On Evaluation of the Optimum Equalization
in Optical Fiber Communication
ShoichiIINO MikioTAKAHARA （Received August 30，1980） AbStract In this paper we evaluate quantitatively the effect of the optimum equalization in comparison with the full cosine roll off（F． C． R．0．）and with Gaussian equalizabion in optical丘ber digital transmission systems・ As a result， the optimum equalization makes the repeater spacing longer by about O．4dB（at 10Mbits／s）than the EC．R．O equalization in the case of the optical舳er having fairly wide bandwidth． Moreover， the effect of improvement of the repeater spacing by the optimum equalization brings us about 2dB of improvement in the case of the optical 舳er having narrower bandwidth than signals． Introduction Recen．tly， various investigations and experim・ ents are being done in the field of the optical 丘ber transmission systems， and the technics and systems in the丘eld are advancing with amazing speed． Now， in the low・bit・rate（below 10Mbits／s）opt− ical fiber transmission systems we have attained to put to practical use， and even in the high−bit− rate ones will have attained in the near future． Generally， it has been said up to this time that the optical fiber has a wide bandwidth， then equalizerless repeater transmission systems may be possible1）． Single−mode・丘ber makes it possible to realize even in a high−bit・rate transmission systems with long repeater spacing 2）， but such asystem can not attain all the sorts of丘ber・ Most part of these systems use the repeater with EC．R．O type equalizer even for relatively low・ bit−rate transmission 3）． In particular， when the optical丘ber has the trade’off between transm． ission loss and bandwidth 4）， it is necessary to equalize the received waveform in order to exte・ nd the repeater spacing・ There are some investigations about the linear optimum equalizations． But， in spite of many investigations of linear optimum equalizations・ there are no investigations which discusse the effect of the optimum equalizations・ Considering the situations mentioned above， it is necessary to study the effect of the optimum equalization． Then， in this paper， we compare the difference between the optimum equalization and others． First of all， we formulate the optimum equaliza・ tion and next， compare the optimu皿equaliza− tion with others． Finally， it is shown that the F．C．R．O equalization is almost agreement with the optimum equalization at low−bit−rate， and the effect of the optimum equalization is granted when the bandwidth of fiber becomes narrower than the signal． R㏄ei▼er ］！】［Odel The system model delt with in this paper is shown in Fig．1． Here， we supPose a step・multi・ mode−fiber as the transmission line and an avala− nche photodiode as the optical detector． To 一 113一
eσ（’） τs（り Rβ 1｛、、 ∼Pα（f） ↑ ↑ ↑
Eq＆Amp
C∫． ら（の ノ）（の：optical power falling on the detector ib（t）：equivallent current noise source of ．Rb iα（t）：equivallent current noise source of Rα θα（彦）：amplifier input voltage noise sourceCT
Rb
RαRT
