Upper Bounds on the Capacity of 2-Layer $N$-Relay Symmetric Gaussian Network

0.39 MB·

by  Satyajit Thakor

1
Upper Bounds on the Capacity of 2-Layer N-Relay
Symmetric Gaussian Network
Satyajit Thakor† and Syed Abbas‡
School of Computing and Electrical Engineering†
School of Basic Sciences‡
Indian Institute of Technology Mandi
email: [email protected]
Abstract
The Gaussian parallel relay network, in which two parallel relays assist a source to convey information to a destination, was
introduced by Schein and Gallager. An upper bound on the capacity can be obtained by considering broadcast cut between the
6 source and relays and multiple access cut between relays and the destination. Niesen and Diggavi derived an upper bound for
1 GaussianparallelN-relaynetworkbyconsideringallotherpossiblecutsandshowedanachievabilityschemethatcanattainrates
0 close to the upper bound in different channel gain regimes thus establishing approximate capacity. In this paper we consider
2 symmetric layered Gaussian relay networks in which there can be many layers of parallel relays. The channel gains for the
r channels between two adjacent layers are symmetrical (identical). Relays in each layer broadcast information to the relays in the
p next layer. For 2-layer N-relay Gaussian network we give upper bounds on the capacity. Our analysis reveals that for the upper
A bounds, joint optimization over correlation coefficients is not necessary for obtaining stronger results.

2
1 I. INTRODUCTION

The capacity of multihop relay networks is largely unknown. Even for a simple relay network set-up [1] in which a relay
]
T assists a source to communicate information to a destination in addition to the direct transmission is unknown for discrete
I memoryless as well as Gaussian case. In [2], Schein and Gallager introduced a new network setup called Gaussian parallel
.
s relay network in which a source communicates information to the destination via two parallel relays. Cut-set outer bounds
c
were derived and a coding scheme called amplify-and-forward (AF) was introduced. In [3], Niesen and Diggavi considered
[
generalization of the Gaussian parallel relay network, called Gaussian diamond network, in which communication is via

