Performance of C-filter based adaptive spread spectrum receiver in non-Gaussian channels

Performance of Gfilter based adaptive spread spectrum receiver in non-Gaussian channels

T.C.Chuah, B.S.Sharif and 0.R.Hinton

Abstract: The performance of an adaptive spread spectrum receiver based on the combination fdter (C-fdter) in the presence of multiuser interference and impulsive channel noise is investigated. For many years, the problem of single user detection in multiuser impulsive channels has been addressed through the use of heuristics in an attempt to reduce the influence of outlying observations. This approach, however ad hoc it may be, is still adopted by many researchers, for reasons of simplicity. C-filters provide an alternative approach in which robust extraction of the desired user signal under both multiuser interference and impulsive noise can be constructed. The Gfilters exploit both rank order and temporal order information of the observation sequence to produce the output. They essentially combine the attractive features of both linear FIR and nonlinear filters of the order statistics type. The retention of the temporal order of the observation sequence is essential for the structure of multiuser interference to be learnt adaptively, while the rank order information is useful for removing outliers. The authors evaluate the performance of the C-fdter receiver via extensive computer simulations, and it is shown that substantial improvement in performance can be achieved with C-filters over linear filters when the noise exhibits long-tailed distributions.

1 Introduction

Multiuser detection [ 11 has been proposed for direct- sequence code-division multiple-access (DS-CDMA) networks to reduce multiple access interference (MAI) and improve channel capacity. The channel noise has often been assumed to be Gaussian thus avoiding mathematical difficulties in algorithm development. However, theoretical considerations supported by experimental evidence [2, 31, confirm that Gaussian distribution poorly describes the noise statistics of many wireless channels that exhibit impulsive properties.

Adaptive interference suppression [4] using least mean square (LMS) based FIR filters [SI attempts to exploit the cyclo-stationarity of the MA1 through adaptive minimum mean squared error (MMSE) learning. They have been shown to significantly outperform the matched filter in near-far environments and they require only a training sequence for the desired user, which is important in scenarios where knowledge of other interfering parameters is unavailable. Since linear FIR filters often perform poorly in impulsive noise [6], a means of robustification becomes necessary.

Signal detection in single-user non-Gaussian channels has gained considerable advance over the last two decades [7], however, this has not been the case for multiuser non- Gaussian channels [8]. The inherent difficulty for the latter lies in the fact that suppressing MA1 that is highly corre- lated with the signal of interest requires linear signal-space

0 IEE, 2001

IEE Proceedrizgs online no. 20010599 DOL 10.1049/ipcom:20010599 Paper fmt received 28th June 2000 and in revised form 26th February 2001 The authors are with the Department of Electrical and Electronic Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK

techniques, while nonlinear filtering is effective for rejecting impulsive noise [8]. The simplest approach to robustness is to reject outliers by heuristic nonlinearities prior to linear detection [9-121. This approach, which is popular due to its simplicity, suffers several drawbacks. For example, an opti- mum solution requires explicit knowledge of the combined MAI-plus-noise statistics, whch is generally unavailable in a realistic scenario. Furthermore, improper use of nonlinearities may result in loss of important signal characteristics leading to performance deterioration.

Recent years have seen considerable interest in develop- ing order statistics filters (OS fdters or L-filters) and their generalisations [ 131 after Turkey introduced the median tilter in 1971 [14]. L-filters are a class of nonlinear filters, which have proved useful in some applications where robust signal processing is required. They can be viewed either as a modification to linear FIR filters where samples are algebraically ordered prior to linear weighting, or as a generalisation of the median filter by using all ordered samples instead of the single median sample. Ordering of the observation samples presents significant information from which robustness can be constructed [15]. While there clearly exist applications where L-filters offer advantages over linear filters, it should also be emphasised that there are many more general signal processing problems where the converse holds [16]. The inclusion of an ordering unit may produce solutions that are distinctly different from that achieved by linear filtering. Linear FIR filters for DS- CDMA receivers attempt to exploit the cyclo-stationarity of the MAI through adaptive training, which implies that the temporal positions of the received samples must be retained to allow the filter to capture the structure of the MAI. On the other hand, L-filters fail to exploit the temporal context of the sequence and do not allow specification of ‘frequency response’ characteristics for MA1 rejection. Therefore the ordering operation should not be performed on the received samples in the context of DS-CDMA.

