ICI Reduction Method for OFDM Systems
Volker Fischer, Alexander Kurpiers and Dominik Karsunke
Institute for Communication Technology
Darmstadt University of Technology
Germany
v.fischer, a.kurpiers @nt.tu-darmstadt.de, [email protected]
O
RTHOGONAL Frequency Division Multiplexing
(OFDM) divides the transmission bandwidth into
many narrow sub-channels which are transmitted in parallel. The increased symbol duration together with the
insertion of a guard-interval mitigates the Inter-SymbolInterference (ISI) caused by time-dispersive fading channels. However, a transmission in a mobile communication environment or through an ionospheric channel is
impaired by both delay and Doppler spread. The time
variations of the channel due to Doppler spread introduce
Inter-Carrier-Interference (ICI) which degrades the performance. Using conventional channel equalization without ICI compensation, an additional noise term has to be
considered when estimating the channel [1]. The effects
of ICI are analyzed in [2] and a bound is given in [3].
The problem of performance degradation due to ICI
arises if an OFDM-based system like the digital terrestrial television (DVB-T) which was originally developed
for stationary reception is used in a mobile environment.
Furthermore, the new OFDM-based Digital Radio Mondiale (DRM) system [4] which uses a high-level modulation
II. S YSTEM M ODEL
For our investigation on ICI we consider a simplified
OFDM base-band system as depicted in Fig 1. In this
X0
X1
X2
..
.
XN
wn
sn
rn
hn
1
OFDM demodulation
I. I NTRODUCTION
system (16 / 64 QAM) also suffers from additional ICI
noise. We show in our paper the performance degradation
of a DRM transmission due to ICI on a fast fading channel
and that this impairment can be significantly reduced by
using the proposed algorithm.
In several papers, methods are described to combat the
ICI effects. One method is to linearly approximate the
channel variation during one OFDM symbol [5] [6]. In
most applications this approximation is possible since the
Doppler spread is usually much smaller than the carrier
spacing. From this approximation an estimate of the channel matrix can be obtained which can be used to cancel
ICI. In our paper, we propose a new method for constructing a linearized model which outperforms the algorithms described in [5] and [6]. Furthermore, a new lowcomplexity method for using the estimated channel matrix
for ICI cancellation is presented.
The paper is structured as follows: Sect. II defines a
system model which is sufficient for analyzing the ICI
effects. The estimation of the channel matrix utilizing a
linear approximation is described in Sect. III and a performance analysis of this method is given in Sect. IV. In
Sect. V, methods for using the estimated channel matrix
for ICI cancellation are presented and simulation results
are shown in Sect. VI.
OFDM modulation
Abstract— Orthogonal Frequency Division Multiplexing
(OFDM) has an increased symbol duration which makes
it robust against Inter-Symbol-Interference (ISI). However,
the longer symbol duration increases the Inter-CarrierInterference (ICI) caused by Doppler spread in time variant
channels. One method to combat the performance degradation due to ICI is to linearly approximate the channel variation during one OFDM symbol and create a channel matrix which can be used for ICI cancellation. This approximation is applicable if the Doppler spread is much smaller
than the carrier spacing, which is usually the case. There
are various possibilities for constructing a linearized model
for the channel matrix. In our paper, we propose a new
model which outperforms the algorithms published in earlier papers. We investigate the effects of ICI on the new
OFDM-based digital radio standard DRM and show that
our proposed algorithm can significantly reduce the performance degradation caused by ICI.
Y0
Y1
Y2
..
.
YN
1
Fig. 1. Base-band OFDM system.
paper, we assume perfect synchronization and the channel impulse response to be shorter than the length of the
guard-interval. Under these assumptions the convolution
of the transmitted signal and channel turns into a circular
convolution and the received signal can be expressed for
one OFDM symbol as
2π
1 N 1N 1 hm n Xl e j N l n m wn ∑
∑
N l 0 m 0
n 0 N 1 rn
(1)
where N is the number of sub-channels, Xl represents
the
transmitted symbol cell on the l th sub-carrier, h m n denotes the channel impulse response at position m and instant n and wn is a sample of white Gaussian noise.
