Communication Theory

Performance Analysis of Trellis Coded Biorthogonal Sequences in a Variable Rate System

Nevio Benvenuto (*) Dipartimento di Elettronica e Informatica Universiti di Padova Via Gradenigo 6A, 35131 Padova - Italy Franco Chiaraluce Dipartimento di Elettronica ed Automatica Universith di Ancona Via Brecce Bianche, 60131 Ancona - Italy Paolo Falcioni (*) Aet hra Telecomunicazioni S . r. 1. Via Matteo Ricci 10, 60020 Palombina Nuova (Ancona) - Italy Sante Andreoli (**) Telettra Telecomunicazioni S.p.A. Via Erasmo Piaggio 71, 66013 Chieti Scalo (Chieti) - Italy

Abstract. For digital communication systems with a given binary modulation scheme and vari- able information rate, a coding technique is described which employs a particular structure of trellis code. It consists of a convolutional encoder followed by a mapper which selects bior- thogonal codewords. Based upon the state diagram of the system, performance is evaluated and then compared to that of more conventional solutions. Beside having good performance, this approach is very modular, i.e. its structure easily adapts to different information rates. Fur- thermore, it allows. the time synchronization system to work at lower signal-to-noise ratios.

1. INTRODUCTION

In some systems, where the modulation scheme is fmed, when the link error rate deteriorates, the problem arises of assuring both a satisfactory system perfor- mance (i.e. a low error rate) and a correct time syn- chronization. Typical examples, in this sense, are given by tactical radio-links and radio equipment in the VHF band [I] .

In this context, in order to reduce the bit error rate, a simple solution would be to double the average ener- gy per information-bit; for example by halving the in- formation rate and using a repetition code. If this would not lower the error rate sufficiently we could reiterate the procedure by further halving the informa- tion rate. The main disadvantage of this approach is that longer sequences of 0’s and 1’s are transmitted as the information rate is reduced. As a consequence, the power spectral density of the transmitted signal be- comes more concentrated at lower frequencies and the

(*) Formerly with the Dipartimento di Elettronica ed Automati- ca. Universith di Ancona, Via Brecce Bianche. 60131 Ancona - Italy.

(**) Currently working as an independent consultant.

scheme becomes less immune to selective fading. Moreover, the signal is prone to be easily detectable, i.e. it has characteristics of ((finger printing>>. Our objective is to synthesize a code which presents

a modular structure as the information rate is lowered. At the same time it should make the time synchroniza- tion system able to work at lower signal-to-noise ra- tios [2]. For simplicity in the implementation, the modulation scheme is a binary phase-shift keying (BPSK) or a minimum-shift keying (MSK).

In this paper a coding technique based on a trellis code is proposed. It consists of a convolutional encoder followed by a mapper in which the latter, instead of choosing a waveform in a L-ary symbol alphabet (L 1 4 ) [3-51, selects a binary sequence belonging to a set of biorthogonal words [6]. We have adopted this solution as we did not want to modify the signal space, remaining in the framework of a two level signaling.

Using biorthogonal words, simple correlation-type demodulators [6] can be employed in the time recov- ery system [7, 81. The trellis encoder, on the other hand, which in its elementary configuration exhibits a cod- ing rate of 2/3, is used to increase the mean distance

Vol. 3 , No. 3 May-June 1992 23 I

12 Nevio Bcnvenulo. Fmnco Chiamlucc, Paolo Falcioni, Sank Andreoli

r + + S + + - + - + + - - + - - + - + - +

+ + - + + -

- - _ _ A1 =

- - .I

between different codewords. In this way the system performance is greatly improved.

This scheme can be adapted to variable input infor- mation rates in the following simple way. Let RO denote the channel transmission rate. When the information rate R is half of Ro, i.e. R = Ro/2, the encoder is in- serted in its elementary configuration. However, when the information rate is further reduced to R = Ro/4, or lower rates, biorthogonal words of suitable length are considered.

