In this section, we study the power allocation scheme over the direct and cross channels in the DCLIC that maximizes its sum-rate. Recall from Corollary 1 that, if the underlying GIC of the DCLIC has strong interference (i.e., and ), then the sum-rate of the DCLIC is known for all values of the remaining channel parameters, and, in particular, for all transmit powers in the second band. Therefore, we pose this problem for the class of DCLICs with strong underlying GIC (see [A1] below). Also recall that the normalized bandwidth () of the second band is shared between the direct channels () and the cross channels (). We denote the fraction of alloted to the direct and cross channels by and , respectively, where and , and . Thus, provides a trade-off between the bandwidths in the direct channels () and the cross channels ().
5.1. Problem Formulation and Solution
For a fixed power allocation
in the mm-wave channels, we denote the sum-rate achievable at
and
in (
14) and (
15) by
and
, and the interference-free sum-rate given by the sum of individual rates in (
12) and (
13) by
, and present them below
where
and
. Furthermore, they satisfy
due to [A2] and [A1]. Therefore, a necessary and sufficient condition for
R to be an achievable sum-rate of the DCLIC is
, for some power allocation (
). The optimization problem that maximizes
R over the transmit powers (
) is then
Note that
is a convex optimization problem, since its objective function
R is linear, the equality constraints (
30) and (
31) are affine, and the inequality constraints (
27)–(
29) are convex. Furthermore,
satisfies Slater’s condition [
35] (Chapter 5.2.3), and thus it can be solved using the Karush–Kuhn–Tucker (KKT) conditions [
35] (Chapter 5.5.3). We relegate the details to
Appendix C.
It is well known that optimal power allocation in parallel Gaussian point-to-point channels follows the Waterfilling (WF) property. Due to this, if the power budget is sufficient small, it is allocated entirely to the “strongest” sub-channel, and, as the power budget is increased, power is allocated to the other “weaker” sub-channels, in addition to the strongest one (see Chapter 3.4.3 in [
32]). In the ensuing discussion, it becomes clear that the optimal power allocation in
(hereby referred to as “the optimal allocation”) has two noticeable properties: a
WF-like property, due to which it assigns power to the cross and direct channels following a WF-like allocation, and a
max-min property, due to which it increases the minimum of the sum-rate constraints.
Due to its WF-like property, the optimal allocation assigns the entire power budget (
P) to only a subset of all the direct and cross channels, depending on channel conditions that indicate whether the direct channels are stronger than the cross channels or vice versa. Moreover, if
P is sufficiently increased, it becomes optimal to allocate power to the remaining set of channels. In addition, since the objective of the problem is to maximize
, the optimal allocation assigns powers in such a way that minimizes or eliminates any difference between
and
. We observe that, due to this property (max-min property), the optimal allocation imposes a maximum limit on the cross-channel powers, which is unlike WF in [
32] where no such limit exists.
We study the optimal allocation by partitioning the entire set of channel parameters and
P into disjoint sets (
), such that the optimal allocation can be classified according to the power levels in the direct and cross channels in each set. Without loss of generality, we present the optimal allocation under
in
Table 1, and study it in detail. In this case, only fours sets,
,
,
and
, are sufficient. The power allocation under
can be readily obtained from
Table 1 by swapping the indices 1 and 2, and the case with
is briefly discussed in
Section 5.3.
For notational convenience, we express the optimal powers and the conditions of the sets
in
Table 1 in terms of the following functions:
and
. We relegate the details of the derivation to
Appendix C.
In the following, we use “the optimal allocation” and interchangeably to refer to the optimal power allocation for under , and discuss some interesting characteristics of .
First, the condition in implies that the direct channels are “stronger” than the cross channels, in the sense that, for sufficiently small P, the sum-rate achieved from allocating P only to both direct channels is larger than that achieved from any other subset of channels. Thus, following its WF-like property, allocates P entirely to the direct channels, i.e., , and zero power to the cross channels, i.e., . This allocation also achieves , which is consistent with the max-min property of .
Second, the condition in implies that the cross channels are “stronger” than the direct channels, in the sense that, for sufficiently small P, the sum-rate achieved from allocating P to both cross channels and a direct channel is larger than that achieved from only direct channels. Note that, unlike in , needs to allocate power to a direct channel in addition to both cross channels in , since allocating P entirely to the cross channels causes an imbalance between and () due to , which violates the max-min property. Therefore, shares P among the cross and direct channels from to preserve . Moreover, the condition ensures that power in the cross channels have not yet reached their maximum limits.