After OFDM demodulation we get the relation between
transmitted and received symbol on a sub-carrier µ as
Hµ µ Xµ Yµ
N 1
∑ Hµ l Xl
l 0
l
µ
W
µ (2)
ICI
with Wµ
∑nN 1
j 2πN nµ
0 wn e
Hµ l
and the channel matrix
1 N 1N 1 ∑ hm n e N n∑
0 m 0
j 2π
N l m n nµ
(3)
The second term in Eq. 2 represents the ICI introduced
by the time variations of the channel. Using conventional
OFDM channel equalization with a one-tap equalizer, this
part is not compensated and acts as an additional noise
term. Thus, the Hµ µ in the first term are the channel estimates for a one-tap equalizer assuming ideal noise reduction. Consequently, by knowing the channel matrix,
an ICI perturbed ideal channel estimation for conventional OFDM channel equalization can be obtained. In our
derivations we assume that we know these estimates for
all sub-channels. In a real system these estimates must be
obtained from training symbol cells (so called pilot cells)
which are known at the receiver. In the DRM system the
pilot cells for channel estimation are scattered on the subcarriers. An example of a pilot pattern for one of the four
DRM modes is shown in Fig. 2. The channel estimates on
the data cells are obtained via interpolation.
transfer function as
N 1
Vl n ∑ hm n e j 2π
N lm
m 0
Using this definition, the channel matrix can be rewritten
as
2π
1 N 1 Hµ l
Vl n e j N n µ l (5)
N n∑
0
The Vl n are the time varying channel weights for each
sub-channel l and time instant n.
In the following, the calculation of the channel matrix
is given for a tap delay line channel model which can be
written as
I 1 (6)
rn ∑ γi n s n di i 0
where sn is the transmitted signal, γi n is the i-th fading
tap with the delay di and I is the number of taps. In this
special case, the calculation of the channel matrix simplifies to
2π
1 I 1 Hµ l
Γi l µ e j N ldi (7)
N i∑
0
2π
where Γi l ∑nN 01 γi n e j N nl is the DFT transformation of γi n .
III. E STIMATION
OF THE
C HANNEL M ATRIX
If the time variation of Vl n in Eq. 5 during one OFDM
symbol is small, a linear interpolation on each sub-carrier
l can be used as an approximation. For
applying the
approximation, at least one value of Vl n during each
OFDM symbol must be known. From a conventional onetap channel estimation
we get an estimate of Hµ µ which is
an average of Vµ n during one OFDM symbol according
to Eq. 5 setting l µ. Hence, we define
Vµ
Hµ µ
1 N 1 Vµ n N n∑
0
frequency (subcarriers)
Fig. 2. Pilot pattern for one of the four DRM modes. Filled dots mark
the pilot cell positions.
For our further derivations we define a time variant
(8)
It can be shown [5] that it is a good
choice to use the value
of V µ as a representation of Vµ n in the middle of the
symbol. The proposed linear model for one OFDM symbol is then
N 1
Vµ n V µ V µ n 2
time
(OFDM
symbols)
(4)
(9)
where V µ is the derivative of V µ . This derivative can
be calculated from the V µ value of the previous symbol
prev next (V µ
) and the next symbol (V µ ) with
V µ next prev Vµ
Vµ
2NS
(10)
where NS is the length of one OFDM symbol including
guard-interval. This corresponds to an average derivation
between the previous, current and next symbol. The proposed linearization is depicted in Fig. 3.
Vµ n
guard
interval
linear
approx.
N 1
2
N
previous
symbol
current
symbol
n
NS
next
symbol
Fig. 3. Linear approximation of Vµ n .
Inserting Eq. 5 in Eq. 2 by utilizing the approximation
from Eq. 9, using the fact that for integer n
N 1
∑e
j 2π
N nk
k 0
and defining ζn as
ζn
1 N 1 j 2π nk
ke N
N k∑
0
N
0
n
n
0 mod N
0 mod N
(11)
XµV µ diag v Ξdiag v N 1
2
n
1
1 e
N 1
∑ XlV l ζl l 0
containing the elements Hµ l . Since the matrix Ξ only depends on OFDM parameters, it is possible to precalculate
it once at initialization.
Our linear model differs from the model proposed in
[6] by the choice of positioning the origin of linearization.
