2012 IEEE International Symposium on Information Theory Proceedings
Generalized Degrees of Freedom of the Symmetric K -user Interference Channel with Feedback ∗
Soheil Mohajer∗ , Ravi Tandon†, H. Vincent Poor† University of California at Berkeley, E-mail:
[email protected] † Princeton University, E-mail:{rtandon, poor}@princeton.edu
Abstract—The symmetric K user interference channel with fully connected topology is considered, in which (a) each receiver suffers interference from all other K − 1 transmitters, and (b) each transmitter has causal and noiseless feedback from its respective receiver. The number of generalized degrees of freedom (GDoF) is characterized in terms of α, where the interference-tonoise ratio (INR) is given by INR = SNRα . It is shown that the number of per-user GDoF of this network is the same as that of the 2-user interference channel with feedback, except for α = 1, for which existence of feedback does not help in terms of GDoF. The coding scheme proposed for this network, termed cooperative interference alignment, is based on two key ingredients, namely, interference alignment and interference decoding.
I. I NTRODUCTION Wireless networks with multiple pairs of transceivers are quite common in modern communications. The broadcast and superposition nature of the wireless medium introduces complex signal interactions between multiple competing flows. Managing such interfering signals is a long standing and fundamental problem in multiple flow wireless communication. The 2-user interference channel [1] is perhaps the simplest example of a multiple flow network. While the exact capacity regions of networks is still unknown except for very specific network topologies, much broader progress has been made in finding approximate solutions [2], [3]. A closely related notion is the generalized degrees of freedom (GDoF) which determines the asymptotic behavior of the capacity region with respect to growth of signal-to-noise ratio (SNR) and interference-to-noise ratio (INR) in various regimes. It is well known that feedback does not increase the capacity of point-to-point discrete memoryless channels. However, feedback can be beneficial in improving the capacity regions of more complex networks. The effects of feedback on the capacity region of the interference channel have been studied in several papers (e.g., [4]). Suh and Tse in [3] have provided an approximate feedback capacity region of the 2 user Gaussian interference. Perhaps, the most interesting part of the result of [3] is the multiplicative gain provided by feedback at high SNR. The gap between the capacity of the channel with and without feedback can be arbitrarily large for certain channel parameters. The key technique here is to use the feedback links to create an artificial path from each transmitter to its respective receiver through the other nodes The research was supported in part by the Air Force Office of Scientific Research MURI Grant FA-9550-09-1-0643 and in part by the National Science Foundation Grant CNS-09-05398.
978-1-4673-2579-0/12/$31.00 ©2012 IEEE
in the network. For instance, the message intended for Rx 1 , can be sent either through the direct link Tx 1 → Rx1 , or the cyclic path Tx 1 → Rx2 → Tx2 → Rx1 . In particular, the advantage of such artificial paths can be clearly understood when the cross links are much stronger than the direct links (e.g., the strong interference regime). A similar improvement would be expected in any network with such artificial cyclic paths. A natural generalization of [3] is the K-user interference channel, in which each transmit signal is corrupted by all the other signals transmitted by other transmitters. This model is appropriate for a network with densely located nodes, in which every receiver suffers interference from every transmitter. Fig. 1 shows the fully connected K-user interference channel (FC-IC) with feedback for K = 3 users. In this paper, we study the FC-IC network with feedback, and for simplicity, we consider a symmetric network topology, in which all the direct links (from each transmitter to its respective receiver) have the same gain, and similarly, the gain of all cross (interfering) links are identical. The number of symmetric generalized degrees of freedom for the K-user FC-IC (without feedback) is characterized by Jafar and Vishwanath in [5]. In this work, we study the impact of feedback on the K-user FC-IC, and determine the GDoF for this case. The main contribution of this paper is to show that feedback can arbitrarily improve the performance of the network. In particular, except for the intermediate interference regime where SNR = INR, the effect of interference from K − 1 users is as if there were only one interfering transmitter in the network. This is analogous to the result of [6], where it is shown that the number of per-user degrees of freedom of the K-user fading interference channel is the same as if there were only 2 users in the network. Beside characterizing the asymptotic behavior of the capacity, the number of GDoF also tells us whether an interference management techniques is (asymptotically) optimal for a certain regime of SNR and INR. In order to get the maximal benefit of feedback, we propose a novel coding scheme, called cooperative interference alignment, which combines two well-known interference management techniques, namely, interference alignment and interference decoding. More precisely, the encoding at the transmitters is such that all the interfering signals are aligned at each receiver. Unlike the standard interference alignment approach in which the aligned interference is usually suppressed, we need to decode aligned interference to be able to remove it from the received signal.