4 N relays. For diamond network with symmetric channel gains an outer bound was derived. The authors [3] also proposed
v
bursty amplify-and-forward scheme and showed constant additive and multiplicative gaps between the achievable rates for the
6
coding scheme and the upper bound for all regimes (parameter choices). This also established an approximate capacity for
8
6 the symmetric Gaussian diamond network. This results were further extended for asymmetric Gaussian diamond network by
6 converting an asymmetric diamond network into (approximately) symmetric parallel diamond networks.
0 The recent work [4] suggests that sub-linear gap (in terms of network nodes) between the cut-set bound and an achievable
.
1 scheme for general relay networks is very unlikely else the cut-set bound is tight in general. However, for certain class of
0 networks a constant factor gap may be a possibility and should be investigated.
6
1 ...
:
v
i ...
X
ar s ... ... ... ... d
...
1 2 L
Fig.1. NetworkwithLlayersofparallelrelays
Figure 1 depicts a generalization of the diamond network called Gaussian layered relay network. This general network has
Llayersofparallelrelaysbetweenthesourceandthedestination.Eachrelayinalayerreceivesinformationfromtherelaysof
previous layers and broadcasts a function of received information to the relays in the next layer. The source node s is at layer
0 and the destination node is at layer L+1. Layered deterministic relay networks were considered in [5]. In layered networks
the information flowing through different paths to a node in the networks reaches at the same time (i.e., with the same delay).
This makes both upper bounding using information theoretic arguments and lower bounding using coding schemes some what
easier by dropping the causality constraint for information transmission.
2
As noted in [6, p. 39], some of the achievable coding schemes are seen from a better perspective once the converse results
are established. In this paper, we focus on converse results (upper bounds) for the network capacity. As a first step towards
characterizingbetterupperboundsonthecapacityofGaussianlayeredrelaynetworks(and,hopefully,establishingapproximate
capacity), in this paper we consider 2-layer N-relay Gaussian network. Diamond network is a sub-network of this network.
For this network we obtain upper bounds on the capacity which involves joint optimization over correlation parameters. Our
analysis reveals a surprising result: joint optimization over correlation parameters across different layers is not necessary to
obtain a tighter cut-set type bounds (29)-(30) for 2-layer N-relay network. This desirable result may lead to characterization
of an approximate capacity of layered relay networks with small multiplicative and additive gaps between the outer bounds
and achievable rates for coding schemes.
The remaining part of the paper is organized as follows. Section II describes the network model and related work. Section
III presents our main results. Conclusion and future directions are discussed in Section IV.
II. NETWORKMODEL
A 2-layer N-relay Gaussian network with N relays in each layer is shown in Figure 2. We adopt most notations from
[3]. The sets of relays are labeled [iN] = i1,i2,...,iN ,i = 1,2. In this context, 12,2k etc., are relays and should not be
{ }
interpreted as numbers 12 or 2 k. Parameters A ,i I,A ,j J are denoted A for simplicity. The transmitted random
i j I,J
× ∈ ∈
variables X have unit average power constraint. The channel gains r are real positive constants. The channels
0,[1N],[2N] 1,2,3
introduce i.i.d. additive white Gaussian noise Z each with zero mean and variance 1. A random variable without
[1N],[2N],3
time index may be viewed as a generic copy.
Z Z
11 21
Y11 X11 √r2 Y21 X21
11 21
√r1 √r3
Z Z
s X0 √r1 1Y212 12 X12 √r2 2Y222 22 X22√r3 ZY33 d
√r1
√r3
ZY1N1N X1N √r2 Z2YN2N X2N
1N 2N
Fig.2. Symmetric2-layerN-realyGaussiannetwork
The source message V is encoded as (X [t])T =g (V). The input to the relay node 1j at time t is
0 t=1 s
Y [t]=√r X [t]+Z [t].
1j i 0 1j
Assume that the relays incur delay of one block length T (or one time instance t). In the next block interval, each of these
relays broadcasts a relay code X [t+T]=g (Y [t]) to the next layer. Each relay 2j in layer 2 receives
1,j 1,j 1,j
N
(cid:88)
Y [t]=√r X [t]+Z [t].
2j 2 1l 2j
l=1
The coded symbols (X [t+T])T =g ((Y [t])T ) are transmitted by relays and
2,j t=1 2,j 2,j t=1
N
(cid:88)
Y [t]=√r X [t]+Z [t]
3 3 2l 3
l=1
