Coded modulation scheme employing turbo
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`codes
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`P. Robertson and T. Worz
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`Indexing terms: Convolutionul codes, Modulation coding
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`The authors present a turbo coding scheme for bandwidth-
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`efficient modulation that outperforms turbo coding with Gray
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`mapping by employing Ungerboeck codes as component codes.
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`The scheme can be decoded iteratively and achieves very good bit
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`error rates at low signal to noise ratios after a small number of
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`iterations. XPSK is used as an example. although the scheme can
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`be extended to other constellations.
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`turbo codes have been introduced [1]
`Intt‘0dm'ti0n.' Recently,
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`which achieve good bit error rates (104 ~ 10 5) at a low SNR.
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`They are of interest in a wide range of telecommunications appli-
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`cations and are composed of two binary component codes. They
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`originally
`proposed
`for binary modulation
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`were
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`Approaches have been undertaken to combine binary turbo codes
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`with higher order modulation (e.g. SPSK,
`l6QAM) and Gray
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`mapping [2]. In contrast, in our approach we have employed two
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`Ungerboeck codes [3] in combination with TCM in their recursive
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`systematic form as component codes in a structure similar to
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`binary turbo codes. We use 8PSK as an example throughout this
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`paper, although the method can easily be extended to other modu-
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`lation formats (e.g.
`l6 and 64 QAM) with similar results. We will
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`begin by presenting the encoder structure, followed by a brief
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`description of the decoder and conclude with simulation results.
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`Encoder: An important property of turbo codes is that they use
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`recursive systematic component codes that are simple and have
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`interleaving between two such component encoders, which results
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`small error event probability. Ungerboeck codes were
`in a
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`designed to provide high spectral and power efficiency through
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`signal set expansion. The motivation for our approach was to
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`replace the binary component codes
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`turbo code
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`schemes with Ungerboeck codes and retain the advantages of
`both.
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`Major differences to classical turbo codes are:
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`(i) The interleaving now operates on pairs of bits (for SPSK)
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`instead of single bits.
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`(ii) To achieve the desired spectral efficiency of Zbit/symbol, the
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`mandatory puncturing of parity information is not quite as
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`straightforward as in the binary turbo coding case.
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`Let the size of the (pseudorandom) interleaver be N, also equal
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`to the number of SPSK symbols per block, and let the number of
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`information bits transmitted per block be ZN. The encoder is illus-
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`trated in Fig. 1, where we have chosen a short interleaving length
`N = 6.
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`sequence of
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`_ _-r1*=1b£'.t>9I:s_ _ _ ,
`!(di ,d2.d3.dz..%.de>=:
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`'-.000HH00_0"___.'
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`_ _ _ _ _ _ _
`BPSK symbols
`i9.s.z.«.i.7]
`ro'§,‘7,‘sTa‘;
`""T'Tj seié-c"io'r' ‘ ' ‘
`I /):?#output
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`even position to even position
`J“:
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`odd position to odd position
`’_/6:‘~3:‘__1
`r
`‘ll,1l,0 ,oi,oo_1o.=(d3,d6,d5,d2,d, ,d,,)
`s5qTie'nc_e_o? ififobit pairs
`1 5,7,o,3,o,z."
`._ _ _ _ _ _ _ J
`sequence of
`SPSK symbols
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`95m
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`Fig. 1 Encoder shown for 8PSK with component ("odes memory three
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`and example of interleaving with N 2 6 shown
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`Underlined letters indicate that symbols or pairs of bits correspond to
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`first encoder
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`ELECTRONICS LETTERS 31stAugust 1995 Vol. 31
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`No. 18
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`Ericsson Exhibit 1027
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`Ericsson v. IV
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`lPR20l4—Ol 149
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`achieved by keeping only the largest term in each of the sums. By
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`taking the logarithm above we obtain simplified detector (optimal
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`for large average signal to noise ratios, SINR), which is
`0
`2
`2
`s,2>
`H1lI1‘I‘
`.—f
`lI1lIl‘T‘
`.A
`kZ:%fl,1,.l no)
`I < gzepifl nu)
`1
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`This detector uses the Euclidean distances between the received
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`signal and the possible (faded) channel symbols only. In all the
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`results below.
`this simplified front—end detector has been used
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`throughout with small performance loss relative to the optimal
`case.
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`Mk)
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`s2
`fv»(k) 2|
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`Performance examples.’ Both uncoded and coded (F lg. 2) schemes
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`are considered and the bit error probabilities estimated by compu-
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`ter simulations. Starting with the uncoded case, all schemes have
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`the same power spectrum as BPSK, which serves as a reference.
