Random-walk-based characterisation of multiple access intederence in a DS/SSMA system

LTriantafillidis and C.G.Guy

Abstract: The problem of calculating the probability of error in a DS/SSMA system has been extensively studied for more than two decades. When random sequences are employed some conditioning must be done before the application of the central limit theorem is attempted, leading to a Gaussian distribution. The authors seek to characterise the multiple-access interference as a random walk with a random number of steps, for random and deterministic sequences. Using results from random walk theory, they model the interference as a K-distributed random variable and use it to calculate the probability of error in the form of a series, for a DS/SSMA system with a coherent correlation receiver and BPSK modulation under Gaussian noise. The asymptotic properties of the proposed distribution agree with other analyses. This is, to the best of the authors’ knowledge, the first attempt to propose a non-Gaussian distribution for the interference. The modelling can be extended to consider multipath fading and general modulation.

1 Introduction

The use of spread-spectrum (SS) concepts for multiple access, and in particular CDMA, has attracted a lot of attention over the last twenty years. The determination of the exact (average) probability of error (PE) for such a sys- tem, under additive white Gaussian noise (AWGN) or more general multipath channels, remains elusive. In a standard DS/SSMA system employing a matched filter receiver, the output of the receiver is not a sufficient statis- tic, because the interference from users other than the one the receiver is matched to is, in general, non-Gaussian. The characterisation of this multiple access interference (MAI) is at the heart of any analysis of the DSiSSMA system and the evaluation of useful system parameters.

The attempts made to provide solutions to the problem could be broadly divided into bounding and approximating techniques. The most simple, but still applicable in many situations of practical interest, is the standard Gaussian approximation (SGA) [I], where the MA1 is approximated by a Gaussian random variable (RV) and the PE is given in terms of the familiar error function. Based on that, an improved Gaussian approximation (IGA) has been pro- posed, where the MA1 is modelled by a Gaussian RV con- ditioned on its variance [2], when random signature sequences are used. Averaging over the variance gives the PE with increased accuracy, even when the SGA fails, although at the cost of an increased computational burden.

.In a different approach [3], the characteristic function (CF) of the MA1 can be numerically integrated and used in conjunction with a series expansion to approximate the PE

0 IEE, 2001 IEE Proceedns onlinr. no. 200 10066 DOL 10.1049/ipcom:20010066 Paper first receivd 1st October 1999 and in revised fonn 25th October 2000 The authors arc with the School of Computer Science, Cybernetics and Elec- tronic Engineering, The University of Reading Whiteknights, Reading RG6 6AY, UK

to any degree of accuracy. The last two methods produce results that agree with ones from lower and upper bounds obtained for the PE. The early moment space bounds [4], integral bounds [5] and vector-based bounding techniques [6] provide accuracy at the expense of varying complexity. Lastly, maximum entropy methods have been applied to determine the actual PDF of the MA1 and test the validity of the SGA. Although the results are accurate with a reduced number of computations, they do not offer much insight into the structure of the MA1 [7]. Furthermore, the general picture is significantly more complicated when the modulation used is complex instead of binary, the channel is not AWGN and when users employ random sequences, for example [&lo].

The aim of this paper is to introduce a family of distribu- tions, arising as limit distributions of a random walk, that will accurately model the MA1 under general conditions, while utilising the requirements and results of previous analyses. The standard model for a DS/SSMA is intro- duced and the general requirements of the MA1 are consid- ered. We then introduce the family of K-distributions, citing previous applications and some physical justification for its use in the CDMA problem.

2 System model

The model under consideration is similar to the models defined in [2, 61. The received signal in this asynchronous binary DSiSSMA system is the sum of M spread-spectrum signals q<(t ~ zl,) plus an AWGN process n(t) which has two sided spectral density Nd2. The spread spectrum signal is given by - zi,) = G!P>b/,(t - @q& - z/,) cos(o,t + h), where bk(t) is the data signal, ak(t) is the spectral spreading signal, qc is the time-delay parameter that accounts for the propagation delay and the lack of synchronism between the transmitters, & is the phase angle of the kth carrier and P is the power of the transmitted signal. Equal power levels are assumed for all the transmitters, although the results can be modified to consider unequal power levels. If a rectangular pulse is defined by pdt) = 1 for 0 5 t < T and