is received by the destination node d which estimates the transmitted source message from the received message via the
decoding function
Vˆ =g ((Y [t])3T ).
d 3 t=2T+1
Note that, due to the layered structure of the network, in any block interval a relay in the second layer receives a corrupted
version of the transmitted codes for the source message for the same block. In other words (Y [t])2T ,j = 1...N are
2j t=T+1
associated with the transmitted code block (X [t])T from the source. Also, the destination node receives a corrupted version
0 t=1
of the transmitted codes for the source message for the same block. In other words (Y [t])3T is associated with the
3 t=2T+1
transmittedcodeblock(X [t])T fromthesource.Assuch,weassumenodelayintherelayoperationsanddropthecausality
0 t=1
constraint for information transmission in the network.
For the symmetric diamond network the capacity is upper bounded as follows. We refer (1) as the ND bound.
3
Theorem 1 (Lemma 7, [3]): For symmetric diamond network with N relays and channel gains r at source broadcast side
1
and r at destination mutiple-access side, the capacity
2
1
C(N,r ) sup min (log(1+(N n)r )+log(1+ψ(n,ρ )r )) (1)
1,2 ≤ρ1∈[0,1)n∈{0,...,N}2 − 1 1 2
where
(cid:16) n(N n)ρ2 (cid:17)
ψ(n,ρ)=n 1+(n 1)ρ − .
− − 1 (N n 1)ρ
− − −
Note that, due to symmetrical gains only N +1 cuts need to be considered from all possible 2N cuts. This bound is a
generalization of the outer bounds for Gaussian parallel 2-relay (symmetric) network in [6] (see also [2]). In particular, [3,
Lemma 7] is a generalization of the broadcast cut-set bound (2.49) and cross cut-set bounds (2.83-84) in [6] (and the simpler
bound (4) in [3] is a generalization of (2.49), the multiple access cut-set bound (2.58) and pure cross cut-set bound (2.68) in
[6]).
III. ANUPPERBOUNDONTHECAPACITYOF2-LAYERN-RELAYGAUSSIANNETWORK
In this section an upper bound on the capacity of 2-layer N-relay network is given. A cut is a set of nodes s 1S 2S
{ }∪ ∪
where 1S [1N],2S [2N]. The cardinalities of 1S and 2S are denoted n and m respectively. Also,
⊆ ⊆
iSc (cid:44)[iN] iS,i 1,2 .
\ ∈{ }
The three cuts s , s [1N] and s [1N] [2N], separating each layer from the previous layer, are referred as broadcast
{ } { }∪ { }∪ ∪
cut, multiple broadcast-access cut and multiple access cut respectively. The set of channels in every cut in the network is a
union of disjoint subset of channels in these three cuts. Accordingly, a cut is regarded to have a broadcast part, a multiple
broadcast-access part and a multiple access part. We start from the cut set bound [7]:
C(N,r ) sup min I(X ;Y X ) (2)
1,2,3 0,1S,2S 1Sc,2Sc,3 1Sc,2Sc
≤X0,[1N],[2N]1S⊆[1N], |
2S⊆[2N]
where, the supremum is subject to the unit transmit power constraint at the nodes. Now,
I(X ;Y X )
0,1S,2S 1Sc,2Sc,3 1Sc,2Sc
|
=h(Y X ) h(Y X )
1Sc 1Sc,2Sc 1Sc 0,[1N],[2N]
| − |
+h(Y Y ,X ) h(Y Y ,X )
2Sc 1Sc 1Sc,2Sc 2Sc 1Sc 0,[1N],[2N]
| − |
+h(Y Y ,X ) h(Y Y ,X )
3 1Sc,2Sc 1Sc,2Sc 3 1Sc,2Sc 0,[1N],[2N]
| − |
h(Y )+h(Y X )+h(Y X )
1Sc 2Sc 1Sc 3 1Sc,2Sc
≤ | |
h(Z ) h(Z ) h(Z )
1Sc 2Sc 3
− − −
=h(Y )+h(Y X )+h(Y X )
1Sc 2Sc 1Sc 3 1Sc,2Sc
| |
h(Y X ) h(Y X ) h(Y X )
1Sc 0 2Sc [1N] 3 [1N],[2N]
− | − | − |
=I(X ;Y )+I(X ;Y X )
0 1Sc 1S 2Sc 1Sc
|
+I(X ;Y X ) (3)
1S,2S 3 1Sc,2Sc
|
From (2) and (3),
C(N,r ) sup min I(X ;Y )+I(X ;Y X )+I(X ;Y X ) . (4)
1,2,3 0 1Sc 1S 2Sc 1Sc 1S,2S 3 1Sc,2Sc
≤X0,X[1N],X[2N]1S⊆[1N],2S⊆[2N]{ | | }
Note that, in (3) the three conditional mutual information terms are associated with the broadcast, the multiple broadcast-
access and the multiple access parts of a cut respectively. The first term I(X ;Y ) in (3), associated with the broadcast part,
0 1Sc
can be further upper bounded as
1
I(X ;Y ) log(1+ 1Sc r ). (5)
0 1Sc ≤ 2 | | 1
4
Now, let us focus on the third term.
I(X ;Y X )
1S,2S 3 1Sc,2Sc
|
=h(Y X ) h(Y X ) (6)
3 1Sc,2Sc 3 [1N],[2N]
(cid:16) | (cid:88) − | (cid:12) (cid:17)
=h √r3 (X2j f2j(X1Sc,2Sc))+Z3(cid:12)(cid:12)X1Sc,2Sc
−
2j∈2S
h(Z )
3
−
(cid:16) (cid:88) (cid:17)
h √r3 (X2j f2j(X1Sc,2Sc))+Z3 h(Z3) (7)
≤ − −
2j∈2S
The equality (6) can also be written as
I(X ,X ;Y X ,X )
1S 2S 3 1Sc 2Sc
|
=h(Y X ) I(Y ;X X ) h(Y X )
3 2Sc 3 1Sc 2Sc 3 [1N],[2N]
| − | − |
and if we ignore the term I(Y ;X X ) then the remaining terms can be further upper bounded by the ND bound. We
3 1Sc 2Sc
|
use the additional information X available across the cut in an attempt to obtain a tighter bound.
1Sc
In (7), f (X ,X ),2j 2S can be any functions. But, to obtain tightest possible upper bound, it is natural to consider
2j 1Sc 2Sc
∈
this function to be the “best” estimator of X given X . If we restrict it to be the minimum mean square error
