A practical power allocation method for multi-user and multi-carrier networks

Abstract

In the majority of studies on wireless networks, it is assumed that the receiver perfectly knows the channel. Since this is not the case for real life applications, the channel must be estimated in those systems. Thus, current cellular wireless communication systems are designed to perform coherent detection, and channel estimator is one of the most important part of a receiver. The channel estimation error must be minimized, because the error to be made during the channel estimation phase significantly affects the performance of the data detector. This can be made possible by allocating required amount of power to the pilot symbols to estimate channel with minimum error as possible as. Hence, this study examines how to allocate the power between pilot and data symbols to minimize the error of the data detector. The long term evolution provides an opportunity to change the radiation emitted in the pilot sub-carriers according to the data sub-carriers. Since this opportunity is expected to be the case for next generation cellular wireless networks, in this study, the effect of channel estimation error on data detection performance of a multi-user and multi-carrier cellular wireless communication system is investigated. The proposed solution is a practical one to approximate to the optimum solution of the resulting optimization problem without solving it. It is seen that the results obtained in computer simulations are very close to the optimum ones.

Appendix

The Lagrangian function of (14) can be obtained as seen in (17) with Karush–Kuhn–Tucker conditions [17, 18], from (18a) to (18j):

$$\begin{aligned} L(P_{trn};\lambda _1, \lambda _2, \lambda _3)=&-\frac{1}{L}\sum _{l=0}^{L-1}\frac{(K-1)\hat{\gamma }_l\sigma _{h}^4 \left( P_{tot}-P_{trn}\right) ^2P_{trn}}{K\Big ((K-1)P_{tot}-(K-2)P_{trn}\Big )^2}\nonumber \\&-\lambda _1\left( P_{tot}-P_{trn}\right) -\lambda _2P_{trn}-\lambda _3(K-2) \end{aligned}$$

(17)

$$\begin{aligned}&-\frac{1}{L}\sum _{l=0}^{L-1}\frac{\left[ (K-2)P_{trn}^2-(2K-1)P_{tot}P_{trn}+(K+1)P_{tot}^2\right] }{K\left[ (K-1)P_{tot}-(K-2)P_{trn}\right] ^3} \nonumber \\&\qquad \times \frac{(K-1)\hat{\gamma }_l\sigma _{h}^4 \left( P_{tot}-P_{trn}\right) }{K\left[ (K-1)P_{tot}-(K-2)P_{trn}\right] ^3}+\lambda _1-\lambda _2=0 \end{aligned}$$

(18a)

$$\begin{aligned}&\quad \frac{1}{L}\sum _{l=0}^{L-1}\frac{\hat{\gamma }_l\sigma _{h}^4 \left( P_{tot}-P_{trn}\right) ^2P_{trn}\left[ 2K(K-1)(P_{tot}-P_{trn})-1\right] }{K^2\left[ (K-1)P_{tot}-(K-2)P_{trn}\right] ^2} -\lambda _3=0 \end{aligned}$$

(18b)

$$\begin{aligned}&\quad \lambda _1 \left( P_{tot}-P_{trn}\right) = 0 \end{aligned}$$

(18c)

$$\begin{aligned}&\quad \lambda _1\ge 0 \end{aligned}$$

(18d)

$$\begin{aligned}&\quad P_{trn}\le P_{tot} \end{aligned}$$

(18e)

$$\begin{aligned}&\quad \lambda _2P_{trn}= 0 \end{aligned}$$

(18f)

$$\begin{aligned}&\quad \lambda _2\ge 0 \end{aligned}$$

(18g)

$$\begin{aligned}&\quad P_{trn}\ge 0 \end{aligned}$$

(18h)

$$\begin{aligned}&\quad \lambda _3(K-2)= 0 \end{aligned}$$

(18i)

$$\begin{aligned}&\quad \lambda _3\ge 0 \end{aligned}$$

(18j)

Since the aim of this study is to present a practical solution, the optimal solution of the problem is not given here. However, it is too hard, but possible to find the optimum solution $P_{trn}^{opt}$ of the problem by using the Karush–Kuhn–Tucker conditions given above.