321 IEE Proc.-Commun., Vol. 148, No. 5, October 2001

The limitations of L-filters have been addressed through the development of combination, or C-filters [17], also referred to as Ll filters in [18]. These filters combine the features of linear FIR and nonlinear L-filters by assigning the output to be a weighted sum of the input samples where the weights depend on both the sample rank and temporal position of the observation sequence in each processing window. In particular, Gandhi and Kassam [17] reported promising results achieved by C-filters for frequency selective filtering of nonstationary signals and in signal deconvolution in non-Gaussian environments. The problem of multiuser detection in non-Gaussian channels has a similar nature to the above problems, where preserv- ing the temporal positions of the received signals allows MA1 exploration, and rank order contains significant information for removing outliers.

2

A coherent synchronous CDMA system employing binary phase-shift keying (BPSK) modulation is considered. After conventional chip-matched filtering and sampling at the chip rate UTc, the discrete time version of the received signal within the ith symbol interval, T, can be written in vector form as

Communication system and noise model

I(

r(2) = A k b k ( 2 ) S k + n(i) (1) k = l

where bold variables denote column vectors in CnN such that r(i) = [rl(i), r2(z),.. , r,,(i)lT, N is the processing gain and NT, = T, K is the number of users. For the kth user, A,, bk(i) E {-1, 1 } denote, the received amplitude and ith information bit, respectively. The normalised signature sequence of the kth user is denoted by sk = ldN, [Ak, 4k, ..., where q)k E {f l} is a signature sequence assigned to user k. n(i) = [nl(i), n2(i), ..., ndi)lT is the white ambient noise vector and is assumed to be a sequence of independ- ent and identically distributed (i.i.d.) random variables. We model the channel noise with zero-mean symmetric a- stable ( S d ) random processes, which have been shown to accurately model impulsive noise processes under broad conditions [19]. They also provide a close approximation for the simple Middleton’s class B noise [2, 191. The impul- siveness of the a-stable family is controlled by the characteristic exponent, a (0 < a 5 2) and the dispersion y ( y > 0) si&ies the scale parameter [19]. Since a-stable processes have no finite variance for characteristic exponent a < 2, we characterise the relative strength between signal and noise by the geometric signal-to-noise ratio (SNR) [20]. For a detailed discussion and motivation behind the use of this novel framework, see [21].

Single-user detection in multiuser channels can be formu- lated as follows:

Hi : .(i) = +Ais1 + A k b k ( 2 ) S k 4- n(i) (2) k#l

Ho : r( i ) = -Ais1 4- A k b k ( i ) S k + n(i) (3) k # l

where HI (Ho) corresponds to the transmission of +1 (or -1) at the ith symbol by the first user, which is assumed to be the desired user.

3

3. I Preliminary notation

Definition and design of Cfilters

For simplicity the symbol index i is dropped and the observation vector is denoted as r = [rl, r2, ...., rNIT where the

322

samples are arranged according to temporal order index and the system processing gain N is assumed to be odd for convenience. The order statistic vector of Y is denoted by YOS = [r(l), r(2), ... r(N)],T where the samples r(l) I r!2) I ... 5 r(N) are arranged according to nondecreasing magnitude.

Define rj = rh) so that the rank Rj of rj is given by

R j = m

and the temporal order F,,, of can be expressed as (4)

(5) + _ ‘ m - - 3

It is assumed that rOS is obtained through stable sorting [18] so as to guarantee one-to-one transformation from Y to yOS even if a tie occurs.

