Z-interference channel with conferencing decoders. First we focus on finding expressions to characterize an achievable rate region. Then we study different ...
An Achievable Rate Region for the Gaussian Z-interference Channel with Conferencing c
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HIEU T. DO, TOBIAS J. OECHTERING, AND MIKAEL SKOGLUND
Stockholm 2009 Communication Theory Department School of Electrical Engineering KTH Royal Institute of Technology IR-EE-KT 2009:073
An Achievable Rate Region for the Gaussian Z-interference Channel with Conferencing 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
Abstract—This paper presents an achievable rate region for a 2-user Gaussian Z-interference channel with a noiseless and bidirectional digital communication link between the receivers. The region is achieved by utilizing the rate-splitting encoding technique, and the decode-and-forward and compress-andforward strategies. In the very strong interference regime, the capacity region is achieved. In the weak interference regime, the asymptotic sum rate is characterized and shown to be possibly unbounded, which is in contrast to a recent result by Yu and Zhou for a similar scenario, however, with a unidirectional communication link between the receivers.
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. In the quest for information theoretic results for the interference channel, the Z-interference channel plays an important role as an idealized special case [2]. Recently, the Z-interference channel with and without cognitive transmitter has received much research attention, for example [3, 4] and references therein. On the other hand, the Gaussian Z-interference with a digital relay link at the receivers’ side has been investigated by Yu and Zhou in [5–7]. In these papers, a unidirectional digital relay link from one receiver to the other is assumed and its effect on the achievable rate region as well on the sum rate is considered. The unidirectional relay link is utilized through a combination of various coding strategies such as rate-splitting and superposition encoding, partial interference forwarding, and well known strategies for the relay channel [8]. Achievable rate regions are found and shown to constitute the capacity region in special cases. In the present paper, we consider the situation in which both receivers can exchange information over noiseless digital links with limited rates. We refer to this scenario as the Gaussian Z-interference channel with conferencing decoders. First we focus on finding expressions to characterize an achievable rate region. Then we study different interference conditions. In the very strong interference regime the capacity region is established. In the weak interference regime we investigate the achievable sum rate and asymptotic sum rate when the conference links have infinite capacities. Our work is motivated by its strong connection to current problems in information theory and wireless communications. This work was supported in part by the Swedish Research Council
In practice, a Z-interference channel with conferencing decoders can serve as a good model for a situation in a wireless ad hoc network where two receivers are close enough to be able to cooperate and one of the transmitters suffers from the shadowing effect. Furthermore, the channel under consideration fits well in the uplink of Wyner-type model [9] for the multi-cell communications in wireless cellular networks where base stations are interconnected through wireline backhaul links, e.g. [10]. Eventually, the result for the Gaussian Zinterference channel with conferencing decoders promises to give insights for the problem of Gaussian interference channel with decoder cooperation. The remainder of the paper is organized as follows: Section II introduces the channel model and formulate the problem. In Section III an achievable rate region for the Gaussian Z-interference channel with conferencing decoders is presented. Section IV shows that the capacity region is achieved when the interference is very strong. Section V studies the rate region and sum rate in the weak interference regime. The asymptotic sum rate is found together with discussions of the result. Finally, Section VI concludes the paper. II. C HANNEL M ODEL AND P ROBLEM F ORMULATION Consider a discrete-time Z-interference channel as in Fig. 1. The noises Z1 and Z2 at the receivers are independent zeroZ1 X1
h11
Y1 Rc2
Rc1
h21 X2
Y2
h22 Z2
Fig. 1.
