ON THE PERFORMANCE OF RATE 1/2 CONVOLUTIONAL-CODES WITH QPSK ON RICIAN FADING CHANNELS

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 39, NO. 2, MAY 1990 161

On the Performance of Rate

1/2

Convolutional

Codes with QPSK on Rician Fading

Channels

Abstract-This paper is concerned with the bit error probability per- formance of rate lL! convolutional codes in conjunction with quaternary phase shift keying (QPSK) modulation and maximum likelihood Viterbi decoding on fully interleaved Rician fading channels. Applying the gen- erating function union bounding approach, an asymptotically tight ana- lytic upper bound on the bit error probability performance is developed under the assumption of using the Viterbi decoder with perfect fading amplitude measurement. Bit error probability performance of constraint length K = three to seven codes with QPSK is numerically evaluated us- ing the developed bound. Tightness of the bound is examined by means of computer simulation. The influence of perfect amplitude measure- ment on the performance of Viterbi decoder is also observed. Finally, performance comparison with rate 112 codes with binary phase shift keying (BPSK) is also provided.

I. INTRODUCTION

IGITAL COMMUNICATIONS over mobile channels of-

D

ten suffer from multipath effects, which result in sig- nal fading. Multipath fading plagues the propagation medium by imposing random amplitude and phase variations onto the transmitted waveform. For satellite-aided mobile communica- tions, the channels can be modeled, in most cases, as non- frequency selective Rician channels for which the fading am- plitude obeys a Rician distribution. It is known that this fad- ing degrades the performance of communication systems. The fade margin for uncoded binary phase shifted keying (BPSK) and quaternary phase shift keying (QPSK) systems on chan- nel impaired by Rician fading has been examined in [l]. To combat Rician fading, convolutional codes with Viterbi de- coding and interleaving could be used. The performance of short constraint length convolutional codes in conjunction with BPSK modulation and various types of maximum likelihood (ML) Viterbi decoding on Rician channels has been studied in detail [2]-[5]. The studies indicate that, of the various types of Viterbi decoder, the one utilizing full channel state infor- mation, channel fading amplitude, and phase, contributes the best performance.

With BPSK modulation, the code redundancy introduced by convolutional coding represents a bandwidth sacrifice in that the ratio of required bandwidth to information rate is in- creased. To avoid this sacrifice in bandwidth efficiency and Manuscript received August 11, 1986; revised September 25, 1989. This work was supported by the National Science Council of the Republic of China under Contract NSC78-0404-E009-32.

The authors are with the Institute of Electronics, National Chiao Tung University, Hsin-Chu, Taiwan, Republic of China.

IEEE Log Number 9034219.

also provide effective error-control protection, convolutional codes with multilevel modulation could be used instead. Error performance of convolutional codes with M-ary PSK modula- tion and Viterbi decoding on RiciadRayleigh fading channels has been studied recently [6], [7]. An analytic upper bound on

the bit error probability performance was developed and eval- uated along with the studies. Simulation results show that the developed bound is somewhat loose for rate 1/2 convolutional codes with QPSK modulation on the mentioned channels.

In this paper, bit error probability performance of rate 1/2 short constraint-length convolutional codes with QPSK modu- lation and ML Viterbi decoding on Rician fading channels will be examined further because: 1) these codes are the most pop- ular convolutional codes, and monolithic IC are commercially

available for Viterbi decoders of constraint length K = six and seven codes; 2) rate 1/2 FEC codes with QPSK have been pro- posed for the second generation Inmarsat maritime satellite communication systems [8]. Our primary interest is in devel- oping a tight performance bound on the bit error probability of convolutional codes with QPSK and in the performance com- parison between convolutional codes with BPSK and QPSK. In Section 11, the system under consideration, including the channel, is described. In Section 111, applying the generation function union bounding approach, an analytic upper bound on the bit error probability performance is developed. For the convenience of derivation, we assume that the Viterbi decoder with perfect fading amplitude measurement is used. In Sec- tion IV, bit error probability performance of constraint length K = three to seven codes are numerically evaluated using the derived bound. Moreover, simulation results are provided for examining the tightness of the analytic bound. The influence of perfect fading amplitude measurement on the performance of Viterbi decoder is also observed. Finally, performance com- parison with rate 1/2 convolutional codes with BPSK is made in Section V.