An interesting characteristic of this solution is that the number of receiver-correlators is equal to the num- ber of pairs of antipodal words in the elementary con- figuration, independently of the information rate R. The only difference is in the integration period which becomes a multiple with respect to the case for R = Ro 12. Increasing the integration period determines also a significant lowering in the threshold value of the signal-to-noise ratio, below which a loss of synchroni- zation takes place [8].

The paper is organized as follows. In Section 2 a detailed description of the proposed system is given, while its performance analysis is carried out in Section 3. In Section 4 two different types of convolutional en- coders (both of rate 2 /3 ) are considered and their per- formance is compared to that of uncoded systems and to convolutional codes without mapping.

2. DESCRIPTION OF THE SYSTEM

The block diagram of the proposed coding system is shown in Fig. 1. It consists of a convolutional en- coder, with rate n /( n + 1 ), followed by a mapper. Let

Convolutional Encoder Mapper -=-El-- Rate n/(n+l)

Fig. 1 - General scheme of the proposed encoder.

Bp be a vector formed of n consecutive input information-bits at timepT( Tis the time for the trans- mission of n bits). C,, is the corresponding coded vec- tor formed of ( n + 1) bits. Moreover, for each input vector C, the mapper selects a biorthogonal sequence X,, called codeword, coincident with one of the rows of a matrix A.

The simplest convolutional encoder we have analysed is shown in Fig. 2 , where D represents a delay of T/ns.

c12) P

-I I

Fig. 2 - Description of the coding technique which includes a con- volutional encoder with rate 2/3 and 4 states.

It is characterized by a rate of 2 / 3 and 4 states (the basic convolutional code of rate 1 / 2 is taken from [9, p. 228, Fig. 6.11). Its state trellis will be shown later (Fig. 4). In this particular case the matrix A is given by

Hence the transmission alphabet consists of N = 8 codewords, each formed of m = 4 symbols. This yields a mapper with a coding rate of 3 /4. In general, matrix A has a dimension (N x m ) when R = Ro/2. However, when the information rate is reduced to R = Ro/2,, q = 2, 3, ..., matrix A is obtained by the matrix A at previous step, .as follows:

In (2) A, - 1 denotes a matrix obtained by scrambling the rows of A, - 1. The simplest choice would be to take A,- I = A,- I . However, this case suffers from some of the drawbacks mentioned in the introduction in regards to a repetition code. Note that A, has dimen- sion N x 2, - m.

Now, the mapper is based on the set partitioning technique suggested by Ungerboeck [31, which max- imizes the distance between all sequences of codewords (&.). The basic idea lies in subdividing the original set in a certain number of subsets, in such a way that the minimum code distance, within each subset, is greater than that of the original set.

The most suitable partition can be determined by considering the following definition of Euclidean dis- tance, d ~ , between two vectors of length m, x = (XI, ..., x m ) and y = (yl, . . . ,ym):

( 3 )

Since in this paper our attention is focused on words formed of binary symbols (+ 1, - 1). it is useful to write d~ in terms of the Hamming distance d H between x and y:

& = 4dH (4)

From (3) and (4), the normalized Euclidean distance (with respect to the energy of each word) is given by

The most appropriate partition of the rows of matrix A is shown in Fig. 3 where each subset is formed by two antipodal codewords. In this way = 2 within each subset, while L& = fi for two codewords belong- ing to two different subsets. The convolutional encoder employed for selecting one of the four subsets ,400, AOI, AIO, AtI is characterized by four states S = (00,01,

232 ETT

Performance Analysis of Trellis Coded Biorthogonal Sequences in a Variable Rate System 13

Step 0

p = P

+ + + + _ _ _ - + - + - - + - +

Step 1

OAl + + - - - - + + + - - + - + + -

+ + + + + - + - + + - - + - - + Step 2 _ _ - - - + - + - - + + - + + -

‘400 A10 01 11

Fig. 3 - Partition of the biorthogonal words which form matrix A.