3.2 FIR and L-filter filters Given the observation vector Y, a linear FIR filter produces an output as follows:

N

j=1

where a = [al, a2, ..., aNIT denotes filter coefficient vector. For the matched fdter receiver, we have a = si, and for the linear MMSE receiver a is allowed to change adaptively. On the other hand, the output produced by an L-filter is given by

N

(7 ) m=l

where b = [b(l), b!,,, ..., h(,IT is the L-filter coefficient vector and the output gven by eqn. 7 is based solely on the order statistic vector rOs without regarding to the temporal order of Y.

3.3 C-filters A C-filter retains the temporal information of Y whilst exploiting the rank order by combining eqn. 6 of the FIR filter and eqn. 7 of the L-filter as follows

N

j=1

N ’ ’

(9) m=l

where c(m, 1) E 31 depends on both rank and temporal order of Y. Therefore the C-filter can be thought of as the rank-order-dependent weighting of the temporal-order data given by eqn. 8 or alternatively, as the temporal-order- dependent weighting of the order statistics data given by eqn. 9. In other words, the C-filter requires N2 coefficients c(m, 1’) but uses only data-dependent selection of N coefficients to produce the output in each window. To express the C-Wter in matrix form, we define C as the N-by-N filter coefficient matrix with entries cmj = c(m, J) , eqns. 8 and 9 can alternatively be expressed as

Yc = w , r T

= l T ( c o P)r (10) where w, E 3iN is a data-dependent weight vector, I is an N-by-1 vector of ones, P is an N-by-N data dependent permutation matrix, and 0 denotes the elementwise (Schur) product. Thus the coefficients in the mth row of C are exclusively reserved for rcn,) while coefficients in the jth

IEE Proc.-Conimun.. Vol. 148, No 5. October 2001

column correspond to r,. The entries pmt of P are defined as

0 otherwise Note that similar to the matrix C, P contains both rank and temporal information of Y, since the mth row of P characterises r(m) while thejth column governs rJ. In fact, P is a permutation matrix since there are exactly N ones and N(N -1) zeros with a single one in each row and column, and given Y, it plays a role in selecting N c(m, t ) coefficients from a restricted set of N2 coefficients. Therefore for different Y, different C coefficients are used to produce the output. Clearly, the total number of permutation matrix P that can arise for different r is A?.

The influence of the outlying components can be reduced by incorporating appropriate trimming to the C matrix. For example, by setting all thecoefficients of the mth row of the C matrix to zero, we remove any influence due to r(m, from the output. Since outlying observations are always located at the low and high ranks of r(m,, their influence can be eliminated by setting c(m, t ) = 0 for m < TI + 1 and m > N - T2, where T, and T2 are integer-valued trimming parameters satisfying the conditions TI I 0, T2 < N and T,

Example: Assume that the system processing gain N = 5, and the observation vector Y = [2, -1 1, 33, 1, 3IT Let T = TI = T2 = 1 to censor the smallest and largest samples, wrT is obtained by

+ T2 < N.

0 0 0 0 0 0 1 0 0 0 c2l c22 c23 c24 c25 0 0 0 1 0

w?=lT[:;: :; :;: :; : d o [ ; 0 0 1 0 0 :: 0" 0" "j = [c31,0,0, c24, CG] (12)

It is easy to see that without incorporating trimming, a C-

with or without trimming, an L-filter is achieved with identical columns in C. For a detailed treatment on the general properties and robustness of C-filters, see [17, 181.

3.4 Variably trimmed C-filters (VTC filters)

filter with identical rows in C becomes an FIR filter, while

In practice, to achieve improved performance over a wide range of underlying noise distributions, the trimming parameters T, and T2 should be allowed to vary adaptively. For channel noise with symmetric distributions, it is convenient to use T = TI = T2. Eqn. 10 can be generalised as

yc = tT(C 0 P). (13) where t = [tl, t2, ..., t,lrcontains trimming information and is called the trimming vector [17]. The mth component t, of t corresponds to r(,), and the assignment of either 1 or 0 to tm determines whether or not the coeficients of the mth row of the C matrix will be used, or in other words, whether or not rem, is to be trimmed. Without setting the coefficients in the first and last row of the C matrix to zero, eqn. 12 could have been achieved by using t = [0, 1, 1, 1, OlT. Furthermore, the definition of t can be generalised to include more general weighting instead of a zero-one weighting vector. The strategy for choosing the appropriate trimming vector t is problem dependent. Our requirement here is to effectively suppress MA1 and recover the desired user's data embedded in heavy-tailed noise. In general, in impulsive noise environments, a large value of T is neces-

IEE Proc -Commun , Val 148, No 5, October 2001

sary to prevent the outliers from contaminating the decision statistics. On the other hand, when the MA1 is strong and the noise exhibits Gaussian behaviour, small or zero trimming is desirable.

