NOVEL 6-PSK TRELLIS CODES

NOVEL 6-PSK TRELLIS CODES Gerhard Fet tweis Teknekron Communications Systems, 2121 Allston Way, Berkeley, CA 94704, USA phone: (510)649-3576, fax: (510)848-885 1, fet t weis@ t cs.com Abstract The use of trellis coded modulation leads to the requirement of a 2-fold increase in the channel alphabet size, e.g. from 4PSK to 8-PSK. It is shown that multi-dimensional 6-PSK trellis codes can be designed which achieve the same spectral efficiency at a reduced channel symbol alphabet size. This allows for easier phase synchronization over &PSK. Examples of rotationally invariant as well as non-invariant codes are presented. 1 Introduction Trellis coded modulation (TCM as introduced by Ungerboeck [l is a well established met b od for combining coding and mo d ulation in one step to achieve high coding gains without changing the bandwidth of the signal. To date, many different trellis codes have been developed for phase shift keying (PSK) modulation as well as quadrature amplitude modultaion (QAM). TCM is applied in many practical systems. For an overview on TCM see [2, 31. The main idea of TCM is to increase the alphabet size of the channel symbols and to use this additional redundancy to trellis code the signal. The typical way of introducing the additional redundancy for coding is achieved by doubling the symbol alphabet size from M to 2M, e.g. from 4PSK to 8-PSK. One important feature of all TCM codes is that they require at least a doublin of the symbol alphabet size, due to the set-partitioning metfod that is used in their derivation. If the channel symbols are grouped into sets of N symbols each, this allows for the development of higher rate codes or features as rotational invariance [4, 51. However, grouping N symbols can also be used to decrease the channel symbol alphabet size. For example for N = 2 the uncoded channel alphabet size is MN = M2. In this case, doubling the channel alphabet size leads to (2M)2 = 4M, a four-fold increase in the symbol alphabet size. However, the minimum required doubling of the symbol alphabet size can be achieved by increasing the channel alphabet e.g. 1.5 times, since (1.5M)2 = 2.25M. The purpose of this paper is to show that a smaller than 2-fold increase in channel alphabet size can be used for the design of practical codes, in particular for 6-PSK modulation, by applying the classical set-partitioning method. It furthermore shall be pointed out that when the symbol alphabet size is not doubled, TCM codes can still have interesting coding gains. The discussion in this paper concentrates on codes with 2bits/symbol only. 2 6-PSK Modulation It is certainly unconventional to use 6-PSK modulated transmission. Hence, this subsection should provide some insight into advantageous features of 6-PSK, which makes it an interesting candidate for the derivation of new 0-7803-09~-2/93/$3.W@1993IEEE 2.1 Phase Noise The uncoded transmission of 2 bits per channel symbol with PSK modulation leads to 4PSK. Therefore, PSK TCM codes for 2-bits/symbol conventionally employ &PSK. Before decoding a TCM code at least two synchronization steps have to be completed, i.e. phase and timin recovery. The decoding delay of TCM with the Viterbi Jgorithm makes it difficult to use decision aided synchronization algorithms. Therefore non-decision aided algorithms are the prime candidates (see e.g. [SI). The timing synchronization performance is hardly affected by the denser constellation of 8-PSK. However, for non decision aided phase synchronization the &PSK constellation translates into a drop of signal-to-noise ratio (S- NR) of 5.3dB versus 4PSK. In case of 6-PSK modulation, see Fig. 1, the SNR drops only by 3dB, a gain of 2.3 db over 8-PSK. This can be of special importance for fading channel conditions. Figure 1: Channel symbol constellation of 6-PSK. 2.2 Demodulation and Quantization The baseband processing of the complex valued transmit signal requires that the signal is decomposed/demodulated into 2 basis vector components. The conventional method of doing so is by decomposition into two orthogonal components, the I-signal (in phase) and Q-signal (quadrature phase), named x- and y-axis in Fig. 1, respectively. For decoding TCM encoded signals in the Viterbi decoder, branch metrics must be formed. These branch metrics are computed as the inner product of the vector of the branch symbol with the received symbol. Due to the constellation of 8-PSK1 this requires the multiplication with 1/& It is evident that this multiplication typically is costly in hardware, since 1/4 needs to be approximated by 12/17 to achieve 2% error. This coefficient translates into a word-length increase by 5 bits.' 'In case the &PSI< constellation is rotated by u/8, this leads to an equivalent situation where it requires the multiplication by tan(u/8) =

