32

EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

�� �� �� � � � � � � �� �� � � � � ��

� � � � � � � � � � �� � � �� � � � � �� � �

� � ���� � �� �� � �� � � � � �! "# �

$ %& $ '( ) *� +, �,- , � . ) "� �! � ' "/ / - � �# � , � " � + � � 0 � , / � � ,

� " 021 � �3 � , � 0 "! � + .24 �� # �

567 89 :;< :; =>? @A >B @ @C

Page 2: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

D

EF GHI J K

L 4 - ! ! � � ��� + � �� � 0 0 " �# 1

L 3 � � 0 , � M� 4 �! , � �� N � 0 0 " �# 1

L O � � � �! P � � � + � �� 3 � � 0 , � M� 4 �! , � �� N Q O P3 4 R � 0 0 " �# 1

L O P3 4 �� � 3 � � 0 , � M� S ! , � �� NL � , - # ,- � � # � +� + "T �� , � � +,

L U "V ' " / 0 ! �W �, � P � ��� + � � � 3 � � 0 , � M � 4 �! , � �� N V �, 1 *� � � 0 � � � � � ,

X Y Q[Z R 4 �! , � ' " �\ # �� � , ) " � � ! +

L P � ��� + � �� 3 � � 0 , � M� 4 �! , � �� N 3 , ! �� � � # " + ,

L ' " � # ! - + � " �

567 89 :;< :; =>? @A >B @ @C

Page 3: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

]

^F H H _ ` a Kb I ` J c d dfe g `h i

L � 1 � V 1 "! � / � , �# � �! + 0 � # , - / j k�l Qfm Ron j k�l k�l Qfm R "Tp q # "- � , +r

, 1 � 0 � �/ � , � M � � � , � " � . 0 "V � � � ! � � 0 " S ! � . # " + + + 0 � # , �

� � , V � � � # " �\ # �� � , +s s s

L t �! / � � 4 �! , � �� N Q t 4 R �! ! "V + , " � " �! ! , 1 � + �� , 1 � + � + , � /

�� � � , � S # � , � " � +� ,vu - 0� - , � N � " � + , 1 � � + , �/ � , � " � " T j kwl Qfm R

567 89 :;< :; =>? @A >B @ @C

Page 4: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

x

c y ` d GI z K{ H G Ke I J | ` d dfe g `h i

L � 1 � 0 � " � " + � N � �! } ~ n � ~�� ��� ~p ~�� � � +- +� � , " � � � 0 , , 1 �

S ! , � # " �\ # �� � , + s

L � V " V � ! ! � � "V � �� � 0 , � M � S ! , � + � � �� , " � - # � � r

U ) � S ! , � �� N rp ���~ n p ���~� ��� � ��� ~ } ~

& U � S ! , � �� N rp � �� ~ n p � �� ~� �� �Y � �~ � � ~ } ~

567 89 :;< :; =>? @A >B @ @C

Page 5: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� K J Ke ` H _ ` a Kb I ` J c y ` d GI z K ^I H G Ke I J | ` d dfe g `h i

U � ,� ~ n � ~�� ��� ~p �~� � . Y n � � � � ~ ��� ~� s 3 + +- / � , 1 � , , 1 � �� 0- ,

# " M � � �� # � / � , �W Y � + V � ! ! � +, �/ � , � � . , 1 � �

� ~ n Y � � ��� ~ � ~ n p q~ � Y � � � � ~� ~� �� � � ~� Q Y � � ��� �¡� � ¢ Rp q~

# �� � � # " � + �� � � � � + � / � � +- � / � � , T " , 1 � " 0 , �/ �! S ! , � +� £- � � # �

p q~� 1 � O � � � �! P � ��� + � �� 3 � � 0 , � M� 4 �! , � Q O P3 4 R + "! - , � " � V �! ! � � � � +� �

