Causal feedback and delay constraint
4.4 Long-term average capacity region per unit energyenergy
A byproduct of the proof of Theorem 1 is that the long-term average capacity region coincides with the standard \ergodic" capacity region of the
N
-slot extension channel, which is frame-wise memoryless (since the power control\correlates" only symbols inside the same frame). The following theorem is an immediate consequence of this fact and of the general theory of capacity per unit cost [14]:
Theorem 4.
The long-term average capacity region per unit energy isU
K;N = [2R
K
+
r2RK+: (
1r
1;
;
Kr
K)2C
K;N() (4.20)Proof.
The proof follows immediately from [14, Theorem 5].In analogy with [14], it is easy to show:
Theorem 5.
The long-term average capacity region per unit energy is the hyper-rectangleU
K;N =nr2RK+ :r
ks
(k)N o (4.21) wheres
(k)N , given bys
(k)N = limk!0 1
k sup2;1;N(k)E
"
N
1XN
n=1
k;nk;n(Sn)#
(4.22) is the
k
-th user single-user long-term average capacity per unit energy.Proof.
See Appendix 4.D.4.4 Long-term average capacity region per unit energy 121
The explicit solution of (4.22) was found in [92]. We report it here in our notation for later use:Theorem 6.
Thek
-th user single-user long-term average capacity per unit energys
(k)N is given by the Dynamic Programming recursions
(k)n = E[maxfs
(k)n;1;
kg] (4.23) forn
= 1;::: ;N
with initial conditions
(k)0 = 0 and where expectation is with respect to kF
(k)(x
). Furthermore,s
(k)N is achieved by the \one-shot" power allocation policy dened by ?k;n =N
k ifn
=n
?k0 otherwise (4.24)
where the random variable
n
?k, function of (k;1;
;
k;N), is dened asn
?k = minnn
2f1;::: ;N
g:k;ns
(k)N;no (4.25)Proof.
Expression (4.23) is the solution of the Dynamic Programming algorithm when the cost function to be maximized is (4.22). For moredetails, see the proof given in [92].
We have nicknamed the optimal policy
? \one-shot" because the whole available energyN
k is spent all at once in a single slot. In fact, in each slotn
2 f1;
;N
g, the transmitter compares the instantaneous fading gain k;n with the time varying thresholds
(k)N;n, if the fading is above the threshold then it transmits on the current slot by using all the available energy otherwise it waits for the next slot. Since the threshold to be used on the last available slot iss
(k)0 = 0, the available energy is used within the required delay ofN
slots with probability one. This feature of optimal policy was already found out in [92], but in this work the authors did not realize that what is the restricted context of an \approximation for low SNR" is actually the general solution to long-term average capacity per unit energy for every nite delayN
. Fig. 4.1 shows a fading realization over a window ofN
= 10 slots. We can this that in this case transmission takes place in slotn
= 6.From a practical implementation point of view, the \one-shot" policy is appealing. It requires virtually no computation, just a comparison of the instantaneous fading amplitude with a threshold. The threshold sequence
f
s
(k)n g1n=0can be pre-computed and stored in memory since it only depends on the fading statistic and not on the instantaneous fading values. Pol-icy (4.24) is \memoryless" in the sense that the only information needed about the past slots of the frame is whether transmission has already took place and it is decentralized, i.e.,n
?k only depends on (k;1;
;
k;N). More-over, when varying the delay requirements fromN
1 toN
2, the threshold1 2 3 4 5 6 7 8 9 10 0
1 2 3
n Fading power
Threshold sequence
Figure 4.1: Fading realization over a frame of
N
= 10 slots.sequence need not to be re-computed, only a dierent \chunk"f
s
(k)n gN2;1 n=0 , instead offs
(k)n gN1;1n=0 , has to be used. Notice also that the number of active users
K
does not aect the value of the thresholds.The behavior of
s
(k)N whenN
grows to innity is given by the following:Theorem 7.
For largeN
, thek
-th user single-user long-term average ca-pacity per unit energys
(k)N tends to thek
-th user single-user ergodic capacity per unit energy given explicitly byNlim!1
s
(k)N = supfkg (4.26) where supfkg= inf fx
0 :F
(k)(x
) = 1g.Proof.
See Appendix 4.E.Remark.
LetC
() be the capacity expressed in nat/dimension as a function of, and letC(E
b=N
0) denote the corresponding spectral eciency in bit/s/Hz as a function of the energy per bit vs. noise power spectral density,E
b=N
0, given implicitly by( ENb0 = log2C()
C
ENb0
= C()log2 (4.27)
The value (
E
b=N
0)min for which C(E
b=N
0)>
0 ,E
b=N
0>
(E
b=N
0)minis given by [85] ENb0
min = log2_C(0) where _
C
(0) is the rst derivative of the4.4 Long-term average capacity region per unit energy 123
capacity functionC
() at = 0. From the proof of Theorem 5, we see immediately that the reciprocal of (E
b=N
0)minfor thek
-th user is its capacity per unit energy (expressed in bit/joule), of the channel, i.e.,E
bN
0
min = log2
s
(k)N (4.28)The \one-shot" policy not only makes the most ecient use of the energy, by maximizing the number of expected correctly received number of bits per joule, but also reduces to the minimum the interference to other users since all the users transmit at minimum
E
b=N
0. Notice that as the delay constraint is relaxed, i.e.,N
grows, the minimum requiredE
b=N
0 lowers down. Of course nothing is for free: the fact that the system works at the minimumE
b=N
0 is because it uses of a large number of degree of free-dom (L
) per information bit, i.e., the system works in the so-called \innite bandwidth regime". As shown recently in [99], information theoretic per-formance in \wideband regime" is not only characterized by the minimum energy per bit since minimumE
b=N
0 alone is unable to give the correct tradeo bandwidth-power. Transmitting at minimumE
b=N
0 implies us-ing of an innitely large bandwidth (innite bandwidth regime) and hence having zero spectral eciency while, by increasing a bitE
b=N
0 from its min-imum value, the required bandwidth for reliable communication is large but nite (wideband regime) as well as the spectral eciency. The analysis on the wideband performance of our system, characterized by causal feedback and delay constraint, will be the topic of next chapter.The non-causal policy achieving long-term average capacity per unit-energy.
At this point is interesting to compare the optimal \one-shot" (causal) policy with the optimal non-causal policy achieving long-term average capacity per unit energy. We consider only the single user case, since we saw that in the multi-user case the long-term average capacity region is the Cartesian product of the single-user long-term average capacities. 2 If we allow the input to depend on the whole CSISN in a non-causal way, it is immediate to show that the optimal policy is \maximum selection" k;n?(nc)=( Nk
jMkj if
n
2M
k0 otherwise (4.29)
where
M
k =fn
: k;n = maxfk;1;
;
k;Ngg (4.30)2The K-user capacity region per unit cost is equal to the Cartesian product of the K single-user capacities per unit cost only if a) every user has an alphabet that contains a symbol of zero cost and b) for a given user, the use of the zero-cost symbol by all the other users corresponds to the most favorable single-user channel seen by the considered user. Those two hypothesis are always satisfy by additive channels.
and j