Approximate Capacity of a Class of Partially Connected Interference Channels

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by  Muryong Kim

1
Approximate Capacity of a Class of
Partially Connected Interference Channels
Muryong Kim, Yitao Chen, and Sriram Vishwanath, Senior Member, IEEE
Abstract—We derive inner and outer bounds on the capacity based on rate-splitting, lattice alignment, and successive de-
region for a class of three-user partially connected interference coding.
channels. We focus on the impact of topology, interference
alignment, and interplay between interference and noise. The
7 representative channels we consider are the ones that have clear B. Related Work
interference alignment gain. For these channels, Z-channel type
1
Lattice coding based on nested lattices is shown to achieve
outerboundsaretighttowithinaconstantgapfromcapacity.We
0
present near-optimal achievable schemes based on rate-splitting the capacity of the single user Gaussian channel in [12], [27].
2
and lattice alignment. The idea of lattice-based interference alignment by decoding
b
the sum of lattice codewords appeared in the conference
e Index Terms—Interference channel, interference alignment,
F nested lattice code, side information graph, topological interfer- version of [4]. This lattice alignment technique is used to
ence management. derive capacity bounds for three-user interference channel in
3
[2], [3]. The idea of decoding the sum of lattice codewords is
1
also used in [13]–[15] to derive the approximate capacity of
I. INTRODUCTION
] the two-way relay channel. An extended approach, compute-
T A. Motivation
and-forward [16], [17] enables to first decode some linear
I
. ThecapacityoftheInterferencechannelremainsoneofthe combinations of lattice codewords and then solve the lattice
s
most challenging open problems in the domain of network equation to recover the desired messages. This approach is
c
[ information theory. The capacity region is not known in alsousedin[7]tocharacterizeapproximatesum-ratecapacity

general, except for a specific range of channel parameters. of the fully connected K-user interference channel.
2
For the two-user scalar Gaussian interference channel, where The idea of sending multiple copies of the same sub-
v
6 the interference alignment is not required, the approximate message at different signal levels, so-called Zigzag decoding,
7 capacity region to within one bit is known [1]. For the appeared in [5] where receivers collect side information and
5 channels where interference alignment is required such as the use them for interference cancellation.
7
K-user Gaussian interference channel [2]–[5], [7], [11] and The K-user cyclic Gaussian interference channel is con-
0
the Gaussian X-channel [9]–[11], a tight characterization of sidered in [6] where an approximate capacity for the weak
.
1
the capacity region is not known, even for symmetric channel interference regime (SNR INR for all k) and the exact
0 k ≥ k
cases. capacityforthestronginterferenceregime(SNR INR for
7 k k
≤
1 Atractableapproachtothecapacityofinterferencechannels allk)arederived.Ourtype4and5channelsareK =3cases
: is to consider partial connectivity of interference links and in mixed interference regimes, which were not considered in
v
analyze the impact of topology on the capacity. Topological [6].
i
X interference management [8] approach gives important in-
r sightson thedegrees-of-freedom (DoF)of partiallyconnected
a C. Main Results
interference channels and their connection to index coding
problems [18]–[25]. It is shown that the symmetric DoF of We consider five channel types defined in Table I and
a partially connected interference channel can be found by described in Fig. 1 (a)–(e). Each channel type is a partially
solving the corresponding index coding problem. connected three-user Gaussian interference channel. Each
conInnetchtiesdpinatpeerrf,erwenececocnhsaindneerlsaacnldascshaorfacttherreiez-euaseprprpoaxritmiaalltye tLraentsumsidtteenroties tshuebjneocitsetovaproiawnecrecboynsNtrkai=ntEE[[ZXk2k2].]W≤itPhoku=tloPss.
capacity regions at finite SNR. We focus on the impact of of generality, we assume that N1 N2 N3.
≤ ≤
interferencetopology,interferencealignment,andinterplaybe- Definition1(sideinformationgraph):Thesideinformation
tween interference and noise. We choose a few representative graph representation of an interference channel satisfies the
topologies where we can achieve clear interference alignment following.
gain. For these topologies, Z-channel type outer bounds are A node represents a transmitter-receiver pair, or equiva-
•
tight to within a constant gap from the corresponding inner lently, the message.
bound. For each topology, we present an achievable scheme There is a directed edge from node i to node j if
•
transmitter i does not interfere at receiver j.
The authors are with the University of Texas at Austin, Austin,
The side information graphs for five channel types are de-
TX 78701 USA (e-mail: [email protected], [email protected], sri-
[email protected]). scribed in Fig. 1 (f)–(j). We state the main results in the
2
Type Channelmodel
II. CAPACITYOUTERBOUNDS
Y1=X1+X2+Z1
1 Y2=X1+X2+X3+Z2 We prove the capacity outer bound in Theorem 1 for each
Y3=X2+X3+Z3
Y1=X1+X2+X3+Z1 channel type. The result is summarized in Table II. The shape
2 Y2=X1+X2+Z2 oftheouterboundregionisillustratedinFig.2.Forallchannel
Y3=X1+X3+Z3 types, we assume P =P =P =P and N N N .
Y1=X1+X3+Z1 1 2 3 1 ≤ 2 ≤ 3
3 Y2=X2+X3+Z2
Y3=X1+X2+X3+Z3
Y1=X1+X3+Z1 A. Channel Type 1
4 Y2=X1+X2+Z2
Y3=X2+X3+Z3 In this section, we present an outer bound on the capacity
Y1=X1+X2+Z1 region of Type 1 channel defined by
5 Y2=X2+X3+Z2
Y3=X1+X3+Z3       
TABLEI Y 1 1 0 X Z
1 1 1
FIVECHANNELTYPES  Y2 = 1 1 1  X2 + Z2 .
Y 0 1 1 X Z
3 3 3
We state the outer bound in the following theorem.
following two theorems, of which the proofs will be given Theorem 3: The capacity region of Type 1 channel is
in the main body of the paper. contained in the following outer bound region:
Theorem 1 (Capacity region outer bound): For the five
channel types, if (R1,R2,R3) is achievable, it must satisfy Rk ≤Ck, k(cid:18)=1,2,3(cid:19) (cid:18) (cid:19)
1 P 1 2P +N
(cid:88) 1 (cid:18) P (cid:19) R1+R2 log 1+ + log 2
Rj log 1+ |K| (1) ≤ 2 N1 2 P +N2
j∈K ≤ 2 minj∈K{Nj} R2+R3 1log(cid:18)1+ P (cid:19)+ 1log(cid:18)2P +N3(cid:19).
≤ 2 N 2 P +N
foreverysubset ofthenodes 1,2,3 thatdoesnotinclude 2 3
K { }
a directed cycle in the side information graph over the subset. Proof: The individual rate bounds are obvious. We pro-
Theorem 2 (Capacity region to within one bit): ceed to sum-rate bounds.