3.5 Minimum mean squared error (MMSE) criterion Let z(i) E SiN2 be a vector defined by

42) = [ a 1 ( i ) , a 2 ( i ) , '", "), ~21(i),~22(2), ...,z ,vw(i)IT (14)

(15)

with

Z m j = r j ( i ) if R j = m ( ' { 0 otherwise

b",j = 1, ..., N. Note that only N out of N2 elements of z are nonzero. Similarly, using the same indexing as in eqn. 14, the N2-dimensional C-filter weight vector c(i) E SiN2 can be defined as

42) = [ C I I ( i ) , C 1 2 ( ~ ) , ..., C l N ( i ) ,

c21(i), C22 (4, . . . , C" (4 (16)

y c ( i ) = C T ( i ) % ( i ) (17)

and the C-fdter output is

If a noise-free training sequence dl(i) is available, and since the impulsive noise comes from the input and not from the training signals, then the MMSE criterion remains valid. Therefore given the observation vector ~ ( i ) , our objective is to find the C-filter coefficient vector c(i) so as to bring the output as close to the desired data as possible in the mean squared error sense, i.e. our goal is to minimise the following objective function

J ( c ) = E[(di(i) - ~ ~ ( i ) ~ ( i ) ) ~ ] (18) where E[.] denotes the expectation operator. The solution to eqn. 18 is given by the well known standard normal equation [7]

c ( i ) = E [ z ( i ) zT ( i ) ] -l [dl (i).(i)] (19) In general, the Wiener solution given by eqn. 19 is difficult to achieve in practice since it requires knowledge of both the correlation matrix E[z(i)zT(i)] and the cross-correlation vector E[d, (i)z(i)), which are unavailable in most cases.

3.6 Adaptive LMS implementation of C-filters A practical approach to implement C-filters is to use the LMS algorithm, which is of stochastic gradient nature [7]. The instantaneous estimation error e(i) is defined as

e ( i ) = & ( i ) - y c ( i )

ci+l(m,j) = cz(m,j) + P e ( i ) ~ m j ( i )

(20)

(21)

and the LMS update equation is given by [17]

where c,(m, 1) is the mjth C-filter coefficient at the ith symbol and ,u > 0 is the adaptation gain and must be chosen to ensure algorithmic stability [7]. The initial values of all C-filter coeficients can be chosen arbitrarily and the trimming or nonlinear weighting operation can be controlled entirely by the trimming vector t in eqn. 13. Proper choice of this trimming vector ensures that all the weight coefficients adversely affected by the presence of outliers, are accumulated in the lower and higher rows of C. They are then given a lower weight when producing the output through the incorporation of the trimming vector. This feature characterises the novel advantage of the C-filters.

323

Since only N of N2 coefficients are updated at each itera- tion, the overall convergence rate of the C-filter is much slower than an FIR or L-filter adapted by LMS algorithm. This increased penalty is offset by an increased robustness in the impulsive channels. The final bit estimate is achieved by zero-thresholding the C-filter output

& ( i ) = sgn(Yc(i)) (22) After the training session, this receiver operates in decision- directed mode by using its own symbol estimates. The overall structure of adaptive receiver based on the VTC filter is shown in Fig. 1. After finite collection the input vector v(i), an impulsivity estimator can be used for online estimation of the underlying noise impulsivity aild dispersion, and the trimming vector t can then be parameterised adaptively to improve performance. In general, the optimal value for the trimming parameters depends on the channel conditions.