11. SYSTEM DESCRIPTION

The block diagram of the system under consideration is shown in Fig. 1. The data bit U ; is encoded into a dibit

w ;

=

(w;,,

w;2) by a rate 1/2 convolutional encoder. The dibit is assigned to (using Gray mapping) a phase

0; E ( 0 , a / 2 , T , 3 ~ 1 2 ) which is corresponding to a signal point x i = exp(j0;) on the normalized signal space. After fully interleaving, the sequencex = { x i } is transformed into a time-domain waveform s ( t ) by the QPSK modulator. The

OO18-9545/90/05OO-0161$01 .OO

0

1990 IEEE

I

Source encoder I

Data ui

-

Convolutional wi,)ti 162

I

QPSK S ( t )

lnterleaver

modulator I

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 39, NO. 2, MAY 1990

I I I I

F

sink I I I I I I I I I I I I

I I

4

Viterbi

I

-yi ,hi

1

Deinterleaver w t d e c i s i o n p

I

decoder demodul at0 I I

U -1

-

I

L

_ _ _ _

- _ _ _

_ _ _ _

-

_ _ _ _ _

_ _ _

_ J

Fully interleaved discrete-time channel Fig. 1. Block diagram of the system under consideration.

transmitted signal s ( t ) can be expressed as

s ( t ) = Re {g(t)ej""}. (1) where WO is the carrier frequency, and g ( t ) is the complex

envelope given by

g ( t ) = d z C X i g o ( t

-iT,).

(2)

I

Here X i represents the interleaved version of x;, g o ( t ) is the

complex envelope of the transmitted signal with duration T , and unit energy, and E , represents signal energy per channel

symbol. Since each data bit corresponds to a channel symbol,

E , equals the signal energy per bit Eb.

Passing through the Rician channel, transmitted signal s( t) is corrupted by multipath fading resultant from the receiv- ing of a stable specular (direct) component and a random diffuse (multipath) component. This fading, imposes random amplitude a ( t ) and phase

4(t)

onto s ( t ) . Besides fading, the transmitted signal is also corrupted by additive white Gaussian noise (AWGN) n ( t ) with double-sided power spectral density

N o / 2 . The received signal r ( t ) hence can be described by

r ( t ) = Re {a(t)ei'#@)g(t)ejwo'}

+

n ( t ) (3) (4)

ye'+ is the specular component and c ( t ) is the diffuse com- ponent. The specular component is a complex quantity whose amplitude y is a fixed deterministic quantity while the phase

$ is uniformly distributed over [-a, a ] . The diffuse compo- nent can be modeled as individual quadrature components that are Gaussian with zero mean and common variance a 2 . The amplitude process a( t) then possesses the Rician distribution

= Re {(yej+

+

c(t))g(t)e'"'}

+

n(t).

where Io(

.

) is the modified Bessel function of the first kind of order zero. We will assume that the fading varies slowly

compared to the signaling rate so that the channel amplitude

a ( t ) is constant during one signal interval of duration T,. Moreover, for the convenience of analysis in the next section, we impose the following normalization

E [ a 2 ] = y2

+

2a2 = 1 (6) so that the received energy per channel symbol is E , and

represents the sum of the corresponding specular and diffuse energy. The Rice factor defined by { = y2/2a2 denotes the ratio of specular to diffuse energy. It is an important channel parameter in describing the channel. The distribution function for a ( t ) can also be completely characterized in terms of { as

f ( a > = 2 a ( l + O e x p {-(I

+

r ) a 2 - a ) 1 0 ( 2 a & F 2 3 ) .