43

p d t ) = 0 otherwise, the kth data signal can be expressed as &(t) = ?& bj/p7(t - j,) where bk is the binary data sequence of the kth transmitter. If the chip waveform w(t) is defined to be time-limited to the interval [0, T,) and normalised to have energy T,, the spectral-spreading signal of the kth transmitter can be expressed as q,(t) = 2;- a&(t -JTJ where U: is a periodic binary sequence of elements from the set {-1, 1 ) . We assume that each bit is encoded with N chips, i.e. T = NT, and that the signature sequence U/' has period N . If the chip waveform is the rectangular pulse function P,(t), the DS/SSMA system has the BPSK signalling format. The signature sequences are, in general, deterministic but here we will treat them as random, with Pr{aj/ = +1> = Pr{uf= -1} = 112.

Because of the symmetry of the model, we need only consider the receiver that is listening to the first transmitter. The properties of an SSMA system and the stationarity of the noise n(t) allow us to restrict attention to time delays modulo T and phase angles modulo 2z. Therefore we model the delay rk as an RV uniformly distributed in [0, T ] and the phase as an RV uniformly distributed in [0, 24, and since relative delays and phases are important, we set q = = 0. The lzth data sequence is modelled as a sequence of independent and identically distributed RVs bj/ such that Pr{b,k = +1> = Pr{bj/ = - 1 ) = 112.

2.1 System analysis The decision statistic for a coherent correlation receiver is given by [2] as

1 M

z:,, = v + m % + C 4 , 1 ( b k , T I , , + k ) (1) [ k=2 where

and I k , l (bk, 7 ! + ) = [&,l ( b k ) 41 cos 4J

B k , l ( b k , .) = blC&,l(.) + bk&J (.)

(2)

(3) The RV q is Gaussian with zero mean and variance NOT 1 4. The vector bk = (b!, , b k ) denotes a pair of consecutive data bits of the lcth signal. The functions RIC,Jr) and R/c,m(r) are the continuous time partial cross-correlation functions of the kth and mth yaveforms and are given by

These can be expressed as Rk,m = .fo"%(t - mm(t)dt and R/c,rn(r) = JrTUk(t - z)am(t)dt.

& , m ( T ) = Ck,m(Y - qk,(r - Y T C )

& , m m = C k , m ( Y ) & h - Y T C )

+ Ck,m(Y + 1 - N ) R & - Y T C ) (4a) and

+ Ck,m(Y + 1)8$(7 - YTC) (4b) where Cl',,(.) is the discrete aperiodic cross-correlation function (DCF) of the two sequences and RIp(r) and RJr) are the continuous-time partial auto-correlation functions of the chip waveform. Using the explicit dependence of the DCF on the sequences we can write

Bk, l (bk! 7 ) =

1 [_I j = Y

N - 1

b k , U $ - , + N U j + bka$-ya; &(s)

(5) I N - 1

b?,a?-y-,+Na; + bkaS_y-la; 8 $ ( S ) j = O j = Y + l

where y,' = LdT,] and qC = rk ~ nT, are uniformly distrib- uted RVs in the obvious intervals.

2.2 Gaussian approximations The rationale behind the SGA is that the overall MA1 con- sists of contributions from many different interferers and the MA1 from each interferer consists of the contributions from many different chips. Then central limit arguments may apply and the overall MA1 in a symbol interval may well be approximated by a Gaussian RV. However, the contributions from different interferers are, in general, dependent and the contributions from different chips of the same interferer are also dependent [ 1 I].

With large M and therefore high PE the SGA seems to be accurate. In fact, as M tends to infinity the MAI, nor- malised by l/d(M - l ) , tends to a Gaussian RV for any value of N . Therefore for a large but finite A4 the normal- ised receiver output statistic can be accurately modelled as a Gaussian RV. However, for a large N and small M the applicability of the CLT is different. The SGA is not only inaccurate in this region, it also becomes very optimistic as N grows. This is because N acts differently on each of the various terms of eqn. 5, [2]. In general, the MA1 cannot be accurately modelled as a Gaussian RV when the delays and phases are random unless the number of simultaneous users is large.