~ ~~~ In the case of 6-PSK transmission the irrational coeficient for multiplication is &/2, see Fig. 1. The rational coefficient which achieves 1% error is 4/7, which leads to only 3 bits additional word-length. Also note that 4 is a simple shift operation. The in-phase and quadrature phase signals are denoted as x-axis and y-axis in Fig. 1. Another way of decomposing 6- PSK signals can be carried out by non-orthogonal decomposition into two vectors that have a?r f 3 phase offset, denoted as x- and z-axis in Fig. 1. In this case only {0,1/2,1} occur as coefficients for computing inner products. That enables an extremely simple branch metric computation with only 1 bit word-length extension and no coefficient quantization error. It is clear that a 60 degree (1/3) decomposition is quite unconventional, however, simple phase and timing recovery algorithms can easily be obtained from standard approaches. 3 Code Construction In this section the code construction is presented. It is to be noticed that the codes which were designed have been derived by systematic reasoning only, and not by an extensive search. 3.1 Set Partitioning Since we here consider codes with 2-bits/symbol only, we consider a symbol alphabet with elements of two consecutive channel symbols, which leads to the 4-dimensional set of 36 elements of 6 x 6-PSK symbols. The 6-PSK symbols are named 0-5 according to Fig. 1. They are divided into the following subsets Figure 2: The set-partitioning of the 2 x 6-PSK symbol alphabet. The way the butterflies were constructed is to provide e- nough transitions within each butterfly for 8 transitions per state. This leads to a trellis diagram construction with only 2 arriving/departing butterflies per trellis node. Furthermore, each butterfly as a trellis diagram on its own has an internal squared free distance of 6. The 4 butterflies {A, B, C, D} that were constructed according to these criteria can be found in Fig. 3. 6x6-PSK sets I- 13 35 51 11 33 A 12 34 C 13 35 51 14 30 52 1531 53 10 54 32 C2 10,32,54 0 2 21,43,05 These sub-sets can be arranged on a set-partitioning tree as shown in Fig. 2. Each subset has an internal minimum squared euclidean free distance of 6. Hence, for trellis codes with a squared free distance of up to 6, this allows to label branches in the trellis code with a sub-set, i.e. each branch has 3 possible parallel transitions. 3.2 Butterfly Construction To design a code with 2 bits per channel symbol, for 6 x 6- PSK this yields 4 bits per code symbol. Each state of the trellis diagram therefore has 16 incoming and outgoing transitions. Since the subsets each comprise 3 elements, and the branches are labelled with one sub-set each, 6 branches need to be entering and leaving each state. The 6 branches allow for 18 transitions. It therefore is necessary to prune 2 transitions from one branch, or 1 transition from 2 branches. The basic idea of how to construct codes is the same as was applied in [7]. The code symbols are arranged into butterflies, which then are used as %uper-branches in a conventional shuffle-exchange trellis. Each node of the shuffle-exchange trellis therefore consists of a number of states determined by the butterflies. 0.4142. 31 53 15 33 55 3551 13 32 54 10 34 50 30 52 14 51 1335 52 14 30 53 15 31 54 10 32 55 11 50 12 22 44 B 23 45 D 24 40 02 20 42 04 42 04 20 44 00 40 02 24 2541 03 21 4305 43 05 21 45 01 41 0325 02 24 40 03 25 41 04 20 42 0521 43 00 22 01 23 107 Figure 3: The four 3-state butterflies A, B, C, and D.

3.3 Trellis Codes The butterflies A, B, C, D} can be used as super-branches in a conventional s 6 uffle-exchange trellis to design new 6 x 6- PSK trellis In case they are used in the 4node shuffleexchange graph of Fig. 4, this leads to a 12-state code referred to as tcl2 with squared free distance of 5 (dzree(tc12) = 5). Figure 6: The rotationally invariant 12-state 6-PSK trellis code. Figure 4: The 12-state 6-PSK trellis code. By using the butterflies in an 8-node shuffle-exchange graph of Fig. 5, a 24state trellis code tc24 is obtained with dzree.(tc24) = 6. Since this is also the minimum free distance within the butterflies as well as on the branches, new butterflies need to be constructed to obtain higher coding gains. Figure 7: The rotationally invariant 24state 6-PSK trellis code. A squared free distance of 5 can be achieved e.g. by using the graph of the 6-state code given in [8], which would result in an 18-state 6-PSK code. It is to be noticed that the same twisting of butterflies needs to be carried out, as described in [I for the rotational invariant butterfly 8-PSK The twisted butterflies are marked with primes. Figure 5: The 24state 6-PSK trellis code. 3.4 Rotational Invariance The codes tc12 and tc24 are n x 120 rotationally invariant. However, to obtain ra x 60 rotational invariance further changes are necessary. One way of achieving this is to change the labeling of the super-branches of the shuffleexchange graph from tc12 to the one presented in Fig. 6, named tcl2r. It can be seen that the free distance now drops to dzree(tcl2r) = 4. In case of tc24, the shuffle-exchange graph of Fig. 7 can be used to obtain rotational invariance. It this case the free distance remains unchanged at diree(tc24r) = 6. 4 Discussion Simulation results of the code performance can be found in Fig. 8 for an additive white gaussian can be seen that the performance of tc12 an T= tc24 differ only It by 0.2dB at a bit-error-rate (BER of This is due to the fact that tc24 has a large num b er of parallel paths with the minimum free distance. The rotational invariant versions have a larger performance difference because of the larger difference of dirh. Their performance is clearly degraded due to the differential encodin /decoding on the symbol-basis which needs to be carried out for rotational invariance [SI. The measured performance gains at a BER of are summarized in the following table..08

