ÏÖ´úÊý×ÖÐźŴ¦Àíѧϰ±¨¸æ(¶þ) ÏÂÔØ±¾ÎÄ

¹¦ÂÊÆ×ÃܶÈPxx(?)???H(?)22?1???p22

z?ej??k?1akz?kµ«ÊÇÖ±½Ó´ÓYule-Walker·½³ÌʽÇó½â²ÎÊýak(k =1£¬2£¬¡­£¬N)ÐèÒª×÷ÇóÄæ¾ØÕóµÄÔËË㣬µ±N´óʱ£¬ÔËËãÁ¿ºÜ´ó£¬²¢ÇÒÿµ±Ä£Ðͽ×ÊýÔö¼ÓÒ»½×£¬¾ØÕóÔö´óһά£¬ÐèҪȫ²¿ÖØÐ¼ÆËã¡£ 1.3.3 Levinson-DurbinËã·¨

Levinson-DurbinËã·¨ÊÇÒ»ÖÖ°´Õս״νøÐеÝÍÆµÄËã·¨£¬¼´Ê×ÏÈÒÔAR(0)ºÍAR(1)Ä£ÐͲÎÊý×÷Ϊ³õʼÌõ¼þ£¬¼ÆËãAR(2)Ä£ÐͲÎÊý£»È»ºó¸ù¾ÝÕâЩ²ÎÊý¼ÆËãAR(3)Ä£ÐͲÎÊý£¬Ò»Ö±µ½¼ÆËã³öAR(p)Ä£ÐͲÎÊýΪֹ¡£ Ò»½×ARÄ£Ð͵ÄYule-Walker¾ØÕó·½³ÌӦΪ

2??1???xx(0) ?xx(1)??1???? ????a?(1) ?(0) 0?11?xx?xx???

?(1?a11)?xx(0)2´ÓÕâ¸ö¾ØÕ󷽳̿ɽâµÃa

11???xx(1)/?xx(0)£¬?21

ÔÙ´Ó¶þ½×ARÄ£Ð͵ľØÕ󷽳̣º

??2???xx(0) ?xx(1) ?xx(2)??1????????xx(1) ?xx(0) ?xx(1) a21?? 0?????? 0????a22????xx(2) ?xx(1) ?xx(0)????2

2 ½âµÃ

a22????xx(0)?xx(2)??xx(1)?/??xx(0)??xx(1)?????xx(2)?a11?xx(1)?/?1????a21????xx(0)?xx(1)??xx(1)?xx(2)?/??xx(0)??xx(1)??a11?a22a11 ??22222

?2?(1?a22)?1222

ÒÔ´ËÀàÍÆµÃµÝÍÆ¹«Ê½£º

akk?????xx(k)??k?1?l?1?ak?1, l?xx(k?l)?/??2k?1

aki?ak?1,i?akkak?1,k?i

?k?(1?akk)?k?1, ?0??xx(0)2222

Óŵ㣺±ÜÃâÁ˾ØÕóµÄÇóÄæ¼ÆË㣬ÇÒ¿ÉËæ×Ž״εÄÔö¼ÓÖð½¥ÕÒµ½Ä£

Ð͵Ä×îÓŽ״Îp

1.3.4 ARÄ£Ð͵ÄÎȶ¨ÐÔÎÊÌâ¼°Æä½×µÄÈ·¶¨

Îȶ¨µÄARÄ£Ð;ßÓÐÒÔÏÂÐÔÖÊ£º ¢Ù?¢Úa¢ÛH21??2??322??2p?0

, pkk?1 k?1, 2,

e(z)µÄËùÓм«µã¾ùÔÚµ¥Î»Ô²ÄÚ

¢Ü×ÔÏà¹Ø¾ØÕóÊÇΪÕý¶¨¾ØÕó

Ò»°ãÀ´½²½×ÊýÔ¤ÏÈÊDz»ÖªµÀµÄ£¬µ±ÎÒÃǵÝÍÆµ½µÚk½×£¬Âú×ãËùÔÊ

ÐíµÄÖµ£¬¾Í¿ÉÑ¡½×Êý p=k ¡£Èç¹ûÐźŵÄÕýÈ·Ä£ÐÍÊÇp½×µÄARÄ£ÐÍ£¬ÔòÓ¦ÓÐa

kk?0 ¼° ?k??p µ±k?p22ʱ

ARÄ£Ðͽ״εÄÈ·¶¨

¢Ù×îÖÕÔ¤²âÎó²î×¼ÔòFPE(k)??¢ÚÐÅÏ¢ÂÛ×¼ÔòAIC(k)?22kN?kN?k

Nln(?k)?2k

1.3.5 ARÆ×¹À¼ÆµÄÐÔÖÊ

¢ÙARÆ×¹À¼ÆÒþº¬×Å×ÔÏà¹Øº¯ÊýµÄÍâÍÆ

ÉèÒª¹À¼ÆÒ»¸öARÆ×£¬ÒÑÖªËüµÄ×ÔÏà¹Øº¯ÊýµÄp£«1¸ö¹À¼ÆÖµ

?xx(0),??xx(1),??xx(p),?

