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� Y T [ R S ] ] ^ T V R Y _ V S T W Q Z V R Q U d U ^ R f Y U e V X Q b Q U U U Q T U S X T S W Q UY T W ] S \ V b Q _ Q X ] V T Y b U d Y X Q b V ] V _ Q W \ [ _ f Q Q T Q X a [ _ f Y _ V U R S T l_ Y V T Y \ b Q _ S _ f Q \ Y _ _ Q X V Q U _ f Q [ f Y Z Q i n b _ f S ^ a f _ f Q V X U ^ U _ Y V T lY \ V b V _ [ R Y T \ Q V ] h X S Z Q W \ [ Q ] h b S [ V T a Q T Q X a [ f Y X Z Q U _ V T a_ Q R f T V t ^ Q U d V _ V U S g g ^ T W Y ] Q T _ Y b V ] h S X _ Y T R Q _ S ] Y c Q Q g k R V Q T _^ _ V b V � Y _ V S T S g _ f Q Y Z Y V b Y \ b Q Q T Q X a [ i s T X Q R Q T _ [ Q Y X U d _ f Q X Qf Y Z Q \ Q Q T Z Y X V S ^ U e S X c U W V U R ^ U U V T a _ f Q _ X Y W Q l S g g U \ Q _ e Q Q TQ T Q X a [ � Q T Q X a [ l Q g k R V Q T R [ Y T W _ f Q _ X Y W V _ V S T Y b h Q X g S X ] Y T R Q] Q _ X V R U U ^ R f Y U W Q b Y [ Y T W U h Q R _ X ^ ] l Q g k R V Q T R [ d � � � � d � � � � � � db Q Y W V T a _ S R S T U _ X Y V T Q W h Y X Y ] Q _ Q X S h _ V ] V � Y _ V S T U V T W V g g Q X Q T _U [ U _ Q ] R S T _ Q u _ U i P f Q T V _ R S ] Q U _ S _ f Q � S V T _ S h _ V ] V � Y _ V S TS g _ f Q R S ] ] ^ T V R Y _ V S T \ Q _ e Q Q T Y _ X Y T U ] V _ _ Q X Y T W Y X Q R Q V Z Q X d_ f Q h X S \ b Q ] V U ] ^ b _ V W V ] Q T U V S T Y b e f V R f ^ U ^ Y b b [ f Y U Y X Y _ f Q XR S ] h b Q u U _ X ^ R _ ^ X Q i � T _ f Q S _ f Q X f Y T W d e f Q T _ f Q ^ U Y a Q S g Q T lQ X a [ V U ^ T W Q X W V U R ^ U U V S T d _ f Q V ] h Y R _ S g _ f Q Z Y X V Y _ V S T S g h S e Q XR S T U ^ ] h _ V S T S Z Q X _ V ] Q T Q Q W U _ S \ Q Q u h b S X Q W d e f V R f V U S T Q S g
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T S _ U ^ h h S X _ Q W \ [ R S T U _ Y T _ h S e Q X U ^ h h b V Q U \ ^ _ f Y Z Q Y R Q X _ Y V TQ T Q X a [ \ ^ W a Q _ i s _ V U Y U U ^ ] Q W f Q X Q _ f Y _ _ f Q _ X Y T U ] V _ _ Q X Y T W _ f QX Q R Q V Z Q X \ S _ f f Y Z Q S T Q U V T a b Q Y T _ Q T T Y d Y T W _ f Q R S ] ] ^ T V R Y _ V S TR f Y T T Q b \ Q _ e Q Q T _ f Q ] U _ Y [ U R S T U _ Y T _ W ^ X V T a _ f Q _ V ] Q V T _ Q X Z Y bS g V T _ Q X Q U _ i � ^ X _ f Q X ] S X Q d e Q Y U U ^ ] Q _ f Y _ _ f Q X Q V U Y R Q T _ X Y bR S T _ X S b ^ T V _ V T _ f Q U [ U _ Q ] e f V R f f Y U h Q X g Q R _ c T S e b Q W a Q Y \ S ^ _Y b b X Q b Q Z Y T _ U [ U _ Q ] h Y X Y ] Q _ Q X U Y U e Q b b Y U _ f Q R f Y T T Q b U _ Y _ Q dU S _ f Y _ V _ V U Y \ b Q _ S � S V T _ b [ S h _ V ] V � Q _ f Q _ X Y T U ] V _ Y T W X Q R Q V Z QU _ X Y _ Q a V Q U _ S Y R f V Q Z Q _ f Q ] Y u V ] Y b h S U U V \ b Q _ f X S ^ a f h ^ _ i r X Y T U l] V _ h S e Q X Q ] h b S [ Q W \ [ _ f Q _ X Y T U ] V _ _ Q X d W Q T S _ Q W e V _ f
ptxd Y T W
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bd Y X Q _ Y c Q T Y U _ f Q _ e S R S T _ X S b Z Y X V Y \ b Q U e f V R f e Q R Y T Y W Y h _
R S T _ V T ^ S ^ U b [ V T _ V ] Q Y T W ] Y a T V _ ^ W Q i s T _ f V U U Q R _ V S T d _ f Q X Q lb Y _ V S T U \ Q _ e Q Q T _ f Q R S T _ X S b Z Y X V Y \ b Q U Y T W _ f Q Y R f V Q Z Y \ b Q W Y _ YX Y _ Q Y U e Q b b Y U _ f Q Q T Q X a [ R S T U ^ ] h _ V S T U Y X Q _ S \ Q Q u h b Y V T Q W i
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Cd Y T W _ f Q X Q R Q V Z Q T S V U Q h S e Q X \ Q
