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The Gaussian Z-interference Channel with Rate-Constrained Conferencing Decoders c

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HIEU T. DO, TOBIAS J. OECHTERING, AND MIKAEL SKOGLUND

Stockholm 2010 Communication Theory Department School of Electrical Engineering KTH Royal Institute of Technology IR-EE-KT 2010:014

The Gaussian Z-interference Channel with Rate-Constrained Conferencing Decoders Hieu T. Do, Tobias J. Oechtering, and Mikael Skoglund School of Electrical Engineering and the ACCESS Linnaeus Center Royal Institute of Technology (KTH), Stockholm, Sweden

Z1

Abstract—We derive achievable rate regions for a 2-user Gaussian Z-interference channel with conferencing decoders. We identify different cases where the rate-limitedness of the conference link from the interference-free receiver to the interfered receiver affects the conferencing strategy as well as the achievable rate region. Furthermore, an outer bound to the capacity region based on cut-set and genie-aided bounds is presented.

h11

X1

This work was funded in part by the ERC, under grant agreement No. 228044, and by the Swedish Research Council.

Rc2

Rc1

h21 X2

Y2

h22

I. I NTRODUCTION The Z-interference channel [1] is a special case of the general 2-user interference channel, where only one receiver suffers from interference. The Z-interference channel with a digital relay link at the receivers’ side has been investigated by Yu and Zhou in [2]. In our recent work [3] we established an achievable rate region of the Gaussian Zinterference channel with conferencing decoders, in which the receivers can exchange messages via conference links orthogonal to the main channel. Such a channel can serve as an information theoretic model for a number of problems in wireless communications such as a wireless ad hoc network where two receivers are close enough to be able to cooperate while one of the transmitters suffers from the shadowing effect, or the uplink of Wyner-type [4] multi-cell wireless cellular networks with base stations interconnected through wireline backhaul links, e.g. [5]. One of the key assumptions in [3] is that the conference link from the interference-free receiver to the interfered receiver has sufficient capacity to accommodate the binning rate for the “private” message of the interference-free receiver, allowing for the successive decoding at the interfered receiver. This assumption somewhat relaxes the mathematical derivations, facilitating the understanding of the conferencing strategy. In practice, however, the capacity of conference links are constrained by many factors, either from the network’s infrastructure or from the wireless medium. Therefore, it is necessary to impose constraints on the capacity of both conference links. In this follow-up paper, we consider arbitrarily rate-limited conference links in both directions and derive achievable rate regions. Further, an outer bound to the capacity region is established. Section II introduces channel model and our coding strategy briefly. Section III focuses on deriving achievable rate regions of the channel. An outer bound to the capacity region is

Y1

Z2 Fig. 1.

Gaussian Z-interference channel with conferencing decoders

derived in Section IV, and numerical examples in Section V illustrate our analytical results. II. C HANNEL M ODEL AND O UTLINE OF S TRATEGY A. Channel Model Consider a discrete-time Z-interference channel as in Fig. 1. h11 , h21 , and h22 denote channel gains between transmitters and receivers. We assume that the channel gains are known to the transmitters and receivers. The two transmitters have average power constraints P1 and P2 . The noises Z1 and Z2 at the receivers are independent zero-mean Gaussian random variables with equal variance of N . The received signals at receiver 1 and 2 are given by Y1 = h11 X1 + h21 X2 + Z1 Y2 = h22 X2 + Z2 . Furthermore, the two receivers can communicate over two noiseless unidirectional digital conference links, having finite rate of Rc1 and Rc2 as depicted in Fig. 1. B. Definition of Conferencing In this paper we define a conference round as follows: based on its received signals receiver 1 sends the cooperative signal to receiver 2. Receiver 2 processes this side information and its received signals and then sends its cooperative signal to receiver 1. In other words the conference take places sequentially, similar to the definition of a conference round in, e.g., [6]. Regarding the related works, recently there are some works on the interference channel with receiver cooperation, e.g., [7, 8]. The work in [7] assumes a cooperation link interfering with

Z1 Rate

III. ACHIEVABLE R ATE R EGIONS

Power

R11 αP1 U1 R01 αP ¯ 1 W1

X1

h11

Y1 Rq R12 R21

¯ 2 W2 R02 βP R22 βP2 U2

h21 X2

Y2

h22

To quantify the impact of Rc2 on the achievable rates, we need to examine the rate constraints from each decoding step in more detail. The different steps taken in the decoding process are described more thoroughly in [3]. For step 1 the rates are limited by

Z2 Fig. 2.

