1
A New Polar Coding Scheme for the Interference Channel Mengfan Zheng, Cong Ling, Wen Chen and Meixia Tao
arXiv:1608.08742v1 [cs.IT] 31 Aug 2016
Abstract We consider the problem of polar coding for the 2-user discrete memoryless interference channel (DM-IC) to achieve the Han-Kobayashi region, the best achievable rate region to date. Existing schemes are designed according to the original description of the Han-Kobayashi region, in which two deterministic functions that map auxiliary random variables to channel inputs are assumed to be known. Inspired by Chong et al.’s compact description of the Han-Kobayashi region, which uses joint distributions of random variables instead of deterministic functions, we find that the Han-Kobayashi region can be achieved in a simpler way. The original Han-Kobayashi strategy requires joint decoding of the intended sender’s private message and both senders’ common messages at each receiver. In this paper, we show that all corner points of the Han-Kobayashi region can be achieved using partially joint decoders, i.e., each receiver only needs to jointly decode two of the three messages, either both senders’ common messages, or the unintended sender’s common message and the intended sender’s private message. Based on this finding and enlightened by Goela et al.’s superposition polar coding schemes for the broadcast channels, we design two types of polar coding schemes, which can be seen as extensions of Carleial’s superposition coding scheme for the DM-IC with more powerful joint decoders, and show that every corner point can be achieved using one of the proposed schemes. The joint decoders and the corresponding code constructions are implemented using the 2-user multiple access channel (MAC) polarization technique based on Arıkan’s monotone chain rule expansions, which has similar complexity to the single-user case. The chaining method which has already been widely used for constructing universal polar codes is adopted to solve the problem of alignment of polar indices. Index Terms Polar codes, interference channel, Han-Kobayashi region, superposition coding, universal polar coding.
I. I NTRODUCTION Polar codes, proposed by Arıkan [1], are the first class of channel codes that can provably achieve the capacity of any memoryless binary-input output-symmetric channels with low encoding and decoding complexity. Since its invention, polar codes have been widely adopted to many other scenarios, such as source compression [2]–[5], wiretap channels [6]–[11], relay channels [6], [12], [13], multiple access channels (MAC) [5], [14]–[17], broadcast channels [18], [19], broadcast channels with confidential messages [10], [20], and bidirectional broadcast channels with common and confidential messages [21]. In these scenarios, polar codes have also shown capacity-achieving capabilities. The interference channel (IC), first initiated by Shannon [22] and further studied by Ahlswede [23], models the situation where m sender-receiver pairs try to communicate simultaneous through a common channel. In this model, it is assumed that there is no cooperation between any of the senders or receivers, and the signal of each sender is seen as interference by the unintended receivers. Although the 2-user discrete memoryless IC (DM-IC) is rather simple in appearance, except for some special cases [24]–[31], determining the capacity region of a general IC remains an open problem. Reference [23] gave simple but fundamental inner and outer bounds on the capacity region of the IC. In [32], Carleial determined an improved achievable rate region for the IC by applying the superposition coding technique of Cover [33], which was originally designed for the broadcast channel. Later, Han and Kobayashi established the best M. Zheng, W. Chen and M. Tao are with the Department of Electronic Engineering at Shanghai Jiao Tong University, Shanghai, China. Emails: {zhengmengfan, wenchen, mxtao}@sjtu.edu.cn. C. Ling is with the Department of Electrical and Electronic Engineering at Imperial College London, United Kingdom. Email:
[email protected]. The corresponding author is M. Tao.
2
achievable rate region for the general IC to date [34]. A more compact description of the Han-Kobayashi region was given in [35]. The idea of the Han-Kobayashi coding strategy is to split each sender’s message into a private part and a common part, and allow the unintended receiver to decode the common part so as to enhance the total transmission rates. To achieve the whole Han-Kobayashi region, it is required that each receiver decodes its intended private message and both senders’ common messages jointly. There are limited studies on the design of specific coding schemes that can achieve the Han-Kobayashi rate region. A low-density parity-check (LDPC) code-based Han-Kobayashi scheme was proposed for the Gaussian IC in [36], which has close-to-capacity performance in the case of strong interference. In [37], a specific coding scheme was designed for the binary-input binary-output Z IC using LDPC codes, and an example was shown to outperform time sharing of single user codes. For polar codes, reference [38] pointed out how alignment of polarized bit-channels can be of use for designing coding schemes for interference networks, and presented an example of the one-sided discrete memoryless 3-user IC with a degraded receiver structure. A polar coding scheme that achieves the Han-Kobayashi inner bound for the 2-user IC was proposed in [39], and [40] used a similar scheme to achieve the Han-Kobayashi region in the 2-user classical-quantum IC. The idea of [39] is to transform the original IC into two 3-user MACs from the two receivers’ perspectives, and design a compound MAC polar coding scheme for them. The achievable rate region of the compound MAC equals the Han-Kobayashi region, and can be achieved by polar codes. This design is based on the original description of the Han-Kobayashi region in [34] (also shown in Theorem 1 in this paper), in which two deterministic functions that map auxiliary random variables to the actual channel inputs are assumed to be known. By ranging over all possible choices of these functions and auxiliary random variables, the whole Han-Kobayashi region can be achieved. However, such an approach can be problematic in practice since one needs to know the functions before designing a code, which can be a very complex task. Our work is inspired by the compact description of the Han-Kobayashi region in [35] (also shown in Theorem 2 in this paper), in which no deterministic functions are required, and only three auxiliary random variables are needed (as a comparison, the original description in [34] needs five). The Han-Kobayashi coding strategy requires a fully joint decoder at each receiver that jointly decodes all its three messages, which is not directly implementable from the compact description since there are no random variables standing for private messages in it. By analyzing the corner points of the Han-Kobayashi region under different conditions, we find that it is possible to achieve all of them using a partially joint decoder. Each receiver can either jointly decode both senders’ common messages first and then the intended sender’s private message, or solely decode the intended sender’s common message first and then jointly decode the rest two. Based on this finding and enlightened by Goela et al.’s superposition polar coding schemes for the broadcast channels [18], we design two types of superposition coding schemes with joint decoders, which can be seen as extensions of Carleial’s in [32], and show that all possible corner points of the Han-Kobayashi region can be achieved using one of the proposed schemes. The whole region can then be achieved by time sharing. The deterministic functions used in [39] can be seen as implicit in our code design, which makes our scheme a simpler approach. In our proposed schemes, joint decoders and the corresponding code constructions are implemented using the 2-user MAC polarization technique based on Arıkan’s monotone chain rule expansions [5], whose complexities in encoding, decoding and code construction are similar to the single-user case. Our construction is simpler than that in [39], which is based on 3-user MAC polarization, because the 2-user MAC polar codes can be constructed using simple permutations (see Proposition 3 in this paper). To deal with non-uniform input distributions, Reference [39] took Gallager’s method by mapping uniformly distributed auxiliary symbols to the actual ones [41, p. 208], the complexity of which will increase drastically when the optimal distribution cannot be approximated by simple rational numbers. In our work, we adopt the approach of [42] by invoking results on polar coding for lossless compression to extend the monotone chain rule expansion method to the non-uniform input case. In this way, one can achieve the non-uniform rate region of a 2-user MAC without any alphabet extension. One crucial point in designing capacity-achieving polar codes for a general multi-user channel is how to properly align
3
the polar indices. One solution for this problem is the chaining method, which has already been used in several areas [9]–[11], [19], [43]. Another way is to add additional stages of polarization to align the incompatible indices, as shown in [44] and used in [39]. In this paper, we adopt the chaining method because it does not change the original polar transformation and thus may be easier to understand. If using the other method, our scheme will still work. The rest of this paper is organized as follows. In Section II, we introduce the 2-user IC model and the Han-Kobayashi region, and analyze the corner points of the region under different conditions. In Section III, we review some background on polarization and polar codes necessary for our code design. In Section IV, we provide an overview of our scheme and analyze its feasibility. In Section V, we present details of our scheme. Section VI concludes this paper with some discussions. Notations: Random variables are denoted by capital letters X, Y , U , V , ... with values x, y, u, v, ... respectively. [N ] is the abbreviation of an index set {1, 2, ..., N }. Vectors are denoted as XN , {X 1 , X 2 , ..., X N } or X a:b , {X a , X a+1 , ..., X b } for a ≤ b, where X k (k ∈ [N ] or k ∈ {a, ..., b}) denotes an element of the vector. GN = BN F⊗n is the generator matrix of polar codes, where N = 2n is the code length with nbeing an arbitrary integer, BN being a permutation matrix known as bit-reversal 1 0 matrix, and F = . 1 1 II. P ROBLEM S TATEMENT A. Channel Model Definition 1: A 2-user DM-IC consists of two input alphabets X1 and X2 , two output alphabets Y1 and Y2 , and a probability transition function PY1 Y2 |X1 X2 (y1 , y2 |x1 , x2 ). The conditional joint probability distribution of the 2-user DM-IC over N channel uses can be factored as N Y N , x ) = PY1 Y2 |X1 X2 (y1i , y2i |xi1 , xi2 ). PY1N Y2N |XN1 XN2 (y1N , y2N |xN (1) 1 2 i=1 N R1
N R2
Definition 2: A (2 ,2 , N ) code for the 2-user DM-IC consists of two message sets M1 = N R1 {1, 2, ..., [2 ]} and M2 = {1, 2, ..., [2N R2 ]}, two encoding functions N N N xN 1 (m1 ) : M1 7→ X1 and x2 (m2 ) : M2 7→ X2 ,
(2)
m ˆ 1 (y1N ) : Y1N 7→ M1 and m ˆ 2 (y2N ) : Y2N 7→ M2 .
(3)
and two decoding functions (N )
Definition 3: The average probability of error Pe of a (2N R1 , 2N R2 , N ) code for the 2-user DM-IC is defined as the probability that the decoded message pair is not the same as the transmitted one averaged over all possible message pairs, o n X 1 (4) Pr m ˆ 1 (Y1N ), m ˆ 2 (Y2N ) 6= (M1 , M2 )|(M1 , M2 ) sent , Pe(N ) = N (R1 +R2 ) × 2 (M1 ,M2 )∈M1 ×M2
where (M1 , M2 ) are assumed to be uniformly distributed over M1 × M2 . B. The Han-Kobayashi Rate Region In the Han-Kobayashi coding strategy, each sender’s message is split into two parts, a private message, which only needs to be decoded by the intended receiver, and a common message, which is allowed to be decoded by the unintended receiver. Each receiver decodes its intended private message and two common messages jointly so that a higher transmission rate can be achieved. In the rest of this paper, we will refer to the two senders and two receivers as Sender 1, Sender 2, Receiver 1 and Receiver 2 respectively. Sender 1’s message, denoted as M1 , is split into (M1p , M1c ), where M1p ∈ M1p , {1, 2, ..., [2N S1 ]} denotes its
4
private message and M1c ∈ M1c , {1, 2, ..., [2N T1 ]} the common message. Similarly, Sender 2’s message, M2 , is split into (M2p , M2c ) with M2p ∈ M2p , {1, 2, ..., [2N S2 ]} and M2c ∈ M2c , {1, 2, ..., [2N T2 ]}. Define the sets W1 ∈ W1 and W2 ∈ W1 as the random variables that carry common messages M1c and M2c respectively, and V1 ∈ V1 and V2 ∈ V1 as the random variables that carry private messages M1p and M2p respectively. Then each encoding function is decomposed into three functions. For xN 1 (m1 ), the three functions are defined as: w1N (M1c ) : M1c 7→ W1N , v1N (M1p ) : M1p 7→ V1N 0
and x1N (W1N , V1N ) : W1N × V1N 7→ X1N .
(5)
Similarly, for xN 2 (m2 ), the three functions are defined as: w2N (M2c ) : M2c 7→ W2N , v2N (M2p ) : M2p 7→ V2N 0
and x2N (W2N , V2N ) : W2N × V2N 7→ X2N .
(6)
The coding strategy described in (5) or (6) is also known as superposition coding, which is proposed by Cover for the broadcast channel [33], and first used in the interference channel by Carleial [32]. Han and Kobayashi proposed the best achievable rate region for the general interference channel to date in [34]. The result is summarized in Theorem 1. Theorem 1 ( [34]): Let P ∗ be the set of probability distributions P ∗ (·) that factor as P ∗ (q, v1 , v2 , w1 , w2 , x1 , x2 ) = PQ (q)PV1 |Q (v1 |q)PV2 |Q (v2 |q)PW1 |Q (w1 |q)PW2 |Q (w2 |q) × PX1 |V1 W1 Q (x1 |v1 , w1 , q)PX2 |V2 W2 Q (x2 |v2 , w2 , q),
(7)
where Q ∈ Q is the time-sharing parameter, and PX1 |V1 W1 Q (·) and PX2 |V2 W2 Q (·) equal either 0 or 1, i.e., they are deterministic functions. For a fix P ∗ (·) ∈ P ∗ , Consider Receiver 1 and the set of non-negative ∗ rate-tuples (S1 , T1 , S2 , T2 ) denoted by Ro,1 HK (P ) that satisfy S1 T1 T2 S1 + T1 S1 + T2 T1 + T2 S1 + T1 + T2 S1 , T1 , T2
≤ I(V1 ; Y1 |W1 W2 Q) = I(X1 ; Y1 |W1 W2 Q) ≤ I(W1 ; Y1 |V1 W2 Q) ≤ I(W2 ; Y1 |V1 W1 Q) = I(W2 ; Y1 |X1 Q) ≤ I(V1 W1 ; Y1 |W2 Q) = I(X1 ; Y1 |W2 Q) ≤ I(V1 W2 ; Y1 |W1 Q) = I(X1 W2 ; Y1 |W1 Q) ≤ I(W1 W2 ; Y1 |V1 Q) ≤ I(V1 W1 W2 ; Y1 |Q) = I(X1 W2 ; Y1 |Q) ≥ 0.
