- Bayesian Equalization for Satellite Communications with Multipath and Burst Digital Transmission

Bayesian Equalization for Satellite Communications with Multipath and Burst Digital Transmission

Qilian Liang

Department of Electrical Engineering University

of Texas at Arlington

Arlington, TX 76019-0016 USA E-mail: qliang@ieee.org

Abstract-In this paper, we present a channel equaliza- tion scheme for time-division multiple access (TDMA) satellite communications with burst digital transmis- sion. We show that each channel s t a t e s of multipath satellite channel follows a Gaussian distribution, which means a Bayesian equalization can b e implemented. T h e parameters of t h e Bayesian equalization are determined using an unsupervised clustering m e t h o d - fuzzy ^c- m e a n s (FCM) m e t h o d . A n e x t r e m e l y small number of training symbols (about 1% of a burst) are used t o determine the category of each channel s t a t e with t h e aid of d a t a mining. Simulation results show that our Bayesian equalizer performs much better than t h e re- cently proposed nearest neighbor classifier-based equal- izer a t moderate t o high signal-to-noise ratio (SNR).

1. INTRODUCTION

With the increasingly mainstream role of multime- dia, laptop PCs, personal digital assistants (PDAs) and personal information assistants (PIAs) require commu- nication techniques with higher data rate. But due to the limited bandwidth of satellite channel, more than one path transmission is required. In the TDMA-based satellite communications, the data are transmitted in burst by burst (e.g.. 8 bursts per frame). The classical equalizers do not perform well for such fading channels with burst transmission because the number of train- ing symbols is quite limited (less than 2% of one whole burst).

There are two types of adaptive equalization: se- quence estimation and symbol detection. Sequence es- timation has very high computation complexity, b e cause channel estimation is needed. Symbol detection is essentially a classification problem, in which the in- put baseband signal is mapped onto a feature space de- termined by the direct interpretation of a known train- ing sequence. Chen, et a1 131 used a radial basis function (RBF) network t o implement a Bayesian equalizer for a time-invariant channel, and demonstrated that it has an identical structure to the optimal Bayesian symbol decision-equalizer. Although they provided a decision- directed clustering algorithm to track the changes of channel states for a timevarying channel, no equal- izer w a ~ designed for such channels. Patra and Mul- grew [lo] observed that the Bayesian decision solution

can be represented using a normalized formula which has an identical structure to afuzzy filter, and used it t o again design a Bayesian equalizer for a timeinvariant channel. In [ll], a nearest neighbor classifier equalizer is used to classi;y the distorted signal for GSM com- mnnications. In [9], a systematic feature space parti- tioning method is proposed to divide the entire feature space into two decision regions using a set of hyper- planes. In [7] 181, a type-2 fuzzy adaptive filter is prc- posed and applied to timevarying nonlinear channel equalization using both transversal and decision feed- back structures. In all these classifier-based approach, channel estimation is unnecessary, which tremendously simplifies this approach, but all of them need a large number of training symbols (more than 10% of a whole burst). In this paper, we focus on the classifier a p proach t o adaptive equalization, and show that each channel state of a satellite channel with multipath follows Gaussian distribution, and apply a Bayesian equalizer to such a fading channel, but use a very small number of training symbols (about 1%).

In Section 11, we introduce an unsupervised- clustering method ^~ fuzzy c-means (FCM), which will be used to cluster channel states blindly. In Section III, we discuss the system model we used in this paper. In Section IV, we show why each channel state of satellite channel with multipath follows Gaussian distribution, and apply a Bayesian equalizer to such a fading chan- nel. In Section V , we evaluated our Bayesian equalizer using simulations and compared it against the nearest neighbor classifier equalizer by 1111. Conclusions and future research directions are given in Section VI.

11. INTRODUCTION TO A N UNSUPERVISED CLUSTERING ^~ Fuzzy C-MEANS FCM clustering is a data clustering technique where each data point belongs t o a cluster to a degree speci- fied by a membership grade. This technique was orig- inally introduced by Bezdek [2] as an improvement on earlier clustering methods. Here we briefly summarize FCM clustering.

Definition 1 (Fuzzy c-Partition) Let

X =

2 1 , .