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`For large SYVR, the bit error probability behaves as FNR 1. The
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`remaining three schemes have repetition codes resulting in 2, 3 or
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`4bits, which are delayed and mapped into QPSK, SPSK and
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`l6QAM signal sets, respectively.
`is found that
`the diversity
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`orders are cl, = 1, 2. 3 and 4. respectively, and that 37VR’“'° shows
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`the asymptotic bit error rate. Note, however, that significant per-
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`formance gains are obtained even at fairly large bit error rates. At
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`the 103 level, 10dB of3'NR is saved with ti, :2. With an (n, k)
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`i.e.
`block code, the diversity order is now of, : n/k,
`the inverse
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`code rate. In Fig. 2, two TCM schemes with eight states are also
`considered. The case with code rate 2/3 and SPSK is taken from
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`[1] and represents state of the art for Zbit/channel use. The new
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`system uses code rate 2/4 with l6QAM. This latter code was
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`designed by hand and is probably not the best possible. In spite of
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`this, an improvement of ~l dB of SW}? is observed as high as the
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`20% error rate.
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`C()m‘lu5i0r1.s‘.‘ Starting from the CBI concept due to Zehavi [1], a
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`diversity increase is obtained in a very simple way by introducing
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`more coded bits. To keep the power spectrum unchanged,
`the
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`channel symbol set must be doubled for each new coded bit. Sig-
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`nificant bit error rate improvements are demonstrated even at high
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`error rates. The proposed simple suboptimal detector performs
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`close to optimal, even at low SNTE.
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`© IEE 1995
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`Electronics Letters Onlinc No: 19951057
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`U. Hansson and T. Aulin (Chalmers University of Technology.
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`Computer Engineering, Telecommunication Theory, S-412 96 Goteborg.
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`Sweden)
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`Email: tor@ce.chalmers.se
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`19 June 1995
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`References
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`l
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`3
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`4
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`1546
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`ZEHAVl.E.:
`‘8—PSK trellis codes for a Rayleigh channel”, IEEE
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`Trt1n.r., 1992, COM—40. pp. 873—884
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`2 WlLSON,S.G., and LEUNG.Y.S.2 ‘Trellis-Coded phase modulation on
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`Rayleigh channels’. Proc.
`l987 Int. Conf. Commun., Seattle. WA,
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`USA. 7-10 June 1987, pp. 2l.3.l~2l.3.S
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`S(‘HLEGEL.('., and COSTELLO. 1).:
`‘Bandwidth efficient coding for
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`fading channels: Code construction and performance analysis’.
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`IEEE J. Se]. Areas Commun., 1989, SAC-7, pp. 135(»l368
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`JELICIC. 111).. and ROY.S.:
`‘Design of trellis coded QAM for flat
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`fading and AGWN channels’, IEEE Tl'ail.S‘., I994, VT-44, pp.
`l92—
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`201
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`‘Detection, estimation and modulation theory’
`5 VAN TREES, H.L.:
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`(John Wiley & Sons, New York, l968), Part I
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`

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`mi
`{
`105k’
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`tbs
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`32
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`Simulation results.’ As examples, we have used SPSK (with N :
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`1024 and 5000) transmitted over an AWGN channel. The compo-
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`nent code we used was the same as the Ungerboeck recursive
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`eight—state code,‘ we found that optimising the ‘punctured’ mini-
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`mum distance yielded the same code in this case. The bit error rate
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`(BER) curves are shown in Fig. 3 for different numbers of itera-
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`tions. Also shown is a Gray mapping scheme as presented in [2],
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`that has the same complexity (number of trellis branches per infor-
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`mation bit) as our four—iteration scheme and the same number of
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`information bits per block; 2048. In all simulations, we used the
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`symbol—by-symbol MAP algorithm [4]. Compared to TCM with
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`64—state Ungerboeck codes, we achieve gains of 1.7 and 0.5dB
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`compared to the Gray mapping scheme, at a BER of 104. The
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`Shannon limit for 2bit/symbol and SPSK is at 2.9dB E,/N0.
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`Cam:/usiuns: We have presented a novel bandwidth—efficient turbo
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`coding scheme that employs Ungerboeck codes as component
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`codes and combines the advantages of both schemes. The codes
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`can be decoded iteratively and achieve very good performance
`after a small number of iterations.
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`© IEE 1995
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`Electronics Letters Online No: 19951064
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`P. Robertson and T. Worz (Institute of Communi'cati0n.i' Technology.
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`German Aerospace Research Esrablishineni (DLR), PO Box 1116, D-
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`82230 Wess/frzg, Germany)
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`5 July 1995
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`References
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`1
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`‘Near Shannon
`BERROU. (1. GL/\VlEUX,A., and THITIMAJSHIMA,P.:
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`limit errorcorreeting coding and decoding: T1irbo—codes’. Proc.