(8) (9) (10) (11) (12) (13) (14) (15)
∗ Similarly, let Ro,2 HK (P ) be the set of non-negative rate-tuples (S1 , T1 , S2 , T2 ) that satisfy (8)–(15) with indices 1 and 2 swapped everywhere. For a set S of 4-tuples (S1 , T1 , S2 , T2 ), let R(S) be the set of (R1 , R2 ) such that 0 ≤ R1 ≤ S1 + T1 and 0 ≤ R2 ≤ S2 + T2 for some (S1 , T1 , S2 , T2 ) ∈ S. Then we have that [ o,2 o,1 ∗ o ∗ RHK (P ) ∩ RHK (P ) (16) RHK = R P ∗ ∈P ∗
is an achievable rate region for the DM-IC. The equations in (8) and (12) are proved in [35], and those in (10), (11) and (14) are direct because (V1 , W1 ) → X1 is a deterministic mapping. The original description for the Han-Kobayashi region in Theorem 1 needs five auxiliary random variables and two deterministic functions to describe. This poses difficulties in IC coding design in practice, since finding such functions and distributions of auxiliary randoms variables that can achieve a required rate pair may be very complex. Reference [35] presented a simplified description for the region, in which
5
only three auxiliary random variables are used and no deterministic functions are needed. Their result is summarized in Theorem 2. Theorem 2 ( [35]): Let P1∗ be the set of probability distributions P1∗ (·) that factor as P1∗ (q, w1 , w2 , x1 , x2 ) = PQ (q)PX1 W1 |Q (x1 , w1 |q)PX2 W2 |Q (x2 , w2 |q).
(17)
For a fix P1∗ (·) ∈ P1∗ , let RHK (P ∗ ) be the set of (R1 , R2 ) satisfying R1 ≤ I(X1 ; Y1 |W2 Q) , a
(18)
R2 ≤ I(X2 ; Y2 |W1 Q) , b
(19)
R1 + R2 ≤ I(X1 W2 ; Y1 |Q) + I(X2 ; Y2 |W1 W2 Q) , c
(20)
R1 + R2 ≤ I(X1 ; Y1 |W1 W2 Q) + I(X2 W1 ; Y2 |Q) , d
(21)
R1 + R2 ≤ I(X1 W2 ; Y1 |W1 Q) + I(X2 W1 ; Y2 |W2 Q) , e
(22)
2R1 + R2 ≤ I(X1 W2 ; Y1 |Q) + I(X1 ; Y1 |W1 W2 Q) + I(X2 W1 ; Y2 |W2 Q) , f
(23)
R1 + 2R2 ≤ I(X2 ; Y2 |W1 W2 Q) + I(X2 W1 ; Y2 |Q) + I(X1 W2 ; Y1 |W1 Q) , g R1 , R2 ≥ 0.
(24) (25)
Then we have that RHK =
[
RHK (P1∗ )
(26)
P1∗ ∈P1∗
is an achievable rate region for the DM-IC. It is proved in [35] that the regions described in Theorem 1 and 2 are equivalent if we impose the following constraints on the cardinalities of auxiliary sets |Wj | ≤ |Xj | + 4 for j = 1, 2 and |Q| ≤ 7, where Wj , Xj and Q respectively are the finite sets on which Wj , Xj and Q are defined. It is also noted that constraints (9), (10) and (13), and their counterparts for the second receiver, are unnecessary. The deterministic functions used in the original description can be seen as contained in the joint distribution of PX1 W1 |Q (x1 , w1 |q) and PX2 W2 |Q (x2 , w2 |q). In this paper, we design our polar coding schemes based on the compact description. C. Corner Points of the Han-Kobayashi Region The purpose of introducing the time-sharing parameter Q in Theorem 1 and 2 is to replace the convexhull operation. In the rest of this paper, we will consider a fixed Q = q, and we will drop the condition Q = q in expressions for simplicity, including a, b, ..., g in (18)–(24). If for any q ∈ Q, the corresponding Han-Kobayashi region is achievable, then the whole region defined in Theorem 2 can be achieved by time sharing. To determine the corner points of the Han-Kobayashi region, we first need to determine the constraint on the sum-rate, i.e., the minimum one among c, d and e in (20), (21) and (22). For this purpose, we have the following result. Let I1 , I(W1 ; Y2 |W2 ) − I(W1 ; Y1 ),
(27)
I2 , I(W2 ; Y1 |W1 ) − I(W2 ; Y2 ).
(28)
Proposition 1: In the Han-Kobayashi region for the interference channel, (1) if I1 ≥ 0 and I1 ≥ I2 , the upper bound for the sum rate is R1 + R2 ≤ c; (2) if I2 ≥ 0 and I2 ≥ I1 , the upper bound for the sum rate is R1 + R2 ≤ d; (3) if I1 < 0 and I2 < 0, the upper bound for the sum rate is R1 + R2 ≤ e. In the rest of this paper, we will refer to these three cases as Case 1, Case 2 and Case 3 respectively. Proof: See Appendix A.
6
Fig. 1.
Corner points of the Han-Kobayashi region.
Next, we can determine the corner points of the Han-Kobayashi region in each case. The result is shown as follows. Denote P1 , I(X1 ; Y1 |W2 ), I(X2 ; Y2 |W1 W2 ) + I(W2 ; Y1 ) (29) P2 , I(X1 W2 ; Y1 ) − I(W2 ; Y2 |W1 ), I(X2 ; Y2 |W1 ) (30) P3 , I(X1 W2 ; Y1 |W1 ) + I(W1 ; Y2 ) − I(W2 ; Y2 |W1 ), I(X2 ; Y2 |W1 ) (31) P4 , I(X1 ; Y1 |W2 ), I(X2 W1 ; Y2 |W2 ) + I(W2 ; Y1 ) − I(W1 ; Y1 |W2 ) (32) P5 , I(X1 W2 ; Y1 ) + I(W1 ; Y1 ) − I(W1 W2 ; Y2 ), I(X2 W1 ; Y2 ) − I(W1 ; Y1 ) P6 , P7 , P8 , P9 , P10 ,
I(X1 ; Y1 |W1 W2 ) + I(W1 ; Y2 |W2 ), I(W1 W2 ; Y1 ) + I(X2 ; Y2 |W1 W2 ) − I(W1 ; Y2 |W2 ) I(X1 ; Y1 |W2 ), I(X2 W1 ; Y2 |W2 ) + I(W2 ; Y1 ) − I(W1 ; Y1 |W2 ) I(X1 W2 ; Y1 |W1 ) + I(W1 ; Y2 ) − I(W2 ; Y2 |W1 ), I(X2 ; Y2 |W1 ) I(X1 ; Y1 |W1 W2 ) + I(W1 ; Y1 ), I(X2 W1 ; Y2 |W2 ) + I(W2 ; Y1 |W1 ) − I(W1 ; Y1 ) I(X1 W2 ; Y1 |W1 ) + I(W1 ; Y2 |W2 ) − I(W2 ; Y2 ), I(X2 ; Y2 |W1 W2 ) + I(W2 ; Y2 ) .
(33) (34) (35) (36) (37) (38)
Proposition 2: In order to achieve the Han-Kobayashi rate region, it is equivalent to achieve the following corner points. If they can all be achieved, other points in the region can then be achieved by time sharing. (1) In Case 1, if I(W2 ; Y1 ) ≥ I(W2 ; Y2 |W1 ), the Han-Kobayashi region reduces to a rectangle formed by lines R1 = 0, R2 = 0, R1 = a and R2 = b, and the corner point is (a, b). Otherwise, there are four sub-cases. • (Case 1-1) If I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) ≥ 0 and I(W1 ; Y2 ) − I(W1 ; Y1 ) ≥ 0, the corner points are P1 , P2 . • (Case 1-2) If I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) ≥ 0 and I(W1 ; Y2 ) − I(W1 ; Y1 ) < 0, the corner points are P1 , P3 , P5 .
7
(Case 1-3) If I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) < 0 and I(W1 ; Y2 ) − I(W1 ; Y1 ) ≥ 0, the corner points are P2 , P4 , P6 . • (Case 1-4) If I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) < 0 and I(W1 ; Y2 ) − I(W1 ; Y1 ) < 0, the corner points are P3 , P4 , P5 , P6 . (2) Case 2 is symmetric to Case 1 by swapping the indices 1 and 2 everywhere in the expressions. (3) In Case 3, the corner points are P7 , P8 , P9 , P10 . Proof: See Appendix B. The shapes of the Han-Kobayashi region in Case 1 and Case 3 are shown in Fig. 1. •
III. P OLAR C ODING P RELIMINARIES In this section, we briefly review some related knowledge on polarization and polar codes. Let (X, Y ) be a pair of random variables with joint distribution PXY , where X ∈ {0, 1} and Y ∈ Y with Y being an arbitrary discrete alphabet. The Bhattacharyya parameter is defined as Definition 4 (Bhattacharyya parameter): q X (39) Z(X|Y ) = 2 PY (y) PX|Y (0|y)PX|Y (1|y). y∈Y
A. Polarization of Binary Memoryless Sources First, let us consider lossless compression of a binary memoryless source. Let X 1:N be N independent copies of a random variable X ∈ X with X ∼ PX , and U 1:N = X 1:N GN . It is shown in [2] that the components of U 1:N polarize in the sense that U j (j ∈ [N ]) are either almost independent of U 1:j−1 and uniformly distributed, or almost determined by U 1:j−1 . Therefore, we can define the following two sets of indices: (N )
HX = {j ∈ [N ] : Z(U j |U 1:j−1 ) ≥ 1 − δN },
(40)
(N ) LX
(41)
j
= {j ∈ [N ] : Z(U |U
1:j−1
) ≤ δN },
−N β
and β ∈ (0, 1/2). It is shown that 1 (N ) lim |HX | = H(X), N →∞ N (42) 1 (N ) |LX | = 1 − H(X). lim N →∞ N Next, consider compression of two correlated sources. Let (X, Y ) ∼ pX,Y be a pair of random variables over (X × Y), where X = {0, 1} and Y can be arbitrary countable set. Consider X as the memoryless source to be compressed and Y as side information of X. This is the case of source compression with β additional side information. Let U 1:N = X 1:N GN . For δN = 2−N with β ∈ (0, 1/2), define the following sets of indices: where δN = 2
(N )
(43)
(N )
(44)
HX|Y = {j ∈ [N ] : Z(U j |Y 1:N , U 1:j−1 ) ≥ 1 − δN }, LX|Y = {j ∈ [N ] : Z(U j |Y 1:N , U 1:j−1 ) ≤ δN }. Also, we have
1 (N ) |HX|Y | = H(X|Y ), N →∞ N 1 (N ) lim |LX|Y | = 1 − H(X|Y ). N →∞ N Furthermore, we have the following inclusion relations, lim
(N )
(N )
(N )
(N )
HX|Y ⊆ HX , LX|Y ⊇ LX .
(45)
(46)
8
B. Polar Coding for Asymmetric Channels Let W (Y |X) be a binary-input discrete memoryless channel (B-DMC). The output alphabet Y can be (N ) (N ) arbitrary. Let U 1:N = X 1:N GN and define the sets HX and LX as in (40) and (41). Consider Y as a (N ) (N ) side information about X as discussed in the previous subsection and define the sets HX|Y and LX|Y as in (43) and (44). Then we can construct a polar code for W as follows [42]. Let I , HX ∩ LX|Y ,
(47)
Fr , HX ∩ LcX|Y ,
(48)
Fd ,
c HX .
(49)
The encoding procedure goes as follows. {uj }j∈I carry information, {uj }j∈Fr are filled with frozen bits (uniformly distributed random bits shared between the sender and the receiver), and the value for {uj }j∈Fd will be calculated as uj = arg max PU j |U 1:j−1 (u|u1:j−1 ). u={0,1}
After receiving y 1:N , the receiver computes the estimate u¯1:N of u1:N as j if j ∈ Fr u , j u¯ = arg maxu∈{0,1} PU j |U 1:j−1 (u|u1:j−1 ), if j ∈ Fd arg max 1:N 1:j−1 ,u ), if j ∈ I u∈{0,1} PU j |Y 1:N U 1:j−1 (u|y
(50)
The rate of such a scheme is R = |I|/N , which satisfies lim R = I(X; Y ).
(51)
N →∞
The block error probability of such a polar code can be upper bounded by X β Pe ≤ Z(U j |Y 1:N , U 1:j−1 ) = O(2−N ).
(52)
j∈I
C. Polar Coding for Multiple Access Channels Let PY |X1 X2 (y|x1 , x2 ) be the transition probability of a binary-input memoryless 2-user MAC. For a fixed product distribution of PX1 (x1 )PX2 (x2 ), the achievable rate region of PY |X1 X2 is given by [45] 0 ≤ R1 ≤ I(X1 ; Y |X2 ) R1 0 ≤ R2 ≤ I(X2 ; Y |X1 ) R(PY |X1 X2 ) , . (53) R2 R + R ≤ I(X , X ; Y ) 1
2
1
2
Polar coding for MACs has been studied in [5], [14]–[17]. Although [17] provides a more general scheme that can achieve the whole uniform rate region of a k-user MAC, in our scheme, we adopt the monotone chain rule expansion method in [5] because it has simple structure and possesses similar complexity to the single-user polar codes. Reference [5] mainly deals with the Slepian-Wolf problem in source coding, but the method can be readily applied to the problem of coding for the 2-user MAC since they are dual problems, which is studied in [14] and used in [39]. However, both [14] and [39] consider uniform channel inputs. Here we generalize it to the non-uniform case. Define U11:N = X11:N GN , U21:N = X21:N GN .