" � U ) ) � $ � + , �/ � , � " � " Tp q~ T "/ � ~ r

�p �~ n ¤ Q¥ R � ~

567 89 :;< :; =>? @A >B @ @C

Page 6: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

¦

� K J Ke ` H _ ` a Kb I ` J c y ` d GI z K ^I H G Ke I J | ` d dfe g `h i

*� , 1 � � " � u # �- + �! § 4 # � +� r

L ¤ Q ¥ Ron j k�l ¨ Q¥ R j � �¨ ¨ Q¥ R n ¢ � jª© ¨ © ¨ Q¥ R j � �¨ ¨ Q¥ R

*� , 1 � # �- + �! § 4 # � +� r

L ¤ Q ¥ Ron ¢ � Y © ¨ © ¨ Y � �«¨ «¨­¬ Q¥ R . V �, 1 ¬ Qm R j ¨ ¨ Qm R¬ ® Qfm R n Y © ¨ © ¨

567 89 :;< :; =>? @A >B @ @C

Page 7: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

¯

� _ c ^ ` J y c y ` d GI z K{ H G Ke I J |

*T V � � + +- / � , 1 � �� � 0 , � , � " � + 0 � � � � + � " , , " " T � +, . V � N� ,

� 0 0 "W �/ � , � ! � r

Lp ���~ n Q ¢ � Q ¢ � � Y R ¥ � � R� � � Y Qp q~ � Y � � � � � ~ R

Lp � �� ~ n �� °�� ° ±³² ´ Qp q~ � Y � � � � ~� ~ R

V 1 �# 1 � + M� � # ! " +� , " , 1 � O P3 4 �� � 0 , � , � " � + , - # , - � � +

p q � Y � � ��� ~� ~ � + � ! � , � � , " r

� ~ n Y � � ��� ~ � ~ n p q~ � Y � � � � ~� ~ � Q Y � � � � �µ� � ¢ Rp q~� � � �

� � N! � N �� ! � �T p �~ ! "V 0 � + +

567 89 :;< :; =>? @A >B @ @C

Page 8: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

· Ke ¸ ge ¹ ` J h K c J ` H a b I b

§ � # "/ 0 � � , 1 � � +- ! , �� N $W # � + + ) � $ T " � �º � � � , # � +� + r

L U ) � V �, 1 " 0 , �/ �» � � �� � � M �� - �! + , � 0 + �» �

L ' ! � + + �# �! U ) � V �, 1 � � " 0 , �/ �» � � N! "� �! + , � 0 + �» �

L ' ! � + + �# �! & U � V �, 1 " 0 , �/ �» � � �� � � M �� - �! T " N� , �� N T �# , "