Coding strategy for the Z-interference channel with conferencing

R01 ≤ I(W1 ; Y1 |W2 ) , I1

(1a)

R02 ≤ I(W2 ; Y1 |W1 ) , I2

(1b)

R01 + R02 ≤ I(W1 , W2 ; Y1 ) , I3 ,

(1c)

and for step 2 the main channel and the receivers transmit their cooperative information simultaneously. More similar to our considerations, the work of Wang and Tse [8], which appears after the initial submission of this paper, considers cooperation links that are orthogonal to the interference channel and also assumes that the conference takes place in a sequential manner. However, their coding strategy is different from ours.

R02 ≤ I(W2 ; Y2 , Yˆ1 |U2 ) + R12 , I4 R22 ≤ I(U2 ; Y2 , Yˆ1 |W2 ) , I5 R02 + R22

(2b) (2c)

The constraints in (1a)–(1c) and (2a)–(2c) essentially come from the capacity regions of the underlying multiple access channels (MACs).

C. Overview of Coding Strategy Both transmitters employ the rate-splitting and superposition coding technique introduced in [9], as depicted in Fig. 2. Specifically, transmitter 1 splits its information into messages W1 and U1 , where W1 is aimed to be decoded immediately by receiver 1, and U1 is to be decoded with an additional support from the conferencing. The two messages are encoded by independent Gaussian codebooks with variance αP1 and (1 − α)P1 , and then superimposed to form the channel input sequence X1n . The codebook for transmitter 2 is similarly drawn with messages W2 , U2 , channel input sequence X2n , power P2 , and the power-splitting factor β. Let R0i and Rii denote the respective rates of Ui and Wi , i = 1, 2, i.e., Ri = R0i + Rii . Outline of Decoding and Conferencing Let Rx1 and Rx2 denote receivers 1 and 2. The following 3 steps constitute the decoding and conferencing process: 1) Rx1 decodes (W1 , W2 ) jointly, treating U1 and U2 as noise. Utilizing the conference link from Rx1 to Rx2, Rx1 bins and forwards W2 [10] with the rate R12 ≤ Rc1 and compresses Y1 into Yˆ1 with the remaining rate Rq = Rc1 − R12 to help Rx2 decode (W2 , U2 ). 2) Rx2 decodes (W2 , U2 ) jointly, with side information from Rx1. Rx2 then uses a binning scheme with rate R21 ≤ Rc2 to provide side information through the conference link from Rx2 to Rx1 to help Rx1 alleviate the interference to U1 . 3) Rx1 decodes U1 with the help from Rx2. The conferencing strategy from Rx2 to Rx1 depends on how large the capacity Rc2 is compared with the rate constraints that the decoding step 2 sets on U2 . Studying this relationship and its effect is the focus of the present paper. 2 2 For brevity we define SNR1 = |h11N| P1 , SNR2 = |h22N| P2 , 2 INR2 = |h21N| P2 , C(x) = 12 log(1 + x), α ¯ = 1 − α, β¯ = 1 − β, where the logarithm is to the base 2.

≤ I(U2 , W2 ; Y2 , Yˆ1 ) + R12 , I6 .

(2a)

R11 ∗ R11

Decoding step 3

∗ ¯ ˜ 22 R22 R22 R

P

R22

Decoding step 2

R02 Fig. 3. Rate constraints in decoding steps 2 and 3 R∗11 ¯ 22 ˜ 22 I(U1 ; Y1 |W1 , W2 , U2 ), R = I(U2 ; Y1 |W1 , W2 , U1 ), R I(U2 ; Y1 |W1 , W2 )

