Lattice Coding and the Generalized Degrees of

RUB

Chair of Communication Systems

Lattice Coding and the Generalized Degrees of Freedom of the Interference Channel with Relay Anas Chaaban and Aydin Sezgin

ISIT 2012

Anas Chaaban, Aydin Sezgin

IC with relay

1

RUB


Outline

1 State of the art


IC with relay

2

RUB


Outline

1 State of the art 2 Definitions


IC with relay

2

RUB


Outline

1 State of the art 2 Definitions 3 Main results

GDoF for α ≥ γ Upper bounds Transmission strategy (FDF)


IC with relay

2

RUB


Outline


GDoF for α ≥ γ Upper bounds Transmission strategy (FDF) 4 Conclusion


IC with relay

2

RUB


State of the art

• Upper bounds [Maric et al 09], [Tian et al. 11], [C et al. 10]


IC with relay

3

RUB


State of the art


• Decode-forward [Sahin et al. 07]


IC with relay

3

RUB


State of the art


• Decode-forward [Sahin et al. 07] • Compress-forward and Compute-forward [Tian et al. 11]


IC with relay

3

RUB


State of the art



• DoF=1 [Cadambe et al. 09],


IC with relay

3

RUB


State of the art



• DoF=1 [Cadambe et al. 09], • Capacity unknown in general,


IC with relay

3

RUB


State of the art



• DoF=1 [Cadambe et al. 09], • Capacity unknown in general,

Contribution GDoF of the symmetric IRC when the interference is stronger than the source-relay channel. Anas Chaaban, Aydin Sezgin

IC with relay

3

RUB


Outline




IC with relay

4

RUB


Definitions

• Full-duplex relay


IC with relay

5

RUB


Definitions

• Full-duplex relay • Causal relay


IC with relay

5

RUB


Definitions

• Full-duplex relay • Causal relay Define α=

log(h2c P ) , log(h2d P )

β=

log(h2r P ) , log(h2d P )


γ=

IC with relay

log(h2sr P ) . log(h2d P )

5

RUB


Definitions

• Full-duplex relay • Causal relay Define α=

log(h2c P ) , log(h2d P )

β=

log(h2r P ) , log(h2d P )

γ=

log(h2sr P ) . log(h2d P )

Definition The GDoF d(α, β, γ) or simply d is defined as d=

CΣ (α, β, γ) . log(h2d P )

lim 1 h2 d P →∞ 2


IC with relay

5

RUB





IC with relay

6

RUB


GDoF

Theorem The GDoF of the IRC with γ ≤ α is given by  2 max{1, β}     2 max{1, γ}    max{1, α, β} + max{1, α} − α d = min 2 max{1, α} + γ − α     2 max{α, β, 1 − α}    2 max{α, 1 + γ − α}


IC with relay

       

.

      

7

RUB


GDoF (with weak γ)

(γ < 1): β = 1.1, γ = 0.2

• GDoF gain for γ ≤ α ≤ β


IC with relay

8

RUB


GDoF (with weak γ)

(γ < 1): β = 1.1, γ = 0.2

• GDoF gain for γ ≤ α ≤ β • There are regimes where the relay does not help.


IC with relay

8

RUB


GDoF (with weak γ)

(γ < 1): β = 1.1, γ = 0.2

• GDoF gain for γ ≤ α ≤ β • There are regimes where the relay does not help. • E.g. α ≥ max{1, β, γ} Anas Chaaban, Aydin Sezgin

IC with relay

8

RUB


GDoF (with strong γ)

(γ > 1): β = 1.4, γ = 1.2

• GDoF gain


IC with relay

9

RUB



(γ > 1): β = 1.4, γ = 1.2

• GDoF gain • Higher GDoF in the very strong interference regime


IC with relay

9

RUB



(γ > 1): β = 1.4, γ = 1.2

• GDoF gain • Higher GDoF in the very strong interference regime • There are regimes where the relay does not help. Anas Chaaban, Aydin Sezgin

IC with relay

9

RUB





IC with relay

10

RUB


Upper bounds

• 2 individual rate constraints from the cut-set bounds,


IC with relay

11

RUB


Upper bounds

• 2 individual rate constraints from the cut-set bounds, • 2 genie-aided upper bounds from [C et al. 2010]


IC with relay

11

RUB


Upper bounds

• 2 individual rate constraints from the cut-set bounds, • 2 genie-aided upper bounds from [C et al. 2010] • 2 new genie-aided upper bounds.


