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¾ðÊó¤òk¥Ó¥Ã¥ÈËè¤Î¥Ö¥í¥Ã¥¯¤Ë¶èÀÚ¤ê, ¥Ö¥í¥Ã¥¯Ëè¤ËÉä¹æ²½¤·¤ÆÁ÷¿®¤¹¤ë¥Ö¥í¥Ã¥¯Éä¹æ¤¬¿ô³ØŪ¤Ë¤É¤¦µ½Ò¤µ¤ì¤ë¤«, ¤½¤·¤Æ¤É¤Î¤è¤¦¤ÊÀ¼Á¤ò»ý¤Ä¤«¤ò¹Í¤¨¤Æ¤ß¤è¤¦.
B¤ò0¤È1¤ÎÆó¤Ä¤Î¸µ¤«¤é¤Ê¤ë½¸¹ç¤È¤¹¤ë¤È, 0¤È1¤«¤é¤Ê¤ëk¥Ó¥Ã¥È¤Î¿®¹æÁ´ÂΤÏB¤Îk¸Ä¤ÎľÀÑBk¤ÈƱ°ì»ë¤Ç¤¤ë.
¿ô»ú¤ò°ì¸Ä¤º¤Ä¤Ð¤é¤·¤Æ³ç¸Ì¤ËÆþ¤ì¤¿¤À¤±¤È»×¤¦¤«¤âÃΤì¤Ê¤¤¤¬, ¤½¤ÎÄ̤ê¤Ç¤¢¤ë.
¤¿¤À, ¤³¤¦¤¹¤ë¤³¤È¤Ç, ¼¡¤Î¤è¤¦¤Ê¡Ö¿ô³Ø¡×Ūɽµ¤¬²Äǽ¤Ë¤Ê¤ë¤Î¤Ç¤¢¤ë.
³Æk¥Ó¥Ã¥È¤Î¿®¹æ¤Ël¥Ó¥Ã¥È¤Î;ʬ¤Ê¾ðÊó¤òÉղä·¤ÆÁ÷¤ë¥Ö¥í¥Ã¥¯Éä¹æ¤ò¹Í¤¨¤ë¤³¤È¤Ë¤¹¤ë¤È, ¤³¤ÎÉä¹æ²½¤Ï¼¡¤Î¤è¤¦¤Ë¥â¥Ç¥ë²½¤µ¤ì¤ë.
¤ËÂФ·,
¤¬
x¤ÎÉä¹æ²½¤µ¤ì¤¿¤â¤Î¤È¤¤¤¦¤ï¤±¤Ç¤¢¤ë(ñ¼Í¤È¤¤¤¦¾ò·ï¤¬É¬ÍפʤΤÏ, °ã¤¦¥Ç¡¼¥¿¤¬Éä¹æ²½¤·¤ÆƱ¤¸¤â¤Î¤Ë¤Ê¤Ã¤Æ¤Ï, Éü¹æ¤¬¤Ç¤¤Ê¤¯¤Ê¤ë¤«¤é¤Ç¤¢¤ë).
¤³¤Î¤è¤¦¤Ë¥â¥Ç¥ë²½¤µ¤ì¤ë¥Ö¥í¥Ã¥¯Éä¹æ¤Ë¤Ä¤¤¤Æ,
k¤ò¾ðÊóĹ(information length),
n¤òÉä¹æĹ(code length),
n-k¤ò¾éĹÀ(redundancy),
k/n¤òÉä¹æ²½Î¨(infomation rate, ¤Þ¤¿¤Ï code rate)¤È¤¤¤¤,
Éä¹æĹn, ¾ðÊóĹk¤ÎÉä¹æ¤ò(n,k)-Éä¹æ¤È¤¤¤¦.
¤Þ¤¿, ñ¼Í¤Ë¤è¤ëBk¤ÎÁü
¤òÉä¹æ(code)¤È¤¤¤¤,
¤Î¸µ¤òÉä¹æ¸ì¤È¤¤¤¦.
°Ê²¼,
¤òC¤È½ñ¤¯¤³¤È¤Ë¤¹¤ë.
