Filter bank based interference suppression for fading channels

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					                FILTER BANK BASEDINTERFERENCE SUPPRESSION
                           FOR FADING CHANNELS

                                           Yuhong Wang and Xiao-Ping Zhang
                                    Department o Electrical and Computer Engineering
                                                f
                                         Ryerson lJniversi@ 350 Victoria Street,
                                           Toronto, Ontario, Canado, M5B 2K3
                                             xzhang, yuwang@ee.ryerson.ca


                        Abstract                                    reduce this sensitivity with cost of system efficiency. In
                                                                      DWMT system, interchannel interference (ICI) is
      In this paper, we propose a compla-valued unitary             minimized by well-designed prototype filter [2], while
filter bank for multicarrier (MC) transmission to                   the intersymbol interference (1st) is not considered and
 suppress the interference introduced by fading channels.           so far only real-valued filter banks are used.
 The filters of proposed filter bank are orthogonal, have                In this paper, we propose a complex-valued unitary
 asymmetric )equency responses and are adaptive to                  filter bank for OFDM system. Unlike to real-valued
 diferent applications. The advantage o the proposed
                                             f                      coefficient filters, complex-valued filters have
filter bank is that it is more suitable to deal with complex-       asymmetric frequency responses and are more suitable to
 valued signals and can be optimized towards various                deal with complex-valued signals which are often present
 objective firnctions. We show that, in terms o the total
                                                   f                in a wireless system. The filters of proposed filter bank
 interference power over two sample fading channels, the            are orthogonal, have asymmetric and can be adaptive to
 MC systems based on proposedfilter banks has superior              different applications. The application adaptability is
 performance over discrete Fourier transform (DFV                   achieved by introducing free parameters into the
 based orthogonal frequency division multiplexing                   coefficients of filter bank by fustly using free parameters
 (OFDM) and discrete wavelet multitone (DWMTJ .                     to produce the Householder parameters of polyphase
                                                                    component matrices of the filter bank, then generating
Keywords: OFDM; I S t ICI; complex-valuedfilter bank.                filter bank with Householder parameters according to
                                                                    Householder factorization algorithm. The values of free
                                                                    parameters can be determined according different
                   1. INTRODUCTION                                  objective functions. As an application, we formulate the
                                                                    normalized sum of IC1 and IS1 power over fading
      Recently,      orthogonal      frequency       division        channels, and take the average interference power as
multiplexing (OFDM) system, which is a special form of               objective function to check the performance of proposed
multicarrier modulation (MCM), has been paid a lot of                filter hank described above. Simulation results show the
attention in various application of high speed wireless              OFDM system based on proposed complex-valued
digital communication systems. In OFDM systems,                     unitary filter bank can significantly reduce the power of
modulation filters form a set of orthogonal basis function           interference compared to DFT-based OFDM and DWMT
so that if the distortion in the channel is sufficientlymild         over two sample fading channels.
(relative to the bandwidth of a subchannel), the data in a
subchannel can be demodulated with a negligible small                          2. UNITARY FILTER BANK DESIGN
amount of interference from the other subchannels [l].
Discrete Fourier transform (DFT) based OFDM and                             Figure 1 is the general block diagram of OFDM
discrete wavelet multitone (DWMT) are two kinds                         systems. In OFDM systems, the transmitting filters
realizations of OFDM systems. DFT-based OFDM is
sensitive to narrow band interference because it bas
                                                                        Po              .
                                                                             (n), (n), .,f M - (n)] form a set of orthogonal
                                                                                        .        l
                                                                        basis functions. To make the filter hank in figure 1 a
significant speckal overlap between subchannels and                     perfect reconstruction (PR) filter bank, the transmitting
the technique called “cyclic prefix” is offen employed to               filters and receiving filters should satisfy the
                                                                        Biorthogonal property [3]. Unitary filter banks are a
CCECE 2003-CCGEI 2003, Montrtal. Mav/mai 2003
0-7803-7781-8/03/$17.00/62003 IEEE           .