Gaussian Z-interference channel with conferencing decoders
mean Gaussian random variables with equal variance of N . h11 , h21 , and h22 denote time-invariant channel gains between transmitters and receivers. We assume that the channel gains
Z1 Rate
R11 αP1 U1
X1
R01 αP ¯ 1 W1
h11
Y1 Rq
R12 R21
h21
¯ 2 W2 R02 βP R22 βP2 U2
X2
Y2
h22 Z2
Fig. 2. decoders
|h11 |2 P1 |h22 |2 P2 , SNR2 = N N |h21 |2 P2 1 INR2 = , C(x) = log(1 + x) N 2 α ¯ = 1 − α, β¯ = 1 − β,
SNR1 =
Power
Coding for Gaussian Z-interference channel with conferencing
are known to the transmitters and receivers. The two receivers can communicate over two noiseless digital conference links having finite capacities of Rc1 and Rc2 , each link is unidirectional. This general rate constraint on the conference links can be modified to reflect different situations, for example, Rc1 = Rc2 means symmetric conference links, Rc1 = 0 or Rc2 = 0 is equivalent to the settings in [5, 6], and a constraint on Rc1 + Rc2 corresponds to a sum capacity constraint on the conference link. The received signals at receiver 1 and 2, respectively, are given by
where the logarithm is to the base 2. Let Rx1 and Rx2 denote receivers 1 and 2, respectively. The decoding process consists of the following 3 steps: 1) Rx1 decodes (W1 , W2 ) jointly, treating U1 , U2 as noise. Rx1 then quantizes Y1 into Yˆ1 and bins and forwards W2 [8] to Rx2 to help Rx2 decode (W2 , U2 ). The rate used for binning W2 is R12 and the rate for Yˆ1 is Rq , and R12 + Rq ≤ Rc1 . 2) Rx2 decodes (W2 , U2 ) jointly with the help of the relay link from Rx1. Rx2 then performs random binning [13] of U2 with an appropriate number of bins and sends the bin index to Rx1 using rate R21 . 3) Rx1 decodes U2 , subtracts it, and decodes U1 without interference. Note that the rate constraints on the conferencing links require Rq ≤ Rc1 and R21 ≤ Rc2 . Next we investigate the decoding process and the resulting achievable rates.
Y1 = h11 X1 + h21 X2 + Z1
B. Achievable Rate Region
Y2 = h22 X2 + Z2 .
To gain additional insight, we first present the achievable rate region in terms of mutual information expressions and subsequently by their corresponding expressions in the Gaussian settings. Theorem 1: Assume Rc2 is large enough to accommodate R21 , the following rate region is achievable1 for the Gaussian Z-interference channel with conferencing decoders R1 ≤ I(W 1 ; Y1 |W2 ) + I(U1 ; Y1 |W1 , W2 , U2 ) R2 ≤ min I(U2 , W2 ; Y2 , Yˆ1 ) + Rc1 − Rq , (1) ˆ I(W ; Y |W ) + I(U ; Y , Y |W ) 2 1 1 2 2 1 2 R1 + R2 ≤ I(W1 , W2 ; Y1 ) + I(U2 ; Y2 , Yˆ1 |W2 ) +I(U1 ; Y1 |W1 , W2 , U2 )
The two transmitters have average power constraints P1 and P2 , respectively, i.e., n
1X E |xi [j]|2 ≤ Pi , n j=1
i = 1, 2
where n is the codeword length. The main contribution of this paper is to characterize an achievable rate region for the channel in Fig. 1. We also study the rate region in very strong and weak interference regimes, and analyze the achievable sum rate when the interference is weak. III. A N ACHIEVABLE R ATE R EGION A. Encoding and Outline of Decoding Steps Both transmitters employ the rate-splitting and superposition coding technique as in the works of Carleial [11] and Han and Kobayashi [12]. Specifically, transmitter 1 splits its information into messages W1 and U1 . The two messages are encoded separately by independent Gaussian codebooks with powers at most αP1 and (1 − α)P1 , and then superimposed to form the channel input X1 . The channel input X2 at transmitter 2 is similarly constructed from messages W2 and U2 , power P2 , and the power-splitting factor β. Let R0i and Rii denote the respective rates of Ui and Wi where i = 1, 2, i.e., Ri = R0i + Rii . The coding scheme is depicted in Fig. 2. For notational simplicity we define
where Yˆ1 is a Gaussian quantized version of Y1 with the quantization rate Rq . For the explicit expression of the rate region, let us define βINR2 f1 (α, β, η, Rq ) = C (2) (1 + βSNR2 )(1 + ∆ + αSNR1 ) f2 (α, β, η, Rq ) ¯ 2 ¯ 2 )INR2 β βη INR2 SNR2 + (1 + 2η β¯ + βη =C , (3) (1 + SNR2 )(1 + ∆ + αSNR1 ) 1 The achievability of a particular rate is in the usual sense that if we allow sufficiently long codewords, it is possible to communicate at arbitrarily low error probability.
