Capacity of OFDM systems in Nakagami-m fading channels: The role of channel frequency selectivity

Abstract

In this paper, we analyze the capacity of orthogonal frequency division multiplexing (OFDM) systems with carrier frequency offset (CFO) in frequency-selective Nakagami-m fading channels. Previous work on this topic has not taken into account the frequency selectivity of the channel. In this work, we have explicitly attributed the effect of channel frequency selectivity, i.e. frequency domain correlations, in evaluating the OFDM system performance in the presence of CFO. A closed-form expression is derived of the probability density function (PDF) of the signal-to-interference-and-noise ratio (SINR) in terms of CFO and channel correlation vector. Capacity is evaluated using numerical integration. The frequency-flat fading scenario and the perfectly frequency-selective fading (uncorrelated subcarriers) scenario form the two extremes, i.e. bounds, of the achievable OFDM capacity in the presence of CFO in Nakagami-m fading channels.

Full text

1
Capacity of OFDM Systems in Nakagami-mFading Channels:
The Role of Channel Frequency Selectivity
C. R. N. Athaudage†, M. Saito†∗and J. Evans†
†ARC Special Research Center for Ultra-Broadband Information Networks (CUBIN)
Department of Electrical and Electronic Engineering
University of Melbourne, Australia
∗Nara Institute of Science and Technology (NAIST)
Nara, Japan
Abstract— In this paper, we analyze the capacity of orthogonal
frequency division multiplexing (OFDM) systems with carrier
frequency offset (CFO) in frequency-selective Nakagami-mfad-
ing channels. Previous work on this topic has not taken into
account the frequency selectivity of the channel. In this work, we
have explicitly attributed the effect of channel frequency selectiv-
ity, i.e. frequency domain correlations, in evaluating the OFDM
system performance in the presence of CFO. A closed-form
expression is derived of the probability density function (PDF)
of the signal-to-interference-and-noise ratio (SINR) in terms of
CFO and channel correlation vector. Capacity is evaluated using
numerical integration. The frequency-flat fading scenario and the
perfectly frequency-selective fading (uncorrelated subcarriers)
scenario form the two extremes, i.e. bounds, of the achievable
OFDM capacity in the presence of CFO in Nakagami-mfading
channels.
I. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) has
recently become a key modulation technique for high data
rate mobile wireless applications [1]. Computationally simple
per subcarrier equalization in multipath frequency-selective
fading channels makes OFDM more attractive compared to
its single-carrier counterpart. The high spectral efficiency
achieved through the use of orthogonal subcarriers is an
additional advantage of OFDM and leads to efficient usage
of the RF spectrum by multiple broadband systems.
A major drawback of OFDM is its relatively high sensitivity
to the carrier frequency offset (CFO) errors, compared to
a single carrier system [2]-[4]. The frequency offset error
is caused by the misalignment in subcarrier frequencies at
the receiver due to fluctuations in receiver RF oscillators
or channel’s Doppler frequency. This frequency offset can
destroy the subcarrier orthogonality of the OFDM signal
introducing inter-carrier-interference (ICI). The ICI results in
severe degradation of the bit-error-rate (BER) performance
of the OFDM systems. Although frequency offset correction
techniques [3] can largely compensate for CFO, any residual
error in frequency synchronization (CFO estimation error)
contributes to the degradation of receiver performance.
In this paper we investigate the effect of the CFO error on
the performance of OFDM systems in multipath frequency-
selective Nakagami-mfading channels. In reported work in
literature [4]-[6] on this topic, ICI noise is either ignored or
it is assumed to be independent of the useful signal, which is
only valid for highly frequency-selective channels, i.e. when
the individual subcarrier responses of the OFDM system are
totally uncorrelated (i.i.d.). Therefore, these results provide
worse case capacity performance, i.e. lower-bound on the
capacity in the presence of CFO. Some recent work [7] shows
that in Rician channels with strong line-of-sight component
the average OFDM capacity with CFO is only negligibly
dependent on subcarrier correlations.
In contrast, in this paper, we provide an analytical technique
of evaluating the capacity of an OFDM system with CFO in
frequency-selective Nakagami-mfading channels.The exact
capacity for a given frequency domain channel correlation
function can be calculated using the proposed technique.
Moreover, by using the frequency-flat fading (FFF) case and
the perfectly frequency-selective fading (PFSF) case as ex-
treme channel conditions, the dependency of achievable capac-
ity on channel correlations in frequency domain is illustrated.
II. INTERFERENCE DUE TO FREQUENCY OFFSET
For an OFDM system, the post-FFT signal y(k)at the kth
subcarrier, where 0≤k≤N−1, can be given as [2]
y(k)=s0h(k)x(k)+
N−1

