ELECTRONICS”
`
`
`~.__
`
`T E
`
`
`
`)0 35 L I 0 AT I O N
`
`7th January 1999 Vol. 35 No. 1
`
`
`
`
`
`
`
`CONTENTS
`pages 1 - 96
`
`ANTENNAS
`Accurate modelling of anti-
`resonant dipole antennas
`using the method of moments
`D.H. Werner and
`Fi..J. Allard (USA)
`Dual-polarised uniplanar conical-
`beam antennas for HIPERLAN
`EM. Ibrahim, N.J. McEwan,
`FLA. Abd-Alhameed and
`RS. Excell (United Kingdom)
`
`CIRCUIT THEOHYSI DESIGN
`Analogue CMDS high-frequency
`continuous wavelet transton-n
`circuit
`E.W. Justh and EJ. Kub (USA)
`Apparent power transducer for
`three-phase three-wire system
`S. Kusui and M. Kogane
`(Japan)
`Efficient and fast iterative reweighted
`least-squares nonrecursive filters
`Yue-Dar Jou, Chaur~Heh Hsieh
`and Chung-Ming Kuo (Taiwan)
`Input switch configuration
`suitable for reil-to-rail
`operation of switched
`oparnp circuits
`M. Dessouky and
`A. Kaiser (France)
`Unified model of PWM switch
`including inductor in DCM
`Su ng—Soo Hong (Korea)
`
`COMMUNICATIONS 8: SIGNAL
`PROCESSING
`Adaptive multiwavelet prefilter
`Yang Xinxing and
`Jiao Licheng (China)
`Decision feedback equalisation
`of coded l-O. OPSK in mobile
`radio environments
`A. Adinoyi, S. A|—Semari
`and A. Zerguine (SaudiArabia)
`Detection algorithm and initial
`laboratory results using
`V—BLAST space-time
`communication architecture
`G.D. Golden, C.J. Foschini,
`Fl.A. Valenzuela and
`RW. Wolniansky (USA)
`
`page
`
`10
`
`11
`
`13
`
`14
`
`
`
`Efficient complexity reduction
`technique in trellis decoding
`algorithm
`Sooyoung Kim Shin
`and Soo ln Lee (Korea)
`Extended complex HBF and
`its application to M-DAM
`in presence of co-channel
`interference
`Ki Yong Lee and
`Souhwan Jung (Korea)
`Fair queueing algorithm with
`rate independent delay for
`ATM networks
`S. Ho, 8. Chan and
`l(.T. Ko (Hong Kong)
`Integrated
`time equaliser
`(or DSICDMA receiver with
`unequal reduced lengths
`V.D. Pham and T.B. Vu
`(Australia)
`Investigation of sensor failure
`with respect to ambiguities
`in linear arrays
`3'
`V. Lefkaditis and A. Manikas
`(United Kingdom)
`Laaming algorithms for minimum
`cost. delay hounded multicast
`routin in dynamic environments
`J. Reeve, P. Mars and
`T. Hodgkinson (United Kingdom)
`Multiple target tracking using
`constrained MAP data association
`Hong Jeong and
`Jeong-Ho Park (Korea)
`Passband flattening and
`broadening techniques for
`high spectral efficiency
`wavelength demultiplexers
`E.G. Churin and
`P. Bayvel (United Kingdom)
`Performance of CDMAIPRMA
`protocol for Nakegami-m
`frequency selective fading
`channel
`R.P.F. Hoefel and
`C. de Almeida (Brazil)
`Tbitls switching scheme
`for ATMIWDM networks
`J. Nir, l. Elhanany
`and D. Sadoi (Israel)
`
`20
`
`24
`
`27
`
`30
`
`
`
`é:'L","(‘..‘x..=]'|ml‘.'
