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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 . - 961 - 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 - 962 - 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)). - 964 -