Third, as P is increased, shares P among all channels in . As P increases, the additional benefits from transmitting in a particular subset of channels (either direct channels as in , or cross and direct channels as in ) begins to diminish. Thus, following its WF-like property, starts sharing P among all channels. In addition, shares P in such a way that preserves .
Finally, if
P is increased sufficiently,
follows the allocation in
, where the cross-channel powers have reached their maximum limits,
. Therefore, as
P increases further, all subsequent increments of
P is allotted to only direct channels. We now say that the cross channels are saturated, in the sense that allocating more power beyond these limits does not improve the sum-rate. Such limits for the cross channels in
is unlike the WF allocation in [
32].
The cross channels become saturated due to the max-min property of . Recall that, in , allocates powers to all channels, which increase as P increases. The increases in and results in the increase of and by an equal margin. However, an increase in only increases , and an increase in only increases . Now, note that, due to its max-min property, preserves in , , and . However, there may exist a gap between and . As and are increased, this gap reduces and finally becomes zero in . At this point, both cross channels become saturated simultaneously. If any more power is allocated to either or , a suboptimal sum-rate will result. Therefore, maintains and diverts all additional power to the direct channels.
Note that once, in , the sum-rates achieved by joint decoding ( and ) become equal to the sum of interference-free user rates (), which is somewhat similar to the behavior of the GIC under very strong interference. At this point, all additional increments in P are allocated to the direct channels to increase the individual user rates.
In addition, note that the sets
form a partition due to their construction using the optimal Lagrange multipliers (defined in
Appendix C). Thus, the conditions in
Table 1 are mutually exclusive.
Moreover, these conditions can be equivalently described in terms of three
critical powers,
, defined by
where
is defined in (
33)–(
35) for
. Specifically, the conditions of
,
, and
are given by
,
, and
, respectively. In addition, we note that the direct channels are “stronger” than the cross channels in the sense specified above, if
. Similarly, if
, the cross channels are “stronger” than the direct channels. Furthermore, if
, the cross channels are said to be “much stronger”, in the sense that
continues allocating power to the cross channels as in
until they become saturated, and only after that it assigns power to both direct channels.
We note from the mutual exclusiveness of and that, if then no exists that satisfies . This shows that, if the direct channels are stronger, the allocation in is suboptimal for any . Similarly, if then no exists such that , and thus the allocation in is suboptimal for any .
5.2. The Waterfilling-Like Nature of the Optimal Power Allocation
Now, we characterize how
adapts the power allocation, as the power budget (
P) increases and crosses the critical thresholds
. We say that
follows the sequence
where
, if
allocates power as in
for sufficiently small
P, and then adapts the powers according to the allocation in
and
as
P increases. In this regard, we note that
follows one of the three sequences, as explained below and illustrated graphically in
Figure 3.
If : follows the sequence (denoted by [S1]). Since the direct channels are stronger, allocates all of P to them as in when P is sufficiently small (i.e., ). However, as P increases, the additional benefit from transmitting only in the direct channels decreases, and thus, when , begins transmitting in both cross and direct channels as in . This allocation follows from its WF-like property, and remains optimal for all . Finally, when , the max-min property of comes into effect, and thus the cross channels become saturated and starts following the allocation in . Note that, in this case, the saturation threshold for P is .
If but is not satisfied: follows the sequence ([S2]). This case is similar to [S1] above, except for the fact that now the cross channels are stronger. Hence, transmits in the cross channels and the direct channel with gain as in when P is sufficiently small (i.e., ). Next, following its WF-like property, starts transmitting in all the direct and cross channels as in when . Finally, when the cross channels become saturated, and follows the allocation in thereon. The saturation threshold for P in this case is .
Note that, whenever , follows [S2] irrespective of how , and compare, except in two cases: (a) , where follows [S3], described next, and (b) , which is infeasible as they violate the mutual exclusiveness of and .
If : follows the sequence ([S3]). In this case, the cross channels are much stronger than the direct channels. Hence, similar to [S2], allocates power to both cross channels and a direct channel as in when P is sufficiently small (i.e., ). As P increases and , the cross channels become saturated, and begins assigning powers to all channels as in . Interestingly, in this case, skips . This shows that, since the cross channels are much stronger, it is optimal to allocate power as in until they become saturated at , beyond which the allocation in becomes optimal.
