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Title
Performance degradation of OFDM systems due to Doppler spreading
Permalink
https://escholarship.org/uc/item/83z4b8pf
Journal
IEEE Transactions on Wireless Communications, 5(6)
ISSN
1536-1276
Authors
Wang, T J
Proakis, J G
Masry, E
et al.
Publication Date
2006-06-01
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California
1422 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006
Performance Degradation of OFDM Systems
Due to Doppler Spreading
Tiejun (Ronald) Wang, Student Member, IEEE, John G. Proakis, Life Fellow, IEEE,EliasMasry,Fellow, IEEE,
and James R. Zeidler, Fellow, IEEE
Abstract— The focus of this paper is on the performance of
orthogonal frequency division multiplexing (OFDM) signals in
mobile radio applications, such as 802.11a and digital video
broadcasting (DVB) systems, e.g., DVB-CS2. The paper considers
the evaluation of the error probability of an OFDM system
transmitting over channels characterized by frequency selectivity
and Rayleigh fading. The time variations of the channel during
one OFDM symbol interval destroy the orthogonality of the
different subcarriers and generate power leakage among the
subcarriers, known as Inter-Carrier Interference (ICI). For con-
ventional modulation methods such as phase-shift keying (PSK)
and quadrature-amplitude modulation (QAM), the bivariate
probability density function (pdf) of the ICI is shown to be a
weighted Gaussian mixture. The large computational complexity
involved in using the weighted Gaussian mixture pdf to evaluate
the error probability serves as the motivation for developing a
two-dimensional Gram-Charlier representation for the bivariate
pdf of the ICI. It is demonstrated that its truncated version
of order 4 or 6 provides a very good approximation in the
evaluation of the error probability for PSK and QAM in the
presence of ICI. Based on Jakes’ model for the Doppler effects,
and an exponential multipath intensity profile, numerical results
for the error probability are illustrated for several mobile speeds.
Index Terms— OFDM, Doppler spreading, ICI, C/I ratio,
Gaussian mixture, two-dimensional Gram-Charlier series.
I. INTRODUCTION
I
N OFDM sy stems, a serial data stream is split into pa rallel
streams that modulate a group of orthogonal sub-carriers.
Compared to single carrier modulation, OFDM symbols h ave
a relatively long time duration, but a narrow bandwidth. Con-
sequently, OFDM is robust to channel multipath dispersion
and results in a decrease in the complexity of equalizers for
high delay spread channels or high data rates. However, the
increased symbol duration makes an OFDM system more
sensitive to the time variations of mobile radio channels.
In particular, the effect of Doppler spreading destroys the
orthogonality of the sub-carriers, resulting in inter-carrier
interference (ICI) due to power leakage among subcarriers.
In several previous publications [1]-[4], the system per-
formance for OFDM was analyzed based on the assumption
Manuscript receiv e d April 15, 2004; revised November 21, 2004; accepted
April 26, 2005. The associate editor coordinating the review of this paper
and approving it for publication was A. Molisch. This work was supported
by the Center for Wireless Communications under the CoRe research grant
core 00-10071 and 03-10148.
The authors are with the Center for Wireless Communications, Uni-
versity of California, San Diego, La Jolla, CA 92093-0407 USA (e-
mail: ronald@cwc.ucsd.edu; jproakis@ucsd.edu; emasry@ucsd.edu; zei-
dler@ucsd.edu).
Digital Object Identifier 10.1109/TWC.2006.04223
that the ICI distribution is Gaussian by invoking the central
limit theorem. In other related papers [5]-[9], efforts have
been mad e to evaluate the effect of ICI by calculating its
average power and comparing it with the p ower of the desired
signal. In [5][6], the carrier to interference (C/I) ratio has
been introduced to demonstrate the effect of the ICI under
various maximum Doppler spreads and different Doppler
spectra. Through numerical evaluations of the C/I ratios, it is
reported in [7] that an OFDM system is robust to frequency
selectivity but quite sensitive to time varying fading channels.
Li and Cimini [8] provide universal bounds on the ICI in
an OFDM system over Doppler fading channels, which are
easier to evaluate and can provide useful insights compared
with the exact ICI expression. Furthermore, the closed-form
expression of ICI power is derived and evaluated in [9], where
the normalized ICI power is rep resented as a function of the
normalized Doppler spread. However, all these papers do not
attempt to determine the underlying probability distribution
function (pdf) of the ICI.
