An iterative intercarrier-interference reduction algorithm for OFDM systems

Abstract

A low-complexity iterative algorithm is proposed for intercarrier-interference reduction in orthogonal frequency-division multiplexing systems. Design examples are presented which demonstrate that the proposed algorithm outperforms several existing algorithms in terms of bit-error-rate performance and computational complexity.

Full text

Page 1

An Iterative Intercarrier-Interference Reduction
Algorithm for OFDM Systems
Y. J. Kou, W.-S. Lu, and A. Antoniou
Department of Electrical and Computer Engineering, University of Victoria
P.O. Box 3055, Victoria, B.C., Canada V8W 3P6
Email:
?ykou, wslu?@ece.uvic.ca, aantoniou@ieee.org
Abstract—A low-complexity iterative algorithm is proposed for
intercarrier-interference reduction in orthogonal frequency-division mul-
tiplexing systems. Design examples are presented which demonstrate that
the proposed algorithm outperforms several existing algorithms in terms
of bit-error-rate performance and computational complexity.
I. INTRODUCTION
The demand for high data rate services over wireless networks
has been increasing very rapidly and there is no slowdown in
sight. These services often require reliable data transmission over
band-limited wireless channels, which experience many degradations,
such as noise, multipath fading and nonlinearities. A physical-layer
technique that has gained much popularity due to its robustness
in dealing with these impairments is orthogonal frequency-division
multiplexing (OFDM). Well known examples of OFDM modulation-
based systems include digital audio broadcasting (DAB) [1], digital
video broadcasting (DVB) [2], and the IEEE 802.11a and 802.11g
standards [3] for wireless local area networks (WLAN).
Unfortunately, there are several drawbacks associated with OFDM
modulation [4]. In a rapidly fading environment, channel variations
within an OFDM symbol duration lead to a loss of orthogonality
in the OFDM subcarrier waveforms and result in intercarrier in-
terference (ICI) which, in turn, degrades the bit-error-rate (BER)
performance of the system [5]-[8]. If not compensated for, ICI
will result in an error floor that increases with Doppler frequency.
However, channel variations also introduce frequency diversity which
can be exploited to improve the system performance [9]. Recently,
a number of algorithms have been proposed to mitigate the effect
of ICI [10]-[12]. An optimal linear pre-filtering algorithm has been
developed in [10] where improved performance was achieved at a
cost of increasd computational complexity. In [11], a linear minimum
mean-square error (MMSE) has been proposed. Since the number of
subcarriers is generally quite large, this algorithm requires intensive
computation. In attempts to reduce the computational complexity, a
decision feedback (DF) algorithm has been derived in [12] where
only signals on several neighbouring subcarriers are used in order
to suppress the ICI for a particular subcarrier. The computational
complexity of this algorithm is reduced at the cost of a slight
degradation of performance.
In this paper, a low-complexity ICI-reduction algorithm based on
an iterative optimization scheme is proposed for OFDM systems
where 4-quadrature-amplitude-modulation (4-QAM) is assumed for
all subcarriers. Design examples are presented which demonstrate
that the proposed algorithm outperforms several existing algorithms
in terms of BER performance and computational complexity. It is also
shown that a better performance can be achieved by the proposed
algorithm at higher Doppler frequencies because of the frequency
diversity introduced by channel variations.
II. SIGNAL MODEL
Consider an
Fig. 1, where
parallel-to-serial, and digital-to-analog converters, respectively, and
the block labeled as “????” represents a power amplifier (PA). The
information bits
the data point and subsymbol at the
tors
to as the frequency-domain and the time-domain
respectively.
?-subcarrier OFDM transmitter as illustrated in
???,
???, and
??? represent serial-to-parallel,
?
? and the modulated symbol
?
? are referred to as
?th subcarrier, respectively. Vec-
?
???
?
??????
?
??
?
?and
?
???
?
??????
?
??
?
?are referred
??????????s,
Amp.
Inverse
DFT
bit
S/P
stream
ModulationP/S
Insertion
CP
channel
DAC
00
x
N−1
N−1
X
N−1
D
xXD0
Fig. 1.An OFDM transmitter.
Mathmetically, the OFDM symbol
inverse discrete Fourier transform (IDFT) as
? can be obtained by using the
?
?
?
?
?
?
??
?
?
??
?
?
?
?
??????
for
???????????
(1)
where
be expressed as
?
? represents the
?th element of
?. In matrix form, (1) can
?
?
??
(2)
where
? is the IDFT matrix whose elements are
?
???
?
???
of the channel impulse response (CIR) is inserted in the begin-
ning of the OFDM symbol before it is transmitted into the chan-
nel. Denoting the transmitted and received signals as
???
?
??????. A cyclic prefix (CP) with length equal to that
?
??
?
?,
??
respectively, the received signal can be written as
?
????
??????
?
??
??
?
??????
?
??
?
?and
?
???
?
??????
?
??
?
?
?
?
??
?
??
?
?
(3)
where
noise (AWGN) variables with zero mean and covariance matrix
?
???
?
??????
?
??
?
?is a vector of additive white Guassian
?
?
??
??
??
?
?, and the channel matrix
?
?? is given by
?
??
?
?
?
?
?
?
???
?
?
???
?
????
?
?
??????
?
...
?
???
...
?
?
...
???
?
...
?
?
?
...
?
?
...
???
...
?
...
???????????
?
?
??
?
?
?
??
?
?
?
?
(4)
where
fading coefficient of the
the CP is only a copy of part of the OFDM symbol
rewritten as
?
?
?for
??????????? and
????????? represents the
?th path at the
?th sample instance. Since
?, (3) can be
?
?
??
?
?
(5a)