L O P3 4

567 89 :;< :; =>? @A >B @ @C

Page 9: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

¼

½¾ h Kb b K ` J ¿ ÀF `e K ½ e e ge Á ½ ¿ ½ Â

� 1 � ) � �� � £- � � � $ " # �� � � V �, , � � � + r

à j � n � Ä } D ~ Å n Æ DÇ � � È � k~Ép � ~Ép � ~ � ~ Ê Q Z R

� 1 � $W # � + + ) � �� � £- � � $ " � + � � S � � � � +r

� à j � n � Ä } D ~ Å� à à j �

n � È � k~ Ép � ~Ép � ~ � ~ Ê Q[Ë R

V 1 � � Ép ~ n p �~ � p ~ s567 89 :;< :; =>? @A >B @ @C

Page 10: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� q

½ ¿ ½ ÁÍÌ Â

� 1 � � W # � + + ) � $ # �� � � �W 0 � + +� � �� , 1 � T "! ! "V �� N T " / r

� Ã j � n ÎÏ Ð � QÉp ~Ép k~ Y R Ñ

n ÎÏ Ð Y´Ò

� ´Ò jÔÓ k Ó kvÕ Q }×Ö D Ø Ù RfÚ Û Ñ QÝÜ R

 p ~ n p �~ � p ~

n Q ¢ � ¤ Q ¥ R Rp �~ � Y � � ¤ Q ¥ R �� ~� ~

n Q ¢ � ¤ Q ¥ R Rp �~ � ¤ Q ¥ R  � ~

� � � � � # " / � +r

� à j � n Î Ï Y дÒ

� ´Ò ¤ Q }×Ö D Ø Ù R Y © ¨ © ¨ ¤ k Q }×Ö D Ø Ù RfÚ Û Ñ

� Î Ï Y дÒ

� ´Ò Q ¢ � ¤ Q }×Ö D Ø Ù R R j k2Þ k2Þ Q } � Ö D Ø Ù R Q ¢ � ¤ k }×Ö D Ø Ù R Ñ Ú Û

567 89 :;< :; =>? @A >B @ @C

Page 11: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� �

½ ¿ ½ Á×ß Â

& � / � � , 1 � , , 1 � $ ) � $ # � � � � � " � � � - 0 �� , " , V " , � / + r

L � à �á� â n ÎÏ Y Ð ã´Ò� ´Ò ¤ Q }äÖ D Ø Ù R Y © ¨ © ¨ ¤ k Q }äÖ D Ø Ù RfÚ Û Ñ # 1 � �# , � �» �� N , 1 �

� " � +� # " � , �� - , � " � å � � � # �� � � �� , � 0 � , � � � + , 1 � � +, �/ � , � " �

�# # - �# � - � � � +, � , � " � � � # " � � �, � " � +

L � �æ ç n ÎÏ Y Ð ã´Ò� ´Ò Q ¢ � ¤ Q }fÖ D Ø Ù R R j k Þ k Þ Q }fÖ D Ø Ù R R Q ¢ � ¤ k Q }fÖ D Ø Ù R R Ñ Ú Û

� 0 � +� � , �� N , 1 � � +, �/ � , � " � � " � +- ! , �� N T "/ , 1 � + � +, � /

M � � � , � " � + Q U � N � " � +� R567 89 :;< :; =>? @A >B @ @C

Page 12: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� D

¿ Ge F h GF e K y h ` b Kb g ¸I J G Ke Kb G

� 1 � �� �! � + � + � + � � + �� + , V 1 � � , 1 � �� 0- , � + V 1 �, � . Y n Æ Dè ¢ s

§ � # " � + �� � , 1 � T "! ! "V �� N +, - # ,- � � / " � � ! + T " , 1 � " 0 , �/ �! � " 0 0 ! �

+ 0 � # , - / rL Q � R +- � + 0 �# � / " � � ! rp q~ n X é ~ V 1 � � 3 � + , �! ! �� � j ê ê Qm R � +

� � � N " � �!

L Q � � R � � # "- 0 ! � � # " �\ # �� � , � � � �/ �# + r j kël kël Qfm R � � � N " � �!

L Q � � � R- � �T " / � � � �/ �# + 0 ! - + 0 "V � � � ! � � 0 " S ! � r

j kwl k l Qm R n j ì ì Qfm R[í V 1 � � j ì ì Qm R � + +# �! � �� � í � + � # " � + , �� ,

� � � N " � �! s

567 89 :;< :; =>? @A >B @ @C

Page 13: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� ]

¿ d Kh I ` H h ` b K g ¸F JI ¸ ge ¹ y a J ` ¹ I h b d HF b d gî Ke y K H ` a de g{ H K

' " � + �� � Q � � � R �� " M� V �, 1 , 1 � +# �! � + 0 � # , - / j ì ì� � �� N �ï � ,

! "V u 0 � + + + 0 � # , - / å �s � sñðò

j ì ì Q }óÖ D Ø Ù R nô

õ öZ �T ÷ Û ÷ùø Û q

ú " , 1 � V � +�û

ü

567 89 :;< :; =>? @A >B @ @C

Page 14: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� x

· Ke ¸ ge ¹ ` J h K ` J ` H a b I b ¸ ge yIý Ke K J G ` H | ge I e i ¹ b

� 1 � - +� "T � N� � � �! S ! , � ¤ Q¥ R V �! ! ! � � � , " � � � +, �/ � , � " � � "  p n p q~ � p ~ V �, 1 # " M � � �� # � / � , �W

Y © k © k n Ú mË þ ÿ m Q ¢ � ¤ R j k�l k l Q ¢ � ¤ R k � Ú mË þ ÿ m ¤ Y © ¨ © ¨ ¤ k

V 1 � � ¤ n ¤ Qm R �� � j kël k l n j k l kël Qfm R �� � � +- ! , + �� $W # � + + ) � $