channel behaviour, phase and timing recovery, as well as the Viterbi decoding. BER 4.1 Ring-Convolutional Codes It shall be pointed out that 6-PSK coded modulation can very well be designed into ring convolutional In this case the traditional binary tree of set-partitioning does not have to hold anymore. The trellis diagram of a simple 2-state 6-PSK trellis code shown in Fig. 9 shows that no classical setpartitioning needs to be applied. To allow for 2 bits/symbol 51 Figure 8: The code performances for an additive white gaussian noise (AWGN) channel (BER: bit error rate). U Figure 9: Two-state &PSK trellis code; a) trellis, b) encoder. code @ simulated BER of asympt. coding gain tc tc;: 23: ddbb : :BB tcl2r 1.8 db 3.0 db tc24r 2.1 db 4.8 db Note that the butterfies are almost catastrophic codes, i.e. their three states only differ by the pruning of one branch. Simulations showed that the survivor depth of Viterbi decoding therefore is about a factor 3 times larger than for conventional When comparing the implementation complexity of these codes with 8-PSK trellis codes it has to be taken into account that the complexity is dictated not only by the number of states, but also by the word-length which is required for the implementation. VLSI design experience2 shows that the gain in reducing the word-length by the 6-PSK modulation with 60 decomposition amounts to a reduction in word-length of approximately 30% over 8-PSK. Therefore, the 12-state code compares with the 8-state 8-PSK code, and the 24state with the 16-state code, respectively. Hence. the decoding complexity is equivalent at comparable asympt,otic coding gains. For making receiver design decisions, the performance results should be compared in simulations which include the 2See e.g. [9] transmission, the code is designed as a rate 1/1 linear code in a ring of size 6 all operations are carried out modulo 6). The rate 0.77 (4-PS k /6-PSK) code is achieved by constraining the input set to take on only 4 values. Note that this strategy cannot only be used for designing ring-convolutional 6-PSK codes, but can also be applied for 2 -PSK modulation. 5 Conclusions Trellis coded 6-PSK modulation can be an interesting alternative over trellis coded 8-PSK, to achieve more phasenoise resistance for non-decision aided phase synchronization. The codes which were presented here have comparable AWGN performance and decoding complexity as 8-PSK Acknowledgement The author would very much like to thank Stefan Fechtel for his discussions and his work on writing the Cossap simulation models tc12, tcl2r, tc24, and tc24r of the 6-PSK trellis

References G. Ungerboeck, Channel coding with multilevel/phase signals, IEEE!&tans. Information Theory, vol. IT-28, pp. 55-67, 1982. G. Ungerboeck, Trellis-coded modulation with redundant signal sets - part I: Introduction, IEEE Communications Magazine, vol. 25, pp. 5-11, February 1987. E. Bi lieri, D. Divsalar, P. J. McLane, and M. K. Simon, Intmfuetion to!lkllis- Coded Modulation with Applications. New York: Macmillan, 1991. ISBN 0-02-309965-8. L. F. Wei, Trellis-coded modulation with multidimensional constellations, IEEE!&tans. on Information Theory, pp. 483-501, July 1987. S. Wilson, Rate 516 trellis coded 8-PSK, IEEE lhns. on Communications, vol. COM-34, pp. 1045-1049, December 1986. M. Oerder and H. Meyr, VLSI implementation of synchronization a1 orithms in a 100 Mbit/s digital revceiver, in Proc. o~ieee Globecom SO, vol. 1, (San Diego), pp. 589-593 (407.1), December 1990. G. Fettweis, New rotationally invariant trellis codes for 8-PSK modulation, in Proc. IEEE Int. Conf. Commun. (ICC), vol. 3, (Boston MA), pp. 1388-1392 (45.6), June 1989. M. Oerder and H. Meyr, Rotationally invariant trellis codes for M-PSK modulation, in Proc. of IEEE Int. Conf. on Communications, ICC 85, (Chicago), pp. 18.1.1-5, June 1985. G. Fettweis and H. Meyr, High-speed parallel Viterbi decoding, IEEE Communications Magazine, pp. 46-55, May 1991. 110