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d(α)
Delay
1
Rx1
Tx1
2 3 1 2
Delay Tx2
K-user/ w. FB
Rx2 1 K
Delay
1 2
Rx3
Tx3
Fig. 1. The 3-user fully connected interference channel with output feedback.
II. P ROBLEM S TATEMENT AND M AIN R ESULT Consider a network with K pairs of transmitter/receivers. Each transmitter Txk has a message Wk that it wishes to send to its respective receiver Rx k . The signal transmitted by each transmitter is corrupted by the interfering signals sent by other transmitters and Gaussian noise, and received at the receiver. This can be mathematically modelled as !√ √ Yk (t) = SNRXk (t) + INRXi (t) + Zk (t), (1) i!=k
where Xk and Yk are the signals transmitted and received by Txk and Rxk , respectively; Zk ∼ N (0, 1) is an additive white Gaussian noise. All transmission powers are constrained to 1, i.e., E[Xk2 ] ≤ 1, for k = 1, . . . , K. We assume a symmetric network, where all the cross links have the same gain (INR), and the gains of the all the direct links (SNR) are identical. There is a perfect feedback link from each receiver to its respective transmitter. Hence, at each time instance, each transmitter generates its transmission signal based on its own message as well as the output sequence observed at its receiver over the past time instances, i.e., Xk (t) = fkt (Wk , Yk (1), . . . , Yk (t − 1)) = fkt (Wk , Ykt−1 ),
where Ykt−1 = (Yk (1), Yk (2), . . . , Yk (t − 1)) indicates the channel outputs observed at Rx k up to time t − 1. A rate tuple (R1 , R2 , . . . , RK ) is called achievable if there exists a family of codebooks with corresponding rates whose average decoding error probability vanishes as the block length grows. We denote the set of all achievable rate tuples by R. In the high SNR regime, the performance of wireless networks is measured in terms of the number of degrees of freedom, that is the pre-log factor in the expression of the capacity in terms of SNR. We consider the GDoF for this network in the presence of feedback. Since the problem is parametrized in terms of two growing factors, namely SNR and INR, we use the standard parameter α (as in [2] and [5]) to capture the growth rate of INR in terms of SNR. More formally, we define α=
log INR , log SNR
K-user/ no FB
(2)
2 3
1
2
α
Fig. 2. The per-user generalized degrees of freedom for the K-user interference channel, with and without feedback.
and the per-user generalized degrees of freedom as "K max(R,...,R)∈R k=1 Rk (SNR, α) 1 lim sup d(α) = . (3) 1 K SNR→∞ 2 log SNR Our goal is to characterize the generalized degrees of freedom of the K-user interference channel with output feedback. The following theorem states the main result of this paper. Theorem 1: For the K-user FC-IC with output feedback, the number of per-user GDoF is given by 1 − α2 α < 1 (weak interference) 1 α=1 (4) dFB (α) = K α α > 1 (strong interference). 2 In order to demonstrate the benefit gained by availability of output feedback, we present the following theorem from [5], which characterizes the GDoF for the FC-IC without feedback. Theorem 2 ( [5], Theorem 3.1): The number of per-user GDoF for the K-user FC-IC without feedback is given by 1−α 0 ≤ α ≤ 12 (noisy interference) 1 2 α 2 ≤ α ≤ 3 (weak interference) α 2 1 − 2 3 < α < 1 (moderate interference) dNo FB (α) = 1 α=1 K α 1 < α ≤ 2 (strong interference) 2 1 α > 2 (very strong interference).