(54)
Let S 1:2N = (S 1 , ..., S 2N ) be a permutation of U11:N U21:N such that it preserves the relative order of the elements of both U11:N and U21:N , which is called a monotone chain rule expansion. Such an expansion can be represented by a string b2N = b1 b2 ...b2N , called the path of the expansion, where bj = 0 (j ∈ [2N ])
9
represents that S j ∈ {U11 , ..., U1N }, and bj = 1 represents that S j ∈ {U21 , ..., U2N }. Then we can expand the mutual information between the receiver and the two users in the following form I(Y 1:N ; U11:N , U21:N ) = H(U11:N , U21:N ) − H(U11:N , U21:N |Y 1:N ) 2N X 1:N 1:N = H(U1 ) + H(U2 ) − H(S j |Y 1:N , S 1:j−1 ) j=1
= N H(X1 ) + N H(X2 ) −
2N X
H(S j |Y 1:N , S 1:j−1 )
j=1
It can be shown by duality from [5] that H(S j |Y 1:N , S 1:j−1 ) (j ∈ [2N ]) polarizes to 0 or 1 as N goes to infinity. Define the rates for user 1 and user 2 as 1 X RU1 = H(X1 ) − H(S j |Y 1:N , S 1:j−1 ) (55) N j:b =0 j
and RU2 = H(X2 ) −
1 X H(S j |Y 1:N , S 1:j−1 ) N j:b =1
(56)
j
respectively. The following proposition shows that the rate pair (RU1 , RU2 ) lies on the dominant face of the achievable rate region of the MAC as N goes to infinity, and given sufficiently large N , arbitrary points on the dominant face of R(PY |X1 X2 ) can be achieved with this method. Although the permutations can have lots of varieties, even non-monotone [17], expansions of type 0i 1N 0N −i (0 ≤ i ≤ N ) are sufficient to achieve every point on the dominant face given large enough N . This property can make code design and construction simpler. Proposition 3 ( [5]): Let (R1 , R2 ) be a given rate pair on the dominant face of the achievable rate region of a 2-user MAC. For any given > 0, there exists N and a chain rule b2N on U11:N U21:N such that b2N is of the form 0i 1N 0N −i (0 ≤ i ≤ N ) and has a rate pair (RU1 , RU2 ) satisfying |R1 − RU1 | ≤ and |R2 − RU2 | ≤ .
(57)
To polarize a MAC sufficiently while keeping the above rate approximation intact, we need to scale the path. For any integer l = 2m , let lb2N denote · · b}2 · · · · · · |b2N ·{z · · b2N}, b|1 ·{z · · b}1 |b2 ·{z l
l
l
which is a monotone chain rule for U11:lN U21:lN . It is shown in [5] that the rate pair for b2N is also the rate pair for lb2N . β Now we can construct a polar code for the 2-user MAC with arbitrary input distributions. For δN = 2−N with β ∈ (0, 1/2), define the following polarized sets of indices: (N )
HSU = {j ∈ [2N ], bj = 0 : Z(S j |S 1:j−1 ) ≥ 1 − δN }, 1
(N ) LSU |Y 1
= {j ∈ [2N ], bj = 0 : Z(S j |Y 1:N , S 1:j−1 ) ≤ δN },
(N )
HSU = {j ∈ [2N ], bj = 1 : Z(S j |S 1:j−1 ) ≥ 1 − δN }, 2
(N ) LSU |Y 2
= {j ∈ [2N ], bj = 1 : Z(S j |Y 1:N , S 1:j−1 ) ≤ δN },
(58)
10
which satisfy 1 (N ) |HSU | = H(X1 ), 1 N →∞ N 1 (N ) 1 X lim |LSU |Y | = lim H(S j |Y 1:N , S 1:j−1 ), 1 N →∞ N N →∞ N j∈S lim
(59) (60)
U1
1 (N ) lim |HSU | = H(X2 ), 2 N →∞ N 1 (N ) 1 X lim |LSU |Y | = lim H(S j |Y 1:N , S 1:j−1 ), 2 N →∞ N N →∞ N j∈S
(61) (62)
U2
where (59) and (61) hold because U1 and U2 are independent. Then we can partition two users’ indices as (N ) (N ) I1 , HSU ∩ LSU |Y 1
F1r ,
(N ) HSU 1
1
∩
(N ) (LSU |Y )C 1
(63)
(N )
F1d , (HSU )C 1
and
(N )
(N )
I2 , HSU ∩ LSU 2
F2r ,
(N ) HSU 2
∩
2
|Y
(N ) (LSU |Y )C 2
(64)
(N )
F2d , (HSU )C 2
respectively. The polarization result can be summarized as the following proposition. Proposition 4 ( [5]): Let PY |X1 X2 (y|x1 , x2 ) be the transition probability of a binary-input memoryless 2-user MAC. Consider the transformation defined in (54). Let N0 = 2n0 for some n0 ≥ 1 and fix a path b2N0 for U11:N0 U21:N0 . The rate pair for b2N0 is denoted by (RU1 , RU2 ). Let N = 2l N0 for l ≥ 1 and let S 1:2N be the expansion represented by 2l b2N0 . Then, for any given δ > 0, as l goes to infinity, we have 1 {1 ≤ j ≤ 2N : δ < H(S j |Y 1:N , S 1:j−1 ) < 1 − δ} → 0, 2N (65) |I2 | |I1 | → RU1 and → RU2 . N N Proposition 3 and 4 can be readily extended from Theorem 1 and Theorem 2 in [5] because of the duality between the 2-user MAC and the Slepian-Wolf coding. Similar to the single-user channel case, the polar coding scheme for the 2-user MAC is for user k (k = 1, 2) to put information bits to {ujk }j∈Ik , fill {ujk }j∈Fkr with frozen bits, and compute bits at position {ujk }j∈Fkd according to the input distribution. The receiver uses a successive cancellation decoder to decode the two users’ information together according to the expansion order. The error performance of such a scheme is upper bounded by X β Pe ≤ Z(S j |Y 1:N , S 1:j−1 ) = O(2−N ), (66) j∈I
where I is the set of information indices with respect to S 1:2N .
11
Fig. 2.
Proposed superposition polar coding scheme for the 2-user DM-IC.
IV. A N OVERVIEW OF O UR N EW A PPROACH In this section, we introduce the main idea of our scheme. We follow the Han-Kobayashi coding strategy to design polar codes for the interference channel. Sender j’s (j = 1, 2) splits its message Mj into a private message, denoted by Mjp , and a common message, denoted by Mjc , and encodes them into one codeword using superposition coding. For a coding scheme designed to achieve corner point Pi (i ∈ {1, 2, ..., 10}), let Rjip and Rjic respectively denote the corresponding private message rate and common message rate for Sender j, and define P1i , (R1ip + R1ic , R2ic ) and P2i , (R1ic , R2ip + R2ic ) as Receiver 1’s and Receiver 2’s receiving rate pairs respectively. Furthermore, define Pci , (R1ic , R2ic ) as the common message rate pair. For a rate pair P, let P(1) and P(2) respectively denote its first and second components. A. Synthesized MACs for Receivers Since the Han-Kobayashi coding scheme requires each receiver jointly decode the intended sender’s private message and both senders’ common messages, it is natural to consider the strategy that decomposes the interference channel into a compound MAC consisting of two 3-user MACs, as [39] suggests. However, 0 0 such an approach requires the functions x1N (·) and x2N (·) in (5) and (6) to be known, which can be problematic in practice, especially when N is large. In this paper, we propose an alternative approach based on the compact description of the Han-Kobayashi region as shown in Theorem 2. According to the characteristics of random variables used in this description, we design our superposition polar coding scheme as illustrated in Fig. 2. Our scheme is designed for a given Q = q, i.e., fixed joint distributions PX1 W1 |Q=q (x1 , w1 ) and PX2 W2 |Q=q (x2 , w2 ). For Sender 1, encoder E1b maps its common message M1c into 0 a sequence U11:N of length N , which goes into a polar encoder to generate an intermediate codeword W11:N (corresponding to auxiliary random variables W1 in Theorem 2). Encoder E1a then maps its private message M1p together with W11:N into U11:N , which goes into another polar encoder to generate the final codeword X11:N . E2b and E2a serve the same purposes for Sender 2. From Fig. 2 we can see some similarity to Goela et al.’s superposition polar coding scheme for the broadcast channel [18]. Indeed, if we remove one sender from this block diagram, it will become the broadcast superposition polar coding scheme of theirs. The main challenge in applying such a scheme to the DM-IC is the design of joint decoders. Since the intended common message will always be decoded before the intended private message at each receiver using such a scheme, the fully joint decoder cannot be realized, and therefore there exist only two types of partially joint decoding orders, each receiver either joint decodes two common messages first and then its private message, or solely decodes its intended common message first and then jointly decodes the rest two. We will show that these two types of decoding orders are sufficient to achieve all corner points of the Han-Kobayashi region. For the purpose of determining the corresponding common and private message rates at each corner points in our scheme, we synthesize the following 2-user MACs. For Receiver 1, since its purpose is to recover M1p , M1c and M2c with Y11:N , define the 2-user MAC PY1 |X1 W2 , with X1 and W2 being the
12
Fig. 3.
Illustration for each receiver’s effective rate region and stand-alone common message rate region.
channel inputs and Y1 the output, as the effective channel seen by Receiver 1. For a given q, PY1 |X1 W2 is defined as X PY1 |X1 W2 (y1 |x1 , w2 ) , PY1 |X1 X2 (y1 |x1 , x2 )PX2 |W2 Q (x2 |w2 , q). (67) x2
Similarly we can define the effective channel PY2 |W1 X2 for Receiver 2: X PY2 |W1 X2 (y2 |w1 , x2 ) , PY2 |X1 X2 (y2 |x1 , x2 )PX1 |W1 Q (x1 |w1 , q).
(68)
x1
The achievable rate regions for these two MACs are 0 ≤ R1 ≤ I(X1 ; Y1 |W2 ) R1 0 ≤ R2 ≤ I(W2 ; Y1 |X1 ) R(PY1 |X1 W2 ) = R2 R1 + R2 ≤ I(X1 W2 ; Y1 ) 0 ≤ R1 ≤ I(W1 ; Y2 |X2 ) R1 0 ≤ R2 ≤ I(X2 ; Y2 |W1 ) R(PY2 |W1 X2 ) = R2 R1 + R2 ≤ I(X2 W1 ; Y2 )
,
(69)
.
(70)
Now we can study the Han-Kobayashi coding problem in PY1 |X1 W2 and PY2 |W1 X2 . In these two MACs, the rate of Xj (j = 1, 2) equals the total message rate of Sender j, while the rate of Wj equals the common message rate of Sender j. Obviously, for i ∈ {1, 2, ..., 10}, P1i and P2i must lie inside R(PY1 |X1 W2 ) and R(PY2 |W1 X2 ) respectively in order to make reliable communication possible. Giving two effective channels only is insufficient to determine the decoding order of the common messages and the private message for each receiver. If we hope to use a partially joint decoder instead of a fully joint one, the following two MACs, PY1 |W1 W2 and PY2 |W1 W2 , will be useful. For a given q and j = 1, 2, define XX PYj |W1 W2 (yj |w1 , w2 ) , PYj |X1 X2 (yj |x1 , x2 )PX1 |W1 Q (x1 |w1 , q)PX2 |W2 Q (x2 |w2 , q), (71) x1
x2
and the achievable rate region of PYj |W1 W2 is 0 ≤ R1 ≤ I(W1 ; Yj |W2 ) R1 0 ≤ R2 ≤ I(W2 ; Yj |W1 ) R(PYj |W1 W2 ) = . R2 R1 + R2 ≤ I(W1 W2 ; Yj )
(72)
13
If there exists a possible common message rate pair Pci (i = 1, 2, ..., 10) that lies inside R(PYj |W1 W2 ) (j = 1, 2), then Receiver j can decode two common messages separate from its private message while still being able to achieve Pji . Otherwise, it will have to jointly decode its private message and the common messages. For this reason, we refer to these two regions as the stand-alone common message rate regions for Receiver 1 and Receiver 2 respectively. The relations between the above four regions are shown in Fig. 3. B. The General Idea of Our Scheme Based on what has been discussed in the previous subsection, we can design polar codes as follows. If Pci ∈ R(PY1 |W1 W2 ) ∩ R(PY2 |W1 W2 ), we first design a MAC polar code for two common messages that achieves rate pair Pci in the compound MAC composed of PY1 |W1 W2 and PY2 |W1 W2 , and then design a point-to-point polar code for Sender j’s (j = 1, 2) private message with Wj1:N as side information. We will refer to this kind of corner points as the Type I points. In the next subsection we will show that P1 , one case of P2 , and P6 are of Type I (see Fig. 4, 5 (a) and 9 for preview). Otherwise, if Pci ∈ / R(PY1 |W1 W2 ) ∩ R(PY2 |W1 W2 ), private messages will have to be jointly decoded with common messages. At first glance, our partially joint decoding strategy does not fit this case. Nevertheless, we find that for each i ∈ {1, 2, ..., 10}, all possible choices of Pci lie inside at least one receiver’s standalone common message region (see Fig. 4–11), meaning that at least one receiver can sequentially decode common messages and its intended private message. And even if all possible choices of Pci lie outside of the other receiver’s (say Receiver j for j = 1, 2) stand-alone common message rate region, we can still find a possible Pci such that Pci (j) ≤ I(Wj ; Yj ) (see Fig. 5 (b), 6, 7, 8, 10 and 11), i.e., there exists coding schemes such that Receiver j can reliably decode its intended common message with only its channel output. This inspires us with another coding strategy. As an example, suppose Pci ∈ R(PY2 |W1 W2 ) \ R(PY1 |W1 W2 ) and Pci (1) ≤ I(W1 ; Y1 ). Under these conditions, the code constructions for M1c , M1p and M2c are jointly designed in such a way that, Receiver 1 can first decode M1c (equivalently W11:N ) with Y11:N only, and then jointly decode (M1p , M2c ) with Y11:N and the decoding result of W11:N , while Receiver 2 can jointly decode (M1c , M2c ) (equivalently (W11:N , W21:N )) with Y21:N only. The code construction for M2p is simply a point-to-point polar code with W21:N as side information. We will refer to this kind of corner points as the Type II points. The other case of P2 and the rest corner points that are not of Type I all belong to Type II. C. Analysis of Each Corner Point In this subsection we decompose each corner point into two common/private message rate pairs for two senders that satisfy our coding requirement. Let us first consider some special cases. As has been discussed in [35], if I(W1 ; Y2 |X2 ) < I(W1 ; Y1 |W2 ) and I(W2 ; Y1 |X1 ) ≥ I(W2 ; Y2 |W1 ), when Sender 1 transmits at a rate above I(X1 ; Y1 |W1 W2 )+I(W1 ; Y2 |X2 ) (which can still be within), its common message rate will exceed Receiver 2’s decoding capability. Under this condition, we will set W1 = ∅, i.e., Sender 1 will only transmit private message. Then the Han-Kobayashi region reduces to R1 R2 R1 + R2 R1 + R2 2R1 + R2 R1 + 2R2 R1 , R2
≤ I(X1 ; Y1 |W2 ) ≤ I(X2 ; Y2 ) ≤ I(X1 W2 ; Y1 ) + I(X2 ; Y2 |W2 ) ≤ I(X1 ; Y1 |W2 ) + I(X2 ; Y2 ) ≤ I(X1 W2 ; Y1 ) + I(X1 ; Y1 |W2 ) + I(X2 ; Y2 |W2 ) ≤ I(X2 ; Y2 |W2 ) + I(X2 ; Y2 ) + I(X1 W2 ; Y1 ) ≥ 0.