(7)

The distribution f ( a ) is sufficiently general, since as the pa-

rameter [ approaches zero we have the Rayleigh channel, while if { approaches infinite the Rician channel reduces to the nonfading Gaussian channel (AWGN channel) with a = 1. The received signal is coherently demodulated by a soft de- cision, optimum demodulator under the assumption of perfect timing recovery and exact tracking of the carrier phase. The output of the deinterleaver, relevant to the transmitted x ; , is the normalized decision variable y ; given by

I

y ; =

{:

-six;

+

N ; .

Here { N , } is an independent identically distributed (iid) se- quence of complex Gaussian variates with zero mean and unit variance. {a;} is also an iid sequence of Rician variates with distribution specified by (5). The decision variable y ; is sup- plied to the Viterbi decoder. In addition t o y = { y i } the

Viterbi decoder with perfect amplitude measurement is also supplied with a = {a;}, the channel amplitude estimates, from a channel estimator. The Viterbi decoder performs ML decod- ing and recovers the transmitted data.

CHEN AND WEI: PERFORMANCE OF RATE 112 CONVOLUTIONAL CODES 163

111. PERFORMANCE BOUND

In determining the bit error probability performance bound of convolutional codes with Viterbi decoding, it is useful to recall the generating function union bounding approach [9]. Therefore, the pairwise error probability between the trans- mitted sequence and the estimated sequence must be first de- veloped. Moreover, for the convenience of deriving the ana- lytic expression of the pairwise error probability, we assume that the Viterbi decoder with perfect amplitude measurement is used. For such an ML Viterbi decoder, the decision rule be- tween possible transmitted sequencesx = {x;} andx’ = { x l }

given the received sequencesy = { y ; } and a = { a ; } is (de- rived in Appendix I)

2 = X I , i f C n ; y ; ( x ;

-xi>*

i

+a;yf(xi - x i )

<

O (9) where 2 denotes the estimated sequence of the decoder and the asterisk denotes the operation of complex conjugate.

We may assume, without lose of generality, that the se- quence x = {x;}, with x; = 1, relevant to the all-zero mes- sage is the transmitted sequence. For the estimated sequence

2 (which differs from the transmitted sequence x in exact k

symbol positions and of these k symbols there are kl symbols

with value * j and k2 symbols with value -l), the uncon-

ditional pairwise error probability P k l k 2 for such estimated

sequence is derived in Appendix 11. It is bounded by

where l ( k l , k2, i ) is the number of sequences with both er- ror pattern ( k l , k2) and i bit error. This information can be determined from the augmented generating function associ- ated with the particular code employed. To deduce the aug- mented generation functions such that it contains a ( k l , k2, i ) ,

the modified state diagram is used and must be properly la- beled. For expository purposes, rate 1/2, constraint length K = 3, convolutional code with Gray mapping into QPSK is given as an example. Figs. 2(a) and 2(b) show the encoder and the state diagram of the code. Fig. 2(c) gives the QPSK signal space. Modified state diagram of the code is illustrated in Fig. 2(d). The branches of this diagram are labeled as ei- ther

D O

= 1,

D

1 , or

D2,

if the relevant transmitted symbol

associated with the particular branch is 1, kj, or - 1, respec-

tively. If the branch transition was caused by an input data “1,” an additional factor Z is introduced to the branch. From this labeled state diagram, a set of state equations is obtained and the augmented generating function is derived. In general, for rate 1/2 convolutional codes with QPSK, the augmented generating function can always be expressed as

T ( D I , D ~ , I )

=CxT~(ki,

k 2 , i ) Z ‘ D f 1 D t 2 . (14)

Partial derivative of T(D1, D2, Z) with respect to Z at I = 1 becomes

i k , k z

where CO depends on the code and the channel used. Z1 and

2 2 are given by Substituting the result of ( 10) into ( 13) and comparing with

(15), we obtain the desired bit error probability upper bound

(1lb) For rate 1/2, k = three to eight optimum convolutional codes [lo], factor CO obtained in Appendix I1 is given below

where Q ( z ) = 1 / f i Jz” e - w 2 / 2 d w .

the bit error probability P b

.