2j 1Sc,2Sc
(MMSE) estimator for approximating X based on X ,X then the covariance matrix for the vector of random variables
2j 1Sc 2Sc
(X f (X ,X ),2j 2S) (note that these random variables are MMSE [8]) can be represented as [9]
2j 2j 2Sc 1Sc
− ∈
Q =Q Q Q− Q (8)
2S·1Sc2Sc 2S,2S − 2S,1Sc2Sc 1Sc2Sc,1Sc2Sc 1Sc2Sc,2S
where,Q isthesub-matrixofthecovariancematrixQ (whichis,bydefinition,positivesemidefinite)forX
A,B [1N][2N] [1N],[2N]
corresponding to the rows for X and the columns for X . Also, Q− is the Moore-Penrose generalized inverse
A B 1Sc2Sc,1Sc2Sc
of the matrix Q . The matrix Q is the generalized Schur complement of Q in Q .
1Sc2Sc,1Sc2Sc 2S·1Sc2Sc 1Sc2Sc,1Sc2Sc [1N][2N]
If Q is invertible, then its generalized inverse is also the inverse of the matrix and hence the generalized Schur
1Sc2Sc,1Sc2Sc
complementreducestotheSchurcomplement.TheMMSEhasbeenusedtoobtainupperboundsonthecapacityofsymmetric
channel models in [10] as well as [3]. A pictorial presentation of the correlation matrix Q and submatrices associated
[1N],[2N]
with a generic cut s 1S 2S is given in Figure 3.
{ }∪ ∪
1S 1Sc 2S 2Sc
1S
[1N]
1Sc
2S
2Sc [2N]
n m
N n N m
− −
Fig.3. SketchofthematrixQ andsubmatricesassociatedwithacut{s}∪1S∪2S.
[1N],[2N]
Again, we continue with the notations in [3]: I means the a a identity matrix, 1 means a b matrix of ones. We drop
a a,b
× ×
the subscripts when the dimension is clear from context. Then, the equation (7) can be further upper bounded as follows.
(cid:16) (cid:88) (cid:17)
h √r3 (X2j f2j(X1Sc,2Sc))+Z3 h(Z3)
− −
2j∈2S
1
log(1+r 1TQ 1) (9)
≤ 2 3 2S·1Sc2Sc
5
The second term I(X ;Y X ) in (3) is associated with the multiple broadcast-access part of a cut.
1S 2Sc 1Sc
|
I(X ;Y X )
1S 2Sc 1Sc
|
=h(Y X ) h(Y X )
2Sc 1Sc 2Sc [1N]
(cid:16)(cid:16) | (cid:88) − | (cid:17) (cid:12) (cid:17)
=h √r2 (X1i f1i(X1Sc))+Z2j :2j 2Sc(cid:12)(cid:12)X1Sc
− ∈
1i∈1S
h(Z :2j 2Sc)
2j
− ∈
(cid:16)(cid:16) (cid:88) (cid:17) (cid:17)
h √r2 (X1i f1i(X1Sc))+Z2j :2j 2Sc
≤ − ∈
1i∈1S
h(Z :2j 2Sc)
2j
− ∈
1
2Sc log(1+r 1TQ 1) (10)
≤ 2| | 1 1S·1Sc
where Q = Q Q Q− Q is the Schur complement, Q ,Q , Q are sub-matrices
1S·1Sc 1S,1S − 1S,1Sc 1Sc,1Sc 1Sc,1S 1S,1S 1S,1Sc 1Sc,1S
of Q and Q− is the Moore-Penrose generalized inverse of Q .
[1N] 1Sc,1Sc 1Sc,1Sc
Note that, “most” of correlation between X and X for all 1i 1S is “taken care of” by subtracting the MMSE
1i 1Sc
∈
estimator. Hence, using independence inequality for entropies (independence bound) is not the worst way of bounding the
term. But, (10) is still crude since correlation among (√r2(cid:80)1i∈1S(X1i−f1i(X1(cid:80)Sc))+Z2j) for all 2j ∈2Sc is not utilized in
upper bounding the term. We now obtain a different bound as follows. Let √r2 1i∈1S(X1i−f1i(X1Sc))=W and without
loss of generality, assume that 2Sc = 21,22,..., 2Sc , then
{ | |}
I(X ;Y X )
1S 2Sc 1Sc
|
=h(Y X ) h(Y X )
2Sc 1Sc 2Sc [1N]
| − |
h((W +Z ):2j 2Sc) h(Z )
2j 2Sc
≤ ∈ −
|2Sc|
(cid:88)
= h(W +Z W +Z ,...,W +Z )
2j 2(j−1) 2|Sc|
|
2j=21
h(Z )
2Sc
−
|2Sc|
(cid:88)
h(Z Z )+h(W +Z ) h(Z )
2j 2(j−1) 21 2Sc
≤ − −
2j=22
=(2Sc 1)h(Z Z )+h(W +Z ) h(Z )
2j 2j(cid:48) 2k 2Sc
| |− − −
1
=(2Sc 1) log4πeσ2+h(W +Z )
| |− 2 2k
1
2Sc log2πeσ2
−| |2
1
=(2Sc 1) log4πeσ2+h(W +Z )
| |− 2 2k
1 1
(2Sc 1) log2πeσ2 log2πeσ2
− | |− 2 − 2
2Sc 1 1
= | |− +h(W +Z ) log2πeσ2
2 2k − 2
1(cid:0)2Sc 1+log(1+r 1TQ 1)(cid:1) (11)
≤ 2 | |− 2 1S·1Sc
where, 2j =2j(cid:48) and 1log2πeσ2 is the differential entropy of Gaussian distribution with variance σ2. Note that the inequality
(cid:54) 2
(11) is valid only for 2Sc 1 (for 2Sc = , I(X ;Y X )=0).
1S 2Sc 1Sc
| |≥ ∅ |
Lemma 1: From (4), (5), (10), (11), we have (12) for any 2Sc and (13) for nonempty 2Sc.
C(N,r )
1,2,3
1(cid:16) (cid:17)
sup min log(1+ 1Sc r )+ 2Sc log(1+r 1TQ 1)+log(1+r 1TQ 1) (12)
≤Q[1N],[2N]1S⊆[1N],2 | | 1 | | 2 1S·1Sc 3 2S·1Sc2Sc
2S⊆[2N]
C(N,r )
1,2,3
1(cid:16) (cid:17)
sup min log(1+ 1Sc r )+ 2Sc 1+log(1+r 1TQ 1)+log(1+r 1TQ 1) (13)
≤Q[1N],[2N]1S⊆[1N],2 | | 1 | |− 2 1S·1Sc 3 2S·1Sc2Sc
2S⊆[2N]
6
Now we focus on the form of the covariance matrix. It can be shown using time-sharing argument and symmetry in the
network that we can restrict our attention to Q of the form
[1N],[2N]
(cid:18) (cid:19)
ρ 1 +(1 ρ )I ρ 1
1 N,N − 1 N 1,2 N,N (14)
ρ 1 ρ 1 +(1 ρ )I
1,2 N,N 2 N,N 2 N
−
without loss in optimality which is equivalent to (15) where, the parameters ρ represent correlation between X and X ,
i il ik
il,ik [iN],i 1,2 , and ρ , represents correlation between X and X , 1l [1N],2k [2N].
12 1l 2k
∈ ∈{ } ∈ ∈
 