For any rate triple (R ,R ,R ) on the boundary of the outer
1 2 3
boundregion,thepoint(R 1,R 1,R 1)isachievable. n(R +R (cid:15))
1− 2− 3− 1 2−
I(Xn;Yn)+I(Xn;Yn)
≤ 1 1 2 2
I(Xn;Yn Xn)+I(Xn;Yn Xn)
D. Paper Organization and Notation ≤ 1 1 | 2 2 2 | 3
=h(Yn Xn) h(Yn Xn,Xn)
The capacity outer bounds are derived in Section II. The 1 | 2 − 1 | 1 2
+h(Yn Xn) h(Yn Xn,Xn)
innerboundsforeachchanneltypeandthecorrespondinggap 2 | 3 − 2 | 2 3
analysis are given in Section III, IV, V, VI, VII, respectively. =h(Xn+Zn) h(Zn)
1 1 − 1
Section VIII concludes the paper. While lattice coding-based +h(Xn+Xn+Zn) h(Xn+Zn)
achievablerateregionsforchanneltypes4and5arepresented (cid:18)1 2 (cid:19) 2 − (cid:18) 1 2 (cid:19)
n P +N n 2P +N
1 2
in Section VI and VII, random coding achievability is given log + log
≤ 2 N 2 P +N
in Appendix. 1 2
Signal x is a coded version of message M with code where the first inequality is by Fano’s inequality, the second
ij ij
rate R unless otherwise stated. The single user capacity at inequality due to the independence of X ,X ,X . The third
ij (cid:16) (cid:17) 1 2 3
receiver k is denoted by C = 1log 1+ P . Let denote inequalityholdsfromthefactthatGaussiandistributionmaxi-
k 2 Nk C mizesdifferentialentropyandthath(Xn+Zn) h(Xn+Zn)
the capacity region of an interference channel. Also, let i 1 1 − 1 2
R is also maximized by Gaussian distribution. Similarly,
and denotethecapacityinnerboundandthecapacityouter
o
R
bound, respectively. Thus, . Let δ denote the
i o k
R ⊂ C ⊂ R n(R +R (cid:15))
gapontherateRk between i and o.Letδjk denotethegap 2 3−
on the sum-rate Rj +Rk bRetweenRRi and Ro. For example, ≤I(X2n;Y2n)+I(X3n;Y3n)
if I(Xn;Yn Xn,Xn)+I(Xn;Yn)
≤ 2 2 | 1 3 3 3
=h(Yn Xn,Xn) h(Yn Xn,Xn,Xn)
i = (Rj,Rk):Rk Lk,Rj +Rk Ljk (2) 2 | 1 3 − 2 | 1 2 3
R { ≤ ≤ } +h(Yn) h(Yn Xn)
o = (Rj,Rk):Rk Uk,Rj +Rk Ujk , (3) 3 − 3 | 3
R { ≤ ≤ } =h(Xn+Zn) h(Zn)
2 2 − 2
thenδk =Uk Lk andδjk =Ujk Ljk.Forsideinformation +h(Xn+Xn+Zn) h(Xn+Zn)
graph, we us−e graph notation of−[23]. For example, = (cid:18)2 3 (cid:19) 3 − (cid:18) 2 3 (cid:19)
1 n P +N n 2P +N
G 2 3
(13),(2),(31) means that node 1 has an incoming edge log + log .
f{rom| node3,t|hat}node2hasnoincomingedge,andthatnode ≤ 2 N2 2 P +N3
3 has an incoming edge from node 1.
3
1 1 1 1 11 11 11 11 1 1
2 2 2 2 22 22 22 22 2 2
3 3 3 3 33 33 33 33 3 3
(a) Type1 (b) Type2 (c) Type3 (d) Type4 (e) Type5
1 1 1 1 11 11 1 1
2 1 2 2 1 2 22 1 22 2 1 2 1
3 3 3 3 33 33 3 3
1 1 1 1
2 3 2 3 2 3 2 3 2 3
2 2 2 2
(f) G1 3 (g) G2 3 3 (h) G3 3 (i) G4 (j) G5
Fig. 1. Five channel types and their side information graphs: G1 = {(1|3),(2),(3|1)}, G2 = {(1),(2|3),(3|2)}, G3 = {(1|2),(2|1),(3)}, G4 =
{(1|2),(2|3),(3|1)},andG5={(1|3),(2|1),(3|2)}.
B. Channel Type 2 n(R +R (cid:15))
1 3
−
I(Xn;Yn)+I(Xn;Yn)
≤ 1 1 3 3
In this section, we present an outer bound on the capacity I(Xn;Yn Xn,Xn)+I(Xn;Yn)
≤ 1 1 | 2 3 3 3
region of Type 2 channel defined by =h(Yn Xn,Xn) h(Yn Xn,Xn,Xn)
1 | 2 3 − 1 | 1 2 3
+h(Yn) h(Yn Xn)
3 − 3 | 3
       =h(Xn+Zn) h(Zn)
Y1 1 1 1 X1 Z1 1 1 − 1
 Y2 = 1 1 0  X2 + Z2 . +h(X(cid:18)1n+X3n(cid:19)+Z3n)−h(cid:18)(X1n+Z3n)(cid:19)
Y 1 0 1 X Z n P +N n 2P +N
3 3 3 1 3
log + log .
≤ 2 N 2 P +N
1 3
We state the outer bound in the following theorem.
Theorem 4: The capacity region of Type 2 channel is
contained in the following outer bound region:
Rk Ck, k =1,2,3 C. Channel Type 3
≤ (cid:18) (cid:19) (cid:18) (cid:19)
1 P 1 2P +N
2
R +R log 1+ + log
1 2
≤ 2 N1 2 P +N2 In this section, we present an outer bound on the capacity
(cid:18) (cid:19) (cid:18) (cid:19)
1 P 1 2P +N3 region of Type 3 channel defined by
R +R log 1+ + log .
1 3
≤ 2 N 2 P +N
1 3
      
Y 1 0 1 X Z
1 1 1
Proof:  Y2 = 0 1 1  X2 + Z2 .
Y 1 1 1 X Z
3 3 3
n(R +R (cid:15)) We state the outer bound in the following theorem.
1 2
−
I(Xn;Yn)+I(Xn;Yn) Theorem 5: The capacity region of Type 3 channel is
≤ 1 1 2 2
I(Xn;Yn Xn,Xn)+I(Xn;Yn) contained in the following outer bound region:
≤ 1 1 | 2 3 2 2
=h(Yn Xn,Xn) h(Yn Xn,Xn,Xn)
1 | 2 3 − 1 | 1 2 3
+h(Yn) h(Yn Xn)
2 − 2 | 2 Rk Ck, k =1,2,3
=h(Xn+Zn) h(Zn) ≤ 1 (cid:18) P (cid:19) 1 (cid:18)2P +N (cid:19)
1 1 − 1 R +R log 1+ + log 3
+h(Xn+Xn+Zn) h(Xn+Zn) 1 3 ≤ 2 N 2 P +N
(cid:18)1 2 (cid:19) 2 − (cid:18) 1 2 (cid:19) (cid:18) 1(cid:19) (cid:18) 3 (cid:19)
n P +N n 2P +N 1 P 1 2P +N
1 2 3
log + log . R +R log 1+ + log .