¿ÉÒԵõ½

p?xx(m)???ak??xx(m?k) , m?p?k?1

??xx(m?k) , m?0???ak??k?1?xx(m)???p2???xx(?k)???,m?0ak????k?1p

¢ÚARÆ×¹À¼ÆÓë×î´óìØÆ×¹À¼ÆµÈЧ ×î´óìØÆ×¹À¼Æ·¨µÄ»ù±¾Ë¼Ï룺

Èç¹û¼ÙÉèÒÑÖª?xx(0), ?xx(1), , ?xxN(), ÎÊÌâÔÚÓÚ°´Ê²Ã´Ô­ÔòÍâÍÆ

?xx(N?1), ?xx(N?2), ÔÚ±£Ö¤×ÔÏà¹Øº¯ÊýµÄ Toeplitz¾ØÕóÊÇÕý¶¨µÄ

Çé¿öÏÂÓÐÎÞÇî¶àÖÖÍâÍÆ·¨£¬BurgÈÏΪÍâÍÆµÄ×ÔÏà¹Øº¯ÊýӦʹʱ¼äÐòÁбíÏÖ³ö×î´óìØ£¬Òò´Ë°ÑBurgÌá³öµÄÕâÖÖ·½·¨³ÆÖ®Îª×î´óìØÆ×¹À¼Æ·¨¡£

ìØÊÇ´ú±íÒ»ÖÖ²»¶¨¶È£¬×î´óìØÎª×î´ó²»¶¨¶È£¬¼´ËüµÄʱ¼äÐòÁÐ×î

Ëæ»ú£¬¶øËüµÄPSDÓ¦ÊÇ×îÆ½·ü(×î°×É«)¡£°´Shannon¶ÔìØµÄ¶¨Ò壬µ±xµÄȡֵΪÀëÉ¢µÄʱ£¬ìØH¶¨ÒåΪH

???pilnpii

µ±xµÄȡֵΪÁ¬ÐøµÄʱ£¬ìر»¶¨ÒåΪ

H???p(x)lnp(x)dx??E?ln(p(x))?͉˕

p(x)Ϊ¸ÅÂÊÃܶȺ¯Êý¡£

xxµ±x(n)ÊǸßË¹Ëæ»ú¹ý³Ì£¬ÇÒÆä¹¦ÂÊÆ×ÃܶÈP??0??0(?)Âú×ã

lnPxx(?)d???

?0??0Ôò¿ÉÒÔÖ¤Ã÷ÆäìØÕý±ÈÓÚ?lnPxx(?)d?

?N2N2?j?n¿ÉÒÔÖ¤Ã÷ʹìØ×î´óµÄ¹¦ÂÊÆ×Ϊpxx(?)?1?

?n?1ane x(n)Ϊ¸ß˹·Ö²¼Ê±×î´óìØÆ×¹À¼Æ·¨ÓëARÄ£ÐÍ·¨Êǵȼ۵ġ£

1.3.6 Ô¤²âÎó²î¸ñÐÍÂ˲¨Æ÷

?(n)¿ÉÓÃx(n)µÄ¸÷¹ýÈ¥ÖµµÄ¼ÓȨ֮°´ÏßÐÔÔ¤²âÀíÂÛ£¬x(n)µÄ¹À¼ÆÖµxp?(n)???ºÍ±íʾ¡£xk?1apkx(n?k)

Îó²îe(n)£¬ÔÚÕâÀïÓÃep(n)±íʾ(pΪ½×Êý)

p?(n)?x(n)?ep(n)?x(n)?x?k?1apkx(n?k)£¬³ÆÎªÏßÐÔÔ¤²âÆ÷µÄǰÏòÎó²î¡£

ÓÉLevinson¹ØÏµÊ½¿ÉµÃaÁîKppk?ap?1, k?appap?1, p? k?app£¬ÕâÀïµÄKP³ÆÎª²¿·ÖÏà¹ØÏµÊý(PARCOR)»ò·´ÉäϵÊý¡£

p?1ep(n)?x(n)??k?1p?1apkx(n?k)?appx(n?p)

Kpx(n?p)

?x(n)???ak?1p?1,k?KPap?1,p?k?x(n?k)???x(n?p)??p?1p?1?x(n)??k?1ap?1,kx(n?k)?Kp?k?1?ap?1,p?kx(n?k)??

?ep?1(n)?Kpbp?1(n?1)

p?1p?1 ÆäÖУ¬bp?1(n?1)?x(n?p)??k?1ap?1,p?kx(n?k)?x(n?p)??k?1ap?1,kx(n?p?k)

pbp(n)?x(n?p)??k?1?(n?p)apkx(n?p?k)?x(n?p)?xp?(n?p)???apkx(n?p?k)xk?1

p?1), x(n?p?2), , x(n)¼Ó

?(n?p)xÊÇÓÉx(n-p)ÒÔºóµÄ¸÷Êý¾Ýx(n?Ȩ֮ºÍµÃµ½µÄ£¬¹Êbp(n)³ÆÎªºóÏòÔ¤²âÎó²î¡£