1 ρ ρ ρ ρ ρ
1 1 12 12 12
··· ···
ρ1 1 ρ1 ρ12 ρ12 ρ12
 ... ... ·.·.·. ... ... ... ·.·.·. ... 
 
ρ1 ρ1 ··· 1 ρ12 ρ12 ··· ρ12 (15)
ρ12 ρ12 ρ12 1 ρ2 ρ2
 ··· ··· 
ρ12 ρ12 ρ12 ρ2 1 ρ2
 ··· ··· 
 ... ... ... ... ... ... ... ... 
ρ ρ ρ ρ ρ 1
12 12 12 2 2
··· ···
Sketch of proof: First fix a cut under consideration and now suppose a matrix of some another form is optimal. Note that
all matrices obtained by permutation mappings σ :[1N] [1N] and σ :[2N] [2N] are also optimal due to symmetry.
1 2
−→ −→
Now, time sharing between all such (N!)2 matrices is also optimal. By time-sharing all the matrices for equal time duration,
the matrix for such time sharing scheme has the form (15).
Positivesemidefinitepropertyofacovariancematrixrestrictstherangeofρ ,ρ ,ρ asfollows(seeAppendixAfordetails).
1 2 12
ρ ,ρ [ 1,1]
1 2
∈ −
(cid:20) (cid:26) (cid:27)(cid:21)
1+(N 1)ρ 1+(N 1)ρ
ρ 1,min − 1, − 2 (cid:44)ζ
12 ∈ − N N
Note that 1TQ 1=ψ(n,ρ ) is already derived in [3] and the valid range for ρ is also given. Now we turn our attention
1S·1Sc 1 1
to deriving expression for 1TQ 1. This involves finding inverse of sub-matrices associated with cuts (see Appendix
2S·1Sc2Sc
B for details) and the Schur complement. We derive the expression for Schur complement as follows (see Appendix C for
details). We derive the expression for 1TQ 1 and is given in (16) (see Appendix D).
2S·1Sc2Sc
1TQ 1=φ(n,m,ρ )=m(cid:0)1+(m 1)ρ m(N n)ρ2 (x+(N n 1)y)
2S·1Sc2Sc 1,2,12 − 2− − 12 − −
m(N n)2(N m)ρ4 (e+(N m 1)f)(x+(N n 1)y)2
− − − 12 − − − −
+2m(N m)(N n)ρ ρ2 (e+(N m 1)f)(x+(N n 1)y)
− − 2 12 − − − −
m(N m)ρ2(e+(N m 1)f)(cid:1) (16)
− − 2 − −
1
where x+(N n 1)y =
− − 1+(N n 1)ρ
1
− −
(1+(N n 1)ρ )
and e+(N m 1)f = − − 1 .
− − (1+(N n 1)ρ )(1+(N m 1)ρ ) (N m)(N n)ρ2
− − 1 − − 2 − − − 12
Note that, If 1S =2S = , then (we take the convention Q =0 , Q =0 and Q =0)
∅,A 1,|A| A,∅ |A|,1 ∅,∅
∅
1TQ 1=0. (17)
2S·1Sc2Sc
Since the derivation of φ() involves (or assumes) inverting Q , we cannot use the function φ() as a repre-
1Sc2Sc,1Sc2Sc
· ·
sentation of 1TQ 1 when for certain values of correlation coefficients the submatrices Q and Q
2S·1Sc2Sc 1Sc,1Sc 1Sc2Sc,1Sc2Sc
corresponding to cuts are not invertible. In particular, for the following cases the matrix Q is not invertible and
1Sc2Sc,1Sc2Sc
we use Moore-Penrose generalized inverse to find generalized Schur complement.
• At the other extreme (compared to the situation in (17)) if 1S 2S =[1N] [2N] then
∪ ∪
1TQ 1=N(1+(N 1)ρ ). (18)
2S·1Sc2Sc 2
−
• If 1S (cid:40)[1N],2S =[2N] and ρ1 =1 then
1TQ 1=N(1+(N 1)ρ Nρ2 ). (19)
2S·1Sc2Sc − 2− 12
7
• If 2S (cid:40)[2N],1S =[1N] and ρ2 =1 then
1TQ 1=0 (20)
2S·1Sc2Sc
• If =1S 2S (cid:40)[1N] [2N] and ρ1 =ρ2 =ρ12 =1, then
∅(cid:54) ∪ ∪
1TQ 1=0. (21)
2S·1Sc2Sc
Now, for these values the function φ() is undefined:
·
ρ =1 (22)
1,2,12
1
ρ = − ,n=0 (23)
1 N 1
−
Note that φ() seems undefiend if we put the values n=N and ρ =1 simulteneously. But, if we evaluate φ() case by case
1
· ·
(cut by cut) basis, i.e., first fixing a cut and thus a value of n (e.g., N) and evaluating the function for this value, and then
evaluate the function with the value of ρ (e.g., 1) then, φ() is in fact defined. Similar is the case with the values (1) ρ =1
1 2
and m=N and, (2) ρ = −1 and m=0. ·
2 N−1
But, for the parameter values (22), we have
lim φ(n,m,ρ =1,ρ =1,ρ )=1TQ 1 (24)
ρ12↑1 1 2 12 2S·1Sc2Sc |ρ1,2,12=1
which follows from (17)-(21). Now, note that for the situation (23), φ() is undefined (the limit is positive infinity) only when
·
n = 0 and m > 0. But n = 0,m = 0 implies broadcast cut and n = 0,m > 0 is practically meaningless since the channels
in the broadcast cut already separates s from d (and hence there is no need to add more channels to the cut). As such, we
can assume that if n=0 then m=0 and thus avoid the situations when φ() is undefined. Actually, it may be assumed that
·
n m without loss of generality, but this assumption is unnecessary (does not lead to undefined function) for most cases
≥
except when m=0.
There may be still more parameter values for which the matrix is not invertible. For examining this values we first make
rank preserving transformation of the matrix (15), as shown in (25). Then we check the paprameter values for which the
submatrices Q and Q are non-invertible.
1Sc,1Sc 1Sc2Sc,1Sc2Sc
 
