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Frequency Domain Equalization for Doubly Selective Channels
YuQin Chen JongYoung Han Jae Moung Kim
river4416@sina.com, fanaticey@hotmail.com, jaekim@inha.ac.kr
Graduate school of Information Technology & Telecommunications Inha University
Abstract- In this paper, a kind of conventional linear frequency-domain equalizer for doubly selective (time-and
frequency-selective) channels is proposed. We use a basis expansion model (BEM) to approximate the doubly selective channels.
The time-varying impulse response of rapidly fading mobile communication channels is expanded over a basis of complex
exponentials that arise due to Doppler effects encountered with multipath propagation. This model allows us to turn a complicated
equalization problem into a simpler one, containing only the BEM coefficients of the doubly selective channels and the frequency
domain equalizer. Both Zero-Forcing (ZF) and Minimum Mean Square Error (MMSE) equalization are considered.
I. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is an
attractive technique for high-speed data transmission in mobile
communications, since it can avoid intersymbol interference (ISI).
The Doppler effect caused by time-variation in conjunction with
ISI give rise to a so-called doubly selective (frequency- and
time-selective) channel. In a doubly selective channel, the channel
variation over an OFDM block destroys the orthogonality between
the subcarriers resulting into so-called inter-carrier interference
(ICI)[1]. In order to mitigate ICI, many techniques, e.g.,
time-domain windowing combined with iterative MMSE
estimation, ICI self-cancellation, have been proposed[2][3]. These
approaches require a high computational complexity or at a price
of sacrificing the spectral efficiency. Moreover, these works
assume perfect knowledge of the TV channel at the receiver,
which is rather difficult if not impossible to obtain.
In this paper, a kind of frequency domain equalization technique
which can compensate for the effects of channel variation in a
multipath fading channel is described by approximating that the
channel impulse response (CIR) is expanded over a basis of
complex exponentials that arise due to Doppler effects
encountered with multipath propagation. This allows us to turn a
large time-variant (TV) problem into an equivalent small
time-invariant (TIV) problem, containing only the BEM
coefficients of the doubly selective fading channel[4].
II.FADING CHANNELS: RANDOM MODLES
In some communication schemes, unpredictable changes in the
medium warrant modeling the TV impulse response (TVIR)
h c(t;τ )as a stochastic process in the time variable t. Since the
components of the multipath signal arise from a large number of
reflections and scattering from rough or granular surfaces, then by
virtue of the central limit theorem, the TVIR can be modeled as a
complex Gaussian process. A simple way to model the fading
channel is wide-sense stationary uncorrelated scattering (WSSUS).
It assumes the channel to be wide sense stationary for a fixed lag
τ and uncorrelated for different lags. The channel spectral density
for a fixed τ is called the scattering function S(ω ;τ );and it
provides a single measure of the average power output of the
channel as a function of the delayτ and the Doppler frequency
ω . The delay-power profile, sometimes also called as multipath
intensity profile, which is defined as the integral ,
∫ ∞
−∞
p ( τ ) = S(
τ , w)
dw
(1)
represents the average received power as function of delay τ ,and
the length of its support is called the multipath spread and is a
measure of the average extent of the multipath. Another function
called Doppler power spectrum
∫ ∞
−∞
S ( w)
= S(
τ , w)
dτ
(2)
is derived from the scattering function and its extent, the Doppler
spread, is a measure of the channel’s time variation[2][5].
s(l)
( )
f tr
c
( l)
sc(t )
(
f ch
c
)
( t;
τ)
vc(t )
rc (l)
f (l)
rec
c
xc (t)
nTs
Fig. 1. Continuous-time TV communication system.
x(n)
Consider the fading communication system model of Fig. 1
before the sampler with the input/output (I/O) relationship
∑ ∞
=
l=
−∞
x c ( t)
s(
l)
hc
( t;
t − lTs
) + vc
( t)
(3)

where subscript c denotes continuous time, h c(t;T) is the
convolution of the spectral pulse f c (tr) (t), the TV impulse response
f c (ch) (t;τ ), the receive-filter fc (rec) (t), s(l) is the sequence of input
symbols, and vc(t) is the noise process. h c(t;T) can be truncated to
an order L, which is common practice in communication
applications. On approach in characterizing the variation of the
impulse response h(n;l) is to consider it as a stochastic process in
the time index n. In communication, tracking the variations of the
channel taps is of importance. To this end, linear TV channel is
assumed in [6].
