IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 44, NO. 2, MAY 1995 29 1

A Comparison of Two Combining Techniques for Equal Gain, Trellis Coded Diversity Receivers

Lars K. Rasmussen and Stephen B. Wicker, Senior Member, IEEE

Abstract-Thllis coded modulation is widely used for digital transmission over fading channels. Classical diversity techniques are also frequently employed to combat fading. In this paper two different strategies for equal gain combining are compared. One scheme is based on an interleaved code combining technique. The alternative scheme is based on averaged diversity combining. The well known transfer function bounding technique for trellis codes is used to obtain expressions for the bit error rate performance of the two treiiis coded diversity receivers over a slowly fading Rayleigh channel. The analysis of interleaved code combining is a straightforward modification of the analysis for multiple trellis coded modulation. The analysis of averaged diversity combining is accommodated through a more involved, novel modification of the branch labeling of the error state diagram. The analytic techniques presented in this paper are supported by simulation results using a TCM scheme based on QFSK modulation and a rate-Y2 linear convolution code.

I. INTRODUCTION

RELLIS coded modulation (TCM) has proven to be T well suited for digital transmission over slowly fading channels [l]. Upper bounds on error event probability and bit error rate have been developed using techniques originally developed for the analysis of convolutional codes [2]. The bounds are based on the generating function for the code, suitably modified for the slowly fading channel [3].

Diversity techniques are commonly used to improve the performance of communication systems that encounter severe channel noise and/or distortion [4]. For example, they have been profitably applied in mobile ratio networks that suffer from fading and multipath interference. Numerous methods for implementing diversity systems have been proposed in the literature. In the past these systems were generally based upon frequency, antenna, or time diversity [5], though recently code diversity has attracted the most attention [6]. One trait common to all of the diversity schemes is that the same message is transmitted over M 2 2 different effective channels. The signals received from the hl channels are combined in the receiver to create a more reliable copy of the transmitted message. There are essentially three classes of combining strategies: selection diversity combining, equal gain combin- ing, and maximum ratio combining [5 ] . This paper considers two equal gain combining techniques called interleaved code

Manuscript received January 31, 1994; revised June 7, 1994. This work

L. K. Rasmussen is with Mobile Communication Research Centre, Univer-

S. Wicker is with the School of Electrical and Computer Engineering,

IEEE Log Number 9408535.

was supported by National Science Foundation Grant NCR-9016275.

sity of South Australia, The Levels. South Australia 5095, Australia.

Georgia Institute of Technology, Atlanta. GA 30332 USA.

4

Fig. 1. General structure of a trellis coded diversity communication system.

combining (ICC) [7] and averaged diversity combining (ADC) [6]. The block diagram for a general trellis coded diversity communication system is shown in Fig. 1.

The paper is organized as follows. In Section 11 the channel model and the two combining techniques are described. Per- formance expressions are derived in Section III and in Section IV an example is provided to illustrate the conclusions.

11. SYSTEM DESCRIPTION

This paper focuses on techniques designed to enhance communications over the frequency-flat Rayleigh fading chan- nel. The fading channel discussed here is simplified through the assumption that shadowing effects and path losses are negligible. Furthermore, the channel is assumed to be perfectly co-phased while the receiver is provided with ideal channel state information. The receiver is assumed to perform ideal coherent detection and to use the Viterbi algorithm for trellis decoding. The channel is slowly fading (Le., the channel amplitude is assumed to be constant over one full signal interval), and as in Divsalar and Simon [3], ideal channel symbol interleaving is assumed.

For M-branch diversity, the received signal on the ith branch is

T i = p ; z + n; (1)

where z is the transmitted signal, n; is the additive white Gaussian noise in the ith branch, and p ; is the fading sample in the ith branch. The fading samples are assumed to be statistically uncorrelated from branch to branch, and Rayleigh distributed.

Interleaved code combining was originally proposed by Chase for use in packet retransmission protocols [7]. The same approach can be applied to diversity reception. The M received diversity signals are regarded as an M-dimensional signal. For a TCM system the corresponding decoding trellis is then modified accordingly. Each trellis transition corresponds to an M-dimensional signal with M identical coordinates. Since the individual signals that comprise each dimension are

0018-9545/95$04.00 0 1995 IEEE

292 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 44, NO. 2, MAY 1995

not weighted as a function of the branch from which they are taken, interleaved code combining, as used here, is a form of equal gain diversity combining. It can be viewed as either a straightforward application of a simple repetition code, or equivalently, as a special case of multiple TCM (MTCM) [8]. The relevant decoding metric (DM) for the ICC decoding trellis is the same as the decoding metric for MTCM.