1 ρ ρ ρ ρ ρ ρ ρ
1 1 1 12 12 12 12
··· ···
ρ1 1 1 ρ1 0 0 0 0 0 0 
 − − ··· ··· 
ρ1 1 0 1 ρ1 0 0 0 0 0 
 −... ... −... ·.·.·. ... ... ... ... ·.·.·. ... 
 
ρ1 1 0 0 1 ρ1 0 0 0 0 
 − ··· − ··· 
ρ1 1 0 0 1 ρ1 0 0 0 0  (25)
 − ··· − ··· 
 0 0 0 0 1 ρ2 0 0 ρ2 1
 ··· − ··· − 
 ... ... ... ... ... ... ... ... ... ... 
 
 0 0 0 0 0 1 ρ2 0 ρ2 1
 ··· ··· − − 
 0 0 0 0 0 0 1 ρ2 ρ2 1
··· ··· − −
ρ ρ ρ ρ ρ ρ ρ 1
12 12 12 12 2 2 2
··· ···
Now,notethatthesubmatricesQ associatedwiththisnewmatrix(25)arenotinvertible(thatis,thesubmatrices
1Sc2Sc,1Sc2Sc
are not full-rank) when ρ =1,ρ =1,ρ = 1 or when ρ = 1,n=N 1,m=N 1. For the first case the limit of
1 2 12 12
− − − −
the function is as follows.
lim φ(n,m,ρ =1,ρ =1,ρ )=0 (26)
1 2 12
ρ12↓−1
For the second case
(cid:18) (cid:19)
1 1
Q1Sc2Sc,1Sc2Sc = 1 −1
−
we have
(cid:16) (N 1)(1+ρ )2(cid:17)
1TQ2S·1Sc2Sc1=(N −1) 1+(N −2)ρ2− − 4 2 (27)
(cid:44)µ(ρ ) (28)
2
whereas φ() as ρ 1 except for N =1 or ρ =1 for which the limit and (27) are zero. Hence, for this special
12 2
· →−∞ →−
case, we must rely on (27) to evaluate the bound. Also, the right hand side of (27) is positive when these two conditions are
satisfied: ρ <1 and ρ > N−5.
2 2 N−1
Combining these results with (12)-(13) render the following bounds.
8
Lemma 2: The capacity of 2-layer N-relay Gaussian network is upper bounded as
1(cid:16)
C(N,r ) sup min log(1+(N n)r )
1,2,3 ≤ρ1,ρ2∈[−1,1),n,m∈{0,...,N}2 − 1
ρ12∈ζ
+(N m)log(1+ψ(n,ρ )r )
1 2
−
(cid:17)
+log(1+φ(n,m,ρ )r ) (29)
1,2,12 3
for N m 0 and
− ≥
1(cid:16)
C(N,r ) sup min log(1+(N n)r )
1,2,3 ≤ρ1,ρ2∈[−1,1),n,m∈{0,...,N}2 − 1
ρ12∈ζ
+N m 1+log(1+ψ(n,ρ )r )
1 2
− −
(cid:17)
+log(1+φ(n,m,ρ )r ) . (30)
1,2,12 3
for N m 1 where, if n=0 then m=0 for both (29) and (30) and if n=N 1,m=N 1 and ρ = 1 then ρ =1.
12 2
− ≥ − − −
For the special case when n=N 1,m=N 1,ρ = 1 and ρ <1 we have
12 2
− − −
1(cid:16)
C(N,r ) log(1+r )
1,2,3 ≤ 2 1
+ sup log(1+ψ(n,ρ )r )
1 2
ρ1∈[−1,1)
(cid:17)
+ sup log(1+µ(ρ )r ) . (31)
2 3
ρ2∈[max{−1,NN−−51},1)
Remark 1: By letting ρ ,ρ [ 1,1), the possibility of 1 ζ is already eliminated. In other words, ρ = 1 or ρ = 1
1 2 1 2
∈ − ∈ (cid:54) (cid:54)
implies ρ =1.
12
(cid:54)
Let, (n∗,m∗) be a pair from 0,...,N 0,...,N such that it minimizes the quantity in (29) or in (30). We call such
{ }×{ }
a pair as minimizer pair.
Note that if we put ρ =0 in φ(n∗,m∗,ρ ) then it reduces to ψ(m∗,ρ ). Specifically,
12 1,2,12 2
sup φ(n∗,m∗,ρ )(=a) sup ψ(m∗,ρ )
1,2,12 2
ρ1,2∈[−1,1),ρ12=0 ρ2∈[−N1−1,1)
(=b) sup ψ(m∗,ρ )
2
ρ2∈[0,1)
where the lower bound for ρ in (a) follows from the range ζ for ρ and (b) follows from [3, Lemma 7]. Moreover, ρ =0
2 12 12
isoneofthevalidvaluestooptimizethefunctionφ(n,m,ρ ).Theparametervalueρ =0alsoenforcesthelowerbound
1,2,12 12
−1 for ρ . Thus,
N−1 1
Lemma 3: For2-layerN-relaynetworks,jointmaximizationovertheparametersρ in(29)-(30)doesnotrenderabetter
1,2,12
bound.
Remark 2: These observations may be of interest - The function φ(n,m,ρ ) is even in ρ suggesting that its values
1,2,12 12
may be restricted to non-negative (or non-positive) range (without loss in optimization). Moreover, ρ =0 is a critical point
12
and is a maxima (see Appendix E).
Now we state upper bounds on the capacity which follows from Lemma 2 and 3.
Theorem 2: For 2-layer N-relay networks, the capacity is upper bounded as
1(cid:16)
C(N,r ) log(1+(N n∗)r )
1,2,3 ≤ 2 − 1
+(N m∗) sup log(1+ψ(n∗,ρ )r )
1 2
−
ρ1∈[0,1)
(cid:17)
+ sup log(1+ψ(m∗,ρ )r ) (32)
2 3
ρ2∈[0,1)
for N m∗ 0 and
− ≥
1(cid:16)
C(N,r ) log(1+(N n∗)r )+N m∗ 1
1,2,3 ≤ 2 − 1 − −
+ sup log(1+ψ(n∗,ρ )r )
1 2
ρ1∈[0,1)
(cid:17)
+ sup log(1+ψ(m∗,ρ )r ) (33)
2 3
ρ2∈[0,1)
9
for N m 1 and for all minimizer pairs (n∗,m∗).
− ≥
Boththebounds,(32)and(33),areequalwhensup log(1+ψ(n ,ρ )r )=1orwhenN m∗ =1.ForN m∗ 2,
ρ1∈[0,1) ∗ 1 2 − − ≥
the bound (32) is tighter in the range sup log(1 + ψ(n∗,ρ )r ) < 1 and the bound (33) is tighter in the range