where h 1 ¯ 2 )INR2 (1 + 2η β¯ + βη ∆= (1 + SNR2 )(22Rq − 1) i ¯ 2 INR2 SNR2 + (1 + αSNR1 )(1 + SNR2 ) , (4) +β βη
then the achievable rate region is given by [ co R(α, β, η, Rq ) ,
(5)
0≤α≤1,0≤β≤1,η∈R,Rq ≤Rc1
where “co” denotes the convex hull operator, η is a real number (can be optimized later), and 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 1+βINR2 +αSNR1 + C (βSNR2 ) + f1 (α, β, η, Rq ) ¯ αSNR ¯ 1 +βINR2 R + R ≤ C 1 2 1+βINR2 +αSNR1 + C(βSNR2 ) +f1 (α, β, η, Rq ) + C(αSNR1 ) (6)
Remark 2: i) ∆ in (4) denotes the ratio between the variance of the auxiliary random variable used to generate the distribution of Yˆ1 and the receiver’s noise variance. ∆ is determined by Rq and the design parameter η. ii) By letting α = 0, we obtain the result of Yu and Zhou in [6]. Proof: We begin with the proof of the achievable rate region in (1), the derivations leading to (6) and (5) will follow later. First notice that in decoding step 1, Rx1 decodes (W1 , W2 ) jointly with U1 , U2 treated as noise. We can interpret this as a Gaussian multiple access channel (MAC). Accordingly, the achievable rates are given by R01 ≤ I(W1 ; Y1 |W2 ) , F1
(7a)
R02 ≤ I(W2 ; Y1 |W1 ) , F2
(7b)
R01 + R02 ≤ I(W1 , W2 ; Y1 ) , F3
(7c)
After decoding (W1 , W2 ), ˆ1 using rate • Rx1 quantizes Y1 (with W1 subtracted) into Y Rq to help Rx2 decode U2 . • Rx1 uses a rate of R12 = Rc1 − Rq to help Rx2 decode W2 as in [7, Lemma 1]. By doing so, any R12 > 0 is always helpful for the decoding of W2 at Rx2. To do quantize and forward, Rx1 first forms a mediate signal [6]: Y¯1 = h21 (U2 + W2 ) + ηh21 W2 + h11 U1 + Z1 ,
(8)
where η ∈ R is the subtracting factor for W2 . Rx1 performs Gaussian quantization as follows: Yˆ1 = Y¯1 + Zq
(9)
where Zq is zero-mean Gaussian with variance σ 2 and independent of Y¯1 . The rate for quantizing Y¯1 to Yˆ1 is stated in [6] (which in turn refers to Cover and Chiang [14, Theorem 2]) as follows: Rq ≥ I(Yˆ1 ; Y¯1 , W2 ) − I(Yˆ1 ; Y2 ) = I(Yˆ1 ; Y¯1 ) + I(Yˆ1 ; W2 |Y¯1 ) − I(Yˆ1 ; Y2 )
(10)
= I(Yˆ1 ; Y¯1 ) − I(Yˆ1 ; Y2 ) = I(Yˆ1 ; Y¯1 , Y2 ) − I(Yˆ1 ; Y2 )
(12)
= I(Yˆ1 ; Y¯1 |Y2 )
(13)
(11)
where (11) follows because Yˆ1 = Y¯1 + Zq , Zq is independent of W2 and Y2 . (12) follows from the fact that Yˆ1 and Y2 are independent given Y¯1 . For maximum efficiency, we choose Yˆ1 such that Rq = I(Yˆ1 ; Y¯1 |Y2 ). In decoding step 2, Rx2 jointly decodes (W2 , U2 ) based on observations Y2 and Yˆ1 with Rx1 acts as a relay. This is equivalent to a MAC with a rate-limited relay who knows W2 perfectly. The fact that W2 is known at the relay will enlarge the capacity region of the MAC by increasing the rate for W2 and the sum rate by exactly R12 bits [5]. Therefore the following rates are achievable: R02 ≤ I(W2 ; Y2 , Yˆ1 |U2 ) + R12 , F4 R22 ≤ I(U2 ; Y2 , Yˆ1 |W2 ) , F5
(14b)
R02 + R22 ≤ I(U2 , W2 ; Y2 , Yˆ1 ) + R12 , F6
(14c)
(14a)
After decoding U2 , Rx2 randomly bins 2nR22 codewords into 2nR21 = 2n(R22 −I(U2 ;Y1 |W1 ,W2 )+2ǫ) bins and transmits the index of the bin containing the codeword of U2 to Rx1 [15, 16]. Rx1 intersects the contents of this bin with the list of codewords jointly typical with its observation Y1 . On average, within a bin there are 2nI(U2 ;Y1 |W1 ,W2 ) random codewords, hence I(U2 ; Y1 |W1 , W2 ) is adequate to resolve them. In step 3 of the decoding process, Rx1 decodes U2 , subtracts it and decodes U1 without interference. The achievable rate is given by R11 = I(U1 ; Y1 |W1 , W2 , U2 ). (15) Note that in this step it is not necessary to perform joint decoding of (U2 , U1 ) since Rx1 is interested in U1 only. To this point we need to combine the rates from (7) and (14). Note that we always have F1 + F2 ≥ F3 and F4 + F5 ≥ F6 , which reduces (7) and (14) to R01 ≤ F1
(16a)
R02 + R22 ≤ min{F6 , F2 + F5 } R01 + R02 + R22 ≤ F1 + F6
(16b) (16c)
R01 + R02 + R22 ≤ F3 + F5 .