(73) (74) (75) (76) (77) (78) (79)
We can see that (76), (77) and (78) can be readily obtained from (73), (74) and (75), thus are redundant.
14
Similarly, when I(W2 ; Y1 |X1 ) < I(W2 ; Y2 |W1 ) and I(W1 ; Y2 |X2 ) ≥ I(W1 ; Y1 |W2 ), we will set W2 = ∅, and the Han-Kobayashi region reduces to R1 ≤ I(X1 ; Y1 ), R2 ≤ I(X2 ; Y2 |W1 ) R1 + R2 ≤ I(X1 ; Y1 |W1 ) + I(X2 W1 ; Y2 ) R1 , R2 ≥ 0. When I(W2 ; Y1 |X1 ) < I(W2 ; Y2 |W1 ) and I(W1 ; Y2 |X2 ) < I(W1 ; Y1 |W2 ), we will set W1 = ∅ and W2 = ∅, and the Han-Kobayashi region reduces to a rectangle R1 ≤ I(X1 ; Y1 ), R2 ≤ I(X2 ; Y2 ) R1 , R2 ≥ 0. The polar coding schemes for the above special cases can also be seen as special (or simplified) cases of our proposed schemes in the next section. Therefore, we will not discuss them in detail. In the following, we assume I(X1 ; Y1 |W2 ) ≥ I(W1 ; Y2 |W2 ) and I(X2 ; Y2 |W1 ) ≥ I(W2 ; Y1 |W1 ). The results of this subsection are directly derived from constraints in Theorem 1 and 2. A more intuitive analysis why our proposed strategy works is presented in Appendix C. 1) Case 1: The conditions of Case 1 are I(W1 ; Y2 |W2 ) − I(W1 ; Y1 ) ≥ 0, I(W1 W2 ; Y2 ) − I(W1 W2 ; Y1 ) ≥ 0.
(80) (81)
If I(W2 ; Y2 |W1 ) − I(W2 ; Y1 ) ≤ 0, as has been discussed in Section II-C, the Han-Kobayashi region becomes a rectangle, and the whole region can be achieved by letting each sender transmit common message only, which is also a special case of our scheme. Thus, we add the following condition I(W2 ; Y2 |W1 ) − I(W2 ; Y1 ) > 0.
(82)
(P1 ) When P1 is a corner point, we have an additional condition of I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) ≥ 0.
(83)
At P1 , Sender 1 transmits at its maximum single-user rate I(X1 ; Y1 |W2 ). Since in our proposed schemes, the intended common message will always be decoded ahead of the intended private message, we have R11c ≤ I(W1 ; Y1 |W2 ). And since R11p ≤ I(X1 ; Y1 |W2 W2 ), we have R11p = I(X1 ; Y1 |W1 W2 ), R11c = I(W1 ; Y1 |W2 ). Since P11 ∈ R(PY1 |X1 W2 ), we have
(84) (85)
R21c ≤ I(W2 ; Y1 ).
From (29) and (8) we have R21p + R21c = I(X1 ; Y1 |W1 W2 ) + I(W2 ; Y1 ), R21p ≤ I(X1 ; Y1 |W1 W2 ). Thus, R21p = I(X1 ; Y1 |W1 W2 ), R21c = I(W2 ; Y1 ). The positions of P1 , P11 , P21 and Pc1 are shown in Fig. 4. Obviously P1 belongs to Type I.
(86) (87)
15
Fig. 4.
Rate decomposition of P1 .
(P2 ) When P2 is a corner point, we have the following additional condition I(W1 ; Y2 ) − I(W1 ; Y1 ) ≥ 0.
(88)
At P2 , Sender 2 transmits at its maximum single-user rate. Similar to the case of Sender 1 at P1 , we have R22p = I(X2 ; Y2 |W1 W2 ), R22c = I(W2 ; Y2 |W1 ). Since P22 ∈ R(PY2 |W1 X2 ), we have
(89) (90)
R12c ≤ I(W1 ; Y2 ).
From (12) we have R12p ≤ I(X1 W2 ; Y1 |W1 ) − R22c = I(X1 W2 ; Y1 |W1 ) − I(W2 ; Y2 |W1 ). Then R12c ≥ I(X1 W2 ; Y1 ) − I(W2 ; Y2 |W1 ) − I(X1 W2 ; Y1 |W1 ) − I(W2 ; Y2 |W1 ) = I(W1 ; Y1 ). And since R12p ≤ I(X1 ; Y1 |W1 W2 ), we have R12c ≥ P2 (1) − I(X1 ; Y1 |W1 W2 ) = I(W1 W2 ; Y1 ) − I(W2 ; Y2 |W1 ). Finally we have, 2c • (a) if I(W2 ; Y1 |W1 ) ≥ I(W2 ; Y2 |W1 ), then I(W1 W2 ; Y1 ) − I(W2 ; Y2 |W1 ) ≤ R1 ≤ I(W1 ; Y2 ). We can choose R12c = I(W1 W2 ; Y1 ) − I(W2 ; Y2 |W1 ),
(91)
R12p
(92)
= I(X1 ; Y1 |W1 W2 ),
then P2 belongs to Type I, as Fig. 5 (a) shows.
16
Fig. 5.
•
Rate decomposition of P2 .
(b) otherwise if I(W2 ; Y1 |W1 ) < I(W2 ; Y2 |W1 ), then I(W1 ; Y1 ) ≤ R12c ≤ I(W1 ; Y2 ). We can choose R12c = I(W1 ; Y1 ),
(93)
R12p
(94)
= I(X1 W2 ; Y1 |W1 ) − I(W2 ; Y2 |W1 ),
then P2 belongs to Type II, as shown in Fig. 5 (b). (P3 ) When P3 is a corner point, we have the following additional condition I(W1 ; Y2 ) − I(W1 ; Y1 ) < 0.
(95)
At P3 , Sender 2 transmits at its maximum single-user rate, so we have
Since P23 ∈ R(PY2 |W1 X2 ), we have
R23p = I(X2 ; Y2 |W1 W2 ), R23c = I(W2 ; Y2 |W1 ).
(96) (97)
R13c ≤ I(W1 ; Y2 ).
(98)
From (12) we have R13p ≤ I(X1 W2 ; Y1 |W1 ) − R23c = I(X1 W2 ; Y1 |W1 ) − I(W2 ; Y2 |W1 ). Thus, R13c ≥ P3 (1) − I(X1 W2 ; Y1 |W1 ) − I(W2 ; Y2 |W1 ) = I(W1 ; Y2 ).
(99)
From (98) and (99) we have R13c = I(W1 ; Y2 ),
(100)
R13p = I(X1 W2 ; Y1 |W1 ) − I(W2 ; Y2 |W1 ),
(101)
which implies We can see that P3 belongs to Type II, as shown in Fig. 6.
17
Fig. 6.
Rate decomposition of P3 .
Fig. 7.
Rate decomposition of P4 .
(P4 ) When P4 is a corner point, we have the following additional condition I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) < 0.
(102)
At P4 , Sender 1 transmits at its maximum single-user rate, and we have R14p = I(X1 ; Y1 |W1 W2 ), R14c = I(W1 ; Y1 |W2 ).
(103) (104)
Similar to the analysis of R13c and R13p , we have R24c = I(W2 ; Y1 ),
(105)
R24p
(106)
= I(X2 W1 ; Y2 |W2 ) − (W1 ; Y1 |W2 ).
We can see that P4 belongs to Type II, as shown in Fig. 7. (P5 ) When P5 is a corner point, the additional condition is (95). From (33) we have R15p + R15c + R25p + R25c = I(X1 W2 ; Y1 ) + I(X2 ; Y2 |W1 W2 ). From (107), (14) and the counterpart of (8) for the second receiver, we have
(107)
18
Fig. 8.
Rate decomposition of P5 .
R25p = I(X2 ; Y2 |W1 W2 ), R15p
+
R15c
+
R25c
(108)
= I(X1 W2 ; Y1 ).
Thus, R25c = P5 (2) − R25p = I(W1 W2 ; Y2 ) − I(W1 ; Y1 ),
(109)
R15p + R15c = I(X1 W2 ; Y1 ) + I(W1 ; Y1 ) − I(W1 W2 ; Y2 ).
(110)
R15p ≤ I(X1 W2 ; Y1 ) − I(W1 W2 ; Y2 ).
(111)
From (12) we have From (110) and (111) we have R15c ≥ I(W1 ; Y1 ). Since P25 ∈ R(PY2 |W1 X2 ), we have R15c ≤ I(X2 W1 ; Y2 ) − P5 (2) = I(W1 ; Y1 ). Finally, we have R15c = I(W1 ; Y1 ),
(112)
R15p
(113)
= I(X1 W2 ; Y1 ) − I(W1 W2 ; Y2 ).
We can see that P5 belongs to Type II, as shown in Fig. 8. (P6 ) Using a similar analysis of P5 , we can show that R16p = I(X1 ; Y1 |W1 W2 ), R16c = I(W1 ; Y2 |W2 ),
(114) (115)
R26p = I(X2 ; Y2 |W1 W2 ), R26c = I(W1 W2 ; Y1 ) − I(W1 ; Y2 |W2 ).
(116) (117)
Obviously P6 belongs to Type I, as shown in Fig. 9.
19
Fig. 9.
Fig. 10.
Rate decomposition of P6 .
Rate decomposition of P7 .
2) Case 3: In Case 3, the conditions are I(W1 ; Y2 |W2 ) − I(W1 ; Y1 ) < 0, I(W2 ; Y1 |W1 ) − I(W2 ; Y2 ) < 0.
(118) (119)
From (35), (36), (37) and 38) we can see that P8 (resp. P10 ) is symmetric with P7 (resp. P9 ) by swapping the roles of two receivers, we will only consider P7 and P9 here. (P7 ) At P7 , Sender 1 transmits at its maximum single-user rate, we have R17p = I(X1 ; Y1 |W1 W2 ), R17c = I(W1 ; Y1 |W2 ). Since P17 ∈ R(PY1 |X1 W2 ), we have
R27c ≤ I(W2 ; Y1 |Q).
From the counterpart of (12) for the second receiver, we have R27p ≤ I(X2 W1 ; Y2 |W2 ) − R17c = I(X2 W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ).
(120) (121)
20
Fig. 11.
Rate decomposition of P9 .
Then R27c = P7 (2) − R27p ≥ I(W2 ; Y1 ). Thus, R27c = I(W2 ; Y1 ),
(122)
R27p = I(X2 W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ).
(123)
We can see that P7 belongs to Type II, as shown in Fig. 10. (P9 ) From (37) we have R19p + R19c + R29p + R29c = I(X1 W2 ; Y1 |W1 ) + I(X2 W1 ; Y2 |W2 ).
(124)
Combining (124) with (12) and its counterpart for the second receiver, we have R19p + R29c = I(X1 W2 ; Y1 |W1 ), R29p + R19c = I(X2 W1 ; Y2 |W2 ). Then we can readily obtain R19p = I(X1 ; Y1 |W1 W2 ), R19c = I(W1 ; Y1 ),
(125) (126)
R29p = I(X2 W1 ; Y2 |W2 ) − I(W1 ; Y1 ), R29c = I(W2 ; Y1 |W1 ).
(127) (128)
We can see that P9 belongs to Type II, as shown in Fig. 11. Now we have shown that for every possible corner point of the Han-Kobayashi region, there exists a common/private message rate-tuple that satisfies the requirement of our proposed coding strategy. In the next section, we will describe the details of coding schemes for Type I and II points.