2 3
≤ 2 N 2 P +N ≤ 2 N 2 P +N
1 2 2 3
4
where Y = 1 Y , Z = 1 Z , N = E[Z 2] = 1,
R3 R3 E[X2] k(cid:48)P =√PNkandk k(cid:48) √Nk k 0 k(cid:48)
k ≤ k
 h h h   1 0 1 
11 12 13 √N1 √N1
R1 R1  h21 h22 h23 = √1N2 √1N2 0 .
R2 R2 h31 h32 h33 0 √1N3 √1N3
(a) Channeltype1 (b) Channeltypes4and5 With the usual definitions of SNRk = h2kNk0Pk and
Fig.2. Theshapeoftheouterboundregion.Theregionsforchanneltypes INR = h2jkPk for j =k as in [1], [6],
2and3looksimilartotheoneforchanneltype1(withchangeofaxis). k N0 (cid:54)
P P
SNR = INR = (4)
1 1
N ≥ N
Proof: 1 2
P P
SNR = INR = (5)
2 2
N ≥ N
n(R +R (cid:15)) 2 3
1 3
− P P
≤II((XX1nn;;YY1nn)X+nI)(+XI3n(;XY3nn;)Yn Xn) SNR3 = N3 ≤INR3 = N1. (6)
≤ 1 1 | 3 3 3 | 2
=h(Yn Xn) h(Yn Xn,Xn) We state the outer bound in the following theorem.
1 | 3 − 1 | 1 3
+h(Yn Xn) h(Yn Xn,Xn) Theorem 6: The capacity region of Type 4 channel is
3 | 2 − 3 | 2 3 contained in the following outer bound region:
=h(Xn+Zn) h(Zn)
1 1 − 1
+h(Xn+Xn+Zn) h(Xn+Zn)
(cid:18)1 3 (cid:19) 3 − (cid:18) 1 3 (cid:19) Rk Ck, k =1,2,3
nlog P +N1 + nlog 2P +N3 . ≤ 1 (cid:18) P (cid:19) 1 (cid:18)2P +N (cid:19)
2
≤ 2 N1 2 P +N3 R1+R2 ≤ 2log 1+ N + 2log P +N
1 2
(cid:18) (cid:19)
1 2P
R +R log 1+
n(≤R2I(+XR2n;3Y−2n(cid:15)))+I(X3n;Y3n) R12+R33 ≤ 21log(cid:18)1+ NP1(cid:19)+ 1log(cid:18)2P +N3(cid:19).
I(Xn;Yn Xn)+I(Xn;Yn Xn) ≤ 2 N2 2 P +N3
≤ 2 2 | 3 3 3 | 1
=h(Yn Xn) h(Yn Xn,Xn)
2 | 3 − 2 | 2 3 Proof:
+h(Yn Xn) h(Yn Xn,Xn)
3 | 1 − 3 | 1 3
=h(Xn+Zn) h(Zn)
2 2 − 2 n(R1+R2 (cid:15))
n+h(X(cid:18)2Pn++XN3n(cid:19)+Z3nn)−h(cid:18)(X2P2n++ZN3n)(cid:19) ≤I(X1n;Y−1n)+I(X2n;Y2n)
log 2 + log 3 . I(Xn;Yn Xn)+I(Xn;Yn)
≤ 2 N 2 P +N ≤ 1 1 | 3 2 2
2 3 =h(Yn Xn) h(Yn Xn,Xn)
1 | 3 − 1 | 1 3
+h(Yn) h(Yn Xn)
2 − 2 | 2
=h(Xn+Zn) h(Zn)
1 1 − 1
D. Channel Type 4 +h(X(cid:18)1n+X2n(cid:19)+Z2n)−h(cid:18)(X1n+Z2n)(cid:19)
n P +N n 2P +N
1 2
In this section, we present an outer bound on the capacity log + log .
≤ 2 N 2 P +N
region of Type 4 channel defined by 1 2
      
Y 1 0 1 X Z
1 1 1
 Y2 = 1 1 0  X2 + Z2 . n(R2+R3−(cid:15))
Y3 0 1 1 X3 Z3 I(Xn;Yn)+I(Xn;Yn)
≤ 2 2 3 3
I(Xn;Yn Xn)+I(Xn;Yn)
This is a cyclic Gaussian interference channel [6]. We first ≤ 2 2 | 1 3 3
show that channel type 4 is in the mixed interference regime. =h(Yn Xn) h(Yn Xn,Xn)
2 | 1 − 2 | 1 2
By normalizing the noise variances, we get the equivalent +h(Yn) h(Yn Xn)
channel given by =h(Xn+3Zn−) h3(Z|n)3
2 2 − 2
 Y1(cid:48)   h11 h12 h13  X1   Z1(cid:48)  +h(X(cid:18)2n+X3n(cid:19)+Z3n)−h(cid:18)(X2n+Z3n)(cid:19)
 Y2(cid:48) = h21 h22 h23  X2 + Z2(cid:48)  nlog P +N2 + nlog 2P +N3 .
Y3(cid:48) h31 h32 h33 X3 Z3(cid:48) ≤ 2 N2 2 P +N3
5
n(R +R (cid:15)) Proof:
1 3
−
I(Xn;Yn)+I(Xn;Yn)
≤≤≤==≤hhIII(((((XXXXY11111n1nnnn),;;(cid:18)+YYX−111nn3Xnh));3(nY++Y1+1nnII)|((ZXXX1n(cid:19)1n333nn),;;−XYY133h3nnn()||ZXX1n12nn))) n(≤≤≤≤=R1IIIIh(((((+XXXXYR1111nnnnn);;;,2YYYX−111nnn2nh(cid:15))));()Y+++Y1nnIII)(((XXXXn222nnn,;;;XYYY221nnnn))||XX31nn))
n 2P +N1 1 − 1 | 1 2
≤ 2 log N1 =h(X1n(cid:18)+X2n+Z1n(cid:19))−h(Z1n)
n 2P +N
1
log
where we used the fact that I(Xn;Yn Xn) = I(Xn;Xn + ≤ 2 N1
3 3 | 2 3 3
Zn) I(Xn;Xn+Zn)=I(Xn;Yn Xn).
3 ≤ 3 3 1 3 1 | 1 where we used the fact that I(Xn;Yn Xn) = I(Xn;Xn +
2 2 | 3 2 2
Zn) I(Xn;Xn+Zn)=I(Xn;Yn Xn).