ÍѸì¤Î¤Þ¤È¤á |
Éä¹æĹ |
= |
Éä¹æ¤Î¥Ó¥Ã¥È¿ô |
¾ðÊóĹ |
= |
¾ðÊó¥Ó¥Ã¥È¤ÎŤµ |
¾éĹÀ |
= |
Éä¹æĹ-¾ðÊóĹ |
Éä¹æ²½Î¨ |
= |
¾ðÊóĹ¡¿Éä¹æĹ |
ÌäÂê 4.2
Îã
, Îã
¤ÎÉä¹æ¤Ë¤Ä¤¤¤Æ,
Éä¹æĹ, ¾ðÊóĹ, ¾éĹÀ, Éä¹æ²½Î¨¤òµá¤á¤Æ¤ß¤è¤¦.
°Ê¾å¤Îµ¹æ¤Î²¼¤Ç, ¥Ö¥í¥Ã¥¯Éä¹æ¤Î¸í¤êÄûÀµ¤Ë¤Ä¤¤¤Æ¹Í¤¨¤Æ¤ß¤è¤¦.¤½¤Î¤¿¤á¤Ë, ¤Þ¤º¥Ï¥ß¥ó¥°µ÷Î¥¤È¤¤¤¦³µÇ°¤òƳÆþ¤¹¤ë.
ÄêµÁ 4.3
¤ËÂФ·,
(¤¹¤Ê¤ï¤Á°Û¤Ê¤ëÀ®Ê¬¤Î¸Ä¿ô)
¤ò
x¤È
y¤Î
¥Ï¥ß¥ó¥°µ÷Î¥¤È¤¤¤¤,
d(
x,
y)¤Çɽ¤¹. ¤³¤ì¤Ïµ÷Î¥¤Î¸øÍý¤òËþ¤¿¤¹
.
Îã 4.4
Î㤨¤Ð, n=4¤Î¾ì¹ç,
d((0,0,0,0), (0,0,0,1))=1,
d((0,1,0,1), (1,1,0,0))=2¤Ê¤É¤È¤È¤Ê¤ë.
¤µ¤Æ, Éä¹æ¸ì
¤òÁ÷¿®¤·¤Æ, ¼õ¤±¼ê¤¬¼õ¿®¤·¤¿¼õ¿®¸ì¤¬
x'¤À¤Ã¤¿¤È¤¹¤ë¤È¤,
¤Ç¤¢¤ë¤«¤é,
¤³¤Î¥Ï¥ß¥ó¥°µ÷Î¥¤ò»È¤¦¤È,
¤Èɽ¸½¤µ¤ì¤ë¤³¤È¤¬Ê¬¤«¤ë.
¤¢¤ëÄÌ¿®¤ò¹Ô¤Ã¤Æ, ¼õ¿®Â¦¤¬
¤È¤¤¤¦¼õ¿®¸ì¤ò¼õ¤±¼è¤Ã¤¿¤È¤¹¤ë. ¼õ¿®Â¦¤ÏÅöÁ³Á÷¿®¸ì¤¬²¿¤Ç¤¢¤Ã¤¿¤«ÃΤë¤è¤·¤â¤Ê¤¤¤«¤é, ¼õ¿®¸ì¤«¤éÁ÷¿®¸ì¤òÅö¤Æ¤ë(Éü¸µ¤¹¤ë)¤³¤È¤¬É¬ÍפȤʤë. ¤³¤ì¤Ë¤Ï¤É¤¦¤¹¤ì¤Ð¤è¤¤¤À¤í¤¦¤«¡©
¤¤¤Þ, ³Æ¥Ó¥Ã¥È¤ËÂФ·¤Æ, ¸í¤ê¤¬À¸¤¸¤ë³ÎΨ¤¬¤É¤ì¤âÅù¤·¤¯ÆÈΩ¤Ëp¤Ç¤¢¤ë¤È¤·¤è¤¦(2¸µÂоÎÄÌ¿®Ï©). p¤ÏÄÌ¿®Ï©¤Î¥Ï¡¼¥ÉŪ¤ÊÆÃÀ¤«¤éÄê¤Þ¤ëÈó¾ï¤Ë¾®¤µ¤¤¿ô¤Ç¤¢¤ë¤¬(¤È¤¤¤¦¤«½½Ê¬¾®¤µ¤¯¤Ê¤±¤ì¤Ð¿®ÍѤΤª¤±¤ëÄÌ¿®Ï©¤È¤Ê¤é¤Ê¤¤), ¤³¤³¤Ç¤ÏÊص¹¾åp=10-a, (a>1)¤È¤Ç¤â¤·¤Æ¤ª¤³¤¦. ¤¹¤ë¤È,
x¤¬ÄÌ¿®Ãæ¤Î¥¨¥é¡¼¤Ç
y¤Ë¤Ê¤ë³ÎΨ
¤Ï,
¤È¤Ê¤ë¤«¤é,
¤ËÂФ·¤Æ,
¤Ç¤¢¤ê,
x'¤Ë¥Ï¥ß¥ó¥°µ÷Î¥¤Ç¶á¤¤°ÌÃ֤ˤ¢¤ë¤â¤Î¤Û¤É
x'¤ÎÁ÷¿®¸ì¤Ç¤¢¤ë³ÎΨ¤¬¹â¤¤(¶Ïº¹¤Ç¤Ï¤Ê¤¯°µÅÝŪ¤Ë¹â¤¤)¤³¤È¤¬Ê¬¤«¤ë.