21
V. P ROPOSED P OLAR C ODING S CHEMES A. Polar Coding Scheme for Type I Points Let Pc be the common message rate pair for a Type I corner point. From the discussion in the previous section we can see that, for all possible corner points of Type I (P1 , P2 of type (a) and P6 ), the private message rate of each sender equals its maximum, i.e., Rjp = I(Xj ; Yj |W1 W2 ) for j = 1, 2. This is also verified in Appendix C. Therefore, if we can show that the rate pair (Pc (1) + I(X1 ; Y1 |W1 W2 ), Pc (2) + I(X2 ; Y2 |W1 W2 )) is achievable for the interference channel, we can say that all Type I corner points are achievable. Since for all Type I points, the corresponding Pc lies on the dominant face of either R(PY1 |W1 W2 ) or R(PY2 |W1 W2 ), we assume in this subsection that Pc is on the dominant face of R(PY1 |W1 W2 ) without loss of generality. The case when Pc is on the dominant face of R(PY2 |W1 W2 ) is similar. ˜ c on the dominant face of R(PY2 |W1 W2 ), which 1) Common Message Encoding: First, choose a point P ˜ c (1) ≥ Pc (1) and P ˜ c (2) ≥ Pc (2), as the target point for conducting is larger than Pc in the sense that P the monotone chain rule expansion in our code design. Let S 1:2N be the monotone chain rule expansion ˜c for achieving Pc in R(PY1 |W1 W2 ), and T 1:2N be the monotone chain rule expansion for achieving P 0 1:N 0 1:N 1:2N j j in R(PY2 |W1 W2 ). Denote the sets of indices in S and S ∈ U2 by SU10 and SU20 with S ∈ U1 0 1:N 0 1:N j 1:2N j respectively, and the sets of indices in T with T ∈ U1 and T ∈ U2 by TU10 and TU20 respectively. −N β Let δN = 2 for 0 < β < 1/2, define the following polarized sets (N ) HSU 0 , j ∈ SU10 : Z(S j |S 1:j−1 ) ≥ 1 − δN , 1 (N ) HSU 0 , j ∈ SU20 : Z(S j |S 1:j−1 ) ≥ 1 − δN , 2 (N ) LS 0 |Y1 , j ∈ SU10 : Z(S j |Y11:N , S 1:j−1 ) ≤ δN , U1 (N ) LS 0 |Y1 , j ∈ SU20 : Z(S j |Y11:N , S 1:j−1 ) ≤ δN , U2 (129) (N ) HTU 0 , j ∈ TU10 : Z(T j |T 1:j−1 ) ≥ 1 − δN , 1 (N ) HTU 0 , j ∈ TU20 : Z(T j |T 1:j−1 ) ≥ 1 − δN , 2 (N ) LT 0 |Y2 , j ∈ TU10 : Z(T j |Y21:N , T 1:j−1 ) ≤ δN , U1 (N ) LT 0 |Y2 , j ∈ TU20 : Z(T j |Y21:N , T 1:j−1 ) ≤ δN . U2
Since two senders’ common messages are independent from each other, we have (N )
(N )
(N )
(N )
(N )
(N )
HSU 0 = HTU 0 = HW1 , HSU 0 = HTU 0 = HW2 . 1 1 2 2 0j 0 1:j−1 (N ) where HWi , j ∈ [N ] : Z(Ui |Ui ) ≥ 1 − δN for i = 1, 2. Define the following sets of indices for Sender 1, (N )
(N )
(N )
C11 , HSU 0 ∩ LS 1
0 |Y1 U1
(N )
, C12 , HTU 0 ∩ LT 1
0 |Y2 U1
,
(130)
,
(131)
and similarly for Sender 2, (N )
(N )
C21 , HSU 0 ∩ LS 2
(N )
0 |Y1 U2
(N )
, C22 , HTU 0 ∩ LT 2
0 |Y2 U2
which satisfy 1 1 |C1 | = Pc (1), N →∞ N 1 1 lim |C2 | = Pc (2), N →∞ N lim
1 2 ˜ c (1) ≥ Pc (1), |C1 | = P N →∞ N 1 2 ˜ c (2) ≥ Pc (2). lim |C2 | = P N →∞ N lim
22
Now we can use the chaining method introduced in [43] to achieve Pc . Suppose the block number is k. Choose an arbitrary subset of C12 \ C11 , denoted as C121 , such that |C121 | = |C11 \ C12 |, and an arbitrary subset of C22 \ C21 , denoted as C221 , such that |C221 | = |C21 \ C22 |. Define the following sets of indices for Sender 1 I1c = C11 ∩ C12 , 1 I1c = C11 \ C12 , 2 I1c = C121 , 0 F1r
=
0 F1d =
(132)
(N ) HW1 \ (I1c (N ) (HW1 )C ,
∪
1 I1c
∪
1 I2c
∪
2 I1c ),
∪
2 I2c ),
and similarly define the following sets for Sender 2. I2c = C21 ∩ C22 , 1 = C21 \ C22 , I2c 2 = C221 , I2c 0 F2r
=
0 F2d =
(133)
(N ) HW2 \ (I2c (N ) (HW2 )C .
The encoding rules are as follows. (1) In the 1st block, for Sender 1: 0i • {u1 }i∈I1c ∪I 1 store common message bits. 1c 0i • {u1 }i∈F 0 ∪I 2 carry uniformly distributed frozen bits. 1r 1c 0i • {u1 }i∈F 0 are computed as 1d 0
0
u1i = arg max PU 0 i |U 0 1:i−1 (u|u11:i−1 ). u∈{0,1}
1
(134)
1
For Sender 2: 0i • {u2 }i∈I2c ∪I 1 store common message bits. 2c 0i • {u2 }i∈F 0 ∪I 2 carry uniformly distributed frozen bits. 2r 2c 0i • {u2 }i∈F 0 are computed as 2d 0
0
u2i = arg max PU 0 i |U 0 1:i−1 (u|u21:i−1 ). u∈{0,1}
2
(135)
2
(2) In the jth (1 < j < k) block, for Sender 1: 0i 0i 0i • {u1 }i∈I1c ∪I 1 , {u1 }i∈F 0 and {u1 }i∈F 0 are determined in the same ways as in block 1. 1r 1d 1c 0i 0i • {u1 }i∈I 2 are assigned to the same value as {u1 }i∈I 1 in the j − 1th block. 1c 1c For Sender 2: 0i 0i 0i • {u2 }i∈I2c ∪I 1 , {u2 }i∈F 0 and {u2 }i∈F 0 are determined in the same ways as in block 1. 2r 2d 2c 0i 1 • {u2 }i∈I 2 are assigned to the same value as I2c in the j − 1th block. 2c (3) In the kth block, for Sender 1: 0i 0i 0i 0i • {u1 }i∈I1c , {u1 }i∈F 0 , {u1 }i∈I 2 and {u1 }i∈F 0 are determined in the same ways as in block j (1 < 1r 1d 1c j < k). 0i • {u1 }i∈I 1 also carry frozen bits. 1c For Sender 2: 0i 0i 0i 0i • {u2 }i∈I2c , {u2 }i∈F 0 , {u2 }i∈I 2 and {u2 }i∈F 0 are determined in the same ways as in block j (1 < 2r 2d 2c j < k). 0i • {u2 }i∈I 1 also carry frozen bits. 2c
23
The common message rates of two senders in this scheme are 1 |C 1 | |I 1 | | k|I1c | + (k − 1)|I1c = 1 − 1c , kN N kN 1 1 k|I2c | + (k − 1)|I2c | |C | |I 1 | R2c = = 2 − 2c . kN N kN
R1c =
(136)
Obviously, lim
N →∞,k→∞
R1c = Pc (1),
lim
N →∞,k→∞
R2c = Pc (2).
(137)
β
2) Private Message Encoding: Let δN = 2−N for 0 < β < 1/2. Define the following polarization sets: 0 0 (N ) HX1 |W1 W2 , j ∈ [N ] : Z(U1j |U11:N , U21:N , U11:j−1 ) ≥ 1 − δN , (138) 0 0 (N ) LX1 |Y1 W1 W2 , j ∈ [N ] : Z(U1j |Y11:N , U11:N , U21:N , U11:j−1 ) ≤ δN , which satisfy 1 (N ) |H | = H(X1 |W1 W2 ), N X1 |W1 W2 1 (N ) |LX1 |Y1 W1 W2 | = 1 − H(X1 |Y1 W1 W2 ). lim N →∞ N lim
N →∞
(N )
(N )
Similarly we can define HX2 |W1 W2 and LX2 |Y2 W1 W2 . Since W2 is independent from W1 and X1 , and W1 is independent from W2 and X2 , we have (N )
(N )
(N )
(N )
HX1 |W1 W2 = HX1 |W1 , HX2 |W1 W2 = HX2 |W2 , 0 (N ) where HXi |Wi , j ∈ [N ] : Z(Uij |Ui 1:N , Ui1:j−1 ) ≥ 1 − δN for i = 1, 2. The private message sets for two senders are defined as follows. (N )
(N )
(N )
(N )
I1p , HX1 |W1 W2 ∩ LX1 |Y1 W1 W2 ,
(139)
(140)
I2p , HX2 |W1 W2 ∩ LX2 |Y2 W1 W2 . The private message rates are then R1p =
1 1 |I1p |, R2p = |I2p |, N N
(141)
which satisfy lim R1p = I(X1 ; Y1 |W1 W2 ),
N →∞
lim R2p = I(X2 ; Y2 |W1 W2 ).
N →∞
(142)
The remaining bits are partitioned into two subsets. For Sender 1, the two subsets are the deterministic set of (N ) (N ) F1d = (HX1 |W1 W2 )C = (HX1 |W1 )C , (143) and the frozen set of (N )
(N )
(N )
(N )
F1r = HX1 |W1 W2 ∩ (LX1 |Y1 W1 W2 )C = HX1 |W1 ∩ (LX1 |Y1 W1 W2 )C .
(144)
For Sender 2, the two subsets are (N )
(N )
(N )
F2d = (HX2 |W2 )C , F2r = HX2 |W2 ∩ (LX2 |Y2 W1 W2 )C .
(145)
Fig. 12 shows the partition of U11:N in this scheme. In each block, Sender 1 generates its final codewords as follows. i • {u1 }i∈I1p store private message bits.
24
Fig. 12.
Graphical representation of the partition for U11:N of Type I corner points.
{ui1 }i∈F1r carry uniformly distributed frozen bits. For i ∈ F1d , we set 0 ) ui1 = arg max PU i |U 0 1:N U 1:i−1 (u|u11:N , u1:i−1 1
(146)
And Sender 2 as follows. i • {u2 }i∈I2p store private message bits. i • {u2 }i∈F2r carry uniformly distributed frozen bits. • For i ∈ F2d , we set 0 ui2 = arg max PU i |U 0 1:N U 1:i−1 (u|u21:N , u1:i−1 ) 2
(147)
• •
1
u∈{0,1}
2
u∈{0,1}
1
1
2
2
3) Decoding: The decoding procedure goes as follows. 0 0 (1) Receiver 1 decodes common messages {u1i } and {u2i } from Block 1 • In Block 1, two senders’ common messages are jointly decoded as 0 i u 1 , 0 0i u¯1 = arg maxu∈{0,1} PU 0 i |U 0 1:i−1 (u|u11:i−1 ), 1 1 arg maxu∈{0,1} PS f1 (i) |Y11:N S 1:f1 (i)−1 (u|y11:N , s1:f1 (i)−1 ), 0 i u 2 , 0 0i u¯2 = arg maxu∈{0,1} PU 0 i |U 0 1:i−1 (u|u21:i−1 ), 2 2 arg maxu∈{0,1} PS f2 (i) |Y11:N S 1:f2 (i)−1 (u|y11:N , s1:f2 (i)−1 ),
to Block k. 2 0 ∪ I1c if i ∈ F1r 0 if i ∈ F1d
(148)
1 if i ∈ I1c ∪ I1c 2 0 ∪ I2c if i ∈ F2r 0 if i ∈ F2d
(149)
1 if i ∈ I2c ∪ I2c
0
where f1 (i) denotes the mapping from indices of U11:N to indices of S 1:2N , and f2 (i) the mapping 0 from indices of U21:N to indices of S 1:2N . 0 0 0 0 2 and {¯ 2 are deduced from {¯ 1 and {¯ 1 in Block • In Block j (1 < j < k), {¯ u1i }i∈I1c u2i }i∈I2c u1i }i∈I1c u2i }i∈I2c j − 1 respectively, and the rest bits are decoded in the same ways as in Block 1. 0 0 1 and {¯ 1 are assigned to the pre-shared value between Sender 1 and the • In Block k, {¯ u1i }i∈I1c u2i }i∈I2c two receivers, and the rest bits are decoded in the same ways as in Block j (1 < j < k). Having recovered common messages in a block, Receiver 1 decodes its private message {ui1 } as i if i ∈ F1r u1 , 0N 1:i−1 i u1 , u1 ), if i ∈ F1d u¯1 = arg maxu∈{0,1} PU1i |U10 N U11:i−1 (u|¯ (150) 0N 0N arg max 1:i−1 1:N ¯1 , u¯2 , u1 ), if i ∈ I1p u∈{0,1} PU i |Y 1:N U 0 N U 0 N U 1:i−1 (u|y1 , u 1
1
1
2
0i
1
0
(2) Receiver 2 decodes common messages {u1 } and {u2i } from Block k to Block 1.
25
•
In Block k, two senders’ common messages are jointly decoded as 0 i u1 , 0 0 uˆ1i = arg maxu∈{0,1} PU 0 i |U 0 1:i−1 (u|u11:i−1 ), 1 1 arg maxu∈{0,1} PT g1 (i) |Y21:N T 1:g1 (i)−1 (u|y21:N , t1:g1 (i)−1 ), 0 i u2 , 0 0 uˆ2i = arg maxu∈{0,1} PU 0 i |U 0 1:i−1 (u|u21:i−1 ), 2 2 arg maxu∈{0,1} PT g2 (i) |Y21:N T 1:g2 (i)−1 (u|y21:N , t1:g2 (i)−1 ),
0 1 if i ∈ F1r ∪ I1c 0 if i ∈ F1d
if i ∈ I1c ∪
1 0 ∪ I2c if i ∈ F2r 0 if i ∈ F2d
if i ∈ I2c ∪
(151)
2 I1c
(152)
2 I2c
0
where g1 (i) denotes the mapping from indices of U11:N to indices of T 1:2N , and g2 (i) the mapping 0 from indices of U21:N to indices of T 1:2N . 0 0 0 0 1 and {ˆ 1 are deduced from {ˆ 2 and {ˆ 2 in Block • In Block j (1 < j < k), {ˆ u1i }i∈I1c u2i }i∈I2c u1i }i∈I1c u2i }i∈I2c j + 1 respectively, and the rest bits are decoded in the same way as in Block k. 0 0 2 and {ˆ 2 are assigned to the pre-shared value between Sender 2 and the • In Block 1, {ˆ u1i }i∈I1c u2i }i∈I2c two receivers, and the rest bits are decoded in the same way as in Block j (1 < j < k). Having recovered common messages in a block, Sender 2 decodes its private message {ui2 } as i if i ∈ F2r u2 , 0N 1:i−1 i u2 , u2 ), if i ∈ F2d uˆ2 = arg maxu∈{0,1} PU2i |U20 N U21:i−1 (u|ˆ (153) 0N 0N arg max 1:i−1 1:N ˆ1 , uˆ2 , u2 ), if i ∈ I2p u∈{0,1} PU i |Y 1:N U 0 N U 0 N U 1:i−1 (u|y2 , u 2
2
1
2
2
4) Performance: The block error probabilities of the two receivers in this case can be upper bounded as X X Pe1 ≤ k Z(S f1 (i) |Y11:N , S 1:f1 (i)−1 ) + (k − 1) Z(S f1 (i) |Y11:N , S 1:f1 (i)−1 ) 1 i∈I1c
i∈I1c
+k
X
Z(U1i |Y11:N , U11:i−1 )
(154)
i∈I1p β
Pe2
= O(2−N ), X X ≤k Z(T g2 (i) |Y21:N , T 1:g2 (i)−1 ) + (k − 1) Z(T g2 (i) |Y21:N , T 1:g2 (i)−1 ) 1 i∈I2c
i∈I2c
+k
X
Z(U2i |Y21:N , U21:i−1 )
(155)
i∈I2p β
= O(2−N ). B. Polar Coding Scheme for Type II Points Let P be a corner point of Type II, Pc be the corresponding common message rate pair, and P1 and P2 be Receiver 1’s and Receiver 2’s rate pairs respectively. Without loss of generality, we only consider the case when Pc ∈ R(PY2 |W1 W2 ) \ R(PY1 |W1 W2 ) and Pc (1) ≤ I(W1 ; Y1 ), where Pc (1) stands for Sender 1’s component in Pc . The case when Pc ∈ R(PY1 |W1 W2 ) \ R(PY2 |W1 W2 ) and Pc (2) ≤ I(W2 ; Y2 ) is just symmetric to this one by swapping the roles of two receivers. 1 1 1 ¯ Choose P = I(X1 W2 ; Y1 ) − P (2), P (2) , which is on the dominant face of R(PY1 |X1 W2 ) and larger 1 c c c ˜ than P , and P = I(W1 W2 ; Y2 ) − P (2), P (2) , which is on the dominant face of R(PY2 |W1 W2 ) and larger than Pc , as the target points for conducting monotone chain rule expansions in our code design. From the discussion in the previous section (also in Appendix C) we can see that, in the case we are discussing (P2 of type (b), P3 and P5 ), the private message rate of Sender 1 is ¯ 1 (1) − I(W1 ; Y1 ), R1p = P (156)
26
while the private message rate of Sender 2 is R2p = I(X2 ; Y2 |W1 W2 ).