Taking the limit as N tends to infinity, while keeping the delays and phases constant we see, [2] or [12], that the CF of the MA1 tends to a Gaussian RV with a random vari- ance, due to the randomness of the delays and phases. This observation leads to the IGA. The MA1 from any individ- ual interferer looks like a mixture of Gaussian RVs since the contribution to each chip of the interferer is affected by the same delay and phase shift. By averaging the total MA1 over the variance q, an accurate approximation for the MA1 and hence for the PE can be achieved, even for rela- tively small N and M. For the PE the resulting expression is

with given by the convolution of the density derived in [12]. The determination of the PDF of the variance is not possible analytically but provided some distributional analysis is carried out, this PDF need not be known [13]. Actually, the first and second moments will be enough. In fact, it is the limitations of the SGA that make the use of the signal-to-noise ratio insufficient for the description of the system [14].

Notice that in the SGA for random sequences, the func- tion of the expectation is evaluated, whereas in [13] and similar papers the correct expectation of the function is found. This interchange between expectation and function greatly improves the SGA. Still, these results are true only for systems using random sequences. One can extend them, by allowing the desired user to use a deterministic sequence but the other users contribute MA1 from random sequences [ 151.

To summarise, the MA1 from an individual interferer tends to a Gaussian RV when N tends to infinity, provided the time delays and the phase shifts are considered con- stant, and the same holds when both M and N tend to infinity with MIN constant. Therefore for random delays and phases the individual MAI, even asymptotically, is not Gaussian. On the other hand, the total MA1 should asymptotically tend to a Gaussian RV as M tends to infin- ity, irrespective of N, a consequence of the CLT. Based on these two requirements and the desire for simplicity and, to

IEE Proc -Commuri I Vol 148, No I , Fehruary 2001 44

some extent, analytical tractability we must look for a PDF that adequately models the individual and the total MAI. In view of the sufficiency of two parameters to provide accurate results in [13-1.51, we would also restrict attention to two parameter distribution families modelling non- Gaussian events.

the walk is random, obeying the negative binomial distribu- tion independently of the steps. After averaging over these fluctuations we have

N=O

3

The concept of a random walk has proved to be of great utility and has been used with success in diverse areas of mathematics, physics and biology. Consider for example the case of radar. The performance of microwave radars operating over the sea is often limited by unwanted returns from the sea surface. when a radar illuminates a large area of the sea it is usually found that the PDF of the envelope of the return signal is Rayleigh. This is a consequence of the CLT since the signal can be thought of as being the vector sum of randomly phased components from a large number of independent scatterers. However, it is possible to illuminate areas of the sea of dimensions comparable to the longer wavelengths of the sea surface, and then large devia- tions from Rayleigh statistics are found [16]. The complex back-scattered field deviates from Gaussian statistics to longer tailed distributions. Various explanations for such returns have been proposed. One important factor seems to be the fluctuations in the number of scatterers contributing to the return signal. Such fluctuations play a role in deter- mining the properties of the return from a small area of sea surface and must be taken into account in any theory.

Similar deviations from the Gaussian are observed in other fields, such as coherent light scattered from a fluctu- ating population of statistically independent and identical scatterers, particles in suspension or thin layers of highly turbulent media which contain focusing elements [17]. In general, we can write the scattered field as

Non-Gaussian scattering and the random walk

N

E(?, t ) = ut(?, t ) cxp~jwt + j$z(?, t ) ] 2=1

where a, (+, t ) is a real factor governing the angular distri- bution of radiation from the ith scatterer and 4, is a phase factor. This equation describes a random walk of a finite number of steps of fluctuating lengths in the complex plane. Without knowledge of the distribution of n,{i’, t), the distribution of E(r’, t ) cannot be found exactly, apart from the Gaussian limit when N .+ E. A class of distributions based on the modified Bessel functions K,t has proved useful, and generally applicable in non-Gaussian situations [18, 191. We proceed to show, essentially repeating [20], how these distributions can be arrived at by a limiting procedure in a random walk.