. . ,

xn be any finite set,

V,,

be the set of real c x n matri- ces, and c be an integer, where 2

5

c

<

n. The Furzy

c-partition space for

X

is the set

Mf. =

U E LLIuik E [O, 11 Vi, k ; ₍₁₎ where

"

X I Q ~ ⁼

^{1 VkandO}

^< X

^ujk

<

ⁿ Vi (2) The row i of matrix U E fit," contains values of the ith

i = l X = l

~ ,-

membership function, u i , in the fuzzy c-partition i'J of X.

Definition 2 (Fuzzy c-Means Functionals) [2] Let J, : M J , x 'RcP

+

'Rf be

where U E M j C is a fuzzy c-partition of

X;

'v

=

( V I , V Z , .

. .

, v c ) E Rep, where vi E ' R p , is the cluster center of prototype uj, 1

5

i

5

^c;

where

1 ) . 11

is any inner product induced norm on UP;

weighting exponential m E [l, m); and, U.k is the mem- bership of x* in fuzzy cluster U;. Jm(U,v) repraients the distance from any given d a t a point t o a cluster weighted by that point's membership grade.

T h e solutions of

min Jm(U,v)

UEM,.,VER=P ( 5 )

are least-squared error stationary points of J,. An infinite family of fuzzy clustering algorithms - one for each m E _(1,m) - is obtained using the necessary conditions for solutions of (5). Bezdek proposed an iterative method [Z] to minimize J,(U,v).

111. SYSTEM MODEL

Satellite channel is often modelled as a Rician Eding channel. Rician fading occurs when there is a strong specular (direct path or line of sight component) signal in addition t o the scatter (multipath) components. The channel gain,

g ( t ) = sr(t) + h ~ ( t ) (6) can be treated as a wide-sense stationary complex Gaussian random process, and g l ( t ) and gQ(1) are Gaussian random processes with non-zero means mr ( t ) and m p ( t ) , respectively; and they have same variance

U : , then the magnitudeof the received complexenvelop has a Rician distribution [12],

where

s2

=

m:(t)

+

^m,$(t)

and I o ( - ) is the zero order modified Bessel function.

This kind of channel is known as Rician fading channel.

A Rician channel is characterized by two parameters, Rician factor A' which is the ratio of the direct path power to that of the multipath, i.e., K

=

s2/202 (121, and the Doppler spread (or single-sided fading band- width) Jd. We simulate the Rician fading using a di- rect path added by a Rayleigh fading generator. The Rayleigh fade generator is based on Jakes' model [5] in which an ensemble of sinusoidal waveforms are added together t o simulate the coherent sum of scattered rays with Doppler spread fd arriving from different direc- tions to the receiver. The amplit,ude of the Rayleigh fade generator is controlled by the Rician factor K The number of oscillators to simulate the Rayleigh fad- ing is 60.

In this paper, the system model we used in our sim- ulations consists of random bits generator, burst (cell) builder, modulator, upsampler by 16, pulse shaping filter (a square root raised cosine filter with roll off factor 0.35), Fiician frequency selective fading channel, matched filter, down-sampler by 16, Bayesian equal- izer, burst extractor, and bit error counter, as shown in Fig. 1. In Fig. 2, we summarize the burst format we used in this paper. Its length is 1013 QPSK sym- bols long, in which 980 symbols are payload. The ran- dom bits generator generates a binary d a t a stream with equally likely zeros and ones, which are for the pay- load bits (1960 bits). The burst builder insert unique word (UW) and guard bits, and makes a complete burst with 2026 bits, snd then the 2026 bits are modulated to 1013 QPSK symbols. The unique word (for training purposes) consists of 13 QPSK symbols, which only oc- cupy 1.28% of a burst. In contrast, GSM uses 16.64%

of a burst for unique word, and IS-54/136 uses 8.64%of a burst for unique word [12]. In our design, one burst takes 5ms, which means the symbol rate is 202.6ks/s, and payload bits rate is 392kbls when it's uncoded.