l=0,l=k
sl−kh(l)x(l)+w(k)(1)
where, x(k)and h(k)are the transmit data symbol and
frequency response for the kth subcarrier, respectively. w(k)
is additive white Gaussian noise (AWGN). The sequence sk
(ICI coefficients) depends on the CFO and is given by [2]
sk=sin π(k+)
Nsin π
N(k+)exp jπ 1−1
N(k+)(2)
where, is the normalized frequency offset, which is the ratio
between the CFO and the adjacent subcarrier spacing.
The decision variable ˆx(k)after per subcarrier equalization
can be formed as ˆx(k)=¯s0¯
h(k)y(k), where it is assumed that
the effective channel ˜
h(k)=s0h(k)is known, i.e. perfectly
estimated. The complex conjugate of a complex number is
denoted by ¯
(.). Thus, ˆx(k)becomes
ˆx(k)=|s0|2|h(k)|2x(k)+¯s0¯
h(k)i(k)+¯s0¯
h(k)w(k)(3)
where, i(k)=N−1
l=0,l=ksl−kh(l)x(l)is the ICI interference
term. In (3), the three terms show the signal, interference, and
channel noise, respectively.
978-1-4244-2644-7/08/$25.00 © 2008 IEEE

2
Note-1 Let uand vare two correlated zero-mean complex
Gaussian R.V.s. The correlation coefficient between uand v
are given by ruv =E{u¯v}. The conditional mean μu|vand
the variance σ2
u|vof ugiven vare given by [8]
μu|v=ruvv
σ2
v
and σ2
u|v=σ2
u−|ruv|2
σ2
v
(4)
where σ2
uand σ2
vare the variances of uand v, respectively.
Also, the conditional power of ugiven vis given by
E|u|2|v=|μu|v|2+σ2
u|v.(5)
The power associated with the three terms in (3) for a
given channel realization h(k)for the kth subcarrier can be
calculated as follows. The signal power Px(k)for the kth
subcarrier becomes
Px(k)=|s0|4|h(k)|4E|x(k)|2=|s0|4|h(k)|4σ2
x(6)
where, σ2
x=E|x(k)|2is the average symbol power, and the
notation E{.}depicts the expected value of a random variable.
The interference power Pi(k)for a given h(k)becomes
Pi(k)=|s0|2|h(k)|2E|i(k)|2|h(k)(7)
where, E|i(k)|2|h(k), the conditional power of i(k)for
agivenh(k)is given by ((8)), where rlk =E
h(l)¯
h(k)
is the channel correlation coefficient between the lth and
the kth (k=l) subcarriers. The constant average power of
each subcarrier is given by σ2
h=E
|h(l)|2. Assuming
σ2
h=1(without loosing generality) and substituting (8) in
(7) gives (9). Similarly, the channel noise power becomes
Pw=|s0|2|h(k)|2σ2
w. Therefore, the signal-to-interference-
and-noise ratio (SINR) becomes (10), where ρ=σ2
x/σ2
wis
the average channel SNR, and 0≤rlk ≤1. Above (10) can
be rewritten as
γ(k)= |s0|2|h(k)|2
ρ−1+λ1(, r)+|h(k)|2λ2(, r)(11)
where,
λ1(, r)=
N−1