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special class of PR filter banks where the receiving filters
are determined by the analysis filters as follows:
      fk   = (-n)                                        (1)
    There are different realizations for the receiving filter
                                                                       r2-')
                                                                       columns may be produced via Gram-Schmidt process in
                                                                                      ways.


bank [h,(n),4(n);..,hM_,(n)] , for example, DFT                        2.2 Parameterization of Householder Parameters
matrix in DFT-based OFDM and extended orthogonal
transform in DWMT. The complex-valued unitary filter                        The Householder parameters are of unit norm,
bank is used in this paper because of its advantages to                therefore each Householder parameter in (5), v, , can be
process complex-valued signals and its adaptability to                 further parameterized as [7]:
different applications.                                                                   I-I
                                                                                                                   (@n,J
                                                                               vn,J= [ ~ s i n ( ~ , , k ) ~ c o s )exp(jv,,, ),
2.1            Householder Factorization        of   Polyphase                            k=O
               Component Matrices                                      when j=O,. ..,M-2, and
                                                                                                M-2
   The class of FIR unitary filter banks has several                            'v.M-1   = nsin(en,k),                             (6)
advantages, they can be completely factorized according                                         k=Q
to Householder factorization algorithm, are easy to                    i.e.,     V,      can be determined by       2(M-1) angle
implement and stable [4].
   The Householder factorization of polyphase                          parameters and therefore if         vohas been     determined,
component matrices of transmitting filters ho(n),                      filter bank ( f , ( n ) , f ; ( n ) ,...,f,,( is determined by
                                                                                                                   n))
    4(n),...,hM-,(n)} for complex-valued unitary filter                2 K I ( - ) angle
                                                                        (-)Ml                         parameters   e",,   and      (on,j,
bank can be summarized as following. The z-transform of                j = 0,1,___, - 2, n = 0,1, _.., - 1 . The length of filters
                                                                                 M                   K
filter h,(n), i = 01 ..., M-1 can be formulated in
                   ,,                                                  f;(n),i=O,l, ..., M - l is N = K M .
polyphase form as:
~     ~~




                      M-l                                              2.3 Design Procedure
           Hi(z) =          z-~H~,~(z")
                      k=O
                                                                         To calculate all coefficients in (5), constant
Defme the polyphase component matrices                H(z) as
                                                                       matrixVoand free parameters@,,,, pn,,in (6) should all
follows:
                                                                       be determined. We propose to determine Vo first
           (H(z)),,t =Hi,&) ,                              (3)
                                                                       according to different applications and then determine the
if    H(2) is a unitary matrix, i.e.                                   values of ffee parameters with numerical optimization
           H~(z-')H(z) I ,
                     =                                      (4)        method towards different objective functions. The non-
where I is the unity matrix, then               H(2) has the           uniqueness in the generation of V, provides flexibility in
Householder factorization                                              the design of M-band complex-valued unitary filter bank.
                      K-1                                              In this paper, we propose Vo to be the M x M DFT
           H(z) = { I [I - vnv,"+ z'vnv,"]}V, .
                    I                                       (5)
                      "=I                                              matrix. The conjugate gradient method is used for filter
    K is the     McMillan degree of      H(Z) , v, is unit-norm        bank optimization. Note that the optimization may fall
                                                                       into a local minimum. Some global optimization methods
Householder parameters and               v," is the transposed         such as adding random interference and simulated
                                                                       annealing may be used. However, in practice, a local
conjugation of v, .           vo
                           is a M x M constant unitary
                                                                       minimum may be satisfactory as well.
matrix and can be selected according to different
applications. As a special case, for filter hanks associated                3. INTERFERENCE SUPPRESSION BASED ON
with     band wavelet transform, Vo may be generated                                 PROPSED FILTER BANKS
by assigning one column as a constant vector
                                                                       3.1 Calculation of Interference Power
           1      1
                               ' which   corresponds to scaling
                                                                          In OFDM system, data are transmitted in blocks, with
filter, then add M-l orthogonal columns to generate                    each block comprising M symbols. The M symbols in a
wavelet filter vectors [5]-[6] . The M-l orthogonal                    block are transmitted simultaneously with each symbol



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assigned to a different one of M subchannels. For a given            It is noted that when increasing the filter length, the
subchannel ml , the IS1 and IC1 introduced to                    complex-valued M-band filter bank bas better
symbolx,,(il) which transmitted in data block il , can           performance in case of averaged power of interference.
                                                                 Figures 4 and 5 show the change of average power of
be expressed respectively as [8]:                                interference with the increase of filter length N over the
               m
                                                                 two sample channels.
   ZSIml= Cam,                   (i) ,                (7)
             i=-
             id1                                                                   4. CONCLUSIONS
                   m       M-1
   K I m 1= C Eam(i)xm(i)
                        ,                             (8)        In this paper we proposed unitary filter banks for OFDM
              i=-nm=O
                       m+ml                                      systems to suppress the effects of interference introduced
                                                                 by fading channels. The unitary filter banks have much
a,(i) in eqn. (8) and (9) is the weight of the                   flexibility which allows the room of adaptability to
contribution from symbol x, (i) , and it is calculated as:       different applications. Simulation results show that the
                                                                 proposed filter bank can significantly reduce the averaged
     a . ( = ? 4, ( j ) I-fh(l)hm,[(il i ) - j -11
           ~,h                       -     ~          (9)        power of interference comparable to DFT-based OFDM
                                                                 and DWMT. It is also shown that better performance can
where  P    is the length of channel impulse response
                                                                 be achieved by using longer filters in the designed unitary
h,,,(n) .    The normalized IS1 and IC1 power at                 filter banks.
                             2
subchannel ml , oml be calculated as:
                  , can
                                                                                     REFERENCES

