Communications and Network, 2009, 01-05
doi:10.4236/cn.2009.11001 Published Online August 2009 (http://www.scirp.org/journal/cn)
The Comparisonal Analysis of the Concept of Rectangular
and Hexagonal Pilot in OFDM
Jeanbaptiste YAMINDI1, Muqing WU2
Broadband Communication Network Laboratory, Beijing University of Posts and Telecommunications, Beijing, China
Abstract: Channel estimation in coherent OFDM by inserting pilot-symbols into two-dimensional time- fre-
quency lattice is a promising candidate for improving channel transmission capacity in future wireless commu-
nication systems. This paper proposes a new Cartesian mathematical concept of the Hexagonal Pilot Pattern,
Hexagonal Pilot with Virtual and Rectangular Pilot Pattern. The main focus of this work is to compare between
the two pilots and by virtue of simulation conclusion which of the two pilots yields a better performance.
Keywords: hexagonal pilot pattern, hexagonal pilot with virtual, rectangular pilot pattern, estimation,
interpolation, extrapolation
1. Introduction
One of the most important techniques applied in Physical
layer of WiMAX is known as Orthogonal Frequency
-Division Multiplexing (OFDM), which is the system for
data transmission and the support for channel estimation
and synchronization.
Many authors proposed the concept of pilot tones to
interpolate the channel response in [1] and [2], recently
some literature proposed methods of pilot symbols apart
from the existing equi-spaced pilot pattern for OFDM
system as described in [4] and [5]. In this paper, we are
focusing on a specific pilot subcarriers by doing the ana-
lytical description and structural pilot of the two different
pilots known as Hexagonal and Rectangular.
The rest of this paper is organized as follows: The
graphical representation of Rectangular and Hexagonal
Pilot Pattern in Section 2, The Simulation results of the
Hexagonal Pilot Pattern, Hexagonal with Virtual and
Rectangular Pilot Pattern in Section 3. The analytical
evaluation and comparison in Section 4 and the paper is
concluded in Section 5.
2. The Rectangular and Hexagonal Pilot Pat-
tern Representative
2.1 The Rectangular Pilot Grid Representation
The rectangular Pilot grid matrix can be expressed by
0
0
f
rect
t
p
p










(1)
where Δpf is the distance in frequency and Δpt is the dis-
tance in time direction and this rectangular matrix can be
written as:
0
0
t
RECT
f
N
VV N




(2)
where subcarriers in the frequency direction and N
the OFDM symbols in the time direction that can be ar-
ranged as a set given by:
f
N
 



112 2
, , ,,...,,
 
tap tap
NN
Pnknknk . (3)
And the number of pilot subcarriers in the input block
is expressed by
det
ft
p
rect
NN
N









 (4 )
The estimation of block contains 



t
t
N
p
number of
OFDM symbols each hosting 




f
f
N
p
pilot positions.
The total number of pilot positions of a rec-
tangular pilot grid is given by
,prect
N
,






 

ft
prect
ft
NN
Npp

(5)
Under the condition of
2,max
1
2(1 )
t
D
NNf and f
N
NCP . (6)
The graphical representation of the rectangular pilot in
the coordinate system is depicted below.

2 THE COMPARISONAL ANALYSIS OF THE CONCEPT OF RECTANGULAR AND HEXAGONAL PILOT IN OFDM
t

Rec
t
an
g
ula
r
Figure 1. The rectangular pilot grid.
2.2 Modification of Rectangular Pilot Pattern
The application of Cognitive Radio for optimum channel
estimation by using pilot pattern can affect the frequency
band by the presence of Licensed User. The pilot pattern
in the presence of Licensed User can be classified ac-
cording to the Bandwidth into Narrowband Licensed
User and Wideband Licensed User.
1) The Narrowband Licensed User: It can exchange
easily the position of the pilot symbols with data parts.
This exchange of the pilot subcarrier is connected to the
data subcarrier of the previous or the next subcarrier and
we assumed that the application of pilot symbols in the
channel estimation process cannot modify the pilot sym-
bol but only one data subcarrier and can lose some
throughput so this narrowband interference is depicted.
2) The Wideband Licensed: There are similarities in
the process of Narrowband and Wideband, the only dif-
ference is on the edge location of the License User signal
exchange of two pilot subcarriers with data subcarriers
and the number of deactivated subcarriers is depending
upon the Licensed User Band.
2.3 The Hexagonal Pilot Grid Representation
The implementation of Hexagonal Pilot is facilitated by the
Virtual Pilot in reducing the Pilot Density by the annulment
of u2 =0 and the Pilot Density becomes dp=| u1 v2| and the
representation of the structure is according to the Hex-
agonal Pilot spacing. The equation of the Hexagonal Pi-
lot is expressed by
t
H
ff
NN
VV NN





