Resource Allocation in Downlink OFDM Wireless Systems with User Rate Allowed Regions

Abstract

In this paper, a novel adaptive resource allocation in downlink orthogonal frequency division multiplexing wireless system is investigated. Different from the conventional algorithms, user rate allowed regions are considered. That is, not only the user minimum rate requirements are to be satisfied, but also the maximum rate limitations are introduced to make resource allocation more reasonable. Given equal power distribution, a three-round subcarrier assignment is proposed at first to achieve a preliminary performance. Then, waterfilling-based power allocation is designed to guarantee user rate allowed regions and enhance system performance further. Numerical results show that the proposed algorithm ensures users achieving data rates within their allowed regions more precisely. In the mean time, the system throughput and transmit power consumption are reasonably balanced compared to the traditional methods when channel-gain-to-noise-ratio is high.

Appendix: Proof that the Three-Round Power Allocation is Optimal

Proof

The original power allocation problem under the given subcarrier assignment is a standard convex optimization which can be expressed as

$$\begin{aligned} \mathrm {maximize}&\underset{k=1}{\overset{K}{\sum }}\underset{n\in D_{k}}{\overset{}{\sum }}\frac{1}{N}\mathrm {log_{2}}\left( 1+p_{n,k}g_{n,k}\right) \nonumber \\ \mathrm {subject\, to}&\underset{k=1}{\overset{K}{\sum }}\underset{n\in D_{k}}{\overset{}{\sum }}p_{n,k}=P_{total}\nonumber \\&\check{P}_{k}\le \underset{n\in D_{k}}{\overset{}{\sum }}p_{n,k}\le \hat{P}_{k}\,\,\mathrm {for\, all}\, k\nonumber \\&p_{n,k}\ge 0\,\,\mathrm {for\, all}\, k,n, \end{aligned}$$

(10)

for which the Karush–Kuhn–Tucker (KKT) conditions that are both sufficiently and necessarily optimal can be derived as

$$\begin{aligned}&p_{n,k}=\frac{1}{\mathrm {ln}2\left( \lambda -\alpha _{k}+\beta _{k}\right) }-\frac{1}{g_{n,k}}\end{aligned}$$

(11)

$$\begin{aligned}&\alpha _{k}\left( \underset{n\in D_{k}}{\overset{}{\sum }}p_{n,k}-\check{P}_{k}\right) =0\end{aligned}$$

(12)

$$\begin{aligned}&\beta _{k}\left( \hat{P}_{k}-\underset{n\in D_{k}}{\overset{}{\sum }}p_{n,k}\right) =0\end{aligned}$$

(13)

$$\begin{aligned}&\underset{k=1}{\overset{K}{\sum }}\underset{n\in D_{k}}{\overset{}{\sum }}p_{n,k}-P_{total}=0\end{aligned}$$

(14)

$$\begin{aligned}&\check{P}_{k}\le \underset{n\in D_{k}}{\overset{}{\sum }}p_{n,k}\le \hat{P}_{k}. \end{aligned}$$

(15)

Obviously, (14) and (15) are satisfied through the three-round power allocation. Hence, we need to find proper $\lambda$, $\alpha _{k}$, and $\beta _{k}$ to prove that (11)–(13) are hold.

For simplicity, we term $\lambda -\alpha _{k}+\beta _{k}$ in (11) the “water level coefficient” for user $k$. In the first round, the water level coefficient corresponding to user $k$ is $\eta _{k}$, as expressed in (5). In the second round, we can split users into two groups, of which one is the set of users that obtain power, denoted by $\fancyscript{G}_{1}$, and the other is the set of users that maintain their minimum levels, denoted by $\fancyscript{G}_{2}$. For $k\in \fancyscript{G}_{1}$ there is $\frac{1}{\theta \mathrm {ln}2}-\frac{1}{g_{k,n}}-\check{p}_{n,k}>0$ according to (7), and for $k\in \fancyscript{G}_{2}$ we have $\frac{1}{\theta \mathrm {ln}2}-\frac{1}{g_{k,n}}-\check{p}_{n,k}\le 0$. Hence, for $k\in \fancyscript{G}_{1}$ we can get $p_{n,k}=\frac{1}{\theta \mathrm {ln}2}-\frac{1}{g_{k,n}}$, which indicates the water level coefficient changes to $\theta$.

In the third round, we split the users into three groups: $\fancyscript{G}_{1}$ for the users that still maintain the minimum power levels, $\fancyscript{G}_{2}$ for the users of power level between the minimum and maximum, and $\fancyscript{G}_{3}$ for the users whose power are lowered as a result of the limitation. For $k\in \fancyscript{G}_{2}$ there is $\frac{1}{\lambda \mathrm {ln}2}-\frac{1}{g_{k,n}}-\check{p}_{n,k}-\varDelta p_{n,k}^{1}>0$ according to (9); hence, we can have $p_{n,k}=\frac{1}{\lambda \mathrm {ln}2}-\frac{1}{g_{k,n}}$, which indicates that the water level coefficient changes to $\lambda$. Since power level of user $k\in \fancyscript{G}_{1}$ still unchanges, it means that the gradient of user rate respect to power in group $\fancyscript{G}_{1}$ is no more than that in group $\fancyscript{G}_{2}$, that is

$$\begin{aligned} \left. \frac{1}{\lambda \mathrm {ln}2}=\frac{g_{n,k}}{1+p_{n,k}g_{n,k}} \right| _{k\in \fancyscript{G}_{1}}\ge \left. \frac{g_{n,k}}{1+p_{n,k}g_{n,k}}= \frac{1}{\eta _{k}\mathrm {ln}2}\right| _{k\in \fancyscript{G}_{2}}. \end{aligned}$$

(16)

Hence, it can be derived that $\lambda \le \eta _{k}$. On the other hand, since power level of user $k\in \fancyscript{G}_{2}$ is increased comparing to the second round, we can have $\theta <\lambda$. Likely, the power level of user $k\in \fancyscript{G}_{3}$ is lowered comparing to the second round, therefore we can also have $\theta >\gamma _{k}$, where $\gamma _{k}$ denotes the corresponding water level coefficient for $k\in \fancyscript{G}_{3}$.

Based on above analysis, there is

$$\begin{aligned} \left. \gamma _{k}\right| _{k\in \fancyscript{G}_{1}}<\left. \lambda \right| _{k\in \fancyscript{G}_{2}}\le \left. \eta _{k}\right| _{k\in \fancyscript{G}_{3}}. \end{aligned}$$

(17)

Hence, $\alpha _{k}$ and $\beta _{k}$ can be found as $\alpha _{k}=\eta _{k}-\lambda ,$ $\beta _{k}=0$ for $k\in \fancyscript{G}_{3}$, $\alpha _{k}=0$, $\beta _{k}=\lambda -\gamma _{k}$ for $k\in \fancyscript{G}_{1}$, and $\alpha _{k}=\beta _{k}=0$ for $k\in \fancyscript{G}_{2}$. It is readily indicated that Eqs. (11)–(13) are hold, which show the optimality of the proposed power allocation. $\square$