A1

DMICC = 1 IJrt - pt2I l2 . (2) r=l

The results derived for MTCM codes apply directly to the case of trellis coded ICC. In this case all symbols attached to a transition branch are identical.

Averaged diversity combining was also originally proposed for use in packet retransmission protocols [6]. In an averaged diversity receiver, the M diversity signals are averaged on a symbol by symbol basis to provide a single, more reliable signal of the same dmensions as the originally transmitted signal. All of the additional signal processing required by the combining operation is thus performed prior to decoding. There is thus no need to modify the decoder itself. The analysis of the system, however, is not as straightforward as for the ICC system. The decoding metric is defined as follows:

(3) DMADC = l/li 1 c (r t - ~1”)1[.

1 = 1

In the analysis of averaged diversity receivers for Rayleigh fading channels, the problem of finding the probability dis- tribution function (pdf) of a sum of Rayleigh distributed random variables is encountered. It is not possible to find a general closed form expression for the resulting pdf. An exact solution has been found for the case of two independent Rayleigh variables by Altman and Sichak [9] and Halpern [lo]. Mason et al. [ l 11 have tabulated the distribution function for a sum of Rayleigh variables with eight or fewer summands. These results were obtained through numerical integration of the convolution formula for the distribution of a sum of independent random variables. Recently Beaulieu [ 121 has devised an infinite series technique to compute the pdf for the sum of independent Rayleigh variables. A very good approximation can be achieved by truncating this infinite series approach. The method has been successfully applied by Beaulieu and Abu-Dayya [13] and by F’ratt and Ingram [14]. However, the technique is computationally intensive, so other approximation techniques have been pursued in the literature. Jakes [5] has made use of a “small argument” approximation for the pdf suggested by Schwartz et al. [4] and by Stein and Jones [15]. Beaulieu [12] has compared the infinite series approach and the small argument approximation using graphical techniques and has found that the approximation is fairly accurate for small values (M 5 8 ) of the argument but inaccurate for medium and large values.

Given the simplified analysis achieved through the small argument approximation, this technique is an attractive alter- native for the analysis of trellis codes used in ADC systems. In this paper the small argument approximation is used to directly modify the branch labeling of the error state diagram.

The standard upper bounding technique for trellis codes is then applied and the result averaged over the approximate pdf of the fading distribution. An approximation for the bit error rate performance based on the upper bounding technique is thereby obtained. The new labeling is developed in the following section.

It can be shown that interleaved code combining and aver- aged diversity combining are conceptually identical (and thus have identical performance) over an additive white Gaussian noise channel when using the Viterbi decoding algorithm. This is seen by comparing the path metrics calculated by the Viterbi decoder for each of the combining methods [ 161. However, this is not true in general. For a slowly fading channel, a straightforward comparison of the decoding path metrics shows that identical performance cannot be expected. For nonlinear channels it has been shown that ICC systems offer better reliability performance than ADC systems [7]. ADC systems, on the other hand, are much easier to implement because they do not require any additional modifications of the decoder. Only the addition of a simple averaging circuit is required. For the ICC system the decoder must be able to handle the M-dimensional signal. An increase in complexity is thus encountered. In the following section performance expressions are derived to support a formal performance comparison of the two combining schemes.

111. PERFORMANCE ANALYSIS FOR COMBINING RECEIVERS

The bit error rate performance of a TCM system over a slowly fading Rayleigh channel can be bounded through the use of the generating function T ( D : I) for the trellis code. The generating function is obtained by regarding the error state diagram of the code as a signal flow graph. The following bound on bit error rate is the result:

where (m-1) is the number of input bits, I and D are counting variables, T ( D 1 I) the transfer function averaged over the fading distribution, Eb is the averaged bit energy and No is the one-sided spectral density of the additive white Gaussian noise. The averaging of the transfer function can be done on the branch level. For the case of a slowly fading Rayleigh channel the branches of the flow graph are labeled with

where, in case of parallel branches, a summation is taken over all output signal sequences assigned to the branch, R is the Hamming distance between the corresponding input sequences, k is the number of modulation signals assigned to each output sequence, 6: is the squared Euclidean distance between output symbols, and the overbar in Dpfst signifies the expected value with respect to the fading distribution. DP:s: can be interpreted as the pairwise error event probability for a specific trellis branch. For slowly fading Rayleigh channels

RASMUSSEN AND WICKER: A COMPARISON OF TWO COMBINING TECHNIQUES FOR EQUAL GAIN, TRELLIS CODED DIVERSITY RECEIVERS 293

the following is obtained

For the ICC case the branch labels are also assigned according to (5). In this case k = M and all of the symbols are identical. The branch labeling for the error state diagram is thus found to be