10, 1 1 ) and has the trellis diagram plotted in Fig. 4. As- sociated to each branch of the trellis there is a pair (a, b ) where ‘a’ denotes the input symbol while ‘b’denotes the selected subset. For example, if at time pT the system is in the state 01, for an input symbol equal to 0 the selected subset is A11 and the following state, at

S

00

01

10

11

time ( p + 1 ) T, will be 00. S

00

01

10

11

Fig. 4 -Trellis diagram of the basic convolutional encoder in Fig. 2.

For each subset, the choice of which codeword is transmitted is left to the information-bit B$), which re- mains uncoded. Let us note that because of the uncoded bit we have parallel transitions in the trellis diagram [ 3 ] .

3. EVALUATION OF THE SYSTEM PERFORMANCE

It is well known that the distance of the minimum length error event, d,, approximately determines the code performance [lo]. From the coding system shown in Fig. 2, it is easy to see that d,,, = 2 due to the presence of parallel transitions.

In order to achieve a tight bound on the bit error rate, we use the property that the proposed coding sys- tem is linear. Furthermore, once defined the codeword weight as

N+ W(X,) = - m / 4

where N, is the number of + 1’s in X,, it is possible to prove that the code is also superlinear (see [l 1 1 for the definition of a superlinear code). We can derive the signal flow graph plotted in Fig. 5 . On the basis of the code superlinearity and its state diagram, the flow graph is characterized by the four states of the state

ND‘(l+Nl

N D‘

Fig. 5 - Signal flow graph for the evaluation of the transfer func- tion. Weights corresponding to ,each branch of the graph are also evidenced.

diagram and by the fact that each branch is labelled by a suitable weight. This is obtained by multiplying the dummy variable D”, where w is the codeword weight (6) , by N‘, c being B,’s Hamming weight. Note that the weight of each branch has been set equal to the sum of the weights of each of the parallel transi- tions.

The transfer function for the graph of Fig. 5 is given by [lo1

T(D,N) = - (N+ 1)* [ 2 d N ( N + 1 ) - 11 +

N[D6N2(N+ 1 ) + 2 d N ( N + 1 ) - I ]

(7) ( N + 1 ) 2

N + D 4 N -

Note that the same result could be obtained by fol-

Finally, based upon the derivative of (7), the bit er- lowing the Zehavi-Wolf’s formulation [12].

ror rate can be overbounded as follows [ l 1 1

exp ( -k- 2 Eb) aT(D,N) I (8 )

where N0/2 denotes the noise double-sided spectral density, Eb the energy per information-bit, and k the number of information-bits per codeword. The upper bound in (8) is used for Pb.

In order to evaluate how performance varies when the information rate is changed, it is useful to express (8) in terms of the ratio between the coded symbol ener- gy E, and NO. This is given by

No aN N-l.D-exp(-kEd4No

(9)

This in (8 ) shows that, for a given Pb, if we double the codeword lenght rn we have a 3 dB improvement in the ratio E,/No. This assures that for a channel with fiied capacity we can have a good quality transmission

Vol. 3, No. 3 May-June 1992 233

14 Nevio Benvenuto, Franco Chiaraluce, Paolo Faleioni, Santc Andreoli

even in the presence of a very noisy channel if we decrease the information rate sufficiently. In this way we can increase the codeword length rn. More impor- tantly, by increasing m we also have an improvement in the time synchronization system. As a matter of fact, it only needs to acquire the codeword synchronization and this can be done more effectively for longer codewords [7]. The advantage is particularly evident in the tracking mode where the gain, in the signal-to- noise ratio of using codewords of length rn with respect to using codewords of length one, is indeed equal to m [81.

4. COMPARATIVE PERFORMANCE ANALYSIS

On the basis of bound (8), we are going to compare performance of the proposed system for three differ- ent convolutional encoders, namely: a) convolutional encoder with rate 2/3 and 4 states,

P ) convolutional encoder with rate 2/3 and 8 states,

y) convolutional encoder with rate 2/3 and 8 states,

already described in Fig. 2;

described in Fig. 6 [9, p. 402, TabIe B-21;

reported in Fig. 7 [13, p. 14, Fig. 3al.