= =

At this point, Rx2 will somehow use the conference link with capacity Rc2 to help Rx1 reduce the interference caused by U2 . Since decoding step 3 (at Rx1) will put additional constraints on the rate of U2 , which are possibly different from the rate constraints that the decoding step 2 (at Rx2) put on U2 , the achievable rate of U2 will be restricted by the tightest constraint. To gain more insight, suppose we ∗ allocate R12 and Rq such that R22 is the rate constraint on ∗ U2 from decoding step 2, i.e., R22 ≤ R22 . Fig. 3 shows the relationship between the rate regions in decoding step 2 and ∗ step 3, with R22 emphasized. The solid pentagon denotes the rate region achievable at Rx1 if Rx1 decodes (U1 , U2 ) without ∗ side information from Rx2. Also note that R11 is the maximum rate of U1 that can be achieved in decoding step 3. To help Rx1, Rx2 performs bin-and-forward [10] on the message U2 . Decoding of (U1 , U2 ) at Rx1 is hence equivalent to a MAC with a digital relay link of rate Rc2 , and the relay (Rx2 in this case) knows U2 perfectly. Yu and Zhou proved in [2] that the constraints on the rate of U2 and on the sum-rate

will be increased by exactly Rc2 bits, i.e., the boundary of the solid pentagon in Fig. 3 would shift to the right by Rc2 bits. However, due to the rate constraints on U2 in decoding step 2, the actual achievable rate region will be the intersection ∗ of the shifted pentagon and the half plane R22 ≤ R22 in Fig. 4. Depending on the value of Rc2 the following cases may happen. R11 ∗ R11

T

Q

(C)(B)

∗ ˜ 22 + Rc2 < R22 ¯ 22 + Rc2 B. Case 2: R ≤R As noted in the third paragraph of Section III, the conference link may help to shift the boundary of the solid pentagon ∗ in Fig. 3 to the right by Rc2 bits. However, since R22 is now the bottleneck as shown by region (B) in Fig. 4, the individual ∗ rate of U2 will be upper bounded by R22 and the achievable rates in decoding step 3 are given by

(A) S Rc2

˜ 22 R

∗ ¯ 22 R22 R

R(α, β, η, Rq ) is a pentagon characterized by    αSNR ¯ 1  R ≤ C + C(αSNR1 ) 1  1+βINR 2 +αSNR1       R2 ≤ min C(SNR2 ) + f2 (α, β, η, Rq ) + Rc1 − Rq ,       ¯ βINR 2 C + C (βSNR ) + f (α, β, η, R ) 2 1 q  1+βINR2 +αSNR1       ¯ αSNR ¯  1 +βINR2  R + R ≤ C 1 2  1+βINR2 +αSNR1 + C(βSNR2 )    +f1 (α, β, η, Rq ) + C(αSNR1 ).

R22

Fig. 4. Rate regions in decoding step 3. R∗11 = I(U1 ; Y1 |W1 , W2 , U2 ), ¯ 22 = I(U2 ; Y1 |W1 , W2 , U1 ), R ˜ 22 = I(U2 ; Y1 |W1 , W2 ) R

R11 ≤ I(U1 ; Y1 |W1 , W2 , U2 ) , I7 R22 ≤

∗ R22

, I8

R11 + R22 ≤ I(U1 , U2 ; Y1 |W1 , W2 ) + Rc2 , I9 ∗ ˜ 22 + Rc2 ([3]) A. Case 1: R22 ≤R

As shown in Fig. 4, the intersection of the shifted pentagon ∗ (region (A)) with the half plane R22 ≤ R22 is a rectangle. In other words, decoding at Rx1 does not limit the rate of U2 , and therefore the maximum rate of U1 is achievable without putting more constraints on the rate of U2 . This case corresponds to the case studied in [3]. The optimal operating point in terms of individual rates as well as sum-rate is point Q in Fig. 4. To achieve point Q, Rx1 performs either joint decoding or successive decoding as described in [3]. The achievable rate region in this case is restated in the next theorem. For more details we refer to [3]. Theorem 1 ([3]): Let us define   βINR2 (3) f1 (α, β, η, Rq ) = C (1 + βSNR2 )(1 + ∆ + αSNR1 ) f2 (α, β, η, Rq )  ¯ 2 ¯ 2 )INR2  β βη INR2 SNR2 + (1 + 2η β¯ + βη , (4) =C (1 + SNR2 )(1 + ∆ + αSNR1 ) where  1 ¯ 2 )INR2 (1 + 2η β¯ + βη 2R (1 + SNR2 )(2 q − 1)  ¯ 2 INR2 SNR2 + (1 + αSNR1 )(1 + SNR2 ) . +β βη (5)

∆=

With the assumption for case 1, the following rate region is achievable for the Gaussian Z-interference channel with conferencing decoders     [ co R(α, β, η, Rq ) , (6)   0≤α≤1,0≤β≤1,η∈R,Rq ≤Rc1

where

“co”

denotes

the

convex

hull

operator,

and

(7a) (7b) (7c)