ρ1∈[0,1) 1 2
sup log(1+ψ(n∗,ρ )r )>1.
ρ1∈[0,1) 1 2
IV. CONCLUSIONANDFUTUREDIRECTIONS
Inthisworkwederivedupperboundson2-layerN-relayGaussiannetworkandshowedthatjointoptimizationofcorrelation
coefficients is not necessary for evaluating a better bound. We are investigating generalization of the upper bounds for general
L-layer symmetric networks. Other directions of future research are to (1) analyze and compare achievable rates for existing
and new schemes, and to characterize the gap between the upper and lower bounds on the capacity, (2) investigate whether
for L-layer N-relay networks the joint optimization of correlation coefficient is necessary, and (3) extend the bounds for the
case of asymmetric gains.
In general, the joint optimization across different layers (in non-layered) networks may render tighter bound. It would be
interesting to characterize a class of networks for which the joint optimization leads to a better cut-set bound.
REFERENCES
[1] E.C.vanderMeulen,“Three-terminalcommunicationchannels,”AdvancesinAppliedProbability,vol.3,no.1,pp.120–154,1971.
[2] B.ScheinandR.Gallager,“TheGaussianparallelrelaynetwork,”inIEEEInt.Symp.Inform.Theory,pp.22–,2000.
[3] U.NiesenandS.Diggavi,“TheapproximatecapacityoftheGaussianN-relaydiamondnetwork,”IEEETrans.Inform.Theory,vol.59,pp.845–859,
Feb2013.
[4] T. Courtade and A. Ozgur, “Approximate capacity of Gaussian relay networks: Is a sublinear gap to the cutset bound plausible?,” in IEEE Int. Symp.
Inform.Theory,pp.2251–2255,June2015.
[5] A.Avestimehr,S.Diggavi,andD.Tse,“Wirelessnetworkinformationflow:Adeterministicapproach,”IEEETrans.Inform.Theory,vol.57,pp.1872–
1905,April2011.
[6] B.Schein,Distributedcoordinationinnetworkinformationtheory. PhDthesis,MassachusettsInstituteofTechnology,Cambridge,2001.
[7] T.M.CoverandJ.A.Thomas,ElementsofInformationTheory. Wiley-Interscience,2006.
[8] T.Kailath,LecturesonWeinerandKalmanfiltering. Springer-Verlang,1981.
[9] R.Muirhead,Aspectsofmultivariatestatisticaltheory. Wiley,1982.
[10] J.Thomas,“FeedbackcanatmostdoubleGaussianmultipleaccesschannelcapacity(corresp.),”IEEETrans.Inform.Theory,vol.33,pp.711–716,Sep
1987.
APPENDIXA
RANGEOFCORRELATIONCOEFFICIENTS
Nowwefindtherangeofρ ,ρ ,ρ forthematrixtobepositivesemidefinite.FormatricesA,B,Cofthesamedimension,
1 2 12
(cid:18) (cid:19)
A C
det =det(AB C2) (34)
C B −
Letting A=ρ 1 +(1 ρ )I , B=ρ 1 +(1 ρ )I and C=ρ 1 , we get
1 N,N 1 N 2 N,N 2 N 12 N,N
− −
det(Q λI)
[1N],[2N]
−
=(cid:2)(1 λ)2+(N 1)(1 λ)ρ +(N 1)(1 λ)ρ +(N 1)2ρ ρ N2ρ2 (cid:3)
− − − 1 − − 2 − 1 2− 12
((1 λ)2 (1 λ)ρ (1 λ)ρ +ρ ρ )N−1
1 2 1 2
− − − − −
The eigenvalues are
λ=1 ρ
1
−
=1 ρ
2
−
=1 Nρ +(N 1)ρ
12 1
− −
=1 Nρ +(N 1)ρ
12 2
− −
Hence, ρ ,ρ must be less than 1 and ρ min 1+(N−1)ρ1,1+(N−1)ρ2 for the matrix to be a positive semidefinite. Thus
1 2 12 ≤ { N N }
the range of the correlation coefficients is ρ ,ρ [ 1,1] and ρ [ 1,min 1+(N−1)ρ1,1+(N−1)ρ2 ].
1 2 ∈ − 12 ∈ − { N N }
10
APPENDIXB
FINDINGINVERSEOFSUB-MATRICESASSOCIATEDWITHCUTSTOFINDSCHURCOMPLEMENTS
The following is the inverse of a sub-matrix of Q corresponding to a cut where A = ρ 1 +(1 ρ )I is
[1N],[2N] 1 n,n 1 n
−
n n and n 0,1,...,N , B = ρ 1 is n m with entries ρ , C = ρ 1 is m n with entries ρ and
12 n,m 12 12 m,n 12
× ∈ { } × ×
D=ρ 1 +(1 ρ )I is m m where m 0,1,...,N .
2 m,m 2 m
− × ∈{ }
(cid:18)A B(cid:19)−1 (cid:18)A−1+A−1B(D CA−1B)−1CA−1 A−1B(D CA−1B)−1(cid:19)
C D = (D CA−−1B)−1CA−1 − (D C−A−1B)−1 (35)
− − −
Remark 3: In this section calculations are with respect to the sets sizes 1S =N n and 2S =N m. Note that letting
| | − | | −
1S =n and 2S =m does not change the set of bounds.
| | | |
1 ρ1 ρ1−1  (n−2)ρ1+1 −ρ1  x y y
A−n×1n =ρρ...1 ρ1... ··.··.··. ρ1...1 =−−((nn−−11))ρρ2121−++...ρ((1nn−−22))ρρ11++11 −−((nn−−11(n))ρρ−21212++)...((ρnn1−−+221))ρρ11++11 ··.··..··=yy... xy... ··.··.··. xy... (36)
1 1
··· ···
 