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where
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?????
?
?
?
?
?
?
?
...
?
?
?
...
?
?
...
???
?
...
?
?
?
...
?
?
???
???
...
?
???
???
...
?
...
???
...
?
...
??????
?
?
?
??
?
??????
?
??
?
?
?
??
?
?
?
?
?
?
?
?
?
?
(5b)
At the receiver, after the removal of the CP, the received signal is
transformed to
transform (DFT) as
?
???
?
??????
?
??
?
?by using the discrete Fourier
?
?
?
?
?
(6)
where
can be expressed as
???
?represents matrix Hermitian. From (2), (5a), and (6),
?
?
?
??
?
?
(7a)
where
?
?
?
?
??
(7b)
and
is unitary,
variations, matrix
at the
?
???
?
??????
?
??
?
?
?
?
?
?. Since the DFT matrix
?
?
? in (7a) is still white Guassian noise. Due to channel
? is not a diagonal matrix. In particular, the signal
?th subcarrier can be written as
?
?
??
? ??
?
?
?
?
??
?
????????
?
???
?
?
??
?
(8)
for
of matrix
signal depends not only on the transmitted signal for this particular
subcarrier but also on the transmitted signals for other subcarriers.
The first and second terms on the right-hand side (r.h.s.) of (8)
represent the attenuated signal and the ICI at the
respectively.
In order to mitigate the effect of the ICI, joint detection (JD) is
required at the receiver. This can be done by inserting immediately
after the DFT a processor that implements a JD algorithm. A general
structure for the OFDM receiver that implements a JD algorithm is
illustrated in Fig. 2. For the signal detection problem in (7), maximum
likelihood (ML) detection [13] can be carried out by solving the
optimization problem
??????????? where
?
? ?? represents the
?????th element
?. It can be seen that for the
?th subcarrier the received
?th subcarrier,
minimize
??
for
?
???
?
?
(9a)
(9b)
subject to
??
?
???????????????
where
modulation scheme of the OFDM system. The problem in (9)
is a combinatorial optimization problem whose solution requires
computational complexity that grows exponentially with the number
of variables. In addition, in OFDM systems over frequency-selective
fading channels, complete information on the CIR is required for ML
detection. In what follows, a low-complexity algorithm is proposed
that can be used to achieve a suboptimal solution of the optimization
problem in (9).
? is the set of the constellation points associated with
III. AN ITERATIVE ALGORITHM FOR ICI REDUCTION
The variables in (9) are complex-valued. If we let
?
?
?
?
??
?
?,
?
the form
?
?
?
??
?
?, and
?
?
?
?
??
?
?, then the norm in (9a) assumes
?
?
?
?
?
?
?
??
?
?
(10a)
S/P
bit
stream
Joint
Detection
DFT
CP
Removal
S/P
ADC
Φ[ ]
N−1
Y
0 Y
0 y
yN−1
Fig. 2. An OFDM receiver.
where
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
(10b)
and the problem in (9) can be expressed as
minimize
?
?
?
?
?
?
?
??
?
?
(11a)
(11b)subject to
?
?
?????
?
?
for
??????????????
where
imaginary components of the modulation constellation. For 4-QAM
modulation, we have
For the optimization problem in (11), the elements of vector
can be determined iteratively. Assume that the number and the set
of indices of the undetermined elements in vector
iteration are denoted as
iteration, we have
define
?
? represents the set of points associated with the real and
?
????????.
?
?
?
? during the
?th
?
? and
?
?, respectively. During the first
?
?
??? and
?
?
?????????????. If we
?
?
?
?
?
???
?
??
?
?
?
?
?
?????
?
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
(12)
and let
???, the problem in (11) can be converted to
minimize
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
(13a)
subject to
?
?
?
?
??????????
for
?????????
?
??
(13b)
Since the constraints in (13b) imply that
in (13) can be relaxed to
?
?
?
?
?
?
?
???
?, the problem
minimize
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
(14a)
subject to
?
?
?
?
?
?
?
?
???
?
(14b)
Note that matrix
can be diagonalized as
?
? is a positive definite matrix and, therefore, it
?
?
?
?
?
?
?
?
?
?
(15)
where
If we let
?
?and
?
?are diagonal and orthogonal matrices, respectively.
?
?
?
?
?
?
?
?
(16a)
(16b)
?
?
?
?
?
?
?
?
and
?
?
?
?
?
?
?
?
?
(16c)
then the problem in (14) can be converted to
minimize
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
(17a)
subject to
?
?
?
?
?
?
???
?
(17b)
Here, we define the Lagrangian
???
?
??
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
??
?
?
?
?
???
?
?
(18)