In the analysis of the ADC system the approximation of the sum of Rayleigh random variables developed by Schwartz et al. [4] is considered. Applying the small argument approxima- tion [ 151 to the pdf of the sum of Rayleigh distributed random variables, a generalized branch labeling of the error state dia- gram for the TCM code is developed. The pdf approximation is described by

where e is the random fading variable, (.)! denotes the factorial operation, the p is defined to be

(9) 1

2M p = - [(2M - 1)!!]1/hf

where (2M-1)!!= (2M- 1) . (2M-3) . . . 3 .1 .Toder ive the approximation, the branch labeling for the transfer function must be modified to accommodate the generalized pdf of (8). The form of the labeling is identical to (5) . However, DP:'? is modified to obtain the generalized labeling.

To evaluate the upper bound in (4) for the nondiversity case and the ICC case, the dummy variable D is set equal to the Bhattacharyya parameter [ 171

D = e x p (-%). In the averaged diversity case the signal-to-noise ratio (SNR) affecting the transmitted signals is increased by each additional diversity branch. The following is thus substituted for D:

The averaging over the fading distribution then proceeds as follows:

Using the equality

Zn . exp ( - -ax2> dz = Za(%+1)/2 (13)

where I?(.) is the gamma function

t(n-l) . exp (4) d t = (n - l)! (14)

the following is obtained:

Applying the definitions for p and the gamma function, the derivation is completed:

and

(17) [(2M - 1)!!]1/M 5 1. M

K =

Using this labeling in the evaluation of the transfer function bound for the bit error rate provides an approximation to the performance for the M-order averaged diversity receiver based on the upper bounding technique. It has been shown that the approximation to the pdf is fairly accurate for small arguments (M 5 8). Usually the number of diversity branches satisfy this constraint so with some considerations the performance approximation can be treated as an upper bound on the bit error rate. Simulation results to support this claim are presented in the following section.

Comparing (7) and (16) it is obvious that the ICC system has a better BER performance than the ADC system. For large SNR's the pairwise emor event probability for a single trellis transition is improved by a factor of

(iL)M= (2M - l)!! .

The improvement factor is monotonically increasing for an in- creasing number of diversity branches. However, the decoding complexity is increasing dramatically for the ICC system. Let XU be the total number of metrics calculations. Then

(19)

For each additional diversity branch another 2" . 2m-1 decod- ing metrics per trellis stage must be calculated (assuming a 2"-state, m - 1 input code). In the ADC scheme only 2" .2"-l metrics are calculated per trellis stage regardless of the order of diversity. The additional complexity is in this case limited

PICC = M .2" .2"-l.

294 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 44. NO. 2, MAY 1995

2 4 6 8 10 I2 14 16

Eb/No (dB)

Fig. 2. Simulation results and the performance bounds on BER for ICC and ADC systems with diversity factor, of 2, 4, and 6. For reference the results for the nondiversity case are included.

to a pre-decoding averaging circuit. There is thus a significant off-set in the trade-off between decoding complexity and per- formance improvement for the ICC scheme. In the following section a simple example is provided to illustrate the difference in performance between the two systems.

IV. EXAMPLE Consider a rate 1/2, 2-state convolutional encoder used in

conjunction with a QPSK signal set as described by Biglieri et al. [17]. This simple TCM system has an overall throughput of 1 b/s/Hz and a minimum free Euclidean distance of &. Fig. 2 shows upper bounds and simulation results for the BER provided by ICC and ADC systems with diversity factors 2,4, and 6. For reference the upper bound and simulation results for a nondiversity system (M = 1) are included. There is a close correspondence between the simulations and the analytically derived expressions. The claim that the BER approximation based on (16) can be regarded as an upper bound for small values of M is supported. The performance improvement provided by the ICC scheme is readily observed. The per- formance gains are on the order of 1 dB over a wide range of SNRs, increasing for an increasing number of diversity branches. The corresponding number of metric calculations is 4 per trellis stage for the ADC system in this example. For the ICC schemes the number of metrics calculations per trellis stage is 4M. A significant increase in decoding complexity is encountered by the ICC system as the diversity order increases. For M 2 6 the performance improvement is heavily off-set by the corresponding decoding complexity increase. For M < 6 the performance improvement is minimal. The ADC scheme is thus the preferred approach for most practical systems.