We have

At this point, the union bound is then used to upperbound

p b

5

7;

i a ( k l , k2, i ) P k , k 2 i kl k i (13) 81 I = I , D I =Z I, Dz = Z ? . (16) d T ( D 1 7 D 2 9

7

p b <CO

Z I and 2 2 are given by ( l l a ) and ( l l b ) , respectively. For

constraint length K = three to eight codes, C O is given by (12).

IV. NUMERICAL AND SIMULATION RESULTS In evaluating the bit error probability performance we uti- lize the analytic bound (16). A computer program has been developed to compute numerically d T ( D 1 , D2, I)/aZ at Z = 1 and hence to determine the analytic bound for the selected convolutional codes.

Due to practical interest, only the rate 1/2, constraint length K = three to seven, optimum codes with Gray mapping into QPSK are evaluated. Figs. 3 and 4 show the computed upper bounds for channel with Rice factor ( = 0 and 10 (severe and medium fading) respectively. It is found that each increment

in K provides an improvement in performance and that the

performance improvement versus K increases with decreasing bit error probability. Typical behavior with various

!:

(from severe fading to nonfading case) is given in Figs. 5 and 6 for K = four and seven codes, respectively. Comparing with the

164 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. VOL. 39, NO. 2, MAY 1990 s 1 State eauations SI = DzI

SA

+ I S 2 5 2 = D l S l + DlS3 S3

=

D i I S i + D 1 I S 3 S:

=

D 2 S 2

Q Augmented generating function

w = ( 1 , l ) w = (0,O) x = - 1 X=*l

-?-

(c) x = - j

w = ( 1 , O )

_-

D l _{1 -2D1 I} 0 ; I

Fig. 2 . (a) Rate 112, K = three, convolutional encoder. ( b ) State diagram of the code. (c) Signal space of QPSK with Gray mapping. (d) Modified state diagram and augmented generating function of the code with Gray mapping into QPSK.

nonfading performance curves, we found that the performance degradation is large, medium, and small for { in the range of

1

<

5, 5

<

t

<

20, and 20

<

t,

respectively.

Tightness of the analytic bound and the influence of channel amplitude measurement on the decoder performance are fur-

ther investigated using Monte Carlo computer simulation. In the simulation, the Viterbi decoder is supplied with infinitely finely quantized receiver output and the decoding depth of the decoder is six times the code memory. Both the Viterbi de- coder with perfect amplitude measurement and that without amplitude measurement are used. The decoding rule for de- coder without amplitude measurement is similar to (9) except that the amplitude weighting factor a, is set to 1. Figs. 7 and

8 show the simulation results and the corresponding upper

bounds for K = four and seven codes with QPSK on chan- nels with

t

= 0 and 10, respectively. The following points are observed from the performance curves.

1) The computed upper bounds are in good agreement with the simulation results for Pb approaching lop4. The generat-

ing function bound is asymptotically tight and the performance can be ascertained accurately for low Pb even in the absence of simulation.

2 ) Viterbi decoder with perfect amplitude measurement

has better performance than that without amplitude measure- ment. About 1-2 dB relative performance improvement can be gained when fading is severe. For medium fading the rela- tive improvement is small, hence the Viterbi decoder without amplitude measurement is good enough in most cases and the generating function bound can be applied as well to evaluate the performance of convolutional codes with such decoder.

V . PERFORMANCE COMPARISON BETWEEN CONVOLUTIONAL CODES

WITH BPSK A N D QPsK

Bit error performance of rate 1/2 convolutional codes with BPSK modulation and ML Viterbi decoding which makes per- fect amplitude measurement has been studied 131. For such

decoder, the generating function bound on the Pb given in

10-2;

CHEN AND WEI: PERFORMANCE OF RATE 112 CONVOLUTIONAL CODES

K : constraint length % = 10

ris

2. c .-

Fig. 3. Computed upper bounds for selected rate 1/2 codes with QPSK on fully interleaved Rician channel with

r

= IO.