Page 3

Using the first-order Karush-Kuhn-Tucker (KKT) conditions, we
have
????
?
??
?
?
?
?
?
???
?
???
?
?
?
??
?
?
?
?
??
(19)
It can be seen from (17) and (19) that the solution of the problem in
(17) can be obtained by solving the following problem
?
subject to
?
????
?
???
?
?
?
?
??
?
?
?
(20a)
(20b)
?
?
?
?
?
?
???
?
On replacing vector
have
?
? in (20b) using the relationship in (20a), we
?
?
?
?
??
?
???
?
?
?
?
??
?
?
?
???
?
(21)
where
further expressed as
?
?
???
?
?
? is a diagonal matrix. Therefore, Eq. (21) can be
?
?
??
?
???
?
?
?
?
???
??
?
??????
?
?
?
???
?
(22)
where
main diagonal of matrix
in (22) is
the following one-dimensional optimization problem
?
?
?
??? and
?
?
??? represent the
?th element of vector
?
?
?and the
?
?, respectively. The only unknown variable
?
?and its value can be numerically determined by solving
minimize
?
?
?
?
?
?
?
??
?
???
?
?
?
?
???
??
?
??????
?
?
?
???
?
?
?
?
?
?
(23a)
subject to
??
?
??
?
??
?
(23b)
where
?
?
??
?
?
min
?????
?
??
?
?
???????
?
?
?
?
max
?????
?
??
?
?
?
?
?
(23c)
and
can be found using, for example, a dichotomous search algorithm.
When
? is a small positive number. The solution of the problem in (23)
?
? is found, vector
?
? can be computed as
?
?
?????
?
?
?
???
?
?
??????
?
for
?????????
?
??
(24)
where
estimzation of the vector
?
?
??? is the
?th element of vector
?
?. Using (16c), an
?
?
? can be obtained as
?
?
?
?
?
?
?
?
?
(25)
In order to obtain the discrete-valued vector, a decision process is
applied where
?
?
?
????
?
?
if
if
otherwise
?
?
?
?????
??
undetermined
?
?
?
??????
(26)
for
then the decision process in (17) becomes the hard decision
?????????
?
?? and
? is the decision threthold. If
???,
?
?
?
? sign
?
?
?
?
?
(27)
and all the elements of vector
If
region where
determined in the present iteration. In such a case, more iterations
are needed and a feedback scheme can be exploited. Denoting the
?
?
? can be determined by using (18).
???, some of the elements of vector
?
?
? may fall into the
?
?
?
?
??????, and, therefore, these elements cannot be
set of indices of the elements of
?
? that have been determined in the
?th iteration as
?
?
?, we have
?
?
??
??
?
??
?
?and
?
?
?
??
?
?
???
?
??
??
?
?
?
??
?
?
?????
?
??
??
?
?
?
??
?
?
?
?
?
?
?
????
?
?
?
?
???
?
?
??
?
?
??
??
?
?
?
?
??
?
?
?
??
?
?
?
??
???
?
?
?
?
??
?
?
?
??
(28)
Based on the updated variables in (28), a similar optimization
problem can be formulated as in (13) for the
can be solved following the same procedure. When all the elements
of vector
terminates. A step-by-step description of the above algorithm is listed
in Table 1.
??? ??th iteration, which
?
? are determined, i.e., set
?
?
?? is empty, the algorithm
TABLE I
AN ITERATIVE OPTIMIZATION ALGORITHM FOR ICI REDUCTION
Step 1
Set
Initialize variables
Step 2
Formulate the optimization problem as in (4).
Find
Compute
Compute
Find
Step 3
If
wise, compute the variables
??? and input the decision threthold
?.
?
?,
?
?,
?
?
?,
?
?
?,
?
?
?,
?
?, and
?
? as in (3).
?
? by solving the problem in (14).
?
?
? using (15) and (16).
?
?
? using (17).
?
?
?and compute
?
?
??.
?
?
?? is empty, output the detected vector
?
? and stop; other-
?
?
??,
?
?
??,
?
?
?
??,
?
?
?
??,
?
?
?
??,
?
Set
?
??, and
?
?
?? using (19).
?????, and repeat from Step 2.
IV. SIMULATIONS
The proposed ICI-reduction algorithm was applied to an OFDM
system where the number of subcarriers was chosen to be
4-QAM was adopted as the modulation scheme for each subcarrier.
The bandwidth of the OFDM system and the carrier frequency were
set to
the CP was set to
are the time durations of OFDM symbols and chips, respectively. A
two-tap Rayleigh fading channel model [12] was assumed where the
Doppler frequency of the channel is denoted as
of the first tap was zero, the delay of the second tap was randomly
generated with uniform distribution from
BER performance of the proposed algorithm was evaluated and
compared with that of several existing algorithms under a variety
of system configurations. For the DF algorithm in [12], the number
of neighbouring subcarriers that were used to suppress the ICI at a
particular subcarrier was taken to be
Example: First we considered an OFDM system where
For the proposed ICI-reduction algorithm with
the BER versus the ratio of energy-per-bit to spectral noise density
(Eb/N0) is plotted as the solid and dash curves in Fig. 3, respectively.
For the sake of comparison, the BER of the DF algorithm in [12] is
plotted in the same figure for various values of
It can be observed that improved performance can be obtained by
the proposed algorithm compared with that of the DF algorithm for
less computation. For example, at the BER level of
and 1.5-dB improvement of Eb/N0 can be achieved by the proposed
algorithm with
?? and
????? kHz and
?
?
?? GHz, respectively. The length of
?
?
?
?
???????
?
??
?
??, where
?
? and
?
?
?
?. While the delay
??
?
???
?
??????
?
?
?
?. The
????.
?
?
?
?
?????.
????? or
?????,
? as dotted curves.
??
??, a 0.5-dB
????? compared with the DF algorithm with
??
? and
?? ??, respectively. It was found out that the CPU time