III. FADING CHANNELS: BASIS EXPANSION MODELS
In this paper, we model the doubly selective channel h(n;v)
using a basis expansion model (BEM). In this BEM, the channel is
modeled as a TV FIR filter, where each tap is expressed as a
superposition of complex exponential basis functions with the
frequencies on a DFT grid, as described next.
Let us first make the following assumptions:
A1) The delay spread is bounded by τ max.
A2) The Doppler spread is bounded by fmax.
Under assumption A1) and A2), it is possible to accurately
model the doubly selective channel h(n;v) for n∈{0,1,..., N −1}
as
(see Fig.2)
h(
n;
v)
=
L
∑ v−l
Q / 2
j 2πqn
/ N
∑e
hq,
l
l=
0 q=
−Q
/ 2
δ (4)
where L and Q satisfy the following conditions:
C1) LT≥τ max;
C2) Q/(NT) ≥2f max;
h− Q /2,0 h− Q /2,1 h− Q /2,2
Q/2, L
j2 Q/2 n/ N
h Q /2,0 h Q /2,1 h Q /2,2 hQ/2, L e π
h −
Fig. 2. BEM channel.
e
− j2 πQ/2
n/ N
where T is the symbol period. In this expansion model, L
represents the discrete delay spread (expressed in multiples of T,
the delay resolution of the model), and Q/2 represents the discrete
Doppler spread (expressed in multiples of 1/(NT), the Doppler
resolution of the model). Note that the coefficients h q,l remain
invariant for n∈{0,1,..., N −1}
and hence are the BEM
coefficients of interest[7].
IV. BLOCK DATA MODEL.
The discrete-time baseband equivalent model of the OFDM
system under consideration is shown in Fig. 3. In the transmitter,
the transmitted high-speed data is first converted into parallel data
of N subcarriers by the serial-to-parallel converter (S/P). After
each symbol is modulated by the corresponding subcarrier, it is
sampled and D/A converted. The sampled symbols, implemented
by an inverse fast Fourier transform (IFFT), can be expressed as
follows:
x
∑ −
=
=
1 N
n
m 0
X
e
j 2πnm
/ N
m
, 0 ≤ n ≤ N
(5)
where xn represents the nth sample of the output of the IFFT[6].
Input Bit
Stream
Output Bit
Stream
S/P
P/S
X m
~
IFFT
Equalizer
Y m
Xm
FFT
yn
P/S
xn
D/A
S/P A/D
AWGN
wn
Multipath
Channel
Fig. 3. A baseband block diagram for an OFDM system.
Assuming that the multipath fading channel consists of L
discrete paths, the received signal can be written as
L−1
∑
y = h x + ω = h x + h x + ... + h x 1+ ωn
n n, l n−l n n,0 n n,1 n−1 n, L−1 n− L+
l=
0
where hn,l and ω n represent the complex random variable for the
lth path of the CIR and additive white Gaussian noise (AWGN) at
time n, respectively. For reason of simplicity, a cyclic extension of
length N G, used to avoid ISI and to preserve the orthogonality of
subcarriers, is not shown in Fig.4 and not used in this paper’s
simulation. The demodulated signal in the frequency domain is
obtained by taking the FFT of y n as
m
N−1 L−1
∑∑
k= 0 l=
0
k
( m−k) l
− j2 πlk/
N
Y = X H e + W
L−1
⎡ 0 − j2 πlm/
N⎤
∑ He l Xm
l=
0
= ⎢ ⎥
⎣ ⎦
N−1 L−1
∑∑
+ XH e + W
k≠ m l=
0
( m−k) − j2 πlk/
N
k l m
= α X + β + W , 0≤m≤ N −1
m m m m
( m k)
where Wm denotes the FFT of wn . Also, H represents the
l
−
FFT of a time-variant multipath channel hn,l
, which is modeled
using BEM. It can be represented by
1
H h e
N −1
( m−k) − j2 π n( m−k)/ N
l = ∑ n, l
N n=
0
N −1
Q /2 ⎡
∑ ∑
j2 πqn/ N⎤ − j2 πn(
m−k)/ N
hql , e e
n= 0 q=−Q/2 1
= ⎢ ⎥
N ⎣ ⎦
m
+
(6)
(7)
(8)

here, αm and β m represents the multiplicative distortion at the
desired subcarrier and the ICI, respectively. If the channel keeps
invariant during a block period, β m will be zero, which means no
ICI.