：total shunt capacitance across the detect、or including the shunt capacitance of the detector and that of the amplifier ：detector biasing resistor ：amplifier input resistor ＝R。／／Rb Fig．1 Receiver equivallent circuit discuss the wave equalization， we must know the input waveform to the detector． Then we need to know the loss characteristics of step．multimo． de・丘ber． These characteristics are proposed by Koyama and Kobayashi as follows 4） L（f）−6（☆㌃一）＋・L〔dB〕 （・）where
the constants are f：baseband frequency 〔MHz〕 L：fiber length 〔Km〕 α：丘ber loss 〔dB／Km〕 0．5≦≦γ∠≦0．6． Next， we supPose some premise conditions re− lated to the waveform equalization． （i） Fiber input waveform is su伍cient narrow ， rectangler pulse， therefore， received pulse waveform is the impulse response of a丘ber． （ii）We descrive the optical power falling on the detector as 1》（t）， 1）（t）＝Σ6／hz）（t一元丁）． （2） − oo Here， hp（t） is the input optical pulse res− ponse which is constrained as follows l：．．h・（・）dt−・ （3） where， bプis the energy transmitted in the ブth epoch． The baud rate is 1／T with T being the signaling Period． （iii） As we use the instantaneous detection， we can not neglect the effect of the timing’ jitter． Then we introduce that effect to the criterion function as the jitter occurs ±θ，Owith probability 1／3， respectively．Formulation of optimum equalization
To determine the optimum equalizer， we had better use the error rate as the criterion funct− ion， but the optimization of the equalizer isl1’t so easy． Therefore we use the total noise power of thermal noise， shot noise at decision point， and intersymbol interference due to the timing・jitter． We de丘ne the equalization which minimize the total noise power under the condition of fixed error rate， the optimum equalizatio11． Considering the total noise power， the criterion function is written as 1）5） F一竺・≦εi≧〔b1，十bl（Σ1−11）〕 ＋（h9η）2＜G舞〔s・＋禦…＋￡多〕 ＋（h2η）2乏き綴，s…3 ＋三㌦茎．．〔h・・t（ゐT一言一） ＋h…（kT）＋ん・誕（〃T＋一㌃）〕 （4）where
the constants are ＜G＞：mean avalanche gain η θh2
k The weighting function de丘ned as follows 11＝1 3 ㊥晋（：tU，Z2−一〕（1＋・…θ・）4ωムーr．．綴曇Σ2婦一r．．豊設烏）
x1− k量．。H・’（k）〔聖鵠） ⑧舞tll緊〕（・＋2…θω）・ ．detector quantum ef五ciency ：electron charge ：absolute temperature ：energy of a photon of frequency J2 ：Boltzman’s constant． 1，， 12， 13， and Σl are ／fcoH・’（・）〔漂（絆． ω24ω （5） 一114 一一and where Hp’（ω）＝Hp（2πω／T） H。。t’（ω）＝H。11t（2πω／T）／T Hp（ω）＝F〔hp（の〕 ＝optical input power pulse transform H。。t（ω）＝F〔h。。．t（め〕 ＝output pulse transform・ Assuming that the signal bo＝0， the criterion function is rewritten as follows F一A2 X1＋Ao 12＋Al 13 ＋票ヨ．．〔h…2（kT−一㌃） ＋h…2（の＋h・’・et2（kT＋5）〕 （6）
where
A・一（h2η）2・τ＝τ（5・＋2美三＋￡き） Al−（h2η）2・乏き袈；， S・A・一（h2η）2・≦磨61・ （7）
We set up transmission function of equalizer as Z（ω）＝Houtt（ω）／Hp’（ω）and represent Z（ω）with sampling values Z（ωの〔i＝1，2，……N〕as follows Z＝〔Z（ω1），Z（ωi），一・Z（c・N）〕 （8） then， the term cocerning with the thermal noise is given by 1‘＝＝ZBZ＊ （9）where
B＝〔bim〕 i，〃z＝1，2，……N bim＝（A。2＋A1ω乞2）4ω i＝m ＝O Zキ〃z∠ω＝ωi一ω仁1 （10）
Z＊＝transeposed conjugate vector． Next， the term concerning with the shot noise is given by五一丁（ZZo＊十ZZo＊） （・・）
where
Zo＝ΣαZκ k＝1 ぱ＝〔ぴ伽〕 i，m＝1，2，……N C伽＝⊥A，〔1＋。1・・i 3 十θ一ゴθω乞〕Hp’（ωの4ω i＝m andωz＝ωle ＝O i≠mor ω乞≠tOiC tUiC ・・2 xk／T k ＝1，2，……N． （12） Finally， the term concerning with the intersy− mbol interference is given by Ii＝ZG Z＊ （13）where
G＝KHp Hp＊K＊ Hp＝〔Hp’（ω1）， Hp’（ω2）ヂ・・…Hp’（ωN）〕 K＝〔Kim〕 ‘，m＝1，2ド…・・NK伽一