rimming e(i)

impulsivity /Evedor estimator i Fig. 1 on vuriabry trimmed C-filier

General schemutic diugm of aduptive spreadspectrum receiver bused

4 Performance evaluation

This Section presents a performance comparison via computer simulations. Throughout the following examples, we consider a CDMA network employing Gold codes of length N = 31 as the spreading sequences. We set up the trimming vector t to incorporate a weighting of 0.01 rather than 0, and therefore the trimming parameter T = TI = T2 signifies the number of components to be weighted by 0.01 at the low and high rank of t , respectively, with the rest of the components set to unity. The trimming weight was set to 0.01 to attenuate outliers while preventing a total loss of the spread spectrum signals that contain useful information to recover data and also to rapidly learn the MA1 structure.

500 1000 1500 symbols

L

3 U

;I a, E

0 500 1000 1500 0

symbols b

Fig. 2 Four equal-power users at a G-SNR 5dB, p = 0.15, T = 1, trimming weight = 0.01 a Linear FIR filter b Variably trimmed C-filter

Convergence behvior ofjilters in Gmsimi noice

Figs. 2 and 3 show the convergence properties of the FIR and VTC filter (with a trimming parameter T = 1)

324

under Gaussian and mild impulsive noise (a = 1.9), respectively. The adaptation gain ,U is set to 0.15 and there are four equal-power users at a G-SNR of 5dB. In Gaussian noise, both the FIR and VTC filters reach a similar steady- state performance although the latter needs a longer training period. However, when the noise deviates slightly from Gaussian statistics (a = 1.9), the FIR filter suffers from severe perturbations by outliers, with an impact on the algorithm’s stability. On the other hand, by using a VTC filter with finite trimming, the influence of outliers is effectively suppressed and the receiver enjoys a smooth learning process.

symbols

0 500 1000 1500 symbols

b Fig.3 a = 1.9 Four equal-power users at a G-SNR 5dB, p = 0.15, T = 1, trimming weight = 0.01 a Linear FIR filter b Variably trimmed C-filter

Convergence behaviour offilters m mild imphive noise

loo r

Fig.4 All users have equal powers -+- matched filter -*- linear FIR filter 4- -0- -0-

BER in %user impulsive chunnel of a = 1.5

variably trimmed C-filter, T = 0 variably trimmed C-filter, T = 1 variably trimmed C-filter, T = 2

Figs. 4 7 compare the bit error rate (BER) between the matched filter, adaptive receivers using an FIR fdter and the VTC fdters with several trimming parameters in an 8-user channel for various noise scenarios. In Fig. 4, the channel noise is hghly impulsive (a: = lS), and all users have equal powers. As expected, the performances of both FIR filter and matched filter degrade substantially under the influence of impulsive noise. On the other hand, by exploiting the rank order information through finite trimming, the influence of outliers is greatly reduced and a

IEE Proc.-Commun., Vol. 148, No. 5 , October 2001

significant performance improvement is achieved using the VTC filters, and the VTC filter with T = 2 has the best performance. It is also observed that the VTC filter with zero trimming fmally converges to the FIR filter as a

1 2 3 4 5

Fig. 5 BER in 8 - m ~ Gaussian channel All users have equal powers -+- matched filter -*- linear FIR filter -0- -0- 4-

variably trimmed C-filter, T = 0 vanably trimmed C-filter, T = I variably trimmed C-filter, T = 2

I I 1

1 2 3 4 5 Fig.6 The power of each interferer is 3 dB above user 1 -+- matched filter -%- linear FIR filter -0- -0- -0-

BER in 8-user Guursian c h ~ l


IO'

10-2

1 0 3

I 0-4

I 0-5 0 2 4 6 8 10 12 14 16 18 20

Fig. 7 The power of each interferer is 3 dB above user 1 -+- matched filter -.*- linear FIR filter -0- -0- -0-