We illustrate the characteristics of
with two numerical examples where we choose the following parameters,
. First, we illustrate an example of [S1] by additionally choosing
, such that the direct channels are stronger than the cross channels in the sense that
and
. In
Figure 4a, we plot the optimal powers against the power budget
P, and note the following: (i) when
,
allocates
P entirely to the direct channels as in
, and thus
and
increase with
P, and
; (ii) when
,
allocates power to all channels as in
, and thus
and
also increase with
P; (iii) finally, when
,
follows
where the cross channels become saturated simultaneously, and all increments of
P are added to
and
.
We depict the resulting constraints
, and
in
Figure 4b. First, note that
preserves
for all
P. However, there exists a gap between
and
in
and
. Specifically, in
, the gap remains constant (
); in
it reduces gradually as
transmits in the cross channels, and, in
it becomes zero as
achieves
, as expected.
Next, we illustrate an example of [S2] with the channel gains,
such that the cross channels are stronger than the direct channels in the sense that
and
. In
Figure 5a, we plot the optimal powers against
P, and observe the following: (i) when
,
follows the allocation in
, and thus
,
and
increase with
P, whereas
; (ii) when
,
allocates power to all channels as in
, and thus
now increases with
P; (iii) finally, when
,
follows
, and thus the cross channels become saturated simultaneously, as expected in [S2]. In
Figure 5b, we plot the sum-rate constraints, and note that
achieves
in all the sets. In addition, the gap between
and
is gradually offset as
transmits in the cross channels in
and
, and it finally becomes zero in
, as expected. We omit an example of [S3] (
), which is somewhat similar to [S2].
Remark 1. Recall that presents the solution for under and [A2] (implying ). If remains unchanged, and is formulated under a more general assumption [A2G]; where , then seven disjoint sets are needed to describe the optimal allocation (now denoted by ). The conditions and power allocations of these sets are found by solving the corresponding KKT conditions in a similar way to that of under [A2] in Appendix C. Hence, their explicit derivation is omitted for brevity. Of the seven sets, the first four sets, , , are counterparts of the corresponding sets in Table 1, in the sense that the power allocation in is similar to that in as discussed below. The remaining three sets are “new” in the sense that the power allocation in these sets do not resemble that in any of the sets in Table 1. The power allocation in the first four sets are similar to that of in that (a) when the power budget (P) is sufficiently small, allocates power to only both direct channels (in ), or both cross channels and one direct channel (in ) depending on whether the direct or the cross channels are “strong” as in or before; (b) as P is increased allocates power to all channels (in ) as in ; and (c) when P is sufficiently large, both cross channels become saturated simultaneously, and allocates all additional increments of P to only the direct channels (in ) as in before. Note that the optimal powers and the critical powers of differ slightly from that of due to the general assumption [A2G]. For conciseness, we omit the conditions, and present the optimal powers in Table 2. Here , and and η are defined in (
39)
and (
40)
belowwith , defined in (
37)
. Note that, under [A2] (), the powers in Table 2 simplifies to that in Table 1. Next, we present the optimal powers and the critical powers of the three new sets in Table 3, where The power allocation for the new sets can be understood by following the WF-like and max-min properties of . For illustration, we explain the allocations in and . In , the cross channels are stronger than the direct channels. Thus, due to its WF-like property, preferably allocates as much of P to the cross channels as possible. However, under [A2G] () and , allocating power to both cross channels results in , and does not increase as much as possible. This results in a suboptimal sum-rate. Therefore, allocates , which increases only , and assigns , which represents the allocation in .
Furthermore, as P increases and , additional benefits from transmitting only in the cross channel with gain reduces, and thus starts allocating a fraction of P to the direct channel with gain , due to its WF-like property. Thus, follows the power allocation in , which remains optimal as long as . The power allocation in can be interpreted similarly.
The important insight is that can be explained with the WF-like and max-min properties as in , and it does not reveal any new fundamental properties.
5.3. Optimum Power Allocation in the Symmetric DCLIC
We briefly discuss the optimum allocation for the symmetric DCLIC, where
,
,
, and
. Due to symmetry, considering only symmetric power allocation of the form
is sufficient, and does not cause loss of generality. Moreover, any feasible (symmetric) power allocation achieves
, rendering the constraint
in (
28) redundant. The sum-rate optimization problem in this case can be formulated and solved as in
, and is omitted here for brevity. In addition, we denote the optimal power allocation for this case by
.