`
`(continued on back cover)
`
`Apple 1103
`
`Apple 1103
`
`

`
`CONTENTS
`(continued from front cover)
`
`ELECTROMAGNETIC WAVES
`Electromagnetic penetration into
`20 multiple slotted rectangular
`cavity: TE-wave
`H.H. Park and H.J. Eom (Korea)
`
`IMAGE PROCESSING
`Encoding edge blocks by partial
`blocks of codevectors in vector
`quantisation
`Hui-Hsun Huang, Chang-Wen Ko
`and Chien-Ping Wu (Taiwan)
`Technique for accurate correspondence
`estimation in object borders and
`occluded image regions
`E. tzquierdo M. (United Kingdom)
`
`INFORMATION THEORY
`Analysis of turbo codes with
`asymmetric modulation
`Young Min Choi and
`Pil Joong Lee (Korea)
`Improved group signature
`scheme based on discrete
`logarithm problem
`Yuh-Min Tseng and
`Jinn-Ke Jan (Taiwan)
`Low density parity check codes with
`semi-random parity check matrix
`Li Ping, W.K. Leung (Hong Kong)
`and Nam Phamdo (USA)
`Non-binary convolutional
`codeojor turbo coding
`Cgfibrrou-and M. Jézéquel (France)
`. ,-JNTEGRATEID OPTOELECTRONICS
`Iflfilfibandwidth monolithic
`i|iPIlnGaAs pinISHBT photoreoeiver
`D. Huber, M. Bitter, T. Morf,
`,C. Bergamaschi, H. Melchior
`and HL Jéickel (switzeriand)
`
`'
`
`LASERS
`1.5).1m lnGaAlAs-strained MOW ridge-
`waveguide laser diodes with hot-
`carrier injection suppression structure
`H. Fuk__ano, Y. Noguchi
`7‘ “and S. Kondo (Japan)
`9.5W CW output power from high
`, hrighmess 980nm lnGaAsIAlGaAs
`tapered laser arrays
`F.J. Wilson. J.J. Lewandowski,
`B.l(. Nayar, D.J. Robbins,
`P.J. Williams, N. Carr and
`F.O. Flobson (United Kingdom)
`Investigation of data transmission
`characteristics of polarisation-
`controlled 850nm GaAs-based
`VCSELs grown on (311)B substrates
`H, Uenohara, K. Tateno,
`T. Kagawa, Y. Ohiso,
`H. Tsuda, T. Kurokawa
`and C. Amano (Japan)
`Low current and highly reliable
`operation at 80°C of 650nm
`5mW LDs for DVD applications
`M. Ohya, H. Fujii,
`K. Doi and K. Endo (Japan)
`Modelocked distributed
`Bragg reflector laser
`H. Fan. N.K. Dutta, U. Koren,
`C.H. Chen and A.B. Piccirilli (USA)
`
`Near room-temperature continuous-
`wave operation of electrically
`pumped 1.55pm vertical cavity
`lasers with InGaAsPIinP bottom
`mirror
`S. Rapp, F. Salomonsson,
`J. Bentell (Sweden),
`I. Sagnes, H. Moussa,
`C. Mériadec, R. Flaj (France).
`K. Streubel and
`M. Hammar (Sweden)
`Record high characteristic
`temperature (1; = 12K) of 1.55pm
`strain-compensated AiGa|nAsl
`AlGaInAs MOW lasers with
`AlAsIAlInAs multiquantum banier
`N. Ohnoki, G. Okazaki,
`F. Koyama and K. lga (Japan)
`Red light generation by sum frequency
`mixing of ErfYb fibre amplifier
`output in OPM LiNbO,
`D.L. Hart, L. Goidberg
`and W.K. Burns (USA)
`
`MICROWAVE GUIDES 8:
`COMPONENTS
`Lumped D0-50GHz amplifier
`using |nP[InGaAs HBTs
`A. Huber, D. Huber,
`C. Bergamaschi,T. Mori
`and H. Jackel (Switzerland)
`RF tunable attenuator and modulator
`using high Tc superconducting filter
`Lu Jian, Tan Chin Yaw, C.K. Ong
`and Chew Siou Teck (Singapore)
`
`NEURAL NETWORKS
`Compact building blocks for
`artificial neural networks
`M. Melendez-Rodriguez and
`J. Silva-Martinez (Mexico)
`
`OPTICAL COMMUNICATIONS
`40C-ibitis single channl dispersion
`managed pulse propagation in
`standard fibre over 5lJ9km
`S.B. Alleston, P. Harper,
`LS. Penketh, I. Bennion and
`N.J. Doran (United Kingdom)
`All-optical ZR regeneration based
`on interferometric structure
`incorporating semiconductor
`optical amplifiers
`D. Wolfson, F-‘.B. Hansen,
`A. Kioch and ((.E. Stubkjaer
`(Denmark)
`Demonstration of time interweaved
`photonic four-channel WDM
`sampler for hybrid analogue-
`digital converter
`J.U. Kang and RD. Esman (USA)
`Dign of short dispersion decreasing
`fibre for enhanced compression of
`higher-order soliton pulses around
`155Dnm
`M.D. Pelusi, Y. Matsui
`and A. Suzuki (Japan)
`Experirnerrtal measurement of
`group velocity dispersion in
`photonic crystal fibre
`M.J. Gander, Fl. McBride.