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ptxd _ f Q X Q R Q V Z Q U V a T Y b l _ S l T S V U Q X Y _ V S m � � � q
γR Y T \ Q e X V _ _ Q T
Y Uγ = |h|2 · ptx/σ
2 i r f Q R f Y T T Q b R Y h Y R V _ [ Y U W Q h Q T W Q T _ S TγY T W _ f Q n o p X Q U S b ^ _ V S T
bV U b S e Q X \ S ^ T W Q W \ [ � �
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),
m � q
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2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)
978-1-4799-2893-4/14/$31.00 ©2014 IEEE 4758
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- � � � � � ( � � � � � � 1 . V 0 � � � � � ( � � � � � � � � � � � � * � �
� � P � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � ( � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � ( � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � ( � � 6 � � � � � � � � � � � � � � � � � � � � � � � � �
(P,R) = (0, 0)� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � ( � � � � � � � � � � ( � �
� � � �0
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( �
( � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
( � � � � � � � � � � � � � � � � � � � � � � �P
� � � � � � ( * � � � � � P
�R
� � ( � � � � � � � � � � � � ( * � � � � � � P � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � W � � � ( � � � � � � � � � � 1 � � � � � ( * � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � �(P1, P2, R) = (0, 0, 0)
� � � � � � � � � � � � � � (
� � � � � � � � � � � � � � � � � � � � � 1 � � � � � � � � � � � �
X A E H B L I J Y B D J N Z I L B J N [ D \ ] \ L ^ \
Z B D J B \ G _
, � � � � � � � � � � � � � � � � � � � � � � ( � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � R � � � � � � � � � � � � � P � � � � � � �P1� � �
P2� � � � � � � � � � � � � � � � � � � � � � � ( � � �
R = g(P1, P2) =
⎧⎨⎩
0, P2 = 0,
log2
(a1+P1
a1+a2·P1
P2
), P2 > a2.
- � � � � � � � � � � � � � �g
� � � � � � � ( � � � � � �P1
� � �P2
� � �P2 >
a2� � � � � � � � � � � � � � � � � � � S � � � � � � � � � � *
|H(g)| = −a1a2
(ln 2)2·
a1a2 + 2P1(a2 − P2)
(a1 + P1)2P 22 (a2P1 + a1P2)2
� � � � � � � � � � � � � � � � ( � O � � � � � � � � � � � �P1
� � �P2
� α
�
2 � � � * �α > 0
�g
� � � � � � � ( � �P1
� � �P1 ∈ (a2α,+∞)
�
3 �g(0, 0) = 0
� � � � � � � �g
� � � � � � � � � � � � � * �α
� � �
� � � � ( * � � � � � � 1 � � � � � � � � � � � � � � � � � � � � �(0, 0, 0)
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �(0, 0, 0)
� � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � O �u
� � � � � �
P1� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � g(u,u/α)
u =
4759
∂g∂P1
∣∣∣P1=u
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (0, 0, 0)
� � � � � � � � � � � � � � u
a1 + u= ln
(a1 + u
a1 + a2α
) � � �� � � � � � � � � � � � � � � � � � �
u � � � � � � � �
α� � � � � � � � � � �� � � �
α ∈ (0,+∞)� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
g� � � � � � � � � � � � � � � � � � � � � � � � � � � � �α
� � � � � ! � � � � � � � � " � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � �g(P1, P2)
�� � �a1 = a2 = 1
� � � � � � � � � � � � � � � � # � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � $ % & '( ) & $ * ) ( + , & � - � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
. / . 0 . . 0 / . 1 . ../ .0 . .0 / .1 . . .