  z z z
nρ12((n−1)ρ12y+ρ12x) nρ12((n−1)ρ12y+ρ12x) ··· z z ··· z
CA−1Bm×m =nρ12((n−1)...ρ12y+ρ12x) nρ12((n−1)...ρ12y+ρ12x) ·.·..·=... ... ·.·.·. ... (37)
z z z
···
where CA−1B is a m m matrix.
×
   
1 z ρ z ρ z u v v
2 2
− − ··· − ···
ρ2 z 1 z ρ2 z v u v
D−CA−1Bm×m = −... −... ·.·.·. −... =... ... ·.·.·. ... (38)
ρ z ρ z 1 z v v u
2 2
− − ··· − ···
 (m−2)v+u −v  e f f
(D−CA−1B)−m1×m =−−((mm−−11))vv22++−((...vmm−−22))uuvv++uu22 −−((mm−−11())mvv22−++2((...)mmv+−−u22))uuvv++uu22 ··.··..··=ff... fe... ··.··.··. fe... (39)
···
B(D CA−1B)−1C (40)
n×n
−
 
  w w w
mρ12((m−1)ρ12f +ρ12e) mρ12((m−1)ρ12f +ρ12e) ··· w w ··· w
=mρ12((m−1...)ρ12f +ρ12e) mρ12((m−1...)ρ12f +ρ12e) ·.·..·=... ... ·.·.·. ... (41)
w w w
···
A−1B(D CA−1B)−1CA−1 (42)
− n×n
     
x y y w w w x y y
··· ··· ···
y x y w w w y x y
=... ... ·.·.·. ...×... ... ·.·.·. ...×... ... ·.·.·. ... (43)
y y x w w w y y x
··· ··· ···
   
x y y w(x+(n 1)y) w(x+(n 1)y) w(x+(n 1)y)
··· − − ··· −
y x y w(x+(n 1)y) w(x+(n 1)y) w(x+(n 1)y)
=... ... ·.·.·. ...× ... − ... − ·.·.·. ... −  (44)
y y x w(x+(n 1)y) w(x+(n 1)y) w(x+(n 1)y)
··· − − ··· −
w(x+(n 1)y)2 w(x+(n 1)y)2 w(x+(n 1)y)2
− − ··· −
w(x+(n 1)y)2 w(x+(n 1)y)2 w(x+(n 1)y)2
= ... − ... − ·.·.·. ... −  (45)
w(x+(n 1)y)2 w(x+(n 1)y)2 w(x+(n 1)y)2
− − ··· −