`J.D.C. Jones, D. Mogilevtsev,
`T.A. Birks, J.C. Knight and
`P.St.J. Russell (United Kingdom)
`
`(continued on inside back cover)
`
`

`
`ELECTRONICS
`LETTIERS
`
`T H E
`
`I N S T I T U T I O N
`
`O F
`
`E L E C T Fl
`
`I C A L
`
`E N G I N E E H S
`
`7 JANUARY 1999
`ELLEAK 35 (1)
`
`VOLUME 35
`1 — 96
`
`NUMBEH1
`ISSN 0013-5194
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`
`the proposed
`
`arzaI_i‘.ri.:.- Some possible attacks against
`Sec'urt't_i'
`scheme are presented below.
`Attack 1: Although the group authority has the knowledge of (r_.
`5,. k,). the group signature cannot be forged without the secret l-tey
`.v, of L’, It is impossible to obtain .v, from _I‘. without being able to
`solve the discrete logarithm problem Moreo\er_ because the gen-
`erator OL, = g“ *1 mod p. It’, e Z1’ and II e Z.,'. both (1. and is have
`the same order q. Therefore. forging (R. S) is as dillicult as break-
`ing ElGamal‘s scheme [P-E. Since the group authority cannot forge
`the group signature. forgery by an adversary is even more difli-
`cult. Thus. the impersonation attack can not be successful.
`Attack 2: The signer L-", can be identified if we can obtain 1', from
`the signature (R. S. /i(m).
`:1. B. C’. 1). EL Since the receiver does
`not know the (r,. 3,. rig) of the group authority. he cannot check the
`equation D” - _t‘,‘
`- E E D“ mod p. Obtaining (r,.
`.\‘,-. A',) from given
`{A. B. C. D. E $ depends on the discrete logarithm.
`Attack 3: The group auth0t'ity may publish the inl'ot1nation (r,. s,.
`_1‘,} for the message i.-rs signature to enable a verifier to check the
`identity of U, This does not damage the anonymity of L»-','s previ-
`ous group signatures because the information (i',.
`.t,. _i',)
`is only
`provided for the specific group signature :R. S. i'i(m]. A. B. C‘. D.
`E. For different messages. U, will have chosen different random
`integers rt and It
`to generate group signatures.
`if an adversary
`wants to obtain (1. It and (:3.
`.\g) from given {.«1. B. C D. E‘ .this is
`as dillietilt as solving the discrete logarithm
`
`DtZmt.w‘r;1i.' The improved schenie preserves the main merits inher-
`ent in most of the Lee-Chang scheme. In the case of a later dis-
`pute. the group authority inu_\- publish the information (rT. KT. )3)
`to enable a vcrilicr to check the identity of the signer. although
`this does not damage the anonyrnity of the other previous signa-
`tures of the signer. Meanwhile.
`the group authority need not
`renew any key of the signer. The reason is that the information
`(r,. .s',. 1;) is only provided for the specific group signature {R S.
`/itm).
`.4. B.
`('. D. E3. C'ompared to the original scheme.
`the
`improved scheme requires some additional cost
`in IEFITIS of com-
`putational time and the size of the group signature. For generating
`a group signature. the signer E’, may precoinpute several diITerent
`:(t,.
`.4. B. (L 1).
`If’, using ti-,.
`.\',l to reduce the real-time computa-
`tional time.
`
`(‘wit-!u.n'wi.s.' We have proposed an improved group signature
`scheme based on the discrete logarithm. in our improvement. a
`group signature can be opened to reveal the identity of the signer.
`the anonymity of the other previous signatures signed by this
`group member are not damaged. Meanwhile. thé group authority
`also need not renew the keys of the signer. We have demonstrated
`some possible attacks against the proposed scheme. Under the dif-
`liculty of comptiting the discrete logarithm problem. vie have
`shown that the improved scheme is secure against these attacks.
