Interference Channels

A_j (4.65)

with which (4.64) implies the last constraint in (4.55).

By now, we have proved the achievability through the existence of a proper power allocation function such that (4.58) and (4.59) are satisfied for every pair (d1, d2) in the outer bound.

4.5.2 Interference Channels

The proposed scheme for MIMO IC is similar to that for BC, with the di↵er-ences that (a) the interferdi↵er-ences can only be reconstructed at the transmitter from which the symbols are sent, and (b) antenna configuration does matter at both transmitters and receivers. Further, as with the broadcast channel, we notice that the region (4.12) given in Theorem 4.2 does not depend on M_k (resp. N_k) when M_k > N₁ +N₂ (resp. N_k > M₁ +M₂), k = 1,2.

Therefore, it is sufficient to prove the achievability for the caseM_kN₁+N₂ andN_k M₁+M₂,k= 1,2, since the achievability for the other cases can be inferred by simply switching o↵ the additional transmit/receive antennas.

Thus, it yields

M_k= min{M_k, N1+N2},

N_k= min{N_k, M₁+M₂}, k= 1,2. (4.66) We also define for notational convenience

N₁⁰ ,min{N₁, M₂}, N₂⁰ ,min{N₂, M₁}. (4.67) Block-Markov encoding

The block-Markov encoding is done independently at both transmitters x₁[b] =x_1c(l_{1,b 1}) +u₁(w_1b), (4.68)

THE TIME-CORRELATED CASE N_j⁰ ⇠_k)-dimensional (resp. ⇠_k-dimensional) subspace of the null space of Hˆjk where⇠k will be specified later on. Therefore, the rank of Hˆ_jk is N_j⁰ interference at Receiver j has power level A_k and the desired signal at Receiver kat power levelA⁰_k.

Interference quantization

Recall that the common messages x_1c(l_{1,b 1}) andx_2c(l_{2,b 1}) are sent with power P. The received signals in blockbare given by

y₁[b] =H₁₁x_1c(l_{1,b 1}) +H₁₂x_2c(l_{2,b 1})

It is readily shown that⌘_1b and ⌘_2b can be compressedseparately up to the noise level with two independent source codebooks of sizeP^N10A2 andP^N20A1, into indicesl_2,b and l_1,b, respectively. For convenience, let us define

d_⌘1 ,N₁⁰A₂, d_⌘2 ,N₂⁰A₁, andd_⌘ ,d_⌘1 +d_⌘2. (4.75) Backward decoding

The decoding starts at block B. Since w_1B and w_2B are both known, the private signals can be removed from the received signals y1[B] and y2[B].

The common messages l_{1,B 1} and l_{2,B 1} can be decoded at both receivers if d_⌘k min{M_j, N₁, N₂}, (4.76) d⌘1 +d⌘2 min{N1, N2}, (4.77) i.e., the common rate pair should lie within the intersection of MAC regions at both receivers for the common messages. At blockb, assuming both l_1,b andl_2,b are known perfectly from the decoding of blockb+ 1,⌘_1b and⌘_2bcan be reconstructed up to the noise level, forb=B 1, . . . ,2. The following MIMO system can be obtained at Receiverk

y_k[b] ⌘_kb

⌘_jb =

H_kk

0 x_kc(l_{k,b 1}) +

H_kj

0 xjc(l_{j,b 1}) +

H_kk

H_jk u_k(w_kb) (4.78)

forj 6=k2{1,2}. Note that this system corresponds to a multiple-access channel from which the three independent messagesl_{1,b 1}, l_{2,b 1}, and w_kb are to be decoded. It will be shown in the Appendix 4.8.1 that the three messages can be correctly decoded if the DoF quadruple (d_⌘1, d_⌘2, d_1b, d_2b) lies within the following region

d_kbN_j⁰A_k+ (M_k N_j⁰ ⇠_k)A⁰_k+⇠_kA⁰⁰_k (4.79) d⌘_k min{Mj, N1, N2} (4.80) d_⌘1 +d_⌘2 min{N₁, N₂} (4.81) d_⌘k +d_kbN_k⁰ + min M_k N_j⁰, N_k N_k⁰ A⁰_k

+N_j⁰A_k (4.82) d_⌘j+d_kbmin{M_k, N_k}+N_j⁰A_k (4.83) d_⌘1 +d_⌘2 +d_kbN_k+N_j⁰A_k. (4.84)

