INTERLEAVED ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING SYSTEM
KG.S.Prasad and K.KS.Hari
Department of Electrical Communication Engineering,
Indian Institute of Science, Bangalore 560 012, India
vprasad@protocol.ece.iisc.ernet.in, hari@ece.iisc.ernet.in
ABSTRACT
A new orthogonal frequency division multiplexing (OFDM)
system for wireless communications over frequency selective fading channels is presented. The proposed interleaved
orthogonal frequency division multiplexing (IOFDM) system promises higher code rate as compared to the conventional OFDM system without bandwidth expansion and with
very slight increase in computational complexity. Simulation results verify that the performance, in terms of bit error
rate, of the proposed IOFDM system is very close to the
conventional OFDM system.
1. INTRODUCTION
In recent years, orthogonal frequency division multiplexing
(OFDM) has been adopted as a standard for applications
like digital audio broadcasting (DAB), digital video broadcasting (DVB) and wireless LANs etc. In order to avoid
inter block interference (IBI) between OFDM blocks arising due to the frequency selective nature of the channel,
either cyclic prefix (CP) or zero padding (ZP) is added to
each OFDM block before transmission [l], [2]. The length
of the CP/ZP should be longer than the channel delay profile to avoid IBI. Due to this CP/ZP, the code rate of the
communication system reduces. In this paper, we propose
a new scheme called the interleaved OFDM (IOFDM) system, which enhances the code rate compared to the conventional OFDM system.
The role of C is to effectively convert a frequency selective
fading channel into a number of flat fading channels.
Let us assume that the frequency selective channel is
FIR in nature and the upper bound on its order is assumed
to be known i.e., h(Z) = 0,VZ $! [0,L]. Let us construct
~ ' ( n of
) ,length N = K
L, as follows
+
:'(n)
=
where H is the channel convolution matrix of size N x N
with the (k,l)th element being h(k - 1).
Define the following z-transforms:
= Cm=OCk(m)Z-m,
K-1
H ( z ) = C,=LO h(m)Z-m
.
the zand Y'(n;z) = CE=Ay'(nN m ) ~ - ~Then,
transform of (3) is given by
Ck(Z>
' T stands for transpose. Underline denotes vectors.
0-7803-7402-9/02/$17.0002002 IEEE
c
K-1
+
+ K - 1)]T = Cg(n).
+
Y'(n;z ) = H(z)X'(n; z)
In an OFDM system [3], [4], the bit stream is mapped to
an information symbol sequence s ( n ) using a modulation
scheme. The sequence s ( n ) is divided into blocks, ~ ( n=)
[ s ( n K ).,. . ,s(nK K - 1)IT,of length K. Then ~ ( n )
is further precoded into another block g(n), of length K ,
through a precoder matrix C such that
[u(nK),. . .,u(nK
+
+
[u(nK),. . . ,u(nK K - l ) , 0,.. . ,O]T
[.'(nN), . . . ,z'(nN N - l)]T.
(2)
The last L elements of the block ~ ' ( nare
) set to zero. This
process is known as zero padding. K and N are called
the OFDM block length and the transmitted OFDM block
length, respectively. The sequence z'(n) is then serially
transmitted through a transmitting antenna. Because of ZP,
the channel induced IBI between the blocks of ._' ( n )is avoided and one can focus at each received block separately. At
the receiver, the received OFDM block, in the absence of
noise, is given by
2. ORTHOGONAL FREQUENCY DIVISION
MULTIPLEXING SYSTEM
u(n) =
=
(1)
= H(z)
+
Ck(Z)S(TZK k).
(4)
k=O
Let us choose K distinct points { p l } L ; ' on the complex z-plane. The matrix C is designed such that
Ck(pl) = d
( k - Z),
V k , l E [0, K - 11,
+
(5)
+
2 c k ( m ) denotes the element present in (m
1 ) s t row and (k 1 ) s t
column of C. Ck(r) denotes the z-transform of ( k 1 ) s t column of C.
111- 2745
+
where u is chosen to impose transmission power constraint3
i.e., E ( g ( n ) N g ( n )=
) E ( ~ ( n ) ~ g (5n Ptot.