V. CONCLUSION

Two equal gain combining strategies, interleaved code com- bining and averaged diversity combining, have been compared for use in trellis coded systems. The performance analysis of the ICC system was recognized as a special case of multiple TCM coding. The corresponding analysis of ADC systems was based on a similar framework. A modification of the branch labeling of the error state diagram for TCM codes was suggested to accommodate the performance evaluation of trellis coded averaged diversity receivers over slowly fading

Rayleigh channels. For a relatively small number of diversity branches, reliable upper bounds on the BER were obtained. These bounds showed that an ICC scheme can be expected to outperform a comparable ADC scheme. However, a simple example showed that the performance improvement is minimal compared to the additional complexity required to support the ICC scheme. For practical solutions the ADC scheme provides comparable performance at a much lower complexity level.

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[2] A. J. Viterbi and J. K. Omura, Principles of Digital Communication und Coding. New York McGraw-Hill, 1979.

[3] D. Divsalar and M. K. Simon. “The design of trellis coded MPSK for fading channels: Performance criteria.” IEEE Trans. Commun., vol. 36, no. 9, pp. 1004-1012, Sept. 1988.

[4] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York McGraw-Hill, 1966.

[5] W. C. Jakes, Microwave Mobile Communications., 1st ed. New York: Wiley, 1974.

[6] L. K. Rasmussen and S. B. Wicker. “Code combining systems based on trellis coded, type-I hybrid-ARQ protocols over AWGN channels and slowly fading channels,” IEEE Trans. Commun., to be published.

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[8] D. Divsalar and M. K. Simon, “Multiple trellis coded modulation (MTCM).” IEEE Trans. Commun.. vol. 36, no. 4, pp. 4 1 M 1 9 , Apr. 1988.

[9] F. J. Altman and W. Sichak, “A simplified diversity communication system for beyond-the-horizontal links,” IRE Trans. Commun. Systems, vol. CS-4, no. 2, pp. 50-55, Mar. 1956.

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[I21 N. C. Beaulieu, “An infinite series for the computation of the com- plementary probability distribution function of a sum of independent random variables and its application to the sum of Rayleigh random variables,” IEEE Trans. Commun.. vol. 38, no. 9, pp. 1463-1474, Sept. 1990.

[I31 N. C. Beaulieu and A. A. Abu-Dayya, “Analysis of equal gain diversity on Nakagami fading channels,” IEEE Trans. Commun., vol. 39. no. 2, pp. 225-234, Feb. 1991.

[14] T. G. Pratt and M. A. Ingram, “Optimal receiver thresholds in a phase and polarization diversity receiver,” J. Lightwave Technol., to be published.

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[16] B. A. Harvey and S. B. Wicker, “Packet combining systems based on the Viterbi decoder,” IEEE Trans. Commun., vol. 42. no. 2-4, pt. 3, pp. 1544-1547, Feb.-Apr. 1994.

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New York McGraw-Hill, 1967.

Lars K. Rasmussen was born in Charlottenlund, Denmark, on March 8, 1965. He received the com- bined B.S. and M.S. degree in electrical engineering in 1989 from the Technical University of Denmark. In 1993 he received the Ph.D. degree in electrical engineering from Georgia Institute of Technology.

Since November 1993 he has been with the Mobile Communication Research Centre at the Uni- versity of South Australia as a Research Fellow. His research interests include mobile data transmission, retransmission enor control protocols, and multiuser CDMA systems for mobile cellular communication.

U S M U S S E N AND WICKER: A COMPARISON OF TWO COMBINING TECHNIQUES FOR EQUAL GAIN. TRELLIS CODED DIVERSITY RECEIVERS

Stephen B. Wicker (S’83-M’83-SM’93) was born in Hazelhurst, MI, on September 25, 1960. He received the B.S.E.E. with High Honors from the University of Virginia in 1982. He received the M.S.E.E. from Purdue University in 1983 and the Ph.D. degree in electrical engineering from the University of Southern California in 1987.

From 1983 through 1987 he was a subsystem and system engineer with the Space and Commu- nications Group of the Hughes Aircraft Company, El Segundo, CA. In September 1987 he joined the

faculty of the School of Electrical Engineering at the Georgia Institute of Technology, where he currently holds the title of Associate Professor. He was named a Visiting Fellow of the British Columbia Advanced Systems Institute in 1992. He has also served as a consultant in telecommunication systems, error control coding, and cryptography for various companies in North America, Europe, and West Asia. His current research interests center on the development of algorithms for emor control, data compression, and data security for digital communication systems. He is the author of Error Control Systems for Digital Communication and Storage (Englwood Cliffs, NJ: Rentice-Hall, 1995) and is co-editor of Reed-Solomon Codes and Their Applications (IEEE Press, 1994). Dr Wicker is the Editor for Coding Theory and Techniques for the IEEE TRANSACTIONS ON COMMUNICATIONS. Dr. Wicker is a member of Eta Kappa Nu, Tau Beta Pi, Sigma Xi, and

Omicron Delta Kappa.

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