(16d)
We readily see that third inequality in (16) is redundant since it is implied by the first and the second inequalities. Finally, by combining (16) with (15) and plug in the appropriate mutual information terms we end up with the general achievable rate region of the Gaussian Z-interference channel with conferencing decoders as in (1).
We proceed by evaluating the mutual information terms in (7) with Gaussian random variables to obtain the following terms for decoding step 1: α ¯ SNR1 I(W1 ; Y1 |W2 ) = C (17a) 1 + βINR2 + αSNR1 ¯ βINR 2 (17b) I(W2 ; Y1 |W1 ) = C 1 + βINR2 + αSNR1 ¯ α ¯ SNR1 + βINR 2 I(W1 , W2 ; Y1 ) = C . (17c) 1 + βINR2 + αSNR1 Recall that we choose the quantization rate for Y1 as Rq = I(Yˆ1 ; Y¯1 |Y2 ), i.e., Rq = h(Yˆ1 |Y2 ) − h(Yˆ1 |Y¯1 , Y2 ) 1 1 = log 2πeσY2ˆ |Y − log 2πeσ 2 1 2 2 2 σY2ˆ |Y 1 = log 12 2 2 σ
(18) (19)
where h(·) denotes differential entropy and the formula for differential entropy of Gaussian random variables are used in (18). To find σY2ˆ |Y , the mean square error when estimating 1 2 Yˆ1 given observation Y2 , we first see that the minimum mean squared error (MMSE) estimator of Yˆ1 is ˆ Y2 , Yˆ1 = RYˆ1 Y2 RY−1 2 RY2 denotes the covariance of Y2 , and RYˆ1 Y2 denotes the ˆ cross covariance of Yˆ1 and Y2 . Yˆ1 can be easily calculated and results in σY2ˆ
1 |Y2
ˆ = E{|Yˆ1 − Yˆ1 |2 } ¯ 2 )|h21 |2 P2 + |h11 |2 αP1 = N + σ 2 + (1 + 2η β¯ + βη ¯ 2P 2 |h21 |2 |h22 |2 (1 + βη) 2 − , 2 |h22 | P2 + N
which makes (19) equivalent to 22Rq − 1 =
¯ 2 INR2 SNR2 + (1 + 2η β¯ + βη ¯ 2 )INR2 N h β βη σ2 1 + SNR2 i
+1 + αSNR1 .