IC with relay

11

RUB





IC with relay

12

RUB


Transmission strategy (FDF)

We split the message into three parts:


IC with relay

13

RUB




• A private message,


IC with relay

13

RUB




• A private message, • A common message,


IC with relay

13

RUB




• A private message, • A common message, • A cooperative public message


IC with relay

13

RUB




• A private message, • A common message, • A cooperative public message • ⇒ x1 = x1,c + x1,p + x1,cp


IC with relay

13

RUB





• The C and P signals are treated the same as in the IC [Etkin et al. 08]


IC with relay

13

RUB





• The C and P signals are treated the same as in the IC [Etkin et al. 08] • The relay only processes the CP message (treats C and P as noise)


IC with relay

13

RUB





• The C and P signals are treated the same as in the IC [Etkin et al. 08] • The relay only processes the CP message (treats C and P as noise) In the following example, we only discuss the CP message


IC with relay

13

RUB



Tx sends a nested-lattice ’cooperative public’ (CP) signal


IC with relay

14

RUB



Relay receives a noisy superposition


IC with relay

14

RUB



Relay decodes the superposition of the CP signals (nested-lattice codes!)


IC with relay

14

RUB



Relay maps the decoded CP superposition to a Gaussian codeword xr


IC with relay

14

RUB



xr (b) is sent in block i = b + 1


IC with relay

14

RUB



Receivers receive a noisy superposition of x1,cp , x2,cp , and xr


IC with relay

14

RUB



Receivers use backward decoding


IC with relay

14

RUB



Receivers know x1,cp (b) + x2,cp (b) in block b


IC with relay

14

RUB



Receivers decode desired (or undesired) CP signal in block b


IC with relay

14

RUB



Receivers use the available info. to remove the CP signals


IC with relay

14

RUB



Receivers then decode x1,cp (b − 1) + x2,cp (b − 1) in block b


IC with relay

14

RUB



and so on...


IC with relay

14

RUB



and so on...


IC with relay

14

RUB



and so on...


IC with relay

14

RUB



and so on...


IC with relay

14

RUB



• If interference is weak (α ≤ 1), the receivers decode the desired CP signal first,


IC with relay

15

RUB




• otherwise, the interfering CP signal is decoded first


IC with relay

15

RUB




• otherwise, the interfering CP signal is decoded first • We can also split the CP message into multiple parts (CP rate splitting)


IC with relay

15

RUB


Comparison

hd = 1, hsr = hr = 1.5, hc = 5. Anas Chaaban, Aydin Sezgin

IC with relay

16

RUB





IC with relay

17

RUB


Conclusion • We have derived new upper bounds on the sum capacity of the IRC,


IC with relay

18

RUB


Conclusion • We have derived new upper bounds on the sum capacity of the IRC, • We have proposed a new cooperative strategy for the IRC (FDF),


IC with relay

18

RUB


Conclusion • We have derived new upper bounds on the sum capacity of the IRC, • We have proposed a new cooperative strategy for the IRC (FDF), • We have characterized the GDoF of the IRC for γ ≤ α, achieved by the FDF scheme,


IC with relay

18

RUB



• The GDoF of the IC is increased by using a relay,


IC with relay

18

RUB




• What is the GDoF of the IRC for α ≤ γ? (ongoing work)


IC with relay

18

RUB




• What is the GDoF of the IRC for α ≤ γ? (ongoing work)

Thank You! Anas Chaaban, Aydin Sezgin

IC with relay

18

RUB



IC with relay

19

RUB


IRC model

• Noises N (0, 1) • Power constraint P (for each node)

• Rates R1 and R2 • Full-duplex relay • Causal relay: Xr (i) = fri (Yr (1), . . . , Yr (i − 1)) • Sum-rate RΣ = R1 + R2 and sum-capacity CΣ


IC with relay

20

RUB


Upper bounds (old) • Cut-set bounds: CΣ ≤ 2C((|hd | + |hr |)2 P ),

CΣ ≤ 2C(h2d P + h2sr P )

• Upper bounds from [C et al. 2010] q CΣ ≤ C P h2d + h2c + h2r + 2|hr | h2d + h2c h2sr P h2d P +C + C h2c P + 1 1 + max{h2d , h2c }P n obtained by giving (m1 , yrn , hc xn 2 + z1 ) to Rx-2 as side information, and starting with

n(RΣ − n ) ≤ I(m1 ; Y1n ) + I(m2 ; Y2n , m1 , Yrn , Scn )


IC with relay

21

RUB


Upper bounds (old) • Upper bounds from [C et al. 2010] h2d P 2 2 CΣ ≤ 2C (|hc | + |hr |) P + 4 max , hr P 1 + h2c P 2 hsr . + 2C h2c obtained by giving (hc X1n + U1n , Xr1 ) to RX-1 and (hc X2n + U2n , Xr1 ) to RX-2 (U=noise) and starting with n(RΣ − n ) ≤ I(m1 ; Y1n , S1n , Xr1 ) + I(m2 ; Y2n , S2n , Xr1 ) (b)