12
34 5 6 7 8 9 : ; < 7 = 8 7 ; > 9 6 7 ; ?
P1
P2
R
@ A B C D � E � � � � " � � � � � � � � � � � � � � � � � � � � � �F � � � � � � � � � � � � � �
X(α1x, x) �
Y (α2y, y)� # � �� � � � � � � � � � � � � � � � � � � " � � � � � � �
X �
Y� � � � � � � �
Z (βα1x+ (1− β)α2y, βx+ (1 − β)y) , 0 ≤ β ≤ 1,
with αZ =βα1x+ (1− β)α2y
βx+ (1− β)y.
# � � � � � � � � � � � � � � � Z
� � � � � � � � � � � � � � � � � � "� � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � �
RZ =1
ln 2·βα1x+ (1− β)α2y
a1 + u(αZ)
� G �� � �
Z� � � � � � � � � � � � H � � � � � � � �
RZ� � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � �
g
Z� � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� ∂2RZ
∂β2 < 0� � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � �
g� � I � � � �� � � � � �
P1 �
P2� J � � � � � K � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
X �Y
� � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � �αZ
� � � � � � � � � � � � � � � � � dαZ
dβ=
(α1 − α2)xy
(βx+ (1− β)y)2,
du
dα= a2 ·
(a1 + u)2
u(a1 + a2αZ).
L � � � � � � � � � � � � � � � � � � � � du(αZ)dβ
� � � � � � � � � � �dRZ
dβ=
1
ln 2
[α1x− α2y
a1 + u(αZ)−
a2(α1 − α2)xy · αZ
(βx+ (1 − β)y)u(a1 + a2αZ)
]d2RZ
dβ2= −
1
ln 2·
a2(α1 − α2)2(xy)2
(βx + (1− β)y)3u(αZ) (a1 + a2αZ)
×
[2−
a2αZ
a1 + a2αZ·(a1 + u(αZ))
2 + u2(αZ)
u2(αZ)
]
� � � � � � � � � � � � � � � � � � M � � � � � � � � � � � � � � � �u
a1 + u= ln
(1 +
u− a2α
a1 + a2α
)<
u− a2α
a1 + a2α,
� � � � � � � � � � � � � � �a2α < u2
2u+a1
� � � � � � � � � � � � � � � � � RZ
� � � � � � � � � � � � � � β
� � � �a2αZ
a1 + a2αZ·(a1 + u(αZ))
2 + u2(αZ)
u2(αZ)
<u2(αZ)
(a21 + 2a1u(αZ) + 2u2(αZ)
)(a21 + 2a1u(αZ) + u2(αZ))u2(αZ)
< 2.
N O P P Q R S P Q T U VQ W T V X
P1 Y Z Q Q [ \ T U ] V Sp
] U \ [ ] V T Qα
T R Q O V YR T \ S V X S V ] U ^ S U V [ S ^ T Q U _ ` a b a _p > u(α) c N T U Z S
gT R R V [ T Z V d e
Z Q U Z ] f S T UP1
] d Q U ^ V X S g h S \ \ T [ S Z V T Q U R P S Z T g S \ i eα
_ V X SR Y T U V S [ Z S P V Q j V X S V ] U ^ S U V d T U S ] V
Q k O R V i S P Q R T V T f S _ e T S d \ T U ^
g(p, p/α)− p ·∂g
∂P1
∣∣∣∣P1=p
> 0.