(20)
Note that for given α and β, Rq and η will determine the variance σ 2 of the auxiliary random variable Zq . For notational 2 brevity we define ∆ , σN , whose explicit value is given in (4). Next we evaluate the mutual information terms in F5 and F6 of (14). F4 is skipped since it does not appear in the expression of the rate region (see (16)). We have I(U2 ; Y2 , Yˆ1 |W2 ) = I(U2 ; Y2 |W2 ) + I(U2 ; Yˆ1 |Y2 , W2 ) = C(βSNR2 ) + f1 (α, β, η, Rq ) ˆ I(U2 , W2 ; Y2 , Y1 ) = I(U2 , W2 ; Y2 ) + I(U2 , W2 ; Yˆ1 |Y2 ) = C(SNR2 ) + f2 (α, β, η, Rq ),
where we defined f1 (α, β, η, Rq ) = I(U2 ; Yˆ1 |Y2 , W2 ) and f2 (α, β, η, Rq ) = I(U2 , W2 ; Yˆ1 |Y2 ). Their values are calculated as follows: f1 (α, β, η, Rq ) = I(U2 ; Yˆ1 |Y2 , W2 ) = h(Yˆ1 |Y2 , W2 ) − h(Yˆ1 |W2 , U2 , Y2 )
(21) ˆ = h(Y1 |Y2 , W2 ) 1 − log 2πe(N + σ 2 + |h11 |2 αP1 ) , (22) 2 where (22) is a consequence of (8) and (9). Similarly, f2 (α, β, η, Rq ) = I(U2 , W2 ; Yˆ1 |Y2 ) = h(Yˆ1 |Y2 ) − h(Yˆ1 |U2 , W2 , Y2 )
(23)
= h(Yˆ1 |Y2 ) − h(Yˆ1 |U2 , W2 ) 1 = log πeσY2ˆ |Y 1 2 2 1 − log πe(N + σ 2 + |h11 |2 αP1 ) . 2 After calculating the MMSE components, we end up with expressions for f1 (α, β, η, Rq ) and f2 (α, β, η, Rq ) as in (2) and (3), respectively. In decoding step 3, (15) is easily computed as R11 = C (αSNR1 ) .
(24)
Substituting each mutual information term in (1) by its corresponding numerical term calculated above, we obtain the pentagon in (6). Finally, the overall rate region in (5) is obtained by taking the convex hull of the union of (6) over all appropriate parameters. IV. V ERY S TRONG I NTERFERENCE R EGIME Corollary 3: In the very strong interference regime, defined by INR2 ≥ (SNR1 + 1) 22Rc1 (SNR2 + 1) − 1 , (25)
the capacity region of the Z-interference channel with conferencing decoders is given by R1 ≤ C(SNR1 ) (R1 , R2 ) : . (26) R2 ≤ C(SNR2 ) + Rc1 Proof: We first prove the converse. Assume (R1 , R2 ) is in the capacity region of the channel, i.e., Rxi can decode Xi at rate Ri , i = 1, 2 (probably with the help of conferencing). Since R1 and R2 are achievable, they must satisfy the cut-set upper bound R1 ≤ C(SNR1 )
(27a)
R2 ≤ C(SNR2 ) + Rc1 .
(27b)
For achievability, by setting α = β = 0, Rq = 0 in the general achievable region (6), and absorbing (25) we obtain R1 ≤ C(SNR1 ) R2 ≤ C(SNR2 ) + Rc1 R1 + R2 ≤ C(SNR1 + INR2 ).
(28a) (28b) (28c)
Now, with the very strong interference condition (25) we can verify that C(SNR1 ) + C(SNR2 ) + Rc1 ≤ C(SNR1 + INR2 ). Therefore, the sum rate constraint (28c) is not active. Since the outer bound in (27a)–(27b) is identical to the achievable rate region in (28a)–(28b) the capacity region is established. Remark 4: i) This result is the same as the result for very strong interference regime in [6]. The capacity region is a rectangle. ii) In this regime, the conference link from Rx2 to Rx1 is not needed since the result in Corollary 3 holds for any Rc2 ≥ 0. V. W EAK I NTERFERENCE R EGIME A. Rate Region and Asymptotic Sum Rate We now focus on the weak interference regime, defined by INR2 ≤ SNR2 , i.e., |h21 |2 ≤ |h22 |2 . This is a common scenario in a cellular network, where the mobile devices are designated to the base station with strongest channel. Corollary 5: In the weak interference regime, defined by INR2 ≤ SNR2 , and assuming that R21 ≤ Rc2 , the achievable region of the Gaussian Z-interference channel with conferencing decoders is given by [ R(α, β) (29) co 0≤α≤1,0≤β≤1
where R(α, β) is a pentagon characterized by αSNR ¯ 1 R ≤ C + C(αSNR1 ) 1 2 +αSNR1 1+βINR ¯ βINR 2 R2 ≤ C 1+βINR2 +αSNR1 + C (βSNR2 )
+f1 (α, β,η ∗ , Rc1 ) ¯ αSNR ¯ 1 +βINR2 R1 + R2 ≤ C 1+βINR + C(βSNR2 ) +αSNR 2 1 ∗ +f1 (α, β, η , Rc1 ) + C(αSNR1 )
(30)
where η ∗ = − βSNR12 +1 . Proof: First note that (21) and (23) imply