Page 4

required by the proposed algorithm with
that required by the DF algorithm with
The performance of the proposed algorithm for the cases where
????? is only 80? of
????.
?
As can be seen, the performance of the algorithm improves with an
increase in the Doppler frequency. For example, while for the case
of
the BER level of
ratio of 25.5 dB is required to achieve the same BER level. This
improvement with increasing
frequency diversity introduced by the higher Doppler spread [9].
?
?
?
???? and
?
?
?
?
????? is plotted in Fig. 4 and 5, respectively.
?
?
?
?
???? an
????? ratio of 27.5 dB is required to achieve
??
??, for the case of
?
?
?
?
????? an
? ????
?
? can be attributed to the increase in
V. CONCLUSIONS
A low-complexity ICI-reduction algorithm based on an iterative
optimization scheme has been proposed. Design examples have been
presented which demonstrate that the proposed algorithm outperforms
several existing algorithms in terms of bit-error-rate performance and
computational complexity. It has also been shown that the proposed
algorithm exploits the frequency diversity introduced by channel
variations and, therefore, improved performance can be achieved at
higher Doppler frequencies.
ACKNOWLEDGEMENT
The authors are grateful to Micronet, NCE Program, and the Natural
Sciences and Engineering Research Council of Canada for supporting
this work.
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10 15202530
10
−3
10
−2
10
−1
DFD (K=5)
DFD (K=15)
Proposed (α=1.0)
Proposed (α=0.9)
Eb/N0 (dB)
BER
Fig. 3. Performance of ICI-reduction algorithms with
?
?
?
?
?????.
1015202530
10
−3
10
−2
10
−1
DFD (K=5)
DFD (K=15)
Proposed (α=1.0)
Propsoed (α=0.9)
Eb/N0 (dB)
BER
Fig. 4.Performance of ICI-reduction algorithms with
?
?
?
?
????.
1015 2025 30
10
−4
10
−3
10
−2
10
−1
DFD (K=5)
DFD (K=15)
Proposed (α=1.0)
Propsoed (α=0.9)
Eb/N0 (dB)
BER
Fig. 5.Performance of ICI-reduction algorithms with
?
?
?
?
?????.