2 ≤ 2 2 1 2 1 | 1
E. Channel Type 5 n(R2+R3 (cid:15))
−
I(Xn;Yn)+I(Xn;Yn)
In this section, we present an outer bound on the capacity ≤I(X2n;Y2n)+I(X3n;Y3n Xn)
region of Type 5 channel defined by ≤ 2 2 3 3 | 1
I(Xn;Yn)+I(Xn;Yn Xn)
≤ 2 2 3 2 | 2
       I(Xn,Xn;Yn)
Y1 1 1 0 X1 Z1 ≤ 2 3 2
 Y2 = 0 1 1  X2 + Z2 . =h(Y2n)−h(Y2n|X2n,X3n)
Y3 1 0 1 X3 Z3 =h(X2n(cid:18)+X3n+Z2n(cid:19))−h(Z2n)
n 2P +N
2
log
This is a cyclic Gaussian interference channel [6]. We first ≤ 2 N
2
show that channel type 5 is in the mixed interference regime.
By normalizing the noise variances, we get the equivalent where we used the fact that I(Xn;Yn Xn) = I(Xn;Xn +
3 3 | 1 3 3
channel given by Zn) I(Xn;Xn+Zn)=I(Xn;Yn Xn).
3 ≤ 3 3 2 3 2 | 2
 YYY123(cid:48)(cid:48)(cid:48) = √√011NN13 √√011NN12 √√011NN23  XXX123 + ZZZ123(cid:48)(cid:48)(cid:48) . n(≤≤=R1IIh(((+XXYR11nnn;;3XYY−11nnn)(cid:15))|X)+2nhI)((+YXnI3n(;XXY3n3nn,;)XY3nn))
1 | 2 − 1 | 1 2
We can see that +h(Yn) h(Yn Xn)
3 − 3 | 3
=h(Xn+Zn) h(Zn)
SNR = P INR = P (7) +h1(Xn+1 X−n+Z1n) h(Xn+Zn)
1 N1 ≥ 1 N3 n (cid:18)1P +N3 (cid:19) 3n − (cid:18)2P1 +N3 (cid:19)
P P log 1 + log 3
SNR = INR = (8) ≤ 2 N 2 P +N
2 N ≤ 2 N 1 3
2 1
P P
SNR = INR = . (9)
3 3
N ≤ N
3 2
We state the outer bound in the following theorem.
F. Relaxed Outer Bounds
Theorem 7: The capacity region of Type 5 channel is
contained in the following outer bound region: For ease of gap calculation, we also derive relaxed outer
bounds. First, we can see that for N N ,
j k
≤
R C , k =1,2,3 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
k k 1 P 1 2P +N 1 2P
≤ 1 (cid:18) 2P(cid:19) log 1+ + log k log 1+ .
R1+R2 log 1+ 2 Nj 2 P +Nk ≤ 2 Nj
≤ 2 N
1
(cid:18) (cid:19)
1 2P Five outer bound theorems in this section, together with this
R +R log 1+
2 3 ≤ 2 N inequality, give the sum-rate bound expression in Theorem 1.
2
R +R 1log(cid:18)1+ P (cid:19)+ 1log(cid:18)2P +N3(cid:19). Next, we can assume that P ≥ 3Nj for j = 1,2,3.
1 3 Otherwise, showing one-bit gap capacity is trivial as the
≤ 2 N 2 P +N
1 3
6
Type OuterboundregionRo RelaxedouterboundregionR(cid:48)o Two-dimensionalcross-sectionofR(cid:48)o
(cid:16) (cid:17)
1 RR21++RRRk32≤≤≤C2211kll,ooggk(cid:16)(cid:16)=PPNN++1,NN12212,3·· 22PPPP++++NNNN2323(cid:17)(cid:17) RR12++RRRk23≤≤≤121212llloooggg(cid:16)(cid:16)(cid:16)NNNPPPk12 ···437733(cid:17)(cid:17)(cid:17) RRAt13s≤≤ommmeiiRnn2(cid:110)(cid:110)∈1122ll[oo0gg,C(cid:16)(cid:16)2NNPP],12 ·· 3377(cid:17)(cid:17)−−RR22,,1212lloogg(cid:16)(cid:16)NNPP13 ·· 4343(cid:17)(cid:17)(cid:111)(cid:111)
2 RR11++RRRk32≤≤≤C2211kll,ooggk(cid:16)(cid:16)=PPNN++1,NN11211,3·· 22PPPP++++NNNN2323(cid:17)(cid:17) RR11++RRRk23≤≤≤121212llloooggg(cid:16)(cid:16)(cid:16)NNNPPPk11 ···437733(cid:17)(cid:17)(cid:17) RRAt23s≤≤ommmeiiRnn1(cid:110)(cid:110)∈1122ll[oo0gg,C(cid:16)(cid:16)1NNPP],11 ·· 3377(cid:17)(cid:17)−−RR11,,1212lloogg(cid:16)(cid:16)NNPP23 ·· 4343(cid:17)(cid:17)(cid:111)(cid:111)
3 RR21++RRRk33≤≤≤C2211kll,ooggk(cid:16)(cid:16)=PPNN++1,NN12212,3·· 22PPPP++++NNNN3333(cid:17)(cid:17) RR12++RRRk33≤≤≤121212llloooggg(cid:16)(cid:16)(cid:16)NNNPPPk12 ···437733(cid:17)(cid:17)(cid:17) RRAt12s≤≤ommmeiiRnn3(cid:110)(cid:110)∈1122ll[oo0gg,C(cid:16)(cid:16)3NNPP],12 ·· 3377(cid:17)(cid:17)−−RR33,,1212lloogg(cid:16)(cid:16)NNPP12 ·· 4343(cid:17)(cid:17)(cid:111)(cid:111)
4 RRR121+++RRRRk332≤≤≤≤C222111klll,ooogggk(cid:16)(cid:16)(cid:16)=P2PPNN++1N+,NN1212N12,13··(cid:17)22PPPP++++NNNN2323(cid:17)(cid:17) RRR112+++RRRRk233≤≤≤≤12121212lllloooogggg(cid:16)(cid:16)(cid:16)(cid:16)NNNNPPPPk112 ····43777333(cid:17)(cid:17)(cid:17)(cid:17) RRRAt232s+≤≤omRmme3iiRnn≤1(cid:110)(cid:110)∈121122lll[ooo0ggg,C(cid:16)(cid:16)(cid:16)1NNNPPP],211 ···733377(cid:17)(cid:17)(cid:17)−−RR11,,1212lloogg(cid:16)(cid:16)NNPP23 ·· 4343(cid:17)(cid:17)(cid:111)(cid:111)
5 RRR211+++RRRRk332≤≤≤≤C222111klll,ooogggk(cid:16)(cid:16)(cid:16)=22PPPN+1NN++,N1122NN1,123·(cid:17)(cid:17)2PP++NN33(cid:17) RRR121+++RRRRk233≤≤≤≤12121212lllloooogggg(cid:16)(cid:16)(cid:16)NNNNPPPPk121 ····43777333(cid:17)(cid:17)(cid:17) RRRAt131s+≤≤omRmme3iiRnn≤2(cid:110)(cid:110)∈121122lll[ooo0ggg,C(cid:16)(cid:16)(cid:16)2NNNPPP],112 ···733377(cid:17)(cid:17)(cid:17)−−RR22,,1212lloogg(cid:16)(cid:16)NNPP13 ·· 4343(cid:17)(cid:17)(cid:111)(cid:111)