`
`{LI lEl.i [999
`Et't't'rt‘rirrtt'.\ Lt'i'.'(‘l'\' (}t.n'mr= No: 1999007!
`
`fit‘? Ot‘mt‘)¢'.r 193943‘
`
`.-l!utt'tmiuIi'z'.v.
`Yuh-Min Tseng and Jinn-Kc Jan (f.lI.\‘I.f[llt'E‘ of Appft'm'
`Nrrri'mmt'
`('lit..rng h'.w'.v1g Utt."t‘c’r.rtt_t‘. Tutrhrtrig. Tuntwi 4(1). Rc,uirfift'<' of
`(iftfmtl
`
`Jinn-Ke .lan: corresponding author
`F.-mail: jitjilllfiititlllflill.l)Cll1l.€(_lLLIW
`
`References
`
`sigtiatttres‘.
`
`Pi-mu
`
`‘Group
`HEYST. F.:
`and
`LH.-\t'\t. n..
`EL'R0(.'Rl'PT'91. 1992. pp. 257 265
`('Ht-'=‘\
`|.. and Pt.:t)t;t<st;.\t. Tr: ‘New group signature seheines‘.
`EL'ROCRl'PT'94, l995. pp. l7l
`lit]
`L'L(i.-\.'\l.~\L. r.:
`‘A public key crypto system and a sign-inure scheme
`based on discrete logarithms‘. IEEE Tm.vi,\.
`lrifl
`lln'ui't.
`l‘Jh‘5. 31.
`(4I. pp. 469-472
`Lt;£. wn. and t'H-‘\.\('j. L‘ C’; ‘Eflieient group signature scheme hasctl
`on the discrete logaritltilf. IE5 I’:-or
`('um,rmr.
`lltfeit.
`i"t-«ti,
`IWPH’.
`l45.li). pp. is H4
`‘Message rceo\‘cr_\'
`\tva1-:it(;.t;. and Rt‘r.Pt*t.t..tt..\:
`schemes based on the discrete logarithm prtibleilf.
`tum‘ (‘r_i'p1ugrctphr. 1996. 7. pp. {il SI
`
`.“l'Hr‘.
`
`tiir ~.ign;nnre
`t'.)t*.\f_<,'.-is. (‘with
`
`Low density parity check codes with semi-
`random parity check matrix
`
`Li Ping. W.K. Leung and Nam Phamdo
`
`A serni-random approach to low deitsity parity check code design
`is shown to achieve essentially the same perfomtance as an
`existing method. but with considerably reduced coniplexity.
`
`[mmu'm'Ifm1.' Recently. there has been revived interest in the low
`density parity check (LDPC) codes originally introduced in 1962
`by Gallager [1]. It has been shown that such codes can achieve
`very good performances (within l.5dB of theoretical limits) with
`modest decoding complexity [2].
`An LDPC code is defined from a randomly generated parit_v
`check matrix H [2]. For the purpose of encoding. it is necessttry to
`transfer H into H._.. the equivalent systematic form of H. which
`can be accomplished by Gaussian elimination. For a rate R = Inn
`(R 2 information length. n = coded length). the size of H is (n In x
`it. When n is large. Gaussian elimination can be costly in terms of
`both memory and the operations involved. Besides. it considerable
`amount of memory is required to store H._.
`in the encoder. which
`is not necessarily sparse even though H is usually designed so.
`In this Letter. we report a tnoditied approach to LDPC code
`design. We adopt a semi-random technique. i.e. onl_v part of H is
`generated randomly. and the remaining part is deterministic. The
`new method can achieve essentially the stone performance as the
`standard LDPC encoding method with significantly reduced coin-
`plexity.
`
`Pr-uposed app:-u:m'i: For simplicity we will only consider binary
`codes. Decompose the codeword c as c = [p. d]'. where p and d
`contain the parity and information bits. respectively. Accordingly.
