An efficient data rate maximization algorithm for OFDM based wireless networks

Abstract

In this paper we present a computationally efficient, suboptimal integer bit allocation algorithm that maximizes the overall data rate in multiuser orthogonal frequency division multiplexing (OFDM) systems implemented in wireless networks. Assuming the complete knowledge of a channel and allowing a subchannel to be simultaneously shared by multiple users we have solved this data rate maximization problem in two steps. The first step provides subchannel assignment to users considering the users’ requests on quality of service (QoS) expressed as the minimum signal-to-noise ratio (SNR) on each subchannel. The second step provides transmit power and bit allocation to subchannels in order to maximize the overall data rate. To reduce computational complexity of the problem we propose a simple method which assigns subchannels to users and distributes power and bits among them. We have analyzed the performance of our proposed algorithm by simulation in a multiuser frequency selective fading environment for various signal-to-noise ratios and various numbers of users in the system. We have concluded that our algorithm, unlike other similar algorithms, is suitable for OFDM wireless networks, especially when signal-to-noise ratio in the channel is low. Also, the results have shown that the total data rate grows with the number of users in the system.

Appendix

1.1 Proof of the Theorem 1

Following the discussion in Sect. 3 and in order to maximize data rate, we have solved the subchannel assignment problem using our theorem (Theorem 1) which states that k users can share the same subchannel m with required SNR value ╬│_j,m only if

$$ \sum\limits_{j=1}^k {\frac{\omega _{j,m} }{\omega _{j,m} +\frac{1}{\gamma _{j,m} }} < 1} $$

(A.1)

Proof

Signal to noise ratio (SNR) for the i-th user on the m-th subchannel can be written as

$$ \left({\frac{S}{N}} \right)_{i,m} =\frac{s_{i,m} \lambda _{i,m} }{\sum\nolimits_{\begin{array}{l} j=1 \\ j\neq i \\ \end{array}}^M {\lambda _{j,m} s_{j,m} \omega _{j,m} +N_0 B_m } } $$

(A.2)

By introducing the userΓÇÖs request on SNR, the problem is then formulated as

$$ \left({\frac{S}{N}}\right)_{i,m}=\frac{s_{i,m} \lambda _{i,m} }{\sum\nolimits_{\begin{array}{l} j=1 \\ j\neq i \\ \end{array}}^M {\lambda _{j,m} s_{j,m} \omega _{j,m} +N_0 B_m } }\ge \gamma _{i,m} $$

(A.3)

One of the optimal power vectors is obtained if

$$ \frac{s_{i,m} \lambda _{i,m} }{\sum\nolimits_{\begin{array}{l} j=1 \\ j\neq i \\ \end{array}}^M {\lambda _{j,m} s_{j,m} \omega _{j,m} +N_0 B_m } }=\gamma _{i,m} $$

(A.4)

$$ \frac{s_{i,m} \lambda _{i,m} }{\gamma _{i,m} }-\sum\limits_{\begin{array}{l} j=1 \\ j\ne i \\ \end{array}}^M {\lambda _{j,m} s_{j,m} \omega _{j,m} } =N_0 B_m $$

(A.5)

This leads to the matrix formulation ${\mathbf{DS}}=N_{0}B_{m}{\mathbf{I}}$ where

$$ {\mathbf{D}}=\left({{\begin{array}{cccc} {\frac{\lambda _{1,m} }{\gamma _{1,m} }}& {-\lambda _{2,m} \omega _{2,m} }& \cdots & {-\lambda _{M,m} \omega _{M,m} } \\ {-\lambda _{1,m} \omega _{1,m} }& {\frac{\lambda _{2,m} }{\gamma _{2,m} }}& \cdots & {-\lambda _{M,m} \omega _{M,m} } \\ \vdots & \vdots & \ddots & \vdots \\ {-\lambda _{1,m} \omega _{1,m} }& {-\lambda _{2,m} \omega _{2,m} }& \cdots & {\frac{\lambda _{M,m} }{\gamma _{M,m} }} \\ \end{array} }} \right) $$

(A.6)

${\mathbf{S}}= [s_{1,m},s_{2,m},{\ldots},s_{M,m}]^{T}$ is the optimal power vector, and I is the unity matrix. After doing two elementary matrix operations (subtracting the first row from the next one, and starting from the second row, multiplying each row with ${\omega _{j,m} }/{({\omega _{j,m} +\frac{1}{\gamma _{j,m} }})}$ and adding it to the first row) the matrix form is reduced. Diagonal elements and elements in the first column are non-zero. The equation with parameter s_1,m is

$$ \left({\omega _{1,m} +\frac{1}{\gamma _{1,m} }} \right)\lambda _{1,m} \left({1-\sum_{j=1}^M {\frac{\omega _{j,m} }{\omega _{j,m}+\frac{1}{\gamma _{j,m} }}} } \right)s_{1,m}=N_0 B_m $$

(A.7)

Power s _1,m must be positive which yields the following condition

$$ \sum_{j=1}^M {\frac{\omega _{j,m} }{\omega _{j,m} +\frac{1}{\gamma _{j,m} }}} < 1 $$

(A.8)

and completes the proof of Theorem 1. Γûí