The generalized degrees of freedom of the K-user interference channel with/without feedback are illustrated in Fig. 2. As derived in [5], the number of GDoF for the K-user no feedback case is similar to that of 2-user case [2], except for α = 1. Similarly, here we show that in presence of feedback, the number of GDoF for the K-user case is the same as that of the 2-user channel [3], except for α = 1. At this particular point, the whole K by K network behaves as a singular network, and the available GDoF = 1 has to be shared between K users. III. A T RANSMISSION S CHEME The encoding scheme we propose for this problem bears some similarity to that of the 2-user case. It is shown in [3] that for the 2-user feedback interference channel, depending on the interference regime (value of α), it is (approximately) optimum to decode the interfering message. Due to existence of the
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feedback, decoding the interference is not only useful for its removal and consequent decoding of the desired message (akin to the strong interference regime without feedback), but also helps for decoding a part of the intended message that is conveyed through the feedback path. However, a fundamental difference here is that in the Kuser case, there are multiple interfering messages received at each receiver. Decoding of all interfering messages would dramatically decrease the maximum rate of the intended message. Our approach to deal with this is to decode the total interference received from all other users as a single message, without resorting to resolving the individual components of the interference. There are two key conditions to be fulfilled that allow us to perform such decoding, namely, (i) interfering signals should be aligned, and (ii) the summation of interfering signals should belong to a codebook of proper size which can be decoded at each receiver. Here, the first condition is satisfied since the network is symmetric. In order to satisfy the second condition, we can use a common lattice code in all transmitters, instead of random Gaussian codebooks. The closedness of lattice codes under summation implies that the total aligned interfering signal observed at each receiver is still a codeword from the same codebook. This allows us to perform decoding by searching over a single codebook, instead of the Cartesian product of all codebooks. Due to the fact that the aligned interference is decoded, we call this coding scheme cooperative interference alignment. A. Lattice Codes: Preliminaries Let Λc be a good quantization lattice [7] with σ 2 (Λc ) = 1/K 2 and G(Λc ) ≈ 1/2πe, and Λf be a good lattice for channel coding. Define Λ &c = KΛc to be the scaled version of Λc by the constant factor K. We construct a codebook C = Λf ∩ Vc , where Vc is the Voronoi cell of the lattice Λ c . We also construct the codebook C & = Λf ∩ Vc& . The following properties are fairly standard in the context of lattice coding: a) Lattice codes C and C & can be used to reliably transmit up 1 to rates R = 12 log(SNR/K) and R& = √ 2 log SNR over a Gaussian channel modelled by Y = SNRX + Z with E[Z 2 ] = 1. b) For any set of codewords c 1 , . . . , cK ∈ C, the summation c0 = c1 + · · · + cK is a codeword in C & . In the rest of this section, we study coding schemes for weak and strong interference regimes, separately. B. Weak Interference Regime α < 1 We consider three messages w k0 , wk1 , and wk2 , for transmitter Txk which will be conveyed to receiver Rx k over two blocks. All sub-messages w ki with the same index i have the same rate, which is denoted by R i , for i = 0, 1, 2. Encoding of wk1 and wk2 is performed using the usual random Gaussian codebooks with block length T and average power 1, which results in codewords c k1 and ck2 . In order to encode w k0 , we use the common lattice code C defined above. Let c k0 be the lattice codeword to which w k0 is mapped, and define c 0 = c10 + c20 + · · · + cK0 .