¤·¤¿¤¬¤Ã¤Æ, ºÇ¤â¸¶»ÏŪ¤Ê¸í¤êÄûÀµ¤ÎÊýË¡¤È¤·¤Æ,
x'¤È¤Î¥Ï¥ß¥ó¥°µ÷Î¥¤¬°ìÈÖ¾®¤µ¤¤C¤Î¸µ¤òÁ÷¿®¸ì¤À¤È¡Ö»×¤¦¡×(ÅöÁ³¥Ï¥º¥ì¤Æ¤¤¤ë¤³¤È¤â¤¢¤ë)¤È¤¤¤¦Éü¹æË¡¤¬¹Í¤¨¤é¤ì¤ë¤Ç¤¢¤í¤¦. ¤³¤ì¤òºÇÌàÉü¹æ(ºÇ¤âÌà¤â¤é¤·¤¤¤â¤Î¤òÂò¤ë¤È¤¤¤¦°ÕÌ£)¤È¤¤¤¦.
¤³¤ì¤Ç¤¹¤Ù¤Æ¤¬¾å¼ê¤¯¤¤¤±¤Ð¤è¤¤¤Î¤À¤¬, »ÄÇ°¤Ê¤¬¤éºÇÌàÉü¹æ¤Ë¤Ï¼¡¤Î¤è¤¦¤Ê¼åÅÀ¤¬¤¢¤ë.
- 1.
- C¤Î¸µ¤òÁíÅö¤¿¤ê¤ÇÄ´¤Ù¤ë¤Î¤Ç, ²¿¤È¤¤¤Ã¤Æ¤â»þ´Ö¤¬¤«¤«¤ë.
- 2.
- °ìÈֶᤤ¤È¤³¤í¤Ë¤¢¤ë¤â¤Î¤¬°ì¤Ä¤À¤±¤È¤Ï¸Â¤é¤Ê¤¤.
¤³¤¦¤¤¤Ã¤¿¼åÅÀ¤ò²þÁ±¤·¤¿Éü¹æ¤¬, º£¤«¤é½Ò¤Ù¤ëºÇ¾®µ÷Î¥Éü¹æ¤Ç¤¢¤ë.
¤³¤ì¤òÀâÌÀ¤¹¤ë¤¿¤á¤Ë, ¼¡¤ÎÎ̤¬Âç»ö¤Ë¤Ê¤ë.
ÄêµÁ 4.5
Éä¹æ
¤ËÂФ·,
¤ò
C¤Î
ºÇ¾®µ÷Î¥¤È¤¤¤¤,
d(
C)¤Þ¤¿¤Ï, ñ¤Ë
d¤È½ñ¤¯.
¤Þ¤¿, ºÇ¾®µ÷Î¥¤¬
d¤Î(
n,
k)-Éä¹æ¤ò(
n,
k,
d)-Éä¹æ¤È¤¤¤¦.