Linnartz and Gorokhov use the beginning of an OFDM
symbol
as the origin which requires the knowledge of
Vµ n at this point. Using their model with our estimates
of V µ and V µ results in a poor performance.
The linear model in [5] proposes the same origin of linearization as we do but instead of using only one linear
function per OFDM symbol it is divided into two regions
with different slopes. As a result, the estimation of the
channel matrix requires more calculations than ours. In
the next section we show that our model not only is a
more efficient realization but also performs better than the
model using two slopes.
n
0 mod N
j 2π
N n
0 mod N
(12)
µ
XµV l ζ0 Wµ (13)
For a mathematical analysis of the linear approximation
proposed in the previous section we follow the approach
of [5]. By using the channel statistics defined in the DRM
standard it is possible to restrict the following equations to
a real valued analysis. For complex valued channel statistics the equations are slightly longer.
The Signal-to-Interference Ratio (SIR) assuming only
one fading path and a noise free system is given in [5] as
Using the following vector and matrix definitions
Ξ
v
v
w
x
y
ζ1 N
ζ1
0
..
.
0
ζ 1
..
.
ζ2 ..
N
ζN ζN ..
.
.
1
2
0
SIR
(14)
V 0 V 1 VN 1 T V 0 V 1 V N 1 T W0 W1 WN 1 T X0 X1 XN 1 T Y0 Y1 YN 1
T
N
2
2
N 1 σEm ρm Ei ∑m
0
ln 1 ε 1 ρ2m σ2Em
σ2Ĥ
with the normalized covariance
(17)
m
E Em Ĥm
σEm σĤm
ρm
(18)
where Ei stands for the exponential integral and
ε
10 6 .
Using Eq. 4 with only one path of
time-variant chan
nel (flat-fading) results in Vl n h n . Substituting this
result in Eq. 9 and together with Eq. 10 we get
Eq. 13 can compactly be written as
y
diag v Ξdiag v x w (16)
IV. P ERFORMANCE A NALYSIS
results in
Yµ
H
useful
part
0
where diag z is a diagonal matrix composed of the elements of vector z. The estimated channel matrix of size
N N follows as
ĥ n (15)
Nc n h Nc h Nc NG h NS Nc 2NS
(19)
where we define the position in the middle of useful part
as Nc N 2 1 and NG is the length of the guard-interval.
The error caused by linear approximation is defined as
e n
rHH m ĥ n h n 50
45
(20)
Since h n is a realization of a stochastic process H n ,
we define the following channel autocorrelation function
E H n m H n 15
rHH 0 1 Nc m rHH 2Nc NG rHH NS NS
1 Nc m 2 rHH 0 rHH 2Nc NS NG (23)
2
2NS
E Em Ĥm
rHH 0 1 Nc m rHH 2Nc NS
1 Nc m rHH NS 2NS
1 Nc m 2 rHH 0 2
2NS
30
rHH Nc m NG rHH NS 10
2
3
4
Doppler spread [Hz]
5
6
Fig. 4. Average SIR of models using one or two slopes for linearization. The OFDM carrier spacing is 46 Hz.
method is to apply an MMSE solution to Eq. 15 [6] to acquire an estimate of the transmitted symbols. This method
requires some matrix multiplications and the calculation
of the inverse of a matrix having the size N N. Since
N can often have large dimension it is necessary to simplify the calculation of matrix inversion. Since the most
energy in the channel matrix is concentrated in the neighborhood of the main diagonal, in [7] only some bands parallel to this diagonal line are used for inversion, the other
elements are set to zero. The effort for the inversion can be
significantly reduced with this method. The influence on
BER performance caused by simplification of the channel
matrix is shown in Fig 5, where we analyze an uncoded
−1
10
complete
q=2
q=4
q=6
q = 10
Nc m rHH Nc NG m rHH 2Nc NS NG 1
(24)
−2
10
BER
35
20
(21)
σ2Em
E Em2
2 rHH 0 rHH Nc m 1 Nc m rHH 2Nc NG rHH NS NS
rHH NS Nc m rHH Nc NG m 1 Nc m 2 rHH 0 rHH 2Nc NS NG (22)
2
2NS
E Ĥm2
40
25
After a straight forward calculation by using Eq. 19 - 21
we get the following averages:
σ2Ĥm
1 slope
1 slope simulation
2 slopes
55
average SIR [dB]
60
−3
10
In our
simulation, we use a Gaussian Doppler model for
rHH m defined in [4] Annex B. Fig. 4 shows the results
using the OFDM parameters of DRM mode B with a spectrum occupancy of 4 5 kHz. A good match is observed
between the analytical result and simulation. Additionally, we plotted the analytical results for using a two slope
linearization. We can see that the average SIR performance of the proposed model is approx. 5 dB better than
the model using two slopes.