Iv. BAYESIAN

EQUALIZATION

^FOR SATELLITE

CHANNEL WITH MULTIPATH A N D BURST TRANSMISSION A . Theoretical Basis

For the system we discussed in Section 111, the matched filter outpht when sampled in t i m e

synchronization can be modeled as

L-1

r(k)

=

g(k,/)s(k - I )

+

^{n(k) (9)}

I=O

where L is the number of multipath, and

n(k)

= w ( k ) + i n a ( k )

(10) is additive white Gaussian noise (AWGN) with mean 0 and variance

ai

in the in-phase and quadrature compw nents. g(k, I ) is the truncated channel gain. For QPSK modulation, s ( k ) E { l , j , -1, -j) are the signal points.

Based on different values of s ( k ) , s(k-1),

. . . ,

s ( 6 - L + l ) , there are N ,

=

4L possible channel states. Assume there are 2 paths ( L = 2), so there are 4*

=

16 channel states. If s ( k ) = 1 and s(k - 1) = 1, then (9) can be expressed as

r(k)

=

g(k, 0)

+

^{g(k, 1)}

+

^{n(k) (11)}

=

[ g r ( k , O ) + g r ( k , l ) + n r ( k ) l

+

jbp(k,O) +g&1) + n & l (12) Since g,(k,O), gr(k, l), and n r ( k ) are Gaussian dis- tributions with mean mr(k,O), mr(k, l ) , and 0, and with variance ^U&, ui1, and ^U:, respectively, so r,(k)

=

g r ( k , O ) + g r ( k , l)+nr(k) isaGaussiandistributionwith mean mr(k, O)+m,(k, 1) and variance U ~ ~ + U ~ ~ + U ~ [l].

Similarly, rp(k)

2

gp(k, O)+gq(k, l)+nq(k) is a Gaus- sian distribution with mean mq(k,O)

+

m q ( k , l ) and variance ^{U $}

+

^U:,

+

^U:

Similarly, it's easy t o show t h a t all the other channel states follow Gaussian distributions. So the received signals of one b u n t can be clustered t o 4L clusters us- ing FCM method introduced in Section 11, and the sig- nals associated with each cluster have Gaussian distri- bution. The mean (time average) and variance of each cluster can be computed, so a Bayesian equalizer can be implemented.

B. Designing the Bayesian Equalizer

B . l Expanding the Unique Word Based on Data Min-

In this paper, we focus on a satellite channel with 2 paths. As shown in Fig. 2, there are 13 QPSK sym- bols (unique words) for training. But there are 42

=

16 channel states for the channel with 2 paths, so it's o b vious that the number. of unique words (training sym- bols) is not enough. Besides, due to the intersymbol interference and AWGN, one channel state should have more than one symh,ol for reliable category determina- tion (which will be discussed in IV-B.2). We propose a method t o expand the number of unique word based on data mining."

a

ing

Multiplying j t o both sides of

(Q),

we obtain,

L -1

j . r ( k )

=

j C g ( k , I ) s ( k - l ) + j . n ( k ) (13)

=

C g ( k , I ) l j . s ( k - I ) ] + j . n ( k ) (14)

It's easy to prove that j

.

n(k) is an AWGN with same mean and variance as n ( k ) , so rj(k) = 3

.

r(k) is equivalent to the received signal if the in- put is j ^. (s(k),s(k

-

I ) , . . . , s ( k - L

+

I)]. Simi- larly, rve can artificially construct the received signal r _ l ( k )

=

- r ( k ) and r - , ( k )

=

- j

.

r(k) if the input patterns are -1

.

[s(k), s ( k - l ) ,

- . . ,

s(k

-

L

+

l)] and - j

.

( s ( k ) , s ( k

-

I ) ,

. . .

, s ( k

-

L

+

l)], respectively. In this paper, we msume there are two paths ( L = 2), and we designed the 13 unique words (in QPSK) as

UW

=

[ I , 1,-1, -j, 1,1, -1, - j , 1 , 1, -1,-j, 11. Based on (9), these 13 unique words can generate 12 out- puts, r(k) ( k

=

1 , 2 . .

.

, 1 2 ) , (corresponding t o 4 chan- nel states with each repeated 3 times). Besides the 12 outputs generated from the unique words, we have arti- ficially generated 36 virtual signals rj(k), r-l(k), and r_j(k), where (k

=

1 , 2 , - . . , 1 2 ) , so we totally have 12 x 4

=

48 training prototypes (corresponding to 16 channel states with each repeated 3 times).