In this paper we focus on providing a statistical analysis
for the ICI in an OFDM system that employs conventional
PSK and QAM signal modulation in a frequency selective,
Rayleigh fading time-varying channel. The channel, which
is assumed to be wide-sense stationary with uncorrelated
scattering (WSSUS), is modeled by a two-dimensional corre-
lation function in time and frequency, representing the time
variations and frequency selectivity of the channel. Each
subchannel is assumed to be frequency flat and, based on
the power series model developed by Bello [10], a two-term
Taylor series expansion is used to model the time variations
in an OFDM symbol. Jakes’ model [14] is used as the m odel
for the Doppler power spectrum and an exponential multipath
intensity profile is the model adopted for the multipath effects.
A cyclic prefix is assumed to remove the effects of inter-
symbol interference. Based on this channel model, the ICI is
expressed as the summation of leakage terms into each of the
subcarriers and its pdf is shown to be characterized statistically
by a b ivariate pdf that is a weighted sum of Gaussian pdfs.
In d eriving the probability of error for the OFDM system
in the presence of ICI, the use of the weighted Gaussian pdf
proves to be computationally intensive. This difficulty serves
as the mo tivation to develop a two-dimensional Gram-Charlier
series to represent the pdf of the ICI. A truncated version of the
Gram-Charlier is u sed in the evaluation of the error probability
for PSK and QAM signal modulations in an OFDM system.
The paper is organized as follows: In Section II we describe
the model for the OFDM system. In Section III we describe
1536-1276/06$20.00
c
2006 IEEE
WANG et al.: PERFORMANCE DEGRADATION OF OFDM SYSTEM DUE TO DOPPLER SPREADING 1423
Fig. 1. Base-band OFDM transmisson model with N subcarriers.
the channel model and use a Taylor series expansion for
the time variations within an OFDM symbol. In Section
IV an expression for the ICI and its power is presented.
Section V provides a thorough analysis of the statistics of the
ICI, its joint probability density, joint moments, and a two-
dimensional Gram-Charlier representation. In Section VI, the
error rate p erformance of BPSK and 16-QAM OFDM systems
are presented and compared. Finally, concluding remarks are
given in Section VII.
II. OFDM S
YSTEM
An OFDM system with N subcarriers is represented in Fig.
1. In an OFDM system that employs M -ary digital modula-
tion, a block of log
2
M input bits is mapped into a symbol
constellation point d
k
by a data encoder, and then N symbols
are transferred by the serial-to-parallel converter (S/P). If 1/T
is the symbol rate of the input data to be transmitted, the
symbol interval in the OFDM system is increased to NT,
which makes the system more robust against the channel delay
spread. Each sub-channel, however, transmits at a much lower
bit rate of
log
2
M
NT
bits/s. The parallel symbols (d
1
, ··· ,d
N
)
modulate a group of orthogonal subcarriers, which satisfy
1
NT
NT
0
exp(j2πf
i
t)exp(j2πf
j
t)dt =
1 i = j
0 i = j
(1)
where f
i
=
i−1
NT
, (i =1, 2, ··· ,N)
Consider the system shown in Fig. 1. The baseband trans-
mitted signal can be represented as
s(t)=
1
√
NT
N
k=1
s
k
e
j2πf
k
t
, 0 ≤ t ≤ NT, f
k
=
k −1
NT
.
(2)
We denote by 2E
s
the average energy for the complex
baseband symbol s
k
.Thens
k
is given by
s
k
=
2E
s
d
k
(3)
where d
k
= d
k,r
+ jd
k,i
, is the signal constellation point
(e.g. BPSK, QPSK, QAM, etc.) with normalized variance
E[|d
k
|
2
]=1. Square M-QAM signal constellations may be
viewed as two independent
√
M-PAM signals on orthogonal
carriers. In this case, the real and imaginary parts d
k,r
and
d
k,i
are statistically independent, identically distributed and
E[d
k,r
]=E[d
k,i
]=0.
