ALTERNATIVE INTERLEAVING SCHEMES FOR
INTERLEAVED ORTHOGONAL FREQUENCY DIVISION
MULTIPLEXING
Karthik Subramanian and K.V.S Hari
Department of Electrical Communication Engineenng
Indian Institute of Science
Bangalore 560012, INDIA
080-293-2745
karthik@dsplah.ece.iisc.emet.in,hari@ece.iisc.emet.in
Absrrucr-We propose a matrix model for the Interleaved Orthogonal Frequency Division Multiplexing System OOFDM)
[31, [41, [SI. Using this model, we show that the interleaving scheme originally proposed for IOFDM is not unique in
terms of preserving system complexity. We further show that
a class of interleaving matrices yields IOFDM systems with
the same complexity as the original IOFDM system. Furthermore, we show that the interleaving scheme does not affect
the BER performance of the system as long as the interleaving mamx is unitary.
: W ~ ~ ' ]E, 1
by the D' blocks*, D' = d i U ~ [ l : W p K...,
{ O ; 1,.. . P - 1). Each component K x 1block is then subjected to a K-point DlT,
as represented by the blocks FE
and the composite PK x 1 block is subjected to interleaving, and finally a zero pad of length L equal to the maximum
channel length is appended to the IOFDM blockJhe size of
the block with zero-pad added is M = P K L. This block
sequence !c(rk) is now converted to serial and then fed to a
single antenna for transmission across the channel.
+
1. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM)[l], [2]
is an effective technique for data transmission over frequency
selective channels. However, due to the redundancy introduced in the form of the zero-pad or the cyclic prefix, OFDM
has a loss in code rate. Interleaved OFDM [31,[41,[51 is an
OFDM system which offers an improvement in code rate
without an increase in the FFT size used, by interleaving several data blocks using a specific scheme before the addition
of the cyclic prefixhero pad. IOFDM was proposed with a
specific interleaving scheme. We propose a simplified matrix model for IOFDM and use this model to show that the
interleaving scheme is not unique in terms of yielding an efficient receiver structure. We further state a result giving a
class of interleaving schemes that yield IOFDM systems with
the same complexity as the original IOFDM system. We also
show that as long as the interleaving matrix is unitary, it does
not affect the BER of the system.
(&Ix PXI
_
I
2. A BRIEFREVIEWOF IOFDM
Figure (1) shows an IOFDM Transmitter. The information sequence s(n) consists of complex numbers obtained by mapping a binary sequence to a constellation, and is blocked into
a vector sequence s(n) of size K x 1'. Next, P consecutive blocks of size K x 1 are blocked to get a composite
block of size P K x 1. This block is pre-processed by multiplying the block elements by scalars, which is represented
0-7803-7651-x103/$17.00
~ m r i u ~
K is the OFDM block size i FFT size, and P is the interleaving facta
Heme the IOFDM block size i s P K x I.
'
-/Yid.
,PX
I,
Figure 2. IOFDM Receiver
To illustrate the interleaving scheme, consider a sequence3
[U,,, U,,. . . a K - - l , bo, bl .,.. bx- ,IT.
( We have consid~
2WK = P $
3Tsmds far manspose. H is the Hermitian operator, an underline denotes
a vector. FK is a K x K DfT mtnx.
Communications-/ /\J 47,
ered P = 2 hcrc ). 'The interleaved sequence is now
[(LO; b u , a ~ , b , ; . . ; O K - I , b1<-1]"'.
If we choose the interleaving factor P to he I , IOFDM reduces to the familiar
OFDM system. At the receiving end, thc serial sequence
is blocked into blocks of size Pfi' x 1, and the zero pad is
strippcd from each block. The blocks are now subjected to a
PI< point DFT. This is represented.hy the V matrix, where
V = FA,!,.R z J ~RKI,
, representing the zero-pad removal.
Equalization is done by simple scalar division as represented
by the Rt blocks. given by
'.",,<+,,
R~ = d i , l , ~ [ H ( p j ~ ~ ) , H ( c J ~ )
VI t (0; A'
l ) ,where H(eJ?*) is the channel frequency
response evaluated at the I"' subcartier frequency. The block
sequence is now subjected to a P point IDFT, de-interleaved
and fed to the detector. This yields an estimate of the information sequence a(.). A complete exposition of lOFDM can
be found in [4].
We define LpX as
Where each &,j is a P x I< matrix of zeros, with a 1 at the
(i,j)"' entry
~
3. MATRIXMODELFOR IOFDM
Figute ( 3 ) shows thc matrix models for the IOFDM transmitter and receiver. s ( ? ~is) the complex modulated information
sequence. which is blocked into a PI( x 1 vector sequence,
~ ( n )G-'
.
is the pre-processor, C ~ isKa PIC x P K block
i the interIDFT comprising fi' x K IDFT hlocks 4 . L p ~ is
7'.
