Nonlinear Channel Equalization in Digital Satellite Systems

Nonlinear Channel Equalization in Digital Satellite Systems

Ching-Hsiang Tseng and Edward J. Powers

Department of Electrical and Computer Engineering Electronics Research Center

The University of Texas at Austin Austin, Texas 78712-1084

U S A

ABSTRACT A new nonlinear equalizer for digital transmission over a nonlinear satellite channel with a

PSK

modulation is presented in this paper. The non- linear channel under consideration is assumed to have a Volterra series representation. By considering the channel equalization as a fixed point problem and tak- ing advantage of the properties of a contraction m a p ping, we show that the transmitted data symbols can be recovered in an iterative manner. The equalizer has a modular structure for easy implementation. The computer simulation shows that the proposed equalizer has an excellent performance.

I.

I n t r o d u c t i o n

Digital transmission over satellite channels has r+

ceived great attention in the recent years [l]. Due to the power-limited nature of the satellite and the in- creasing demand for high-speed data transmission, ef- ficient use of the power and the bandwidth has become one of the major challenges for designing a digital satel- lite link.

In a digital satellite system, nonlinear distortion in- troduced by the saturation effect of the satellite am- plifier is a major factor which limits the performance of the system. In order to avoid the undesired channel nonlinearities, one common approach is to introduce an input backoff, that is, to operate the amplifier at a level which is a few decibels below the saturation point.

However, driving the amplifier at or near the satura- tion point provides higher power efficiency and is desir- able for satellite systems. As the need for high-speed data transmission becomes more and more demand- ing, high-level modulation to improve the bandwidth efficiency is desirable. In such cases, the nonlinear dis- tortion problem becomes even more critical.

In the presence of channel nonlinearities, the per- formance of the traditional linear equalizer has been found to be inadequate, thus various nonlinear tech- niques have been proposed. One approach is to use

a nonlinear equalizer which has a matching Volterra structure of the nonlinear channel [2]. This approach, although it does improve the signal t o distortion ra- tio at the output of the equalizer, lacks a theoretic basis. Another approach is to design a nonlinear pre- distorter, based on the theory of a p t h order inverse, a t the transmitting end to compensate for the nonlin- earities of the channel [3]. This approach, although it does compensate for the nonlinearites up to p t h order, might introduce nonlinearities higher than p t h order.

A more recent approach is to use a neural network to design the equalizer 141. However, this approach faces learning problems and the current result indicates that there is room for improvement.

In this paper, we present an iterative nonlinear equalization scheme to combat the nonlinear distortion in the digital satellite channel. In this approach the nonlinear channel is expressed in terms of the Volterra series [5]. By taking advantage of the Volterra struc- ture and considering the nonlinear equalization a fixed point problem, we derive a successive approximation approach to remove the channel nonlinearities. This method is easy to design and rather flexible in the sense that the performance can be improved by adding additional processing units without changing the origi- nal structure. The computer simulation shows that the proposed equalizer can effectively remove the intersym- bo1 interference (linear and nonlinear) with rather few processing units.

11. Volterra Series Representation of a Nonlinear Channel

A discrete bandpass nonlinear satellite channel can be modeled by a complex Volterra series [SI, as shown in the following:

N-1

1639 0-7803-0917-0/93$03.00 8 1993 IEEE

N-1 N-1 N-1

i=O j = i k=O

+....

= V(Z,, ^{Zn-11- * *}

,

^Zn--N+l) (1) where x, and 9, are the transmitted and the received complex data symbols respectively, and h l ( . ) , h3(.),

. . .

are the Volterra kernels of the nonlinear satellite channel. The V(x,,,xn-~,.

. .

^,z,,-N+I) denotes that the Volterra series is a function of ^2, to ^z,,-N+~. Note that due to the bandpass nature of the nonlinear chan- nel, only odd-order terms survive in (1) [6].

The PSK is one of the most popuIar modulation scheme used in digital satellite communications. For an M-ary PSK, the transmitted data symbols take val- ues from the set { S k : k = 1 , .

. . , M }

with Sk having the following form:

s k = A&'k (2)

27r(k

-

1/2) (3)

$k = M

where A and ^$k are the amplitude and the phase of ^Sk respectively. It has been shown that the PSK is rather insensitive to certain kind of nonlinearities. This prop- erty enables a PSK satellite channel to be modeled by a low-order (3rd or 5th) Volterra series.

111. The Fixed Point Problem

Since the proposed method approaches the channel equalization problem by considering it a fixed point problem, we review some properties of a fixed point in the following.

A fixed point problem is a matter of solving an equa- tion with the following form:

2: = T ( z ) (4

1

A point xf which satisfies (4) is called a fixed point of the transformation T since it is invariant under

T.

A fixed point of a system is not necessarily unique, or it may not even exist. However, a classical method, which is based on the contraction mapping theorem, can be used to find the fixed point if the transforma- tion T i s a contraction mapping, where the contraction mapping is defined as follows [7]:

Definition. Let S be a subset of a normed space

X

and let T be a transformation mapping S into S. Then T is said t o be a contraction mapping if there is an a , 0 I Q

<

1 such that IIT(z1)

-

T(z2)Il I a112:1

-

~ 2 1 1

for all ^z1,22 E S .