Once the encoding process is performed, the signal transmitted by Tx k in the first block (of length T ) is formed as 1 ' ' INR − 1 1 ck0 + ck1 . xk1 = INR INR Therefore, the signal received at Rx k can be written as ! √ √ yk1 = SNRxk1 + INR xi1 + zk1 i!=k
'
' SNR SNR (INR − 1)ck0 + ck1 = INR ! ! INR √ + INR − 1 ci0 + ci1 + zk1 . i!=k
i!=k
This received signal is sent to the transmitter Tx k over the feedback link. Having x k1 and yk1 , Txk can compute K ! √ √ √ ˜ k = yk1 − ( SNR − INR)xk1 = INR xi1 + zk1 y i=1
K K ! ! √ = INR − 1 ci0 + ci1 + zk1 . i=1
Recall that c0 =
"
i=1
˜ k if ci0 ∈ C & . So it can be decoded from y ( ) INR − 1 1 R0 ≤ log . (5) 2 K(K + 1)
In the second block, having c 0 decoded, Tx k transmits ' ' INR − 1 1 c0 + ck2 , xk2 = INR INR which together with other transmit signals results in + *' ' ! √ SNR SNR yk2 = + K −1 INR − 1c0 + ck2 + ci2 +zk2 . INR INR i!=k
Receiver Rxk first decodes c0 treating everything else as noise. This is possible as long as .√ / √ (INR − 1) SNR + (K − 1) INR 2 1 . (6) R0 ≤ log 2 K(SNR + KINR)
After decoding and removing c 0 from the received signal, Rx k can decode the Gaussian codeword c k2 , provided that ( ) SNR 1 R2 ≤ log 1 + . (7) 2 KINR " Next, the decoder uses c 0 to remove the interference i!=k c0i from yk1 in order to decode c k0 and ck1 . Rxk can compute 3 2' √ √ SNR −1 INR − 1ck0 yk1 − INR − 1c0 = INR ' ! SNR ck1 + + ci1 + zk1 , INR i!=k
1 We may assume INR ≥ 1, since we are interested in asymptotic analysis in which SNR, INR → ∞.
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from which ck0 and ck1 can be decoded provided that /2 .√ √ SNR − INR 1 (INR − 1) R0 ≤ log , 2 K(SNR + KINR) R1 ≤
( ) SNR 1 log 1 + . 2 KINR
which can be used for decoding c k . The effective noise power and signal power would be (8)
K(K − 1)γ 2 INR + 1 = (K 2 INR + 1)/(KINR + 1) < K, √ √ √ √ γ 2( SNR+(K−1) INR)2 ( INR− SNR)2 ≥γ 2 (INR−SNR)2,
(9)
respectively. Hence, c k with any rate below ( ) (INR − SNR)2 1 Rsym = log 1 + 4 K(KINR + 1)
It only remains to choose R 0 , R1 , and R2 that satisfy all constraints in (5)–(9). It is easy to verify that the choice of 6 ( ) INR − 1 1 " R0 = min log , 2 K(K + 1) 2 37 √ √ (INR − 1)( SNR − INR)2 1 log , 2 K(SNR + KINR) ( ) SNR 1 R1" = R2" = log 1 + 2 KINR
satisfies all the constraints, and so R sym = (R0" + R1" + R2" )/2 can be simultaneously achieved for all the K pairs of transmitters/receivers. Hence, we get Rsym dFB (α) ≥ lim 1 SNR→∞ log SNR 8 2 9 1 1 α = min α, α + (1 − α) = 1 − . 2 2 2 C. Strong Interference Regime (α > 1) The coding scheme for the strong interference regime is simpler than the last case. Each transmitter has a message w k , which is mapped to a codeword c k from a Gaussian codebook of rate Rsym , and sent over the first block. Upon receiving yk1 from the feedback link, Tx k removes its own signal, and resends the residual over the second block. At the end of the second block, each receiver uses its two received signals to remove the interference and decode its intended message. Formally, having y k1 from the feedback link, Tx k sends
can be reliably decoded. This rate together with the fact that α > 1 results in a lower bound on d FB (α) given by dFB (α) ≥
1 4 log INR 1 2 log SNR
=
α . 2