l=0,l=k
|sl−k|21−|rlk|2(12)
and
λ2(, r)=
N−1

l=0,l=k
|sl−k|2|rlk|2.(13)
The N−1element vector rconsists of the channel correlation
coefficients rl,k for l=0,1,...,N −1except for l=k.For
a Nakagami-mchannel |h(k)|2is gamma distributed. Above
(11) indicates that the distribution of γ(k)is not only depends
on the normalized CFO () but also on the subcarrier corre-
lation structure induced by the frequency selective channel
given by r. The main focus of this paper is to reveal the
dependence of OFDM system performance, e.g. capacity, on r
in the presence of CFO. In terms of frequency selectivity of the
channel two extreme cases can be considered as special cases:
(a) Frequency flat fading channel- this corresponds to a zero
delay-spread channel, and (b) Perfectly frequency selective
channel - this corresponds to a very high delay-spread channel.
A. Frequency Flat Fading (FFF)
For frequency flat fading subcarrier channel responses be-
come perfectly correlated thus rl,k =1,∀k, i.e. r=1N−1.
Thus λ1(, r)=0and λ2(, r)=N−1
l=0,l=k|sl−k|2=
1−|s0|2. Therefore, the SINR for the frequency flat fading
becomes
γ(k)= |s0|2|h(k)|2
(1 −|s0|2)|h(k)|2+ρ−1.(14)
B. Perfectly Frequency Selective Fading (PFSF)
For perfectly frequency selective channel, subcarrier chan-
nel responses become zero correlated thus rl,k =0,∀k, i.e.
r=0N−1. Thus λ1(, r)=N−1
l=0,l=k|sl−k|2=1−|s0|2and
λ2(, r)=0. Therefore, the SINR for the frequency flat fading
becomes
γ(k)= |s0|2|h(k)|2
1−|s0|2+ρ−1.(15)
III. DENSITY FUNCTION OF SINR
In this section we derive a closed-form expression the den-
sity function (PDF) of SINR in Nakagami-mfading channels.
The post-equalized SINR γ(k)in (11) can be rewritten as
γ(k)= a|h(k)|2
b|h(k)|2+1 (16)
where,
a=|s0|2
ρ−1+λ1(, r)and b=λ2(, r)
ρ−1+λ1(, r).(17)
For a Nakagami-mfading channel, |h(k)|follows the Nak-
agami distribution. Let v=|h(k)|2then vwill be gamma
distributed - thus for unit-power subcarriers, i.e. E{v}=1,
the PDF of vbecomes
pv(v)= mm
Γ(m)vm−1e−mv (18)
where, Γ(.)is the gamma function and mis a parameter
denoting the severity of channel fading, m≥1/2. The well-
known Rayleigh fading corresponds to m=1.
Note 2: The density py(y)of the random variable Y=f(X)
(function of X) can be found in terms of the density px(x)of
the R. V. Xas follows: To find py(y)for a specific ysolve
the equation y=f(x), denoting its real roots by x1,x2,...,
xN. Then, py(y)is given by [8]
py(y)= px(x1)
|f(x1)|+px(x2)
|f(x2)|+...+px(xN)
|f(xN)|(19)
where f(x)is the derivative of f(x).
Using (19), the PDF of γ(k)can be calculated as follows:
γ=f(v)= av
bv +1 and f(v)= a
(bv +1)
2(20)
where, the subcarrier index kis dropped for simplicity. Also,
0≤γ≤a/b as v≥0.γ=f(v)has only one real root at
v1=γ/(a−bγ), and therefore using (19) the PDF of γ(k)
becomes
pγ(γ)= pv(v1)
|f(v1)|
=mm
Γ(m)
aγm−1
(a−bγ)m+1 exp −mγ
a−bγ (21)

3
E|i(k)|2|h(k)=
N−1

l=0,l=k
|sl−k|2|μh(l)|h(k)|2+σ2
h(l)|h(k)σ2
x
=
N−1

l=0,l=k
|sl−k|2|rlk|2|h(k)|2
σ4
h
+σ2
h−|rlk|2
σ2
hσ2
x(8)
Pi(k)=|s0|2|h(k)|2σ2
x
N−1