BER in 8-user mild impulsive chamel of a = 1.8


special case. Fig. 5 shows the BER performance under Gaussian noise with perfect power control. It can be seen that the VTC filters with finite training only incurs slight performance loss relative to the FIR filter. A similar result is also observed in Fig. 6 where the noise is Gaussian and the near-far effect is taken into account, with the power of each interferer 3dB above the first user. It can be seen that VTC filter is not very sensitive to the presence of MAI, whch is due to retaining the temporal order of the received signals, thus allowing the statistics of MA1 to be learnt. Fig. 6 also shows marginal performance degradation for all receivers. This can be attributed to the fact that in these correlation-type receivers, the weak user does not benefit from the concept of strong user MA1 reconstruction and cancellation. Fig. 7 plots the BER under mild impulsive noise having a =1.8 with the same MA1 as in Fig. 6. In this case, the VTC filters with finite training continue to exhibit robust performance over the linear counterparts, even in the presence of both MAI and impulsive noise, and the VTC filter with T = 1 has the best performance.

A study of BER as a function of user population is useful for illustrating the channel capacity increase offered by the VTC filters. Figs. 8-11 show the BER performance of the first user as a function of the number of active users in a CDMA network operating at a G-SNR of 5dB under the same noise scenarios and using similar trimming vectors as in Figs. 4-7. In all cases, the adaptive spread spectrum receiver based on the VTC filter outperforms both linear FIR and matched filter receivers when the noise is impulsive. Under Gaussian noise, the VTC filter receivers main- tain a comparable performance with the linear FIR filter receivers even if MA1 is dominant. It can be seen from Fig. 8 that when the channel noise is highly impulsive, the VTC filters with finite trimming could deliver communications to 31 users with a BER performance better than that offered by both FIR and matched fdter receivers support- ing only a single user. Similar observation is also found in Fig. 11 when the noise statistics deviate slightly from Gaus- sianity. In other words, the performance gain acheved by the Gfilter receiver indirectly translates into channel capacity increase in multiuser channels.

10-3 ' 5 10 15 20 25 30

Fig. 8 a= 1.5 All users have equal powers -+- matched filter -%- linear FIR filter -0- -0- -0-

BER against user populution at G-SNR of 5dB in hpukive channel of

variably trimmed C-fiter, T = 0 variably trimmed C-filter, T = 1 variably trimmed C-filter, T = 2

IEE Proc.-Commun., Vol. 148, No. 5, October 2001 325

5 Conclusions

We have investigated the performance of the C-filters for implementing robust CDMA receivers. The C-filters have their foundations in order statistics and present many attractive properties in which the problems of robust extraction of desired user signal in multiple access channels contaminated with impulsive noise can be addressed. They essentially combine the features of linear FIR and nonlinear L-filters. The preservation of temporal order in the context of DS-CDMA is crucial since it enables the statistics of the MA1 to be explored while the rank order information contains significant information from which robust estimation can be constructed. The disadvantages of C-filters include large storage requirements and a longer training period.

Simulation results demonstrated that the adaptive receiver based on the VTC filters with finite trimming exhibits significant performance improvement over both FIR filters and matched filters in impulsive noise while maintaining a comparable performance to that of the FIR filter in Gaussian channels even if MA1 dominates.

l o ’ I 10-2

I 0-3

10-4

10-5

10-6 5 10 15 20 25 30

Fig.9 All users have equal powers -+- matched filter -*- linear FIR filter -0- -0- -0-

BER against user population ut G-SNR of 5dB in Gaussian chunnel


10-1 [

Fig. 10 The power of each interferer is 3dB above user 1 -+- matched filter -$- linear FIR filter 13- -0- -0-

BER against user populutwn at G-SNR of 5dB in G m s h channel


l 0 I

10-2

4 0-3 I 5 10 15 20 25 30

BER agaht user poplution at G-SNR of 5dB in mild impulsive Fig. 1 1 channel of a = 1.8 The power of each interferer is 3 dB above user 1 -+- matched filter -*- linear FIR filter + -0- -0-


6 References

1

2

VERDU, S.: ‘Multiuser detection’ (Cambridge University Press: Cam- bridge, UK, 1998) MIDDLETON, D.: “on-Gaussian models in signal processing for telecommunications: New methods and results for class A and class B noise models’, IEEE Trans. In$ Theory, 1999,45, pp. 112S1149