刀??D．〔・xp（−」2π（k−4．）・） 十exp（一元2πんω∂ ＋・・p（づ2π（k＋￡）・・）い一m ＝O i≠m． （14） Then， according to Eqs．（9），（11），（13）， we get the criterion function as follows F−ZBZ・＋…㌃（ZZ・・＋Z2子）＋ZGZ＊・（・5） Constraining the equalizer output value equal to l at t＝0， that is， hout（0）＝1， this constraint condition is shown byT−（ZEHp＊十ZELIp＊）一・ （・6）
where
E＝〔eim〕 i，〃z＝二1，2，……N eim＝ d tu／2z i＝〃z ＝O i≠m． （17） Therefore， we may detertnine Z minimizing Eq． （15）under the condition given by Eq．（16）． The criterion function is written as follows ・−ZBZ＊＋÷（ZZo＊十ZZo＊）＋ZGZ＊ ＋Z〔；（ZEH・・＋蹴・）一・〕 （・8）where
λ：Lagrange multiplier・ To get solution Z， we may solve（2N十1）dim・ entional simultaneous equations getting from partial derivati皿s of Eq．（18）withλ， and Z㍑ （ωz），Z−（ωの， where ZR（ωの， ZAT（ωD are real part and imaginary part of Z（ωの， respectivly． Example and Considerations We assume the following two equalized wave一 forms to compare withwaveform
the optimum equalized一115一
（i） full’cosine−roll−off wave H…’（∫）一；〔・一・i・（誓一芸）〕 」；βぐ∫K1；β 一・・ぐ∫K1；β ＝O elsewhere （19） （ii） Gaussian wave H…t’（∫）−Vltu・xp（一薔）（・・） where u＝1／ん伽 （T）． Farther about（i）we consider only the caseβ＝：1， about（ii）only u＝ 100． To compare the relative e伍ciencies of the opt・ imum equalization with those of others， we con． sider now the operation of a typical receiver． In all our caluculations the following Parameter va・ lues shown in Table l are taken． In］Fig・2the repeater spacing for a error rate of 10−g is graphed as a function of bit rate， and improvement gain for a error rate of 10−g at the various bit rates are plotted in］Fig．3． In Fig．2it can be seen that the effect of the Table 1 used parameters oPtimum equalization aPPea「s near 20 Mbits／s at α＝2dB／km and it apPears remarkably near 50 Mbits／s atα＝4dB／km． However， in both cases the effect of optimum equalization are not so remarkable at lower bit rate． Therefore， when optical fiber having fairly wider bandwidth than signals， the improvement by the optimum equa・ （ ㊦ ￡ よ 三 べ ぷ
3
8
2 1 ／ ／／＼＼
30 50 Blt rate（Mbit／s） Fig●3 1mprovement gain vs bit rate characteristics． 2 1 H θ＝一 α＝2dB／km 6Mbit／s ＜G2＞＝＜G＞2・3−10dBm
O．75 0．85μm10pF
lMΩ
100KΩ
O．55 Optitnum F．C． R．0 E6
の ．庄 冨 3 甘 8 δ k ￡ 吉 ∩ A．P．D fiber input power qUantUm e缶CienCy optical wavelengthCT
Rα R， γ Optimum F．C．R．O Gaussian II θ＝一一 Error rate 10一9 1．0 t／T Fig．4 Equalized waveform， 1、＝25 L＝10km E き の ．E 罵 詩 k 3 8 合 匡 ） α＝4dB／km 10 30 50 Bit rate〔Mbit／s〕 Fig．2 Repeater spacing vs bit rate characteristics． ハ ご E o 0．5 0 Fig．5＼
II θ＝＿ α＝2dB／km 6Mbit／s Optimum−一一
e．C．R．O 0．5 1．0 1．5 f Frequency response of equalized waveform． 一 116 一1ization is not remarkable． But in the case of optical fiber having narrower bandwidth than signals， the effect of the optimum equalization is remarkable． Moreover， those results are reco− gnized qualititatively． We see from Fig．3that the optimum equali・ zation brings us over 2dB of improvement than other equalizations． In Figs．4，5the optimum equalized waveform and its frequency response are sketched． As a result， the optimum equalized waveform shows a steep leading edge in according for repeater spa− cing to be shortened． In this paper， though we assume hout（0）＝1 so that the amplitude of the