`we decompose H into H = [HIE H"]. Then
`
`(ii
`tHP.H“)(p) _ 0
`(I
`In the proposed method. H" is constructed in some deterministic
`form. Empirically. we found the t'ollov.-ring a good choice (recall
`that HI‘ must be a square matrix [3]):
`l
`l
`
`I)
`
`l
`
`H“:
`
`._
`
`It
`
`_
`1
`
`1
`
`(‘ll
`
`We adopt the following rules to create H“. Let t be 21 preset integer
`constrained by (it (divides u-A’ and (ii) it-k divides kt. Partition H‘
`(which has it-A rows) into I equal sub-blocks as
`
`H“ ;
`
`Hdl
`
`g
`Hd:
`
`t3}
`
`-I. we randomly‘ create exactly
`t
`in each sub-block H“; .t : l. I »
`one element 1 per column and kt.-(rt-Ir) is per row. The partition in
`cqn. 3 is to best increase the recurrence distance of each bit in the
`encoding chain (see below) and. intuitively. reduces the correlation
`during the decoding process. The resultant H‘ has a eoltimn
`weight of r and it row weight ol‘A'tx‘(n—-/t) (the weight ofa vector is
`the number of is aiming its elements).
`Based on come.
`I and 2.
`r :p,: can easily be calculated from a
`givctt d '-
`it/,: as
`
`,.,
`
`2"n-,',ir,
`
`;.....t
`
`,. —,.,
`
`.
`
`.-
`I §“;.-pr.
`
`[Hull] 3;
`(Ii
`
`the above
`('omparcd with the standard LDPC code design [2].
`method has several advantages. First. the encoding process in eqn.
`4 is much simpler than a full Gaussian elimination. Secondly. a
`random H" can be singular. which causes additional programming
`complexity in realising a specified rate. On the other hand. HI‘
`in
`con. 2 is always non-singular so the new method can realise any
`given rate directly and precisely. Thirdly.
`it requires very little
`memory to store H“ in the encoder if H“ is sparse (this can be
`ensured using small I).
`
`ELECTRONICS LETTERS
`
`7th January 1999
`
`Vol. 35 No.
`
`1
`
`l
`
`I-J
`
`3
`
`4
`
`‘J:
`
`38
`
`

`
`decoders passing the result of their work to each other. at each
`iterative step.
`
`X
`systematic
`part
`
` data
`
`
`
`
`
`interleaving l
`
`Y1
`
`redundancy
`
`I il
`
`0.6
`
`1 .0
`
`1.3
`
`1.4
`EblND.dB
`
`2.2
`
`95:4
`
`I Pt'l'/(iH'Hit.'Ji('{'.\ u/7 LDPC ('nde.t'gei2w'a.wr1' ht‘ w1i1.!-rmidmi: pm-i'(_|
`Fig.
`rlm'1'r ittn-rt't'\'t’.r um‘: it = JUIJHO
`’ l‘?
`R '
`l
`R —' 3 i‘_JlJ'
`
`Fig.
`
`Y2 —-—-J‘B550
`I Tuu-u't'mt'n.it'mi .'m'/In (‘uric rt.-‘II: _t:e'itc'mroi'.v .'5.
`I3’
`
`l contains the simulated performances ot‘
`Siiiirtlatiuii mu/_1'.‘ Fig.
`the proposed encoding method for various rates (I 3. 12. 233)
`using I I -l. The decoding algorithm lollows that in [2]. The results
`are essentia1l_\
`the same as those obtained using fully random H.
`
`It has been shown that a semirandom approach to
`("wit-i’n.\r'n.Ii:
`l.l)P(' code desit_:n can achieve essentially the same perfonnance
`as the existing method with considerably reduced complexity.
`
`IEE 1099
`.\.
`.’fi’t'r'ri'rmit'.\
`.l'.t'm-r.\ Unltm- -\’n.'
`
`i9WU(Jofi
`
`33 .-‘\"oiwnZm' I998
`
`l.i Ping and W.K. Lettng |[)e,r:ui't.-nwi! of Ht-'c'.'i'ruii’t‘ EH_Ei'l1[’€Hilig.
`ll'tft‘t'r.\.'.‘_t' HI‘ Hring ,'\-rll!_§,"‘
`Ifruttg Kant!)
`I’-n1.:iI; eclipitigmcityu.edu.hk
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`_‘.?."t'J. {Witt
`
`("fry
`
`References
`
`I
`
`2
`
`3
`4
`
`'l.ow density parity check codes’. IRE T'rc.tm'. In/I
`ti.-\Ll..-\t'vl-R. R :..:
`'J"h¢'ri1'1',
`I962. IT-8, pp. ll 38
`.\t.tt-mt. l)..|.( .. and t\'I-.-\I., R.M.; ‘Near Stiaiirioii limit performance of
`low density parity check codes‘, E/t'r‘fl't)rt
`I.m..