A geometric interpretation of the definition of the contraction mapping is that, for any two points, the distance between them after the transformation is no greater than the distance before the transformation.

The contraction mapping theorem is summarized in the following:

Theorem 1 IfT is a contraction mapping on a closed subset S of a Banach space, then

(a) T h e m i s a unique point xf E S satisfying T ( s f ) =

(b) zf = lim,,+mT"(xo), where xo w a n arbitmry xr.

point in

S.

(4 l l ~ n -zrll I

G l l z i 0"

-zoll,

where xn+i = T ( x , ) . Note that Theorem l(a) guarantees the uniqueness of the fixed point, Theorem l(b) tells us how to find the fixed point, and Theorem l(c) provides a guideline on how fast we can find the fixed point.

IV.

Nonlinear Equalization

Now, let's go back to our channel equalization p r o b lem. Each time we receive a data symbol, say Y k , at the receiver, we need to identify the transmitted data symbol Z k . Let's assume that the data symbols t r a n s mitted before Z k are already known. By substituting them into (l), identifying the current transmitted data symbol becomes a problem of solving a singlevariable function described as follows:

Y k =

v(z)

(5)

This can be considered as a fixed point problem by defining a transformation

T

as follows [8]:

T ( s ) = Z + P [ Y k

-

V ( 4 l (6) where ,B is a constant. Note that solving (5) is equiv- alent to finding the fixed point of (6) since at x = xk we have

T(Xt) = x k -/- p [ Y k

-

V ( z k ) ]

= x k (7)

Now the remaining question is whether T is a contrac- tion mapping or not. If it is, according to the the- orem, there is a unique fixed point and we can find the fixed point by Theorem l(b). In the Appendix, we show that, for the channel equalization problem we are interested in, there exists a

p

which makes T a contraction mapping. Therefore, we ^can apply the theorem t o design the nonlinear equalizer. The block diagram of the proposed equalization scheme is shown

in Fig. 1. We see that the equalizer is a combination of many processing units. Each processing unit performs the same transformation T ( . ) . For example, in the i-th processing unit, the following calculation is performed:

Z¶+l = . z i + P [ Y k -V(.zi)l (8) According to Theorem l(b), the sequence {z,,} con- verges to the fixed point zk.

n o m Theorem l(c) we know that the value a con- trols the convergence rate of the sequence {zn}. For fast convergence, a should be as small as possible. The smallest value for a is zero, which leads t o (see the Ap- pendix) the following:

(9)

V. Computer Simulation

To demonstrate the effectiveness of the proposed nonlinear equalizer, we consider a nonlinear satellite channel which has the same Volterra series representa- tion as the one described in [2]. The Volterra kernels of the channel, after taking into account the properties of the PSK, are shown in Table. 1. In order t o design an equalizer which has the fastest convergence rate, we need to find the optimum value for

p.

In this study, we varied both the real and the imaginary parts of

p

from -1.0 t o 1.0 and calcuated the ratio 7, where

Note that the smaller y is, the faster the convergence will be. The result is shown in Fig. 2, where we indicate that the minimum of 7 is equal t o 0.0625 and is located at

p

^{= 0.65}

-

0.32. This result is in close agreement with the estimated optimum value of

p

obtained by (9), where ,B = 0.64

-

0.342. This justifies the contraction mapping analysis described in the Appendix.

Next, we test the performance of the proposed non- linear equalizer. The transmitted data symbols are

&PSK modulated with an amplitude one. 500 data symbols were transmitted through the nonlinear chan- nel and the scatter plot of the received data symbols is shown in Fig. 3. Note that the transmitted data symbols have been scaled and rotated, and the data symbols in each cluster have been spread. The scatter plot of the data symbols obtained after the first pro- cessing unit is shown in Fig. 4, where the major rota- tion and scaling have been rectified. The data symbols obtained after the second processing unit, as shown by the scatter plot in Fig. 5, shows that the spreading of the data symbols in each cluster has almost been elim- inated completely. The output of the third processing

unit, as scatter-plotted in Fig. 6, shows nearly perfect 8-PSK constellation.

VI. Conclusion

In this paper, we present a new method t o design an equalizer for nonlinear digital satellite channels. This method is based on the concept of successive approx- imation. By considering the channel equalization as a k e d point problem and taking advantage of the Volterra series representation of the nonlinear chan- nel, we show that the classical contraction mapping theorem can be applied to solve the problem. This equalizer features great flexibility in the sense that ad- ditional processing units can be added to improve the performance without changing the original structure of the equalizer. The performance of the equalizer is tested through computer simulation. The result shows that the equalizer has a fast convergence rate and is very effective in removing channel nonlinearities.

