Wireless Channel Parameters Maximizing TCP Throughput

(1)

HAL Id: inria-00070358

https://hal.inria.fr/inria-00070358

Submitted on 19 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Wireless Channel Parameters Maximizing TCP Throughput

François Baccelli, Rene L. Cruz, Antonio Nucci

To cite this version:

François Baccelli, Rene L. Cruz, Antonio Nucci. Wireless Channel Parameters Maximizing TCP Throughput. [Research Report] RR-5652, INRIA. 2005, pp.43. �inria-00070358�

(2)

ISRN INRIA/RR--5652--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Wireless Channel Parameters Maximizing TCP Throughput

François Baccelli — Rene L. Cruz — Antonio Nucci

N° 5652

Août 2005

(3)

(4)

(*"+,'-./,"

02143652798;:=<>?3A@B@BCED=DF:

∗ ^GIH CJ5KCML!NPO?1RQ2S

† ^GT 5;U48;52:=8WVXQ2@B@E:

‡

YZ\[#]X^-_`acb dBeJfhgi[#]X^#fkj#lm]X]-n\o\pqj&rso9gif

tvuwlyxz^g{Y|k}P_

|{rs~\~6lmuzg\^uz^&jZ\^#uwjZ\^oAv9mb lsgsm9b 9~rm^&f

"#Ry\s9 ^-j&lmo\fzpq\^#ur"fwpo\mq^Y{_tfw^#fwfzpqlmogiuwrRm^&uzfwpo\/r¡ pquw^#q^#fwf!jZro\o\^&F¢

pgiZ£rXj#lmo\fzgirso9g!fzpqmorgwl¡o\lspqfw^uirygipql£¤¥dE¦¨§|k©krg{giZ\^uw^#j&^&pªm^#u&« ^j&lmo\fzpq\^#u{gwZ\^

~\uwlm¬q^&]ls®2\^#gw^&uw]¯pqopqo\?gwZ\^lm~gip]¡r°giuwrso\fz]Xpfwfwplmo^&o^&uwe~°^#u¬\pg&¢gil±]Xry²Jpq]¯pq³&^

Y{_tWgiZ\uzlmn\mZ\~ngR«dE~°^#j&p´\jRrsem¢pogiZ\^jRrsfz^ Z\^#uw^\pquz^&jgfz^&µBn\^&oj&^?fz~\uw^&rs£fz~°^#j¶

giuwn]·]¯lJ\n¸rgwpqlmo"pfn\fw^#¹l4s^&ukr´J²^#¹¬\rso\ pgiZ/jZrso\o^&¥¢\ º^»´o\giZ\^lm~Jgipq]Xrs

~\uwlEj&^#fwfwpo\"9rpqo

m gwZrg!]Xry²Jpq]¯pq³#^&fY{_tgwZ\uwlsn\mZ\~\nJgR«¦¼ogwZ\^?j&rsfw^- Z^&uw^giZ\^#uw^

pqf!r"ZpqmZfzpqmorKgwlo\lmpqfz^?uirygipql¢° º^j&lmo\fzpq\^#ugiZ^fwj&^#orsuzpql" Z\^#uw^±rsr~gipªm^?]¯lJE¶

n\¸rgwpqlmo¹pqf!nfw^&Ml4m^#ur¡´J²^#½¬\rso\ pgiZjZro\o\^&F¢rso\´o\gwZ\^lm~gwpq]Xrs2fheE]?¬°ls

rsq~\Z\rs¬°^g!fzpq³&^

M gwl]¡ry²Jp]Xp³&^Y{_t¾giZ\uzlmn\mZ~\ngR«¿%lJjÀ/j&lE\^&frs~~\qp^&£gwl^RrsjZ

~rsjÀs^gk®Álsu®Álmuz %rsuw/^&uzuwlmuj#lmuwuz^&jgipqlso/j&rso¹rsqfzl¯¬6^»n\fz^&Â¢6rso\/pqogwZrgjRrfw^» %^j&lsoJ¶

fwpq^&ugiZ\^xzlmpqo9g{lm~gwpq]¯pq³&rgiplmo"ls®2giZ\^j&lE\pqoXuwrgi^gil¯]¡rÃ²p]Xp³&^»Y{_tÄgiZuwlmn\sZ\~\ng&«

Åpqorqes¢s º^\pfwj&nfwfºgwZ\^{pqfwfzn\^l®rsfwfzpqmopqo\»gruwm^gdJ¦¨§k|Äsrqn\^#f&«¦¼o"lsuw\^#uPgwlj&rsuwuhe

lmnglmnursorsªeJfzpqf#¢J º^ls¬grspor?Y{_tÆgwZ\uwlmnmZ\~\ng®Álsuw]?n\qr-pqo"gw^&uz]Xfl®KgwZ\^»~rsjÀ^#g

giuiro\fw]¯pqfzfwpqlso¹^#uwuwlsu~\uzlm¬rs¬\pqpªg¼e

p ro\£giZ^giuwrso\fz]Xpfwfwplmo/jRrs~rj&pg¼e

C¢6 ZpqjZpfkls®

pqo\\^#~°^#o\\^#oBg?pqo9gi^#uw^&fhgR«¦¼olsn\u-Y{_tÇ]XlE\^&F¢KgwZ\^¡ po\\l4 Èfwpq³#^¡pf-j&nJgpqo½Zrs®%®Álmu

^RrsjZÉ~rsjÀ^#g»gwuirso\fz]Xpfwfzpqlmolmfwf#¢ÂrsoÉrsqfzl"j&ngpoZrsª®v Z\^&o^#m^#u»giZ\^- po\\l4 fwpq³#^

∗ ÊÁËKÌÍÊÁÎÏqÐ°Ë2ÑmÒsÓ#ÔÃÕÖm×ØyÙ&Ú\Û¸ÜyÕÃ×R×Ý4Þ#ÞyÙßRÝÖsÚ\Û¸Ó&Ô

† ^àá ^Ñ4âºÒ ^à Ñ4ÎIÒÔ×Ô&ãÃä4ßãm×RÚå°ÛÁÝ&å#ã

‡ æÂçÃèêéÂëì4íFçÃî#ïÂð;ë#é6ðÍèñíFç»ÑRòÃï¥èñó&íÎæÍôAÒ à Ñ4ÎIÒÃðÍçyõhóí¥çÃèöéÂð°î#ïF÷ðÂëé;èñóyèøí¥èêëíFõzùBúAûõèêéÂóÃîððÍèñíFç Ë2ëï¥ìyézÒ à Ñ4ÎPÒÕÖ&ãm×R×4ÙßÖyÕ#Ô&ãmÚÛü×Øiý

(5)

^²Jj&^#^&\f¡gwZ\^giuiro\fw]¯pqfzfwpqlso j&rs~rsj#pg¼e

C«WYZ\pf¯®Álsuw]?n\qrpqfgiZ\^#oWn\fz^&Wgwl½jZ\rsuirsj¶

gi^&uzpq³#^gwZ\^lm~gip]¡rÂ~\uwlEj&^#fwfzpqo\X9rspqolmugiZ^lm~Jgipq]Xrs°fzeE]?¬6lmKrsq~\Z\rs¬°^g{fwpq³#^-rfgwZ\^

fwlmngiplmo¡ls®2r-fzpq]¯~\q^´²J^&/~6lmpqo9g^#µBnrgiplmo¡gwZrg\^#~°^#o\\flmo"gwZ\^ puw^&^&fzfjZrso\o^&

~rsuir]X^gi^&uzfrso\giZ\^~ruirs]¯^#gw^&uwfl®2gwZ\^Y{_tÄj&lmo\o^&j#gwpqlmo;«

&('*),+.-0/ 212s _3»a¹¢vrsr~gipªm^¡]¯lE\n\¸rygipqlsoÂ¢~\uwlEj&^#fwfwpo\9rspoÂ¢¬qlEjÀMj#lJpqo\\¢

fwpqsorsÂgilXo\lmpfw^rso\£po9gi^&uh®Á^&uz^&o\j#^-uirygipql¢~°l4 º^&uj#lmo9giuzlm¥¢A¬rso pqgwZ£fzZrsuwpo\\¢Aj&lsoJ¶

m^&fhgipqlsoj&lmo9giuzlm¥¢sj&lmo\s^&fzgwpqlmo±rRslmpqro\j&^¢Brs\\pªgips^pqoj&uw^&rsfw^%]?n\gwpq~\pqj&rgipªm^I\^&j#uw^&rsfw^

rsqmlsuwpgwZ\]/¢Y{_tgiZuwlmn\sZ\~\ng&¢;lm~gwpq]¯pq³&rgiplmoÂ¢fzgwlJjZ\rsfzgwpqj?~\uzlJj#^&fzf&¢Ífzgirgiplmorsuhe/\pqf¨¶

giuwp¬\ngwpqlmoÂ¢\a¹^&qpo¡giuwrso\fz®Álsuw]/«

(6)