`I997. 33.
`(tab. pp.
`457 458
`.T Li: ‘Digital commuttictttions‘ (Mc(ira\\'-Ilill. 1995)
`I'RrMKI.H.
`t't;it;RsoN. W w.. and wt.t non. I .1,
`it : ‘I--Irror-correcting codes‘ (MIT
`Press. Cettnbt'idge. Massaiclittsetts. 1972) End I.‘t.l|'l.
`
`Non-binary convolutional codes for turbo
`coding
`
`C. Berrou and M. Jézéquel
`
`The authors con.~.ider the use of non-hinary L‘Dl1\'0lllli0l‘l£ll codes in
`turbo ending.
`[I
`is
`s'l‘|o\\n
`that Ltttatertittr}
`codes can he
`advatttageotts.
`both
`from perfomiance
`and
`L‘tm1plc\‘il_\'
`standpoints. but that l‘tlgl1tfT-t\l'Lllfl'tI0tlC:~ may not bring further
`intprovement.
`
`least
`Ir1!rou'trcrr'on.' Tttrbo codes are error correcting codes with at
`two dimensions ti.e. each datum is encoded at least
`twice). The
`decoding of turbo codes is based on an llCl".lli\"t.‘ procedure using
`the concept ofextrinsic information. Fig.
`I gives an example 0|" it
`two—dimensionetl turbo code built from a parallel concatenation of
`two identical recursive systematic convolutional (RSC) codes with
`generators
`l5.
`l3 (octal notation). The global
`(non-iterative)
`decoding of such a code is too complex to he envisa_t_1cd because of
`the very large number of states induced by the intcrlcavcr. /\n iter-
`atire procedure is therefore used.
`the two codes being decoded
`alternately in
`their own dimensions and the two associated
`
`B."mIr_t‘ ("ruler rt't‘.\'tt'.\' qimft'rmri‘_i' t1u1t*.r.' Fig. la represents a block of
`si/c R encoded by the code of Fig.
`l. This block is seen as a two-
`dimensional vi‘: X \-‘k block and for simplicity we consider that the
`intcrlcax-cr is at regular one:
`the sequence is encoded lirst by C],
`following the horizontal or linewise dimension. and secondly by
`C}. following tlte vertical or columnwisc dimension, The dashes on
`both dimensions sytnbolise the path error packets at the output of
`the two decoders. at a particular step of the iterative process.
`These packets do not contain only erroneous decisions but they
`indicate where a wrong path has been chosen either by the
`decoder of ( '1 or by the decoder of (3. This corresponds to a car
`tain path error density per dimension. which is the same in both
`dimensions it‘ the component codes are identical.
`
`
`
`__._._._._..4.-,«.._
`
`
`
`Fig. 2 Putfz cJ‘I'm‘ [):..‘tItn'.\ in .fur‘i'Ira t!'t‘t'ri(fiIl}3
`ti Binary codes
`fr Quaternary codes
`
`-b
`
`systematic part
`
`
`
`
`
`.,.,.,_.._,,.___~F._;_,_—_L_.._...‘
`
`Fig. 3 H-smrt» £fl((Ht’.’l'I!t'.'i'Vl.‘
`um'e' trtifr _Qi'm-i‘r:ror.t Ii [3
`
`f‘('(‘u'F'.l‘it'(.‘
`
`.i‘_t‘.SIt'I)i£II.l(‘
`
`t‘uiirm'ttIi-‘min.’ (REC)
`
`redundancy
`
`The performance of turbo decoding is strongly dependent on
`the path error density per dimension. Obviously. the more numer-
`ous and the longer the horizontal and vertical dashes in the square
`box are. the harder the convergence to the correct codeword is.
`Now For each component code. replace the binary code of Fig.
`l
`by the quaternary code of Fig. 3. The data are thus encoded and
`interleaved in couples. The size of the block is /</'2 couples and the
`square box now has the dimensions ‘M12 x V/<12 (Fig. 212). When a
`decoder selects a path in the decoding trellis. the same amount of
`intortnation is used in the cases of both binary and quaternary
`codes. tlterefore with halt‘ the number of transitions in the case of
`a quaternary code. giving path error packets which are halt" the
`
`ELECTRONICS LETTERS
`
`7th January 1999 Vof. 35
`
`No. 1
`
`39