The encoding scheme for this case is based on sharing the network between the users, where in time block k, Tx k sends at rate Rk = 12 log(1 + SNR), while other transmitters are silent. This scheme clearly achieves d FB (1) = 1/K per user. IV. T HE G AUSSIAN N ETWORK : A N U PPER B OUND In this section we prove the converse part of Theorem 1. To this end, we derive an upper bound on the sum-rate of the network. The essence of this bound is that in the strong interference regime, given all the messages except for two of them, the output signal of any of the respective receivers is not only sufficient to decode its own message, but can be also used to decode the other missing message. Similarly, in the weak interference regime, although one receiver cannot completely decode the message of the other transmitter, it receives enough information to partially decode that message. We first define z˜it = zit − z2t for i = 3, 4, . . . , K and t = 1, . . . , T . Then, we can write T (R1 + R2 ) ≤ H(W1 ) + H(W2 )
= H(W1 |W[2:K] ) + H(W2 |W[3:K] )
T |W[2:K] ) ≤ I(W1 ; y1T y2T , z˜[3:K]
T + I(W2 ; y2T , z˜[3:K] |W[3:K] ) + 2T %T
T T = h(y1T |y2T , z˜[3:K] , W[2:K] ) − h(y1T y2T , z˜[3:K] |W[1:K] )
i=1
i!=k
SNR→∞
Rsym = lim 1 SNR→∞ 2 log SNR
D. Moderate Interference Regime (α = 1)
K ; : ! √ √ √ xk2 =γ yk1 + ( INR − SNR)xk1 =γ( INR ci + zk1 ),
√ where γ = 1/ KINR + 1 guarantees the transmit power constraint. Then, we have .√ /√ . / ! √ SNR + (K − 1) INR INR ck + ci yk2 = γ i!=k ! √ √ zi1 + zk2 . + γ SNRzk1 + γ INR
lim
T |W[3:K] ) + 2T %T , + h(y2T , z˜[3:K]
(10)
where %T vanishes as T grows, and short-hand notation W [2:K] is used to denote (W2 , W3 , . . . , Wk ). We can bound each term in (10) individually. First note that T T |y2T , z˜[3:K] , W[2:K] ) I(y1T ; y[3:K]
" Applying zero-forcing at Rx k to remove i!=k ci , we obtain .√ / √ ˜ k = yk2 − γ SNR + (K − 1) INR yk1 y .√ / √ √ √ =γ SNR + (K − 1) INR ( INR − SNR)ck ! √ √ − γ(K − 1) INRzk1 + γ INR zi1 + zk2 i!=k
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= (a)
=
T !
t−1 T I(y1T ; y[3:K]t |y2T , z˜[3:K] , W[2:K] , y[3:K] )
t=1 T ! t=1
=
T ! t=1
(11)
t−1 T I(y1T ; y[3:K]t |y2T , z˜[3:K] , W[2:K] , y[3:K] , x[2:K]t )
t−1 T I(y1T ; y[3:K]t |y2T , z˜[3:K] , W[2:K] , y[3:K] , x[2:K]t ) = 0
where (a) holds since xjt = fjt (Wj , yjt−1 ). The last equality is due to the fact that for j ≥ 3, we have √ √ yjt − y2t = ( SNR − INR)(xjt − x2t ) + z˜jt , which implies that yjt can be deterministically recovered from (y2t , x2t , xjt , z˜jt ), and hence, each term in the summation is zero. From (11) we can bound the first term in (10) as T T T h(y1T |y2T , z˜[3:K] , W[2:K] ) = h(y1T |y[2:K] , z˜[3:K] , W[2:K] )
=
as
> 1 1 α α? max{1, α} + (1 − α)+ = max 1 − , . 2 2 2 2 For the √ moderate "Kregime (α = 1), we have SNR = INR, and Yk = SNR i=1 Xi + Zk , i.e., all the received signals are the same (except for their additive noise part). In this case, the sum rate can be upper bounded as follows: dFB (α) ≤
T
T T h(y1T |y[2:K] , z˜[3:K] , W[2:K] , xT[2:K] )
! √ √ √ ≤ h( SNRxT1 + z1T |y2T − SNRxT2 − INR xTj ) j>2 √ √ = h( SNRxT1 + z1T | INRxT1 + z2T ) ( ) SNR T T (12) ≤ log 1 + + log(2πe). 2 1 + INR 2 In order to bound the second term in (10) we can write / . T T h y[1:2] , z˜[3:K] |W[1:K] t=1
= (b)
=
t=1
=
T ! t=1
TK log(2πe), 2
t=1
T : /; . ! < = t−1 h y1t , z˜[2:K]t − h y[1:K]t |W[1:K] , y[1:K] = , x[1:K]t t=1 * T ! t=1
+ K ! = < h (y1t ) + h z˜kt ) − h(z[1:K]t k=2
lim
1 2K
SNR→∞
log(1 + KSNR) 1 = . 1 K 2 log SNR
This completes the proof of the theorem. V. C ONCLUSION
= < h z[1:2]t , z˜[3:K]t
h(z[1:K]t ) =
T : /; . ! < = t−1 h y[1:K]t − h y[1:K]t |W[1:K] , y[1:K]
dFB (1) ≤
/ . t−1 t−1 h z[1:2]t , z˜[3:K]t |y[1:2] , z˜[3:K] , W[1:K] , x[1:K]t
t=1 T !