B.2 Determine t h e Mean, Variance, and Cluster Cate-

We use FCM method to cluster the 1013+36

=

1049 symbols (in which 1013 symbols are the input to the equalizer, and 36 symbols are artificially-generated) into c

=

16 (2 paths with QPSK modulation) clusters, and each cluster has the mean vi, (i

=

1 ; 2 , . - - ,16) obtained from FCM algorithm The FCM method also generates U, a 16x 1049matrixin this application. Ev- ery symbols rh (k = 1 , 2 , .

.

. ,1049) has 16 membership grades uiir E U (i

=

1 , 2 , . .

.

,16) and uili

=

1 corresponding to the 16 clusters. Based on the 16 val- ues of uilr (i = 1 , 2 , .

. .

, 1 6 ) for each k , we can deter- mine which cluster this symbol belongs to based ou the maximummembership in U;* (i

=

1 , 2 , .

. .

, 1 6 ) .

So

the 1049 symbols can be clustered to 16 cluster using this hard decision. We compute the variance of each cluster based on this decision. In Fig. 3, for illustration pur- pose, we scattered one received burst (1013 symbols) and 36 constructed symbols using dotted point, and 16 centers (corresponding t o 16 channel states) using cir- cles, when Rician fading K

=

12dB, ^{f d}

=

2 0 H z for both paths and Eb/No

=

7 d B .

Based on t h e cluster to which the training symbols (48 symbols in total) have been assigned (based on the

I=O L-1

I d

A .

a h

gory

maximum membership), we can conclude the category (1, j , -1, or -j according to s ( k ) ) of each cluster. Be- cause of the channel fading, ISI, and AWGN, 3 training patterns for each channel state may be clustered to dif- ferent clusters, we use majority logic t o determine ,each cluster category. There are 4 clusters with cate!:ory

“ l ” , 4 clusters with category “j”, 4 clusters with cate- gory “-l”, and 4 clusters with category “-j”.

B.3 Computation Formula

We apply Bayesian equalization t o every received symbol in one burst using the following rule: the signal r(k) isai ( i = 1 , 2 , 3 , 4 ) a n d a i € { l , j , - l , - j ) i f

p(r(k)ls(k)

=

a i )

>

p(r(k)ls(k)

=

01) Val

#

ai (15) where

To compute p ( r ( k ) l s ( k ) = a i , s ( k - 1)

=

^aj), let r A

=

[rI(k),rQ(k)lT>

p ( r ( k ) l s ( k ) = a i , s ( k - l ) = a j ) = p ( r l a i , a . i )

where mij A

=

[mfj, mZlT and Cij

=

diag{a: +U:,.:

+

U:) are the mean vector (2 x 1) and covariance matrix (2 x 2 ) of [r,(k),rQ(k)lT obtained via FCM clustering.

v.

SIMULATIONS

We compared our Bayesian equalizer with a nearest neighbor classifier (NNC) equalizer [ll] for equalization of mobile satellite channel .with 2 paths. The nearest- neighbor (NN) rule, and its extension, the K-NN al- gorithm [4] (if the number of training prototypes is N, then K

=

is the optimal choice for K ) , are non- parameteric classification algorithms, that have been extensively applied t o many pattern recognition prob- lems. Recently, Savazzi, et al. [Ill applied a NNC which used the K-NN algorithm t o channel equalization for mobile radio communications and achieved good. per- formance. In our example, we totally have N

=

48 training symbols (12 symbols are from the provided unique words and 36 symbols are expanded using d a t a mining), which means K

= J;i8

FS 7 . The NNC q u a l - izer classify the category of r(k) based on the categories of its 7 nearest neighbors from the 48 trainingsynibols.

We studied two Rician fading channels: one with Riciau factor K

=

12dB, doppler shift

fs =

20H;:; and

the other one with Rician factor IC

=

9 d B , doppler shift fd

=

10Hr. For both channels, the symbol rate is 202.6ks/s, i.e., the information (payload) bit rate is 392kbls.