N T k T d ] [ d e ] R i S j Q [ S _ W S ] P P d e V X S [ S d ] V T Q Uln(1 + x) < x
j Q [x > 0
] U \ Q i V ] T U V X ] Vα < p2
a2(2p+a1)
_ W X T Z X ^ T f S R
a1a2 + 2p(a2 −p
α) < a1a2 + 2a2p− 2p2 ·
a2(2p+ a1)
p2
= −a1a2 − 2a2p < 0.
l X T R k S ] U R _ V X S m S R R T ] U k ] V [ T hH(g)
T R U S ^ ] V T f S \ S g U T V S ] V] U e P Q T U V Q O V R T \ S V X S V ] U ^ S U V [ S ^ T Q U _ R O ^ ^ S R V T U ^ V X S R V [ T Z V Z Q U YZ ] f T V e Q j V X S R O [ j ] Z S Q f S [ V X T R [ S ^ T Q U cn T V X V X S ] i Q f S P [ Q Q j _ W S ] R R O [ S V X ] V V X S [ S Z Q U R V [ O Z V S \P Q W S [ Y [ ] V S R O [ j ] Z S _ W X T Z X T R Q i V ] T U S \ i e k ] o T U ^ V ] U ^ S U V d T U S Rj [ Q k V X S Q [ T ^ T U V Q V X S R O [ j ] Z S \ S V S [ k T U S \ i eg
j Q [ ] d d \ T [ S Z YV T Q U Rα ∈ (0,+∞)
_ T R Q f S [ ] d d Z Q U Z ] f S c p ] R S \ Q U V X T R _ V X SQ P V T k ] d R Q d O V T Q U V Q q r s Z ] U i S ] Z X T S f S \ f T ] t d ^ Q [ T V X k u c v P YV T k ] d T V e Q j V X S ] d ^ Q [ T V X k Z ] U i S O U \ S [ R V Q Q \ ] R V X S R V [ ] T ^ X V d T U SZ Q U U S Z V T U ^ ] U e V W Q P Q T U V R Q U V X S P Q W S [ Y [ ] V S R O [ j ] Z S W Q O d \d T S i S d Q W _ Q [ Z Q T U Z T \ S W T V X _ V X S R O [ j ] Z S \ O S V Q T V R Z Q U Z ] f T V e cw j V X S V W Q P Q T U V R X ] f S \ T j j S [ S U V
αf ] d O S R _ V X S U V X S Z Q U U S Z V T U ^
4760
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
α ← A1/A2�
�� � � � �
u(α)� � � � � � � � � �
� �A1/T > u(α)
� � � � � � � � � � � � � �
�� � � � � � � � � � � � � � � � � � � �
[0, T ] � � �P ∗1 = A1/T
�P ∗2 = A2/T � �
� � � � � � � � � � � ��
� � � � � � � � � � � � � � � �A1/u(α)
� � � � � � �P ∗1 = u(α)
�P ∗2 = u(α)/α
� � � � � � � � � � � � � � � � �
� �
� � � � � � � � � � � � � � � � � � � � ��
� � � � � � � � � � � � � � �� �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � ��
� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � α
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! " � � � � � � � � � � � � � � � � �
α� � � � � � � � � �
� � � � � � � � � � � � � � �
� �
� � � � � � � � � � � � � � � �
� � � � �
�� � �
�� � � � � � �
� � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � �
# � � � � � � � � � � � � � � � �
�� � � � � �
α ! � � � � � � � � � � � � � � � � �� �
��
�� � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � �
��
� � � � � � � � � ��
� � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
�� � � � � � � � � �
�� � ! $
� � � � � �
� � �
� � g
�
� � � � � � � � � �
� � � � P1
� �P2
� � � � � � � � � � � � � � � � � � � � �
�� � � � � � � � �
T � � � � � � � � � � �
� � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � �� � � � � � � � �
� � � � � � � � � � � � �
�� � � % � � � � � � � & ! � �
� � � � � � � � � � � � � � � �p∗tx
� � � � � � � � � � % � � � � � � � � � b∗ �
� � � � � � �
� � � � � � � � � � � � P ∗1
� �P ∗2 !
� � � � � � � � � � � � � � � � � �
" � � � � � � � � � � � � � � ��
� � � � � � � � � � � � � � �
� � � ��
� � � � � � � � � � � � � � �ρ∗
� � � � � � � � � � � � � � � � �
� � � � � � � � �
�� � � � � � � � � � � � � � � � � �
a1�a2
�A1
� �A2 ! "
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