()

§klsn\f#gwn\\plmo\f¡\rso\f"j#^#grsuhgipj&q^/q^¹&¬pg"ñn\o^fw^&fzfwplmo Y{_tÈfzn\u

n\oÄjRrorsuirspql ^£¬rso\^£~rfwfiroBgw^\lmoo&^^g"\^¹uirs~~°lmuhg¡fzpqmo\rsfwn\uX¬\uwn\pªg¡^g

pqo9gi^#uz®&uz^&o\j#^£j#lmo\fzgirso9grsnÆ~°lspqo9g¡^¹u&j&^#~giplmoÄ^g¡o\lsn\f¡jZ\^#uwjZ\lso\f½#gi^#uw]¯pqo\^#u

&o\^#uwmp^?\^?giuwrso\fz]Xpfwfwplmo~ru»¬\pgµEnpv]¡ry²Jp]Xpfw^j&^±&¬pgR«!3^#nJ²Éfwj&oruwpqlsf»fwlmo9g

#gin\p #f¯~qn\f±fw~!&j#p´µBn\^#]X^#o9g"/j&^&n\p#øn\o\^/]XlE\n\qrgiplmo$%#girsq^#]X^#o9g±\^¹fz~°^#j#gwuw^

~rsu%f&µBn\^#o\j&^»\puw^#j#gi^¢~6lmn\uºq^&µBn\^#;o\lsn\f%gi^#uw]¯pqo\lmof%¸rrsq^#n\uºls~gip]¡rs^\n"9rspo

\^?giuwrspgw^&]¯^&o9gR¢Â^gj&^#qn\pñn\o^]XlE\n\qrgiplmors\rs~grygips^s¢Íl'&olmn\f#gw^&uz]Xpo\lmo\f!qr

grspqq^lm~gip]¡rq^\^Pñrs~\Zrs¬6^#g&«(ÍlmuzfwµBn\^ºq^&fK])#giZlJ\^#f2^ºj#lJrm^º~\rsu¬\qlEj&fK\^Ig¼eJ~6^

Å}P_Æfwlmo9gºngipqpf*&^#f&¢Bo\lmnfºjZ\^#uwjZ\lso\f+]Xry²Jpq]¯pqfz^&uv^{&¬\pªgº^¸r»fz^&fwfzpqlmo¡Y{_t ~\rsu

n\o\^lm~gwpq]¯pqfwrgiplmo¹j#lmoyxzlmpo9gi^\ngrnJ²\^j#lJ\rsm^?^g\^&f~rsuwrs])#giuz^&f!^&fj&lmnjZ\^&f

~\Z9eJfzpqµBn\^#f#mlEµBn&f¡j&pª¶ü\^&fzfwn\f#« §{lmn\f¡]¯lmo9giuzlmo\fXrsn\fzfwpj#lm]¯]X^#oBgXgi^&uz]Xpo\^&u¯q^#f

uirs~\~6lmuhgiffzpqmo\rsJfzn\u¬\uzn\pg^#gvpo9gi^&uh®&uz^&o\j#^rso\fq^%j#lmo9gi^²Jgw^~rsuhgipqj#n\qp^&uK\n±_3»a/-«

§klsgwuw^rso\rseEfw^¡fw^®Álmo\\^¡fwn\u^"jRrsj&n\º^²J~\pqj&pªgi^X\^¸r£lmpPfzgrygipqlso\orspuw^¡n&¬\pªg

#ñno\^¹fz^&fzfwpqlsoWY{_t,É¸rsµBn\^#qq^¡^&fzgXl.-°^&uhgi^n\o\^/jRrs~\rsj&pªg/\^giuwrso\fw]¯pqfzfwplmo\^

C

¬\pgwf{~\rsufw^&j#lmo\\^^#gkµBn\pfwn\¬pg{n\ogrsnJ²/^~6^&uhgi^\^~rsµBn\^#gwfk^

p«0(r®Á^&o1giuw^\^

Y{_t ^#fzg%\pEpqf&^{~rsuº\^&nJ²XfzlmpgP^&o¡jRrsfº\^~6^&uhgi^{\^~rµEn^#gif%fwlmpªgº^#o"jRrf%#ñrgwgw^&poBgw^

\^£qr½jRr~rsj&pªg/

C«2(r½®Álmuw]-n\q^^²J~\pqj#pgi^/lm¬gw^&oBn\^~6lmn\uXq^/&¬pgX]Xl4em^#o #øn\o\^

gi^&q^fz^&fzfwpqlsoo\lmnf/~6^&uz]X^g/\^ÉjRrsuwrsj#g*&uwpfw^#u/^9rspo^Égwuirspªgi^#]X^#oBglsn¾qrgirspqq^

#êrq~\Zr¬°^glm~gwpq]XrsnJ²Xj&lm]¯]¯^fwlsqngwpqlmo\fº#&µBnrgwpqlmo\f^»~°lmpo9g%´²J^o\^»#~°^#o\rso9g

µEn^^&f{jRrsuwrg/#uwpqfhgipµEn^&f\n/jRrso\rsÍuirspql^#g{\^j#^&q^&f\^qr±fz^&fzfwpqlso¹Y{_tK«

3 / Ã + 54 X _3»a¹¢\]XlE\n\qrgiplmo±rsr~grgwps^s¢mrspqo¯\^gwuirspªgi^&]¯^&o9g&¢Ej#lJ\rsm^~\rsu

¬\qlEjs¢9uwrs~\~6lmuzgIfzpqmorfwn\uIpqo9gw^&uz®#uw^#o\j&^k^#gP¬\uwnpgR¢9j#lmo9giu6mq^\^{~\n\pfwfiro\j&^¢J~ruzgrm^\^

¬rso\\^¯~rsfzfirso9gw^s¢j#lmo9giu*6sq^±\^¯j&lmo\s^&fzgwpqlmo;¢Jpªgi^#]X^#oBg\^Xj&lmo\s^&fzgwpqlmo;¢rsmlmuzpgiZ]X^

\^&fPrsj#j&uwlspqfwfz^&]¯^&o9gifPqpoRrpquw^#f^#gI^{qr»&j#uwlmpfwfiro\j&^]?n\ªgip~\qpjRrgwpm^¢&¬\pªgº^{Y{_tK¢

lm~gip]Xpfirgwpqlmo;¢~\uwlEj&^#fwfzn\f/rs &rgilmpuw^¢\pqfhgiuwp¬\ngwpqlmo\fXfzgirgiplmo\orspuw^¢gwuirsofz®Álmuz]7#^\^

a¹^&qpqo;«

(7)

±Í2+),P#+

_%^&qn\¸ru2 pquz^&^&fwfvo^#g¼ ºlmuwÀBfI %^#uw^{lmuwpmpqorqe!\^&fzpqmo\^#¯gwlfwn\~\~6lmuhgvmlmpj&^s¢9 Z\pjZ±Zrsf

fzgiuzpqo\s^&o9g±\^&qrReuw^#µEnpquw^#]X^#o9gif&« ¦¼ogiZ^&fw^/o\^g¼ %lmuzÀBf&¢r~6lÃ º^&u±j#lmo9giuwlsrsqmlsuwpgwZ\]

pqfn\fw^#½gwl¹]Xrspqo9girspqoÉrgrsuzm^#gfzpqmorgil/o\lmpqfz^±uirgwpql½¤¥dJ¦¨§k|©®Álmu^&rsjZ n\fz^&u#«"YZ\^

~°l4 º^&uj&lsoBgwuwlmrsqmlsuwpgwZ\] rsrs~Jgif?gil®=rfzg]?ngip¶¼~rygiZ½®=r\pqo\gwZrg±ruwpqfz^&f\n^"gil

