Interleaved orthogonal frequency division multiplexing system

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 selec- tive fading channels is presented. The proposed interleaved orthogonal frequency division multiplexing (IOFDM) sys- tem promises higher code rate as compared to the conven- tional OFDM system without bandwidth expansion and with very slight increase in computational complexity. Simula- tion 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 broad- casting (DVB) and wireless LANs etc. In order to avoid inter block interference (IBI) between OFDM blocks aris- ing 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 pro- file 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) sys- tem, which enhances the code rate compared to the conven- tional

OFDM

system.

2. ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING SYSTEM

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 ( n ) = [u(nK), . . .

,

u(nK

+

^{K - 1)]T = Cg(n).} (1) ' T stands for transpose. Underline denotes vectors.

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) = [u(nK), . . .

,

u(nK

+

^{K - l ) , 0,. . .}

,

^O]T

[.'(nN),

. .

.

,

z'(nN

+

^{N - l)]T.}

= (2)

The last L elements of the block ~ ' ( n ) are 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 avoi- ded and one can focus at each received block separately. At the receiver, the received OFDM block, in the absence of noise, is given by

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:

and Y'(n;z) = CE=Ay'(nN

+

m ) ~ - ~ . Then, the z- transform of (3) is given by

h( m) Z-m C k ( Z > = Cm=OCk(m)Z-m, K-1 H ( z ) =

C,=O

^L

Y'(n; z ) = H(z)X'(n; z)

K-1

= H ( z )

c

Ck(Z)S(TZK

+

^{k). (4)}

k=O

Let us choose K distinct points { p l } L ; ' on the com- plex z-plane. The matrix C is designed such that

C k ( p l ) = d ( k -

Z),

V k , l E [0, K - 11, ( 5 )

0-7803-7402-9/02/$17.00 02002 IEEE 111

-

²⁷⁴⁵

2 c k ( m ) denotes the element present in (m

+

^{1 ) s t} row and (k

+

^{1 ) s t}

column of C. C k ( r ) denotes the z-transform of ( k

+

1 ) s t column of C.

where u is chosen to impose transmission power constraint³ i.e., E ( g ( n ) N g ( n ) ) = E ( ~ ( n ) ~ g ( n ) )

5

^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 zeros being { p l } L & [2]. For pl,Vl E [0, K - 11,

K-1

= A Y'(n;pl) = H ( p l )

c

Ck(pl)s(nK+k).

y ' ( n ; z ) l

%=Pi k=O

( 6 )

vz

^E [O, K - 11. (7) Using ( 5 ) , ( 6 ) is simplified as

Y'(n; p l ) = uH(p1)s(nK

+ Z),

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.

3.1. IOFDM Transmitter

The block diagram in Fig. 1 describes the baseband model of an IOFDM transmitter section. Let us consider two con- secutive symbol blocks g ( 2 n ) and g(2n

+

^1). The corre- sponding precoded blocks are t~( 271) and g( 2n

+

^1).

Idea:

Construct ~ ( n ) , of length M = 2K

+

^L, by inter- leaving the elements of g(2n) and 4 2 n

+

¹⁾ and padding L zeros as follows

:(n) ⁼ [u(2nK),u((2n

+

1)K),u(2nK

+

^l), . . . , 0 , .

. .

^101'

[ z ( n M ) ,

. . .

, z ( n M

+

^{M - l)]'.}

~ ( 2 n K

+

^{K -} 1 ) , ~ ( ( 2 n

+

l ) K

+

^{K - 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 .

3.2. IOFDM Receiver

The block diagram in Fig. 2 describes the baseband model of an IOFDM receiver section. Let us assume that the chan- nel 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.

- s ( 2 n + 1 ) 9 2 n + 1 ~ -

S/P : Serial to Parallel conversion. P/S ; Parallel to Serial conversion. ZP : Zero Padding.

Fig, 1. Transmitter section of IOFDM system

gn) ^$)

bi;

2 x 1 Z x l

b = l / a 2 x 1 - 1 2 x 1 -I

Fig. 2. Receiver section of IOFDM system (3), the received IOFDM block, in the absence of noise, is given by

The z-transform of (9) is given by

- dn) = H d n ) . (9)

M-1

Y ( n ; z ) = H ( z )

c

^{z ( n M}

+

^{m ) z -}

m=O K-1

= H ( z )

( c

^{4 2 n K}

+

m ) ~ - ~ ~

+

m=O

) )

K-1

z-l

c

^u((2n

+

^{l ) K}

+

^m)z-Zm

= H ( z )

c

^C@)s(2nK

+

^k)

+

m=O

(1:;

(10) Let us choose 2K distinct points { r,}::;' on the com-

K- 1

2-l

c

Ck(Z2)S((2?2

+

^{l ) K}

+

^k)

.

k=O

plex z-plane such that

For z = T I

,

VZ E [0, K - 11,

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Using (5) and (1 l), (12) can be simplified as follows Y ( n ; p y 2 ) = U H ( p y 2 ) s ( 2 n K

( +

1)

+

(

p [ 1 / 2 s ( ( 2 n

+

l ) K

+

^{1 )}

1

. ₍₁₃₎

1

Similarly, for z = TK+~,VZ E [0, K - 11, we can obtain Y ( n ; #) = u H ( - p : / 2 ) s ( 2 n K

+

^{I ) -}

p 1 1 / 2 s ( ( 2 n

+

^{l ) K}

+

¹⁾

.