� Ã j � n ÎÏ Q Y © k © k Y R

567 89 :;< :; =>? @A >B @ @C

Page 15: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� �

½ ¿ ½ I J G i K� ¿ h ` b K

¤ Qfm Ron � Y

¢ � Q ¢ � � Y R m � � Q�� R

� à j ��� � � n � Æ DÇ � Æ DèË � � Æ Dè

� Ë Æ Dè Q[Z � � Æ Dè R Î Ï Qí R Û � � Æ Dèþ QË � � Æ Dè R � # , �� � Æ DèË � � Æ Dè , �� Q þ Û R

567 89 :;< :; =>? @A >B @ @C

Page 16: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� ¦

½ ¿ ½ I J G i K �� ¿ h ` b K

¤ Qm Ron Z �

Z � m � � ¢ Q�� R

, 1 � $ ) � $ �W 0 � + + � " � T " & U � # � +� r

� à j �� � � n � Æ DÇ Z�

Z �

� Ë Æ Dè Î Ï Qí R Û �� þ

Z � Z � � # , �� Z � Z � , �� Q þ Û � R

U ) �n & U � �T Z � n � Æ Dè s567 89 :;< :; =>? @A >B @ @C

Page 17: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� ¯

½ ¿ ½ � J G i K� g J�� � `F b ` H ^ � ` b K

¤ Q ¥ Ron ¢ � j © ¨ © ¨ Q¥ R j � �¨ ¨ Q ¥ R Q�� R

, 1 � $ ) � $ �W 0 � + + � " � +� � �� N r

� à j � à� � n �á� �

Zÿ Ë þ

Ú mm Q Z Æ DÇ �

ZÆ Dèí á á j ì ì Qm R R� � Q�� R

n Æ DÇ Ë Û ��

á� �

ZZ � � Ò�� Ò�í á á �D ÙÞ

567 89 :;< :; =>? @A >B @ @C

Page 18: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� ¶

½ ¿ ½ h g ¹ d `e `I b g J

§ � # "/ 0 � � , 1 � / �� �/ - / $ ) � $ �# 1 �� M� � � � U ) � .& U � QV �, 1

" 0 , �/ �» � � 0 � �/ � , � + å � R �� � O P3 4 s 4 � N- � Z 0 ! " , + , 1 � / �� �/ - /

$ ) � $ # - M � + Q � + � T - � # , � " � "T , 1 � � �� � V �� , 1 Û q Rs

567 89 :;< :; =>? @A >B @ @C

Page 19: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� ¼

EMSEmin

f0

EMSE

min

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09−70

−60

−50

−40

−30

−20

−10

0

10

20

30

Optimized LMS

LMS with individual stepsize

Optimized RLS

Non−causal WF

4 � N- � Z r � à j � � ��� # - M� + T " ï � , ! "V u 0 � + + M � � � , � " � +

567 89 :;< :; =>? @A >B @ @C

Page 20: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

D q

� � �! "$# %& ' ( )�* + ) %, ' +- . / + " ( '0 %1 '# ( %2 '- 354

' (6 7- / % " %- / %- (8 9 :<; =1 '# ( %2 > �? @ AB ? - CD � /? E,

F G HI J K F GL M G

N G J OQP G F GL R G S�T U

VW XY XZ[ \ M G M P ]^ J _a` G ]cb

dW X e Y fgh Xi j kZ W f Vl f X kl ji ml Xn J o K p _ q V jl W l W X kl ml X k X rts Xu v X

jb X b F Gcwx u X k fh s l j f u j kl f s k Xl W Xy z mh { f Y jl W i b

|}~ �� ��� �� ��� �� �� � ��

Page 21: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

�I

� �� > �� � E? � B C ��� � �? �B �� . / � � CB 0 ? 1 B E C? 2 B � 354

� B C 6 7� /? �? � /? � C8 9 :<; =1 B E C? 2 > �? @ AB ? � CD � /? E � �<� �

dW Xy z mh { f Y jl W i v f u k j kl k f� l W X � fh h f V ju { kl X e kZ

� �u jl j mh j k ml j f u Z k Xh X vl �n � � k Xl � J  

� dW Xy kl X e Z

¡ f i e s l XZ _ Sn p �n ] U J[ ¢£ ¤ \h f {¥ £ S�¦ p¨§ U ©¦ ^

� dW X z kl X e Z

k fh ª XZ �n ] HI J m Y { i m« £ _ Sn p �n ] U

� �� v f u ª XY { X¬ �l XY i ju ml X � fl W XY V j k X ju vY Xi Xul � m u ¬ Y Xl s Y u l f

y ­ kl X e b® j kl W X ju v f i eh Xl X ¬ ml m m u ¬ ¯ J o5° p F q j k m v f i eh Xl