TABLEII
CAPACITYOUTERBOUNDS
capacity region is included in the unit hypercube, i.e., R A. Preliminaries: Lattice Coding
j
(cid:16) (cid:17) ≤
1log 1+ P <1. For P 3N ,
2 Nj ≥ j LatticeΛisadiscretesubgroupofRn,Λ= t=Gu:u
1 (cid:18) 2P(cid:19) 1 (cid:18) P (cid:19) 1 (cid:18)Nj (cid:19) Zn whereG Rn n isarealgeneratormatri{x.Quantizatio∈n
log 1+ = log + log +2 ×
} ∈
2 Nj 2 Nj 2 P with respect to Λ is QΛ(x) = argminλ Λ x λ . Modulo
1 (cid:18) P (cid:19) 1 (cid:18)7(cid:19) operation with respect to Λ is M (x) =∈ [x(cid:107)] m−od(cid:107)Λ = x
log + log Λ −
≤ 2 Nj 2 3 QΛ(x). For convenience, we use both notations MΛ(·) and
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) [] mod Λ interchangeably. Fundamental Voronoi region of Λ
1 P 1 P 1 4
log 1+ log + log . i·s (Λ) = x : Q (x) = 0 . Volume of the Voronoi region
2 Nj ≤ 2 Nj 2 3 V { Λ(cid:82) }
of Λ is V(Λ) = dx. Normalized second moment of
(Λ)
The resulting relaxed outer bounds R(cid:48)o are summarized in Λ is G(Λ) = σ2(ΛV) where σ2(Λ) = 1 (cid:82) x 2dx.
Table II. V(Λ)2/n nV(Λ) (Λ)(cid:107) (cid:107)
Lattices Λ , Λ and Λ are said to be nested if ΛV Λ Λ .
1 2 2 1
⊆ ⊆
For nested lattices Λ Λ , Λ /Λ =Λ (Λ ).
2 1 1 2 1 2
III. INNERBOUND:CHANNELTYPE1 ⊂ ∩V
Webrieflyreviewthelatticedecodingprocedurein[12].We
RαThiseodreefimne8d: bGyiven α = (α0,α2) ∈ [0,1]2, the rate region Vus(eΛn)e=ste(d2lπaettSic)en2s.ΛT⊆heΛtrtawnsimthitσte2r(Λse)n=dsSx,=G([Λt)+=d]2π1meo,danΛd
1 (cid:18)1 α (1 α )P (cid:19) over the point-to-point Gaussian channel y=x+z where the
R1 ≤ 2log+ 2−−α00 + (α0+−α(cid:18)2)P0 +N2(cid:19) tchoedetrwaonrsdmtit∈poΛwte∩r 1V(xΛ)2, t=heSdiatnhderthsiegnnaolisde z∼ Unif((0V,(NΛI))),.
+1log 1+ α0P The code rate is givnen(cid:107)b(cid:107)y R= 1 log(cid:16)V(Λ)(cid:17). ∼N
2 N1 n V(Λt)
1 (cid:18) α P (cid:19) After linear scaling, dither removal, and mod-Λ operation,
2
R log 1+
2 ≤ 2 α P +N we get
0 2
(cid:18) (cid:19)
1 1 P
R log+ +
3
≤ 2 2 α (α +α )P +N
− 0 0 2 3 y(cid:48) =[βy d] mod Λ=[t+ze] mod Λ (10)
−
where log+()=max 0,log() . And,
· { · }
(cid:32) (cid:33)
(cid:91) where the effective noise is ze = (β 1)x + βz1 and its
R= CONV α Rα vMaMriaSnEcescσae2lin=g fn1acEt[o(cid:107)rzeβ(cid:107)2=] =S(β −plu1g)g2−eSd+inβ, 2wNe.gWetitσh2th=e
S+N e
is achievable where CONV() is convex hull operator. βN = SN .Thecapacityofthemod-Λchannel[12]between
· S+N
7
t and y is where t Λ (Λ ) and t Λ (Λ ) are lattice
11 c 1 3 c 3
∈ ∩V ∈ ∩V
codewords. The dither signals d and d are uniformly
1 1 1 11 3
I(t;y) = h(y) h(yt) distributed over (Λ ) and (Λ ), respectively. To satisfy
n n1 − n1 | power constraintsV, we1chooseVE[ 3x11 2] = nσ2(Λ1) = (1
= h(y) h(z mod Λ) α )nP, E[ x 2] = α nP, E[ (cid:107)x 2(cid:107)] = α nP, E[ x 2] −=
n − n 1 (cid:107) 10(cid:107) 1 (cid:107) 2(cid:107) 2 (cid:107) 3(cid:107)
1 1 nσ2(Λ3)=nP.
h(y) h(z)
Withthechoiceoftransmitsignals,thereceivedsignalsare
≥ n − n
1 1 given by
= logV(Λ) h(z)
n − n
(cid:18) (cid:19) y =x +x +x +z
1 S 1 11 2 10 1
= log
2 βN y2 =[x11+x3]+x2+z(cid:48)2
(cid:18) (cid:19)
= 1log 1+ S y3 =x3+z(cid:48)3.
2 N
where x = [x +x ] is the sum of interference, and z =
= C f 11 3 (cid:48)2
x +z and z = x +z are the effective Gaussian noise.
10 2 (cid:48)3 2 3
where I() and h() are mutual information and differential The signal scale diagram at each receiver is shown in Fig. 3
· ·
entropy, respectively. For reliable decoding of t, we have (a).
the code rate constraint R C. With the choice of lattice At the receivers, successive decoding is performed in the
pσa2r(aΛmt)eter2s,πeσβ2N(Λ,t) ≥ βN,≤G(Λt) = 2π1e and V(Λt)n2 = froeclleoiwveirng2,oardnedr:rexc1e1iv→er 3x2on→lyxd1e0coadtersecxe3i.ver 1, xf → x2 at
G(Λt) ≥ Note that the aligned lattice codewords t + t Λ ,
1 (cid:18)V(Λ)(cid:19) and t = [t + t ] mod Λ Λ (Λ )1.1We 3sta∈te thce
R = log f 11 3 1 ∈ c ∩ V 1
n V(Λt) relationship between xf and tf in the following lemmas.