-I I

tb"' ?.

Fig. 6 - Description of the coding technique (3, which includes a con- volutional encoder with rate 2/3 and 8 states.

n P

Fig. 7 - Description of the coding technique y, which includes a con- volutional encoder with rate 2/3 and 8 states.

All three encoders are followed by a mapper which selects biorthogonal codewords of length rn as described previously. In all cases a BPSK modulation scheme has been used. It should be noted that in cases a and P parallel transitions are possible, so that d, = 2, while in case y it is seen that d,,, =

In Fig. 8 the upper bound (8) on bit error rate Pb is reported for the three coding systems above described. As a reference, the P b of an uncoded BPSK system (curve 6) is also reported. From this figure, at Pb = lo-' it is noticeable a 3 dB improvement of case a! with respect to the uncoded one. When the number of encoder states is increased to 8 (curve /3), a further

[lo].

I ,

'b

10-2

10-6

10-8

0 2 4 6 8 1 0

Eb/NO (dB)

Fig. 8 - Upper bound on Pb, as expressed by (8), for a convolutional encoder with: a) rate 2/3 and 4 states, 8) rate 2/3 and 8 states, and y) rate 213 and 8 states. All three encoders are followed by a map- per with rate 3/4. 6) BPSK uncoded modulation.

slight improvement is obtained. This is due to the cor- responding increase in the mean distance of the error events. A further improvement can be achieved by me- ans of code y because it eliminates parallel transitions.

For completeness, in Fig. 9 we compare curves P and y of Fig. 8 with the performance of a convolutional code with rate 1/2 and 8 states (the same convolution- al code of P but without the direct path) without the final mapper (curve 6). All codes in this figure are there-

1

'b

10-2

10-6

10-8

I

0 2 4 6 8 1 0

Eb/No (dB)

Fig. 9 - Upper bound on pb for a convolutional encoder with rate 1/2 and 8 states without the mapper (curve L). Curves 0 and y are the same reported in Fig. 8.

fore characterized by the same overall rate 1/2. It is possible to note that performance of system c is about 1 dB better than that of system /3 but only 0.3 dB bet- ter than that of system y (comparison is made at Pb = lo-'). SO we can say that insertion of the map- per, at the expense of a slight performance degrada- tion, makes a simpler realization possible both of the demodulation and the synchronization recovery system.

Finally, we have removed the convolutional encoder and analysed performance of both a biorthogonal and orthogonal code (141 with rate 1/2. To this purpose, codewords of length m = 4, in the biorthogonal case, and m = 2, in the orthogonal one, have been assumed.

ETT 234

Performance Analysis of Trellis Coded Biorthogonal Sequences in a Variable Rate System I5

Results are reported in Fig. 10. Note that, as expect- ed, when the convolutional encoder is absent we have a penality of 1.5 dB in the biorthogonal case and more than 3.5 dB in the orthogonal case. This confirms how the best results are obtained by combining the positive effects of a biorthogonal code together with those of a convolutional code.

1 I

'b

10-2

10-6

10-8

0 2 4 6 8 1 0

Eb/No (dB)

Fig. 10 -Bit error rate for a: A) orthogonal ad p) biorthogonal code, both with rate 1/2. Curve y is the same reported in Fig. 8.

5 . CONCLUSIONS

A coding scheme which employs a set of bior- thogonal words in the framework of a trellis code has been analysed. This scheme allows us to obtain signifi- cant improvements with respect to both uncoded sys- tems and systems without the convolutional encoder. Furthermore, time synchronization is made more robust by the adoption of biorthogonal waveforms.

When the useful information rate decreases, the pro- posed system is able to exploit the available channel capacity, without increasing the encoder or decoder complexity. Obviously, by making the convolutiona1

code more sophisticated, e.g. by increasing the num- ber of encoder states, a further improvement in per- formance can be obtained.

Acknowledgment

The authors wish to thank G. Guidotti of Elasis for his valuable comments, and the referees for their care- ful revision.

Manuscript received on November 8, 1990

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