f1 (α, β, η, Rq ) ≤ f2 (α, β, η, Rq ). Look at the constraint on R2 in (6), we have: ¯ βINR 2 C + C (βSNR2 ) 1 + βINR2 + αSNR1 ¯ βINR2 + βSNR2 (1 + INR2 + αSNR1 ) =C 1 + βINR2 + αSNR1 ¯ βSNR2 + βSNR2 (1 + INR2 + αSNR1 ) ≤C 1 + βINR2 + αSNR1 ≤ C(SNR2 ),
(31)
and (33) will directly simplify (6) to the following: αSNR ¯ 1 R ≤ C + C(αSNR1 ) 1 1+βINR +αSNR 2 1 ¯ βINR2 R ≤ C + C (βSNR2 ) + f1 (α, β, η, Rq ) 2 1+βINR 2 +αSNR1¯ αSNR ¯ + βINR 1 2 R + R ≤ C 1 2 1+βINR +αSNR 2 1 +C(βSNR2 ) + f1 (α, β, η, Rq ) + C(αSNR1 ) (34) It is interesting to see that when the interference is low, the rate R12 does not affect the achievable rate region of the Gaussian Z-interference channel with conferencing decoders, and therefore Rx1 uses all the conference link of rate Rc1 to describe U2 to Rx2, i.e., Rq = Rc1 . Moreover, for given power splitting factors α and β, we can maximize the achievable rate by maximizing f1 (α, β, η, Rc1 ) over η. By some simple calculations we readily obtain that f1 (α, β, η, Rc1 ) is maximized when η ∗ = − βSNR12 +1 . Substituting η ∗ (βSNR2 + 1) = −1 into (4) to obtain βINR2 1 1 + αSNR1 + (35) ∆ = 2Rc1 2 −1 1 + βSNR2 which, when evaluating (3), results in f1 (α, β, η ∗ , Rc1 ) =C
! βINR2 22Rc1 − 1 . (36) 22Rc1 (1 + βSNR2 ) (1 + αSNR1 ) + βINR2
We now shift our analysis to the achievable sum rate and see how much one can gain when the conference links have infinite capacity in the weak interference regime. Without conferencing, the sum capacity of the Gaussian Z-interference channel is given by [17, 18] SNR1 Csum (0) = C(SNR2 ) + C (37) 1 + INR2 In the expression for the achievable rate region in (30), sum of the constraints on R1 and R2 is larger than or equal to the constraint on the sum rate R1 + R2 . Hence for given α and β, the maximum achievable sum rate is given by ¯ α ¯ SNR1 + βINR 2 + C(βSNR2 ) 1 + βINR2 + αSNR1 + f1 (α, β, η ∗ , Rc1 ) + C(αSNR1 ) (38)
Rsum (α, β) =C
(32) (33)
where (32) and (33) follow from the low interference condition INR2 ≤ SNR2 and the fact that 0 ≤ β ≤ 1, respectively. (31)
By considering the first-order derivative of Rsum (α, β) with respect to α and β, it can be proved that Rsum (α, β) is always a non-decreasing function of α and is a non-decreasing function of β when the interference is weak (INR2 ≤ SNR2 ). Therefore Rsum (α, β) is maximized when α = β = 1, which is not surprising since the weak interference does not allow Rx1 to decode messages from Rx2. As a result, we have the
maximum achievable sum rate as follows Rsum = C(SNR2 ) + C(SNR1 ) INR2 (22Rc1 − 1) +C 22Rc1 (1 + SNR2 )(1 + SNR1 ) + INR2 SNR1 INR2 = Csum (0) + C 1 + SNR1 + INR2 INR2 (22Rc1 − 1) +C . 22Rc1 (1 + SNR2 )(1 + SNR1 ) + INR2 Hence, the sum rate gain by having conferencing is SNR1 INR2 δRsum = C 1 + SNR1 + INR2 INR2 (22Rc1 − 1) +C 22Rc1 (1 + SNR2 )(1 + SNR1 ) + INR2
Therefore, we easily see that point A in Fig. 3 is achieved with (α, β) = (0, 1), point B is achieved with (α, β) = (1, 0), and point C is achieved with (α, β) = (1, 1). Furthermore point C is the maximum sum rate point. By time sharing between point C and A we obtain the subregion of the achievable rate region when there is conferencing between decoders. Since the rate R2 at point C is strictly larger than the rate R2 at point B we can conclude that the time-sharing subregion, and hence the full achievable region in (29), encloses the achievable rate region when the relay link is unidirectional, which is given in Theorem 1 of [6]. In the figure we can also see that the improvement in achievable sum rate between having conferencing (point C) and having only one directional relay link (point A) is rather pronounced for this particular setting of channel parameters. Note that the rate Rc2 ≈ 3.135 bits is enough for the conferencing to be successful.