Appendix

Consider a set S of complex numbers where 0 5

llzll < ^M

for all z E S. It is clear that S is a closed subset of the normed complex space C. Let 5 1 and ^{5 2} be two points in S. By using (6), we have

llT(.l)

-

T(52)II = 1151

-

2 2

-P[V(.1)

-

V(.2)lll (11) By substituting (1) in ( l l ) , we have

llT(z1) ^-

T(52)ll = 111.

-

.2

-P(.1 -

.2>[hl(O)

+ 41

IN1

-

P[hl(O)

+ 4H.1 -

.2)11

.11.1

-

X2ll (12)

=

<

111 - P[hl(O)

+dII

where E denotes the contribution from the higher-order terms of the Volterra series. Define a as follows:

a = 111

-

P[hl(O)

+ dl1

(13) Assume that

1 1 ~ 1 1

is smaller than Ilhl(O)II (which is true in the

PSK

case since the nonlinear terms are rela- tively weak compared to the linear terms), then for

p

= l/hl(O), we have

In fact, if

11.~11

is much smaller than Ilhl(O)ll, than a 0, which is the smallest value for ^a.

1641

Let 5 2 be equal t o zeros, then we have

IIT(a)II 5 QllZlll 5

(15)

We see that

T

does map S into S . This concludes that there exists a ,L9 which makes

T

a contraction mapping.

Acknowledgement

This is supported in part by the Joint Services Elec- tronics Program Contract AFOSR F49620-92C-0027.

References

[l]

R. M.

Gagliardi, Satellite Communications, 2nd ed., New York: Van Nostrand Reinhold, 1991.

[2] S. Benedetto and E. Biglieri, “Nonlinear equaliza- tion of digital satellite channels,” IEEE J. Select.

Areas Commun., vol. SAC-1, no. 1, pp. 57-62, Jan.

1983.

[3] E. Biglieri, S. Barberis, and M. Catena, “Anal- ysis and compensation of nonlinearities in digi- tal transmission systems,” IEEE J. Select. Areas

Commun., vol. 6 , no. 1, pp. 42-51, Jan. 1988.

[4]

N.

Benvenuto, M. Marchesi,

F.

Piazza, and A.

Uncini, “Nonlinear satellite radio links equalized using blind neural networks,” Proc. Int. Con$

Acoust. Speech. Signal Processing, pp. 1521-1524, Toronto, July 1991.

[5]

M.

Schetzen, The Volterra and Wiener Theories of Nonlinear System. New York: Wiley, 1980.

[6] S . Benedetto, E. Biglieri and V. Castellani, Digital Ransmission Theory, Prentice-Hall, Inc., 1987.

[7]

D.

G. Luenberger, Optimization by Vector Space Methods, New York: Wiley, 1968.

[8]

I.

W. Sandberg, “On the properties of some s y s tems that distort signals -

I,” B.

S. T . J., _{vol. 42,} pp. 2033-2046, Sept. 1963.

I

Figure 1: The block diagram of the proposed nonlinear equalizer.

Imaginary Axis

Figure 2: The contour plot of the ratio y achieved by various values of

0.

1.5 1 0.5

0 -0.5

-1 -1.5

-1.5

-1

-0.5 0 0.5 1 1.5

Figure 3: The scatter plot of the received data symbols before the equalizer.

1642

1.5

I I

1

0.5

0

... ; ... .; ... .:. ...

. ; .

...

..

^{I .}

... .: ... .: ... _:. ...

.:_. ... .%.-.*. ...

:

;*y>

1

0.5

_._

...

...

o...

". ..

...

...

-0.5 -1 .*+ .!:

^{~ ...} : ...

.*.

*

:$

... ^,..*. ,- ...

...

...

...*...

: I

-0.5

- 1

+$:. ...

?'

.

. :&

...

.i;.,..

...

;.

...

... .j.. .... .:$E.. ^.my. .. .:. ..

:.g.;. A.

... j ...

. . . ! I

-1.5 I

-1.5 -1 -0.5 0 0.5 1 1.5

Figure 4: The scatter plot of the received data symbols after the first processing unit.

1.5 1

0.5

0

-0.5

-1

...

* : e

...

: e m:

...

: a w :

...

... ? . . _ _ _ ; . . .

r.

...

-1.5

-1.5

-1

-0.5

0

0.5

1 1.5

Figure 5: The scatter plot of the received data symbols after the second processing unit.

...

..

...

...

...

...

...

. .

...

...

-I.J

-1.5 -1

-0.5

0

0.5

1 1.5

Figure 6: The scatter plot of the received data symbols after the third processing unit.

hI(0) = 1.22+j0.646 hi(1) = 0.063 -jO.OOl hi(2) = -0.024

-

j0.014 hi(3) = 0.036 +j0.031

~~

h3(0, 0,2) = 0.039

-

j0.022

h3(0, 0,l) = 0.035

-

j0.035 h3(o, 0,3) = -0.040

-

jO.009 h3(1, 1,o) = -0.010

-

j0.017 h3(3,3,0) = 0.018 - j0.018

hs(O,O, 0, 1,l) = 0.039

-

j0.022

Table 1: The Volterra kernels of the nonlinear digital satellite channel used in the simulation.

1643