(157)
One can easily verify that these two properties also hold for the other case (P4 , P7 and P9 ) by swapping indices 1 and 2 everywhere in the expressions. ¯ 1 in R(PY1 |X1 W2 ), and T 1:2N be the Let S 1:2N be the monotone chain rule expansion for achieving P ˜ c in R(PY2 |W1 W2 ). Denote the sets of indices in S 1:2N with monotone chain rule expansion for achieving P 0 0 1:N j j 1:N by SU1 and SU20 respectively, and the sets of indices in T 1:2N with T j ∈ U11:N S ∈ U1 and S ∈ U2 0 β (N ) (N ) (N ) and T j ∈ U21:N by TU10 and TU20 respectively. Let δN = 2−N for 0 < β < 1/2, and let HW1 , HW2 , HSU 0 , (N )
(N )
(N )
(N )
(N )
(N )
(N )
(N )
(N )
2
HTU 0 , HTU 0 , HX1 |W1 , HX2 |W1 W2 , HX2 |W2 , LT 0 |Y2 , LT 0 |Y2 , LS 0 |Y1 and LX2 |Y2 W1 W2 be defined in the same U1 U2 U2 1 2 forms as in the previous subsection. Define the following extra polarized sets 0 0 (N ) LW1 |Y1 , j ∈ [N ] : Z(U1j |Y11:N , U11:j−1 ) ≤ δN , (158) 0 (N ) LSU |Y1 W1 , j ∈ SU1 : Z(S j |Y11:N , U11:N , S 1:j−1 ) ≤ δN . 1
(N )
(N )
Note that HX2 |W1 W2 = HX2 |W2 since W1 is independent from W2 and X2 . Let (N )
(N )
(N )
(N )
C10 , HW1 ∩ LW1 |Y1 , C100 , HW1 ∩ LT
0 |Y2 U1
(N )
(N )
I1p , HX1 |W1 ∩ LSU (N )
1
, |Y1 W1 ,
(N )
C21 , HW2 ∩ LS
, 0 |Y1
(159)
U2
C22 , I2p ,
(N ) HW2
∩
(N ) LT 0 |Y2 , U
(N ) HX2 |W1 W2
2
(N )
∩ LX2 |Y2 W1 W2 ,
which satisfy 1 0 |C1 | = I(W1 ; Y1 |Q) ≥ Pc (1), N →∞ N 1 00 ˜ c (1) ≥ Pc (1), |C1 | = P lim N →∞ N 1 1 ¯ 1 (2) = Pc (2), lim |C2 | = P N →∞ N 1 2 ˜ c (2) = Pc (2), lim |C2 | = P N →∞ N 1 lim |I2p | = I(X2 ; Y2 |W1 W2 ). N →∞ N lim
I1p is the information bit set for U11:N in this scheme, which is different from the MAC polar coding scheme described in Section III-C. This is because W11:N is superimposed on X11:N in our case. Since W11:N is decoded before X11:N , we need to consider W11:N as side information while encoding and decoding ¯ 1 in Receiver 1’s effective U11:N . In other words, we first determine the permutation S 1:2N that achieves P channel PY1 |X1 W2 , but then use this permutation together with the knowledge of W11:N to determine the code construction for U11:N . We will show that the rate of |I1p | satisfies the requirement of (156).
27 (N )
Let BSU
1
(N )
|Y1 W1
, (HSU
1
(N )
|Y1 W1
∪ LSU
1
|Y1 W1 )
C
, we have
1 (N ) 1 (N ) |I1p | = |HX1 |W1 ∩ LSU |Y1 W1 | 1 N N 1 (N ) (N ) (N ) = |HX1 |W1 ∩ (HSU |Y1 W1 ∪ BSU |Y1 W1 )C | 1 1 N 1 (N ) (N ) (N ) C = |HX1 |W1 ∩ (HSU |Y1 W1 ) ∩ (BSU |Y1 W1 )C | 1 1 N 1 (N ) 1 (N ) (N ) ≥ |HX1 |W1 ∩ (HSU |Y1 W1 )C | − |BSU |Y1 W1 | 1 1 N N 1 (N ) 1 (N ) 1 (N ) = |HX1 |W1 | − |HSU |Y1 W1 | − |BSU |Y1 W1 |. 1 1 N N N Since 1 (N ) 1 X |HSU |Y1 W1 | = lim H(S j |Y11:N W11:N S 1:j−1 ) 1 N →∞ N N →∞ N j∈S lim
U1
= lim
N →∞
1 X 1 H(S j |Y11:N W11:N S 1:j−1 ) H(S 1:2N |Y11:N W11:N ) − N N j∈S 0 U2
1 1 H(S 1:2N W11:N |Y11:N ) − H(W11:N |Y11:N ) N →∞ N N 1 X − H(S j |Y11:N W11:N S 1:j−1 ) N j∈S
= lim
0 U2
1 1 H(S 1:2N |Y11:N ) + lim H(W11:N |Y11:N S 1:2N ) − H(W1 |Y1 ) N →∞ N N →∞ N 1 X − lim H(S j |Y11:N W11:N S 1:j−1 ) N →∞ N j∈S
= lim
0 U2
1 X H(S j |Y11:N S 1:j−1 ) N →∞ N j∈S
= H(X1 W2 |Y1 ) − H(W1 |Y1 ) − lim
(160)
0 U2
¯ 1 (1) = H(X1 W2 |Y1 ) − H(W1 |Y1 ) − H(W2 ) − I(X1 W2 ; Y1 ) + P ¯ 1 (1), = H(X1 ) − H(W1 |Y1 ) − P and
1 (N ) |HX1 |W1 | = H(X1 |W1 ), N →∞ N lim
(161)
1 (N ) |BSU |Y1 W1 | = 0, 1 N →∞ N lim
we have 1 ¯ 1 (1) |I1p | = H(X1 |W1 ) − H(X1 ) − H(W1 |Y1 ) − P N →∞ N ¯ 1 (1) = H(X1 W1 ) − H(W1 ) − H(X1 ) + H(W1 |Y1 ) + P ¯ 1 (1) − I(W1 ; Y1 ), =P (162) 1:N 1:N 1:2N 1 ¯ is on the dominant face of where (160) holds because H(W1 |Y1 S ) = 0, (161) holds because P R(PY1 |X1 W2 ), thus X ¯ 1 (2) = H(W2 ) − lim 1 P H(S j |Y11:N S 1:j−1 ) N →∞ N j∈S lim
0 U2
¯ 1 (1), = I(X1 W2 ; Y1 ) − P and (162) holds because H(X1 W1 ) = H(X1 ). Now we can use the chaining method to achieve P.
28
1) Common Message Encoding: For Sender 1, if Pc (1) = I(W1 ; Y1 ) (e.g., P2 of type (b) and P5 ), let C11 = C10 . Otherwise (e.g., P3 ) choose C11 ⊂ C10 such that |C11 | = N Pc (1). Since lim N1 |C10 | > Pc (1) if N →∞
Pc (1) < I(W1 ; Y1 ), we can alway find such a subset given sufficiently large N . C11 is the information bit ˜ c = Pc (e.g., P3 and P5 ), let C12 = C100 . set of Sender 1’s common message for Receiver 1. Similarly, if P 2 00 2 Otherwise (e.g., P2 of type (b)) choose C1 ⊂ C1 such that |C1 | = N Pc (1). C12 is the information bit set of Sender 1’s common message for Receiver 2. Since C11 and C12 are usually not aligned in the general case, we will use the chaining strategy to achieve the maximum rate. Although N1 lim |C11 | = N1 lim |C12 |, for a finite N , their sizes may be different. If N →∞
N →∞
|C12 | ≥ |C11 |, we choose an arbitrary subset of C12 \ C11 , denoted as C121 , such that |C121 | = |C11 \ C12 |. Otherwise we choose an arbitrary subset of C11 \ C12 , denoted as C112 , such that |C112 | = |C12 \ C11 |. Let I1c = C11 ∩ C12 , ( C11 \ C12 , 1 = I1c C112 , ( C121 , 2 I1c = C12 \ C11 ,
(163) if |C12 | ≥ |C11 | otherwise
(164)
if |C12 | ≥ |C11 | otherwise
(165)
(N )
0 1 2 F1r = HW1 \ (I1c ∪ I1c ∪ I1c ), (N )
0 F1d = (HW1 )C .
(166) (167)
For Sender 2, similar to the case of Sender 1, if |C22 | ≥ |C21 | for a given N , we choose an arbitrary subset of C22 \ C21 , denoted as C221 , such that |C221 | = |C21 \ C22 |. Otherwise we choose an arbitrary subset of C21 \ C22 , denoted as C212 , such that |C212 | = |C22 \ C21 |. Let I2c = C21 ∩ C22 , ( C21 \ C22 , 1 I2c = C212 , ( C221 , 2 I2c = C22 \ C21 ,
(168)
(N )
if |C22 | ≥ |C21 | otherwise
(169)
if |C22 | ≥ |C21 | otherwise
(170)
(1)
(2)
0 F2r = HW2 \ (I2c ∪ I2c ∪ I2c ), (N )
0 F2d = (HW2 )C .
(171) (172)
Let k be the number of blocks used. The encoding procedures for two senders’ common messages can be expressed in the same form as the Type I scheme, and the common message rates are also R1c =
1 |C11 | |I1c | |C 1 | |I 1 | − , R2c = 2 − 2c , N kN N kN
(173)
with lim
N →∞,k→∞
R1c = Pc (1),
lim
N →∞,k→∞
R2c = Pc (2).
2) Encoding for Sender 1’s Private Message: I1p is the information bit set for U11:N . The rest bits are partitioned into two sets as follows: (N )
(N )
F1r = HX1 |W1 ∩ (LSU (N )
F1d = (HX1 |W1 )C .
1
C |Y1 W1 ) ,
(174)
Fig. 13 shows the partition for U11:N . Sender 1 generates its final codewords in the same way as in the Type I scheme.
29
Fig. 13.
Graphical representation of the partition for U11:N of Type II corner points.
3) Encoding for Sender 2’s Private Message: The same as Type I scheme. 0 4) Decoding: Receiver 1 decodes Sender 1’s common message {u1i } from Block 1 to Block k. • In Block 1, Sender 1’s common message is decoded as 0i 0 2 if i ∈ F1r ∪ I1c u1 , 0 1:i−1 0 0 ), if i ∈ F1d u¯1i = arg maxu∈{0,1} PU10 i |U10 1:i−1 (u|u1 (175) 0 1:i−1 arg max 1:N 1 ), if i ∈ I1c ∪ I1c u∈{0,1} PU 0 i |Y 1:N U 0 1:i−1 (u|y1 , u1 1
1
1
0
{ui1 }
Sender 1’s private messages and Sender 2’s common message {u2i } are jointly decoded as i if i ∈ F1r u 1 , 0 1:N 1:i−1 i u1 , u1 ), if i ∈ F1d (176) u¯1 = arg maxu∈{0,1} PU1i |U10 1:N U11:i−1 (u|¯ 0 arg max 1:f (i)−1 1:N 1:N 1 ), if i ∈ I1p ¯1 , s u∈{0,1} PS f1 (i) |Y 1:N U 0 1:N S 1:f1 (i)−1 (u|y1 , u 1 1 0 2 0 i ∪ I2c if i ∈ F2r u 2 , 0 0 0 if i ∈ F2d (177) u¯2i = arg maxu∈{0,1} PU 0 i |U 0 1:i−1 (u|u21:i−1 ), 2 2 1:N 1:f2 (i)−1 1 arg maxu∈{0,1} PS f2 (i) |Y11:N S 1:f2 (i)−1 (u|y1 , s ), if i ∈ I2c ∪ I2c where f1 (i) denotes the mapping from indices of U11:N to indices of S 1:2N , and f2 (i) the mapping 0 from indices of U21:N to indices of S 1:2N . 0 0 0 0 2 and {¯ 2 are deduced from {¯ 1 and {¯ 1 in Block • In Block j (1 < j < k), {¯ u1i }i∈I1c u2i }i∈I2c u1i }i∈I1c u2i }i∈I2c j − 1 respectively, and the rest bits are decoded in the same way as in Block 1. 0 0 1 and {¯ 1 are assigned to the pre-shared value between Sender 1 and the • In Block k, {¯ u1i }i∈I1c u2i }i∈I2c two receivers, and the rest bits are decoded in the same way as in Block j (1 < j < k). Receiver 2 decodes from Block k to Block 1 in the same way as in the Type I scheme. 5) Performance: The block error probability of Receiver 2 in this case can be upper bounded in the same form as in the Type I scheme, while that of Receiver 1 is upper bounded as X X 0 0 0 0 Z(U1i |Y11:N , U11:i−1 ) Pe1 ≤ k Z(U1i |Y11:N , U11:i−1 ) + (k − 1) 1 i∈I1c
i∈I1c
+k
X i∈I1p β
= O(2−N ).