= I(m1 ; S1n |Xr1 ) + I(m1 ; Y1n |Xr1 , S1n ) + I(m2 ; S2n |Xr1 ) + I(m2 ; Y2n |Xr1 , S2n )

= h(S1n |Xr1 ) − h(S1n |Xr1 , m1 ) + h(Y1n |Xr1 , S1n ) − h(Y1n |Xr1 , S1n , m1 ) + h(S2n |Xr1 ) − h(S2n |Xr1 , m2 ) + h(Y2n |Xr1 , S2n ) − h(Y2n |Xr1 , S2n , m2 ),


IC with relay

22

RUB


Upper bounds (new)

• New bound: CΣ ≤ C(2h2sr P ) + C(h2d P + h2c P ) + C +

h2d −1 h2c

obtained by giving Yrn and (Yrn , m1 ) as side information to Rx-1 and Rx-2, respectively, and starting with n(RΣ − n ) ≤ I(m1 ; Y1n , Yrn ) + I(m2 ; Y2n , Yrn , m1 ) = I(m1 ; Y1n , Yrn ) + I(m2 ; Y2n , Yrn |m1 ) = I(m1 ; Yrn ) + I(m1 ; Y1n |Yrn ) + I(m2 ; Yrn |m1 ) + I(m2 ; Y2n |m1 , Yrn ) = I(m1 , m2 ; Yrn ) + I(m1 ; Y1n |Yrn ) + I(m2 ; Y2n |m1 , Yrn ),


IC with relay

23

RUB


Upper bounds (new)

• New bound: CΣ ≤ 2C

2 ! hd h2c + 1− + 2C(2h2sr P ) h2sr hc

obtained by giving the genie side information (hc X1n + Z2n , Yrn ) and (hc X2n + Z1n , Yrn ) to Rx-1 and Rx-2, respectively, and starting with n(R1 − 1n ) ≤ I(m1 ; Y1n , S1n , Yrn ) = h(Y1n , S1n , Yrn ) − h(Y1n , S1n , Yrn |m1 ) = h(S1n , Yrn ) + h(Y1n |S1n , Yrn ) − h(Y1n , S1n , Yrn |m1 ),


IC with relay

24

RUB


FDF: Achievable rates of the weak-interference FDF Theorem P (k) The sum-rate RΣ = 2(Rp + Rc + Rcp ) is achievable where Rcp = K k=1 Rcp and Rp ≤ C

h2d Pp 1 + h2c Pp

!

h2d Pc

Rc ≤ C

,

1 + h2d Pp + h2c Pp

!

h2d Pc + h2c Pc h2c Pc , 2Rc ≤ C Rc ≤ C 1 + h2d Pp + h2c Pp 1 + h2d Pp + h2c Pp   2 P (k) h 1 cp (k) sr P −  Rcp ≤ C +  (i) K 2 1 + 2h2sr i=k+1 Pcp + Pc + Pp   (k) h2d Pcp (k)   P Rcp ≤ C (i) K 2 P (k) + h2 P (2) P + P + P + h 1 + (h2d + h2c ) c p r cp cp c r i=k+1 ! ! (1) (2) h2r Pr h2r Pr Rcp ≤ C . +C (2) 1 + (h2d + h2c )(Pc + Pp ) 1 + h2 Pr + h2 P + h2 P r

d

∀k ∈ {1, . . . , K} where Pp + Pc +

!

!

c

PK

k=1


(k)

(1)

Pcp = P , Pr IC with relay

(2)

+ Pr

≤ P , and K ∈ N. 25

RUB


FDF: Achievable rates of the strong-interference FDF Theorem P (k) The sum-rate RΣ = 2(Rc + Rcp ) is achievable where Rcp = K k=1 Rcp and Rc ≤ C

!

h2d Pc 1 + h2d Pp + h2c Pp

! ! h2d Pc + h2c Pc h2c Pc , 2Rc ≤ C Rc ≤ C 1 + h2d Pp + h2c Pp 1 + h2d Pp + h2c Pp   (k) h2 Pcp 1 (k) Psr −  Rcp ≤ C +  (i) K 2 2 1 + 2hsr i=k+1 Pcp + Pc   (k) h2c Pcp (k)   P Rcp ≤ C (i) K 2 P (k) + h2 P (2) P + P + h 1 + (h2d + h2c ) c r cp cp r i=k+1 d ! ! (1) (2) h2r Pr h2r Pr Rcp ≤ C . +C (2) 1 + (h2d + h2c )Pc 1 + h2 Pr + h2 P + h2 P r

∀k ∈ {1, . . . , K} where Pc +

c

d

PK

k=1

(k)

(1)

Pcp = P , Pr


(2)

+ Pr

IC with relay

≤ P , and K ∈ N. 26