
very little loss relative to the MMSE estimator, and the ZF
estimator degrades by approximately 4dB in array gain.
0 5 10 15 20 2510
-3
10-2
10-1
100
SNR (dB)
P E R
QPSK 3/4 SM PRE 4x2 Perf H
16QAM 3/4 SM PRE 4x2 Perf H
QPSK 3/4 SM PRE 4x2 MMSE H
16QAM 3/4 SM PRE 4x2 MMSE H
QPSK 3/4 SM PRE 4x2 LR uni
16QAM 3/4 SM PRE 4x2 LR uni
QPSK 3/4 SM PRE 4x2 ZF H
16QAM 3/4 SM PRE 4x2 ZF H
Fig. 10: PER performance of 4x2 SM PRE QPSK and 16QAM 3/4 rate
systems with MMSE, LR, and ZF channel estimation
Fig. 11 demonstrates the PER performance of a 2 2×
open-loop SM system using different channel estimators. As
expected, a similar performance loss to the SM PRE system
(Fig. 10) is observed.
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 3810
-3
10-2
10-1
100
SNR (dB)
P E R
QPSK 3/4 SM 2x2 Perf H
16QAM 3/4 SM 2x2 Perf H
QPSK 3/4 SM 2x2 MMSE H
16QAM 3/4 SM 2x2 MMSE H
QPSK 3/4 SM 2x2 LR uni
16QAM 3/4 SM 2x2 LR uni
QPSK 3/4 SM 2x2 ZF H
16QAM 3/4 SM 2x2 ZF H
Fig. 11: PER performance of 2x2 SM QPSK and 16QAM 3/4 rate systems
with MMSE, LR and ZF channel estimation
VI. CONCLUSIONS
This paper has studied the impact of channel estimationerror on the performance of a linear precoding Mobile
WiMAX system. Three different channel estimators were
implemented: 1) an MMSE estimator with a perfect channel
covariance matrix, 2) an LR estimator with an estimated
channel covariance matrix, and 3) a ZF estimator where the
correlation factor is ignored. Spatial Multiplexing and
Dominant Eigenbeamforming systems with 2 2× and 4 2×
antenna configurations were studied. Results have shown
that a linear precoding Mobile WiMAX system using
MMSE channel estimation maintains a very good
performance with only a 0.2dB array gain degradation
compared to the ideal system with perfect channel
knowledge. The LR channel estimator, based on a uniform
channel correlation, only incurred a small loss of
performance (i.e., 0.5dB) relative to the MMSE, even
though it used incorrect channel statistics. Finally, a ZF
estimator with no channel statistic information degraded
system performance by 4-5dB.
VII. ACKNOWLEDGEMENTS
The authors would like to thank the Centre for Communications Research for providing a range of high-
performance computing facilities.
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