¬ ml m b

|}~ �� ��� �� ��� �� �� � ��

Page 22: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� �

� �� > �� � E? � B C ��� � �? �B �� ± ² � � CB³ ? ´ B E C? µB � 354

� B C 6 7� ²? �? � ²? � C8 9 :<; = ´ B E C? µ > �? @ AB ? � CD � ²? E � �·¶ �

dW X u X { ml j ª Xh f { ­h j¸ Xh jW f f ¬ � s u vl j f u f� l W X fg k XY ª X¬ ¬ ml m j k { j ª Xu g ¹ Z

º¼» ½¾ Sh f {À¿ �ÁL h f {À Rà _ L Ä Å _Æ I Ç

GÈ I S§ GÊÉ K § GÆ I U S§ GÊÉ K § GÆ I UP U

L ¿ Æ �ÁÇ

GÈ I © N G É OP G § G © �

Ë X u v X� f Yl W � d Y S�b U j kl W Xl Y m v X f e XY ml f Y b

|}~ �� ��� �� ��� �� �� � ��

Page 23: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� Ì

� �� > �� � E? � B C ��� � �? �B �� ± ² � � CB³ ? ´ B E C? µB � Í5Î

� B CÏ Ð� ²? �? � ²? � C8 9 :<; = ´ B E C? µ > �? @ AB ? � CD � ²? E �

m u ¬ Z_ Sn p �n G U J ½

¾ Sh f {À¿ �ÁL h f { Rà _ L Ä Å _Æ I ÇGÈ I

[ ¢£ ¤ S§ GÉ K § GÆ I U S§ GÉ K § GÆ I UP U

L ¿ Æ �ÁÇ

GÈ I[ ¢£ ¤ © N G É OP G § G © �

|}~ �� ��� �� ��� �� �� � ��

Page 24: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

�Ñ

� �� > �� � E? � B C ��� � �? �B �� ± ² � � CB³ ? ´ B E C? µB � Í5Î

� B C Ï Ð� ²? �? � ²? � C8 9 :<; = ´ B E C? µ > �? @ AB ? � CD � ²? E � �·Ò �

dW X X« e X vl ml j f u kl X e ju l W j k k v Xu m Y j f Y X rs jY X kl W X v f i e s l ml j f u f� l W X

rts m ul jl j X kZ�F G Ó Ç J[ ¢£ Ô o F G ©° q �[ ¢£ Ô o F G F P G ©° q m u ¬ [ ¢£ Ô o F G F P GÆ I ©° q b dW X k X

rts m ul jl j X k m Y X X m k jh ¹ v mh vs h ml X ¬ s k ju { mÕ mh i m u ki f fl W XY �

VW XY X ¯ j kl W X v f i eh Xl X ¬ ml m ¯ J o F G p° G q � m u ¬ F G j kl W X i j k k ju {

¬ ml m b

|}~ �� ��� �� ��� �� �� � ��

Page 25: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� Ö

×Ø ÙÛÚ ÜÝ Þß à

á Xl jul j mh v f u ¬ jl j f u

Õ mh i m u âh l XY ju { m u ¬ f u X ­ kl X e ki f fl W ju {

㧠ä Ó äÆ å J ãæ ä 㧠äÆ å Ó äÆ å�N G Ó GÆ I J O�ç ä 㧠ä Ó äÆ å

è ä J é ä Ó äÆ å O ä S O�ç ä é ä Ó äÆ åëê ä L ¿ ìí UÆ å

î äÆ å J é äÆ å Ó äÆ å ãæç ä é Æ åäÆ å Ó äÆ å㧠ä Ó ä J 㧠ä Ó äÆ å è ä Sï ä É ãï ä Ó äÆ å U㧠äÆ å Ó ä J 㧠äÆ å Ó äÆ åL î äÆ å S 㧠ä Ó ä É㧠ä Ó äÆ å U