Fig. 4. Computed upper bounds for selected rate 1/2 codes with QPSK on fully interleaved Rician channel with { = 0.

165

Fig. 5. Computed upper bounds for rate 1/2, K = four, optimum code with QPSK on fully interleaved Rician channel with { = 0, 2 , 5 , 10, 20, x.

2

-& in dB

Computed upper bounds for rate 1/2, K = seven, optimum code with QPSK on fully interleaved Rician channels with { = 0, 2, 5, 10, 20,

, No

Fig. 6. x.

166 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 39, NO. 2, MAY 1990

-

Eb in dB No

Fig. 7. Comparison of computer upper bounds and simulation results for selected rate 112 codes with QPSK on fully interleaved Rician channel with {

= o .

1 O-'L

-

: analytic bound : no amplitude measurement measarement

-**-

: perfect amplitude \

-s

= 10 1

OPL

1 L 1 7

-

Eb in dB No

Fig. 8. Comparison of computer upper bounds and simulation results for selected rate 112 codes with QPSK on fully interleaved Rician channel with

= 10.

Fig. 9. Performance comparison (using upper bounds) between the selected rate 112 codes with QPSK and that with BPSK on fully interleaved Rician channel with = 0. [3] is corrected here as where 1 2Nol+{ 1f-- 1+-- 2Nol+{ 2NolSi- Z = Eb 1

[

Eb 1

J

* (19) Here d denotes the minimum free distance of the code em-

ployed. This analytic bound is also asymptotically tight. Per- formance comparison between the selected codes with QPSK modulation and that with BPSK are shown in Figs. 9 and 10 for channels with { = 0 and 10, respectively. Simulation results for K = four and seven codes with QPSK and that with BPSK are also provided in Fig. 1 1 and 12, respectively, for channel with { = 10. It is found that, for severe fading, the performance of rate 1/2 convolutional codes with QPSK is worse than that with BPSK. However, for medium fading, relative performance degradation is slight. Moreover, when code constraint length is increased, the relative degradation is improved.

CHEN A N D WEI: PERFORMANCE OF RATE 1/2 CONVOLUTIONAL CODES

le

in dB No

Fig. 10. Performance comparison (using upper bounds) between the se- lected rate 1/2 codes with QPSK and that with BPSK on fully interleaved Rician channel with { = 10.

4Q- : no amplitude measurement -0-+ : perfect amplitude measurement

-

Eb in dB No

Fig. 1 1 . Performance comparison (using simulation results) between K = four, rate 1/2, code with QPSK and that with BPSK on fully interleaved Rician channel with

<

= 10.

167

-

in dB

Fig. 12. Performance comparison (using simulation results) between K =

seven, rate 1/2, code with QPSK and that with BPSK on fully interleaved Rician channel with

r

= 10.

VI. CONCLUSION

In this paper, the bit error probability performance of rate 1/2 short constraint length optimum codes in conjunction with QPSK modulation and maximum likelihood Viterbi decod- ing on fully interleaved Rician fading channels has been stud- ied. By applying the generating function union bounding ap- proach, we have developed an analytic upper bound on the bit error probability performance under the assumption of us- ing Viterbi decoding which makes perfect channel amplitude measurement. Bit error probability performance of selected short constraint length codes was evaluated numerically using the developed bound. We also provided computer simulation results for examining the tightness of the developed bound and for observing the influence of perfect amplitude measurement on the performance of Viterbi decoder. Results indicate that the derived bound is asymptotically tight and the performance of Viterbi decoder without amplitude measurement is compa- rable to that with perfect measurement in most cases, except in the severe fading channels. In comparison with the same codes with BPSK modulation, rate 1/2 convolutional codes with QPSK modulation and Viterbi decoding can provide ef- fective error-control protection without sacrificing bandwidth efficiency and the relative performance degradation is slight when channel fading is not severe and code constraint length is sufficiently long.