I S the zero-pad
leaving inatrix, and TZP= [ I f K Ozxl,li]
addition matrix. Thus we have at the output of the zero-pad
block:
L p x thus defined is the interleaving matrix corresponding
to the original IOFDM interleaving scheme. The signal after passing through the linearly convolutive channel with
impulse-response h ( n ) and being corrupted with additive
white Gaussian noise, is represented as
-dn.) = H : ( T I )
+ y(n)
(5)
H is the channel-convolution matrix [I], ?(TI.) is the noise
component and ~ ( nis)the signal at the receiver's front-end.
The operations performed on the received signal can he written as follows:
Where Rzp = [ I ~ KO p K x L ] is the zero-pad removal matrix, and FPK is a PK-point DFT matrix. We know that the
product H = RZPH TZP.is n circulant matrix [I],and can
-
If
be factorized as H = F,,
'fJ(71)
D F P K [ 6 ] ,and hence we have
= D FPKLPK C P Ka ( l l ) -td ( 7 L )
(9)
"(PK-1
D = d i q ] [ H ( l ) H ( c j % ) , .._,H ( r J w ) ]is a diagonal
matrix and is easily inverted by scalar division. The issue now
is to efficiently invert the product A = FPK LPX C I J X .
which has the following structure:
A=
IDo
K .
Do
...
D1
DI
.. .
DP-I
WK-"KD p - I
(K-IJXI
1
Where D, := diag(1, I.Vj,f<, . . .,LVpK
). We make use
of the following fact to factorize A intvan easily invertible
product: The matrix Z =
TENCON 2003 / 148
We know that
can be
FPK LPK C P KG-I = L' CEp L
factorized as Z = L'r C g p L, where L is the original interleave matrix defined in (4). and C g P is a P K x PI< block
DFT, comprising K blocks of P x P DFT's. That is, the matrix Z is merely a re-indexedversion ofthe block DFT matrix
C;.p.
(14)
This equation is now re-written as
Defininx G = dioy[Do.D1,. . . , D P - ~ ]we
. see that A =
Z G = LT C a p L G. In other words,
FpK
Lph. C P K = L'" cpp L G
# , L G ~ ( n+)$(n)
v(n) = D LT C
(10)
(11)
~(11)=
G-',q(n),the vector infomiation sequence
~ ( 1 % )in the following way, that
yields an efficient receiver structure:
Since
e(,,.)can he recovered from
-
cKp
(DL~)-'
=L ~ '
We want to find the class of matrices Lpx such that G in
the equation above is diagonal. Now, consider the product
F"9 K L$ti C g p . It can he seen that this product can be
written as
(12)
It can he seen that the matrix equation above corresponds to
the IOFDM receiver structure shown in the previous section.
Where, J is a diagonal matrix given by
4. INTERLEAVING SCHEMES PRESERVING
SYSTEM COMPLEXITY
From the previous section, we see that at the receiver:
).(
U
= FPKLPK C P Ks(n)+D-'
P O
J = [ :0
FPKRzr $n) (13)
The signal ~ ( ncan
) be recovered from this equation as long
as the matrix LPK is invertible. In other words, the interleaving scheme is not unique. Permutation matrices are attractive because of their low complexity of implementation.
For any choice of LPK that is invertible, we can always find
a pre-processing matrix G - ' that will ensure the block-DFT
structure at the receiver. For the IOFDM system, G-' is a diagonal matrix, resulting in P K complex multiplications. In
the worst case, a full preprocessing matrix G-' can have a
complexity of (PI<)' multiplications, which is a considerable increase in complexity. Hence this motivates us to look
for interleaving matrices L ~ I that
( result in a pre-processing
matrix G-' with low complexity of multiplication. A diagonal or an antidiagonal G-' would be a good low-complexity
solution, since this would result in only P K multiplications
for the preprocessing operation. However, an antidiagonal
G-' cannot be used in the current IOFDM framework, since
the precoding matrix C is a (forward) block-diagonal. Hence
we concentrate on interleaving matrices that result in a diagonal pre-processing matrix. We immediately notice that
the use of the original interleaving matrix L P K as defined in
equation.(3.3) results in a pre-processing matrix G - ' that is
diagonal.
We state our question as What is the class qfmatrices LPK
that re.salts iii U ~iagonalpre-pmcessingmatrix G-'for. the
IOFDM system Y .
J':
.
.
0.
0
...
.'.,
._
...