Z(S f1 (i) |Y11:N , S 1:f1 (i)−1 )
(178)
30
VI. C ONCLUSION R EMARKS In this paper, the problem of designing polar codes for the general 2-user DM-IC is considered. By adding two types of joint decoders in Carleial’s superposition coding scheme for the IC [32], we have shown that our schemes can achieve all corner points of the Han-Kobayashi region. The reason why this approach works is the fact that the fully joint decoder required by the Han-Kobayashi coding scheme that jointly decodes all three messages of a receiver can be replaced with two types of partially joint decoders that handle only two of them jointly. By extending the monotone chain rule expansion based 2-user MAC polarization method to non-uniform channel input distribution case, our proposed scheme can deal with arbitrary input distribution directly. Besides, our proposed scheme enjoys the major advantage that no deterministic functions which map auxiliary random variables to actual channel inputs are needed to be known, which also makes our schemes simpler and more practical. The chaining method we used and the polar alignment technique used in [39] both make polar coding schemes lengthy, since both methods result in increasing of block length. It is shown in [46] that the non-universality of polar codes is a property of the successive cancellation decoding algorithm. Under maximum likelihood (ML) decoding, polar codes are universal. This makes us wonder if there exist decoding algorithms which maintain universality while still enjoying low complexity. If there do, our proposed scheme may be further simplified as well as polar coding schemes for some other multi-user channels. APPENDIX A From (18) and (20) we have e − c = I(X1 W2 ; Y1 |W1 ) + I(X2 W1 ; Y2 |W2 ) − I(X1 W2 ; Y1 ) − I(X2 ; Y2 |W1 W2 ) = I(X1 W2 ; Y1 |W1 ) − I(X1 W2 ; Y1 ) + I(W1 ; Y2 |W2 ) = I(X1 W2 ; Y1 |W1 ) − I(X1 W1 W2 ; Y1 ) + I(W1 ; Y2 |W2 ) = I(W1 ; Y2 |W2 ) − I(W1 ; Y1 ) = I1 .
(179) (180)
(179) holds because I(X1 W2 ; Y1 ) = H(Y1 ) − H(Y1 |X1 W2 ) = H(Y1 ) − H(Y1 |X1 W1 W2 ) = I(X1 W1 W2 ; Y1 ). Similarly, e − d = I(W2 ; Y1 |W1 ) − I(W2 ; Y2 ) = I2 .
(181)
Then from (180) and (181) we have d − c = I1 − I2 = I(W1 W2 |Y2 ) − I(W1 W2 |Y1 ). From (22), (23) and (24) we have f + g − 3e = I(X1 W2 ; Y1 ) + I(X1 ; Y1 |W1 W2 ) − 2I(X2 W1 ; Y2 |W2 ) + I(X2 ; Y2 |W1 W2 ) + I(X2 W1 ; Y2 ) − 2I(X1 W2 ; Y1 |W1 ) = I(X1 W2 ; Y1 ) − I(W1 ; Y2 |W2 ) − I(X2 W1 ; Y2 |W2 ) + (X2 W1 ; Y2 ) − I(W2 ; Y1 |W1 ) − I(X1 W2 ; Y1 |W1 ) = I(W1 ; Y1 ) + I(W2 ; Y2 ) − I(W1 ; Y2 |W2 ) − I(W2 ; Y1 |W1 ) = −I1 − I2 ,
(182)
31
where (182) holds because I(X1 W2 ; Y1 ) = I(X1 W1 W2 ; Y1 ) = I(X1 W2 ; Y1 |W1 ) + I(W1 ; Y1 ) and I(X2 W1 ; Y2 ) = I(X2 W1 W2 ; Y2 ) = I(X2 W1 ; Y2 |W2 ) + I(W2 ; Y2 ). From (20), (23) and (24) we have f + g − 3c = −2I(X1 W2 ; Y1 ) + I(X1 ; Y1 |W1 W2 ) + I(X2 W1 ; Y2 |W2 ) − 2I(X2 ; Y2 |W1 W2 ) + I(X2 W1 ; Y2 |) + I(X1 W2 ; Y1 |W1 ) = −2I(X1 W2 ; Y1 ) + 2I(X1 ; Y1 |W1 W2 ) + I(W1 ; Y2 |W2 ) − I(X2 ; Y2 |W1 W2 ) + I(X2 W1 ; Y2 ) + I(W2 ; Y1 |W1 ) = −2I(W1 W2 ; Y1 ) + I(W1 ; Y2 |W2 ) + I(W1 W2 ; Y2 ) + I(W2 ; Y1 |W1 ) = 2I(W1 ; Y2 |W2 ) − 2I(W1 ; Y1 ) − I(W2 ; Y1 |W1 ) + I(W2 ; Y2 ) = 2I1 − I2 . Similarly, f + g − 3d = 2I2 − I1 . From the above analysis we can see that, if I1 ≥ 0 and I1 ≥ I2 , then c ≤ d, e, 1/3(f + g), which means R1 + R2 ≤ c is the tightest upper bound on the sum rate. Similarly, if I2 ≥ 0 and I2 ≥ I1 , R1 + R2 ≤ d is the tightest upper bound on the sum rate. If I1 < 0 and I2 < 0, then e < c, d, 1/3(f + g), which means R1 + R2 ≤ e is the tightest upper bound on the sum rate. APPENDIX B (1) In Case 1, if a + b − c ≤ 0, i.e., I(X1 ; Y1 |W2 ) + I(X2 ; Y2 |W1 ) − I(X1 W2 ; Y1 ) − I(X2 ; Y2 |W1 W2 ) = I(W2 ; Y2 |W1 ) − I(W2 ; Y1 ) ≤ 0,
(183)
where (183) is because I(X2 ; Y2 |W1 ) = I(X2 W2 ; Y2 |W1 ), the Han-Kobayashi region reduces to a rectangle formed by lines R1 = 0, R2 = 0, R1 = a and R2 = b. Otherwise, if I(W2 ; Y2 |W1 ) − I(W2 ; Y1 ) > 0, the analysis is as follows. Let Pa1 , (a, c − a) be the intersection point of line R1 = a and R1 + R2 = c, Pa2 , (a, f − 2a) the intersection point of line R1 = a and 2R1 + R2 = f , Pa3 , (a, 12 (g − a)) the intersection point of line R1 = a and R1 + 2R2 = g, Pb1 , (c − b, b) the intersection point of line R1 + R2 = c and R2 = b, Pb2 , (g − 2b, b) the intersection point of line R1 + 2R2 = g and R2 = b, and Pb3 , ( 12 (f − b), b) the intersection point of line 2R1 + R2 = f and R2 = b. To compare their positions, we have 1 Pa3 − Pa1 = (0, (g + a − 2c)), 2 Pa2 − Pa1 = (0, f − a − c), 1 Pb3 − Pb1 = ( (f + b − 2c), 0), 2 Pb2 − Pb1 = (g − b − c, 0).
32
From (18), (20) and (24) we have 1 (g + a − 2c) = −I(X2 ; Y2 |W1 W2 ) + I(X2 W1 ; Y2 ) + I(X1 W2 ; Y1 |W1 ) 2 + I(X1 ; Y1 |W2 ) − 2I(X1 W2 ; Y1 ) = I(W1 W2 ; Y2 ) − I(W1 ; Y1 ) − I(W2 ; Y1 ) ≥ I(W1 W2 ; Y2 ) − I(W1 ; Y1 ) − I(W2 ; Y1 |W1 ) = I(W1 W2 ; Y2 ) − I(W1 W2 ; Y1 ). Because in Case 1, I(W1 W2 ; Y2 ) − I(W1 W2 ; Y1 ) = I1 − I2 ≥ 0, we know that Pa3 can not be an corner point. From (19), (20) and (23) we have 1 (f + b − 2c) = I(X1 ; Y1 |W1 W2 ) + I(X2 W1 ; Y2 |W2 ) + I(X2 ; Y2 |W1 ) 2 − I(X1 W2 ; Y1 ) − 2I(X2 ; Y2 |W1 W2 ) = I(W1 ; Y2 |W2 ) + I(W2 ; Y2 |W1 ) − I(W1 W2 ; Y1 ) ≥ I(W1 ; Y2 |W2 ) + I(W2 ; Y2 ) − I(W1 W2 ; Y1 ) = I(W1 W2 ; Y2 ) − I(W1 W2 ; Y1 ). Similar to the case of Pa3 , we know that Pb3 also can not be an corner point. From (18), (20) and (23) we have f − a − c = I(X1 ; Y1 |W1 W2 ) + I(X2 W1 ; Y2 |W2 ) − I(X2 ; Y2 |W1 W2 ) − I(X1 ; Y1 |W2 ) = I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ). If I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) ≥ 0, then Pa1 (= P1 ) is a corner point (see Case 1-1 and 1-2 in Fig. 1). Otherwise, if I(W1 ; Y2 |W2 ) − I(W1 ; Y1 |W2 ) < 0, Pa2 (= P4 ) is a corner point (see Case 1-3 and 1-4 in Fig. 1). In this case, the intersection point of line R1 + R2 = c and 2R1 + R2 = f , P6 , (f − c, 2c − f ), is also a corner point (see Case 1-3 and 1-4 in Fig. 1). From (19), (20) and (24) we have g − b − c = I(X2 W1 ; Y2 ) + I(X1 W2 ; Y1 |W1 ) − I(X2 ; Y2 |W1 ) − I(X1 W2 ; Y1 ) = I(W1 ; Y2 ) − I(W1 ; Y1 ). If I(W1 ; Y2 ) − I(W1 ; Y1 ) ≥ 0, then Pb1 (= P2 ) is a corner point (see Case 1-1 and 1-3 in Fig. 1). Otherwise, if I(W1 ; Y2 ) − I(W1 ; Y1 ) < 0, Pb2 (= P3 ) is a corner point (see Case 1-2 and 1-4 in Fig. 1). In this case, the intersection point of line R1 + R2 = c and R1 + 2R2 = g, P5 , (2c − g, g − c), is also a corner point (see Case 1-2 and 1-4 in Fig. 1). (2) Because Case 2 and Case 1 are symmetric by swapping the roles of two receivers, now we consider Case 3. Since a + b − e = I(W1 ; Y1 |W2 ) + I(W2 ; Y2 |W1 ) − I(W2 ; Y1 |W1 ) − I(W1 ; Y2 |W2 ) ≥ I(W1 ; Y1 ) + I(W2 ; Y2 ) − I(W2 ; Y1 |W1 ) − I(W1 ; Y2 |W2 ) = −I1 − I2 > 0, we know that the Han-Kobayashi region will not reduce to a rectangle in this case. Let Pe1 , (a, e − a) be the intersection point of line R1 = a and R1 + R2 = e, Pe2 , (a, f − 2a) be the intersection point of line R1 = a and 2R1 + R2 = f , and Pe3 , (a, 12 (g − a)) be the intersection point of line R1 + 2R2 = g and R1 + R2 = e. To compare their positions, we have 1 Pe3 − Pa1 = 0, (g + a − 2e) , 2 Pe2 − Pa1 = (0, f − a − e).
33
From (18), (22), (23) and (24) we have 1 (g + a − 2e) = I(X2 ; Y2 |W1 W2 ) + I(X2 W1 ; Y2 ) − I(X1 W2 ; Y1 |W1 ) 2 + I(X1 ; Y1 |W2 ) − 2I(X2 W1 ; Y2 |W2 ) = I(W2 ; Y2 ) − I(W1 ; Y2 |W2 ) + I(W1 ; Y1 |W2 ) − I(W2 ; Y1 |W1 ) ≥ I(W2 ; Y2 ) − I(W1 ; Y2 |W2 ) + I(W1 ; Y1 ) − I(W2 ; Y1 |W1 ) = −I1 − I2 > 0, and f − a − e = I(X1 W2 ; Y1 ) + I(X1 ; Y1 |W1 W2 ) − I(X1 ; Y1 |W2 ) − I(X1 W2 ; Y1 |W1 ) = I(W2 ; Y1 ) − I(W2 ; Y1 |W1 ) ≤ 0. Thus, Pe2 (= P7 ) is a corner point. Similarly, the intersection point of line R2 = b and R1 + 2R2 = g, P8 , (g − 2b, b), is also a corner point (see Case 3 in Fig. 1). Let P9 , (f − e, 2e − f ) be the intersection point of line R1 + R2 = e and 2R1 + R2 = f , and P10 , (2e − g, g − e) be the intersection point of line R1 + R2 = e and R1 + 2R2 = g. Then they are the other two corner points in Case 3 as shown in Fig. 1. APPENDIX C We present analysis of the common/private message rate-tuples at each corner point of the HanKobayashi region from a different angle in this appendix, which provides a more intuitive explanation why our coding strategy works. We only consider Case 1-1, 1-4 and 3 since they contain all possible types of corner points. Case 1-2, 1-3 and 2 can be analyzed similarly. Fig. 14 is an illustration of the original Han-Kobayashi region (Theorem 1) of Case 1-4. The green and blue polyhedrons with thin lines (mostly covered by the thick ones) respectively represent the achievable rate regions of the two 3-user MACs, and the polyhedron with thick lines of both colors represents the compound region, i.e., the Han-Kobayashi region. The area with cyan dash line border on plane T1 T2 is their compound area for common messages. The two regions on plane T1 T2 with light green and light blue borders are the so-called stand-alone common message (SACM) rate regions, which are projections of two pentagons A and B (marked gray in Fig. 14, standing for the achievable common message rate regions of two senders when the private message rates are maximum) on plane T1 T2 respectively. In this case, the corner points are P3 , P4 , P5 and P6 . For i = 3, 4, 5, 6, let Pci be the corresponding common message rate pair, which is the same as in Section IV-C, and let Q1i and Q2i be the corresponding rate triples with respect to Receiver 1 and Receiver 2 respectively. Draw a line parallel to S1 S2 through Pci , then Q1i and Q2i are the intersection points with surfaces of achievable rate regions of the two 3-user MACs. Let R1 = Q1i (S1 ) + Q1i (T1 ) and R2 = Q2i (S2 ) + Q2i (T2 ), where Q1i (S1 ) means Q1i ’s projection on S1 , etc. Obviously Q1i (T1 ) = Q2i (T1 ) and Q1i (T2 ) = Q2i (T2 ). Then finding the corner points of the Han-Kobayashi region amounts to finding maximal extreme points P = (R1 , R2 ). One can easily verify that for a corner point Pi , either Q1i or Q2i must lie on the edges of pentagon A or B, otherwise we can always find larger points nearby. This explains why Pcj always lies on the edge of one receiver’s SACM region since the SACM regions are projections of A and B on plane T1 T2 , and therefore at least one receiver can first jointly decode two common messages and then decode the private message. In Case 1-4 as Fig. 14 shows, I(W1 ; Y2 ) ≤ I(W1 ; Y1 ) and I(W2 ; Y1 ) ≤ I(W2 ; Y2 ), thus we have c Q3 (T1 ) ≤ I(W1 ; Y1 ), Qc5 (T1 ) ≤ I(W1 ; Y1 ) and Qc4 (T2 ) ≤ I(W2 ; Y2 ). Therefore P3 , P4 and P5 are of Type II. And since Pc6 lies also on the edge of the other receiver’s SACM region, P6 is of Type I. Next
34
Fig. 14.