]Xlm¬pqqpªg¼e?ls®Ân\fw^#uwflmuIgiZ^fwlsn\uwj#^&fls®fzjRrgzgi^#uwpqo\¢Jfzl-giZ\rgrj&lmofzgroBg¬\pgºuirgw^ro\r

uw^&µBn\puw^#]¡rÃ²p]?n\]¬pgk^&uwuzlmuuwrgi^pqf{]¡rpqo9grspo\^&®Álmuk^RrsjZj#lmo\o\^#j#gwpqlmoÂ¢6 pgwZ£ql4

giuiro\fw~6lmuzgIqrgi^#o\j#es«YZBn\f&¢B®Álmu^²rs]X~q^s¢m Z^&o"r»n\fz^&uI^&oj&lmn\o9gw^&uwfºr!®=rs\po\»jZrso\o^&

j&lmo\pgiplmoÂ¢gwZ\^giuiro\fw]¯pqfzfwpqlso~°l4 º^&u±pqf¬6lJlsfzgi^#ÆfzlgiZrg±gwZ\^"mlmpj&^j&lsoBs^&uzfirgwpqlmo

jRrso/j&lmo9gwpqoBn\^pqouz^RrsÂgwpq]¯^s«

YZ^&fw^±fheJfhgi^#]Xf»ZrRs^¬°^#^&oÉrsr~gi^#giljRruwuze¹rgirrsfk º^&¥«» ´J²^#ÉjRrs~rj&pg¼e

jZrso\o\^#]XrRe ¬6^ rsqlEjRrgw^&Ä®Álmu/r rgir n\fz^&u#« ^½rsuz^½m^#o\^&uwrsqªeÄpoBgw^&uz^&fzgw^& po

lm~gip]Xp³&po\¡giZ^XjZro\o\^&P~rsuwrs]¯^#gi^#uwfpqoMlmuz\^&ugil/~\uwl4Epq^¯gwZ\^X¬6^&fhg-~6^&uh®Álmuw]Xrso\j#^

®Álmu%giZ^»rgr?nfw^&u#«PÅAlsufwp]X~\pqj#pg¼e±Z\^#uw^! º^rfwfwn]X^!gwZrgrrgir?n\fw^#uj&lmuzuw^&fz~°lso\\f

gil¯r¯fzpqo\sq^!Y{_tÄj#lmo\o\^#j#giplmoÂ« ^rfwfwn]X^giZryggiZ\^jZrsoo\^&pqfrqqlEjRrygi^&rXdJ¦¨§k|

Z\pqjZ£ pqÂ¬6^-]XrspoBgirspqo^&gil¡rXj&lso\fzgirso9ggruwm^gkrsn\^s« ^®ÁlEj&n\f{lmo/giZ\^jRrsfz^-ls®

r£qlmoqpªm^& Y{_tj&lso\o\^&jgiplmoÂ«ÉYZ\^¡girsuwm^g¯dJ¦¨§k|Èrsqn^¡]Xr4e½¬6^rsrs~Jgi^&l4s^&u

gipq]¯^!gilXuw^#fw~6lmo\¹gwl¡n\fz^&uk]Xlm¬pqqpªg¼em¢rso\¹ º^-j#lmo\fzpq\^#u{gwZ\^-uz^&mp]X^! Z\^#uw^giZ\^Y{_t

j&lmo\o^&j#gwpqlmouz^RrsjZ\^#f!rfzgw^RrsJefzgrygi^»¬6^#g¼ º^&^&o£n\~°\rgi^#f{l®2gwZ\^»grsuzm^#gkdJ¦¨§k|rsqn\^«

Îj&lmofwpq^&uÍúwfhggiZ\^PjRrsfzÎls®_3!a¹-¢rso\j&lso\fwp\^&uÍgiZ^P~uwlm¬\^&] l®lm~Jgipq]¯pq³#pqo\

giZ\^¯~\uwlEj&^#fwfzpqo\¹9rspqo

m ^rso j&lE\po\uirgw^

ρ gil£]Xry²Jpq]¯pq³#^?Y{_t giZ\uzlmn\mZ~\ngR«"¿ºe rsyxzn\fhgipqokgiZ^%~\uzlJj#^&fwfzpqo\9rspqorso\-j&lE\pqo\{uirygi^s¢ %^Pgiuir\^¶ül.-±giZ\^º¬\pgKgiuwrso\fw]¯pqfzfwplmo

uirgw^

C rso\É~rsjÀ^#ggiuwrso\fw]¯pqfzfwplmo^&uzuwlmu»uwrgi^

p«¿ºe]XrsÀBpqo"giZ\^±~\uzlJj#^&fwfzpqo\9rpqo

m ¸ruwm^" º^¹po\j&uz^Rrsfz^giZ\^^&o^&uwe ~°^#u¯¬pgXrso\ fwlÉgiZ\^~rjÀs^#gXgiuwrso\fw]¯pqfzfwplmoqlmfzf

~\uwlm¬\rs¬\pqpg¼e

p pqfºfw]XrsqF«kl4 º^#s^&u&¢\gwZ\^!¬\pªgºgwuirsofw]¯pqfwfzpqlmoXuirgw^

C pqf%~uwlm~6lmuzgwpqlmo\rs\gil

1/m^«

^Xrsqfzl/j#lmo\fzpq\^#ugwZ\^Xj#lmo9gi^²Jgls®r/Z\pmZ dJ¦¨§| jZro\o\^&Pl4m^#u-r´²J^& ¬\rso\\p¶

pqgwZÂ«¦¼ogiZpqfj&rsfw^¢B º^fzgwn\e±rrs~gwpm^]¯lE\n\¸rygipqlsoÂ¢4 Z\^&uz^gwZ\^fzeE]?¬6lmrq~\Zr¬°^g

fwpq³#^

M pqfrsrs~gw^&Â«¦¼oMgiZ\pfj&rsfw^¯giZ\^XuirR giuwrso\fw]¯pqfzfwplmo½uwrgi^

C pqf~\uzlm~°lsuzgiplmors

gil log₂(M)¢¬\ng¯giZ\^¹~\uwlm¬\rs¬\pqpg¼e ls®r½~\rsjÀs^g^&uzuwlmu

p muzl4 f¡µBn\pqjÀBªe pgiZ

M^«

^´o\£gwZ\^-lm~Jgipq]XrsÂrsn\^l®vgiZ\^-rsq~\Z\rs¬°^gfzpq³#^

M po£lmuz\^&u{gil¡]Xry²Jpq]¯pq³#^Y{_t giZ\uzlmn\mZ\~ngR«

YKl/n\o\\^#uwfzgirso\ gwZ\^¯orginuw^±ls®%gwZ\^¯lm~gip]Xp³Rrgwpqlmo/ %^Xj&lmofwpq^&u&¢Kpgpfn\fw^®Án\vgil

giZ\po\Àls®Eg¼ ºlk^²Jgwuw^#]X^#f&«gKlmo\^v^²Egiuw^#]X^I º^Pj&rso]¡rsÀ^vgiZ\^I¬\pªgz¶ügwuirsofw]¯pqfwfzpqlmokuirgw^

C

(8)

Z\pqmZ;¢4¬\nggiZ\^v~\rsjÀs^g2gwuirso\fz]Xpfwfzpqlmo!^#uwuzlmuÍuirgw^

p pqs¬6^v¸rsuzm^s«KtrjÀs^#gKgiuwrso\fz]Xpfwfwplmo

^&uwuzlmuwf pqPs^&o\^#uirseÉjRrn\fw^¡gwZ\^"Y{_t*~\uwlgilEj&lmIgiluz^&\nj&^"pªgif po\\l4 )fwp³&^¢Irso\

pqoÉgin\uzo½\^#j&uz^Rrsfz^¡giZ\uzlmn\mZ~\ngR«¡ggwZ\^XlgiZ\^#u^²Jgwuw^#]X^± %^XjRrsoM]XrsÀs^±giZ\^¯~rsjÀ^#g

giuiro\fw]¯pqfzfwpqlsoM^#uwuzlmuuirgw^

p m^#uzeMfw]Xrs¥¢rggiZ^X^²~6^&ofw^"ls®%uw^#\n\j&^# giuwrso\fz]Xpfwfwplmo uirgw^

C«YZ\^¬pggwuirsofw]¯pqfwfzpqlmo£uirgw^

C ngip]¡rygi^&ªeqp]XpªgifgiZ^Y{_t pqo\l4 fzpq³&^¢ fwpqoj&^?¬\n -;^#u!l4m^&ul4 f» pqqlEj&j#n\u Z\^#oÉgwZ\^- po\\l4 Çm^#gwf!fwn! ¯j#pq^&o9gwe¹¸rsuzm^s¢;rso\

Y{_t½ pqj&nJgPgwZ\^ pqo\lÃ Wfwp³&^pqo±Zr®¨¢muz^&fw~6lmo\pqo\»gil»gwZ\^j&lmo\s^&fzgwpqlmogwZrgIlEj&j&nuwf

rggiZ\^¬\n -;^#u&«IYZBn\f&¢AY{_tWgiZuwlmn\sZ\~\ng{pqffz]¡rqÂryggiZ\pf^²Egiuz^&]¯^rf º^&¥«