¼õ¿®¸ì
¤ËÂФ·¤Æ,
¤È¤¤¤¦½¸¹ç¤ò¹Í¤¨¤ë. ¤¹¤ë¤È, d¤¬C¤ÎºÇ¾®µ÷Î¥¤Ç¤¢¤ë¤³¤È¤«¤é,
Ǥ°Õ¤Î
¤ËÂФ·¤Æ,
¤«¤Þ¤¿¤Ï
Cx'¤ÏÍ£°ì¤Ä¤Î¸µ¤«¤é¤Ê¤ë. ¤³¤Î¤È¤, ¼õ¿®¸ì
¤ËÂФ·¤Æ,
¤Ë¤è¤Ã¤Æ¼õ¿®¸ì¤«¤é¤Î¸í¤êÄûÀµ¤ò¹Ô¤¦Éü¹æË¡¤òºÇ¾®µ÷Î¥Éü¹æ¤È¤¤¤¦(²¼¿Þ»²¾È).
ºÇ¾®µ÷Î¥Éü¹æ¤òÍѤ¤¤¿¸í¤êÄûÀµ¤Ë¤ª¤¤¤Æ¤Ï, Éü¹æ¤òÄü¤á¤ë¾ì¹ç¤ò½ü¤¤¤Æ,
Í£°ì¤Ä¤ÎC¤Î¸µ¤Ø¤ÈÉü¹æ¤µ¤ì¤ë. Éü¹æ¤Î¸õÊä¤ò¥Ï¥ß¥ó¥°µ÷Î¥¤¬d/2̤Ëþ¤Î
°ÌÃ֤ˤ¢¤ë¤â¤Î¤Ë¹Ê¤ë¤³¤È¤Ç, Éü¹æ»þ´Ö¤¬ÀáÌ󤵤ì, ¤Þ¤¿Æ±»þ¤Ë, Éü¹æ¤Ç¤¤ë
¾ì¹ç¤Ë¤ÏÍ£°ì¤Ä¤ËÄê¤Þ¤ë¤³¤È¤¬Êݾڤµ¤ì¤ë¤³¤È¤Ë¤Ê¤ë. Éü¹æ¤Ç¤¤Ê¤¤¾ì¹ç¤Ï
Äü¤á¤ë¤Ê¤ó¤Æ¤¤¤¤²Ã¸º¤Ê¤³¤È¤Ç¤É¤Ê¤¤¤¹¤ó¤Í¤ó, ¤ÈÊ°¤ê¤ò´¶¤¸¤é¤ì¤ëÊý¤â
¤¤¤ë¤À¤í¤¦¤¬, Éü¹æ¤Ç¤¤Ê¤¤¾ì¹ç¤Ï, ºÆÁ÷¤òÍ׵᤹¤ë¤Ê¤ê²¿¤Ê¤êÊ̤ÎÊýË¡¤ò
ºÎ¤ì¤Ð¤è¤¤¤À¤±¤Î¤³¤È¤Ç¤¢¤ë. °Ê¸å, ËܹƤǤÏ, ºÇ¾®µ÷Î¥Éü¹æ¤Î¤ß¤ò¹Í¤¨¤ë¤³¤È¤Ë¤¹¤ë
.
Ãí°Õ 4.6
ºÇÌàÉü¹æ, ºÇ¾®µ÷Î¥Éü¹æ¤È¤â¤Ë, ¸í¤êÄûÀµÇ½ÎϤ˴ؤ·¤Æ¤Ï,
BnÆâ¤Î
C¤ÎÅÀ¤Î»¶¤é¤Ð¤êÊý(¸ß¤¤¤ËÎ¥¤ì¤Æ¤¤¤ì¤Ð¤¤¤ë¤Û¤É¤è¤¤)¤Î¤ß¤Ë°Í¸¤·, Á´Ã±¼Í
¤ÎÄê¤áÊý¤Ë¤Ï̵´Ø·¸¤Ç¤¢¤ë¤³¤È¤ËÃí°Õ¤·¤Æ¤ª¤³¤¦. ¤â¤Á¤í¤ó,
¤âÉä¹æ²½¤ÈÉü¹æ¤Î¥×¥í¥»¥¹¤òô¤¦Âç»ö¤Ê¼ÌÁü¤Ç¤Ï¤¢¤ë¤¬, º£¸å, Éä¹æ¤òµÄÏÀ¤¹¤ë¤È¤, ¤·¤Ð¤·¤Ð
¤Î¤³¤È¤Ï˺¤ì¤Æ
C¤Î¤ß¤ò¹Í¤¨¤ë¤³¤È¤Ë¤Ê¤ë.
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Mitsuru Kawazoe
2001-11-14