3 BLACKARD, K.L., RAPPAPORT, T.S., and BOSTIAN, C.W.: Measurements and models of radio frequency impulsive noise for

indoor wireless communications’, IEEE J. Se/. Areas Conmun.. 1993, 11, pp. 991-1001 MADHOW, U,, and HONIG, M.L.: ‘MMSE interference suppression for direct-sequence spread-spectrum CDMA’, IEEE Truns. Com- mun., 1994,42, pp. 3178-3188

5 HAYKIN, S.: ‘Adaptive filter theory’ (Prentice Hall, Englewood Cliffs, NJ, 1991)

6 PITAS, I., and VENETSANOPOULOS, A.N.: ‘Nonlinear digital filters-principles and applications’ (Kluwer, Boston, 1990)

7 KASSAM, S.A., and POOR, H.V.: ‘Robust techniques for signal processing’, Proc. IEEE, 1985, 73, pp. 433483

8 POOR, H.V.: ‘Non-Gaussian signal processing problems in multiple- access communications’. Proceedings of 1996 USC/CRASP Workshop on Non-Gaussian signulprocess, Ft. George Meade, MD, USA, 1996 AAZHANG, B., and POOR, H.V.: ‘Performance of DYSSMA communications in impulsive channels-Part I: Linear correlation receivers’, IEEE Trans. Commun., 1987, COM-35, pp. 1179-1187

10 AAZHANG,. B.? and POOR, H.V.: ‘Performance of DSiSSMA communications in impulsive channels-Part 11: Hard-hmting correlation reveivers’, IEEE Trans. Commun., 1988, COM-36, pp. 88-96

1 1 AAZHANG, B., and POOR, H.V.: ‘An analysis of nonhnear direct sequence correlators’, IEEE Trans. Commun., 1989, 37, pp. 723-731

12 BATALAMA, S.N., MEDLEY, M.J., and PSAROMILIGKOS, .N.: ‘Adaptive robust spreadspectrum receivers’, IEEE Trans. Commun., 1999,47, pp. 905-917

13 BOVIK, A.C., HUANG, T.S., and MUNSON, D.C.: ‘A generaliza- tion of median filtering using linear combinations of order statistics’, IEEE Trans. Acoust., Speech Signal Process., 1983, ASSP-31, pp. 1342-1 350

14 TURKEY, J.W..: ‘Exploratory data analysis’ (Addison Wesley, Read- ing, MA, 1977)

15 DAVID, H.A.: ‘Order statistics’ (Wiley, New York, 1981) 16 LONGBOTHAM, H.G., and BOVIK, A.C.: ‘16,, and,: ’Theory of

order static filters and their relationship to linear FIR filters’: IEEE Trans. Acoust. Speech Signul Process., 1989, 37, pp. 275-281

17 GANDHI, P.P., and KASSAM, S.A.: ‘Design an d performance of combination filters for signal restoration’, IEEE Trans. Signal Process., 1991,39, pp. 15241540

18 PALMIERI, F., and BONCELET, C.G.: ‘LI-Filters- a new class of order statistic filters’, IEEE Truns. Acoust. Speech Signal Process.,

19 NIKIAS, C.L., and SHAO, M.: ‘Signal Processing with alpha-stable distributions and applications’ (Wiley, New York, 1995)

20 CHUAH, T.C., SHARIF, B.S., and HINTON? O.R.: ‘Nonlinear decorrelator for multiuser detection in non-Gaussian impulsive environments’, Electron. Lett., 2000, 36, pp. 92e922

21 CHUAH, T.C., SHARIF, B.S., and HINTON, O.R.: ‘Robust signal processing for CDMA communications in non-Gaussian noise’. Pro- ceedings of IMA international conference on Mathematical signal processing, Warwick, UK, 2000

4

9

1989, ASSP-37, pp. 691-701

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Performance of C-filter based adaptive spread spectrum receiver in non-Gaussian channels