3, I The K-distribution After N steps in an n-dimensional random walk the result- ing displacement is given by

N

A=XCZ, (6) j=1

If the steps are independent and identically distributed the CE of A simplifies to Cdu) = (eJ‘ta)N, and for an unbiased walk, the direction of the vector being uniformly distrib- uted, the CF becomes

N C ( u ) = uW/2--lI ( 2 7 P N / 2 (a7’l”J,,/2-l (a.)) (7 )

with the bar denoting expectation over the distribution of the step length a. Now, suppose that the number of steps in

IEE Proc -Conzinim , I’ol 148 No I . F ~ h i u o i y 2001

where P(N) is the probability of there being N steps in the walk and is given by

(9) Here, a is a parameter characterising the clustering implicit in the negative binomial distribution, and Ili is the mean number of steps. Scaling the step lengths by (N)’” and letting w go to infinity we find

The corresponding limiting distribution of A is given by

where b = dT (11)

This is quite a general result that does not depend on the distribution of the step lengths. The resulting distribution is called the K-distribution and depends on the parameters a and b. It can also be seen that the distribution of the ampli- tude is the same irrespective of 11, the dimension of the space.

3.2 The MA/ as a random walk Let us consider the case of one interfering user, say the kth, and examine its interference, equation, eqn. 2. Using eqn. 4, the interference can be cast in the form

N - I

j = O

and sIc = R,,,(zk)cos41t, ,8,, = R,(z,)cos@lc. The basic assumption we will make is that the terms a:

are independent and identically distributed. According to [6, 121 this seems to be necessary even when random sequences are used. The next assumption is that the period of the sequences, i.e. the number of steps in the random walk, is not fixed but fluctuates according to a negative binomial distribution, given by eqn. 9, with parameters N and a. This distribution was chosen because it describes well correlated events such as the number of trials needed to get a certain number of successes in independent Bernoulli trials [21] and also allows the distributions derived to be put in a closed form. In fact, it is an exact stationaiy solution for the population in a birth-death- immigration process [19, 221.

45

With these assumptions and according to the discussion in Section 3.1, we model the interference from user k as a RV obeying the K-distribution

with bk = fl (13)

when a is taken to infinity it can be seen from eqn. IO that the CF of the distribution approaches a Gaussian

with mean and variance that depend on the time delays and the phase shifts. In fact

j (15)

After averaging over the sequences and the data bits and setting n = I the CF is asymptotically

s; + 5; + 2a;a$-,sk& a2 = - { + s^E + 2bkb"~ia"skSk j = 0

c ~ ~ ~ ' * h [ R ~ , i r h ) + ~ ~ , i r ~ ) ]

C,(u) = e- 2 (16)

cos2 01 [ l - Z q +2+

Cm(u) = e- (17)

For a rectangular pulse and T, = 1 it reduces to

2 U

as derived in [12], for the interference from one user. This agreement is reassuring.

'The limiting procedure is equivalent to letting the number of steps in the walk, or the period of the signature sequences in the CDMA situation, go to infinity. The dif- ference with the derivation in [12], [6] or [23] is that the application of the CLT there depends on the averaging with respect to the receiving sequence. Here, the Gaussian limit is reached naturally. In the aforementioned papers, dealing only with random sequences, the receiving sequence is kept constant while averaging over the transmitting one(s). Then averaging over the receiving sequences is per- formed. This is done because the RV used in these papers, which are exactly the steps in our random walk, are thought to be dependent. The model proposed in this Sec- tion and the limiting equalities, eqns. 14-17 suggest other- wise. For very long sequences the negative binomial distribution degenerates into a Poisson, and the first assumption we made about the statistical independence of the steps must be true.

In fact, we can prove that for random sequences the steps in the random walk are independent. In this case, because the data bits are random we can absorb them into the random sequences Then eqn. 126 takes the form

We can further simplify the above expression by recalling that y is also a random variable having the uniform distri- bution in [0, I] and then eqns. 18 and 12a take the very simple form

N - l

j=O

(19) Since both ajk and ajfl contain the same random variable, a:, it is not obvious that the terms in eqn. 19 are statisti-

46

cally independent. Using probabilistic arguments, we will show that if {xi} and {yi} are statistically independent ran- dom binary sequences and z, w are arbitrary constants then xjyjz and xjyk'kw are also statistically independent random variables, provided, j z k.