(7)
And graphically it can be represented as below.
In both of the two different types of Pilot, the In-
ter-Symbol Interference is still a big challenge during the
process of pilot transmission. This challenge can be re-
solved by the following techniques.
Hexa
g
onal
V2
V2 V1
V1
Figure 2. The hexagonal pilot grid.
2.4 Modification of Hexagonal Pilot Pattern
The process of exchanging the position of the pilot sub-
carrier with the data subcarrier connected to the previous
subcarrier and the next subcarrier is done by exchanging
the pilot subcarrier with the data subcarrier which lies in
the previous subcarrier or in the next subcarrier. By do-
ing this we can still be able to keep the pilot symbols to
be used in the channel estimation process, but on the
other side we will have to sacrifice one data subcarrier
and thus lossing some throughput. This process is almost
similar as in Rectangular process, the only difference, at
the receiver, the received pilot symbols are processed
using the iterative algorithm in order to obtain the virtual
pilot symbol. This iterative algorithm introduces the
concept of interpolation option which is applied via sim-
ple averaging between the received shifted pilot symbols
and the received pilot symbols. The received shifted pilot
symbols in the next subcarrier or extrapolating the virtual
pilot symbols on the position of the Licensed User by the
effect of the received pilot symbols.
2.5 The Virtual Pilot
The Virtual Pilot symbols are iterative of the interpola-
tion of contiguous pilot to the position of the virtual. It is
defined at first by the pilot spacing in frequency axis,
which is smaller than the channel coherence bandwidth,
and the second is pilot spacing in time axis, which is
smaller than the channel coherence time on the initial
computation of the received pilot symbols for each
OFDM. The OFDM frames during each time slots, and
subcarriers in the frequency domain respectively. Note
that this operation must be processed only at the pilot
positions as:

,
,,
,
ˆ,

 



 
kn
kn kn
kn
N
HH kn
S
(8)
Sk,n is the transmitted symbol, the indexes k, and n in-
dicates the position of each pilot symbol within the
OFDM frame. The symbol W is the set of pilot positions
within the OFDM sub frame.

THE COMPARISONAL ANALYSIS OF THE CONCEPT OF RECTANGULAR AND HEXAGONAL PILOT IN OFDM 3
pilots in time direction, number of subcarriers and the
number of OFDM symbols per frame.
At the positions close to Licensed User's carriers. The
estimation process itself is divided into two parts assume
that it we can assume that it can also be divided into
more than two depending on the number of Licensed
User that exist.
Figure 3. Interpolations processes, (a) the DFT principle for fre-
quency domain interpolation, (b) linear interpolation process over
the time domain.
The first part is the upper part and the second is the
lower part. The part of the spectrum which is occupied in
the middle is discarded and not included in the channel
estimation calculation. By deploying the virtual pilot
concept on certain case for good understanding let us
illustrate by band less than 2x distance of pilots, the edge
effect of the sliding window can be avoided, hence better
channel estimate can be obtained, while the Channel
Transfer Function at the position of the Licensed User
are discarded.
These Interpolations processes are illustrated by the
figure above.
By the initial computation of the channel the value can
be linearly interpolated i.e. the value of the virtual pilot
symbols is expressed by
,,1,
1
ˆˆˆ
2
nknk nk
HHH
 



1


.
(9)
Where and 1 are initial estimates re-
spectively of the previous and the next pilot at this point
,1
ˆnk
H
,
ˆnk
H

The iterative algorithm processing of pilot symbols is
expressed as
for n′=1: ( p′−1) N f
for k′=1: ( q′−1) Nf
,,1,
1
ˆˆˆ
2
nknk nk
HHH
 



1
end
end
1, 2,...c
f
N
pN





and 1,2, ...
s
t
N
qN



Where , ,
f
Nt N
s
N, are respectively the distance
between pilots in frequency direction, distance between
c
NFigure 4. Transmitted modified hexagonal pilot pattern for the
wideband Licensed User case.
Figure 5. Receiver side hexagonal pilot pattern with virtual pilot symbols obtained from the iterative processing algorithm.