                                                                  [l] S.D. Sandberg and M.A. Tzannes, “Overlapped
                       1
where operator means absolute value.                                 discrete multitone modulation for high speed copper
                                                                     wire communication,” IEEE Journal on Selected
   We take the averaged normalized interference power
                                                                     Areas in Communications, vol. 13, no.9, pp. 1571-
over M subchannels as our objective function, which is
                                                                      1585, Dec 1995.
defined as:
                                                                  [2] H. S. Malvar, “Extended lapped transforms:
                                                                     properties, applications, and fast algorithms,” IEEE
                                                                     Trans. Signal Processing, vo1.40, no.11, pp. 2703-
3.2 Numerical Simulations for a Fading Channel                       2714, Nov. 1993.

   System simulation is done for two sample fading                [3] P.P. Vaidyanathan, “Filter banks in digital
channels h,(n)= S(n)+0.5e’“‘66(n-1) and h,(n)=                       communications,” IEEE Circuits and System
                                                                     Magazine, vol.1, no. 2, pp. 4-25,2000.
 S(n)+OSS(n-1)+0.3S(n-2).
     Figure 2 shows the simulation results of PAY,
                                                 over the         [4] P.P. Vaidyanathan, Multirate Systems and Filter
fist channel, for DFT-based OFDM systems with and                    Banks. Englewood Cliffs, NJ: Prentice-Hall, 1992.
without prefvt, DWMT and complex-valued M-band
filter bank which is designed according to the method             [5] H. Zou and A.H. Tewfik, “Discrete orthogonal M-
described. The prefix inclusive DFT-based OFDM                       band wavelets decompositions,” Proc. Of ICASSP’92,
system has better performance over prefix non-inclusive              vol. 4, pp 605 -608, May 1992.
system and DWMT, with cost of system efficiency
decreased by a factor of M / ( M + k ), where k is the            [6] P. Steffen, P. Heller, R. Gopinath, and C.S. Bunus,
length of cyclic prefut and no less than the length of                “Theory of regular m-band wavelet bases,” IEEE
channel. The length of filters, N, for DFT-based filter               Trans. Signal Processing, vol. 41, no.12, pp. 3497 -
bank, DWMT and M-band complex-valued wavelet                          3511 Dec 1993.
transform are M, 2M and ZM, respectively. Figure 3
shows the results over the second channel for same                [7] X.-P. Zhang, M. Desai and Y.-N. Peng, “Orthogonal
circumstances as in Figure I . In both experiments, the               complex filter banks and wavelets: some properties
complex-valued M-band filter bank demonstrates                        and design,” IEEE Trans. Signal Processing, vo1.47,
superior performance in interference reduction than DFT-              no. 4, pp. 1039 -1048, Apr. 1999.
based OFDM and DWMT.



                                                             - 963 -
[XI Y. Wang and X.-P.Zhang, "Error performance
   analysis of general OFDM systems with MPSK
   coding over multipath channels", submitted to ISSPA
   2003.




                            Figure 1. Multirate filter bank based communication system




                                                               c -18
                                                               L
                                                                          ~




                                                                   0

                                                                   L
                                                               n
                                                                   g-24   ~




                                                                   E
                                                                   p-26~                                      L
                                                                   m
                                                                              5   6   7   8   9    10   1;   12
                  number of subchannel                                             number of subchannel
Figure 2. Average power of interference for DFT-based        Figure 4. Average power of interference for new OFDM
filter bank, DWMT and the new OFDM system based on           system based on complex-valued unitary M-band filter
complex-valued    unitary     M-band     filter  bank        bank          with    different     length  of   filter
(h,(n)=S(n)+0.5e'"'66(n-1)).                                 ( h , ( n ) =S(n)+0.5e'"'66(n-1)).




                                                                              5   6   7   8   9    1 0 1 1    I2
                  number of subchannel                                            number of subchamel
Figure 3. Average power of interference for DFT-based        Figure 5. Average power of interference for new OFDM
filter bank, DWMT and the new OFDM system based on           system based on complex-valued unitary M-band filter
complex-valued    unitary     M-band     filter  bank        bank with different length of filter
(h,(n)=S(n)+OSS(n-l)+0.36(n-2)).                             h,(n) = S(n) + 0.56(n - 1) + 0.36(n - 2)).




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