é ä Ó ä J S�ð É è ä O�ç ä U é ä Ó äÆ å

é äÆ å Ó ä J é äÆ å Ó äÆ åL î äÆ å S é ä Ó ä É é ä Ó äÆ å U S�ñ U

|}~ �� ��� �� ��� �� �� � ��

Page 26: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� ò

×Ø ÙÛÚ ÜÝ Þß à

i f ¬ Xh ó m Y m i Xl XY k m ¬ m el ml j f u

ôöõ G ÷I J ø ôöõ GÆ I ÷I L  �ù ú o 㧠ä Ó ä㧠ç ä Ó ä q S�û   U

ôüõ G ÷� J ø ôüõ GÆ I ÷� L  �ù ú o 㧠äÆ å Ó äÆ å 㧠ç äÆ å Ó äÆ å q S�û û U

ôýõ G ÷I � J ø ôýõ GÆ I ÷I � L  �ù ú o 㧠ä Ó ä㧠ç äÆ å Ó ä q S�û ¾ U

þõ G ÷ J ø þõ GÆ I ÷L û

ãÿ ä H å J ûþõ G ÷ S ôöõ G ÷I É ôöõ G ÷I � ô Æ õ G ÷� S ôöõ G ÷I � UP U S�û � U

ãæ ä H å J ôöõ G ÷I � ô Æ õ G ÷�

é ä H å Ó ä J ãæ õ ä H å ÷ä H å é ä Ó äãæç ä H åL ãÿ ä H å S�û � U

|}~ �� ��� �� ��� �� �� � ��

Page 27: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� �

� Ý à �Ø � ÞÝ Ú � �

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000020

22

24

26

28

30

32

34

36

38

40

discrete time

MS

E (

dB)

MSE Kalman filterMSE adaptive Kalman filterMMSE

|}~ �� ��� �� ��� �� �� � ��

Page 28: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� �

� �� �Ý � � × � � ÞÝ �� � ÝØ Þ� ÜÝ � Ù × ÞØ Ý � � � Ü �Ú � Þ

� j m { f u mh K � S�û U kl ml X k e m v X i f ¬ Xh Z

F G HI J K F GL M G

° G J OQP G F GL R G S�û � U

� X v m u V Y jl Xl W X � fh h f V ju { ¡ f i e f u Xul ­ V j k X i f ¬ Xh Z

� G H I ��� J ù � � G �� L � G �� S�û � U

N G J � G ���� G �� L�

� È �� G ���� G ��� L ! G S�û " U

� f Y # J û$ $ $ % � VW XY X % j kl W Xh Xu {l W f� l W X âh l XY b

|}~ �� ��� �� ��� �� �� � ��

Page 29: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

� &

� �� �Ý � � × � � ÞÝ �� � ÝØ Þ� ÜÝ � Ù × ÞØ Ý � � � Ü �Ú � Þ

� X v m u V Y jl XN G É

�� È �

�� G Ó GÆ I ��� G �� J � G ��'� G �� L�

� È �( � G ���� G �� L ! G

) f Y X m vW âh l XY v f i e f u Xul � N G m u ¬ ! G m Y X s e ¬ ml X¬ m k � fh h f V k

N+* G J N G �

� È ��� G Ó GÆ I ��� G ��

m u ¬

!* G J�

� È �( � G ��,� G �� L ! G

b

|}~ �� ��� �� ��� �� �� � ��

Page 30: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

Ì �

� �� �Ý � � × � � ÞÝ �� � ÝØ Þ� ÜÝ � Ù × ÞØ Ý � � � Ü �Ú � Þ

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000020

22

24

26

28

30

32

34

36

38

40

discrete time

MS

E (

dB)

1 Adaptive Kalman filter via CW−EM2 Adaptive Kalman filter via EM3 Kalman filter MMSE

1

2

1

1

2 2

3

|}~ �� ��� �� ��� �� �� � ��

Page 31: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

ÌI

- Ú � �Ø � �Ý Ú � � � . � Ü � � � � ÞÝ �� �

� dW X v f u ª XY { X u v X k e X X¬ f� l W X e Y f e f k X¬ mh { f Y jl W i ju m Y m u ¬ f i

l ji X ­ ª m Y ¹ ju { X u ª jY f u i Xul j k m e e Y f« ji ml Xh ¹ m k � m kl m kl W X f u X

kW f V u g ¹ v f u ª Xul j f u mh Õ mh i m u âh l XY ju { S ¸ u f V u e m Y m i Xl XY k U b

