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Page 1: 0 1 2 3 6 4 5 7 8 9 3 2 5 A B C F D G E H I K L N J O M R ... Z [\] ^ Z B D J \ G _, ( R P P 1 P 2 ( R=g(P 1,P 2)= ⎧ ⎨ ⎩ 0,P 2 =0, log 2 a1+P1 a1+ 2· P1 P2,P>a.-g ( P 1 P

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w x y z { | z } ~ � � r f X S ^ a f h ^ _ ] Y u V ] V � Y _ V S T d S h _ V ] Y b R S T l_ X S b d Q T Q X a [ l Q g k R V Q T R [

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_ f Q R Q T _ X Y b g S R ^ U Q U S g Q T Q X a [ f Y X Z Q U _ V T a U [ U _ Q ] U � � � � � � i r f V UV U _ S U Y [ d V T U _ Q Y W S g S h _ V ] V � V T a Y U R Y b Y X S X Z Q R _ S X h Y X Y ] Q _ Q X de Q S h _ V ] V � Q S T Q S X ] S X Q g ^ T R _ V S T U S g _ V ] Q i s T _ f V U e S X c d e QV T Z Q U _ V a Y _ Q _ f Q � S V T _ S h _ V ] V � Y _ V S T S g _ X Y T U ] V _ h S e Q X Y T W _ f Qn o p X Q U S b ^ _ V S T \ S _ f Y U g ^ T R _ V S T U W Q k T Q W S T _ f Q _ V ] Q V T _ Q X Z Y b

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[0, T ]d e f Q X Q \ S _ f U V W Q U Y X Q

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

_ f Q n o p X Q U S b ^ _ V S T Q ] h b S [ Q W Y _ _ f Q X Q R Q V Z Q X d W Q T S _ Q W e V _ f

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

� Q _ _ f Q R S T U _ Y T _ R f Y T T Q b R S Q g k R V Q T _ W ^ X V T a[0, T ]

\ Qh ∈

Cd Y T W _ f Q X Q R Q V Z Q T S V U Q h S e Q X \ Q

σ2 i P V _ f _ X Y T U ] V _ h S e Q X

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 \ [ �   �

f(γ, b) = log2

(1 + γ

1 + γ · 2−2b

),

m � q

e f Q X Q _ f Q _ X Y T U ] V U U V S T \ Y T W e V W _ f f Y U \ Q Q T T S X ] Y b V � Q W e V _ f lS ^ _ b S U U S g a Q T Q X Y b V _ [ i s T _ f Q Y g S X Q ] Q T _ V S T Q W h Y h Q X V _ e Y U

2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)

978-1-4799-2893-4/14/$31.00 ©2014 IEEE 4758

Page 2: 0 1 2 3 6 4 5 7 8 9 3 2 5 A B C F D G E H I K L N J O M R ... Z [\] ^ Z B D J \ G _, ( R P P 1 P 2 ( R=g(P 1,P 2)= ⎧ ⎨ ⎩ 0,P 2 =0, log 2 a1+P1 a1+ 2· P1 P2,P>a.-g ( P 1 P

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ρ =

∫ T

0

f (γ, b) dt.

� � � � � � �ρ

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� � �γ(t)

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� � �P2

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P1 = ξ · ptx =ξσ2

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{0, b = 0,

cσ2 · 22b�= a2 · 2

2b, b > 0.

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� � �P2

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� � �

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@ A B C D E F G C H F B I J K L I L M J B L E N

O � � � � ( � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( � �

A1� � �

A2� � � � � � � ( � � � � � � � � � � � � � � � � � � � � � � � �

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maxγ≥0,b≥0

∫ T

0

f (γ, b) dt

s.t. W1 = P1(γ), W2 = P2(b),� R �

W1(0) = 0, W2(0) = 0,

W1(T ) ≤ A1, W2(T ) ≤ A2,

� � � W1

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� �b = 0

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� �

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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

Page 3: 0 1 2 3 6 4 5 7 8 9 3 2 5 A B C F D G E H I K L N J O M R ... Z [\] ^ Z B D J \ G _, ( R P P 1 P 2 ( R=g(P 1,P 2)= ⎧ ⎨ ⎩ 0,P 2 =0, log 2 a1+P1 a1+ 2· P1 P2,P>a.-g ( P 1 P

∂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

Page 4: 0 1 2 3 6 4 5 7 8 9 3 2 5 A B C F D G E H I K L N J O M R ... Z [\] ^ Z B D J \ G _, ( R P P 1 P 2 ( R=g(P 1,P 2)= ⎧ ⎨ ⎩ 0,P 2 =0, log 2 a1+P1 a1+ 2· P1 P2,P>a.-g ( P 1 P

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

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

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