Int. J. Communications, Network and System Sciences, 2010, 3, 380-384
doi:10.4236/ijcns.2010.34048 blished Online April 2010 (http://www.SciRP.org/journal/ijcns/) Pu
On Channel Estimation of OFDM-BPSK and -QPSK over
Generalized Alpha-Mu Fading Distribution
Neetu Sood, Ajay K. Sharma, Moin Uddin
National Institute of Technology, Jalandhar, India
Received January 9, 2010; revised February 11, 2010; accepted March 18, 2010
Abstract
This paper evaluates the performance of OFDM-BPSK and -QPSK system in α-µ distribution. A fading
model which is based on the non-linearity present in the propagation medium is utilized here for generation
of α-µ variants. Different combinations of α and µ provides various fading distributions, one of which is
Weibull fading. Investigations of channel estimation schemes gave an idea of further reducing the BER as to
improve the performance of OFDM based systems. In flat fading environment, channel estimation is done
using phase estimation of the transmitted signal with the help of trained symbols. Final results show the im-
provement in BER. However, the amount of results improved depends upon the amount of trained symbols.
The more trained symbols will result into more improved BER.
Keywords: OFDM, Fading Distribution, Weibull Fading, Nakagami Fading, Channel Estimation, Training
Symbols
1. Introduction
In recent years, OFDM have been studied very widely
and deeply in wireless communication systems because
of bandwidth efficiency and its robustness to channel
fading and Inter Symbol Interference (ISI). OFDM sys-
tem is capable of mitigating a frequency selective chan-
nel to a set of parallel fading channels, which need rela-
tively simple processes for channel equalization.
There exists a large number of distribution schemes to
describe the statistics of mobile radio signal. A key as-
sumption in the theoretical explanation of the Rayleigh,
Rician, Nakagami and Weibull distribution was that the
statistics of the channel do not change over the small
(local) area under consideration. However, to describe
the long-term signal variations lognormal distribution is
being used [1]. These distributions are helpful in precise
designing of wireless systems to make the systems more
robust to noise.
Rayleigh and Rician fading channels have already
been studied and employed in OFDM systems in fre-
quency selective and flat fading environment. Nakagami-
m distribution is another useful and important model to
characterize the fading model. A threshold value of m is
calculated for both frequency and flat fading environ-
ment in [2]. There exist many other fading models in
literature, which have been proposed for better fitting of
data while aiming at non-linearity of channel. So our
motivation behind this paper to explore the non-linear
fading environment. One of the interesting models that
we could find in literature is α
μ
distribution [3],
which provides the generalized model for fading distri-
bution. Depending upon the value of α and
μ
, this
model can be utilized for the generation of Nakagami-m
and Weibull variants. However, it also treats One-Sided
Gaussian, Rayleigh and Negative Exponential distribu-
tions as its special cases. The generalized fading model
using three parameter generalized gamma distribution
describing all forms of multipath fading and shadowing
in wireless systems is analyzed in [4].
This paper is organized as follows: In Section 2,
OFDM system model is discussed. Section 3, describes
the generalized model of α
μ
distribution. Flat fading
channel model to use in OFDM systems is described in
Section 4. In Section 5, channel estimation technique is
discussed. The analysis of OFDM system without esti-
mation has been done in Section 6, while results with
estimation have been presented in Section 7. Finally Sec-
tion 8 concludes the paper.
2. OFDM Model
A Complex base band OFDM signal with N sub-carriers,

N. SOOD ET AL. 381
is expressed as
0
1
2
0
()



Njπkf t
i
k
stDe 0 (1) tT
For each OFDM symbol, the modulated data se-
quences are denoted by . Here,
denote the sub-carriers spacing and is set to
(0),(1),... (1)DD DN
0
f
0
f1 T, the condition of orthogonality. After IFFT, the
time-domain OFDM signal can be expressed as:
0
2
1
0
1
S(n) N



jπkf n
NN
i
k
De (2)
After IFFT, the modulated signal is up-converted to
carrier frequency and then the following signal is
produced and transmitted through channel:
C
f
0
1
2( )
0
() Re





C
Njπkff t
i
k
xtDe  0tT

(3)
()
x
trepresents the final OFDM signal in which sub-car-
riers shall undergo a flat fading channel.
3. The
μ
α Distribution
The α
μ
distribution is a general fading distribution
that can be used to represent various fading model. This
distribution deals with non-linearity of propagation me-
dium [5]. Fading signal with envelope , an arbitrary
constant parameter and a root mean value
r
0α

ˆα
α
rEr
)
shall have its probability density function
, which is written as: (pr
1
( )exp()
ˆ
()