In the general case where the multipath cannot be regarded as
time-invariant during a block period, (7) can be expressed in
vector form as:
Y = H X + W
(9)
where Y=[Y 0,…,Y N-1] T , X=[X 0,…,X N-1] T , W=[W 0,…,W N-1] T , and
⎡a0,0 a0,1La0, N −1
⎤
⎢
⎥
a1,0 a1,1 a1,
N −1
H
⎢ L
=
⎥
⎢M O M ⎥
⎢ ⎥
⎢a a a ⎥
⎣ N−1,0 N−1,1 N−1, N−1⎦
here, a m,k in (10) is defined as
( m−k) − j2 π k( L−1)/ N
+ HL1 e ,0 ≤(
m, k) ≤ N −1.
* * * *
= + (14)
⎤
⎦
(10)
( m−k) ( m−k) − j2 π k/ N
amk , = H0 + H1 e + K (11)
−
In this paper, we assume the multipath fading channel is
slowly-varying, so most energy concentrate in the dominant
adjacent subcarriers, the ICI terms which do not significantly
affect Y m in (10) can be ignored. And then by transforming the
matrix H of order N× N to a block-diagonal matrix H * of order
×
(N-q)(q+1) (N-q)(q+1). We obtain:
where,
⎡Aq/2
0 ⎤
⎢
⎥
A
*
q /2+ 1
H =
⎢ ⎥
⎢ O ⎥
⎢ ⎥
⎢⎣0AN−− 1 q/2⎥⎦
⎛an−q/2, n−q/2 L an−q/2,
n L
0 ⎞
⎜ ⎟
⎜an− q/2+ 1, n−q/2 ⎟
⎜ M M ⎟
⎜ ⎟
⎜ann , −q/2
O
⎟
An
= ⎜ ⎟
⎜ O
an+
1, n+ q/
2 ⎟
⎜ M M ⎟
⎜ ⎟
⎜ an+
q/2− 1, n+ q/2⎟
⎜
⎟
0
an+ q/2, n a ⎟
⎝
L L n+ q/2, n+ q/2
⎠
(12)
(13)
We can locate some of the unused subcarriers at the first q/2 and
the last q/2 IFFT grids to get the estimation of all symbols. Then,
the input-output relationship of the multipath channel is expressed
in a vector form as:
where,
Y H X W
*
T
X ⎡xq/2
xq/2+ 1 L xN−1
−q/2
= ⎣
xn = ⎡
⎣Xn−q/2 Xn− q/2+ 1 L X ⎤ n+ q/2⎦
*
T
Y ⎡yq/2 yq/2+ 1 L y ⎤ N−1 −q/2
= ⎣ ⎦
yn = ⎡
⎣Xn−q/2 Xn− q/2+ 1 L X ⎤ n+ q/2⎦
*
T
W ⎡wq/2 wq/2+ 1 L w ⎤ N−1 −q/2
= ⎣ ⎦
wn = ⎡
⎣Wn−q/2 Wn− q/2+ 1 L W ⎤ n+ q/2⎦
T
T
T
(15)
V. FREQUENCY DOMAIN EQUALIZER
A. Zero-Forcing Equalizer
Zero-Forcing equalization compensates both multiplicative
distortion and ICI by multiplying the inverse of , which is the
*
estimated value of H , to (14). The resulting signal can be
expressed as:
* 1 * − % % (16)
X = H Y
where X % is the estimated signal, and
H%
* −1
−1
⎡ A%
0 0 ⎤
⎢ ⎥
−1
⎢ A%
1 ⎥
=
⎢ O ⎥
⎢ ⎥
−1
⎢0 A% ⎣
⎥ N−− 1 q⎦
Also, (16) can be rewritten as
1
X% −
= A% y , q/2≤ n≤ N −1
−q/
2.