¦¼oÉlmuz\^&ukgil"´o\gwZ\^lm~gwpq]Xrslm~6^&uirygipqo"~°lspqo9gR¢6 %^j#lmo\fwp\^&urX]XlE\^&K®Álmu!roJ¶

rseE³&po\¹gwZ\^Y{_tÇgwZ\uwlsn\mZ\~\nJg? Z\^&uz^~rsjÀ^#g±qlsfwfw^#f\n\^¡gwlgiuwrso\fw]¯pqfzfwplmo ^&uzuwlmuzf

rso\M~rsjÀs^gqlsfwfw^#fn\^¯gwl¹j#lmo\m^#fzgwpqlmo½^#m^#o9gifrsuz^±\pqfhgipqomn\pqfzZ\^&;« ^Xj#lmo\fwp\^&ur

n\pq/]¯lJ\^#Â Z\^&uz^-giZ^ po\\l4 fwp³&^pqf{j&ngkpqo/Zrsª®®Álmu^&rsjZ~rsjÀ^#gqlmfzfk\n^gilXr

giuiro\fw]¯pqfzfwpqlso^&uwuzlmu»lmu¬\n-°^&ul4m^#u"l4 ^m^#oBg&«¡YZ\^±~rjÀs^#g-qlmfzf»~\uwls¬rs¬\pqpg¼e¹pqf

p^¢

rso\X º^!rfwfwn]X^{giZrgºrj&lmo\s^&fzgwpqlmoX^#s^&o9gZrs~~°^#o\fº Z\^#o¡giZ^k pqo\l4 fwp³&^{uw^RrjZ\^&f

rrsn\^{gwZrgP]XrgijZ\^#fºgwZ\^k¬pgz¶¥giuwrso\fw]¯pqfzfwplmouirgw^s¢

C¢Elmo¯giZ\^{jZrso\o^&¥«YZ\pqfP]XlE\^#

pqfr~\~\uwls~\uwpqrgi^®Álmu%fz]¡rqA¬\n-°^&uzf&«IYZ\pqf%]¡rsÀ^&ffz^&o\fz^! pgwZ\pqoX^s«ø\«giZ\^»_3»a¹ j&lsoJ¶

gi^²EgI Z\^#uw^kgwZ\^¬\pgh¶ügwuirso\fz]Xpfwfzpqlmouwrgi^pqfvfz]¡rq¥¢sj&lm]¯~rsuz^&±gil» ZrgºpqfIrRrspqqrs¬\^lmo

pquw^#qpo\^o^#g¼ ºlmuwÀBf&¢ro\ Z\^&uz^¸rsuzm^\l4 o\qpo\À¹¬n-;^#uwf ºlmn\pq]¯~\eqrsuws^-|YYKf

rso\"Z\^#o\j&^~°lElmuIY{_t ~6^&uh®Álmuw]Xrso\j#^rf%Y{_tgiZ\uzlmn\mZ\~ngpfºÀBo\l4 o"gil-¬°^{uwlmnmZ\e

pqo9m^#uwfw^#e~uwlm~6lmuzgwpqlmo\rs°gwlX|2YY»«

ÅlmuPgiZ\pfº]¯lE\^&F¢E º^´o\r®Álmuw]?n¸r»®ÁlmuºgwZ\^Y{_t giZ\uzlmn\mZ\~ngrsfr®Án\o\jgipqlso¡ls®

p ^rso\ C¢\ Z\pjZ¹pfls®2pqo\\^#~°^#o\\^#oBgkpqo9gw^&uw^#fzg&«PYZpqfpqfrm^#o\^&uwrsqp³Rrgwpqlmo"l®giZ^ º^&

ÀEolÃ o$#¨fwµBnruw^-uzlJlg&%®Álmuw]-n\¸r/¤=^«ñ«%fw^#^('*)Ã&+¸©®ÁlmuY{_tÄgiZ\uzlmn\mZ\~ngk Z^&uw^~rsjÀ^#g

giuiro\fw]¯pqfzfwpqlso¹^#uwuwlsuwf!ro\£¬\n -;^#ul4m^#u"l4 frsuz^o\lgfz^&~rsuwrgi^#e/]¯lJ\^#q^#Â« ^-rsqfzl

lm¬grpqo"®Álmuz]?n\¸rf%®ÁlmugiZ^~uwlm¬r¬\pqpg¼eX\^&o\fzpg¼e"®Áno\j#gwpqlmols®KgiZ\^Y{_tW po\\l4 fzpq³&^«

^´o\¹gwZrg{Y{_tWgiZuwlmn\sZ\~\ngkjRrso¹¬°^»®=rspuwªe"fw^#o\fwpªgipªm^gil±giZ\^~\Z9eEfwpjRrsK¸rRem^#u

~rsuir]X^gi^&uzf&¢kp¥«ø^s«giZ\^M~\uzlJj#^&fwfzpqo\ 9rspo

m lmufheJ]-¬°lmrq~\Zr¬°^gfwpq³#^

M^«·YZ\pf

fwn\ms^&fzgwf»pg]¡rRe¹¬°^±rsJsroBgirsm^#lmn\fgwlrsrs~ggiZ\^#fw^~rsuwrs]¯^#gi^#uwf»rj&j&lsuw\po\"gil¡gwZ\^

^&o9Epquwlso\]X^#o9gR«¦¼oX~rsuzgwpqj#n\¸rsu#¢sgiZ^#e±jRro¡¬6^rs\rs~gi^#¡rj&j&lsuw\po\gwlgwZ\^grsuzm^#g%dJ¦¨§k|

rso\giZ\^uzlmn\o\J¶¥giuzpq~gip]X^±¤=|YY{©pqolmuz\^&ugwl¯]Xry²Jpq]¯pq³#^!Y{_tÆgwZ\uwlmnmZ\~\ng&«

^"rqfwl¹\pqfzj&n\fzf-giZ^¡~\uzlm¬\q^#]ls®rsfzfwpqso\]X^#o9g?ls®r/grsuzm^#g?dJ¦¨§|rsqn^&f&¢rso\

fw~6^&j&pª´jRrqeX\pfwj&nfwfgiZ\^jRrfw^ls®vr? puw^&^&fzf{j#^&qn\qrsu_3»a/\l4 o\pqo\ÀA«

¿º^#®Álmuz^¡\^#fwj&uzpq¬\po\£lsn\u-]¯lJ^&Prso\rorseEfzpqfpqo ]¯lmuz^X\^grsp¥¢ %^¯´uwfhg\pfwj&nfwf

fwlm]¯^»uw^#¸rgw^& %lsuwÀA«

(9)

#+ +

a£rso9e±rsngwZ\lmuzfvZrRm^{j&lmo\fzpq\^#uw^#±gwZ\^m^&o^&uir\~\uwls¬\q^#] ls®°~°^#uz®Álmuz]¡ro\j&^ls®AgwZ\^Y{_t

~\uwlsgwlJj#lmÂl4m^#u{ puw^#q^&fzfkpqo\ÀBf#«P*j#lm]X~\rsuwpfwlmols®2sruwpqlsn\frs~\~\uzl9rsjZ\^#f{gwl¯gwZ\^~\uwls¬J¶

q^&]pqfmpªm^#o¹po 'ø+F«

`!o\^¬\uwrso\jZls®uw^&qrgi^#/ ºlmuzÀ/pfj&lmo\j#^&uzo\^& pªgiZgiZ\^~\uwls¬\q^#] ls®v\^gi^&uz]Xpo\pqo

Z\^#gwZ\^&u»~rjÀs^#glmfwfz^&f»ruw^?\n^-gil"j#lmo\m^#fzgwpqlmolmuk\n\^-gwl"giuwrso\fw]¯pqfzfwplmo£^#uwuwlsuwf&¢;fwl

giZrg"Y{_t·jRrso slpqo9gilj#lmo\m^#fzgiplmo rRmlmprso\j#^]XlE\^lso\e Z\^&o ~rjÀs^#g/lmfwfz^&f

rsuw^\n\^gilMj&lso\m^&fhgiplmoÂ« fz^&j#lmo\W¬uirso\jZÆls® ºlmuzÀj&lmo\fzpq\^#uwf±giZ\^¹rs~\~uwl9rsjZ ls®

fw~\pgwgwpqo\gwZ\^PY{_tÉj#lmo\o\^#j#gwpqlmorgKgiZ\^I pquz^¶¼pqo\^y pquz^&^&fwf;pqo®Áuwrsfzgwuwn\jgin\uz^º¬6lmn\o\\rsuzes¢

fwl-gwZrggwZ\^!Y{_t j&lmo\o^&j#gwpqlmopf%pfwlmqrgi^#X®Áuwls])gwZ\^!~rjÀs^#ggwuirsofw]¯pqfwfzpqlmoX^&uzuwlmuzflmo

giZ\^- puw^#q^&fzf»jZrso\o^&¥« gwZ\pquzÉrs~\~\uzl9rsjZgwllm~gip]Xp³&po\XY{_t~°^#uz®Álmuz]¡ro\j&^?l4s^&u

pquw^#q^#fwfjZrsoo\^&fkpfgil±lm~gwpq]¯pq³&^kgiZ\^qpo\À¯¸rRem^#u®ÁlsuY{_tW~°^#uz®Álmuz]¡ro\j&^«