V. ICI C ANCELLATION
In the previous sections, a linear model for estimating
the channel matrix was derived. There exist different possibilities to utilize this matrix for ICI cancellation. One
−4
10
20
25
30
35
SNR
40
45
50
Fig. 5. Influence of simplification of estimated channel matrix on
BER using uncoded modulation.
DRM system1 transmitted over a two path fading channel2 . The parameter q denotes the number of bands arranged next to the diagonal line used for matrix inversion.
As seen, the BER performance depends strongly on the
number of bands. Since even with using only two bands
1 DRM
mode B, 10 kHz bandwidth
5 as defined in [4]
2 Channel
(q 2) the effort is too high for a real-time implementation, we had to search for a better solution.
Similar to the idea in [8] we use the estimated channel
matrix in Eq. 2 to subtract the ICI component from the
received signal. This requires knowledge of the transmitted data cells. At the pilot cells, the transmitted symbol is
known but the data cells must be estimated. We propose
to use an MMSE equalization of the received symbols to
estimate these cells. This is a sub-optimal approach since
the received symbols are ICI perturbed. The resulting ICI
cancellation scheme is as follows
pilot cells N 1
N 1
Yl Hl l
Yµ Yµ ∑ Hµ l H 2 1 SNR ∑ Hµ l cl l l
l 0
l 0
l µ p l 0
l µ p l 1
(25)
3
where p l indicates the pilot positions , cl are the known
pilot cells and SNR is the Signal-to-Noise Ratio. The
ICI cancelled Yµ can now be used in a conventional onetap equalizer. Simulations showed that this scheme gives
comparable results to the method using simplified inversion of the channel matrix with q 4 but is much more
efficient from a computational point of view.
data cells
VI. S IMULATION
In our simulation, we use DRM system parameters 4
given in Annex A of the DRM standard. To get the worst
ICI effects we consider the DRM channel 5 which is the
fastest fading channel of the defined DRM channels suitable for the commonly used DRM mode B. It is a two
path fading channel having a common Doppler spread on
both paths of 4% of the carrier spacing. In Fig 6, the
−1
ideal
ideal with ICI
ideal with ICI and ICI comp.
Wiener
Wiener and ICI comp.
10
−2
BER
10
−3
10
−4
10
18
19
20
21
SNR [dB]
22
23
Fig. 6. BER performance for DRM channel 5.
BER performance for ideal channel estimation is plotted
3 If
l is a pilot cell then p l 1, otherwise p l 0
mode B, 10 kHz bandwidth, 64 QAM, code rate R 0 6
4 DRM
as a reference. The ICI perturbed ideal channel estimation gives a performance bound for the conventional channel estimation which is approx. 0 7 dB above the results
for ideal channel estimation at an BER of 10 4 . Applying our ICI cancellation scheme to the noise free ICI perturbed ideal channel estimation results in nearly identical
performance as ideal channel estimation. It shows that
the linear model is a very good approximation of the real
channel matrix since ICI is cancelled almost completely.
The thicker curves show the performance of ICI cancellation applied to a real channel estimation based on pilots
as depicted in Fig. 2 and using a Wiener channel estimation method described in [9]. The performance gain of our
scheme at a BER of 10 4 is approx. 0 5 dB which compares very well with the 0 7 dB ICI effect on ideal channel
estimation.
VII. C ONCLUSION
In our paper, we presented a new method of building
a linear model for the estimation of the channel matrix
which outperforms models proposed in previous papers.
We evaluated a method for cancelling the ICI by calculating the inverse of a matrix which turned out to be not
practically viable. An alternative cancellation scheme was
proposed which performs well and does not require a matrix inversion. Using our ICI cancellation scheme on the
digital radio standard DRM can significantly reduce the
performance degradation due to ICI noise.
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