l=0,l=k
|sl−k|2|rlk|2|h(k)|2+1−|rlk|2.(9)
γ(k)= Px(k)
Pw(k)+Pi(k)=ρ|s0|2|h(k)|2
1+ρ|h(k)|2N−1
l=0,l=k|sl−k|2|rlk|2+ρN−1
l=0,l=k|sl−k|2(1 −|rlk|2)(10)
where, 0≤γ≤a/b. It is important to note that pγ(γ)=0for
γ>G, where G=a/b =|s0|2/λ2(, r).Forafrequency flat-
fading channel a=ρ|s0|2and b=ρ(1−|s0|2).Foraperfectly
frequency selective channel a=|s0|2/(1 −|s0|2+ρ−1)and
b=0.
IV. OFDM CAPACITY WITH CFO
In this section we derive an analytical expression for the er-
godic capacity1of an OFDM system with CFO in a Nakagami-
mfading channel with the effect of the frequency selectivity of
the channel explicitly accounted for. Using the density function
of SINR derived in (21) the ergodic capacity C=C(, r)can
be given as
C=∞
0
log2(1 + γ)pγ(γ)dγ
=IG
0
γm−1log2(1 + γ)
(a−bγ)m+1 exp −mγ
a−bγ dγ (22)
where,
I=I(m, a)= amm
Γ(m).(23)
Note that the capacity Cis function of both CFO and channel
correlation structure as the parameters aand bare dependent
on and r. Capacity in a Rayleigh fading channel can be
obtained by setting m=1in (22) to give
C=aG
0
log2(1 + γ)
(a−bγ)2exp −γ
a−bγ dγ (24)
For a perfectly frequency-selective channel b=0and G=∞,
thus the capacity can be calculated using
C=mm
amΓ(m)∞
0
γm−1log2(1+γ)exp−mγ
adγ. (25)
V. N UMERICAL RESULTS
In this section we present numerical results obtained using
the analytical expressions derived in Sections III and IV for
the density function of the SINR and the ergodic capacity of
an OFDM system with CFO.
1The capacity evaluated here is when the receiver uses conventional per
subcarrier equalization, i.e. capacity limited by ICI noise and AWGN.
A. Density Function of SINR
Fig. 1 shows the density function (PDF) pγ(γ)of SINR
for the Rayleigh fading case in frequency flat fading (FFF)
when ρ=20dB. The PDF for CFO values of =8%,10%,
and 12% are shown separately. As can be seen from Fig. 1,
the probability of lower SNR values increases with increasing
values of CFO. Fig. 2 shows the density function (PDF) pγ(γ)
of SINR for the Rayleigh fading case in perfectly frequency
selective fading (PFSF) when ρ=10dB. Again, the PDF for
CFO values of =8%,10%, and 12% are shown separately.
As can be seen from Fig. 2, the probability of lower SNR
values (close to γ=0) increases with increasing values of
CFO. Also, comparison of Fig. 1 and Fig. 2 for the same
CFO value reveals that PDF for PFSF is much worse than
that for FFF case.
0 5 10 15 20 25 30 35 40 45 50
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
γ
PDF pγ(γ)
ε=8%
ε=10%
ε=12%
Fig. 1. The density function (PDF) pγ(γ)of SINR γfor =
8%,10%,and12% and m=1(Rayleigh fading) in frequency flat
fading (FFF) channel.
B. Capacity with CFO
Fig. 3, Fig. 4, and Fig. 5 show the average OFDM capacity
versus average channel SNR curves numerically evaluated
using the capacity expressions derived in Section IV, for

4
0 50 100 150
0
0.01
0.02
0.03
0.04
0.05
0.06
γ
PDF pγ(γ)
ε=8%
ε=10%
ε=12%
Fig. 2. The density function (PDF) pγ(γ)of SINR γfor =
8%,10%,and12% and m=1(Rayleigh fading) in perfectly
frequency selective fading (PFSF) channel.
m=0.5, 1, and 2, respectively. Capacity for CFO values
of =5%,10%, and 20% and for the channel conditions of
FFF and PFSF fading are shown separately. As can be seen
from Fig. 3, Fig. 4, and Fig. 5, a significant capacity gap exists
for the FFF and PFSF channel cases for the high SNR values.
Note that at low SNR values the AWGN noise dominates over
the ICI noise. Also, it is observable that the capacity gap is
more for large CFO () and deeper fading (low m).
10 12 14 16 18 20 22 24 26 28 30
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Average channel SNR (ρ in dB)
Capacity (bits/s/Hz)
ε=5%, FFF
ε=10%, FFF
ε=20%, FFF
ε=5%, PFSF
ε=10%, PFSF
ε=20%, PFSF
Fig. 3. Average OFDM Capacity versus average channel SNR for
m=0.5(deeper fading than Rayleigh) and normalized CFO values
of =5%,10%,and20%.
VI. CONCLUSIONS
In this paper, we demonstrated the role of channel frequency
selectivity in determining the capacity of an OFDM system
in the presence of CFO in Nakagami-mfading channels. A
closed-form expression is derived for the probability density
function of the signal-to-interference-and-noise ratio in terms
of CFO and channel correlation vector. Numerical results show
that the channel frequency selectivity plays an important role
specially in the high SNR and deeper fading conditions.
10 12 14 16 18 20 22 24 26 28 30
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
Average channel SNR (ρ in dB)
Capacity (bits/s/Hz)
ε=5%, FFF
ε=10%, FFF
ε=20%, FFF
ε=5%, PFSF
ε=10%, PFSF
ε=20%, PFSF
Fig. 4. Average OFDM Capacity versus average channel SNR
for m=1(Rayleigh fading) and normalized CFO values of
=5%,10%,and20%.
10 12 14 16 18 20 22 24 26 28 30
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Average channel SNR (ρ in dB)
Capacity (bits/s/Hz)
ε=5%, FFF
ε=10%, FFF
ε=20%, FFF
ε=5%, PFSF
ε=10%, PFSF
ε=20%, PFSF
Fig. 5. Average OFDM Capacity versus average channel SNR for
m=2(shallower fading than Rayleigh) and normalized CFO
values of =5%,10%,and20%.
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