In this case,
can be described by the power allocation in four disjoint sets,
, which are counterparts of the sets
in
Table 1. The conditions and optimal powers in these sets are given in
Table 4, and they can be derived following the same procedure as in
Appendix C, and thus is omitted here.
Moreover, the critical powers are now given by
where
.
has the same WF-like and max-min properties as that of . In fact, a rudimentary inspection of the conditions reveals that follows one of the three possible sequences of sets depending on the channel gains, as before:
If (direct channels are “stronger”): follows the sequence . It transmits only in the direct channels as in when , then transmits in all channels as in when , and finally starts following the allocation in when .
If (cross channels are “stronger”): follows the sequence . It transmits only in the cross channels as in when , then transmits in all channels as in when , and finally follows when .
If (cross channels are much “stronger”): follows the sequence . It follows when , and then follows when , while skipping altogether.
Thus, allocates all of P to either both direct or both cross channels if P is small, and then shares P among all channels as P increases. In addition, when P is increased beyond the saturation threshold, the cross channels become saturated. Note that, due to symmetry, needs to transmit in both cross channels only in to preserve , unlike in , where additionally transmits in a direct channel.
We note the following, which are revealed due to symmetry. First, if
, the direct channels are considered to be stronger than the cross channels. Similarly, if
, the cross channels are considered to be stronger. Moreover, in the special case of
, neither the direct nor the cross channels are stronger than the other, and thus allocating
P entirely to only direct or cross channel would be suboptimal. Therefore,
transmits in all channels as in
when
. Finally, the condition
implies that the cross channels are much stronger than the direct channels. Thus, it is optimal to transmit only in the cross channels as in
until they become saturated, at which point the allocation in
becomes optimal. Note that these conditions follow trivially from the corresponding conditions of [S1], [S2], and [S3] in
Section 5.2 by applying symmetry.
Furthermore, due to symmetry, one can now directly observe that the conditions of the sets in
Table 4 are mutually exclusive. For example, consider the conditions of
,
and
, and note that they violate the second condition of
(since
), the first condition in
(since
), and the second condition in
(trivially follows from
).
In
Figure 6, we depict the set of cross and direct channel gains (
and
) of a symmetric DCLIC with parameters
, and partition it depending on whether the direct or the cross channels are stronger.
5.4. Discussion and Insights
Now, we discuss some insights obtained from the power allocation in
in
Table 1.
First, if the underlying GIC in the first band has very strong interference, i.e., and , allocates the power budget (P) entirely to direct channels as in for all values of P. Note that, under very strong interference, we have for any feasible power allocation. Therefore, the sum-rate is maximized by maximizing , which is achieved by allocating all of P to the direct channels.
Second, if the transmit powers in the underlying GIC of the strong CLIC ( and ) are small, i.e., , then allocating P entirely to the direct channels as in , is approximately optimal for all values of P, in the sense that, the difference between the sum-rates achieved with the allocation in and that in is very small. Specifically, since , the gap between and at is very small. As P is sufficiently increased, starts transmitting in all channels as in . However, since the gap is very small, only a small fraction of P needs to be allocated to the cross channels to offset the gap, and thus almost all of P is redirected to the direct channels. The resulting allocation closely resembles that in .
We can attribute this behavior of to the fact that the GIC with strong interference () and small transmit powers () approaches the very strong interference regime (, ), where allocating P entirely to the direct channels as in is optimal for all P.
Third, if the transmit powers in the underlying GIC of the strong CLIC are large, i.e., , allocates power in all channels as in for all but a relatively small range of P. Recall that even if transmits in the direct or cross channels as in or for small P, eventually when or , allocates power in all channels as in , where the gap between and starts to reduce. However, if , this gap is very large, and thus very large P is needed to offset the gap as in . Therefore, for all moderate P, allocates power to all channels following .
Fourth, if the cross channel gains in the second band are very large, allocating P entirely to the direct channels as in is approximately optimal for all P, in the sense described in the second point above. This follows since with very large and , needs to allocate only a very small fraction of P to close the gap between and , and thus redirects the remaining of P to the direct channels. The resulting allocation thus closely resembles that in .
Finally, when , the DCLIC can be approximated by the DLIC, and thus allocating P entirely to only the direct channels is approximately optimal. On the other hand, when , the DCLIC can be approximated by the CLIC where allocating P entirely to the cross channels is approximately optimal. However, since the cross channels do not increase , the sum-rate becomes limited to when P is sufficiently large, and does not increase anymore.