Corner points of the Han-Kobayashi region in Case 1-4.
Fig. 15.
Corner points of the Han-Kobayashi region in Case 1-1.
consider the case when I(W1 ; Y2 ) > I(W1 ; Y1 ). As an example, consider Case 1-1 as shown in Fig. 15. In 0 this figure, P1 is of Type I, and P2 is of Type II (see Fig. 4 and 5 (b)). Let P2c be the upper corner point of 01 02 R(PY2 |W1 W2 ), Q2 and Q2 be the corresponding rate triples of two receivers, and P02 be the corresponding rate pair of the system. If P02 in this case is another corner point, then it can not be achieved by our 0 0 scheme. Q21 in this case lies on the dominant face of R(PY1 |W1 W2 V1 ). If we move P2c left along the edge
35
Fig. 16.
Corner points of the Han-Kobayashi region in Case 3.
0
0
of R(PY2 |W1 W2 ) until it reaches Pc2 , the corresponding new P02 will be invariant since Q22 (S2 ), Q22 (T2 ) 0 0 0 0 and Q21 (S1 )+Q21 (T1 )+Q21 (T2 ) are all invariant. Thus, Pc2 and P2c together with the points in between in this case represent one same corner point. This accords with our analysis in Section IV-C for P2 of type 0 (b). We can similarly show that Pc1 and P1c together with the points in between in this case also represent 0c a same corner point, where P1 = (I(W1 ; Y2 |W2 ), Pc1 (T2 )). The case when I(W2 ; Y1 ) > I(W2 ; Y2 ) can be similarly analyzed. The analysis for Case 3 is similar to that for Case 1-4, so we only present a graphical illustration as shown in Fig. 16 without extra explanation. Now we have shown from another point of view that in any case we can always find proper common/private message rate-tuples suitable for our proposed coding scheme to achieve all corner points of the Han-Kobayashi region. Next we consider the private message rates of two senders. For a corner point P of Type I, the corresponding rate triples of two receivers, Q1 and Q2 , lie on pentagons A and B respectively, therefore the private message rates are I(X1 ; Y1 |W1 W2 ) and I(X2 ; Y2 |W1 W2 ), which accords with our conclusion in the beginning of Section V-A. For a corner point P of Type II, without loss of generality, suppose the suitable common message rate pair Pc lies on the edge of R(PY2 |W1 W2 ). Then the corresponding rate triple of Receiver 2, Q2 , will lie on pentagon B, which means the private message rate of Sender 2 is I(X2 ; Y2 |W1 W2 ). Since Pc (T1 ) ≤ I(W1 ; Y1 ), the corresponding rate triple of Receiver 1, Q1 , will lie on the bevel face of R(PY1 |W1 W2 V1 ) on which the sum rate of S1 and T2 are maximum, i.e., S1 + T2 = I(X1 W2 ; Y1 |W1 ) (see Q13 and Q15 in Fig. 14, Q12 in Fig. 15, and Q18 and Q110 in Fig. 16). Let ¯ 1 = (I(W1 ; Y1 ), Q1 (T2 ), Q1 (S1 )). Q ¯ 1 is on the intersection line of this bevel face and the dominant face Q ¯ 1 is a possible rate decomposition of P ¯ 1 defined in Section V-B, of R(PY1 |W1 W2 V1 ). We can see that Q ¯ 1 (T2 ) + Q ¯ 1 (S1 ), which ¯ 1 (2) = Q ¯ 1 (T2 ) = Q1 (T2 ) and P ¯ 1 (1) + P ¯ 1 (2) = I(X1 W2 ; Y1 ) = Q ¯ 1 (T1 ) + Q since P 1 1 1 1 1 ¯ (S1 ) = P ¯ (1) − Q ¯ (T1 ) = P ¯ (1) − I(W1 ; Y1 ), and means Sender 1’s private message rate is Q (S1 ) = Q (156) is verified. R EFERENCES [1] E. Arikan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3051–3073, 2009.
36
[2] ——, “Source polarization,” in 2010 IEEE International Symposium on Information Theory, 2010, pp. 899–903. ´ ´ ERALE ´ [3] S. B. Korada, “Polar codes for channel and source coding,” Ph.D. dissertation, ECOLE POLYTECHNIQUE FED DE LAUSANNE, 2009. [4] S. B. Korada and R. L. Urbanke, “Polar codes are optimal for lossy source coding,” IEEE Transactions on Information Theory, vol. 56, no. 4, pp. 1751–1768, 2010. [5] E. Arikan, “Polar coding for the Slepian-Wolf problem based on monotone chain rules,” in 2012 IEEE International Symposium on Information Theory Proceedings (ISIT), July 2012, pp. 566–570. [6] M. Andersson, V. Rathi, R. Thobaben, J. Kliewer, and M. Skoglund, “Nested polar codes for wiretap and relay channels,” IEEE Communications Letters, vol. 14, no. 8, pp. 752–754, 2010. [7] H. Mahdavifar and A. Vardy, “Achieving the secrecy capacity of wiretap channels using polar codes,” IEEE Transactions on Information Theory, vol. 57, no. 10, pp. 6428–6443, 2011. [8] O. O. Koyluoglu and H. El Gamal, “Polar coding for secure transmission and key agreement,” IEEE Transactions on Information Forensics and Security, vol. 7, no. 5, pp. 1472–1483, 2012. [9] E. S¸as¸o˘glu and A. Vardy, “A new polar coding scheme for strong security on wiretap channels,” in 2013 IEEE International Symposium on Information Theory Proceedings (ISIT), July 2013, pp. 1117–1121. [10] T. Gulcu and A. Barg, “Achieving secrecy capacity of the wiretap channel and broadcast channel with a confidential component,” in Information Theory Workshop (ITW), 2015 IEEE, April 2015, pp. 1–5. [11] Y. P. Wei and S. Ulukus, “Polar coding for the general wiretap channel with extensions to multiuser scenarios,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 2, pp. 278–291, 2016. [12] M. Karzand, “Polar codes for degraded relay channels,” in Proceedings of the International Zurich Seminar on Communications, 2012, pp. 59–62. [13] R. Blasco-Serrano, R. Thobaben, M. Andersson, V. Rathi, and M. Skoglund, “Polar codes for cooperative relaying,” IEEE Transactions on Communications, vol. 60, no. 11, pp. 3263–3273, 2012. [14] S. Onay, “Successive cancellation decoding of polar codes for the two-user binary-input mac,” in 2013 IEEE International Symposium on Information Theory Proceedings (ISIT), July 2013, pp. 1122–1126. [15] E. S¸as¸o˘glu, E. Telatar, and E. Yeh, “Polar codes for the two-user multiple-access channel,” IEEE Transactions on Information Theory, vol. 59, no. 10, pp. 6583–6592, Oct 2013. [16] E. Abbe and I. Telatar, “Polar codes for the m-user multiple access channel,” IEEE Transactions on Information Theory, vol. 58, no. 8, pp. 5437–5448, Aug 2012. [17] H. Mahdavifar, M. El-Khamy, J. Lee, and I. Kang, “Achieving the uniform rate region of general multiple access channels by polar coding,” IEEE Transactions on Communications, vol. 64, no. 2, pp. 467–478, 2016. [18] N. Goela, E. Abbe, and M. Gastpar, “Polar codes for broadcast channels,” IEEE Transactions on Information Theory, vol. 61, no. 2, pp. 758–782, 2015. [19] M. Mondelli, S. H. Hassani, I. Sason, and R. L. Urbanke, “Achieving Marton’s region for broadcast channels using polar codes,” IEEE Transactions on Information Theory, vol. 61, no. 2, pp. 783–800, 2015. [20] R. A. Chou and M. R. Bloch, “Polar coding for the broadcast channel with confidential messages: A random binning analogy,” IEEE Transactions on Information Theory, vol. 62, no. 5, pp. 2410–2429, 2016. [21] M. Andersson, R. Schaefer, T. Oechtering, and M. Skoglund, “Polar coding for bidirectional broadcast channels with common and confidential messages,” IEEE Journal on Selected Areas in Communications, vol. 31, no. 9, pp. 1901–1908, September 2013. [22] C. E. Shannon, “Two-way communication channels,” in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, 1961, pp. 611–644. [23] R. Ahlswede, “The capacity region of a channel with two senders and two receivers,” The Annals of Probability, vol. 2, no. 5, pp. 805–814, 1974. [24] A. Carleial, “A case where interference does not reduce capacity,” IEEE Transactions on Information Theory, vol. 21, no. 5, pp. 569–570, 1975. [25] H. Sato, “The capacity of the Gaussian interference channel under strong interference,” IEEE Transactions on Information Theory, vol. 27, no. 6, pp. 786–788, 1981. [26] S. T. Chung and J. M. Cioffi, “The capacity region of frequency-selective Gaussian interference channels under strong interference,” IEEE Transactions on Communications, vol. 55, no. 9, pp. 1812–1821, 2007. [27] N. Liu and S. Ulukus, “The capacity region of a class of discrete degraded interference channels,” IEEE Transactions on Information Theory, vol. 54, no. 9, pp. 4372–4378, 2008. [28] R. Benzel, “The capacity region of a class of discrete additive degraded interference channels,” IEEE Transactions on Information Theory, vol. 25, no. 2, pp. 228–231, 1979. [29] H. F. Chong and M. Motani, “The capacity region of a class of semideterministic interference channels,” IEEE Transactions on Information Theory, vol. 55, no. 2, pp. 598–603, 2009. [30] A. E. Gamal and M. Costa, “The capacity region of a class of deterministic interference channels,” IEEE Transactions on Information Theory, vol. 28, no. 2, pp. 343–346, 1982. [31] M. Costa and A. E. Gamal, “The capacity region of the discrete memoryless interference channel with strong interference,” IEEE Transactions on Information Theory, vol. 33, no. 5, pp. 710–711, 1987. [32] A. Carleial, “Interference channels,” IEEE Transactions on Information Theory, vol. 24, no. 1, pp. 60–70, 1978. [33] T. Cover, “An achievable rate region for the broadcast channel,” IEEE Transactions on Information Theory, vol. 21, no. 4, pp. 399–404, 1975. [34] T. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Transactions on Information Theory, vol. 27, no. 1, pp. 49–60, 1981.
37
[35] H. F. Chong, M. Motani, H. K. Garg, and H. E. Gamal, “On the Han-Kobayashi region for the interference channel,” IEEE Transactions on Information Theory, vol. 54, no. 7, pp. 3188–3195, 2008. [36] S. Sharifi, A. K. Tanc, and T. M. Duman, “Implementing the Han-Kobayashi scheme using low density parity check codes over Gaussian interference channels,” IEEE Transactions on Communications, vol. 63, no. 2, pp. 337–350, 2015. [37] ——, “LDPC code design for binary-input binary-output z interference channels,” in 2015 IEEE International Symposium on Information Theory (ISIT), 2015, pp. 1084–1088. [38] K. Appaiah, O. O. Koyluoglu, and S. Vishwanath, “Polar alignment for interference networks,” in Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on, 2011, pp. 240–246. [39] L. Wang and E. S¸as¸o˘glu, “Polar coding for interference networks,” CoRR, vol. abs/1401.7293, 2014. [Online]. Available: http://arxiv.org/abs/1401.7293 [40] C. Hirche, C. Morgan, and M. M. Wilde, “Polar codes in network quantum information theory,” IEEE Transactions on Information Theory, vol. 62, no. 2, pp. 915–924, 2016. [41] R. G. Gallager, Information theory and reliable communication. Wiley: New York, 1968. [42] J. Honda and H. Yamamoto, “Polar coding without alphabet extension for asymmetric models,” IEEE Transactions on Information Theory, vol. 59, no. 12, pp. 7829–7838, 2013. [43] S. H. Hassani and R. Urbanke, “Universal polar codes,” in 2014 IEEE International Symposium on Information Theory, 2014, pp. 1451–1455. [44] E. S¸as¸o˘glu and L. Wang, “Universal polarization,” IEEE Transactions on Information Theory, vol. 62, no. 6, pp. 2937–2946, 2016. [45] T. M. Cover and J. A. Thomas, Elements of information theory. John Wiley & Sons, 2012. ´ ´ ERALE ´ [46] E. S¸as¸o˘glu, “Polar coding theorems for discrete systems,” Ph.D. dissertation, ECOLE POLYTECHNIQUE FED DE LAUSANNE, 2011.