The statement is trivial for x = 0 or y = 0. So we will assume that x z 0, y z 0 a n d j z k. Let P(xiyjz = a, xiyl'w = 6) be the joint probability that xiyjz = a and a, xiylp = 6, with la1 = IzI and 161 = /wI. From the theorem of total prob- ability, we have that

P ( z i y j x = a,xiykw = b) = P(x:iyjz = a, xiykw = b, x; = 1)

Because the {xi} and {yi} are independent and random binary sequences, the above becomes

+ P(x ;y j z = a,x;ykw = b,xi = -1)

P(x ;y j z = a, X;ykw = b) = P(yjZ = a)P(ykw = b)P(ai = 1)

+ P ( y j z = -a)P(ykw = -b)P(ai = -1)

and since xi equals +1 or -1 with equal probability

and so

= P(y,x = a)P(ykw = b)

P ( z , y , z = a ) P ( x , ykw = b) = P(y,z = U ) P ( y k w = b) A similar calculation will give

Therefore

P (xn9, = a , x,yk W = b ) = P(Z,y, = a ) P(ZtykW = b ) which means that x,y,z and x,ykw are independent. Direct application of this shows that the steps in the random walk, eqn. 19, are independent and since they are also uniformly bounded, the CLT can be applied. As noted earlier, the K- distribution model also reduces to the ordinary CLT in this case.

3.3 More than one user Interference from more than one user, say A4 > 2, can be written in the form

N - 1 M N - 1

I t o t = a j , t o t = a; (20 j = O j = O k=2

where, using eqn. 12b, we have

IEE Proc.-Cominun., Vol. 148, No. I , Fehsumy 2001

Under the same assumptions as before, we will allow N to fluctuate according to a negative binomial distribution and arrive at

2a and b =

Taking the square of eqn. 21 and averaging over the vector of M - 1 data bits and the interfering sequences we find that

M M

= + = cos2 $ k [ n $ , ( T k ) + k , $ ( T k ) ]

k = 2 k=2

(23) This means that in the limit the CF tends to the Gaussian

agreeing with eqn. 57 in [ll].

3.4 The error probability The calculation of the PE is best facilitated by the use of the CF method introduced in [3]. Following the procedure explained there, we express the PE as the integral

where @(U) is the CF of the total interference and s = Nd 4Eb. Using the expression for the CF of the K-distribution that models the interference we can write

Note that this is conditioned by the time delays and the phase shifts appearing in q, the variance of the interference, given by eqn. 22. Expanding the term in brackets we obtain

. a

(27) 1

= - - &$"Pjn)(s) 2 o

where Pl(n)(s) is the nth derivative of the probability of error with just one user over an AWGN channel and the constants are given by

(n + i ) r (a + n) ?/n= (&r 7 a a ) r ( n )

Now averaging over q needs to be performed. Following the arguments in [13], for simplicity, rather than utilising the actual density of q, we will treat it as a Gaussian RV characterised by a mean p and a variance V. Notice that p = (A4 - 1)/3. The errors introduced by this simplification are very small when the system has more than five active users [13]. This will allow us to write

where the nth moment of I/ can be expressed as

(29) and 2 = J," e-(W-P)*/2 Vdv is a normalisation factor.

form the integrations in eqn. 26, as The result can be put in closed form, if we directly per-

l a 2 o

Pe(s) = - - C ( n ; p, V)s- l /2-n

1 3 x IFI [ + n, 5, -'] 4s (30)

with the constants given by

The incomplete gamma function appearing in eqn. 31 comes from the integrations in eqn. 29.

This is not particularly useful for numerical evaluations. Alternatively, we may express it as

1 P&) = - - D ( n ; p, v)e-ts-2n+l/"[n - 1; SI

2 o (32)

where W is a polynomial in s of degree n - 1, arising from the derivatives of the PE without interference, which is just an error function. The constants are

In the limit when a tends to infinity, the whole calculation reverts back to the one found in [13], before the approxi- mations are made. Alternatively, we may utilise only the mean and variance of + and the three-term expansion in [13] applied to eqn. 27 to give

1 +s (P - hv)"] (34)

3.5 Discussion After their introduction in [24], the class of K-distributions has been found to provide an excellent model for the amplitude statistics of the scattered radiation in a wide vari- ety of experiments involving scattering from turbulent media. Subsequently, they have been used to model devia- tions from the Gaussian limit, and they can be derived as limit distributions in a random walk problem with a ran- dom number of steps.