0
0
]
Jp-'
(17)
and J is J =1[
,l.i;;i<, ,:;W
. . . , W)$'']'.
Inserting this into equation (3.15), we get
LPK = L Cpti J L" L G CFK
= L C ~ ~ J G C $ ~ = L C ~ ~ D ' C ~ ~ . ( I
The matrices J and G are both diagonal, and hence their
product D' = J G is diagonal as well. L is the original
interleaving matrix used in the IOFDM system [4].
Result: -To get a diagonal pre-processirlg matrix, the interleaving matrix LIJKmust be of the form
r~',o
...
o
1
Where the matrices D'I,1 E {O, 1,.. . , ( P - 1))are diagonal
matrices. In other words, the interleaving matrix should be
-
.
Communications4 /149
a permuted P K x P K block diagonal matrix, with each of
the IC x K diagonal blocks being a circulant matrix. Such
an interleaving matrix will yield an IOFDM system with the
same complexity as the original IOFDM system.
by each symbol. The noise vecLor at the output of the deinterleaver is given by
q(n)
= L'Clf, L D-' Fpti RCP~ ( n )
where the cyclic-prefix removal matrix RCP is given by
RCP = [ O F K x f , I $ K x P K ] ' .
The total energy in the vector RCP~ ( n E,
) . is given by
Ret?iarkr
1. The factorization F j j L
~ ~ CPK
K G-' = L' C g p L
is not unique. The factorization is valid for P-circular shifted
versions of the matrix L. However this does not affect the
. validity of the result in equation (3.18) in any way. This is
because, LT C z P L1 = LT C g p Lz. for all L1 and Lr
which are P-circular shifted versions of L. The equality is
due to the block-diagonal symmetric structure o f C 2 ,
2. The choice of interleaving scheme does not affect the performance of the system in any way. In the next section, we
see that the BER of the lnterleavkd OFDM system is independent of the interleavingscheme used. This is bome out by
simulations as well.
3. We see that for LPK to be a permutation matrix, each
of the constituent blocks of the block diagonal matrix
CPK D' Cpfcmust be a (circulant) pennutation matrix. Le,
theelementsofeachdiagonalhlockD1,l E { O , l , . . . , K - l }
should be the K t h roots of unity, in some order. (This is because the eigenvalues of a circular-shift permutation matrix
of order IC are the h't.hroots of unity in some order.)
5 . PROBABILITY OF ERRORFOR IOFDM
Etu = I
((RGPU(7L))w (RCPw ( ~ L ) ) )
Since the matrix Rep is unitary, the expression on the RHS
above becomes
8 ( ( ~ ( n ()~)( ~
n ) ) )= p (PIC
L ) Nu WS,which is
the same as the energy in the vector before the removal of
the cyclic prefix. However, the average noise PSD has now
changed, since the number of symbols in the vector has become P K from P K + L. The noise PSD in each subcarrier
is given by No
All matrices in the matrix product
in the noise vector under consideration are unitary, except
for the diagonal matrix D-' . This vector represents equalization by scalar division for each subcarrier. Hence, the
noise PSD after equalization in each subcarrier is given by
Rqq(L)= No
The average number ofhad bits per
received vector of PIC symbols (after processing at the receiver) is given by the sum of the bad hits in each subcarrier.
Hence the total number ofbad bits per block of P K symbols
is given by
Pbrr.,c
Thus we obtain
an expression for the probability of error (averaged over several blocks of PI( symbols each) as:
+
w.
&!$&,
Xft-')
(m).
We define Pbcr,c($)to be the Probability of Error expresEa being the
sion for the signal constellation c at the SNR
energy per bit, and h'o being the noise power spectral density
in WattslHz. We know that the Probability of Error of a conventional OFDM system using IC subcarriers can he derived
%,
(w)
[7], with paw,,
as pep,lm = CL;' pbe?,c
as defined above. For the sake of convenience, we assume in
this section that in the IOFDM system, the received vector is
processed and ML detected only after the de-interleaving at
the receiver. In other words, the positions of the ML detector,
and the de-interleaver are interchanged. This does not affect
the performance of the system, as we shall see. The received
vector after processing now becomes
~ ( I I=
)
+
~ ( n ) L'r C z p L
D-' FPKRzp ~ ( n )
It can be seen from the equation above that the encrgyper bit
I is unchanged at the receiver. However, the noise energy at the
receiver is no longer No. The noise vector under consideration is L7' C z , L D-' FpK RZP~ ( T L ) .We analyse the
noise vcctor for both the cyclic prefix and the zero-padding
cases.