APPENDIX I

For ML Viterbi decoder with perfect amplitude measure-

ment, the likelihood function is denoted by Pr (y

b,

a ) , where

168

y is the received sequence of decision variables, x is the trans- mitted sequence, and a is the measured fading amplitude se- quence. The ML decoder compares all the likelihood functions and decides in favor of the maximum. Hence the decision rule between transmitted sequences x = {xi} and x’ = {xi‘} given the received sequencesy = {yi} anda = {a;} is

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 39, NO. 2 . MAY 1990

by (23) we have

C a i y ; ( l

-a;)*

+ a ; y t ( l

- a ; )

k

i = l

where where x^ denotes the estimated sequence of the decoder. Be- cause of statistical independence, this is equivalent to

With y ; specified by (8), the conditional probability density function p ( y ; la;, x;) is given by

(22) For rate 1/2 convolutional codes with Gray mapping into QPSK, we have lxil = IX:~. Further simplification of (21) leads to the desired result

X^ = X I , i f C a;y;(x;

-xi>*

+

a;yf(x;

-xi> <

o

i

(23) where the asterisk denotes the operation of complex conjugate. This is the form of decision rule that will be used in Appendix I1 to derive the pairwise error probability. Another equivalent form is given below:

X^ = X I if

E

a;[Re _{( y ; )} Re (Xi)

i

where Re(#) and Im(#) represent the real and imaginary parts of the variable

#,

respectively. With a; = 1, (24)

becomes the one usually employed in the conventional ML Viterbi decoder (decoder without amplitude measurement).

APPENDIX I1

If the estimated sequence X^ differs from the transmitted sequence x = {x;}, with

x;

= 1, (corresponding to the all- zero message) in exact k symbol positions and there are k l

symbols with value & j and k2 symbols with value - 1, then

Here N,; and N,; are the quadrature com onents of Gaussian noise N;. Hence random variable U =

EFLl

MI

+C:=l

Mm is also Gaussian distributed with mean

vu

and variance U :

given by

vu

= ( e a ; + 2 ~ a ; m = l k 2

)

(27a)

No

The conditional pairwise error probability for the incorrect sequence with error pattern ( k l

,

k2) is just

The unconditional pairwise error probability can then be de- termined by averaging over the random variables of a with the result

(29) Because (29) is hard to evaluate, further manipulation is re- quired. Making use of the inequality Q( &)

<

1 /2 exp (-x/2)

for

x

2

0 in (29), we obtain

k k

where, by definition, q1 = a; and qm = a;. Since am is Rician distributed, q m is a noncentral chi-square-

CHEN AND WEI: PERFORMANCE OF RATE 1/2 CONVOLUTIONAL CODES 169 distributed random variable with characteristic function

(31) Recalling the normalization constraint (6) and the definition of parameter

r

= y 2 /2a2, (30) can be simplified as

where

1

’ 2 N o l + { j

(33b) When (32) is used for deriving the bit error probability per- formance bound, a loose bound will result. In order to obtain a tighter bound, all the error pattern associated with a convo- lutional code are further examined. A computer program has been developed for this purpose. For rate 1/2 codes with con- straint length K = three to eight, the minimum values of kl and k2 denoted by dl and d2, respectively, have been found and listed in Table I.

Making use of the inequality Q ( d m )

5

Q(&) exp (-y/2) for x , y

2

0 in (29), we obtain

TABLE I

MINIMUM VALUES OF k , AND k2 AND MINIMUM FREE DISTANCE OF RATE 112, K = THREE TO EIGHT CONVOLUTIONAL CODES Constraint Length ( K 1 Minimum Value of k , ( d i ) Minimum Value (d2) of k2 Minimum Free Distance ( 4 5 6 7 8 10 10

Hence the result given below is used instead.