We can observe the trade-off between rate R11 and R22 in ∗ ∗ Fig. 4: the respective maximum individual rates R11 and R22 for U1 and U2 (point Q) are not simultaneously achievable as ∗ in case 1. To achieve R11 one must reduce the rate R22 (move ∗ to point S), and to obtain the rate R22 one must reduce the rate R11 (move to point T ). Any point between S and T are achievable by joint decoding or time-sharing. Theorem 2: With the assumption for case 2, the following rate region is achievable for the Gaussian Z-interference channel with conferencing decoders     [ co R(α, β, η, Rq ) , (8)   0≤α≤1,0≤β≤1,η∈R,Rq ≤Rc1

where “co” denotes the convex hull operator, and R(α, β, η, Rq ) is a pentagon characterized by    αSNR ¯ 1  + C (αSNR1 ) R ≤ C 1  1+βINR    2 +αSNR1     ¯ βINR  2  R2 ≤ min C 1+βINR2 +αSNR1 + f1 (α, β, η, Rq )         +C (βSNR ) , C (SNR ) + f (α, β, η, R ) + R − R 2 2 2 q c1 q          ¯ αSNR ¯ 1 +βINR2 R1 + R2 ≤ min C 1+βINR + f1 (α, β, η, Rq ) 2 +αSNR1      αSNR ¯ 1  +C (βSNR ) + C (αSNR ) , C 2 1  1+βINR +αSNR 2 1  
  ¯ +C βSNR R − R  2 + f3 (α, β, η, Rq ) + c1 q      ¯ αSNR ¯  1 +βINR2  +C (αSNR1 + βINR2 ) + Rc2 , C 1+βINR  +αSNR 2 1       +C (αSNR1 + βINR2 ) + Rc2 ,

with f1 (α, β, η, Rq ), f2 (α, β, η, Rq ) defined in (3) and (4), respectively. f3 (α, β, η, Rq ) is defined in the following   ¯ (1 + η)2 βINR 2 f3 (α, β, η, Rq ) = C (9) ¯ (1 + βSNR 2 )(1 + ∆ + αSNR1 )

Proof: We first derive the overall achievable rate region by combining rate constraints in decoding steps 1, 2 and 3, i.e., the rates in (1a)–(1c), (2a)–(2c), and (7a)–(7c). Note that I3 ≤ I1 + I2 I6 ≤ I4 + I5 ,

(10a) (10b)

and that (7b) is implied by (2b) and (2c). Collecting the involved equations and replacing Rii by Ri − R0i , i = 1, 2 we have: R01 ≤ I1 , R02 ≤ I2 , R01 + R02 ≤ I3 , R02 ≤ I4 , R2 − R02 ≤ I5 , R2 ≤ I6 , R1 −R01 ≤ I7 , R1 +R2 −R01 −R02 ≤ I9 . Next we apply the Fourier-Motzkin algorithm [11] to eliminate R01 . R01 has the following lower and upper bounds: R01 ≤ I1 , R01 ≤ I3 −R02 , R01 ≥ R1 −I7 , R01 ≥ R1 +R2 −R02 −I9 . Combining the bounds with the inequalities not containing R01 we have R1 ≤ I1 + I7 , R1 + R2 − R02 ≤ I1 + I9 , R1 + R02 ≤ I3 + I7 , R1 + R2 ≤ I3 + I9 , R02 ≤ I2 , R02 ≤ I4 , R2 − R02 ≤ I5 , R2 ≤ I6 . Similarly, by eliminating R02 we obtain 2R1 + R2 ≤ I1 + I3 + I7 + I9 R1 + R2 ≤ I3 + I5 + I7

(11a) (11b)

R1 + R2 ≤ I1 + I2 + I9 R2 ≤ I2 + I5

(11c) (11d)

R1 + R2 ≤ I1 + I4 + I9 R2 ≤ I4 + I5

(11e) (11f)

R1 ≤ I1 + I7 R1 + R2 ≤ I3 + I9

(11g) (11h)

R2 ≤ I6 .