YZ^-rs~~\uwl9rjZ º^grsÀ^pogiZ\pf~rs~6^&u®=rsqfpqo9gwl?giZ\^»giZpquwj&rgi^#mlmuzes«

;lsuw³&psrso\|{rsl ')R +9j&lso\fwp\^&uz^&»giZ\^v^ -;^#j#gKls®Ej&lsuwuw^#¸rgw^&»^#uwuwlsuwfÍlmoY{_t¹gwZ\uwlmnmZJ¶

~\ngR«v_%lmuwuz^&¸rygi^&X^&uzuwlmuzfºg¼eE~\pqj&rsqªelEj&j&nu%\n\^{gil]?n\gwpª¶ü~rgiZ?®=r\pqo\«¦¼o¡lmnuº]¯lJ^&¥¢

%^-rsfwfzn\]X^?r¯´²J^&MdE¦¨§|¤ÁgiZ\rso\ÀBfkgwl¡~6lÃ º^&uj&lmo9giuzlm©¢Afwl"^#uwuwlsuwfj&rso¬6^-]¯lJ^&q^#

rsfpqo\\^#~°^#o\\^#oBg&«

_%Z\rsfwÀyrsu#¢ (rsÀBfwZ]¡rso;¢rso a£r\Z\l4 'ø +Pj&lmo\fzpq\^#u»giZ\^¯n\fw^¯ls®%qpqoÀ¹qr4es^&u!|

lÃs^&uIgiZ\^ puw^&^&fzfPjZ\rso\o\^#AgwlZpq\^~rjÀs^#gºgiuwrso\fw]¯pqfzfwplmo^&uzuwlmuzf&«¦¼o±giZ\pfIjRrsfz^s¢EY{_t

~\uwp]¡rsuzpqªeuz^Rrsjgif2gwl¬\n -;^#u2l4m^#u"l4 fvlmoem«K¦¼o('ø +F¢spªg2pf2fzn\mm^#fzgw^&-gwZrgKgiZ\^º~\Z9eJfzpqj&rs

¸rRem^#u%~rsuwrs]¯^#gi^#uwfºfwZ\lmnq"¬6^lm~Jgipq]¯pq³#^&¯fwl-gwZrgºgiZ\^!¬n-;^#u%l4m^#u"l4 ¾~\uwls¬rs¬\pqpg¼e

q

gipq]¯^&fºgiZ\^¬rso pqgwZ¹^&¸rRe~uwlE\n\j#gfzµBnrsuw^#£pf^&µBnrsÍgil±lmo^s«

(ÍpnÂ¢ lE^&jÀ^&¥¢krso\ YKl4 fwq^e '*) ) +j&lmo\fzpq\^#uw^# gwZ\^~\uzlm¬\q^#] l®-rrs~gwpqo\ gwZ\^

j&lE\pqo\uwrgi^

ρ gilgiZ^½jZro\o\^&j#lmo\\pªgipqlso\f&¢ pgwZgwZ\^Élm¬Exz^&jgipªm^Éls®?]Xry²Jpq]¯pq³#pqo\

Y{_t½giZ\uzlmn\mZ~\ngR«2¦ügv rfºrquw^&rseo\lsgw^&±pqo '*) ) +\gwZrgvls~gip]¡rsBrsqn^&fvls®6lm~6^&uwrgipo\

~rsuir]X^gi^&uzf®ÁlmugiZ^jZ\rso\o\^#®ÁlsuY{_tW %^#uw^\p-°^&uz^&o9ggiZro/gwZ\lmfw^»®Álmu3!t2«

¦¼o '*)4&+¥¢ (ÍpnÂ¢;Zlmnrso\pqrso\orsÀBpfkn\fz^&Éfwp]?n\qrgiplmogil"fhgin\e¹j&uzlmfwf¨¶¼qr4es^&u!ls~J¶

gipq]¯pq³&rgiplmoX pªgiZ\po"giZ\^rs\rs~gipªm^]¯lJ\n¸rgwpqlmoXfw^#gzgipo\\«

YZ^º]Xrspqoo\l4m^&ªg¼els®EgiZ\^I~\uz^&fw^#o9g~r~°^#uKpfÂgiZ\^v®=rsjgÍgiZrygpg~\uzlÃEp\^&fKrsorsoreBgipj

®Áuirs]¯^# ºlmuwÀ¯®ÁlmugiZpqfrsr~grgwpqlmo;«

dE^#m^#uirsvuz^&j#^&o9g~rs~6^&uwf»~uwl4Epq\^±rso\rseBgipjRrsK®Álmuw]-n\¸rsf{®ÁlmugwZ\^?giZ\uzlmn\mZ~\ngls®ºr

¸rsuzm^Ij&lmq^&jgiplmols®j&ls]X~6^#gwpqo\Y{_tl4 f2 pgwZ-¬6lsgiZj&lmom^&fhgipqlso?rso\giuwrso\fw]¯pqfzfwplmo

^&uwuzlmu!lmfwfz^&f¯¤=fz^&^ 'ñ+IrsogiZ\^uz^#®Á^&uz^&o\j#^&f»giZ^&uw^#pqoA©«YZ^]¡rspo\p-;^#uw^&oj&^? pgwZ 'ê&+

rso\ uz^&qrgi^# ~rs~6^&uzfpfgwZrg %^¡rsuw^¯Z\^#uw^X®ÁlEj&nfwpqo¹lmoMgiZ\^Xuirgw^Xl®r !Y{_t

(10)

j&lmo\o^&j#gwpqlmoj#lmo\fhgiuirpqo\^#?gil»uw^#]¡rpqo

C rso\fwn¬Exz^&jgvgil»¬6lsgiZuwrso\\lm]

giuiro\fw]¯pqfzfwpqlsoÄ^&uzuwlmulmfwfz^&frso\ gwlqlmfzfw^&f"gwZrglEj&j#n\u" Z\^#o¾pªgif"uwrgi^uz^RrsjZ\^#flmu

^²Jj&^#^&\f

C^«

YKl»giZ\^¬6^&fhgPls®6lmn\uvÀBo\l4 q^#\m^s¢9gwZ\pqfvfz^#gwgwpqo\¢mgiZ\^rsfzfwlEj&pqrgi^#±®Álsuw]?n\qrsf2®ÁlsuvY{_t

giZ\uzlmn\mZ\~ng{rso\giZ^rsorsªeEgwpqj!jZ\rsuirsjgi^#uwpq³&rgiplmo\fl®2gwZ\^»lm~gwpq]Xrs6 puw^#q^&fzfjZrso\o^&

~rsuir]X^gi^&uzfgiZrygruw^»\^#uwps^&/pqogiZ^~uw^&fz^&o9gk~rs~6^&u{rsuw^rqÂo\^ «

( (Ç"2+-,'./,"

¦¼oXgiZpqfIfw^#j#giplmoÂ¢9 º^!rsoreE³&^giZ^{gwZ\uwlmnmZ\~\ngºls®Ârfzpqo\m^Y{_t l4 l4m^&uºr puw^#q^&fzf

jZrso\o\^#¥«Æ}IrsjZÄ~rsjÀ^#gXj&lmo9girspqo\f

L ¬\pªgif#¢rso\ Z\^#oWpg±pfgiuiro\fw]¯pgzgi^& l4m^#u¯gwZ\^

jZrso\o\^#¥¢pªg¯pqf±qlmfhg± pgiZÆ~\uwls¬rs¬\pqpg¼e

p^« ^¹rsfwfzn\]X^giZ\rg¡~\rsjÀs^g¡giuwrso\fz]Xpfwfwplmo

^&uwuzlmuwfrsuw^pqo\^&~6^&o\\^#o9gR«YZpqfpfuz^Rrsfzlmors¬\^s¢6fwpo\j&^ %^rfwfwn]X^-gwZrg{giZ\^jZrso\o^&

Zrsf!r¡´J²^#fwpmorsgil"po9gi^&uh®Á^&uz^&o\j#^?~\qnfo\lmpfw^uirgwpql

γ^« (Krygi^&uklmoÂ¢°pqoÉdJn¬J¶hdJ^#j#gwpqlmo

J¢ %^/ pq{j#lmo\fzpq\^#u¡Z\l4 giZ\^¹~rsjÀ^#ggiuwrso\fz]Xpfwfwplmo qlmfzf¡~\uzlm¬rs¬\pqpªg¼e

p ^&~6^&o\\f lmo γ ^rso\ lgiZ\^#u~\Z9eEfwpjRrsP¸rRes^&u~rsuir]X^gi^&uzf&« ^¡rfwfwn]X^¯pggrÀs^&f