4 THE COMPARISONAL ANALYSIS OF THE CONCEPT OF RECTANGULAR AND HEXAGONAL PILOT IN OFDM
By deploying this scheme, we can still estimate the
channel optimally despite of the unloaded subcarriers
caused by the Licensed User.
At the receiver, the received pilot symbols are proc-
essed using the Iterative algorithm to obtain the virtual
pilot symbols like in the case of the Narrowband Inter-
ference. The scheme itself can be seen in Figure 5.
3. The Simulation Result
In this section we are going to do the comparison be-
tween the Hexagonal Pilot Pattern with the Rectangular
Pilot Pattern and the Hexagonal Pilot with Virtual sym-
bols with the Rectangular Pilot Pattern with Virtual Pilot
symbols distribution. For this simulation, we consider the
different parameters such as subcarriers per OFDM
symbol, OFDM symbols per estimator frame and the
distance of subcarriers in the frequency direction and the
time direction, the total number of pilot subcarrier posi-
tions is excessive to be directly employed in the estima-
tion of all the channel transfer factors.
Figure 6. The performance comparison of hexagonal pilot pattern and
the rectangular pilot pattern in function of BER in Eb/N0 in dB scale.
Figure 7. The performance comparison of Hexagonal Pilot with
Virtual symbols compared with Rectangular Pilot Pattern with
Virtual Pilot symbols in function of BER in SNR in dB scale.
4. The Comparisonal Analysis
This section presents the main goal of our work by mak-
ing the analytical evaluation and comparison of the
Hexagonal Pilot Pattern and Rectangular Pilot Pattern.
Analysis of the effect of having several scattered in-
terferences and the effectiveness of the proposed hex-
agonal pilot pattern scheme to combat the interferences
effects. Here we analyze the effect of increasing number
of interferers by using the iterative and exploiting the
channel coherence bandwidth property.
The Hexagonal Pilot Pattern is more complicated to be
implemented using the 2xl-D Wiener filter. The imple-
mentation of Hexagonal Pilot Pattern is easy due to the
technique of Hexagonal Pilot Pattern with the Virtual
Pilot symbols.
The Hexagonal Pilot Pattern is an optimum sampling.
It is the main method which requires 13.3% less samples
when compared to the Rectangular Pilot Pattern that
represents the same signal and gives a better BER per-
formance, and therefore the Hexagonal Pilot Pattern per-
forms better and more efficient sampling mechanism
when compared to the Rectangular Pilot Pattern as
shown in Figure 6.
The performance comparison of the Hexagonal Pattern
with virtual pilots at high SNR outperforms significantly
the other three, the Rectangular, Hexagonal, and Hex-
agonal with Virtual pilots and interpolation/extrapolation
process.
Rectangular Pattern with Virtual Pilots outperforms
the Rectangular and Hexagonal Pilot Pattern at high SNR,
while at low SNR performance degradation occurs
caused by the error of interpolating the noisy initial
channel estimates.
5. Conclusions
The simulation shows that Hexagonal Pilot design is bet-
ter than the rectangular and other pilot design patterns.
This better performance is due to the effect of increasing
number of interferers by using the iterative and exploit-
ing the channel coherence bandwidth property of Hex-
agonal Pilot. The technique of combining with Virtual
Pilot symbols in the low SNR space is due to the error
caused by interpolation of the heavily noisy pilots and
the Cognitive Radio is the best support of perform chan-
nel estimation. The Hexagonal Pilot is become the most
useful in the system of Wireless Network communication
thank to the compatibility of the Hexagonal Pilot and
Network configuration.
6. Acknowledgments
Special thanks to Engineers from Beijing University of
Posts and Telecommunications: Mr. Fenta Adinew, Mr.
Mala Umar, my professor Wu Mu-qing and Beijing
University of Posts and Telecommunications for the as-
sistance.

THE COMPARISONAL ANALYSIS OF THE CONCEPT OF RECTANGULAR AND HEXAGONAL PILOT IN OFDM 5
REFERENCES
[1] YIN Q Y, ZHANG Y, DING L. A new space diversity technology.
Transaction of Xi’ an Jiaotong University. 2005, 39(6): 551-557.
[2] SENDONARIS A, ERKIP E, AAZHANG B. Increasing uplink
capacity via user cooperation diversity. IEEE Int. Symp. on
Information Theory, Cambridge, MA, Aug. 1998, 156.
[3] HASNA M O, ALOUINI M S. Harmonic mean and end-to-end
performance of transmission systems with relays. IEEE
Transactions on Communications, January 2004, 52(1).
[4] HASNA M O, ALOUINI M S. Performance analysis of two-hop
relayed transmission over Rayleigh fading channels. In Proc.
IEEE Vehicular Technology Conf. (VTC’02), Vancouver, BC,
Canada, Sept. 2002, 1992-1996.
[5] GRADSHTEYN I S, RYZHIK I M. Table of Integrals, Series,
and Products.San Diego, CA: Academic Press, fifth ed., 1994.
[6] ABRAMOWITZ M, STEGUN I A. Handbook of mathematical
functions with formulas, graphs, and mathematical tables. NY:
Dover Publications, New York, 9th Edition, 1970.
[7] IKKI S, AHMED M H, Performance analysis of cooperative
diversity wireless networks over nakagamim fading channel. IEEE
Communications Letters, April 2007, 11(4).
[8] SIMON M K, ALOUINI M S. Digital communication over fading
channels: A unified approach to performance analysis. NEW York:
Wiley, 2000.
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