THE TIME-CORRELATED CASE Now, let us fix

d_kb ,N_j⁰A_k+ (M_k N_j⁰ ⇠_k)A⁰_k+⇠_kA⁰⁰_k (4.85)

d_⌘j ,N_j⁰A_k (4.86)

from which we can reduce the region defined by (4.79)-(4.84). First, we remove (4.79) that is implied by (4.85). Second, (4.80) is not active as it is implied by (4.86) and (4.81). Third, (4.81) is implied by (4.84) and (4.85). Finally, from (4.86), (4.83) is equivalent to d_kb min{M_k, N_k} that is implied by (4.85). Therefore, by letting B ! 1, we have the following counterpart of Lemma 4.1 for interference channels.

Lemma 4.2 (decodability condition for IC). Let us define AIC,

8<

:(A₁, A⁰₁, A⁰⁰₁, A₂, A⁰₂, A⁰⁰₂) A_k, A⁰_k, A⁰⁰_k2[0,1]

A⁰_k ↵_j A_kA⁰_k, A⁰⁰_k A⁰_k,

⇠_kA⁰⁰_kN_k⁰(1 A_j), k6=j 2{1,2} 9=

; (4.87)

DIC, (d₁, d₂) |d_k2[0,min{M_k, N_k}], 8k2{1,2} (4.88) and

f_A-d : AIC!DIC (4.89)

(A_k, A⁰_k, A⁰⁰_k)7!d_k,N_j⁰A_k+ (M_k N_j⁰ ⇠_k)A⁰_k+⇠_kA⁰⁰_k,

8k6=j2{1,2} (4.90) where

⇠_k,⇢

(Mk N_j⁰)⁺ (Nk N_k⁰)⁺, if Ck holds

0, otherwise. (4.91)

Then (d₁, d₂) =f_A-d(A), for some A2AIC, is achievable with the proposed scheme, if

d_⌘1 +d₁N₁, (4.92)

d_⌘2 +d₂N₂. (4.93)

where we recall d⌘1 ,N₁⁰A2 andd⌘2 ,N₂⁰A1.

Proof. It has been shown that with (4.86) and (4.90), only (4.82) and (4.84) are active. With ⇠_k defined in (4.91), we can verify that M_k N_j⁰ ⇠_k = minn

M_k N_j⁰, N_k N_k⁰o

. Thus, from (4.86), (4.90), (4.91), and the last constraint in (4.87), it follows that (4.82) always holds. Finally, the only constraint that remains is (4.84) that can be equivalently written as (4.92) and (4.93).

Definition 4.5 (achievable region for IC). Let us define IA^IC,

⇢

(A1, A⁰₁, A⁰⁰₁, A2, A⁰₂, A⁰⁰₂)2AIC

(d1, d2) =f_A-d(A1, A⁰₁, A⁰⁰₁, A2, A⁰₂, A⁰⁰₂),

d_k N_k⁰  ^N_N^k0

k A_j, k6=j2{1,2}

)

(4.94) and the achievable DoF region of the proposed scheme

Id^IC ,f_A-d(IA^IC), 8<

:(d₁, d₂)

(d₁, d₂) =f_A-d(A₁, A⁰₁, A⁰⁰₁, A₂, A⁰₂, A⁰⁰₂), (A1, A⁰₁, A⁰⁰₁, A2, A⁰₂, A⁰⁰₂)2AIC,

d_k N_k⁰  ^N_N^k0

k A_j, k6=j2{1,2}

9>

=

>;. (4.95)

Achievability analysis

The analysis is similar to the BC case, i.e., it is sufficient to find a function f_d-A : O^ICd ! AIC where Od^IC denotes the outer bound region defined by (4.12), such that

(d₁, d₂) =f_A-d(f_d-A(d₁, d₂)), and (4.96)

f_d-A(d₁, d₂)2IA^IC. (4.97)

Now, we define formally the power allocation function.