L/C fw^#j&lmo\f gilgiuiro\fw]¯pg^RrjZ ~rsjÀ^#glmoMgiZ\^¯jZrsoo\^&F«YZ\^¡~ruirs]¯^#gw^&u

C pfjRrsq^&ÉgiZ^Xuwr4 giuiro\fw]¯pqfzfwpqlsojRrs~rj&pg¼erso\Zrsfn\o\pªgifls®¬\pgwf fz^&j«

^!o\l4 j&lso\fwp\^&u%r n\p"]XlE\^#A®ÁlsuºgiZ^!Y{_t \lÃ «P`!n\uº]XlE\^#°pf%pqo¯gi^#uw]¯f%ls®

giZ\^~\rsuirs]¯^#gw^&uzf

C ^rso\ ppqfwj#n\fwfz^&¯rs¬6l4m^s«v_%lmo\fwp\^&uruirso\lm]*~\uwlEj&^#fwf

X(t)^gwZrg

]XlE\^&fgwZ\^pqo\fhgrso9girso\^&lsn\f?giZ\uzlmn\mZ~\ng®ÁlmugiZ\^Y{_t \lÃ «½YZ\^pqofzgroBgirso\^#lmn\f

giZ\uzlmn\mZ\~ngvpqfrsfwfzn\]X^#gil»¬6^~\uwls~°lmuhgiplmors9gil!gwZ\^Y{_tM po\\l4 Æfwpq³#^s«`»nu]XlE\^#

®ÁlmugiZ\^eEor]Xpj&fIls®

X(t) pfºrfI®Álsqql4 f#« (Í^#g

R ¬6^giZ\^uzlmn\o\±giuzpq~±\^&qrRe/¤Frsfzfwn\]¯^&

j&lmo\fhgrso9g¯Z\^#uw^4©¢ºpo n\o\pgwfls®{fw^#j&lmo\f&«

Z\^&o giZ^&uw^pfo\lÉqlmfzf&¢Prggip]X^

t^¢ X(t)

pqo\j#uw^Rrfw^&fkrgkuirgw^

L/R²^«%¦ü® X(t) uw^&rsjZ\^&f

C¢6r¯j&lmo\s^&fzgwpqlmo/^#s^&o9g!lEj&j#n\uwf#¢°j&rsn\fzpqo\

X(t) gil¬6^uw^#\n\j&^# pqoZrsª®gil£gwZ\^"rsqn^

C/2«MtrsjÀ^#g¯qlmfzfw^#f±rsuw^"]¯lE\^&^& ¬9e rMgip]X^poJ¶¼Zlm]Xlsm^&o\^#lmn\f¯t2lspqfwfzlmoÄ~\uzlJj#^&fzf&¢ Z\^#uw^¹gwZ\^£uwrgi^/ls®»lmfwf¯rg¯gip]X^

t ^pf λ(t) = pX(t)/L«YZ\pfuz^/^#j#gwf-giZ\^¯®=rsj#ggwZrggiZ\^Xuirgw^Xls®%~rsjÀ^#glmfwfpqfZ\pmZ\^&u Z\^&o±giZ\^~\rsjÀs^g%uwrgi^pfZ\pqsZ\^&u#« Z\^#o¯gwZ\^&uz^{pfvr»~rsjÀ^#gºqlmfzfvrgvgip]X^

t¢mgiZ\^rsn\^

ls® X(t) pfuw^&n\j&^#£pqoZr®¨«

`®vj#lmn\uzfw^rX]¯lmuw^uw^´o\^#]XlE\^#Â %lmnq¹girsÀs^pqo9gilXrsj#j&lmn\o9g{giZ\^~\uw^#fw^#o\j&^?l®vr

¬\n-°^&urso\gwZ\^?®=rsjg!giZryg»qlmfzfw^#f»lmo\egirsÀs^~\qrsj&^ Z\^&oÉgiZ\pf!¬\n-°^&u»l4s^&u"\lÃ f#«-¦¼o

(11)

dJ^&jgipqlso£¢ º^ pqÍfwZlÃ *¬9e/fwp]?n\qrgiplmo"giZryg{gwZ\^j&lmo\j#qn\fzpqlmofklm¬Jgrspo\^&®Áuwlm] lsn\u

fwpq]¯~\p´^#"¬\n-°^&uzq^&fzfk]¯lE\^&;rsuw^fzgwpq°rsqp"®ÁlmugiZpqfuw^´o\^#£]¯lJ\^#¥«

^úwfhg!\^ó\^?fzlm]Xô\lsgirgiplmoÂ« (Í^#g

α= pR²/L²«$3k^#´o\^

Λ0 = 1 rso\¹®Álmu

l ≥1

\^#´o^

Λ_l = Yl

j=1

−4 2^2j−1

. ^¤ ^)Ã©

ÅAlsu

l ≥1 ^rso\ n ≥2 ^{\^#´o^}

al,n = Λ_l−1exp(−(2^2(l⁻¹⁾−1)α(C/2ⁿ)²/2)

1 + ( 4

2^2l−1) exp(−(3)2^2(l−1)α(C/2ⁿ)²/2)

. ^¤Fm©

YZ^®Álmql4 pqo-pqf~uwl4m^#£pqok~\~°^#o\\p² J«êJ«

'/ '

f(x) ^!"!, ^#!2

! ,2$ 2

X(t) ^$! ^, ^!% f ^'&

( *)

df(x) dx =

−αxf(x) ^! C/2< x≤C

−αxf(x) + 4αxf(2x) ^! 0≤x < C/2 , ^¤=9©

α = pR²/L² ^+! f(x) ⁺ ^2$ x = C/2 ^,

f((C/2)⁺)−f((C/2)⁻) = f(C⁻) . ^¤ÁB©

-!

f(x) ^/. ⁰ f(x) =

Xn

l=0

V_n−lΛle⁻²^2l^αx²^/2 ^¤Fm©

!

C/2ⁿ⁺¹ < x < C/2ⁿ ^!2 n ≥ 0²¹ ^! V₀, V₁, . . .

2 ! 0*) !

V1 =V0(1 + (1/3)e^−3αC²^/8) , ^¤=9©

(12)

Vn= Xn

l=1

al,nV_n−l ^®Álmu n >1 , ^¤ ^m©

,

X∞

m=0

Vm

"Z C/2^m C/2^m+1

exp(−αx²/2)dx

#

=

" _∞ X

k=0

2^−kΛk

#−1

. ^¤ ^©

¦üg{pfpqo9gi^#uw^&fhgipo\gil±o\lsgw^gwZrgpo¹rq;j&rsfw^#fk^²rs]Xpo\^&fzl¯®=rsu#¢giZ^j#lmo\fhgrso9gif

Vn

rs~\~6^Rrsugwl¯j#lmo9m^#uwm^gil¯rj&lmo\fhgrso9gkrsf

n → ∞^«

$

Åpquzfzgj#lmo\fwp\^&u"gwZ\^ÉjRrsfz^ Z\^&uz^

p = α = 0^« ^¦¼o gwZ\pqfj&rsfw^Épg"jRrso¬°^s^&uzp´^#

giZrg Pn

l=0V_n−lΛ_l = 0 ^®Álmurq n > 1 rso\ÉZ\^#o\j&^

f(x) pqfrn\o\pª®Álmuw] \pqfhgiuwp¬\ngwpqlmo lmo [C/2, C]«YZpqf±pqf±^²J~6^&jgi^&Æfwpo\j&^/pqogiZpqfjRrfw^giZ\^giuwrÃxz^&jgilmuhe ls®

X(t) ^pf±r

firR gilElsgiZ/~rgzgi^#uworsuzeEpqo¯¬6^#g¼ º^&^#o

C/2 ^ro\ C^«

§{^²Eg±j&lso\fwp\^&u-gwZ\^jRrfw^¡ Z\^#uw^

C rs~\~\uzl9rsjZ\^#f¯po´o\pªg¼em«¦¼o gwZ\pqf?j&rsfw^¢Ipg?j&rso

¬°^m^#uwpª´^&ÉgiZryg

Vn

gw^&o\\fgil

V0

®Álsursq

n^« k^&oj&^¯pqogiZ\pf»jRrsfz^¯ º^¯ZrRm^

f(x) = V0(P_∞

l=0Λle⁻²^2l^αx²^/2) ®Álmukrs

x >0«YZ\pqf{pqf{~\uw^#j&pfw^&ªegiZ\^fir]X^uw^#fwn\ªglm¬Jgrspo\^&

pqo 'ø +F«YZ\pfpqf2^²J~°^#j#gw^&¯fwpqoj&^^#gwgwpqo\

C rs~~\uwl9rjZpqo´opg¼epqf^#µBn\prs^&o9ggilZrRJpo\

giuiro\fw]¯pqfzfwpqlsoqlmfzfw^&flmoem¢J Z\pqjZ/j&lmpo\j&p\^&f pªgiZ~\uz^#Epqlmnf{]¯lE\^&f&«