In a DS/SSMA system with random signature sequences, the multiple access interference from an individual user can

41 IEE Proc.-Comnirm., Vol. 148, No. I . Fe1iruar.y 2001

be considered as a random walk in one dimension with steps given by eqns. 12b or 21. It can be seen from the structure of the interference term that, when deterministic sequences are used, the steps in this walk are not independ- ent and the contribution to each chip is affected by the same delay and phase shift. Therefore, the effective number of steps for which the contribution is non-zero will be less than N. We have, therefore, allowed the steps to be inde- pendent and identically distributed and let the number of steps fluctuate. Of the several choices for the distribution of N, the negative binomial appears to be the simplest one for which the random walk problem can be solved in a closed form. We have presented no rigorous proof of the validity of such a model, but the fact that the K-distribution, which models the interference, tends to a Gaussian when N goes to infinity and the agreement of the resulting expressions for the characteristic function with the ones derived by a different method is very encouraging. Moreover, the K- distribution has the desirable asymptotic properties discussed in Section 2.2 and its moments are greater than the Gaussian ones.

Of the two parameters needed to specify the K-distribu- tion, one has been calculated through the variance of the steps of the random walk. To complete the statistical char- acterisation, the shape parameter a, can be calculated by several methods. One can use maximum likelihood meth- ods (ML) that give asymptotically efficient estimates but are computationally expensive [25]. Estimation techniques based on moments have also been proposed [26], whch can be used in conjunction with ML [27] to provide the lowest variance of parameter estimates when compared with non- ML methods. Also, several methods exist for the numerical evaluation of the modified Bessel functions, for instance [28] which uses saddle point integration. The average number of steps in the walk, or the effective period N of the sequences can also be determined by measurements on the distribution itself [lS].

Finally, the probability of error is put in a power-series form, which can be used to get numerical results. Note that the evaluation of the integrals involved is only made possi- ble by assuming a Gaussian density for the variance of the steps of the random walk. The result will therefore be only an approximation for a small number of users. Prelimiiiary results, with data from M-sequences and Kasami sequences, indicate that the approximations described in this paper for the MA1 are in good agreement with simula- tion results. A future paper will describe this in more detail.

4 Conclusion

A distribution has been derived to model the multiple access interference in a DS/SSMA system with random sig- nature sequences, through the consideration of a certain random walk problem. This family of K-distributions has the desired asymptotic properties, lead to closed-form solu- tions for the probability of error and, to the best of our knowledge, appears for the first time in the spread-spec- trum literature. One parameter, a, is required to complete the characterisation of the MAI, for the determination of which fast and accurate methods exist. The theoretical results presented will provide the same accuracy or better than the improved Gaussian approximation, with little increase in computational burden.

5 References

1 PURSLEY, M.: ‘Performance evaluation of phase-coded spread-spec- trum communication-part 1: system analysis’, IEEE Truns. Cornu",

MORROW, R., and LEHNERT, J.: ‘Bit-to-bit error dependence in slotted DS/SSMA packet systems with random signature sequences’, IEEE Trans. Cornnzun., 1989, 37, pp. 1052-1061 GERANIOTIS, E., and PURSLEY, M.: ‘Error probability for direct- sequence spread-spectrum multiple access communications-part 11: approximations’, IEEE fians. Co~nniun., 1982, COM-30, pp. 985-995 YAO, K.: ‘Error probability of asynchronous spread spectrum multi- ple access communication systems’, IEEE Trcms Cornmun., 1977,

PURSLEY, M., SARWATE, D., and STARK, W.: ‘Error probabil- ity for direct-sequence spread-spectrum multiple-access conununica- tions-part 1: upper and lower bounds’, IEEE Truns. Commun., 1982,

LEHNERT, J., and PURSLEY, M.: ‘Error probabilities for binary direct-sequence spread-spectrum communications with random signa- ture sequences’, IEEE Truns. Comniun., 1987, COM-35, pp. 87-97 SOLMS, F., VAN ROOYEN, P., and KUNICKI, J.: ‘Maximum entropy performance analysis of spread-spectrum multiple-access com- munications’ in SKILLING, J., and SIBISI, S. (Eds.): ‘Maximum entropy and Bayesian niethods’, pp. 101-108 LEHNERT, J., and PURSLEY, M.: ‘Mullipath diversity reception of spread-spectrum multiple-access communications’, IEEE Truns. Corn-

