In the previous section we considered a general Gaussian broadcast channel with parallel subchannels and users, and we showed how to derive convex formulations for inner and outer bounds on the SPCGS rate region. In this section we provide exact convex formulations for three particular instances of the general problem, namely, the 2-user case and the case of users with (independent) particular messages only, and the SPCGS sum rate point of the general -user -subchannel case. (For the first two cases, the SPCGS rate region is known to be the capacity region [17, 21].) Using these convex formulations, optimal power loads and partitions for these three cases can be obtained using efficient interior point techniques.
5.1. Optimal Power Allocation for the 2-User Case
For this case, the capacity region was shown in  to be the same as the SPCGS rate region. Similar to the general case considered in Proposition 1, the boundary of the 2-user SPCGS rate region is parameterized by power loads and partitions. Although the optimal values of these parameters can be determined using the indirect Lagrange multiplier search technique provided in , in this section we provide a (precise) convex formulation that enables us to determine those loads and partitions directly, and in a computationally efficient manner.
Recall that in our notation the degradedness condition on each subchannel implies that . Let , , be the set of subchannels on which User is the stronger user. Using Proposition 1 and the logarithmic substitutions: and , we formulate the weighted sum rate optimization problem as
where , and is the power partition associated with the stronger user on the th subchannel. In order to transform this optimization problem into a convex form, we perform the variable substitutions
and . Using these variable substitutions, and the equivalent constraints in (14), the optimization problem in (25) can be reformulated as
The formulation in (27) is in the form of a convex geometric program and the optimal values of and , , can be efficiently found. Once and have been computed, one can use (26) to find the power loads and the power partitions .
5.2. Optimal Power Allocation for the Broadcast of Particular Information to users
The capacity region for the case in which only particular information is to be transmitted to users over parallel channels was considered in [21–23]. In  the concept of utility functions was introduced. Using the properties of these functions and a search for a Lagrange multiplier, optimal power loads and power partitions were determined algebraically. In this section we will present an alternative efficient numerical technique for determining these loads and partitions through the solution of a convex optimization problem. (This technique is similar to that presented in  and was developed independently.) Using our notation for the rate of particular information of User , , the capacity region is the closure of all points of the form 
where , and . In order to simplify the notation, we will use to denote and to denote . Finding each point on the boundary of the capacity region and the corresponding power loads and partitions is equivalent to solving the following optimization problem for a given set of weights that satisfy :
In its current form, the formulation in (29a)–(29e) is not convex. The key to casting (29a)–(29e) in a convex form is the change of variables
To begin with, we note that this substitution is one-to-one. That is, once the problem is solved in terms of the variables , one can readily obtain the required power partitions . We now examine the constraints in (29a)–(29e). The set of constraints in (29b) can be rewritten as
Observe that because each subchannel is degraded, the constant is greater than or equal to zero. Hence, (31) is in the form of a posynomial constraint, and can be easily incorporated in a geometric program. In order to account for the constraints (29c), (29d), and (29e), we observe that from (30) we have
where we will use the convention that . The set of constraints in (29e) can now be expressed as
This constraint is also in a posynomial format. Finally, we observe that the constraints in (29c) and (29d) can be merged together. In particular, the variables can be eliminated. Using (30), this will lead to the following constraint:
Using these transformations, the weighted sum rate optimization problem in (29a)–(29e) can be recast in the following convex format:
Once (35) has been solved, one can use (32) and (29c) to obtain the required power loads and partitions.
5.3. Optimal Power Allocation for SPCGS Sum Rate Maximization
In Section 3.1 we expressed the points on the boundary of the SPCGS rate region of a -user -subchannel broadcast channel as the solution of the optimization problem in (16a)–(16f). As discussed in Section 3.1, that problem is not convex for general values of the weights . However, for the case in which all the weights are equal, the objective in (16a)–(16f) corresponds to the sum of the common and particular SPCGS rates. We will now show that finding the power loads and partitions that maximize this sum rate can be cast a (convex) geometric program. In order to do that, we observe that the constraints in (16a)–(16f) that bound the sum rate can be extracted from (16c) by setting equal to . It can be shown that in the problem of maximizing the sum rate only these constraints and the constraints in (16d)–(16f) can be active. That is, the constraints in (16a) and (16b) and the constraints in (16c) that correspond to do not constrain the optimal solution to the sum rate optimization problem. In order to see that, we observe that solving (16a)–(16f) with these constraints removed results in a relaxation of the optimization problem. This relaxation yields an upper bound on the maximum sum rate. However, the solution of the relaxed problem provides power allocations that satisfy the power constraints in (16d)–(16f) and achieve this upper bound on the maximum sum rate. Hence, the maximum sum rate that can be achieved by superposition coding and Gaussian signalling, and the corresponding power allocations, can be obtained by solving the relaxed problem.
We now provide an explicit formulation of the relaxed problem in a convex form. In order to do that, let the sum rate be equal to , and note that by setting to be equal to in (16c), we have . Hence, the relaxed problem can be expressed as
In order to cast the optimization problem in (36a)–(36e) in a convex form, we use the transformation in (30) to write the constraints in (36b) in a posynomial form as
Noting from (11) and (30) that is equal to , the constraints in (36c)–(36e) can be easily transformed into posynomial inequality constraints using the same technique that was used to formulate (35).
In addition to casting the SPCGS sum rate in a convex form, it is also possible to show that by setting all the particular rates equal to zero, one can cast the problem of maximizing the common SPCGS rate as a GP. This can be done by removing the constraints in (16b) and (16c) and solving the resulting GP directly.