For each channel, we ran our simulationsfor different Eh/No values. At each Eh/No value, we ran the simula- tions for 5000 bursts, and obtained the average bit error rate (BER) for the FCM-based Bayesian equalizer and NNC-based equalizer. The performances of the two equalizers in both channels are plotted in Figs. 4 and 5. Observe that our Bayesian equalizer performs much better than the NNC equalizer at moderate t o high SNR (EalNo

>

7 d B ) in both channels. At low SNR

( E b I N o

<

7 d B ) , NNC equalizer performs better than the Bayesian equalizer, because we only have 13 sym- bols unique words, and each channel state only has 3 training symbols after we expanded the unique words, and the training symbols could be clustered to a wrong cluster at low SNR.

VI. CONCLUSIOWS

We have proposed a channel equalization scheme for time-division multiple access (TDMA) satellite com- munications with burst digital transmission. We show that each channel state of multipath satellite chan- nel follows a Gaussian distribution, which means a Bayesian equalizer can be implemented. The param- eters of the Bayesian equalizer are determined using an unsupervised clustering method ^~ fuzzy c-means (FCM) method. An extremely small number of train- ing symbols (about 1% of a burst) are used to de- termine the category of each channel state with the aid of data mining. Simulation results show that our Bayesian equalizer performs much better than the r e cently proposed nearest classifier-based equalizer at moderate to high SNR.

REFERENCES

[I] C . Ash. The Probability Tutoring Book, IEEE Pleas, New York. pp. 205-206.1993.

[2] J. C. Bezdek. P o t t e m Recognition with Fuzzy Ohjectim Function Algorithms, Plenum Press, New York. 1981.

[3] S. Chen. B. M u l g n w , and S. McLaughlin, ’’ A clustering technique for digital communications channel equalization using radial basis function network ,” IEEE Tronr. Neural Nctmuorkr, d. 4, pp. 570-579. July 1993.

[4] R. 0. Dudaand P. E. Hart. Pattern Clossificotien and Scene Anolysis, John Wilty & Sons: Inc. USA, 1973.

[5] W. C. Jakes. Micmwouo M o b i l e Communication, New York, NY: IEEE Press. 1993.

(61 B . J . Kim, and D. C. Cox, “Blind equalization for short burst wireless communications.” IEEE Trans. on Vehicular Technology, vol. 49, no. 4, pp. 12351247, July 2000.

[7] Q . Liang. Fnding Channel Equoliiation and Video Tmf- Jic Classification Using Noniincor Signol Processing Tech- niqucr, Ph.D dissertation, University of Southem California, Los Angeles. CA, May 2000.

[E] Q. Limg and J. M. Mendel, "Equalization of nonlinear timc- varying channels using type-2 fuzzy adaptive filters." IEEE Tront. Fuzzg Systems. vol. 8 , no. 5, pp. 551-563, Oct 2000.

[9] J . Moon and T. Jcon, "Sequence detection for binary IS1 channelsusingrignal apacepartitioning." I E E E Trans. Com- municotionr, vol. 46, no. 7, pp. 891-901, July 1998.

"Efficienl architecture for Bayesian equalization using fumy filters," IEEE Trons. Cir- cuits and S p t r m a 11, vol. 45, no. 7, pp. 812-820, July 1998.

(111 P. Savazzi, L. Favalli, E. Costamasna, and A. Mecocci, "

A suboptimal approach to channel equalization baPcd on the nearest neighborrule (n IEEE J. Selected Areo?i in Commu- nications, vol. 16, no. 9, pp. 1640-1648. Dec 1998.

[I21 G. L. Stuber. Principles of M o b i k Communicotions. 2nd Edition, Kluwer Academic Publishers, Norwell, MA. 2001.

[IO] S . K. Pat= and B. Mulgrew.

Fig. 1. System model we used in our simulation.

G Payload Payload G

I

^{1 0 1 490}

I

^{131 490}

I

¹⁰¹

Fig. 2. Burst format we used in this paper

.C

R g . 3 The centers of 16 channel states (denoted by circles) o b t i n e d via FCM clustering when E s j N o = 7dB, K = 1ZdR.

and fd = 2 0 H ~

Fig. 4. The performances of nearest neighbor classifier (NNC) equalizer and our Bayesian equalizer for satellite c h a n n e h i t h 2 paths and Rician factor K = lZdB and / a = 20Hx.

Fig. 5. The perfomaneea of nearest neighbor clarsifier (NNC) equalizer and our Bayesian equalizer for datellitechannel with 2 paths and Rician factor K = 9dB and / d

=

10Hz.