Long-term average capacity region

constrained channels [1] and denition of capacity region per unit cost given in [14].

Denition 2.

A rate

K

-tuple ^R?2R_K+ is long-term average

-achievable if for all

>

0 there exist

L

such that for

L

L

variable-rate (

N;L;

^R

;

;

)-codes can be found with

R

k

> R

_?k ^;

for

k

= 1

;

;K

. A rate

K

-tuple is achievable if it is

-achievable for all 0

< <

1. The long-term average capacity region

C

K;N() is the closure of the convex hull of all long-term

achievable rate

K

-tuples.

Denition 3.

A

K

-tuple^r?2R_K+ is a long-term average

-achievable rate per unit energy if for all

>

0 there exist an energy vector = (

1

;

;

K) such that for ¹ variable-rate (

N;L;

^R

;

=

(

NL

)

;

)-codes can be found with (

LNR

k)

=

k

> r

_?k^;

for

k

= 1

;

;K

. A rate

K

-tuple is achievable if it is

-achievable for all 0

< <

1. The long-term average capacity region per unit-energy

U

K;N is the set of all long-term achievable rate

K

-tuples per

unit-energy.

In this setting, it is meaningful to study the largest achievable long-term average rate region, subject to the short-term power constraint (4.3). More-over, in the energy-limited case investigated here, a meaningful system de-sign criterion is to look for the largest achievable long-term average capacity per unit energy(bit/joule). Next, in analogy with [13, 14], we characterize the long-term average capacity region and the long-term average capacity per unit energy for our system. We also give limiting theorems for large delay

N

.

4.3 Long-term average capacity region

We have the following result:

Theorem 1.

The long-term average capacity region is given by

C

K;N() = ^[

2;_K;N()

R2RK+ : ^8Af1

;

;K

^g

X

k^2A

R

k E

"

N

1

XN

n=1log 1 +^X

k^2A

k;n

k;n(^Sn)

!#)

(4.7) where expectation is with respect to the channel state^SN and where ;K;N() is the set of feasible causal short-term power allocation policies =^f

k;n:

1For two vectors ^a and ^b, the notation ^{a b} means that the dierence ^a;b has nonnegative components.

k

= 1

;

;K;n

= 1

;

;N

^g dened as

;K;N()=

(

2RKN+ : 1

N

XN

n=1

k;n(^Sn)

k

)

(4.8) where

k;n(^Sn) indicates the causality constraint.

Proof.

The achievable part easily follows by constructing random codes with entries ^xk;n ^N(

0 ;

k;n

I

) such that the variances satisfy (4.8) and the rates satisfy all the inequalities in (4.7). The converse part follows by showing that the capacity of the

N

-slot extension channel is achieved by Gaussian input distributions in the form of those used in the achievable part of the theorem. For details see Appendix 4.A.

Remark.

The region

C

K;N() is convex in , in fact, by applying Jensen's inequality it is straightforward to see that if ^{R(a) 2}

C

K;N() and

R(b) 2

C

K;N() then, for every

² [0

;

1] we have

^R(a)+ (1^;

)^{R(b) 2}

C

K;N(). For this reason in Theorem 1 the convex-hull operation is not needed.

For a given power policy in ;K;N(), let ^P() be the set of long-term average rates achievable by applying. Theorem 1 states that the long-term average capacity region

C

K;N() is the union of all the polymatroids ^P()

C

K;N() = ^[

2;_K;N()

P() (4.9)

Such formulation of

C

K;N() is not useful unless we can determine its clo-sure set. We explicitly characterize the boundary surface of the

C

K;N(), following the approach of [13], as the closure of the convex-hull all

K

-tuples

R2RK+ that solve

max

R2C_K;N()

XK

k=1

k

R

k (4.10)

for some= (

1

;

;

K)^2R_K+. As in [13], the optimization in (4.10) can be written as

max

2;_K;N() max

R2P()

XK

k=1

k

R

k (4.11)

Since^P() is a polymatroid, the solution of the inner maximization in (4.11) is attained by one of the (at most)

K

! vertices of ^P(). Such vertex is uni-vocally determined by the entries of the vector: it is the one correspond-ing to the decodcorrespond-ing order

K

;

K^;1

;

;

1 where is the permutation

4.3 Long-term average capacity region 117

of ^f1

;

2

;

;K

^g that orders in decreasing order, i.e.,

¹

>

>

K. Hence, for any policy we have

max maxi-mization (4.10) into the maximaxi-mization of the right hand side (RHS) of (4.12) over the power policies ²;K;N(). Due to the causal nature of the chan-nel state information, the maximization of (4.12) with respect to , and hence the solution of (4.10), is obtained by Dynamic Programming. We have the following:

Theorem 2.