≤
. / T (Rk − %T ) ≤ I y[1:K] ; W[1:K]
T (K − 1)T ≤ log (1 + KSNR) + 2 2 which implies
T / . ! t−1 t−1 ≥ h y[1:2]t , z˜[3:K]t |y[1:2] , z˜[3:K] , W[1:K] , x[1:K]t t=1 T !
k=1
≤
T / . ! t−1 t−1 h y[1:2]t , z˜[3:K]t |y[1:2] , z˜[3:K] , W[1:K]
=
K !
(13)
where (b) is due to the facts that the channels are memoryless and the noise at time t is independent of all the signals and noises in the past.
We have characterized the GDoF for the symmetric K-user fully-connected interference channel with output feedback, by introducing a coding scheme based on alignment and decoding of the interference. This result shows that, except for the moderate regime (SNR = INR), the system performance is as if there is only one source of interference in the network. ACKNOWLEDGEMENT The authors wish to thank Abolfazl S. Motahari for helpful discussions.
Finally, the third term in (10) can be bounded as follows:
R EFERENCES
T T h(y2T ,˜ z[3:K] |W[3:K] ) ≤ h(y2T ) + h(˜ z3T ) + · · · + h(˜ zK )
T (K − 2) log(4πe) ≤ T h(y2 ) + 2 /; : . √ √ T ≤ log (2πe) 1 + ( SNR + (K − 1) INR)2 2 T (K − 2) log(4πe) (14) + 2 where in (c) we used the fact that E[˜ z i2 ] = 2. Substituting (12), (13), and (14) in (10), we get . / √ √ 1 R1 + R2 ≤ log 1 + ( SNR + (K − 1) INR)2 2 ( ) SNR 1 K −1 . + log 1 + + 2 1 + INR 2 (c)
A similar bound can be derived for each rate pair of form "KR k + Rk+1 . Summing up all such bounds, we can bound i=1 Ri
[1] T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Trans. Information Theory, vol. 27, pp. 49– 60, Jan. 1981. [2] R. H. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Trans. Information Theory, vol. 54, no. 12, pp. 5534–5562, Dec. 2008. [3] C. Suh and D. Tse, “Feedback capacity of the Gaussian interference channel to within 2 bits,” IEEE Trans. Information Theory, vol. 57, no. 5, pp. 2667–2685, May 2011. [4] G. Kramer, “Feedback strategies for white Gaussian interference networks,” IEEE Trans. Information Theory, vol. 48, no. 6, pp. 1423–1438, 2002. [5] S. Jafar and S. Vishwanath, “Generalized degrees of freedom of the symmetric Gaussian K-user interference channel,” IEEE Trans. Information Theory, vol. 56, no. 7, pp. 3297–3303, July 2010. [6] V. Cadambe and S. Jafar, “Interference alignment and degrees of freedom of the K-user interference channel,” IEEE Trans. Information Theory, vol. 54, no. 8, pp. 3425–3441, August 2008, . [7] R. Zamir, S. Shamai (Shitz), and U. Erez, “Nested linear/lattice codes for structured multiterminal binning,” IEEE Trans. Information Theory, vol. 48, no. 6, pp. 1250–1276, June 2002.
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