(11i)

Taking the conditions (10a)–(10b) into account we can verify that (11a), (11c), and (11f) are redundant. Therefore, R1 ≤ I1 + I7 R2 ≤ min{I2 + I5 , I6 }

(12a) (12b)

R1 + R2 ≤ min{I3 + I5 + I7 , I1 + I4 + I9 , I3 + I9 }. (12c) The proof is completed by evaluating the mutual information terms with Gaussian random variables. ∗ ¯ 22 + Rc2 < R22 C. Case 3: R Similarly to situation 2 above, the sum-rate and the rate constraint of U2 are enhanced by Rc2 bits. In this case, however, the shifted pentagon is in the left of the delimiter line ∗ R22 = R22 . Therefore the rate constraints in decoding step 3 are the bottleneck of R22 , which are shown in the following, and the rate region is denoted by region (C) in Fig. 4 R11 ≤ I(U1 ; Y1 |W1 , W2 , U2 ) = I7

(13)

R22 ≤ I(U2 ; Y1 |W1 , W2 , U1 ) + Rc2 , I8′

(14)

R11 + R22 ≤ I(U1 , U2 ; Y1 |W1 , W2 ) + Rc2 = I9

(15)

Theorem 3: With the assumption in case 3, the achievable rate region for the Gaussian Z-interference channel with conferencing decoders is given below     [ co R(α, β, η, Rq ) , (16)   0≤α≤1,0≤β≤1,η∈R,Rq ≤Rc1

where “co” denotes the convex hull operator, and R(α, β, η, Rq ) is a pentagon characterized by    αSNR ¯ 1  R1 ≤ C 1+βINR + C (αSNR1 )  +αSNR   2 ¯ 1      βINR2  R ≤ min C + f1 (α, β, η, Rq )  2 1+βINR  2 +αSNR1     ¯  βINR 2  + C (βINR2 ) + Rc2 , +C (βSNR2 ) , C 1+βINR2 +αSNR  1 
  ¯  C βSNR 2 + f3 (α, β, η, Rq ) + Rc1 − Rq + C (βINR2 )     +R , C (SNR + f2 (α, β, η, Rq ) + Rc1 − Rq  c2 2 )    ¯ αSNR ¯ 1 +βINR2 + f1 (α, β, η, Rq ) R1 + R2 ≤ min C 1+βINR  2 +αSNR1       αSNR ¯  1  +C (βSNR ) + C (αSNR ) , C 2 1  1+βINR2 +αSNR1  
 ¯  +C βSNR + f (α, β, η, R ) +  2 3 q  Rc1 − Rq    ¯  αSNR ¯ 1 +βINR2   +C (αSNR + βINR ) + R , C 1 2 c2    1+βINR2 +αSNR1     +C (αSNR1 + βINR2 ) + Rc2

Proof: As in case 2, applying the Fourier-Motzkin elimination algorithm [11] while paying attention to the constraints in (10a)–(10b) and I9 ≤ I7 + I8′ we readily obtain R1 ≤ I1 + I7 R2 ≤ min{I2 + I5 , I2 + I8′ , I4 + I8′ , I6 }

(17) (18)

R1 + R2 ≤ min{I3 + I5 + I7 , I1 + I4 + I9 , I3 + I9 }. (19) The final step is the calculation of the Gaussian mutual information terms. IV. O UTER B OUND

TO THE

C APACITY R EGION

Proposition 4: For the Gaussian Z-interference channel with conferencing decoders with conference links of rates Rc1 and Rc2 , the capacity region is outer bounded by   R1 ≤ C(SNR1 )    R2 ≤ min{C(SNR2 + INR2 ), C(SNR2 ) + Rc1 }  R1 + R2 ≤ min C(SNR1 + SNR  1 SNR2 ),  2 + INR2 + SNR    C (SNR1 + INR2 ) + C SNR2 + Rc2 1+INR2

Proof: The proof follows from the bounding techniques for the interference channel, namely cut-set bound and genieaided bound [12]. We apply a loosened cut-set upper bound [13, Theorem 15.10.1] by directly maximizing the mutual information term corresponding to each cut, see Fig. 5. • For cut (1) we easily obtain R1 ≤ C(SNR1 ). • Cut (2) yields R2 ≤ max I(X2 ; Y1 , Y2 |X1 ). By some p(x1 ,x2 )