(18)
n n n
Finally, the transmitted symbols are estimated by selecting
the elements in the middle of X .
n
%
B. MMSE Equalizer
H %
*
(17)
Using the fact that the ICI power is localized to the dominant
adjacent subcarriers, only a few neighborhood subcarriers can be
used for equalization without much performance penalty. We find
the solution for each desired subcarrier individually[4]. The
problem of MMSE is to find the equalizer coefficient vector
g m=[g m,0,…,g m,q-1] which minimizes the mean-squared error
2
ˆ { m − m }
E X X
where ˆ T
X m = gmy and , y
m y ⎡Y , Y ⎤
where Hm=A m.
The MMSE solution is
m ( m− q/2) L m is then
( m+ q/2)
= ⎣ ⎦
ym = Hmx+ w (19)
m
g R R −
= (20)
1
m X ,
mym ym
assuming H is known, x is a zero-mean i.i.d. random vector with
variance
2
σ x , and w is an AWGN vector with variance
{ } 2
H H
X mym m m
2
σ w ,then
R = E X y = σ xh (21)
m
where hm is the q/2 th column of the matrix H m, i.e.,
Also, we can get
hm = ⎡
⎣am− q/2, q/2 , L , a ⎤
(22)
m+ q/2, q/2⎦
H 2 H 2
{ } σ σ
R = E y y = H H + wIq (23)
ymm m x m m
After inserting (21) and (23) into (20), the q-tap equalizer vector
g m becomes
T

−1
2
H ⎛ H σ ⎞ w
m = m ⎜ m m + 2 q⎟
σ x
g h H H I
⎝ ⎠
where I q is the q-by-q identity matrix. And moreover, we have
N 1
2
σ x ( 1 m
m 0
−
∑
=
MMSE = −gh
m)
(24)
(25)
By reducing the number of equalizer’s taps, the calculation
complexity is reduced largely.
VI. SIMULATION RESULT
In our simulations, we investigate the performance of both the
ZF equalizer and MMSE equalizer. A three-path BEM channel
was considered. Other parameters are listed below:
Tab. 1. Simulation parameter.
Doppler spread fmax
100 Hz
Delay spread τmax 65.1 µ s
Block size N 768
Symbol/sample period T 21.7 µ s
Discrete Doppler spread Q/2=
⎡⎢ fmax NT ⎤⎥
2
Discrete delay spread L=
max /T τ ⎡⎢ ⎤⎥
3
Note that the maximum Doppler spread of 100Hz corresponds
to a vehicle speed of 60 km/h and a carrier frequency of 1.35GHz.
BER
BER
1.00E+00
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1.00E+00
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
Ebn0(dB)
ICI+no equalization
ZF(tap=3)
ZF(tap=5)
no ICI+ZF equalization
Fig.4. BER performance of the Zero-Forcing equalizer.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Ebn0(dB)
ICI+no equalization
MMSE(tap=3)
MMSE(tap=5)
no ICI+MMSE equalization
Fig. 5 BER performance of the MMSE equalizer.
Fig.4 and Fig.5 illustrate the BER performance of ZF and
MMSE equalization respectively. The number of subcarrier is
1024, and 3-tap, 5-tap ZF and MMSE equalizers are considered.
The figures show that the 5-tap equalizer has better performance
than 3-tap one. So we can change the number of equalizer’s tap to
get the tradeoff between the complexity and performance.
VII. CONCLUSION
In this paper, we have utilized the basis expansion model (BEM)
to get a good comprise between complexity and performance in
conventional frequency domain equalizer. Both ZF and MMSE
equalization are considered.
This research was supported by University IT Research Center
Project (INHA UWB-ITRC), Korea
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