and the asymptotic gain as Rc1 → ∞ and Rc2 → ∞ (to ensure R21 ≤ Rc2 ) is SNR1 INR2 ∞ δRsum =C 1 + SNR1 + INR2 INR2 (39) +C (1 + SNR2 )(1 + SNR1 )
B. Numerical Example Fig. 3 shows achievable rate regions of the Gaussian Zinterference channel in the weak interference regime with i) no relay link, using the Han-Kobayashi scheme, ii) a unidirectional relay link of 2 bits from the interfered user to interference-free user [6], and iii) a subregion of the achievable rate region in (29) when Rc1 = 2 bits. The rate regions in cases i) and ii) are showed in Fig. 5 in [6]. The subregion iii) is achieved in the following manner: by similar arguments as in the proof of Theorem 1 in [7] we can show that for a given α, the union over 0 ≤ β ≤ 1 of the pentagons (30) is defined by R1 ≤ C(SNR1 ) R2 ≤ C(SNR2 ) + f1 (α, 1, η ∗ , Rc1 ) together with the lower right corner points (R1 , R2 ) of the pentagons, specified by α ¯ SNR1 R1 = C + C(αSNR1 ) (40) 1 + βINR2 + αSNR1 ¯ βINR 2 R2 = C + C (βSNR2 ) 1 + βINR2 + αSNR1 +f1 (α, β, η ∗ , Rc1 ). (41)
A 3.5
C A’
3
Time−sharing part
2.5 R2 (bits)
which can be unbounded depending on values of SNR1 , SNR2 and INR2 (provided that INR2 ≤ SNR2 ). This is in contrast to results in [6], where it is shown that the sum capacity gain for the Gaussian Z-interference channel with a digital relay link from the interfered receiver to the interference-free user is bounded by half a bit even though the relay link has infinite capacity.
4
2 1.5 1 H−K rate region without relay Achievable rate region with unidirectional relay link Achievable rate region with conferencing
0.5 0
0
0.5
1
1.5 2 R1 (bits)
2.5
B 3
3.5
Fig. 3. Achievable rate regions of the Gaussian Z-interference channel with SNR1 = SNR2 = 20dB, INR2 = 15dB
It is interesting to investigate how the overall achievable rate region described by (29) would look like. To see that in Fig. 4 we plot the curves drawn by the lower corner points specified in (40) and (41) when β moves from 0 to 1, each curve for a fixed value of α. It can be seen that when α = 0, the curve has a small part which exceeds the time-sharing segment AC, whereas this phenomenon does not happen for other values of α. C. Discussion So far we somewhat relaxed the problem by assuming that the condition R21 ≤ Rc2 is always satisfied. By doing so, in the weak interference analysis, we could set Rq = Rc1 to maximize the achievable rates. In fact, we can see that R21 is an increasing function in Rq , therefore increasing Rq may cause R21 to exceed Rc2 so that Rc2 is not sufficient to describe U2 to Rx1 via random binning. As a simple solution, we may choose Rq = min{Rqt , Rc1 } where Rqt is the rate that
4 A
3.5
C
3
α=0 α=0.1
R2 (bits)
2.5 α=0.2 2 1.5 1 Achievable rate region with unidirectional relay link Achievable rate regions with conferencing
0.5 0
0
0.5
1
1.5 2 R1 (bits)
2.5
B 3
3.5
Fig. 4. Overall achievable rate regions of the Gaussian Z-interference channel with SNR1 = SNR2 = 20dB, INR2 = 15dB
makes R21 = Rc2 . Discussions on this issue will be provided in our upcoming work. VI. C ONCLUSION We presented an achievable rate region for the Gaussian Z-interference channel with conferencing decoders, which is shown to outer bound the rate region in the case when there is a unidirectional relay link from the interfered receiver to the interference-free receiver. The achievable region is shown to be the capacity region when the interference is very strong. In the weak interference regime, the sum rate is found and it is shown that the asymptotic sum rate can be unbounded, depending on the SNR’s and INR’s. Our scheme can be extended to the interference channel with conferencing at decoders in different ways, for example, a simple extension would be that Rx2 also performs a hybrid decode-forward and quantize-forward strategy. The analysis of other scenarios of conferencing/cooperation is one subject in our ongoing work. R EFERENCES [1] M. Costa, “On the Gaussian Interference Channel,” IEEE Trans. Inf. Theory, vol. 31, no. 5, pp. 607–615, Sep. 1985.
