Blind MIMO Equalization under Paraunitary constraint

This paper introduces a new Blind Source Separation algorithm for convolutive mixtures. In addition to separate sources, this algorithm respects the paraunitary property of model considered, obtained after whitening observations. In order to respect this property, authors introduce a new model for equalizer, wisely factorized in 3 filters. After a presentation of theoretical results, a numerical algorithm is then derived. This algorithm is based on the solution of a polynomial system, containing some values of output cumulant multi-correlations. Simulations and performances of the numerical algorithm are presented in the last section. KEY WORDS : MIMO, Blind Equalization, Source Separation, Paraunitary matrix, Tensor, Cumulants, Sylvester matrix "! # $ % '& ( )* + # , % .0/2103 46587 9;:=<?>%@ 4BADCECEAGFH9JI89LK >NMPOQ1SR*TVUXWY4BA;Z FV< 7[: \^]`_Yacbd]feXa`ghbXiNjlknmpo*\#irqtstu vDacbdgwedx^yNirq zD{}|`~Eugh€ i ƒ‚Q„'…`…`… bda | eXi†€ irqN\'|f~Egha`uirq o ˆ‡Š‰‹„'‰coˆŒ#‚ …fnŽf…nk%m=a‹px gh]‚Bst‘#eXg =a`ug qH’“iE€^i ”`oQŒ#b•]`‘^~Ei‹– jlknmJ—$ir=acb˜e+—*—$™š„'…`…‹„#™›k`œf™NŒ“— #žEŸp ¡ž‹¢H£=žE¤†¥E¦'§© ̈a¦`¦f ̈ «V¬`­¡®¡ ̄d°n±‹® 2*3f ́wμ}¶ ·c¶^ ̧ 1o ́»n1⁄4 1˜1⁄2`3⁄4`¿ À ̧ μ}·0»f ̧ Á Â$Ãl ́»^3⁄4ÅÄ`1⁄2c¿f1 Àr ̧ Äa ̧ ¶ ·‹1 ·E1⁄4 ́1⁄2» ·cÃƐ1⁄2c1 ́h1⁄4 3fÇÉÈÊ1⁄2c1 À 1⁄2c»aËc1⁄2Ã¿`1⁄4 ́ː ̧ÌÇÅ ́hÍa1⁄4 ¿f1˜ ̧-μ ÎÐϕ» ·c3⁄4 3⁄4` ́h1⁄4 ́1⁄2» 1⁄4 1⁄2 μ˜ ̧ ¶^·‹1 ·E1⁄4˜ ̧ μX1⁄2¿f1 Àr ̧ μ Ñ^1⁄4˜3f ́wμt·‹ÃlÆc1⁄21˜ ́1⁄4˜3fÇ 1 ̧ 옶^ ̧-À¡1⁄4 μ01⁄4˜3f ̧N¶ ·‹1 ·‹¿f» ́h1⁄4 ·‹1 Ò ¶f1 1⁄2c¶# ̧ 1˜1⁄4dÒ 1⁄2‹È*1⁄4 3f ̧NÇÅ1⁄2`3⁄4` ̧ Ã$Àr1⁄2» μX ́w3⁄4` ̧ 1˜ ̧-3⁄4YÑD1⁄2Ó`1⁄4 ·c ́»f ̧-3⁄4 ·‹ÈÔ1⁄4˜ ̧ 1=ÁQ3f ́1⁄4˜ ̧ »f ́l»fÆ01⁄2cÓ μ˜ ̧ 1 ËE·E1⁄4˜ ́l1⁄2c»^μ Îpϕ» 1⁄2c1 3⁄4` ̧ 1D1⁄4 1⁄2+1 ̧ 옶# ̧ À¡1⁄4©1⁄4˜3 ́lμ=¶f1 1⁄2c¶# ̧ 1˜1⁄4dÒcы·c¿`1⁄4˜3 1⁄2c1 μ ́l»n1⁄4˜1 1⁄2a3⁄4f¿ Àr ̧ ·J»f ̧ ÁÕÇÅ1⁄2`3⁄4` ̧ ÃÐÈÊ1⁄2c1% ̧-֐¿^·‹Ãl ́× ̧ 1-Ñ+ÁQ ́wμ˜ ̧ ÃlÒØÈÙ·cÀr1⁄4˜1⁄21˜ ́l× ̧-3⁄4ƒ ́l»ÛÚ Ü Ãh1⁄4 ̧ 1 μ Î Ý ÈÔ1⁄4˜ ̧ 1H· ¶f1˜ ̧-μX ̧ »n1⁄4 ·E1⁄4 ́1⁄2» 1⁄2cÈV1⁄4 3f ̧ 1⁄21˜ ̧ 1⁄4˜ ́wÀ ·cÈ1˜ ̧-μX¿ Ãh1⁄4 μ Ñ*· »a¿fÇÅ ̧ 1 ́lÀ ·‹Ãˆ·‹ÃlÆc1⁄21˜ ́1⁄4˜3 ÇÞ ́lμ 1⁄4 3f ̧ »%3⁄4` ̧ 1˜ ́lËc ̧-3⁄4YÎ}2*3f ́lμ ·‹ÃlÆc1⁄2c1 ́1⁄4˜3fÇß ́wμ*Ó ·μX ̧-3⁄4H1⁄2»N1⁄4 3f ̧Ìμ˜1⁄2cÃl¿`1⁄4˜ ́l1⁄2c» 1⁄2‹È=·Å¶^1⁄2ÃÒa»f1⁄2ÇÅ ́l·cà μ˜Ò`μd1⁄4 ̧ Ç ÑnÀ 1⁄2c»n1⁄4 ·c ́» ́»fƚμX1⁄2Ç ̧VËE·‹Ãl¿f ̧ μP1⁄2‹Èp1⁄2c¿f1⁄4˜¶f¿`1⁄4QÀr¿ ÇпfÃw·‹»n1⁄42ǚ¿fÃ1⁄4˜ ́àáÀ 1⁄2c1 1˜ ̧ Ãl·‹1⁄4˜ ́l1⁄2c» μ Î Äa ́lÇп Ãl·‹1⁄4˜ ́l1⁄2c» μ“·‹» 3⁄40¶# ̧ 1˜ÈÊ1⁄2c1 Ǜ·‹» À ̧ μY1⁄2cÈn1⁄4 3f ̧P»a¿fÇÅ ̧ 1˜ ́wÀ ·cз‹ÃlÆc1⁄21˜ ́1⁄4˜3 NJ·‹1 ̧o¶f1 ̧ μ˜ ̧ »n1⁄4 ̧ 3⁄4 ́l»N1⁄4 3f ̧tÃw·cμX1⁄4 μX ̧-À¡1⁄4˜ ́l1⁄2c»DÎ â2ãäcåQæn ̄dçf­Eèêé Ï é ë Ñ}Â$à ́l» 3⁄4 ìPÖn¿ ·‹Ãl ́l× ·E1⁄4 ́1⁄2»pÑPÄa1⁄2¿f1 À ̧ Äa ̧ ¶ ·‹1 ·E1⁄4 ́1⁄2»pÑoí}·‹1 ·‹¿ »f ́h1⁄4 ·‹1 Ò Ç›·‹1⁄4˜1 ́hÍYÑ 2D ̧ » μX1⁄21 Ñ^î2¿ ÇпfÃw·‹»n1⁄4 μ Ñ ÄaÒaÃlËc ̧ μX1⁄4˜ ̧ 1QǛ·E1⁄4˜1 ́Í