) ) ( 5 ) enables
proper data recovery at the receiver. C can be constructed
by considering Ck ( z ) to be an FIR filter of order K - 1 with
[2]. For pl,Vl E [0,K - 11,
zeros being { p l } L &
- s ( 2n+1)92n+1~
S/P : Serial to Parallel conversion. P/S ; Parallel to Serial conversion. ZP : Zero Padding.
y'(n;z)l
c
K-1
A
= Y'(n;pl)= H ( p l )
Fig, 1. Transmitter section of IOFDM system
Ck(pl)s(nK+k).
k=O
%=Pi
(6)
gn)
Using (5), (6) is simplified as
+
vz E [O, K
Y'(n;p l ) = uH(p1)s(nK Z),
bi;
Zxl
2x1-1
2x1
- 11. (7)
Using (7), s ( n ) can be estimated given the knowledge
of the channel state information (CSI). The code rate of the
OFDM system, denoted by 7,is 7 = K / N = K / ( K
L). A new scheme is presented in the next section which
inproves the code rate.
+
3. INTERLEAVED ORTHOGONAL FREQUENCY
DIVISION MULTIPLEXING SYSTEM
In this section, the transmitter and receiver sections of the
proposed IOFDM system are presented.
b=l/a
(3), the received IOFDM block, in the absence of noise, is
given by
dn) = H d n ) .
(9)
The z-transform of (9) is given by
c
(c
c
(1;:c
c
Y ( n ;z ) = H ( z )
3.1. IOFDM Transmitter
+
Idea: Construct ~ ( n of
) , length M
= 2K
leaving the elements of g(2n) and 4 2 n
L zeros as follows
+m)z-
K-1
+
= H(z)
+
K-1
z-l
+ L, by inter-
0 , . . . 101'
= [ z ( n M ) ., . . , z ( n M
+m
) ~+ - ~ ~
u((2n+ l ) K + m)z-Zm
m=O
+ 1) and padding
~ ( 2 n K K - 1 ) , ~ ( ( 2+nl ) K
42nK
m=O
= H(z)
+
C@)s(2nK
K- 1
[u(2nK),u((2n 1)K),u(2nK l),. . . ,
+
z(nM
m=O
The block diagram in Fig. 1 describes the baseband model
of an IOFDM transmitter section. Let us consider two consecutive symbol blocks g(2n) and g(2n 1). The corresponding precoded blocks are t~(271) and g(2n 1).
=
I
Fig. 2. Receiver section of IOFDM system
M-1
:(n)
$)
2x1
2-l
+K - l),
+
+ k) +
Ck(Z2)S((2?2 l ) K
k=O
)
+ k) .
)
(10)
+ M - l)]'.
(8)
Because of interleaving, we call it as IOFDM system. The
IOFDM block length is 2K and the transmitted IOFDM
block length is M .
Let us choose 2K distinct points { r},:;'
plex z-plane such that
For z = T I ,VZ E [0, K - 11,
3.2. IOFDM Receiver
The block diagram in Fig. 2 describes the baseband model
of an IOFDM receiver section. Let us assume that the channel is constant over a transmitted IOFDM block. Similar to
~~
3 E ( ) and 3t stand for expectation and complex conjugate transpose,
respectively. Ptot denotes the total power allowed.
111 - 2746
on the com-
Using (5) and (1 l), (12) can be simplified as follows
hble 1. GSM
Twelve 1 h TU Model
Relative Avg. Relative
TimeGs)
Power(&)
Y ( n ;p y 2 ) = U H ( p y 2 ) s ( 2 n K + 1) +
(
1
+ l)K + 1 )
p[1/2s((2n
. (13)
Similarly, for z = TK+~,VZE [0,K - 11, we can obtain
Y ( n ;#)
= uH(-p:/2)
(s ( 2 n K + I )
+
p11/2s((2n l)K
1
+ 1) .
(14)
Y ( n ;-p:l2)IT and rewrite (13) and (14) for a given 1 as
[O,K - 11,
-4.0
0.2
0.4
-3.0
0.0
0.4
-6.0
0.6
-2.0
-3.0
-5.0
0.6
-4.0
0.8
0.0
0.0
1.2
1.4
1.8
2.4
3.0
3.2
5.0
Inordertodecodes(n), wedefiney_l(n) = [ Y ( n ; ~ i / ~ ) ,
%(n)= ~ f i i % ( n ) , Vl E
0.0
0.8
-
iannel Mc els
Twelve 1 h HT Model
Relative Avg. Relative
TimeGs)
Power(&)
0.0
- 10.0
0.2
-8.0
2.0
2.4
15.0
15.2
15.8
17.2
20.0
-7.0
-5.0
-6.0
-9.0
-1 1.0
-10.0
-4.0
-8.0
-9.0
-10.0
-12.0
-14.0
(15)
3.3. An example of an IOFDM system
where
Let us choose K distinct points { p l } E ; ' = {ejy}E;'.