1 (cid:18) (2πeS)n2 (cid:19) Lemma 1: The following holds.
log
≤ n (2πeβN)n2
1 (cid:18) S (cid:19) [xf −df] mod Λ1 =tf
= log .
2 βN where d =d +d .
f 11 3
Thus, the constraint R C can be satisfied. By lattice Proof:
≤
decoding [12], we can recover t, i.e.,
[x d ] mod Λ
f f 1
−
QΛt(y(cid:48))=t, (11) =[MΛ1(t11+d11)+MΛ3(t3+d3)−df] mod Λ1
with probability 1−Pe where =[MΛ1(t11+d11)+MΛ1(t3+d3)−df] mod Λ1
=[t +d +t +d d ] mod Λ
Pe =Pr[QΛt(y(cid:48))(cid:54)=t] (12) =[t11+t1]1mod3Λ 3− f 1
11 3 1
is the probability of decoding error. If we choose Λ to be
=t
f
Poltyrev-good [27], then P 0 as n .
e
→ →∞
Thesecondandthirdequalitiesareduetodistributivelawand
B. Achievable Scheme the identity in the following lemma.
Lemma 2: For any nested lattices Λ Λ and
We present an achievable scheme for the proof of any x Rn, it holds that 3 ⊂ 1
Theorem 8. The achievable scheme is based on rate- ∈
splitting, lattice coding, and interference alignment. Message
[M (x)] mod Λ =[x] mod Λ .
M1 1,2,...,2nR1 is split into two parts: M11 Λ3 1 1
∈ { } ∈
1,2,...,2nR11 and M10 1,2,...,2nR10 , so R1 = Proof:
{ } ∈ { }
R + R . Transmitter 1 sends x = x + x where
11 10 1 11 10
x and x are coded signals of M and M , respec- [M (x)] mod Λ
11 10 11 10 Λ3 1
tively. Transmitters 2 and 3 send x2 and x3, coded signals =[x λ3] mod Λ1
poafrtMicu2la∈r, x{111,2a,n.d..x,32naRre2}latatnicde-Mco3ded∈si{g1n,a2l,s....,2nR3}. In =[M−Λ1(x)−MΛ1(λ3)] mod Λ1
We use the lattice construction of [14], [15] with the lattice =[MΛ1(x)−λ3+QΛ1(λ3)] mod Λ1
partition chain Λ /Λ /Λ , so Λ Λ Λ are nested =[M (x)] mod Λ
c 1 3 3 ⊂ 1 ⊂ c Λ1 1
lattices. Λc is the coding lattice for both x11 and x3. Λ1 and =[x] mod Λ1
Λ areshapinglatticesforx andx ,respectively.Thelattice
3 11 3
signals are formed by where λ =Q (x) Λ , thus Q (λ )=λ .
3 Λ3 ∈ 1 Λ1 3 3
Lemma 3: The following holds.
x =[t +d ] mod Λ (13)
11 11 11 1
x =[t +d ] mod Λ (14) [t +d ] mod Λ =[x ] mod Λ .
3 3 3 3 f f 1 f 1
8
(α¯0+1)PNe2 . The capacity of the mod-Λ channel between
(1−α0)P x x +x x P t(αf¯0a+n1d)Py+2(cid:48)Nies2 1
11 11 3 3
α P 1
2 I(t ;y )
x x x n f 2(cid:48)
2 2 2
α P !"#$$%&’()*%’+’ 1 (cid:18)V(Λ )(cid:19)
0 1
log
x10 x10 ≥ n 2h(ze2)
(cid:18) (cid:19)
1 α¯ P
0
RX1 RX2 RX3 = log
2 β N
2 e2
(a) Channeltype1 (cid:18) (cid:19)
1 α¯ (α¯ +1)P +α¯ N
0 0 0 e2
= log
PP 2 (α¯0+1)Ne2
(cid:18) (cid:19)
x2+xx3 xx2 xx3 = 1log α¯0 + α¯0P
α1PP !"#$$%&’()*%’,’ 2 α¯0+1 Ne2
xx1 xx1 xx1 1 (cid:18) α¯0 α¯0P (cid:19)
= log +
2 α¯ +1 (α +α )P +N
RRXX11 RRXX22 RRXX33 0 0 2 2
=C
f
(b) Channeltype2
For reliable decoding of t at receiver 2, we have the
PP xxx3 xxx3 xxx3 PPP icmodpeliersattehactonRstra=int1Rlo1g1(cid:16)=V(n1Λf2l)o(cid:17)g(cid:16)VVC((ΛΛ1c))+(cid:17) 1≤loCgf(cid:16).VT(hΛi2s)(cid:17)als=o
αP xx xx xx (cid:16) (cid:17)3 n (cid:16) V(Λc) ≤ f n (cid:17) V(Λ1)
x1 x2 x1+x2 12log β2PNe2 = 12log α¯01+1 + (α0+αP2)P+N2 . By lattice
RX1 RX2 RX3 decoding, we can recover the modulo sum of interference
RX1 RX2 RX3
RX1 RX2 RX3 codewords t from y . Then, we can recover the real sum
f 2(cid:48)
(c) Channeltype3 xf in the following way.
Recover M (x ) by calculating [t + d ] mod Λ
Fig.3. Signalscalediagram. • Λ1 f f f 1
(lemma 3).
Subtract it from the received signal,
•
Proof: y2−MΛ1(xf)=QΛ1(xf)+z(cid:48)2(cid:48) (18)
[tf +df] mod Λ1 where z(cid:48)2(cid:48) =x2+x10+z2.
Quantize it to recover Q (x ),
=[MΛ1(t11+t3)+df] mod Λ1 • Λ1 f
=[t11+t3+df] mod Λ1 QΛ1(QΛ1(xf)+z(cid:48)2(cid:48))=QΛ1(xf) (19)
=[MΛ1(t11+d11)+MΛ1(t3+d3)] mod Λ1 with probability 1−Pe where
=[M (t +d )+M (t +d )] mod Λ
=[x Λ1+x11] mo1d1Λ Λ3 3 3 1 Pe =Pr[QΛ1(QΛ1(xf)+z(cid:48)2(cid:48))(cid:54)=QΛ1(xf)] (20)
11 3 1
is the probability of decoding error. If we choose Λ to
=[x ] mod Λ 1
f 1 be simultaneously Rogers-good and Poltyrev-good [27]
with V(Λ ) V(Λ ), then P 0 as n .
1 c e
≥ → →∞
Receiver 2 does not need to recover the codewords t and Recover xf by adding two vectors,
11 •
t but the real sum x to remove the interference from y .