ô\l4 pqqn\fhgiuirygi^giZ\^±rs¬6l4mûz^&fwngif! pªgiZfzlm]XôBn\]¯^&uzpqj&rs^²\r]X~\^&f#«?dJn~\~°lsfw^

C = 5^¢ R= 100^rso\ L= 40«2¦¼oÅpmn\uw^ )m¢sgwZ\^%\^#o\fwpªg¼e

f(x)l®giZ\^po\fzgirso9grso^&lmn\f giZ\uzlmn\mZ\lsngpf%~\lsgwgw^&¡s^&uwfzn\f

x ®Álmu%g¼ ºl\p-°^&uz^&o9grsqn^&f%l®ÍgiZ\^»~rjÀs^#glmfwfº~\uwls¬J¶

rs¬\pqpg¼e

p^{«2YZ\^q^®}gv~\qlgvfwZ\l4 fIgiZ\^j&rsfw^

pfw]Xrs¥«¦¼ogiZ\pfvjRrsfz^s¢Bo\lsgw^giZrgvgwZ\^&uz^pqfvr

Jpfwp¬\q^\pfwj&lsoBgwpqoBn\pªg¼e±pqoXgwZ\^{\^&o\fzpg¼e¡rg

x=C/2¢Euw^^&jgipo\-giZ^Ixzn\]¯~\fI®Áuwlm]

C ^gil C/2 Z\^#o¹giZ^po\fzgirso9grso^&lmn\fgwZ\uwlmnmZ\~\ng{Z\pªgifgwZ\^¬°lmno\rsuhe

C«PYZ^uzpqmZ9g~\lsg fwZ\l4 fgiZ\^jRrfw^

p ¸rsuzm^s«¦¼ogiZpqfjRrfw^giZ\^~rsjÀ^#gkqlmfzf~\uwls¬rs¬\pqpg¼eXpqfZpqmZ^#o\lmn\mZ fwlXgwZrg{pg{pfkn\oqpqÀ^&ªe"giZryg{gwZ\^pqo\fhgrso9gro\^&lmnfkgiZuwlmn\sZ\~\ngkpqfo\^&rsu{gwZ\^¬°lmno\rsuhe

C¢°rso\/giZBn\fkgwZ\^&uz^-pfko\l¯Epqfwp¬\q^\pfwj#lmo9gipqoBn\pªg¼epqo/giZ\^\^&ofwpg¼e£rg

x =C/2^{«{§{lsgi^}

giZrg{po£rsÍjRrsfz^&f#¢

f(x) pfo\^Rrsuze³#^&uzl¡ Z\^&o

x pqfj&lmfw^gil¯³&^#uwl\¢A Z\pjZ£pqf{¬6^&jRrn\fw^

~rsjÀs^g-lmfwf»¬6^&j#lm]¯^&frsoMn\o\qpÀs^#e¹^#m^#o9g- Z\^#oMgwZ\^±pqo\fhgrso9gro\^&lmnfgwZ\uwlmnmZ\~\ngpf

o\^Rrsu

x= 0^«

(13)

0 0.1 0.2 0.3 0.4

1 2 3 4 5

x

0 0.2 0.4 0.6 0.8 1

1 2 3 4 5

x

Å2pqmn\uz^ )'"Itvuwls¬rs¬\pqpg¼e 3^#o\fwpªg¼els®¦¼ofzgroBgirso\^#lmn\f{Y{_tÄYZ\uwlmnmZ\~\ng "P}²rs]¯~\^&f

YZ\^-]¯^RrsogiZuwlmn\sZ\~\ngls®Y{_t pqf{mpªm^&o¹¬9e

T CP(C, p) =RC

0 xf(x)dx«{dJ~6^&j#p´¶

T CP(C, p) =

" _∞ X

k=0

4^−kΛ_k

# _∞ X

m=0

V_m

Z C/2^m C/2^m+1

¦¼oÄgiZ\^/®ÁlmqlÃ po\ÉgiZ\^#lmuw^#] º^lm¬girspqoWr j&lmfw^#Ä®Álmuw] ^&µBnrygipqlsoW®Álmu

T CP(C, p)^¢

Z\pqjZ^Rrs\fºgilr-fzpq]¯~\q^krs~\~uwlR²Jpq]XrgiplmoX®Álmu

T CP(C, p)^« ^{(Í^g} f(u)b ¬6^gwZ\^a¹^#qqpo giuiro\fz®Álmuz]·ls®

f(x)^"

fb(u) = Z C

0

(14)

pgiZ

u≥1^« ^{(Í^#g} Γ(u) ¬6^»giZ\^a¹^#qqpo¡giuwrso\fh®Álmuw]l®

e^−x^{¢p¥«ø^s«}

Γ(u) = Z _∞

0

e^−xx^u−1dx .

§klsgw^gwZrg

Γ(·) pqfgwZ\^ %^#qÀBo\l4 oW9r]X]Xr¹®Án\o\jgiplmoWro\ firgwpqfz´\^&f±giZ\^pq^&o9gipªg¼e

Γ(u+ 1) =uΓ(u)«IÅlmurs

l≥0¢\\^#´\o\^

Πl(u) = Yl

k=0

1−2^−u−2k

. ^¤))Ã©

YZ^®Álmql4 pqo-pqf~uwl4m^#£pqok~\~°^#o\\p² J«ñ«

'/ ' 3 ' - / .!'!2 $!

!

f(x) ^$! ^! ^,2 ² ^/. ⁰

f(u) =b P

l≥0Πl(u)C^u

αC² 2

l

Γ(u/2) Γ(u/2+l+1)

P

l≥0Πl(1)C ^αC₂²l Γ(1/2) Γ(1/2+l+1)

. ^¤)4m©

2

T CP(C, p) = P

l≥0Πl(2)C²

αC² 2

l 1 (l+1)!

P

l≥0Πl(1)C ^αC₂²^l ^√π Γ(1/2+l+1)

. ^¤)R9©

' \ . ( "/ !!2 2

0 ( 0 $! ) , $! 2

. 2 0 . "/ !

*) . . ** *) !

! "/ ! ,

# *2 *

(15)

/ / 4 4Á\ ) z / / /$3 ' - / /

2 '&

T CP(C, p) = 3C

4 − pR²C³ L²

11

256 − p²R⁴C⁵ L⁴

497 491520 + 925667

377487360

p³R⁶C⁷

kpqsZ\^&ulmuz\^&u^²J~rso\fzpqlmo¹rsuw^^&rsfwpe"ls¬grspo\^&n\fzpqo\±^«ñ\«Pa¹rs~\q^«

kf^²J~6^&jgi^&;¢giZ\^ksrqn\^#f%ls®ÂgwZ\^gwZ\uwlmnmZ\~\ngrsuw^kpqo\fz^&o\fzpgwpm^{gil-gwZ\^rsn\^kl®

C

p®\gwZ\^~rsjÀ^#gIlmfwf~uwlm¬r¬\pqpg¼e

ppffzn! ¯j&pq^#o9gieZ\pmZÂ«¦¼o?®=rj#g º^ZrRm^gwZ\^%®Álmql4 pqo

j&lmuzlmqqrsuzes«*fwÀ^#gijZ£ls®2gwZ\^~\uwlEls®Kpqf~uwl4Epq\^#¹po/k~\~6^&o\\p² J«êJ«

/ / 4 4Á\

) 2

p > 0 ^.