III. C
HANNEL MODEL
We consider a frequency selective randomly varying chan-
nel with impulse response h(t, τ). Within the n arrower band-
width of each sub-carrier, compared with the coherence
bandwidth of the channel, the sub-channel is modeled as a
frequency nonselective Rayleigh fading channel. Hence, the
channel impulse response h
k
(t, τ) for the k
th
subchannel is
denoted as
h
k
(t, τ)=β
k
(t)δ(τ) (4)
where the process {β
k
(t), −∞ <t<∞} is a statio nary,
zero mean complex-valued process described as follows: It is
assumed that the processes {β
k
(t), −∞ <t<∞},k=
1,...,N, are complex-valued jointly stationary and jointly
Gaussian with zero means and cross covariance function
R
β
k
,β
l
(τ):=E[β
k
(t + τ)β
∗
l
(t)],k,l=1,...,N. (5)
For each fixed k, the real and imaginary parts of the process
{β
k
(t), −∞ <t<∞} are assumed independent with
identical covariance function. We further assume that the
correlation function R
β
k
,β
l
(τ) has the following factorable
form
R
β
k
,β
l
(τ)=R
1
(τ)R
2
(k − l) (6)
which has been freque ntly used in the literature,
e.g.,[4][15][16], and which is sufficient to represent the
frequency selectivity and the time-varying effects of the
channel. R
1
(τ) gives the temporal correlation for the process
{β
k
(t), −∞ <t<∞} which is seen to be identical
for all k =1,...,N. R
2
(k) represents the correlation in
frequency across subcarriers. We assume in this paper that
the corresponding spectral d ensity ψ
1
(f) to R
1
(τ) is given
by the Doppler power spectrum, modeled as in Jakes [14],
i.e.,
ψ
1
(f)=
1
πF
d
·
1−(
f
F
d
)
2
|f|≤F
d
0 otherwise
(7)
where F
d
is the (maximum) Doppler bandwidth. Note that
R
1
(τ)=J
0
(2πF
d
τ) (8)
1424 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006
0 1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ατ
s(τ)/α
multipath intensity profile
−2 −1 0 1 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f/Δ f
|Ψ
2
(kΔf)|
correlation function in frequency
Fig. 2. Multipath delay profile and frequency correlation function.
where J
0
(τ) is the zero-order Bessel function of the first
kind. In order to specify the correlation in frequency across
subcarriers, we adopt an exponential multipath p ower intensity
of the form S(τ)=αe
−ατ
,τ>0,α > 0 where α is
a parameter that controls the coherence bandwidth of the
channel. The Fourier transform of S(τ) yields
ψ
2
(f)=
α
α + j2πf
(9)
which provides a measure of the correlation of the fading
across the subcarrier s as illustrated in Fig. 2. Then R
2
(k)=
ψ
2
(Δfk) where Δf =
1
NT
is the frequency separation
between two adjacent subcarriers. The 3dB bandwidth of
ψ
2
(f) is defined as the coherence bandwidth of the channel
and easily shown to be f
coherent
=
√
3α
2π
.
The channel model described above is suitable for modeling
OFDM signal transmission in mobile radio systems, e.g.,
cellular systems and broadcasting systems. For example, in
DVB-CS2 with 2000 subcarriers, the symbol duration NT is
500μs. In contrast, the delay spread of many fading channels
is much smaller, which make it reasonable to view each
subcarrier as a flat fading channel. However, compared with
the entire OFDM system bandwidth W =1/T , the coherence
bandwidth f
coherent
is usually smaller, f
coherent
<W,
especially in an outdoor wireless communication environment.
Hence, the channel is frequency-selective over the entire
OFDM bandwidth.
We now turn our attention to modeling the time variations
of the channel within an OFDM symbol interval. For most
practical mobile radio fading channels, the time-varying ef-
fects in the channel are sufficiently slow, i.e., the coherence
time is always much larger than the interval of an OFDM
symbol [17][18]. For such slow fading channels, we use the
two terms Taylor series expansion, first introduced by Bello
[10], to represent the time-varying fading response β
k
(t) as
the following form:
β
k
(t)=β
k
(t
0
)+β
k
(t
0
)(t − t
0
),t
0
=
NT
2
, 0 ≤ t ≤ NT.
(10)
Therefore, the impulse response of the k
th
subchannel is
expressed as
h
k
(t, τ)=β
k
(t)δ(τ)=[β
k
(t
0
)+β
k
(t
0
)(t − t
0
)]δ(τ). (11)
Since R
1
(τ) of (8) is infinitely differentiable, all mean-square
derivatives exist an d thus the differentiation above is justified.
We use this model for the time variations of the channel within
an OFDM symbol.