Definition 4.6 (power allocation for IC). Let us define _k, k6=j2{1,2}, as

k,min

⇢

1,Mj dj

⇠_k . (4.98)

THE TIME-CORRELATED CASE

Then, we define f_d-A : O_d^IC!AIC:

(d₁, d₂)7! (A₁, A⁰₁, A⁰⁰₁),f₁(d₁, d₂),

(A₂, A⁰₂, A⁰⁰₂),f₂(d₁, d₂) (4.99) where f_k, k6=j2{1,2}, such that (4.90)is satisfied, and that

• when M_k=N_j⁰: A⁰⁰=A⁰_k=A_k = _M^dk

k;

• when M_k> N_j⁰, d_k <(M_k N_j⁰) _k+N_j⁰( _k ↵_j)⁺:

A_k= (A⁰_k ↵_j)⁺, A⁰_k=A⁰⁰_k< _k; (4.100)

• when M_k> N_j⁰, d_k (M_k N_j⁰) _k+N_j⁰( _k ↵j)⁺, and _k <1:

A_k= (A⁰_k ↵_j)⁺, A⁰_k> A⁰⁰_k= _k; (4.101)

• when M_k> N_j⁰, d_k (M_k N_j⁰) _k+N_j⁰( _k ↵j)⁺, and _k = 1:

A⁰_k=A⁰⁰_k= 1. (4.102) First, one can verify, with some basic manipulations that,f_d-A(O^IC_d )✓ AIC. Second, (4.96) is satisfied by construction. Finally, it remains to show that (4.97) holds as well, i.e., the decodability condition in (4.94) is satisfied.

Since the region O^IC_d depends on whether the conditionC_k holds, we prove the achievability accordingly.

Neither C₁ nor C₂ holds (⇠₁ =⇠₂ = 0)

For any (d₁, d₂)2O_d^IC, we can define (A₁, A⁰₁, A⁰⁰₁, A₂, A⁰₂, A⁰⁰₂),f_d-A(d₁, d₂) which implies, in this case,

d_j =N_k⁰A_j+ (M_j N_k⁰)A⁰_j, j 6=k2{1,2}. (4.103) Applying this equality on the constraints in the outer boundO^IC_d in (4.12), we have

d_k

N_k⁰  min{max{M1, N2},max{M2, N1}}

N_k⁰

(Mj N_k⁰)A⁰_j

N_k⁰ A_j, (4.104)

d_k

N_k⁰  min{M_k, N_k} N_k⁰

min{M_k, N_k} N_k N_k⁰

+(Mj N_k⁰)(A⁰_j ↵k) +N_k⁰Aj

M_j

+

, (4.105)

fork6= j2{1,2}, where the first one is from the sum rate constraint (4.12c) whereas the second one is from the rest of the constraints in (4.12). The final step is to show that either of (4.104) and (4.105) implies the last constraint in (4.94).

• When Mj =N_k⁰, (4.105) implies the last constraint in (4.94) because

min{M_k,N_k} N_k

Nk 0;

• When M_j > N_k⁰ and d_j M_j N_k⁰↵_k, we haveA⁰_j = 1 according to the mapping f_d-A defined in Definition 4.6, since _j = 1. Hence, the right hand side (RHS) of (4.104) is not greater than ^N_N^k₀

k A_j, which implies the last constraint in (4.94);

• When M_j > N_k⁰ and d_j < M_j N_k⁰↵_k, we have A_j = (A⁰_j ↵_k)⁺ according to Definition 4.6 with _j = 1. Since ^min{Mk_N^{,Nk} Nk}

k 0, we

can show that

min{M_k, N_k} N_k

N_k +(M_j N_k⁰)(A⁰_j ↵_k) +N_k⁰A_j M_j

+

min{M_k, N_k} N_k

N_k +A_j (4.106)

with which (4.105) implies the last constraint in (4.94).

C_k holds (⇠_k>0, ⇠j = 0)

In this case, it is readily shown, from (4.90) and (4.91), that

d_k=N_jA_k+ (N_k M_j)A⁰_k+⇠_kA⁰⁰_k, (4.107)

d_j =M_jA_j. (4.108)

Applying the mappingd_j =M_jA_j on (4.12c) results in d_k

N_k⁰  min{M_k, N_k}

N_k⁰ A_j (4.109)

THE TIME-CORRELATED CASE that always implies _N^dk₀

k  ^N_N^k0

k Aj. Due to the asymmetry, we also need to prove that _N^dj

j 1 A_k. Therefore, the final step is to show that it can be implied by at least one of the constraints in (4.12), together with (4.107) and (4.108). the latter into (4.12c), we obtain

d_j

N_j  min{M_k, N_k} M_k+N_j N_jA_k

N_j (4.114)

1 A_k. (4.115)

Thus, the last constraint in (4.94) is shown in all cases. By now, we have proved the achievability through the existence of a proper power allocation function such that (4.96) and (4.97) are satisfied for every pair (d₁, d₂) in the outer bound.

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