Clim→∞T CP(C, p) = r2

α

√1 π

Π_∞(2)

Π_∞(1) = 1.309

√pR/L, ^¤)4m©

YZ^!uw^#fwn\ªg%po_%lmuzlmqqrsuze-j&lmpo\j&p\^&fP pgiZ 'ñ+¥¢B Z\pqjZ¡pqf%^²J~°^#j#gi^#fwpo\j&^!^#gzgipqo

C rs~\~\uzl9rsjZ

∞ pqf^&µBn\pªrsq^#oBggwl¯j#lmo\fzpq\^#uwpqo¯uwrso\\ls]giuwrso\fz]Xpfwfwplmoqlsfwfw^#flmo\es«

.& $»#+ + 2# X¾»

¦¼ogwZ\pqffz^&jgipqlso º^j&lmofwpq^&ulm~gwpq]¯pq³&rgiplmoXl®2gwZ\^! pquz^&q^#fwfjZro\o\^&Í~\rsuirs]¯^#gw^&uzfgil

]¡ry²Jp]Xp³&^!giZ^gwZ\uwlmnmZ\~\ngls®gwZ\^Y{_t fw^#fwfzpqlmo£gwZrgk~rsfzfw^&fkgiZ\uzlmn\mZ¹pgR« ^-j&rso

T CP(C, p) ^rsfr

®Án\o\j#gwpqlmoXls®

C ^rso\ p«YZ\^k pquw^#q^#fwfPjZrso\o\^#;~ruirs]¯^#gw^&uwfP pq\^#gw^&uw]¯pqo^

C ^rso\ p^¢

rso\ %^ pqfzZ/gwl±]¡ry²Jp]Xp³&^

T CP(C, p) pªgiZuw^#fw~6^&jgkgwl±gwZ\^&fz^~\rsuirs]¯^#gw^&uzf&«

^ pqq%j&lmo\fzpq\^#ug¼ %l½uw^&spq]¯^&f&¢IgiZ^¹lÃ dJ¦¨§|uz^&mp]X^ro\ÆgwZ\^Z\pqmZWdJ¦¨§k|

uw^&spq]¯^s«KÅlmuÂ^²rs]¯~\^s¢4giZ\^IlÃ dJ¦¨§k| uw^#mpq]¯^vrsuzpqfw^#fpqo!giZ^Pj#lmo9gi^²Jg2ls®Jj&lE\^¶ü\pEpfwpqlso

m pqfn\fz^&

gilrsÃxzn\fzg»gwZ\^-giuwrso\fz]Xpfwfwplmorygr"uirygi^s¢;lmu^#µEnprsq^#o9gie/giZ^-giuwrso\fw]¯pqfzfwplmo^&o^&uwe

n\fw^# ~°^#u-¬\pªgR«¸rsuzm^±~\uwlEj&^#fwfwpo\9rpqoMpqfm^&o\^#uirseÉo\^#^&\^# gil¹j&lm]¯~°^#o\firygi^±®Álmu

ql4 dJ¦¨§|« ^»fwZrq;j#lmo\fzpq\^#u%giZ\^»~uwlm¬\^&]Èls®Klm~gwpq]¯pq³&po\giZ\^»~\uzlJj#^&fzfwpqo9rspqoXgil

]¡ry²Jp]Xp³&^Y{_t giZ\uzlmn\mZ\~ngR« ^fzZrsÍrsqfzlj&lmofwpq^&ugiZ\^n\fz^ls®^#uwuzlmuj&lmuzuw^#j#giplmo

j&lE\^&f#¢Arso\?xzlmpqo9giªe¡ls~gip]Xp³&^kgwZ\^j&lE\pqo¯uwrgi^ro\/gwZ\^~\uwlEj&^#fwfzpqo\¯9rspoÂ«

(16)

Z^&o½gwZ\^XdJ¦¨§| pfZ\pqmZ;¢Ípgpqf~6lmfwfzpq¬\^?giluw^#qpqrs¬\e¹fw^#o\ ]¯lmuz^±gwZrsoÉlmo\^¯¬\pªg

~°^#uIgwuirsofw]¯pgwgw^&¯fzeE]?¬6lm¥«2¦ü®ÂgiZ^koBn\]?¬6^&uPls®;~°lmfzfwp¬\q^rsn\^&fPls®Â^&rsjZ¡fheE]?¬°ls6pf

M^¢

giZ\^#o

M pqfjRrqq^#giZ\^rs~\Zrs¬6^#gfzpq³&^«ºYZ^oBn\]?¬6^&uls®¬\pgwf~°^#ufzeE]?¬6lmpqf

log₂M^«

Çqrsuws^dJ¦¨§| m^#o\^&uwrsqªe¡^#ors¬\^&f{n\fw^ls®rso¹rsq~\Z\rs¬°^gfwp³&^muw^&rgi^#ugiZrsog¼ ºl¯ pªgiZ

ql4 ~\uzlm¬rs¬pqqpªg¼e l®»fzeE]?¬6lm^&uzuwlmu#« ^¹j#lmo\fwp\^&u±giZ^¹~\uzlm¬\^&] ls®!rsyxzn\fhgipo\MgwZ\^

rsq~\Z\rs¬°^gfwp³&^pqolmuz\^&ugwl¯]Xry²Jpq]¯pq³#^Y{_tWgwZ\uwlmnmZ\~\ng&«

¦¼ogwZ\^?o\^²Egfwn\¬\fz^&jgipqlsoÂ¢; º^~\uw^#fw^&o9grX puw^&^&fzf!jZrso\o^&]¯lJ^&¥¢A Z\pjZeJp^&\f

®Álmuw]?n¸rsf%®Álsu

p ^rso\ C rsf{r?®Án\o\j#gwpqlmols®KgiZ^ puw^#q^&fzfjZrso\o\^#~rsuir]X^gi^&uzf»«

$

^?´uzfzgj&lso\fwp\^&ur]XlE\^#rs~\~uwlm~\uzp¸rgw^?®Álmu»rql4 dJ¦¨§|ryg»giZ\^uz^&j&^#ps^&u&¢KpqogwZ\^

j&lmo9gi^²Egls®r¯_3»a¹Ço\^#g¼ ºlmuzÀ6«

/ 1

' / 3 4Á

4 ' - /

Í

^¯ pqqvj#lmo\fzpq\^#ugiZ^¡jRrfw^¡l®j&lE\^¶ü\pEpfwpqlso½]?ngip~\q^²Jpqo\ pgwZ \puw^#j#gfw^&µBn\^#o\j&^

fw~\uz^Rrs fw~6^&jgiuwn] ]XlE\n\qrgiplmoÂ«É¦¼o gwZ\pqfj&rsfw^¢º º^/rsfwfzn\]X^gwZrg±fwpmorsfrsuz^/fz^&o9g

pgiZ/\puw^&jgfz^&µBn\^#o\j&^?fz~\uw^&rsfz~°^#j#giuzn\] ]XlE\n\qrgiplmoÂ¢E pgiZ¹rXjZpq~£n\uirgwpqlmo/ls®

Tc

]XlE\n\qrgiplmofwjZ\^#]X^«}PrjZÄ¬\pg¯giuwrso\fz]Xpªgwgi^#ÆpfX^#o\j&lE\^#Äpqo9gil

m jZ\p~\fXnfwpqo r fw~\uz^Rrs\po\fw^#µEn^&o\j#^s¢m Z\^#uw^

mpfKj&rsq^&!gwZ\^ * «2YZ\^I¬\pªgÍgiuwrso\fw]¯pqfzfwplmo uirgw^»pqfgiZBn\f

C= 1

mT_c. ^¤)R9©

YPeE~\pjRrsem¢º]¡roBe n\fw^#uwf"]XrRe giuwrso\fw]¯pg¯rygXgwZ\^¹fwrs]¯^/gwpq]¯^s¢PjRrsn\fzpqo\½pqo9gi^#uz®Á^#uw^&oj&^

rg^RrsjZ/n\fz^&uwfuz^&j&^#ps^&u&«P}IrsjZ/n\fw^#uZrsf{ruw^&j#^&pªm^&u Z\pjZ¹j#lmuwuz^&qrgi^#fgiZ\^»po\j&lm]¯pqo

fwpqsorsJ pªgiZ?gwZ\^fw~\uz^Rrspqo\»fz^&µBn\^&oj&^{n\fw^#¡rggiZ\^gwuirso\fz]Xpªgwgw^&u&«(Í^#g

γ ¬6^giZ\^fzpqmor gilolmpqfz^%~\n\f2po9gi^&uh®Á^&uz^&o\j#^uirgwpql?¤¥dJ¦¨§k|©rggwZ\^%lsngi~\nJgls®\gwZ\^%j#lmuwuz^&qrgilmu#« ^%fzZrs

\^#´o^

γ ]¡rygiZ\^#]¡rgwpqj&rsqªe Z\^&o/ %^\pfwj&nfwfkrsfwfzpqmo\]¯^&o9gls®dJ¦¨§k|srqn\^#f¸rgw^&u&«

¦ü®A º^]XlE\^#JgiZ^pqo9gi^#uz®Á^#uw^&oj&^rggwZ\^j&lsuwuw^#¸rgwlmuKlmngi~ngPrsf rsn\fzfwp¸roÂ¢sgiZ^~\uwls¬J¶

rs¬\pqpg¼e¯ls®r¯¬pg^#uwuwlsu{pf

Z\^&uz^

Q(·) pqf!giZ^"_3!Åls®PgiZ\^¯³&^#uwl/]X^&rso½no\pg»rsuzp¸rso\j#^ rn\fwfzp¸rsoM\^&ofwpg¼es¢2pF«ñ^«

Q(x) = (1/√

2π)R_∞

x e⁻^x²^/2dx^«