3 f 2 M (x )+Q (x )=x . (21)
Sincex =M (x )+Q (x ),wefirstrecoverthemodulo Λ1 f Λ1 f f
f Λ1 f Λ1 f
part and then the quantized part to cancel out xf. This idea Wenowproceedtodecodingx2fromy2−xf =x2+z(cid:48)2.Since
appearedin[17]asanachievableschemeforthemany-to-one x2 isacodewordfromani.i.d.randomcodeforpoint-to-point
interference channel. channel, we can achieve rate up to
The mod-Λ1 channel between tf and y2(cid:48) is given by R 1log(cid:18) α2P (cid:19). (22)
2
≤ 2 α P +N
y =[β y d ] mod Λ (15) 0 2
2(cid:48) 2 2− f 1
At receiver 1, we first decode x while treating other
=[x d +z ] mod Λ (16) 11
f f e2 1
− signals x + x + z as noise. The effective noise in the
=[t +z ] mod Λ (17) 2 10 1
f e2 1 mod-Λ channel is z =(β 1)2x +β (x +x +z )
1 e1 1 11 1 2 10 1
NwohteereththaeteEff[ecxtfive2n]o=ise(zα¯e02+=1()βn2P−,1)axndf+thβe2(exff2e+ctxiv1e0+nozi2s)e. wwhitehrevaNriea1nc=e σ(αe210=+αn12E)[P(cid:107)z+e1−(cid:107)N21].=Fo(rβ1re−lia1b)le2α¯d0ePco+dinβg12,Nthe1e
variance σe22(cid:107)= n1(cid:107)E[(cid:107)ze2(cid:107)2] = (β2−1)2(α¯0+1)P +β22Ne2 rate R11 must satisfy
where Ne2 = (α0 + α2)P + N2. With the MMSE scaling 1 (cid:18)σ2(Λ1)(cid:19) 1 (cid:18) α¯0P (cid:19)
factor β2 = (α¯0(+α¯01+)P1+)PNe2 plugged in, we get σe22 =β2Ne2 = R11 ≤ 2log β1σe21 = 2log 1+ (α0+α2)P +N1
9
wheretheMMSEscalingparameterβ = α¯0P .Similarly, C. The Gap
1 α¯0P+Ne1
we have the other rate constraints at receiver 1:
We choose the parameter α = N2, which is suboptimal
(cid:18) (cid:19) 0 P
1 α P
R log 1+ 2 (23) but good enough to achieve a constant gap. This choice of
2
≤ 2 α0P +N1 parameter, inspired by [1], ensures making efficient use of
(cid:18) (cid:19)
1 α P signalscaledifferencebetweenN andN atreceiver1,while
R log 1+ 0 . (24) 1 2
10 ≤ 2 N1 keepingtheinterferenceofx10atthenoiselevelN2atreceiver
2. By substitution, we get
At receiver 3, the signal x is decoded with the effective
3
noise x2+z3. For reliable decoding, R3 must satisfy 1 (cid:18) P N2 (cid:19)
T = log c + −
1 (cid:18) P (cid:19) 1 2 11 α2P +2N2
R3 ≤ 2log 1+ α2P +N3 . (25) +1log(cid:18)1+ N2(cid:19) (33)
2 N
In summary, 1
(cid:18) (cid:19)
1 α P
• x11 decoded at receivers 1 and 2 T2 = 2log 1+ 2N2 (34)
(cid:18) (cid:19) 2
1 (1 α )P (cid:18) (cid:19)
R11 ≤T1(cid:48)1 = 2log 1+ (α0+−α2)P0 +N1 T3 ≥ 12log c3+ α P +PN +N . (35)
(cid:18) (cid:19) 2 2 3
1 (1 α )P
0
wxherdeeRcco111d1e=≤d a(T1t(−1(cid:48)1r(cid:48)1−αec0=α)e0Pi)v2P+ePlro1g= 21c−−11αα+00. (α0+−α2)P +N2 SanindScteca3rαti=n0g=2−frNNoP1m22/P∈≥(cid:2)o0,f12r31.o(cid:3)m, itTafobllleowIIs, wtheatcca1n1e=xp12r−−esNNs22t//hPPe≥two25-,
10 R
• dimensional outer bound region at R as
(cid:18) (cid:19) 2
1 α P
R10 ≤T10 = 2log 1+ N0 (26) (cid:26)1 (cid:18) 2P(cid:19) (cid:27)
1 R min log 1+ R ,C
1 2 1
≤ 2 N −
x decoded at receivers 1 and 2 1
• 2 (cid:26)1 (cid:18) P 7(cid:19) 1 (cid:18) P 4(cid:19)(cid:27)
R2 ≤T2(cid:48) = 12log(cid:18)1+ α0Pα2+PN1(cid:19) (27) ≤min(cid:26)21log(cid:18)N1 ·23P(cid:19)−R2,2lo(cid:27)g N1 · 3
R2 ≤T2(cid:48)(cid:48) = 12log(cid:18)1+ α0Pα2+PN2(cid:19) (28) R3 ≤min(cid:26)12log(cid:18)1P+ N72(cid:19) −R2,1C3 (cid:18) P 4(cid:19)(cid:27)
min log R , log .
• x3 decoded at receivers 2 and 3 ≤ 2 N2 · 3 − 2 2 N3 · 3
(cid:18) (cid:19)
1 P
R T = log c + Depending on the bottleneck of min , expressions, there
3 ≤ 3(cid:48) 2 3 (α +α )P +N {· ·}
0 2 2 are three cases:
(cid:18) (cid:19)
1 P
R3 ≤T3(cid:48)(cid:48) = 2log 1+ α2P +N3 (29) • R12lo≤g(cid:0)217l(cid:1)og(cid:0)R74(cid:1) 1log(cid:16)N3 7(cid:17)
Notewthhaetre0c3 =c(1−α0P)P1,+Pc =+2−c1α0=. 1, and 1 c 1. •• R22 ≥ 124lo≤g(cid:16)NN232≤· 472(cid:17). N2 · 4
≤ 11 ≤ 2 11 3 2 ≤ 3 ≤ (cid:16) (cid:17)
Putting together, we can see that the following rate region is At R = 1log α2P 7 , the outer bound region is
achievable. 2 2 N2 · 4
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
1 P N 4 1 P 4
R1 ≤T1 =min{T1(cid:48)1,T1(cid:48)(cid:48)1}+T10 =T1(cid:48)(cid:48)1+T10 R1 min log 2 , log
≤ 2 α P · N · 3 2 N · 3
R T =min T ,T =T 2 1 1
2 ≤ 2 { 2(cid:48) 2(cid:48)(cid:48)} 2(cid:48)(cid:48) (cid:26)1 (cid:18) P 4(cid:19) 1 (cid:18) P 4(cid:19)(cid:27)
R3 ≤T3 =min{T3(cid:48),T3(cid:48)(cid:48)} R3 ≤min 2log α P · 3 ,2log N · 3 .