� ¡ f u k j¬ XY ju { K m u ¬ _ m ¬ j m { f u mh i ml Y j v X k �l W X v f i eh X« jl ¹ f� l W X

Õ mh i m u âh l XY j kh ji jl X¬ l f / S % � U m u ¬ l W X0 ¬ m el j ª XÕ mh i m u

âh l XY W m kl W X k m i X f Y ¬ XY f� v f i eh X« jl ¹ b

� dW X v f u ª XY { X u v X f� l W X ¡ f i e f u X ul ­ � j k X m ¬ m el j ª XÕ mh i m u âh l XY

k e X X¬ j k kh f V XY l W m u � f Y l W X m ¬ m el j ª XÕ mh i m u âh l XY f Y

v f u ª X ul j f u mh Õ mh i m u âh l XY mh { f Y jl W i k b

� dW X v f i eh X« jl ¹ f� l W X ¡ f i e f u Xul ­ � j k X m ¬ m el j ª XÕ mh i m u âh l XY j k

h ju X m Y ju % �l W X m ¬ m el j ª X âh l XY f Y ¬ XY

� ó XY k e X vl j ª X kZ l f m u mh ¹21 Xl W X v f u ª XY { Xu v X e Y f e XYl j X k f� m ¬ m el j ª X

Õ mh i m u âh l XY m u ¬ ¡ f i e f u Xul ­ � j k X m ¬ m el j ª XÕ mh i m u âh l XY b

|}~ �� ��� �� ��� �� �� � ��

Page 32: EURECOM€¦ · ò ×Ø ÙÛÚ ÜÝ Þ ß à i f ¬ X h ó m Y l k e j u ôöõ G ÷ I J ø Æ I L  ù ú o ã § ä Ó ç ä Ó q Sû U ôüõ G ÷ J ø Æ I L  ù ú o ã §

Ì �

3 � 4� � � �� �

56 57 58 9 5:

\û^ d b á m ¬ j¸ j � m u ¬ � b d b z b áh f v ¸<; = m ¹ X k j m u 0 ¬ m el j ª X ) jh l XY ju {

ó Y ju v j eh X k m u ¬ ó Y m vl j v mh 0 e e Y f m vW X k �; > ?@A BDCE FE G HJIK I G C I �

á X el X i g XY ¾     � b

\ ¾^ d b á m ¬ j¸ j � m u ¬ � b d b z b áh f v ¸<; = m ¹ X k j m u 0 ¬ m el j ª X ) jh l XY ju { 0 l

h ju X m Y v f kl �; L > > > MNO A @ NA C O P MA Q GO PR K E C I @ @A G Q � S á s g i j k k j f u T

¾     � b

U � V d b á m ¬ j¸ j � m u ¬ � b d b z b áh f v ¸<; W XY ¡ XZ []\ ^«_ ` ¹ = a ¹ ^b_ ac

0 ¬ a [ `_ ª ^ )_ \ ` ^d_ c e Y _ `f gc ¬ ^ [ ^c ¬ ^c `0 h S û T )_ \ ` ^ d i X ^j k_ ^c `

l Xm ^\ b n; L > > > MNO A @ NA C O P MA Q GO P R K E C I @ @A G Q n S áop Z _ b b_ X c Trq s s �t

U � Vvu t á am _ w_ nu t l ^ f m _ ac m � tu t l t á\ X k w t; � _ c m XY x [ `_ Z _ 1 a `_ X c

gb b o ^b _ c h ^ ko d b_ y ^ W ^ a b `{z | } ao d ^b 0 m a [ `_ y ^ )_ \ ` ^d_ c e ac m

u d a k w_ c e n; L G R K E C I I~ A G QE H N� I� @A PE �O K FE G HJIK I G C I E G MA Q GO P@

��� �� ��� �� ��� �� �� � ��