IOFDM with Zero Padding
We now examine the probability of error for an IOFDM system using zero-padding. The noise vector under considerationnow becomesg(n) = L" C KL~D-'FPKRZPg ( r i ) .
where Rzp, the overlap-add matrix is given by Rzp =
[IpI< Izp] where Izp is a matrix comprising thb first L
columps of the identity matrix I P K . The overlap-add matrix RZP is not unitary. The average energy in the ~ e c t o r
Rap ~ ( nis )then given by 8 ( ( R ; r p ~ ) ~ ( R ; r p g
Defin)).
ing 9 = [wow' . . . wpx+f,-1IT, we see that the average energy in the vector Rzp &I is given by
IOFDM with c.vclicprefrr
Let 11he the number of bits per symbol in the constellation
used. Then the total energy in the noise vector w(n)that
is added to the signal at the receiver front-end is given by
p (PK L ) Nu
where W8 is the bandwidth occupied
+
The expectation of the cross terms is zero because the w ; s are
uncorrelated. As in the earlier section, the noise PSD in each
subcarrierafter equalization becomes Rq,l(l)= No
m.
TENCON 2003 / 150
The probability of error for the IOFDM system with zero
padding is then given by
This expression is the same as that for the Pe of an IOFDM
system using the cyclic prefix.
Effect of the Inlevleaving Scheme on P,:
We see that the Probability of error of the IOFDM system
is not affected by the choice of the “interleaving” matrix, as
long us rhe nrutrix is unitary. Since any pemiutatiqn matrix
is unitary, the stacked data vectors at the transmitter can be
interleaved in anyfashion without qfecting the bit-error YOtio of the IOFDMsystem. This is borne out by the simulation
results as well. If the “interleaving” matrix chosen is not unitary, it will affect the effective signal-to-noise ratio ($), and
hence the Probability of error as well.
Compari.son of P, between IOFDM and OFDM:
The probability of error for OFDM and IOFDM are given below (assuming that we know the exact P, of the constellation
used.)
*
,0-1
I
1
0
,I
Io
z
1D
U
J10
E b W dB
Figure 4 BER comparison, IOFOM with direrent interleave
schemes
that the choice of the interleaving scheme does not affect the
system performance as long as the interleaving matrix LS unitary Simdation results have been presented to hear out the
same.
REFERENCES
Zhengdeo Wang, Georgios E. Giannakis, “Wircless
Multicarrier Communications - where Fourier meets
-Shannon”, IEEE Signal Processing Magazine. 2 9 4 5 ,
May 2000.
S. B Weinstein, Paul M. Ehert. “Data Transmission
by Frequency-Division Multiplexing Using the Discrete
Fourier Transfonn”, IEEE Trurisucrion,s on G~mmnriication Technoloo, COM-19, 628-634, October I971
We see that the probability of error of an IOFDM system with
Zi’ subcarriers and an interleaving factor of P is exactly the
same as that of an OFDM system with P K subcamers. Furthermore, for K s , the Pis are approximately the same
for both OFDM and IOFDM systems with A’ subcarriers. So
for large P K (with K
), the probability of error of an
IOFDM system is very close to that of an OFDM system.
V.G.S Prasad. K.VS Hari, “Interleaved Orthogonal Frequency Division Multiplexing System” in Proc. lC4SSP
2002, Orlando, FL. May 2002,111.2745-2748,2002.
V.G.S Prasad, Miiltiple-lirput Mri/tiple-Oiitput k h niques for interleaved Orthogonal Frequency Division
Mdfiplexing, Master’s Thesis, Depanment of Electrical Communication Engineering, Indian lnstiture ofScience, January 2002.
V.G.S Prasad, K.V.S Hari, “Interleaved Orthogonal Fre-
6 . S I M U L A T I ORESULTS
N
In figure (4) we see the results of a comparison of the BER‘s
ofOFDM, IOFDM with the original interleaving scheme, and
IOFDM with two different interleaving schemes. In the first
scheme, no interleaving is done and the blocks are inerely
stacked. In the second one, a randomly generated permutation matrix was used for the interleaving scheme. The simulations were performed for the GSM HT channel.
I. CONCLUSIONS
It has been shown that a class of interleaving schemes exist
for IOFDM which yield systems with the same complexity as
the original IOFDM system. Furthermore; it has been shown
quency Division Multiplexing (IOFDM) System” IEEE
Transactions on Signul Processing, To Appear.
Gene H. Golub, Charles van Loan, Matrix Conrpiitulions, 3rd ed. Baltimore. M D Johns Hopkins University
Press, 1996.
Xiang-Gen Xia, “Precoded and Vector OFDM robust to
Channel Spectral Nulls”, IEEE Transactions on Communications, vol. 49, no. 8, pp. 1363-1374. August
2001.