Co

(35) Since q =

C$_I

a i is a noncentral chi-square-distributed random variable with 2d2 degree of freedom, distribution function of q is given by

C

(34) Applying (34) to derive the performance bound will result a tight bound. However, the evaluation of C is still a problem.

where I d ( . ) is the modified Bessel function of the first kind of order d . We then have

(37) The integration in (37) can be computed numerically.

REFERENCES

[I] F. Davarian, “Fade margin calculation for channels impaired by Rician fading,” IEEE Trans. Veh. Techno/., vol. VT-34, pp. 41-44, Feb. 1985.

J. W . Modestino and S. Y. Mui, “Convolutional code performance on the Rician fading channel,” IEEE Trans. Commun.. vol. COM-24, pp. 592-606, June 1976.

J. G. Dunham and K.-H. Tzou, “Performance bounds for convolu- tional codes on Rician fading channels,” in P m . I n t . Conf. Com- mun., 1981, vol. 1, pp. 12.4.1-12.4.5.

J. Hagenauer, “Viterbi decoding of convolutional codes for fading- and burst-channels,’’ in Proc. I n t . Zurich Seminar Dig. Commun.,

Zurich, 1980, pp. (3.2.1-(3.2.7.

R. Schweikert and J. Hagenauer, “Channel modeling and multipath compensation with forward error correction for small satellite ship earth station,” in Pm. Sixth I n t . Conf. Dig. Satell. Commun.,

Phoenix, AZ, 1983, pp. Xll-32-XII-38.

Y.-L. Chen and C.-H. Wei, “Performance bounds of rate 112 convolu- tional codes with QPSK on Rayleigh fading channel,” Electron. Lett., vol. 22, pp. 915-917, Aug. 1986. [2] [3] [4] [5] [6]

I

1

170 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 39, NO. 2, MAY 1990

[71 - , “Performance evaluation of convolutional codes with MPSK on Rician fading channels,” Inst. Elec. Eng. P m . , pt. F., vol. 134, no. 2, pp. 166-173, Apr. 1987.

A. Ghasis and P. Branch, “Future development of the INMARSAT system,” in Proc. 3rd Inst. Elec. Eng. Int. Conf. Satellite Syst. f o r Mobile Commun. and Navigation, 1983, pp. 207-211.

A. J. Viterbi, “Convolutional codes and their performance in com- munications,” IEEE Trans. Commun. Technol., vol. COM-19, pp. 751-771, Oct. 1971.

K. J. Larsen, “Short convolutional codes with maximum free distance for rate 1/2, 1/3, and 1/4,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 371-372, May 1973.

[8]

[9]

[lo]

Yuh-Long Chen (S’85-M’86) was born in Taichung, Taiwan, Republic of China, on November 14, 1950. He received the B.S.E.E. degree from National Chiao Tung University, Hsin-Chu, Taiwan, in 1974, the M.S.E.E. degree from National Tai- wan University, Taipei, Taiwan, in 1978, and the Ph.D. degree in electronics engineering from Na- tional Chiao ’hng University in 1987.

From 1974 to 1976 he served in Chinese Navy. Since July 1978 he has been with Chung Shan Insti- tute of Science and Technology, Lung-Tan, Taiwan.

His research interests include digital modulation techniques and channel cod- ing applications, in particular for mobile communication channels.

Che-Ho Wei (S’73-M’76-M’79-SM’87) was born in Taiwan, Republic of China, in 1946. He received the B S.E.E. and M S.E.E. degrees from National Chiao Tung University, Hsin-Chu, Taiwan in 1968 and 1970, respectively, and the Ph.D. degree from the University of Washington, Seattle, in 1976.

From 1976 to 1979 he was an Associate Profes- sor at National Chiao Tung University, where he is now a Professor of the Institute of Electronics. His research interest areas include active and digital fil- ter design, digital signal processing, and adaptive filtering.