IV. E
XPRESSION FOR THE ICI AND IT S POW E R
Let s(t) be the baseband signal transmitted over the channel
with impulse response h(t, τ) as modeled above. Then the
baseband received signal with additive noise may be expressed
as
r(t)=h(t, τ)s(t)+n(t)=
1
√
NT
N
k=1
β
k
(t)s
k
e
j2πf
k
t
+n(t)
(12)
where denotes convolution and n(t) is the additive noise,
which is modeled as a Gaussian process with zero mean
and spectrally flat within the signal bandwidth, with one-
sided spectral density N
0
watts/Hz. By using the Taylor series
expansion for β
k
(t) as given in (11), we obtain
r(t)=
1
√
NT
N
k=1
β
k
(t
0
)s
k
e
j2πf
k
t
+
1
√
NT
N
k=1
β
k
(t
0
)(t − t
0
)s
k
e
j2πf
k
t
+ n(t). (13)
The received signal in a symbol interval is passed through
a p arallel bank of correlators, where each correlator is tuned
WANG et al.: PERFORMANCE DEGRADATION OF OFDM SYSTEM DUE TO DOPPLER SPREADING 1425
100 200 300 400 500 600
20
22
24
26
28
30
32
34
36
Doppler frequency in Hz
C/I (dB)
from Equation (25)
Reference [9]
Reference [5],[6]
490 495 500 505 510
21.5
21.55
21.6
21.65
21.7
21.75
21.8
21.85
21.9
21.95
Doppler frequency in Hz
C/I (dB)
from Equation (25)
Upper bound from Reference [8]
Lower bound from Reference [8]
(a) (b)
Fig. 3. C/I ratio curves of an OFDM system N = 256 subcarriers, subcarrier distance Δf =7.81KHz, and carrier frequency f
c
=2GHz.
to one of the N subcarriers. The output of the i
th
correlator
is
d
i
=
1
√
2E
s
1
√
NT
NT
0
r(t)e
−j2πf
i
t
dt. (14)
Substituting (13) into (14), we obtain
d
i
=
1
√
2E
s
NT
NT
0
1
√
NT
N
k=1
β
k
(t
0
)s
k
e
j2π(f
k
−f
i
)t
dt
(1)
+
1
√
2E
s
NT
NT
0
1
√
NT
N
k=1
β
k
(t
0
)(t − t
0
)s
k
e
j2π(f
k
−f
i
)t
dt
(2)
+
1
√
2E
s
NT
NT
0
n(t)e
−j2πf
i
t
dt
(3)
(15)
The first term yields
N
k=1
β
k
(t
0
)d
k
1
NT
NT
0
e
j2π(f
k
−f
i
)t
dt
= β
i
(t
0
) d
i
. (16)
The second term yields
N
k=1
β
k
(t
0
)d
k
1
NT
NT
0
(t − t
0
)e
j2π(f
k
−f
i
)t
dt
=
N
k=1
k=i
β
k
(t
0
)d
k
j2π(f
k
− f
i
)
=
NT
2πj
N
k=1
k=i
β
k
(t
0
)d
k
k −i
.
(17)
Finally, the additive noise term is
n
i
=
1
√
2E
s
1
√
NT
NT
0
n(t)e
−j2πf
i
t
dt (18)
where n
i
is a complex Gaussian noise with zero mean and
variance N
0
/E
s
. Thus we have
d
i
= β
i
(t
0
)d
i
desired_signal
+
NT
2πj
N
k=1
k=i
β
k
(t
0
)d
k
k −i
ICI
+n
i
. (19)
In Section V we establish the statistical properties of the
ICI term. Here, we obtain the C/I ratio and we compare the
result with those obtained in [5],[6],[8], and [9], which are
based on different models for the time variations.
From Equation (19), the average power of the desired signal
is
C = E[|β
i
(t
0
)d
i
|
2
]=E[|β
i
(t
0
)|
2
]E[|d
i
|
2
]=1. (20)
Since R
β
k
,β
k
(τ)=R
1
(τ) is infin itely differentiable, all
(mean-square) derivatives of the process {β
k
(t), −∞ <
t<∞} exist. In particular, the first-order derivative
process {β
k
(t), −∞ <t<∞} is a zero mean complex-
valued Gaussian p rocess with correlation function E[β
k
(t +
τ)(β
k
(t))
∗
]=−R
1
(τ) (identical for all k) with corresponding
spectral density
ψ
3
(f)=
⎧
⎨
⎩
(2πf)
2
πF
d
·
1−(
f
F
d
)
2
|f|≤F
d
0 otherwise
(21)
Then,
E[|β
k
(t)|
2
]=
∞
−∞
ψ
3
(f)df
=
F
d
−F
d
(2πf)
2
πF
d
1 − (
f
F
d
)
2
df =2π
2
F
d
2
. (22)