•

manipulations of the mutual information and using the fact that Gaussian distribution maximizes conditional differential entropy given covariance constraints, e.g. [12], we conclude that I(X2 ; Y1 , Y2 |X1 ) is maximized with X2 ∼ N (0, P2 ). Accordingly R2 ≤ C(SNR2 + INR2 ). For cut (3) we have R1 +R2 ≤ max I(X1 , X2 ; Y1 , Y2 ). p(x1 ,x2 )

Similarly to cut (2), the mutual information term is maximized with Gaussian inputs X1 and X2 and we

end up with a bound on the sum-rate: R1 + R2 ≤ C(SNR1 + SNR2 + INR2 + SNR1 SNR2 ). • For cut (4), it is easy to see that R2 ≤ C(SNR2 ) + Rc1 . Note that these bounds can be alternatively derived using Fano’s inequality. Next we derive a genie-aided-type bound on the sum-rate. n n Let V21 and V12 be output sequences of the conference links at Rx1 and Rx2, respectively. By Fano’s inequality we have n(R1 + R2 − ǫn ) n n ≤ I(X1n ; Y1n , V21 ) + I(X2n ; Y2n , V12 )

reasonable in the weak interference regime [2, 3]. As illustrated, with a given Rc1 different Rc2 ’s will produce different achievable rate regions, which are analyzed in Section III. It can be seen that the individual rate constraints on R2 are the same for case 1 and case 2 and larger than that for case 3. The reason is explained in Fig. 4 and Section III, namely the last decoding step puts no additional constraints on the individual rate of U2 for case 1 and case 2, while putting a tighter constraint for case 3. However, case 2 has a smaller sum-rate constraint than case 1.

n n , Y1n ) |Y1n ) + I(X2n ; Y2n , V12 ≤ I(X1n ; Y1n ) + I(X1n ; V21

4.5

2

−

2h(Z1n ),

of (Y1n , Y2n ) because X1n

n(R1 + R2 − ǫn )
n ≤ log 2πe(h211 P1 + h221 P2 + N ) + nRc2 2    n h2 P2 + log 2πeN 1 + 2 22 − n log(2πeN ) 2 h21 P2 + N   SNR2 = nC (SNR1 + INR2 ) + nC + nRc2 . 1 + INR2 Combining the individual bounds we complete the proof. (3)

Z1

h11

X1

Y1

X2

Fig. 5.

Y2

h22 (2)

Rc2

Rc1

h21

(4)

3 2.5 2

R =5 bits (case 1) c2

R =3 bits (case 2) c2

1

where (20) is due and n I(X1n ; V21 |Y1n ) ≤ and X2n are independent and that conditioning reduces entropy. Since Gaussian distribution maximizes differential entropy and conditional differential entropy given the covariance constraints [12], the right hand side of (22) is maximized by i.i.d. Gaussian inputs X1n and X2n . Finally, the MMSE estimator of h22 X2 + Z2 given observation h21 X2 + Z1 leads to

(1)

3.5

1.5

(22) n to V12 is a function n H(V21 ). (21) follows

4

(20) (21) R (bits)

n ≤ I(X1n ; Y1n ) + H(V21 ) + I(X2n ; Y2n , Y1n ) ≤ I(X1n ; Y1n ) + nRc2 + I(X2n ; Y2n , Y1n |X1n ) = h(Y1n ) − h(Y1n |X1n ) + nRc2 + h(Y2n , Y1n |X1n ) − h(Y2n , Y1n |X1n , X2n ) = h(Y1n ) + nRc2 + h(Y2n |Y1n , X1n ) − h(Z1n , Z2n ) = h(Y1n ) + nRc2 + h(h22 X2n + Z2n |h21 X2n + Z1n )

Z2

Cuts for the Z-interference channel with conferencing decoders

V. N UMERICAL E XAMPLES Fig. 6 shows achievable rate regions for various values of Rc2 given a set of parameters: α = β = 1, η = −1, Rq = Rc1 = 3 bits, and SNR1 = SNR2 = 25 dB, INR2 = 15 dB (weak interference condition). This set of parameters is

R =1 bit (case 3) c2

0.5 0

0

0.5

1

1.5

2 2.5 R1 (bits)

3

3.5

4

4.5

Fig. 6. Achievable rate regions for the Gaussian Z-interference channel with conferencing, SNR1 = SNR2 = 25 dB, INR2 = 15 dB, α = β = 1, η = −1, Rc1 = 3 bits, and different Rc2 .

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