9 KAVEHRAD, M.: and RAMAMURTHI, B.: ‘Direct-sequence spread spectrum with DPSK modulation and diversity for indoor wireless communications’, IEEE Trc1n.s. Conimwz., 1987, CONI-35, pp. 224236

IO JALLOUL, L., and HOLTZMAN, J.: ‘Performance analysis of DS/ CDMA with noncoherent M-ary orthogonal modulation in multipath fading channels’, IEEE, J. Sel. Areus Conmniun., 1994, pp. 862-870

11 LOK, T., and LEHNERT, J.: ‘An asymptotic analysis of DS/SSMA communication systems with general linear modulation and error con- trol coding’, IEEE Truns. Commun., 1998, COM-44, pp. 87CL-881

12 MORROW, R., and LEHNERT, J.: ‘Packet throughput in slotted ALOHA DS/SSMA radio systems with random signature sequences’, IEEE Trans. Commun., 1992, pp. 1223-1230

13 HOLTZMAN, J.: ‘A simple, accurate method to calculate spread- spectrum multiple-access error probabilities’, IEEE Truns. Commein., 1992, pp. 461464

14 HOLTZMAN, J.: ’On calculating error probabilities’. Proceedings of IEEE 2nd international symposium on Spreud spctrum techniques and upplications, Yokohama, Japan, 1992

15 MORROW, R.: ‘Accurate CDMA BER calculations with low com- putational complexity’, IEEE Truns. Commun., 1998, pp. 1413-1417

16 JAKEMAN, E., and PUSEY, P.: ‘A model for non-Rayleigh sea ccho’, IEEE Truns. Antennus Propug., 1976, AP-24, pp. 80G814

17 JAKEMAN, E., and PUSEY, P.: ‘Significance of K-distributions in scattering experiments’, Plzys. Rev. Lett., 1978, 40, pp. 546-550

18 WATTS, T., HOPCRAFT, K., and FAULKNER, T.: ‘Single meas- urements on probability density functions and their use in non-Gaus- sian light scattering’, J. Phys. A, Math. Gen., 1996, pp. 7501-7517

19 JAKEMAN, E.: ‘On the statistics of K-distributed noise’, J. Phys. A, Muth. Gen., 1980, 13, pp. 3148

20 JAKEMAN, E., and TOUGH, R.: ‘Generalised K distribution: a sta- tistical model for weak scattering’, J. Opt. Soc. Ani. A, Opt. Imuge Sci,, 1987, 4, (9), pp. 17661772

21 VIVEROS, R., BABASUMBRAMANIAN, K., and BALAKRISH- NAN, N.: ‘Binomial and negative binomial analogues under corre- lated Bernoulli trials’, J. Am. Stat. Assoc., 1984, 48, pp. 243-247

22 TOUGH, R.: ‘A Fokker-Planck description of K-distributed noise’, J. Pliys. A, Muth. Gen., 1987, 20, pp. 561-567

23 GERANIOTIS, E., and GHAFFARI, B.: ‘Performance of binary and quaternary direct-sequence spread-spectrum multiple-access systems with random signature sequences’, IEEE Trans. Commztn., 1991, 39, pp. 713-723

24 PUSEY, P.N., in CUMMINS, H.Z., and PIKE, E.R. (Eds.): ‘ Photon correlation spectroscopy and velocimetry’ (Plenum, New York, 1977)

25 JOUGHIN, : ‘Maximum likelihood estimation of K distribution for SAR data’, IEEE fianx Geosci. Remote Sens., 1993, 29, pp. 989-999

26 RAGHAVAN, R.: ‘A method for estimating parameters of K-distrib- uted radar clutter’, IEEE Trans. Aerosp. Electron. System., 1991, 27, pp. 238-246

27 ISKLANDER, D.: ‘A method for estimating the parameters of the K distribution’, IEEE Truns. Signal Process., 1999, 47, pp. 1147-1 151

28 GORDON, S., and RITCEY, J.: ‘Calculating the K-distribution by saddlepoint integration’, IEE Proc., Rudur Sonnr Nuvig., 1995, 142, pp. 162-165

1977. COM-25, pp. 795-799 2

3

4

COM-25, pp. 803-809 5

COM-30, pp. 975-983 6

7

8

INWZ., 1987, COM-35, pp. 1189--1198