[2] G. Kramer, “Review of Rate Regions for Interference Channels,” in Communications, 2006 International Zurich Seminar on, pp. 162–165. [3] N. Liu and A. Goldsmith, “Superposition Encoding and Partial Decoding Is Optimal for a Class of Z-interference Channels,” in Information Theory, 2008. ISIT 2008. IEEE International Symposium on, Toronto, ON, Jul. 6–11, 2008, pp. 564–568. [4] N. Liu, I. Mari´c, A. J. Goldsmith, and S. Shamai (Shitz), “Bounds and Capacity Results for the Cognitive Z-interference Channel,” in Information Theory, 2009. ISIT 2009. IEEE International Symposium on, Seoul, Korea, Jun. 2009, pp. 2422–2426. [5] L. Zhou and W. Yu, “Gaussian Z-Interference Channel with a Relay Link: Achievable Rate Region and Asymptotic Sum Capacity,” in Proc. International Symposium on Information Theory and Its Applications (ISITA 2008), Dec. 2008. [6] W. Yu and L. Zhou, “Gaussian Z-Interference Channel with a Relay Link: Type II Channel and Sum Capacity Bound,” in Information Theory and Applications Workshop, UCSD, CA., Feb. 2009. [7] ——, “Gaussian Z-Interference Channel with a Relay Link: Achievable Rate Region and Asymptotic Sum Capacity,” IEEE Trans. Inf. Theory, Submitted 2008. [Online]. Available: http://www.comm.utoronto.ca/ ∼weiyu/z relay.pdf [8] T. Cover and A. El Gamal, “Capacity Theorems for the Relay Channel,” IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572–584, Sep. 1979. [9] A. D. Wyner, “Shannon-theoretic Approach to a Gaussian Cellular Multiple-access Channel,” IEEE Trans. Inf. Theory, vol. 40, no. 6, pp. 1713–1727, Nov. 1994. [10] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai (Shitz), “Local Base Station Cooperation Via Finite-capacity Links for the Uplink of Linear Cellular Networks,” IEEE Trans. Inf. Theory, vol. 55, no. 1, pp. 190–204, Jan. 2009. [11] A. Carleial, “Interference Channels,” IEEE Trans. Inf. Theory, vol. 24, no. 1, pp. 60–70, Jan. 1978. [12] T. Han and K. Kobayashi, “A New Achievable Rate Region for the Interference Channel,” IEEE Trans. Inf. Theory, vol. 27, no. 1, pp. 49– 60, Jan. 1981. [13] T. Cover, “A Proof of the Data Compression Theorem of Slepian and Wolf for Ergodic Sources (corresp.),” IEEE Trans. Inf. Theory, vol. 21, no. 2, pp. 226–228, Mar. 1975. [14] T. M. Cover and M. Chiang, “Duality Between Channel Capacity and Rate Distortion With Two-sided State Information,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1629–1638, Jun. 2002. [15] S. C. Draper, B. J. Frey, and F. R. Kschischang, “Interactive Decoding of a Broadcast Message,” in Proc. Allerton Conf. Communications, Control and Computing, Monticello, IL, Oct. 2003. [16] A. Steiner, A. Sanderovich, and S. Shamai (Shitz), “The Multi-session Multi-layer Broadcast Approach for Two Cooperating Receivers,” in Information Theory, 2008. ISIT 2008. IEEE International Symposium on, Jul. 6–11, 2008, pp. 2277–2281. [17] I. Sason, “On Achievable Rate Regions for the Gaussian Interference Channel,” IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 1345–1356, Jun. 2004. [18] R. H. Etkin, D. N. C. Tse, and H. Wang, “Gaussian Interference Channel Capacity to Within One Bit,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5534–5562, Dec. 2008.