2 3
where
(cid:18) (cid:19) Depending on the bottleneck of min , expressions, we
T = 1log c + (1−α0)P consider the following three cases: {· ·}
1 11
2 (α +α )P +N
0 2 2
+1log(cid:18)1+ α0P(cid:19) (30) • αN2P ≥αNP3 N
2 N • 2 ≤ 2 ≤ 3
1 α P N .
1 (cid:18) α P (cid:19) • 2 ≤ 2
T2 = 2log(cid:18)1+ α0P2+N2 (cid:19) (31) 1lCogas(cid:16)eα2i)Pα27P(cid:17) i≥s N3: The outer bound region at R2 =
1 P 2 N2 · 4
T log c + . (32)
3 3
≥ 2 (α0+α2)P +N3 (cid:18) (cid:19) (cid:18) (cid:19)
1 P N 4 1 P 4
Thus, Theorem 8 is proved. R1 log 2 ,R3 log . (36)
≤ 2 α P · N · 3 ≤ 2 α P · 3
2 1 2
10
For comparison, let us take a look at the achievable rate Let us take a look at the achievable rate region. The first
region. The first term of T is lower bounded by term of T is lower bounded by
1 1
T1(cid:48)(cid:48)1 = 121lloogg(cid:18)(cid:18)c211++Pα−2PPα−+2PN2(cid:19)N2 2(cid:19) ((3378)) T1(cid:48)(cid:48)1 = 121lloogg(cid:18)(cid:18)c211++Pα−2PPN−+2N(cid:19)2N2 2(cid:19) ((5523))
≥ 12 (cid:18)5P (cid:19)3α2P ≥ 12 (cid:18)5P (cid:19)3N2
> log . (39) > log . (54)
2 3α2P 2 3N2
We get the lower bounds:
We get the lower bounds:
T =T +T (55)
1 1(cid:48)(cid:48)1 10
T1 =T1(cid:48)(cid:48)1+T10 (40) 1 (cid:18) P (cid:19) 1 (cid:18) N (cid:19)
1 (cid:18) P (cid:19) 1 (cid:18) N (cid:19) > log + log 1+ 2 (56)
> log + log 1+ 2 (41) 2 3N2 2 N1
2 3α2P 2 N1 1 (cid:18) P (cid:19)
1 (cid:18) P N (cid:19) > log (57)
> log 2 (42) 2 3N1
2 3α2P · N1 1 (cid:18)1 P (cid:19)
1 (cid:18)1 P (cid:19) T3 log + (58)
T3 log + (43) ≥ 2 2 α2P +N2+N3
≥ 2 2 α2P +N2+N3 1 (cid:18) P (cid:19)
1 (cid:18) P (cid:19) > log . (59)
> log . (44) 2 3N3
2 3α P
2 (cid:16) (cid:17)
For fixed α and R = 1log α2P , the following two-
For fixed α2 and R2 = 12log(cid:16)α2N2P2(cid:17), the two-dimensional dimensional2rate regio2n is a2chievab2lNe.2
achievable rate region is given by (cid:18) (cid:19) (cid:18) (cid:19)
1 P 1 P
R log , R log . (60)
(cid:18) (cid:19) (cid:18) (cid:19) 1 3
1 P N 1 P ≤ 2 3N ≤ 2 3N
R log 2 , R log . (45) 1 3
1 3
≤ 2 3α2P · N1 ≤ 2 3α2P In all three cases above, by comparing the inner and outer
bound regions, we can see that δ 1log(cid:0)3 4(cid:1) = 1, δ
Cas(cid:16)eii)N2 (cid:17)≤α2P ≤N3:TheouterboundregionatR2 = 1log(cid:0)2 7(cid:1) = 0.91 and δ 11lo≤g(cid:0)23 4(cid:1) =· 31. Theref2or≤e,
1log α2P 7 is 2 · 4 3 ≤ 2 · 3
2 N2 · 4 wecanconcludethatthegapistowithinonebitpermessage.
(cid:18) (cid:19) (cid:18) (cid:19)
1 P N 4 1 P 4
R log 2 , R log . (46) IV. INNERBOUND:CHANNELTYPE2
1 3
≤ 2 α P · N · 3 ≤ 2 N · 3
2 1 3 Theorem 9: Given α [0,1], the region is defined by
1 α
∈ R
Now, let us take a look at the achievable rate region. We (cid:18) (cid:19)
1 α P
1
have the lower bounds: R log 1+
1
≤ 2 N
1
T1 > 21log(cid:18)3αP2P · NN21(cid:19) (47) R2 ≤ 12log+(cid:18)12 + α1PP+N2(cid:19)
1 (cid:18)1 P (cid:19) 1 (cid:18)1 P (cid:19)
T3 ≥ 2log 2 + α2P +N2+N3 (48) R3 ≤ 2log+ 2 + α1P +N3 ,
(cid:18) (cid:19)
1 P (cid:0)(cid:83) (cid:1)
> 2log 3N . (49) and R= CONV α1Rα is achievable.
3
For fixed α and R = 1log(cid:16)α2P(cid:17), the two-dimensional A. Achievable Scheme
achievable ra2te region2 is gi2ven by2N2 For this channel type, rate splitting is not necessary. Trans-
R 1log(cid:18) P N2(cid:19), R 1log(cid:18) P (cid:19). (50) m1,i2t,s3ig.nIanlpxakrtiiscualacro,dxe2dasnigdnxal3oafreMlkat∈tic{e1-c,o2d,e.d..s,i2gnnRalks}u,skin=g
1 3
≤ 2 3α P · N ≤ 2 3N the same pair of coding and shaping lattices. As a result,
2 1 3
the sum x +x is a dithered lattice codeword. The power
Case iii) α P N : The outer bound region at R = 2 3
12log(cid:16)αN22P · 742(cid:17) is≤ 2 2 Eal[loxca3tio2n]=santiPsfi.eTshEe[(cid:107)rexc1e(cid:107)iv2e]d=siαgn1anlPs ,arEe[(cid:107)x2(cid:107)2] = nP, and
(cid:107) (cid:107)
1 (cid:18) P 4(cid:19) 1 (cid:18) P 4(cid:19) y1 =[x2+x3]+x1+z1
R log , R log . (51)
1 3
≤ 2 N · 3 ≤ 2 N · 3 y =x +x +z
1 3 2 2 1 2
For this range of α , the rate R is small, i.e., R = y3 =x3+x1+z3.
12log(cid:16)αN22P · 74(cid:17) ≤ 12l2og(cid:0)74(cid:1) < 12, a2nd R1 and R3 are c2lose The signal scale diagram at each receiver is shown in Fig. 3
to single user capacities C and C , respectively. (b). Decoding is performed in the following way.
1 3