₍₁₄₎

Inordertodecodes(n), wedefiney_l(n) = [ Y ( n ; ~ i / ~ ) , Y ( n ; -p:l2)IT and rewrite (13) and (14) for a given 1 as

%(n) = ~ f i i % ( n ) , Vl E [O,K - 11, (15) where

%(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

Remarks:

0 Compared to the conventional OFDM system, which considers K distinct points { p l } L ; ' on the complex z-plane, the IOFDM system considers 2K distinct points {r,};E;' on the complex z-plane. This im- plies that we need additional constraints on channel's impulse response

H

(2).

0 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).

0 The code rate of the IOFDM system, denoted by Q I ,

is ^{Q I} = 2 K / M = 2 K / ( 2 K

+

^{L ) .} The ratio, Q I / 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.

0 The IOFDM system requires additional computation in terms of a '2 x 2 matrix inversion operation' for every two symbols.

Twelve 1 Relative TimeGs)

0.0 0.2 0.4 0.6 0.8 1.2 1.4 1.8 2.4 3.0 3.2 5.0

hble 1. GSM h TU Model Avg. Relative

Power(&) -4.0 -3.0 0.0 -2.0 -3.0 -5.0 -7.0 -5.0 -6.0 -9.0 -1 1.0 -10.0

iannel Mc Twelve 1 Relative TimeGs)

0.0 0.2 0.4 0.6 0.8 2.0 2.4 15.0 15.2 15.8 17.2 20.0

els h HT Model

Avg. Relative Power(&)

- 10.0 -8.0 -6.0 -4.0

0.0 0.0 -4.0 -8.0 -9.0 -10.0 -12.0 -14.0

3.3. An example of an IOFDM system

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 u =

a.

Using ( l l ) , the 2K distinctpoints

{ T ~ } ; E ; ~

^are

as follows

jd K - 1 2K-1 - jd K - 1

{ T V Z } ~= ~

ie

~ ^{}1=0 7} { T m } m = K - {-. }1=0 *

517) Using (15)' we can get %(n) = [Y (n; e j g ) , Y (n; -ej%)IT as follows

-I y ( n ) =

dRfiia(n),

^{V1 E [0, K - 11,} (18) where

4. SIMULATION RESULTS

In our simulation, the available channel bandwidth is as- sumed 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, de- noted 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 to- tal 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 mu- tipath Rayleigh fading simulator [5] and QPSK modulation

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1 OD 1 1

t I

a

5

10”

I

+ IOFDM: K.64, fD=200Hz

f). OFDM: K.64. fDz20Hz

loo

1 I

1 o - ~

+ IOFDM: K.64, fD=200Hz

-O - OFDM: K.64, fD=20H~

Fig. 3. Performance comparision of IOFDM system and OFDM system at different fading rates in TU 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 = 2 0 H z and fo = 200Hz in HT chan- nel 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 interleaving the elements of two consecutive OFDM blocks has been presented. Simula- tion 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].

Fig. 4. Performance comparision of IOFDM system and OFDM system at different fading rates in HT channel model.

6. REFERENCES

B.Muquet, M.de Courville, P.Duhame1 and G.B.Giannakis, “OFDM with Trailing Zeros Versus OFDM with Cyclic Prefix: Links,Comparisions ans Application to the HIPERLAN/2 system,” Proc. of ICC, New Orleans, LA, pp. 1049-1053, June 2000.

Z.Liu, G.B.Giannakis, S.Barbarossa and A.Scaglione,

“Transmit-antennae space-time block coding for gen- eralized OFDM in the presence of unknown multi- path,” IEEE Journal on Selected Areas in Communi- cations, vol. 19, no. 7, pp. 1352-1364, July 2001.

L.J.Cimini, Jr., “Analysis and simulation of a digi- tal mobile channel using Orthogonal Frequency Divi- sion Multiplexing,” IEEE Transactions on Communi- cations, vol. COM-33, no. 7, pp. 665-675, July 1985.

J.A.C.Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Communications Magazine, pp. 5-14, May 1990.

S.A.Fechte1, “A novel approach to modeling and ef- ficient simulation of frequency selective fading radio channels,” IEEE Journal on Selected Areas in Com- munications, vol. 11, no. 3, pp. 422-431, April 1993.

V.G.S.Prasad and K.V.S.Hari, “Generalized Inter- leaved Orthogonal Frequency Division Multiplexing system,” under preparation.

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