`ICC‘93. May 1993. PD. 1064-1070
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`‘Turbo—codes and high
`2 GOFF,S.L., GLAVlEUX.A. and BERROU,C.:
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`spectral efficiency modulation‘. Proc. lCC’93, May 1994, pp. 645—
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`649
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`‘Channel coding with multilevel/phase signals’.
`3 UNGERBOECK,G.:
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`IEEE Trans, 1982, IT-28, pp. 55~67
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`No. 18
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`11,) 2 (00, 01, ll, 10, 00,
`In our example, the sequence ((1,, dz,
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`ll) of information bit pairs is encoded in an Ungerboeck style
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`encoder to yield the SPSK sequence (0, 2, 7, 5, I, 6). The informa-
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`tion bit pairs are interleaved and encoded again. We de—interleave
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`the output symbols of the second encoder to ensure that the order-
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`ing of the two information bits partly defining each symbol corre-
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`sponds to that of the first encoder. Finally, we transmit the first
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`symbol of the first encoder,
`the second symbol of the second
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`encoder, the third symbol of the first encoder, the fourth symbol
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`of the second encoder etc. Thus each information bit pair is con-
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`tained in only one SPSK symbol, and the parity bit is alternately
`chosen from the first and second encoder (bold, not—bold, bold,
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`.,.). Furthermore, the kth information bit pair exactly determines
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`two of the three bits of the kth symbol x,. Additionally. we need
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`cause each symbol to appear once and only once at the encoder
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`output. It is easily seen that the interleaver must map even posi-
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`tions to even positions and odd positions to odd positions.
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`Decoder: The iterative decoder is similar to that used for binary
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`turbo codes [1], except that there is a difference in the nature of
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`the information passed from one encoder to the other, and in the
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`treatment of the first decoding step. A further problem is the fact
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`that each decoder alternately sees the noisy output symbol of its
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`corresponding encoder and that of the other encoder. The infor-
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`mation bits, ie the systematic bits that partly resulted in the map-
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`ping of each of these symbols, are correct in the sense of being
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`identical in the outputs of both encoders. However, this is not true
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`of the parity bits, since these belong to the other encoder every
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`other symbol. We have indexed these symbols with ‘*’ and some-
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`times call these symbols ‘punctured’.
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`Fig. 2 Complete decoder
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`* denotes ‘puncturec‘ symbols for corresponding component decoder
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`it can be shown that the
`In the binary turbo coding scheme,
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`output of the component decoder can be split into three additive
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`parts: when in the logarithmic or log-likelihood ratio domain, the
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`systematic component 5, the a priari component a and the extrinsic
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`component 2. Here we can only split the output into two different
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`components: a priori and extrinsic and systematic (e&.r). Each
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`component decoder must now pass the second component to the
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`next decoder. The complete iterative decoder is depicted in Fig. 2
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`where the first decoder sees a ‘punctured’ symbol (output by the
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`second encoder: ‘*«mode’), i.e. the corresponding symbol from the
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`first encoder was not transmitted. The first decoder now ignores
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`this symbol, indicated by the position of the upper switch, as far
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`as the direct channel input is concerned. The only input for this
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`step in the trellis is the a priari information a from the other
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`decoder, which contains the systematic information 3. The output
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`of the first decoder is the sum of this a priori information a and
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`the newly computed extrinsic information e. The latter is given to
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`the second decoder as its a priori
`information. The variables
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`passed between our decoders are vectors of four log—likelihoods;
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`one for each possible information group value. The second
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`symbol
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`sees
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`encoder, hence it can compute e&s which is used as the a priori
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`input a of the first decoder in the next iteration. The setting of the
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`ELECTRONICS LETTERS 31st August 1995 Vol. 31
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`switches will alternate from one pair of bits to another, but not
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`from one decoding iteration to the next.
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`The above does not apply to the first decoding of the first
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`decoder if it is in ‘*-mode’ because no (1 priori information from
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`the second decoder is available. In this case the a priori informa-
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`tion must be calculated from the received symbol by calculating
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`the probability of the information group :11, taking into account
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`the unknown parity output bit of the other encoder. This calcula-
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`tion is indicated by ‘metric s’ in Fig. 2.
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`‘Fm I
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`5
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`.-.—wm1
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`Fig. 3 TTCM for SPSK, 2 bit/symbol
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`A N = 5000 (10000 bits), 18 iterations
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`ON:
`5000 ([0000 bits), 4 iterations
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`EN:
`1024 (2048 bits). 4 iterations
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`<>N=
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`2048 bits, Gray mapping, 4 iterations