Dene for

n

= 1

;

;N

the Dynamic Programming recursion

S

n(

P

1

;

;P

K;) = E

Proof.

Recursion (4.13) and the optimal power policy (4.16) follow easily from the general theory of Dynamic Programming [98] when the cost func-tion to be maximized is given by the RHS of (4.12) and the system evolves,

from slot

n

to slot

n

+ 1, according to (n

;

^P)^! (n+1

;

^{P ;bp}_n). Details

can be found in Appendix 4.B.

Note that

XK

k=1

k

R

^bk;N(

;

) = 1

N S

^N(

N

1

;

;N

K;) (4.17) In [92], the authors computed numerically the recursion (4.13) for

K

= 1 in the Rayleigh fading case.

Remark.

In contrast with [13], the convex-hull operation in (4.14) is needed since the rates ^RbN(

;

) might not be continuous functions of . Consider, as an example, the case of

N

= 1. The region

C

K;1() coincides with the ergodic capacity region of a fading channel without CSI at the transmitters, the dominant face of which is an hyper-plane in

K

dimensions.

Due to the polymatroid structure of

C

K;1(), the solution (4.15) is one of the (at most)

K

! vertices of the dominant face. Hence, as varies in ^R_K+, the set of^RbN(

;

) contains at most

K

! points. It is clear that the convex hull operation is needed here.

From Theorem 2, by solving recursion (4.13) for

n

= 1, we see that the optimal solution is

p

^bk;1 =

P

k for every and for every . Hence, from (4.16) with

N

= 1, we have that

^bk;1 =

k for all

k

, i.e., the optimal solution for

N

= 1 is constant power allocation. From (4.16), we also see that

^bk;N =

N

k^;PN^;1

j=1

^bk;j which means that, on the last available slot, all the remaining energy is used regardless of the fading value, which is sensible since the remaining energy cannot be used on the next frame.

Due to the heavy notation of Theorem 2, it might not be so straightfor-ward to understand how to construct the optimal policy ^b and the role of the recursion (4.13). Hence, we give some explanation. To characterize the boundary surface of

C

K;N(), rst we need to solve the recursion (4.13) in or-der to determine the optimal values^f

p

^bk;n(;

;

^P)⁸

k

= 1

;K

^gfor

n

1.

Then, for a given delay

N

, we built up the optimal power policy (4.16) by considering the \length-

N

window" of optimal values ^f

p

^bk;n(;

;

^P) ⁸

k

= 1

;K

^g for

n

=

N;N

^;1

;

;

1. On the rst slot (

n

= 1), the optimal policy ^b1 = [

^b1;1

;

;

^bK;1] is derived from ^f

p

^bk;N(;

;

^P) ⁸

k

= 1

;K

^g computed forequal to the actual fading values 1 = [

1;1

;

;

K;1] and for power ^P equal to the total available energies

N

= [

N

1

;

;N

K], i.e., for each user

k

= 1

;

;K

bk;1(^S1;

;

) =

p

^bk;N(1

;N

;)

On the second slot (

n

= 2), the optimal policy ^b2 = [

^b1;2

;

;

^bK;2] is derived from^f

p

^bk;N^;1(;

;

^P)⁸

k

= 1

;K

^g. Given the fading realization

2 = [

1;2

;

;

K;2] and the remaining energies

N

^;b1(^S1;

;

), the optimal power allocation is

bk;2(^S2;

;

) =

p

^bk;N^;1(2

;N

^;b₁(^S1;

;

);)

4.3 Long-term average capacity region 119

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