μαμ α
αμ α
αμ rr
pr μ
rμr
ˆ
(4)
Weibull and Nakagami-m distribution can be easily
derived from α
μ
distribution as its special cases.
By setting, Equation (4) shall reduce to Weibull
probability distribution function as:
1μ
1
( )exp()


αα
pr αβrβr (5)
where .
ˆ
α βr
Here, by varying the value of different curves of
pdf can be plotted.
α
From Weibull distribution by setting , the Ray-
leigh distribution can be obtained as:
2α
2
2
() exp()
2

rr
pr γ
where .
22
ˆ/2γr
Now, if we put 1

α in Weibull distribution, it shall
reduce to Negative exponential distribution represented
as:
() exp()

pr δδr (7)
where 1
ˆ

δr
So by keeping the value of and varying the
value of it has generated Rayleigh and Negative ex-
ponential distribution. Whereas if we keep
1μ
α
2

α and
vary the value of
μ
, we shall be able to represent this

α
μ
distribution as Nakagami-m distribution
In such a case
21 2
2
()exp( )
()



μμ
μ
μrr
pr μ
μ (8)
By setting 1/2

μ, one-sided Gaussian distribution
can be obtained as:
2
2
2
()exp( )
ˆ 2

r
pr πr2
ˆ r
(9)
However, for

α
μ
distribution the envelope
r
can
be written as:

1
22
1








N
α
ii
i
rxy
(10)
where, i
x
and are in-phase and quadrature elements
of multipath components represented by symbol .
i
y
i
It was interesting to find that in Equation (10)
shall reduce to the envelope equation of Rayleigh fading
distribution [6] described as:
2α
22
1










N
ii
i
rxy
(11)
Same concept has been shown in Equations (5) and (6)
that by putting 2

α, Weibull distribution converts to
Rayleigh distribution, hence introducing the non-linear-
ity into propagation medium. However, at different val-
ues of different fades can be generated. α
4. Channel Model
In this paper, the sub-channel spacing is equal to inverse
of time period, so that the produced parallel fading sub-
channels have flat fading characteristics.
Here

α
μ
distribution has been utilized for genera-
tion of Weibull distribution by setting and vary-
ing the value of .
1μ
α
2
γ
(6)
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N. SOOD ET AL. 382
In flat fading environment, the base-band signal at the
input of receiver is as described as follows: ()yt
()()*() ()yt xtrtnt (12)
where, ()
x
t denotes the base-band transmitted signal,
is the Weibull distributed channel envelope and is
the additive white Gaussian noise with zero mean.
()rt
()nt
5. Channel Estimation
Channel estimation in frequency selective has different
approach then compared with flat fading environment. A
comparative study using Minimum Mean Squared Error
(MMSE) and Least square (LS) estimator in frequency
selective fading environment has been presented in [7].
The channel estimation based on comb type pilot ar-
rangement is studied using different algorithms by bahai
et al. [8]. A novel channel estimation scheme for OF-
DMA uplink packet transmissions over doubly selective
channels was suggested in [9]. The proposed method uses
irregular sampling techniques in order to allow flexible
resource allocation and pilot arrangement. In flat fading
environment, estimation of the channel using trained
sequence of data has been studied and implemented in
[10]. He presented the channel estimation in flat fading
environment using some trained data. Channel phase was
estimated during each coherence time. Then pilot data of
some required percentage of data length (referred as
training percentage in simulation) is inserted into the
source data. It is used to estimate the random phase shift
of the fading channel and train the decision to adjust the
received signal with phase recover. The results obtained
showed the great variation in BER for with and without
estimation curves. It is clear from literature reviewed that
phase estimation using training symbol can be imple-
mented in flat fading environment to improve the per-
formance of system.
In this paper, we have implemented the above de-
scribed phase estimation technique in flat fading for
Weibull fading distribution on OFDM system.
6. Results without Estimation
OFDM-BPSK and -QPSK signal is simulated in MAT-
LAB environment by choosing total number of sub-car-
riers 400, IFFT length 1024 by using guard interval of
length 256.
The results presented in Figures 1 and 2 are simulated
by varying the value of and keeping Here,
the BER values have been obtained for varying over
a range of 1 to 7, however, improvement in BER was not
significant for higher values of , So range has been
kept from 1 to 7, both for OFDM-BPSK and -QPSK
system.
α1.μ
α
α
From the simulations, it has been verified that the re-
sults for 2α