For this choice of { p l } E ; ' , the precoder C becomes an
IDFT matrix with the element c k ( r n ) = & e j v and
+
+
+
%(n) = [ s ( 2 n ~I ) , s ( ( 2 n 1 ) Z~) I T . It is easy to see
that H l is full rank if H ( p ; I 2 ) # 0 and H ( - p : / " ) # 0. The
estimate of 3 (n) can be obtained as
u=
a.Using (ll), the 2K distinctpoints { T ~ } ; Eare; ~
as follows
{
T
V
Z
}
~
jd K - 1
}1=0
=~ ie
~
2K-1
7
{Tm}m=K
- {-.
jd K - 1
}1=0
*
517)
Using (15)' we can get %(n)= [Y(n; e j g ) ,Y (n;-ej%)IT
as follows
Remarks:
0
0
0
0
yI( n ) =
Compared to the conventional OFDM system, which
considers K distinct points {pl}L;' on the complex
z-plane, the IOFDM system considers 2K distinct
on the complex z-plane. This impoints {r,};E;'
plies that we need additional constraints on channel's
impulse response H (2).
+
In presence ofnoise i.e., & ( n ) = u f i l % ( n ) %(n),
where 2(n)denotes the appropriate noise vector, the
estimate h(n)= :fi;'g(n).
The code rate of the IOFDM system, denoted by Q I ,
is QI = 2 K / M = 2 K / ( 2 K L). The ratio, QI/Q,
is greater than one, which is very desirable. It is also
observed that the redundancy due to ZP is less by a
factor of 2.
+
The IOFDM system requires additional computation
in terms of a '2 x 2 matrix inversion operation' for
every two symbols.
dRfiia(n),
V1 E [0,K - 11,
(18)
where
4. SIMULATION RESULTS
In our simulation, the available channel bandwidth is assumed to be 1 M H z . The entire bandwidth is divided into
K = 64 subchannels. To make the subcamers orthogonal
to each other, the symbol duration on each subcamer, denoted by T,,is T, = 64ps. The duration of the IOFDM
block is 1 2 8 ~ s .A ZP of duration T, is used to avoid IBI
due to the frequency selective nature of the channel. The total duration of the transmitted IOFDM block is 1 2 8 p T,,
where T,has been chosen according to channel model. The
twelve path GSM TU and HT channel models (see Table 1)
are used. We have chosen T, = 711s in TU channel model
and T, = 20ps in HT channel model. The direct form mutipath Rayleigh fading simulator [5] and QPSK modulation
111 - 2747
+
1OD
1
loo
1
I
t
I
1
a5
1o - ~
10”
+ IOFDM: K.64, fD=200Hz
f). OFDM: K.64. fDz20Hz
I
Fig. 3. Performance comparision of IOFDM system and
OFDM system at different fading rates in TU channel
model.
+ IOFDM: K.64, fD=200Hz
-O - OFDM: K.64,
fD=20H~
Fig. 4. Performance comparision of IOFDM system and
OFDM system at different fading rates in HT channel
model.
scheme are used. It is assumed that perfect CSI is available
at the receiver.
The average Bit Error Rate (BER) is considered as the
performance measure. To gain insights into the average
BER performance of the OFDM system and the IOFDM
system, we have taken 3000 OFDM blocks, of length K =
64,per trial and averaged over 50 trials. In effect, for each
Ea/N, value considered, the BER values are obtained based
on 19.2 x
lo6 bits.
In Fig. 3, the average BER performance of the proposed
IOFDM system is compared with the conventional OFDM
system at fading rates fD = 20Hz and fD = 200Hz in TU
channel model. In Fig. 4, the average BER performance of
the IOFDM system is compared with the OFDM system at
fading rates fo = 20Hz and fo = 200Hz in HT channel model. In both cases, the performance of the IOFDM
system is very close to the OFDM system.
5. CONCLUSION
A new OFDM system based on interleavingthe elements of
two consecutive OFDM blocks has been presented. Simulation results verify that the code rate of the proposed IOFDM
system is higher than the conventional OFDM system for no
loss in performance. Generalizing this idea by considering
more than two OFDM blocks is under progress [6].
I11 - 2748
6. REFERENCES
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V.G.S.Prasad and K.V.S.Hari, “Generalized Interleaved Orthogonal Frequency Division Multiplexing
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