are same that are obtained by using
Rayleigh fading distribution.
It has been observed that if we plot BER curves by
changing the value of
μ
there is no change in these
curves. This is because of fact that in the envelope Equa-
tion 10, the variable parameter is which varies the
fading variants and
α
μ
has no role to change this fading
envelope and hence no change in BER values.
05 1015 20 2530
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Channel SNR (db)
BER
BER Vs.SNR Using Weibull channel for BPSK
alpha=1
alpha=1.5
alpha=1.85
alpha=2
alpha=2.5
alpha=3
alpha=3.5
alpha=4
alpha=4.5
alpha=5
alpha=6
alpha=7
(dB)
BER vs. SNR Using Weibull channel for BPSK
Figure 1. BER vs. SNR for OFDM-BPSK system.
05 10 1520 25 30
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Channel SNR (db)
BER
BER Vs.SNR Using Weibull channel for QPSK
alpha=1
alpha=1.5
alpha=1.85
alpha=2
alpha=2.5
alpha=3
alpha=3.5
alpha=4
alpha=4.5
alpha=5
alpha=6
alpha=7
(dB)
BER vs. SNR Using Weibull channel for QPSK
Figure 2. BER vs. SNR for OFDM-QPSK system.

N. SOOD ET AL. 383
To explore the other side of α
μ
, by keeping the
value of fixed and varying the value of
α
μ
, we are
able to have BER curves for other distributions. In Fig-
ure 3, Negative exponential distribution has been plotted
as special case where and
1α
μ
can vary. Here, it
has been plotted for fixed value of 1. Rayleigh distribu-
tion is having and 2α
μ
can vary. Here, it has been
plotted with .One sided distribution has been plot-
ted with and .
1μ
2 α1/2μ
BER varies in the range of 10-1 to 10-5 for OFDM-
BPSK and -QPSK for SNR of 0 to 25 dB. In case of
OFDM-BPSK the BER value of 10-5 is obtained at SNR
of 10dB. However, for -QPSK case the BER of 10-5 is
obtained at SNR of 20 dB.
Comparison between Negative exponential value, Ray-
leigh and one sided distribution results clearly reveals the
fact that in α
μ
distribution the variation in value of
can change the value of BER, however by changing
the value of
α
μ
, there is no impact upon the BER. Re-
sults obtained without estimation technique has been
presented in [11].
7. Results with Estimation
Trained symbols are added to source signal as discussed
in Section 5. The percentage of such symbol may be
varied depending upon the system response to the trained
sequence. We have analyzed the results for various per-
centage values of trained sequence. We have plotted new
graphs with value of and varying value of train-
ing sequence over the range from 10% to 50%. Results
7α
05 10152025 30
10
-3
10
-2
10
-1
10
0
Channel SNR (db)
BER
BER Vs.SNR Using diffrent distribution for BPSK
-ve exponential
Rayleigh
one sided distribution
(dB)
BER vs. SNR Using diffrent distribution for BPSK
Figure 3. BER vs. SNR for negative, rayleigh and exponen-
tial distributions.
for OFDM-BPSK and -QPSK have plotted in Figures 4
and 5 respectively.
In depth analysis of these graphs shows that BER de-
creases, if the training percentage is increased. In Figure
5, if we evaluate the reading obtained at SNR = 10 db,
BER has decreased from 0.0208 to 0.001 for training
percentage of 10 to 50. This means, for the same value of
SNR and , different training percentage has resulted
into different values of BER. More trained sequence will
results into lesser errors. The same has been depicted
from Figure 4.
α
0 2 46 810 12
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Channel SNR (db)
BER
BER Vs.SNR Using Weibull channel for BPSK with Channel estimation
Trg
P
ctg=10%
Trg
P
ctg=25%
Trg
P
ctg=50%
(dB)
BER vs. SNR Using Weibull channel for BPSK with Channel estimation
Figure 4. BER vs. SNR for OFDM-BPSK with channel esti-
mation.
024681012 14 1618
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Channel SNR
(
db
)
BER
BER
V
s.
SNR
U
s
i
ng
W
e
ib
u
ll
c
h
anne
l
f
or
QPSK
w
ith
Ch
anne
l
es
ti
ma
ti
on
TrgP ctg=10%
TrgP ctg=25%
TrgP ctg=50%
(dB)
BER vs. SNR Using Weibull channel for QPSK with Channel estimation
Figure 5. BER vs. SNR for OFDM-QPSK with channel
estimation.
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N. SOOD ET AL. 384
8. Conclusions
This paper, presents performance analysis of OFDM
system with generalized fading model of α
μ
distri-
bution with and without estimation. The non-linearity
added in propagation medium has been clearly shown in
simulated results, since the BER has significantly re-
duced by varying from 1 to 7. However, higher val-
ues of can be used for further reductions in BER. It is
clear from the simulations that the result shows signifi-
cant improvement by applying the phase estimation us-
ing trained symbols.
α
α
9. References
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